Introduction to global analysis minimal surfaces in Riemannian manifolds 9781470429508, 1470429500

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GRADUATE STUDIES I N M AT H E M AT I C S

187

Introduction to Global Analysis Minimal Surfaces in Riemannian Manifolds John Douglas Moore

American Mathematical Society

Introduction to Global Analysis Minimal Surfaces in Riemannian Manifolds

GRADUATE STUDIES I N M AT H E M AT I C S

187

Introduction to Global Analysis Minimal Surfaces in Riemannian Manifolds

John Douglas Moore

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed (Chair) Gigliola Staffilani Jeff A. Viaclovsky 2010 Mathematics Subject Classification. Primary 58E12, 58E05; Secondary 46T10, 53C20, 55P62.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-187

Library of Congress Cataloging-in-Publication Data Names: Moore, John D. (John Douglas), 1943- author. Title: Introduction to global analysis : minimal surfaces in Riemannian manifolds / John Douglas Moore. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Graduate studies in mathematics ; volume 187 | Includes bibliographical references and index. Identifiers: LCCN 2017028964 | ISBN 9781470429508 (alk. paper) Subjects: LCSH: Global analysis (Mathematics) | Minimal surfaces. | Riemannian manifolds. | Manifolds (Mathematics) | Morse theory. | AMS: Global analysis, analysis on manifolds – Variational problems in infinite-dimensional spaces – Applications to minimal surfaces (problems in two independent variables). msc | Global analysis, analysis on manifolds – Variational problems in infinite-dimensional spaces – Abstract critical point theory (Morse theory, LjusternikSchnirelmann (Lyusternik-Shnirelman) theory, etc.). msc | Functional analysis – Nonlinear functional analysis – Manifolds of mappings. msc | Differential geometry – Global differential geometry – Global Riemannian geometry, including pinching. msc | Algebraic topology – Homotopy theory – Rational homotopy theory. msc Classification: LCC QA614 .M66 2017 | DDC 516.3/62–dc23 LC record available at https://lccn. loc.gov/2017028964

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Contents

Preface

vii

Chapter 1. Infinite-dimensional Manifolds

1

§1.1. A global setting for nonlinear DEs

1

§1.2. Infinite-dimensional calculus

2

§1.3. Manifolds modeled on Banach spaces

17

§1.4. The basic mapping spaces

25

§1.5. Homotopy type of the space of maps

33

§1.6. The α- and ω-Lemmas

39

§1.7. The tangent and cotangent bundles

40

§1.8. Differential forms

44

§1.9. Riemannian and Finsler metrics

49

§1.10. Vector fields and ODEs

53

§1.11. Condition C

55

§1.12. Birkhoff’s minimax principle

60

§1.13. de Rham cohomology

63

Chapter 2. Morse Theory of Geodesics

71

§2.1. Geodesics

71

§2.2. Condition C for the action

76

§2.3. Fibrations and the Fet-Lusternik Theorem

81

§2.4. Second variation and nondegenerate critical points

85

§2.5. The Sard-Smale Theorem

91

§2.6. Existence of Morse functions

95 v

vi

Contents

§2.7. Bumpy metrics for smooth closed geodesics

100

§2.8. Adding handles

108

§2.9. Morse inequalities

114

§2.10. The Morse-Witten complex

118

Chapter 3. Topology of Mapping Spaces

125

§3.1. Sullivan’s theory of minimal models

125

§3.2. Minimal models for spaces of paths

131

§3.3. Gromov dimension

138

§3.4. Infinitely many closed geodesics

145

§3.5. Postnikov towers

148

§3.6. Maps from surfaces

155

Chapter 4. Harmonic and Minimal Surfaces

169

§4.1. The energy of a smooth map

169

§4.2. Minimal two-spheres and tori

178

§4.3. Minimal surfaces of arbitrary topology

188

§4.4. The α-energy

204

§4.5. Morse theory for a perturbed energy

216

§4.6. Bubbles

225

§4.7. Existence of minimal two-spheres

239

§4.8. Existence of higher genus minimal surfaces

249

§4.9. Unstable minimal surfaces

256

§4.10. An application to curvature and topology

267

Chapter 5. Generic Metrics

277

§5.1. Bumpy metrics for minimal surfaces

277

§5.2. Local behavior of minimal surfaces

281

§5.3. The two-variable energy revisited

292

§5.4. Minimal surfaces without branch points

308

§5.5. Minimal surfaces with simple branch points

318

§5.6. Higher order branch points

334

§5.7. Proof of the Transversal Crossing Theorem

347

§5.8. Branched covers

349

Bibliography

357

Index

365

Preface

This book is devoted to giving the foundations for a partial Morse theory of minimal surfaces in Riemannian manifolds. It is based upon lecture notes for graduate courses on “Topics in Differential Geometry”, given at the University of California, Santa Barbara, during the fall quarter of 2009 and again in the spring of 2014, but it also includes several topics not treated in these courses. It might be helpful to start with a description of the goal of our presentation. Morse theory is concerned with the relationship between the critical points of a smooth function on a manifold and the topology of that manifold. It might have developed in three main stages. The first stage in our fictional history would have been finite-dimensional Morse theory, which relates critical points of proper nonnegative functions on finite-dimensional manifolds to the homology of these manifolds via the Morse inequalities. The foundations were laid in Marston Morse’s first landmark article on what is now known as Morse theory [Mor25], but Morse quickly turned his attention to problems from the calculus of variations, which ultimately became part of the infinite-dimensional theory. Nevertheless, finite-dimensional Morse theory became one of the primary tools for studying the topology of finite-dimensional manifolds and had many successes, including the celebrated h-cobordism theorem of Smale [Mil65], crucial for the classification of manifolds in high dimensions. Modern expositions of finite-dimensional Morse theory often construct a chain complex from the free abelian group generated by the critical points of a “generic” function, the boundary being defined by orbits of the gradient flow which connect the critical points. The homology of this chain complex, called the

vii

viii

Preface

Morse-Witten complex, is isomorphic to the usual integer homology of the manifold. What might have been the second stage, the Morse theory of geodesics, formed the core of what Morse [Mor34] called “the calculus of variations in the large”. Motivated to some extent by earlier work on celestial mechanics by Poincar´e and Birkhoff, Morse studied the calculus of variations for the length function or action function on the space of paths joining two points in a Riemannian manifold (or the space of closed paths in a Riemannian manifold), the critical points of these functions being geodesics. His idea was to approximate the infinite-dimensional space of paths by a finite-dimensional manifold of very high dimension and then apply finite-dimensional Morse theory to this approximation. This approach is explained in Milnor’s classical book on Morse theory [Mil63] and produced many striking results in the theory of geodesics in Riemannian geometry, such as the theorem of Serre [Ser51] that any two points on a compact Riemannian manifold can be joined by infinitely many geodesics. It also provided the first proof of the Bott periodicity theorem from homotopy theory. A Morse theory of closed geodesics, representing periodic motion in certain mechanical systems, was also constructed. One might regard the Morse theory of geodesics as an application of topology to the study of ordinary differential equations, in particular, to those equations which like the equation for geodesics arise from classical mechanics. Palais and Smale were able to provide an elegant reformulation of the Morse theory of geodesics in the language of infinite-dimensional Hilbert manifolds [PS64]. They showed that the action function on the infinitedimensional manifold of paths satisfies “Condition C”, a condition replacing “proper” in the finite-dimensional theory, and they showed that Condition C is sufficient for the development of Morse theory in infinite dimensions. This became a standard approach to the Morse theory of geodesics during the last several decades of the twentieth century, and we will describe it in some detail later. One might regard the third stage of Morse theory as encompassing many strands, but our viewpoint is to focus on techniques for applying Morse theory to nonlinear elliptic partial differential equations coming from the calculus of variations in which the domain is a two-dimensional compact surface. Morse himself hoped to apply the ideas of his theory to what is arguably the central case—the partial differential equations for minimal surfaces—and he focused on the case in which the ambient space is Euclidean space. The first steps in this direction were taken by Morse and Tompkins, as well as Shiffman, who established the theorem that if a simple closed curve in Euclidean space R3 bounds two stable minimal disks, it bounds a

Preface

ix

third, which is not stable. This provided a version of the so-called “mountain pass lemma” for minimal disks in Euclidean space. The results of Morse, Tompkins, and Shiffman suggested that Morse inequalities should hold for minimal surfaces in Euclidean space with boundary constrained to lie on a given Jordan curve and, indeed, such inequalities were later established under appropriate hypotheses (as explained, for example, in [JS90]). But the most natural extension of the Morse theory of closed geodesics in Riemannian manifolds to partial differential equations would be a Morse theory of closed two-dimensional minimal surfaces in a general curved ambient Riemannian manifold M , instead of the ambient Euclidean space used in the classical theory of minimal surfaces with boundary. The generalization to ambient Riemannian manifolds with arbitrary curvature introduces complexity and requires new techniques. Unfortunately, if Σ is a connected compact surface, it becomes awkward to extend the finite-dimensional approximation procedure—so effective in the theory of geodesics—to the space Map(Σ, M ) of mappings from Σ to M . One might hope for a better approach based upon the theory of infinite-dimensional manifolds, as developed by Palais and Smale, but a serious problem is encountered: the standard energy function E : Map(Σ, M ) −→ R, used in the theory of harmonic maps and parametrized minimal surfaces, fails to satisfy the compactness condition, the Condition C which Palais and Smale had used as a foundation for their theory, when Map(Σ, M ) is completed with respect to a norm strong enough to lie within the space of continuous functions. To get around this difficulty, Sacks and Uhlenbeck introduced the αenergy [SU81], [SU82] in which α > 1 is a parameter, a perturbation of the usual energy which is defined on the completion of Map(Σ, M ) with respect to a Banach space norm which is both weak enough to satisfy Condition C and strong enough for the α-energy to be a C 2 function on a Banach space having the same homotopy type as the space of continuous maps from Σ to M . We can express the fact that such a completion exists by saying that the α-energy lies within “Sobolev range”. The α-energy approaches the usual energy as the parameter α in the perturbation goes to one, and we can therefore say that the usual energy on maps from compact surfaces is “on the border of Sobolev range”, approachable by Morse theory via approximation. Using this approximation via the α-energy, Sacks and Uhlenbeck were able to establish many striking results in the theory of minimal surfaces in Riemannian manifolds, including the fact that any compact simply connected Riemannian manifold contains at least one nonconstant minimal two-sphere, which parallels the classical theorem of Fet and Lyusternik stating that any

x

Preface

compact Riemannian manifold contains at least one smooth closed geodesic. But they also discovered a serious obstruction: The phenomenon of “bubbling” as α → 1 prevents full Morse inequalitites from holding for compact parametrized minimal surfaces in Map(Σ, M ) in complete generality. A somewhat different approach to existence of parametrized minimal surfaces in Riemannian manifolds was developed at about the same time by Schoen and Yau [SY79], using Morrey’s solution to the Plateau problem for minimal disks bounded by a Jordan curve in a Riemannian manifold and arguments based upon a “replacement procedure”. Their approach provided striking theorems relating positive scalar curvature to the topology of three-manifolds, including a step toward the first proof of the positive mass theorem of general relativity. The Schoen-Yau replacement procedure can also be used to obtain many of the existence results of Sacks and Uhlenbeck, and, indeed, an alternate treatment of many of their theorems is provided by Jost [Jos91]. Yet other proofs of these theorems were developed using heat flow (see Struwe [Str90] or Hang and Lin [HL03]). An even more general approach to minimal surfaces is provided by geometric measure theory [Mor08], which constructs minimal varieties of arbitrary dimension and codimension via generalized surfaces such as integral currents and varifolds, and then attempts to show that the resulting generalized surfaces are regular. But regularity cannot always succeed because according to results of Ren´e Thom, not every homology class is represented by a smooth submanifold [Tho54] and much work has focused on the special dimensions in which regularity can be established. In this book we restrict our attention to minimal surfaces of dimension two for which a parametrized theory often provides additional information not directly accessible via geometric measure theory, such as the genus of the surface, and this additional information is crucial for many of the applications we will describe. Moreover, the Sacks-Uhlenbeck perturbation is available in the mapping context, and it provides a suggestive link with classical Morse theory, and perhaps the clearest insight into bubbling, the phenomenon observed as the perturbation is turned off, which turns out to be the main technical difficulty in establishing Morse inequalities for (two-dimensional) closed minimal surfaces in Riemannian manifolds. Moreover, analogs of bubbling appear in the study of other nonlinear partial differential equations, such as the Yang-Mills equation on four-dimensional manifolds, suggesting links between theories that can sometimes be exploited. Indeed, the theory of two-dimensional minimal surfaces can be placed in a broader context—that of nonlinear partial differential equations which lie

Preface

xi

on the border of Sobolev range and are conformally invariant or closely related to conformally invariant equations. These equations include the YangMills equations on four-dimensional manifolds from the standard model for particle physics, the anti-self-dual equations used so effectively by Donaldson, the Seiberg-Witten equations, and Gromov’s equations for pseudoholomorphic curves. It is useful to develop a common technology for studying these equations and a common schema for setting up theories: First one needs to develop a transversality theory using Smale’s generalization of Sard’s theorem from finite-dimensional differential topology. This generally shows that in generic situtations, solutions to the nonlinear partial differential equation form a finite-dimensional submanifold of an infinitedimensional function space. The tangent space to this submanifold is studied via the linearization of the nonlinear partial differential equation at a given solution, and often the dimension of the tangent space is obtained by application of the Atiyah-Singer index theorem (which reduces to the Riemann-Roch theorem in the case of parametrized minimal surfaces). Next one develops a suitable compactness theorem, which in Donaldson’s theory provides solutions localized near points, in analogy with bubbling. Finally, one uses topological and geometric methods to derive important geometric conclusions (for example, existence of minimal surfaces under various topological conditions) or to construct differential topological invariants of manifolds (in the Seiberg-Witten or Donaldson theories). The reader can refer to [DK90] for a definitive treatment of Donaldson theory, which might be regarded as a model for the other theories. Although bubbling implies that the most obvious extension of the Morse inequalities to minimal surfaces in Riemannian manifolds cannot hold in general, it also suggests a framework for analyzing how the Morse inequalities fail and to what extent a remnant of the Morse inequalities might still be retained. Of course, other difficulties also need to be controlled. When constructing a minimax critical point, one must allow for variations in the conformal structure on the surface, thought of as an element of Teichm¨ uller space or moduli space, and a sequence of harmonic maps may degenerate as the conformal structure moves to the boundary of moduli space. Moreover, branched coverings of a given minimal surface count as new critical points within the space of functions, although they are not geometrically distinct from the covered surface. We call these the “sources of noncompactness” for the two-variable energy E : Map(Σ, M ) × T −→ R, in which T represents Teichm¨ uller space. Note that this energy is invariant under various groups, an action of the group of complex automorphisms of

xii

Preface

the domain Σ and an action of the mapping class group on Teichm¨ uller space, suggesting use of equivariant Morse theory as described by Bott [Bot82]. One might suspect that a procedure for constructing minimal surfaces that might fail in several different ways is too flawed to be of much use. However, we argue that a different perspective is more productive—since the minimax procedure for a given homology class does not always yield nontrivial minimal surfaces, one should divide homology classes into various categories depending upon which of the possible difficulties can arise. This approach appears ambitious at first, but then one realizes that the topology of Map(Σ, M ) has a rich internal structure that can compensate for the many ways in which minimax sequences might fail to converge. Our purpose here is to provide the foundation for a theory which might explain when sequences of α-energy critical points for minimax constraints converge to minimal surfaces without bubbling as α → 1, a theory which should ultimately have important applications similar to those found in the Morse theory of geodesics. It has long been known that the extension of Morse theory to infinitedimensional manifolds is not really necessary for the study of geodesics. Bott expressed it this way in his 1982 survey on Morse theory [Bot82]: “I know of no aspect of the geodesic question where [the infinite-dimensional approach] is essential; however it clearly has some aesthetic advantages, and points the way for situations where finite-dimensional approximations are not possible....” On the other hand, finite-dimensional approximations suitable for studying the α-energy when α > 1 appear to break down as α → 1. This suggests that for any partial Morse theory of minimal surfaces, in contrast to closed geodesics, calculus on infinite-dimensional manifolds should play an essential conceptual and simplifying role. Our presentation in this book therefore starts with an extension of finitedimensional calculus to calculus on infinite-dimensional manifolds. We assume the reader is familiar with the basics of finite-dimensional differential geometry, including geodesics, curvature, and the tubular neighborhood theorem. We also assume some familiarity with basic complex analysis, the foundations of Banach and Hilbert space theory, and the willingness to accept results from the linear theory of elliptic partial differential operators, in particular, the theory of Fredholm operators on Sobolev spaces. All of these topics are treated very well in highly accessible sources, to which we can refer at the appropriate time. The first chapter gives an introduction to global analysis on infinitedimensional manifolds of maps. The second chapter studies the theory of geodesics on Riemannian manifolds using the action integral J : Map(S 1 , M ) → R

Preface

xiii

and owes much to the beautiful work of Bott, Gromoll, Klingenberg, and Meyer. This leads to the question of determining the topological invariants of the space Map(S 1 , M ) and motivates the exposition in Chapter 3 of Sullivan’s theory of minimal models, which provides an efficient means of calculating rational cohomology of Map(S 1 , M ). Sullivan and Vigu´e-Poirrier were able to use this theory to provide a striking extension of an earlier theorem of Gromoll and Meyer: Most compact smooth manifolds have infinitely many geometrically distinct smooth closed geodesics for arbitrary choice of Riemannian metric. More generally, Sullivan’s theory provides an algorithm for calculating the rational cohomology of Map(Σ, M ), when Σ is a compact manifold of arbitrary dimension and M is a nilpotent manifold. In Chapter 4, we turn to the theory of minimal surfaces in Riemannian manifolds and discuss well-known theorems of Sacks, Uhlenbeck, Schoen, and Yau which use minimal surfaces to elucidate the topology of threedimensional manifolds. We also describe Uhlenbeck’s Morse theory for the αenergy and its perturbations, which provide examples of energy functions for which complete Morse inequalities hold, relating critical points to rational cohomology of Map(Σ, M ), when Σ is a compact smooth surface. Just as a generic choice of proper function f : M → R on a smooth manifold M is necessary for Morse inequalities on finite-dimensional manifolds, so a generic choice of Riemannian metric on a compact manifold M is needed to simplify the relationships between the topology of M and the types of minimal surfaces within M . Chapter 5 gives a proof of the Bumpy Metric Theorem from [Moo06], which describes the generic behavior: It states that for generic choice of Riemannian metric on a smooth manifold M of dimension at least three, closed prime minimal surfaces f : Σ → M have no branch points and are as nondegenerate as allowed by their invariance under the group of complex automorphisms of Σ. This is complemented by a transversality theory which in accordance with Whitney’s theorems implies that for generic metrics minimal surfaces f : Σ → M are imbeddings when M has dimension at least five and immersions with transverse crossings when M has dimension four. We believe that the Bumpy Metric Theorem will be useful in establishing partial Morse inequalities for parametrized minimal surfaces of given genus when the dimension of the ambient manifold is sufficiently large. Such partial Morse inequalities require additional techniques not treated in this book, and we hope to return to this topic in a later publication. We intend to apply the Bumpy Metric Theorem to area-minimizing minimal surfaces in four-manifolds with generic metrics in a subsequent article [Moox], using the twistor degrees studied by Eells and Salamon [ES64].

xiv

Preface

We have left out many topics from the theory of minimal surfaces which are treated very well in existing sources. Readers unfamiliar with the rich theory of minimal surfaces in Euclidean space R3 should seek other sources, such as the excellent books of Osserman [O69] and Colding and Minnicozzi [CM11]. I would like to thank the students in my classes, and in particular Casey Blacker, Robert Ream, and Changliang Wang, for questions and remarks that helped correct minor errors and smooth out the exposition in several places. I am also grateful to anonymous readers of earlier versions of this book for their helpful suggestions and to Xianzhe Dai for his encouragement. Doug Moore Santa Barbara, CA, June 2017

Chapter 1

Infinite-dimensional Manifolds

1.1. A global setting for nonlinear DEs Linear differential equations are often fruitfully studied via techniques from linear functional analysis, including the theory of Banach and Hilbert spaces. In contrast, the proper setting for an important class of nonlinear differential equations is a nonlinear version of functional analysis, based upon infinitedimensional manifolds modeled on Banach or Hilbert spaces. The theory of such manifolds was developed by Eells, Palais and Smale among others in the 1950s and 1960s, and has proven to be extremely useful for understanding many of the nonlinear differential equations which arise in geometry, such as: (1) the equation for geodesics in a Riemannian manifold, (2) the equation for harmonic maps from surfaces into a Riemannian manifold, or for minimal surfaces in a Riemannian manifold, (3) the Yang-Mills equations for connections in principal bundles over a Riemannian manifold, (4) the equations for pseudoholomorphic or J-holomorphic curves in a symplectic manifold, (5) and the Seiberg-Witten equations. In all of these examples, the solutions can be expressed as critical points of a real-valued function (often called the action or the energy) defined on an infinite-dimensional manifold, such as the function space manifolds from one finite-dimensional manifold to another, as described in the following 1

2

1. Infinite-dimensional Manifolds

pages. In favorable cases, a gradient (or “pseudogradient”) of the action or energy can then be used to locate critical points (solutions to the differential equations) via what is often called the method of steepest descent. For this procedure to work, the topology on the function space must be strong enough for the action or energy to be differentiable, yet weak enough to force convergence of a subsequence of a sequence which is tending toward a minimum (or to a minimax solution for a given constraint). The two conflicting conditions often select a unique acceptable topology for the space of functions. In the most favorable circumstances, the topology is strong enough so that it lies within the space of continuous functions, a space which has been studied extensively by topologists. The function space is then often homotopy equivalent, that is equivalent in the sense of homotopy theory, to the space of continuous functions. In favorable cases, this makes existence questions within the theory of nonlinear differential equations accessible by topological methods. An example of this is the Morse theory of geodesics, which became a foundation for interaction between geometry and topology during the twentieth century. In the case of the Yang-Mills equations or the Seiberg-Witten equations over a four-dimensional base manifold, it is useful to reverse the logic. Instead of topology shedding light on existence of solutions to partial differential equations, it is the space of solutions to these equations that enables one to distinguish between different smooth structures on a given topological four-manifold. This illustrates that at a very fundamental level, topology and nonlinear partial differential equations are closely related, and underscores the importance of developing a general theory of partial differential equations founded upon infinite-dimensional manifolds.

1.2. Infinite-dimensional calculus Our first topic is the theory of infinite-dimensional manifolds. We refer to the excellent presentations of Lang [Lan95], Smale [Abr63] or Abraham, Marsden and Ratiu [AMR88] for further elaboration of topics only briefly introduced in the following pages. It was pointed out by Eells, Smale, Palais, Abraham, Lang and others in the 1960s that several variable calculus could be developed not just in finite-dimensional Euclidean spaces, but also with very little extra work within the context of infinite-dimensional Hilbert and Banach spaces, and that this should have important implications for the solution of nonlinear differential equations, particularly those equations coming from the calculus of variations. Gradually, infinite-dimensional calculus became part of the standard tool chest for global analysis. Most of the proofs in the Banach

1.2. Infinite-dimensional calculus

3

space setting are straightforward modifications of the proofs in Rn , so we will go very rapidly over these proofs. Definition. A pre-Hilbert space is a real vector space E together with a function ·, · : E×E → R which satisfies the following inner product axioms: (1) x, y = y, x, for x, y ∈ E. (2) ax, y = ax, y, for a ∈ R and x, y ∈ E. (3) x + y, z = x, z + y, z, for x, y, z ∈ E. (4) x, x ≥ 0, for x ∈ E, equality holding only when x = 0. These axioms simply state that x, y is a positive-definite symmetric bilinear form on E. Given  such a pre-Hilbert space, we can define a map  ·  : E → R by x = x, x. We can regard E as a metric space by defining the distance between elements x and y of E to be d(x, y) = x − y. Definition. A Hilbert space is a pre-Hilbert space which is complete in terms of the metric d. An example of a finite-dimensional Hilbert spaces is Euclidean space Rn , with the inner product ·, · being defined by (x1 , . . . , xn ), (y1 , . . . yn ) = x1 y1 + · · · + xn yn . An example of an infinite-dimensional Hilbert space is the space L2 ([0, 1], R) of equivalence classes of L2 functions studied in basic analysis courses. To construct it, one starts with defining an inner product ·, · on the space C ∞ ([0, 1], R) = { C ∞ functions f : [0, 1] → R } by



1

φ, ψ =

φ(t)ψ(t)dt. 0

We let L2 ([0, 1], R) denote the equivalence classes of Cauchy sequences from C ∞ ([0, 1], R), two Cauchy sequences {φi } and {ψi } being equivalent if for each  > 0 there is a positive integer N such that (1.1)

i, j > N

⇒

φi − ψj  < .

Equivalence classes of Cauchy sequences form a vector space, and we define an inner product ·, · on L2 ([0, 1], R) by {φi }, {ψi } = lim φi , ψi . i→∞

Finally, we check that with this inner product, L2 ([0, 1], R) is a Hilbert space.

4

1. Infinite-dimensional Manifolds

We say that L2 ([0, 1], R) is the completion of C ∞ ([0, 1], R) with respect to this inner product ·, ·. This process of completion is virtually the same as that often used to construct the real numbers from the rationals. Definition. A pre-Banach space is a vector space E together with a function  ·  : E → R which satisfies the following axioms: (1) ax = |a|x, when a ∈ R and x ∈ E. (2) x + y ≤ x + y, when x, y ∈ E. (3) x ≥ 0, for x ∈ E. (4) x = 0 only if x = 0. A function  ·  : E → R which satisfies the first three axioms is called a seminorm on E. If, in addition, it satisfies the fourth axiom it is called a norm. As in the case of Hilbert spaces, we can make E into a metric space by defining the distance between elements x and y of E to be d(x, y) = x − y. Definition. A Banach space is a pre-Banach space which is complete in terms of the metric d. Of course,  every Hilbert space is a Banach space with norm  ·  defined by x = x, x. Let C 0 ([0, 1], R) = { continuous functions f : [0, 1] → R }, and define (1.2)

 ·  : C 0 ([0, 1], R) → R by

f  = sup{|f (t)| : t ∈ [0, 1]}.

Then  ·  makes C 0 ([0, 1], R) into a Banach space with a norm that does not come from an inner product. More generally, we can consider the space C k ([0, 1], R) = { functions f : [0, 1] → R which have continuous derivatives up to order k }, a Banach space with respect to the norm (1.3) ·k : C k ([0, 1], R) → R defined by



f k = sup

k 

 |f (i) (t)| : t ∈ [0, 1] ,

i=0

where f (i) (t) denotes the derivative of f of order i. When 1 ≤ p < ∞ and p = 2, the spaces Lp ([0, 1], R) studied in basic analysis courses are Banach spaces which are not Hilbert spaces. To construct these spaces, one starts with defining a function  ·  on the space

1.2. Infinite-dimensional calculus

5

C ∞ ([0, 1], R) of C ∞ functions f : [0, 1] → R by  1 1/p φ = |φ(t)|p dt , 0

which is shown to be a norm by means of the Minkowski inequality. This agrees with the norm previously defined on L2 ([0, 1], R) when p = 2. Just as in the construction of L2 ([0, 1], R), one can use this norm to define Cauchy sequences, and let Lp ([0, 1], R) be the set of equivalence classes of Cauchy sequences, where two Cauchy sequences {φi } and {ψi } are equivalent if for each  > 0 there is a positive integer N such that (1.1) holds. The basic properties of Lp spaces are treated in standard references on functional analysis; thus for the H¨older and Minkowski inequalities, for example, one can refer to Theorem III.1 of [RS80]. Each Banach space E has a metric d : E × E → R defined by d(e1 , e2 ) = e1 − e2 , which makes E into a metric space, and we say that a subset U of E is open if p ∈ U ⇒ B (p) = {q ∈ E : d(p.q) < } ⊂ U, for some  > 0. A map T : E1 → E2 between Banach spaces is continuous if T −1 (U ) is open for each open U ⊂ E2 , or equivalently, if there is a constant c > 0 such that T (e1 )2 ≤ ce1 1 ,

for all e1 ∈ E1 ,

where  · 1 and  · 2 are the norms on E1 and E2 , respectively. We let L(E1 , E2 ) be the space of continuous linear maps T : E1 → E2 , a Banach space in its own right under the norm T  = sup{T (e1 ) : e1 ∈ E1 , e1  = 1}. It is easily shown that L(E1 , E2 ) is the same as the space of linear maps from E1 to E2 , which are bounded in terms of this norm. In particular, we can define the dual of a Banach space E to be E  = L(E, R). We say that a Banach space is reflexive if (E  ) is isomorphic to E. It is proven in analysis courses that Lp ([0, 1], R) is reflexive when 1 < p < ∞ but L1 ([0, 1], R) and C 0 ([0, 1], R) are not. Banach spaces and continuous linear maps form a category, as do Hilbert spaces and continuous linear maps. The subject known as functional analysis is concerned with the categories of Hilbert spaces, Banach spaces and more general spaces of functions, and is one of the major tools in studying linear partial differential equations. Key theorems from the theory of Banach spaces include the Open Mapping Theorem, the Hahn-Banach Theorem and the Uniform Boundedness Theorem. These theorems are discussed in

6

1. Infinite-dimensional Manifolds

[RS80], [Roy88] and [Rud86]; we will need to use the statements of these theorems and their consequences. The Open Mapping Theorem states that if T : E1 → E2 is a continuous surjective map between Banach spaces, it takes open sets to open sets. Thus if T is a continuous bijection, its inverse is continuous. The Hahn-Banach Theorem implies that if e is a nonzero element of a Banach space E, then there is a continuous linear function λ : E → R such that λ(e) = 0. A bilinear map B : E1 × E2 → F is said to be continuous if there is a constant c > 0 such that B(e1 , e2 ) ≤ ce1 1 e2 2 ,

for all e1 ∈ E1 and all e2 ∈ E2 .

One of the useful consequences of the Uniform Boundedness Theorem is that such a Bilinear map is continuous is and only if B(·, e2 ) : E1 → F

and

B(e1 , ·) : E2 → F

are continuous for each e1 ∈ E1 and each e2 ∈ E2 . The three theorems are almost trivial to prove for finite-dimensional Banach spaces, but the proofs are more subtle for infinite-dimensional Banach spaces and rely on the axiom of choice. Definition. A Banach algebra is a Banach space E together with an associative product μ : E × E → E such that if x · y = μ(x, y), then x · y ≤ xy,

for x, y ∈ E.

For example, the Banach space C 0 ([0, 1], R) with norm given by (1.2) is a commutative Banach algebra with unit when μ is the ordinary multiplication of functions. We now turn to the question of how to develop differential calculus for functions defined on Banach spaces. It is actually the topology, or the equivalence class of norms on the Banach space, that is important for calculus, two norms  · 1 and  · 2 on a linear space E being equivalent if there is a constant c > 1 such that 1 x1 < x2 < cx1 , for x ∈ E. c Lang [Lan95] calls a vector space E a Banachable space if it is endowed with an equivalence class of Banach space norms. However, most other authors do not use this term, and we will usually use the simpler term Banach space, it being understood, however, that we may sometimes pass to an equivalent norm in the middle of an argument, when it is the underlying vector space with its topology, the “topological vector space”—not the norm itself—that is important.

1.2. Infinite-dimensional calculus

7

Definition. Suppose that E1 and E2 are Banach spaces, and that U is an open subset of E1 . A continuous map f : U → E2 is said to be differentiable at the point x0 ∈ U if there exists a continuous linear map T : E1 → E2 such that f (x0 + h) − f (x0 ) − T (h) lim = 0, h h→0 where  ·  denotes both the Banach space norms on E1 and E2 . We will call T the (Fr´echet) derivative of f at x0 and write Df (x0 ) for T . Just as in ordinary calculus, the derivative Df (x0 ) determines the linearization of f near x0 , which is the affine function f˜(x) = f (x0 ) + Df (x0 )(x − x0 ) which most closely approximates f near x0 . If f is differentiable at every x ∈ U , a derivative Df (x) is defined for each x ∈ U and thus we have a set-theoretic map Df : U → L(E1 , E2 ). If this map Df is continuous, we can also ask whether it is differentiable at x0 ∈ U . This will be the case if there is a continuous linear map T : E1 → L(E1 , E2 ) such that Df (x0 + h) − Df (x0 ) − T (h) = 0, lim h h→0 in which case we write D 2 f (x0 ) for T and call D 2 f (x0 ) the second derivative of f at x0 . Note that D 2 f (x0 ) ∈ L(E1 , L(E1 , E2 )) = L2 (E1 , E2 ) = {continuous bilinear maps T : E1 × E1 → E2 }, which can also be made into a Banach space in an obvious way. We say that a function f : U → E2 , where U is an open subset of E1 , is (1) C 0 if it is continuous. (2) C 1 if it is continuous and differentiable at every x ∈ U , and Df : U → L(E1 , E2 ) is continuous. (3) C k for k ≥ 2 if it is C 1 and Df : U → L(E1 , E2 ) is C k−1 . (4) C ∞ or smooth if it is C k for every nonnegative integer k. With the above definition of differentiation, many of the arguments in several variable calculus can be transported without difficulty to the Banach space setting, as carried out in detail in [Lan95] or [AMR88]. For example, the Leibniz rule for differentiating a product carries over immediately to the infinite-dimensional setting.

8

1. Infinite-dimensional Manifolds

Proposition 1.2.1. Suppose that B : F1 × F2 → G is a continuous bilinear map between Banach spaces, that U is an open subsets of a Banach space E and f : U1 −→ F1 , f2 : U2 −→ F2 are C 1 maps. Then e → g(e) = B(f1 (e), f2 (e)): is a C 1 map, and Dg(x0 )h = B(Df1 (x0 )h, f2 (e)) + B(f1 (x0 )Df2 (x0 )h). Sketch of proof. To simplify notation, we write g(x) = B(f1 (x), f2 (x)) = f1 (x) · f2 (x). Then g(x + h) − g(x) = (f1 (x + h) − f1 (x))f2 (x + h) + f1 (x)(f2 (x + h) − f2 (x)), and hence f1 (x + h) − f1 (x) f2 (x + h) − f2 (x) g(x + h) − g(x) = f2 (x + h) + f1 (x) . h h h The Proposition follows by taking the limit as h → 0. QED The following lemma is called the chain rule. Proposition 1.2.2. If U1 and U2 are open subsets of Banach spaces E1 and E2 and g : U2 −→ E3 f : U1 −→ U2 , are C 1 maps, then so is g ◦ f : U1 → E3 , and D(g ◦ f )(x0 ) = Dg(f (x0 ))Df (x0 ),

for x0 ∈ U1 .

Sketch of proof. We have f (x0 + h) = f (x0 ) + Df (x0 )h + o(h), where the symbol o(h) stands for an element in E2 such that o(h)/h → 0 as h → 0. Similarly, g(f (x0 ) + k) = g(f (x0 )) + Dg(f (x0 ))(k) + o(k). Setting k = Df (x0 )h + o(h) yields g(f (x0 + h)) = g(f (x0 )) + Dg(f (x0 ))(Df (x0 )h + o(h)) + o(k). One checks without difficulty using continuity of g that an o(k) expression, where k is a bounded linear function of h, is also o(h), and hence g(f (x0 + h)) = g(f (x0 )) + Dg(f (x0 ))Df (x0 )h + o(h), which gives the desired conclusion. QED

1.2. Infinite-dimensional calculus

9

By induction, one immediately shows that the composition of C k maps is C k and the composition of C ∞ maps is C ∞ . Another familiar theorem from several variable calculus in finite dimension is the “equality of mixed partials”. To state the infinite-dimensional version, we let E and F be Banach spaces, U an open subset of E. Suppose that f : U → F is a C 2 map, so that for x0 ∈ U , D 2 f (x0 ) ∈ L(E, L(E, F )) = L2 (E, F ). Of course, a very important case is the one where F = R, the base field of real numbers. Proposition 1.2.3. D 2 f (x0 ) is symmetric; in other words, for h, k ∈ E.

D 2 f (x0 )(h, k) = D 2 f (x0 )(k, h),

Sketch of proof. First note that by the Hahn-Banach theorem, it suffices to show that if λ : F → R is any continuous linear functional, then D 2 (λ ◦ f )(x0 ) = λ◦D 2 f (x0 ) is symmetric, because if D 2 f (x0 ) is not symmetric, the same will be true of λ ◦ D 2 f (x0 ), for some linear function λ. This reduces the proof to the case where F = R. Next note that via the chain rule,  1 f (x + h) − f (x) = (Df )(x + th)hdt, 0

and by iteration, 

1

(Df )(x + th + k)hdt f (x + h + k) − f (x + k) − f (x + h) + f (x) = 0  1 1 = (D(Df )(x + th + sk)(k))hdsdt. 0

0

Interchanging h and k yields  1 (Df )(x + tk + h)kdt f (x + h + k) − f (x + h) − f (x + k) + f (x) = 0  1 1 (D(Df )(x + sk + th)(h))kdsdt. = 0

0

Since the left-hand sides of the last two expressions are equal, so are the right-hand sides, and hence  1 1 [(D(Df )(x + th + sk)(k))h − (D(Df )(x + th + sk)(h))k]dsdt = 0. 0

0

Since D(Df ) is a continuous function, this can only happen if D 2 f (x)(k, h) = D 2 f (x)(h, k), for all x, h and k, which is exactly what we needed to prove. QED

10

1. Infinite-dimensional Manifolds

More generally, if f : U → F is a C k -map, then D k f (x0 ) = D(D k−1 f )(x0 ) ∈ L(E, Lk−1 (E, F )) = Lk (E, F ), and by an induction based on the previous lemma, we see that in fact D k f (x0 ) ∈ Lks (E, F ) = {T ∈ Lk (E, F ) : T is symmetric}. By symmetric we mean that T (hσ(1) , . . . , hσ(k) ) = T (h1 , . . . , hk ), for all permutations σ in the symmetric group Sk on k letters. It is often useful to have an explicit formula for the higher derivatives of a composition. The following such formula comes from §3 of [Abr63]. If p and k are positive integers with k ≤ p and (i1 , . . . , ik ) is a k-tuple of positive integers such that i1 + · · · + ik = p and i1 ≤ i2 ≤ · · · ik , we define integers σkp (i1 , . . . , ik ) recursively by requiring that σ11 (1) = 1 and p−1 σkp (i1 , . . . , ik ) = δi11 σk−1 (i2 , . . . , ik ) +

k 

σkp−1 (i1 , . . . , ij + 1, . . . ik ),

j=1

where δi11 is 1 if i1 = 1 and 0 otherwise. Proposition 1.2.4. Suppose that U and V are open subsets of Banach spaces E and F , respectively, and that f : U → V and g : V → G are C p maps, where G is a third Banach space. Then D p (g ◦ f ) =

p 

(D k g ◦ f )Pkp (f ),

k=1

where Pkp (f ) =



σkp (i1 , . . . , ik )(D i1 f, . . . , D ik f ),

the sum being taken over all p-tuples of positive integers such that i1 + · · · + ik = p and i1 ≤ i2 ≤ · · · ≤ ik . The proof of Proposition 1.2.5 is by induction on p starting with the case p = 1, which is an immediate application of the chain rule. For p ≥ 2, one obtains the formula for σkp and the expression for D p (g ◦ f ) by applying the chain rule and the Leibniz rule for differentiating a product. Definition of integration along a path. In order to put Taylor’s theorem in the Banach space setting, we need to define the integral of a continuous map γ : [0, 1] → E into a Banach space E. The definition is particularly easy if E is a reflexive Banach space; in this case, we just set  1  1 γ(t)dt = e, where λ(e) = λ ◦ γ(t)dt, for λ ∈ E ∗ . (1.4) 0

0

1.2. Infinite-dimensional calculus

11

A definition of the integral for general Banach spaces can be found in §1.4 of [Lan95]. Let U be an open subset of a Banach space E. Following [AMR88], we ⊂ E × E such that define a thickening of U to be an open subset U , and (1) U × {0} ⊂ U ⇒ x + th ∈ U , for t ∈ [0, 1]. (2) (x, h) ∈ U Proposition 1.2.5 (Taylor’s Theorem). If a map f : U → F is of class C r , there exist continuous maps φk : U → Lks (E, F ),

→ Lr (E, F ), and R : U s

for 1 ≤ k ≤ r

˜ is a thickening of U , such that R(x, 0) = 0 and where U f (x + h) = f (x) + φ1 (x)h +

1 1 φ2 (x)(h, h) + · · · + φr (x)(h, . . . , h) 2! r! + R(x, h)(h, . . . , h).

Here φk (x) = (D k f )(x), for 1 ≤ k ≤ r. Sketch of proof. We first reduce to the case where F = R by applying the Hahn-Banach Theorem. One then uses the fundamental theorem of calculus to establish the first order Taylor formula  1 (Df )(x + th)hdt (1.5) f (x + h) = f (x) + 0  1 [(Df )(x + th) − (Df )(x)]hdt. = f (x) + (Df )(x)h + 0

Using the fundamental theorem of calculus again, one next obtains  1 1 [(D 2 f )(x + sth)](h, h)tdsdt f (x + h) = f (x) + (Df )(x)h + 0

0

1 = f (x) + (Df )(x)h + (D 2 f )(x)(h, h) 2!  1 1 [(D 2 f )(x + sth) − D 2 f (x)](h, h)tdsdt. + 0

0

Continuing in the same manner, we find that 1 2 (D f )(x)(h, h) 2! 1 + · · · + (D k f )(x)(h, . . . , h) + R(x, h)(h, . . . , h), k! where R(x, h) ∈ Lks (E, F ) depends continuously on x and h and R(x, 0) = 0. QED f (x + h) = f (x) + (Df )(x)h +

12

1. Infinite-dimensional Manifolds

Example 1.2.6. We suppose that the domain of the function is the Banach space E = Lp (S 1 , RN ), the completion of the space C ∞ (S 1 , RN ) of smooth RN -valued functions on S 1 with respect to the Lp -norm  1/p p φLp = |φ(t)| dt , for φ ∈ Lp (S 1 , RN ). S1

Here S 1 is regarded as the quotient of the interval [0, 1] obtained by identifying the points 0 and 1, and possesses the standard measure dt with respect to which S 1 has measure one. A useful tool for dealing with functions in the older inequality which states: if φ ∈ Lp , ψ ∈ Lq and Lp spaces is the H¨ 1 1 1 + = where p, q, r ≥ 1, p q r

then φψ ∈ Lr and φψLr ≤ φLp ψLq .

Using this inequality and the chain rule, it is not difficult to show that when p ≥ 2, the function  (1 + |φ(t)|2 )p/2 dt f : E −→ R defined by f (φ) = S1

is C 2 and that its first and second derivatives are given by the formulae  (1 + |φ(t)|2 )p/2−1 φ(t) · ψ(t)dt Df (φ)(ψ) = p S1

and

 2

(D f )(φ)(ψ1 , ψ2 ) = p S1

(1 + |φ(t)|2 )p/2−1 ψ1 (t) · ψ2 (t)dt



+ p(p − 2)

S1

(1 + |φ(t)|2 )p/2−2 (φ(t) · ψ1 (t))(φ(t) · ψ2 (t))dt.

To verify these formulae, we apply the chain rule to f = h ◦ g, where h is integration over S 1 and g : E → L1 (S 1 , RN )

by

g(φ)(t) = (1 + |φ(t)|2 )p/2 .

To check that g is C 1 , we let h : RN → R by

h(u) = (1 + u · u)p/2

and apply Taylor’s formula (1.5) to obtain, for a fixed choice of t, (1.6) h(φ(t) + ψ(t)) = h(φ(t)) + (Dh)(φ(t))ψ(t)  1 [(Dh)(φ(t) + sψ(t)) − (Dh)(φ(t))]ψ(t)ds, + 0

where Dh(φ(t)) = p(1 + |φ(t)|2 )p/2−1 φ(t).

1.2. Infinite-dimensional calculus

13

It then follows from (1.6) that (1.7)

g(φ + ψ)(t) = g(φ)(t) + T (φ)(t)ψ(t)  1 + [T (φ + sψ) − T (φ)](t)ψ(t)ds, 0

where T (φ)(t) = Dh(φ(t)). If (1/p) + (1/q) = 1, one can show that the H¨ older inequality provides a continuous map T : Lp (S 1 , RN ) → Lq (S 1 , RN ), and hence given any ε > 0, we can make [T (φ + sψ) − T (φ)]Lq < ε by choosing ψLp is sufficiently small. Thus the integral term in (1.7) is o(ψ) and the derivative of g is Dg(φ)(ψ) = T (φ)ψ = (1 + |φ|2 )p/2−1 φ · ψ. Moreover, continuity of T implies that g : E → L1 and f : E → R are C 1 . A similar argument using the second order Taylor expansion shows that f is C 2 . On the other hand, if f were C 3 in the case N = 1, one could verify that the third derivative of g would be given by the formula (D 3 g)(ψ1 , ψ2 , ψ3 ) = 2p(p − 2)(1 + |φ|2 )p/2−2 φψ1 ψ2 ψ3 + p(p − 2)(p − 4)(1 + |φ|2 )p/2−3 φ3 ψ1 ψ2 ψ3 , and if p < 3, for a suitable smooth choice of φ, we could choose ψ1 , ψ2 and ψ3 in Lp such that the product ψ1 ψ2 ψ3 is not in L1 . This implies that f cannot be C 3 when 2 < p < 3. We have seen that many of the basic results of differential calculus of several variables extend with little change to the infinite-dimensional context. The following theorem is somewhat deeper: Theorem 1.2.7 (Inverse Function Theorem). If U1 and U2 are open subsets of Banach spaces E1 and E2 with x0 ∈ U1 , and f : U1 → U2 is a C ∞ map such that Df (x0 ) ∈ L(E1 , E2 ) is invertible, then there are open neighborhoods V1 of x0 and V2 of f (x0 ), and a C ∞ map g : V2 → V1 , such that and g ◦ f = idV1 . f ◦ g = idV2 −1 Moreover, Dg(f (x)) = [Df (x)] , for x ∈ V1 . Sketch of proof. We can assume without loss of generality that x0 = 0 ∈ U1 and f (0) = 0 ∈ U2 . We can assume, moreover, that E1 = E2 and Df (0) is the identity map by replacing f by Df (0)−1 ◦ f . Define h : U1 → E1 by

14

1. Infinite-dimensional Manifolds

h(x) = x − f (x). Then Dh(0) = 0, and by continuity of Dh there exists δ > 0 such that 1 x ≤ δ ⇒ x ∈ U1 and dh(x) < . 2 If x, y < δ, then it follows from the chain rule that

 1



d

(h(tx + (1 − t)y))dt

h(x) − h(y) =

0 dt

 1



Dh(tx + (1 − t)y)(x − y)dt

=

0  1  1 ≤ Dh(tx + (1 − t)y)dt x − y < x − y, 2 0 so h decreases distances. More generally, if y ∈ E1 and y < δ/2, we can define the map hy : U1 → E1 by hy (x) = h(x) + y. Then x ≤ δ

⇒

hy (x) ≤ y + h(x)
0, there is a N such that i, j ≥ N implies that xi −xj k <  for all k, then there is an element x ∈ E such that xi − xk converges to zero for all k. Of course every Banach space is a Fr´echet space, but C ∞ ([0, 1], R) with the collection of norms defined above is a Fr´echet space which is not a Banach space. Given a Fr´echet space E with seminorms { · k : k ∈ Z, k ≥ 0}, we can then define a bounded metric ∞  1 x − yk d : E × E −→ R by d(x, y) = , 2k 1 + x − yk k=0

which one can check is complete. This metric defines a topology, so we can talk of continuous maps f : E1 → E2 from the Fr´echet space E1 to the Fr´echet space E2 , and we have a category consisting of Fr´echet spaces and continuous linear maps. Definition. Suppose that E1 and E2 are Fr´echet spaces, and that U is an open subset of E1 . A continuous map f : U → E2 is said to be continuously differentiable on U if there exists a continuous map Df : U × E1 → E2 such that f (x + ty) − f (x) , Df (x)y = lim t→0 t

1.3. Manifolds modeled on Banach spaces

17

where t ranges throughout R − {0}, it being understood that the limit on the right-hand side exists for all x ∈ U and all y ∈ E1 . It is proven in Hamilton’s survey article [Ham82], Part I, 3.2.3 and 3.2.5, that if f : U → E2 is continuously differentiable, the map y → Df (x)y is linear. Thus, if E1 and E2 are Banach spaces, the above definition agrees with the definition previously given. We could develop much of the infinite-dimensional calculus and the theory of infinite-dimensional manifolds in the category of Fr´echet spaces, and in fact this is carried out in [Ham82], but the Inverse Function Theorem does not hold for Fr´echet spaces. Moreover, in the existence theory for solutions to nonlinear partial differential equations, it is often convenient to first prove existence in a given Banach space and then prove regularity using bootstrapping, just as we did in the proof of the Inverse Function Theorem. This technique seems particularly well-adapted to Banach spaces. For these reasons, we prefer to think of C ∞ ([0, 1], R) as the intersection of a “chain” of Banach spaces, · · · ⊆ C k+1 ([0, 1], R) ⊆ C k ([0, 1], R) ⊆ C k−1 ([0, 1], R) ⊆ · · · ⊆ C 0 ([0, 1], R). For proving theorems, we will usually work in a suitable Banach spaces so that we can use theorems like the Inverse Function Theorem. However, the statements of theorems are sometimes more elegant when phrased in terms of a Fr´echet space, or a chain of Banach spaces.

1.3. Manifolds modeled on Banach spaces Definition. Let E be a separable Banach space. A connected smooth manifold modeled on E is a connected second countable Hausdorff space M together with a collection A = {(Uα , φα ) : α ∈ A}, where each Uα is an open subset of M and each φα : Uα −→ φα (Uα ) ⊂ E is a homeomorphism onto an open subset φα (Uα ) of E such that:  (1) {Uα : α ∈ A} = M. ∞ (2) φβ ◦ φ−1 α : φα (Uα ∩ Uβ ) → φβ (Uα ∩ Uβ ) is C , for all α, β ∈ A.

A nonconnected Hausdorff space is called a smooth manifold or a Banach manifold if each component is a connected smooth manifold modeled on some Banach space. (We allow different components to be modeled on different Banach spaces.) The smooth manifold is called a Hilbert manifold if each component is modeled on a Hilbert space.

18

1. Infinite-dimensional Manifolds

We say that A = {(Uα , φα ) : α ∈ A} is the atlas defining the smooth structure on M, and each (Uα , φα ) is one of the charts in the atlas. Remark. We could define Fr´echet manifolds by simply replacing the phrase “Banach space” by “Fr´echet space” in the above definition. Let M and N be smooth manifolds modeled on Banach spaces E and F , respectively. Suppose that M and N have atlases A = {(Uα , φα ) : α ∈ A} and B = {(Vβ , ψβ ) : β ∈ B}. Then a continuous map F : M → N is ∞ said to be smooth if ψβ ◦ F ◦ φ−1 α is C , where defined, for α ∈ A and β ∈ B. It follows from the chain rule that the composition of smooth maps is smooth. In this way we obtain a category whose objects are smooth manifolds modeled on Banach spaces and whose morphisms are smooth maps between such manifolds. As in the case of finite-dimensional manifolds, a diffeomorphism is a smooth map between manifolds with smooth inverse. We will often identify two smooth manifolds if there is a diffeomorphism from one to the other. Later we will construct invariants (such as de Rham cohomology) that will often enable us to determine that two infinite-dimensional manifolds cannot be diffeomorphic. Of course, the simplest example of a smooth manifold modeled on E is an open subset U of E in which the atlas is {(U, idU )}. However, the examples of most interest to us will be function spaces. Key Example. Suppose that M n is a complete Riemannian manifold of finite dimension n, which we can assume is isometrically imbedded in Euclidean space RN in accordance with the celebrated Nash imbedding theorem [N56]. Suppose that Σ is a compact smooth manifold of finite dimension m. Then C 0 (Σ, RN ) = {continuous maps f : Σ → RN } is a Banach space, and we claim that the subspace C 0 (Σ, M ) = {continuous maps f : Σ → M } ⊆ C 0 (Σ, RN ) is an infinite-dimensional Banach manifold, which is modeled on the space of continuous sections of the pullback f ∗ T M of the tangent bundle of M to Σ. To define the model space, we let f ∗ T M = {(p, v) ∈ Σ × T M : f (p) = π(v)}, where π : T M → M is the usual projection, which is the total space of a vector bundle over Σ, which we call the pullback bundle. For simplicity, we often denote this bundle by f ∗ T M . A continuous section X of f ∗ T M can

1.3. Manifolds modeled on Banach spaces

19

be identified with a continuous map X : Σ → TM

such that X(p) ∈ Tp M,

for p ∈ Σ.

Theorem 1.3.1. If Σ and M are finite-dimensional smooth connected manifolds, with Σ compact and M having a complete Riemannian metric, then C 0 (Σ, M ) is a smooth manifold, the various components modeled on the Banach spaces C 0 (f ∗ T M ) = {continuous sections of f ∗ T M }, for choice of smooth map f : Σ → M . Here the norm on C 0 (f ∗ T M ) is XC 0 = sup{|X(p)| : p ∈ Σ}, the | · | on the right being length as defined by the Riemannian metric on M. To construct the charts on C 0 (Σ, M ) we use the exponential map of M : Definition of exponential chart. For given choices of a smooth map f : Σ → M and ε > 0, we set Vf,ε = {X ∈ C 0 (f ∗ T M ) : XC 0 < ε}, and Uf,ε = {g ∈ C 0 (Σ, M ) : dM (g(p), f (p)) < ε for all p ∈ Σ}, where dM : M × M → R is the distance function on M defined by the Riemannian metric. We choose ε > 0 sufficiently small that for each p ∈ Σ, expf (p) : {v ∈ Tf (p) M : |v| < ε} −→ {q ∈ M : dM (q, f (p)) < ε} is a diffeomorphism. We then define a bijection ψf,ε : Vf, → Uf, by ψf,ε (X)(p) = expf (p) (X(p)), and its inverse φf,ε : Uf, → Vf, by φf (g)(p) = exp−1 f (p) (g(p)). We say that (Uf,ε , φf,ε ) is an exponential chart of radius ε for C 0 (Σ, M ) centered at the smooth map f : Σ → M . Proof of Theorem 1.3.1. (Our proof follows the presentation in Abraham [Abr63].) First note that since C 0 (Σ, RN ) is second countable and Hausdorff, so is its subspace C 0 (Σ, M ). As atlas we take A = {(Uf,ε , φf,ε ) : (Uf,ε , φf,ε ) is an exponential chart} . Since smooth maps f : Σ → M are dense in C 0 (Σ, M ), the union of the elements of A cover C 0 (Σ, M ). Thus to verify that C 0 (Σ, M ) is a smooth

20

1. Infinite-dimensional Manifolds

manifold, we need only check that φf2 ,ε2 ◦(φf1 ,ε1 )−1 is smooth where defined, when (Uf1 ,1 , φf1 ,1 ), (Uf2 ,2 , φf2 ,2 ) ∈ A. Let U be the open subset of the total space of f1∗ T M on which the composition (expf2 (p) )−1 ◦ expf1 (p) is defined, and note that U ∩ Tf1 (p) M is convex with compact closure for each p ∈ Σ. Define g : U → (total space of f2∗ T M ) by g(p, v) = (p, (expf2 (p) )−1 ◦ expf1 (p) (v)),

for p ∈ Σ, v ∈ Tf1 (p) M .

We can think of g as a family of smooth maps p → g˜p : Tf1 (p) M ∩ U → Tf2 (p) M, and since Σ is compact and U has compact closure in each fiber, all derivagp ) are bounded on U . Noting that U is an open neighborhood of tives D k (˜ the image of the zero-section, we set ˜ = {X ∈ C 0 (f ∗ T M ) : X(Σ) ⊆ U } U 1 ˜ → C 0 (f ∗ T M ) by ωg (X) = g ◦ X. To finish the proof that and define ωg : U 2 0 C (Σ, M ) is a smooth manifold, it suffices to show that ωg is smooth. This is a special case of what is sometimes called the “ω-lemma”: Lemma 1.3.2. Suppose that E1 and E2 are finite-dimensional vector bundles over the compact smooth manifold Σ and that U is a bounded open neighborhood of the image of the zero section in E1 whose restriction to each fiber of E1 is convex. If g : U → (total space of E2 ) is a smooth map which takes the fiber of E1 over p to the fiber of E2 over p (for each p ∈ Σ), and ˜ = {σ ∈ C 0 (E1 ) : σ(Σ) ⊆ U }, U ˜ → C 0 (E2 ), defined by ωg (σ)(p) = g(σ(p)) for p ∈ Σ, is then the map ωg : U also smooth. As in the example above, we can think of g as providing a smooth collection of maps p → g˜p : (E1 )p ∩ U −→ (E2 )p , where (E1 )p and (E2 )p are the fibers of E1 and E2 at p ∈ Σ. Proof. It is straightforward to show that ωg is continuous. The idea for proving that ωg is C 1 is to use the first-order pointwise Taylor expansion for each g˜p to determine a corresponding Taylor expansion on sections. Suppose

1.3. Manifolds modeled on Banach spaces

21

˜ . It follows that σ and η are C 0 sections of E1 with σ and σ + η lying in U from Taylor’s theorem for any given p ∈ Σ that gp (σ(p))η(p) + R(σ, η)(p)(η(p)), g˜p ((σ + η)(p)) = g˜p (σ(p)) + D˜ where (1.8)



1

R(σ, η)(p)(η(p)) =

 [D˜ gp ((σ + tη)(p)) − D˜ gp (σ(p))]dt η(p).

0

We can write this on the section level as ωg (σ + η) = ωg (σ) + T (σ)(η) + R(σ, η)η, where T (σ)(η)(p) = D˜ gp (σ(p))η(p), and R(σ, η)η is the remainder term given by (1.8). Note that gp (σ(p))η(p)| ≤ sup |D˜ gp (σ(p))|ηC 0 , T (σ)(η)C 0 = sup |D˜ p∈Σ

p∈Σ

so T (σ) is a continuous linear map from a neighborhood of 0 in C 0 (E1 ) to C 0 (E2 ), and it extends to a linear map from C 0 (E1 ) to C 0 (E2 ). To see that ωg has a derivative at σ given by Dωg (σ) = T (σ),

(1.9)

Dωg (σ)(η)(p) = D˜ gp (σ(p))η(p),

we need to estimate the error term R(σ, η)η. But uniform continuity of Dg ¯ shows that over U  1 sup [D˜ gp ((σ + tη)(p)) − D˜ gp (σ(p))]dt ≤ (constant)ηC 0 , p∈Σ

0

so we get the needed estimate R(σ, η)ηC 0 ≤ (constant)η2C 0 = o(η), ¯, so T (σ) is indeed the derivative. Since D˜ gp is uniformly continuous on U it is relatively straightforward to show that Dωg is continuous, establishing that ωg is C 1 . We would like to extend this argument to higher derivatives, and for this we factor the derivative given by (1.9): Recalling that g : U → (total space of E2 ), we define a corresponding map

: U → (total space of L(E1 , E2 )) Dg

by setting Dg(v) = D˜ gp (v),

for v ∈ (E1 )p ∩ U .

as a “partial derivative” of g in which the point p ∈ Σ (We can regard Dg is held fixed.) We can then define 0 ˜ ωDg  : U → C (L(E1 , E2 )) by

ωDg gp (σ(p)).  (σ)(p) = D˜

22

1. Infinite-dimensional Manifolds

Using the fact that C 0 (Σ, R) is a Banach algebra, we can show that the map A : C 0 (L(E1 , E2 )) → L(C 0 (E1 ), C 0 (E2 )) defined by

A(T )(σ)(p) = T (p) · σ(p),

is continuous linear, and this provides the desired factorization, Dωg = A · ωDg . The first-order Taylor expansion argument presented in the preceding para is C ∞ , ω  is C 1 , from which we conclude graph now shows that since Dg Dg that Dωg is C 1 , and hence ωg is C 2 . Now we use an induction on k ∈ N: k ωDg  is C

→

Dωg is C k

→

ωg is C k+1 .

This then implies that ωg is C ∞ , and proves Lemma 1.3.2 and Theorem 1.3.1. QED The above lemma also has the following consequence: Theorem 1.3.3. If g : M → N is a smooth map, then the map ωg : C 0 (Σ, M ) → C 0 (Σ, N )

defined by

ωg (f ) = g ◦ f,

is also smooth. To prove the “moreover” part of the theorem, we need to show that if f1 : Σ → M and f2 : Σ → N are smooth, then φf2 ,ε2 ◦ ωg ◦ φ−1 f1 ,ε1

is smooth where defined.

The proof of this is a straightforward application of Lemma 1.3.2. Unfortunately, the manifolds C 0 (Σ, M ) are not sufficient for constructing a global theory of partial differential equations. We need to be able to differentiate elements in our function spaces. Thus we need to start off with a somewhat stronger Banach space than the space C 0 (Σ, R) of continuous real-valued functions on Σ. Thus for k ∈ N, we are led to consider the space C k (Σ, R) of real-valued functions on Σ which have continuous derivatives up to order k, a Banach space with respect to the norm (1.10)

f C k = sup{f (p) + Df (p) + · · · + D k f (p) : p ∈ Σ}.

In fact, it is easily checked that if f, g ∈ C k (Σ, R), then (1.11)

f gC k ≤ (constant)f C k gC k ,

where (constant) denotes a positive constant, so after possibly replacing  · C k with an equivalent norm, C k (Σ, R) is in fact a Banach algebra.

1.3. Manifolds modeled on Banach spaces

23

More generally, we can consider the space C k (Σ, RN ) of RN -valued functions on Σ which have continuous derivatives up to order k, which is also a Banach space with norm defined by (1.10). The Banach algebra condition (1.11) ensures that we can define a continuous multiplication μ : C k (Σ, L(RN , RM )) × C k (Σ, RN ) −→ C k (Σ, RM ) by simply multiplying in the range. If M is an n-dimensional Riemannian manifold which is isometrically imbedded in RN , we let C k (Σ, M ) = {f ∈ C k (Σ, RN ) : f (p) ∈ M for all p ∈ Σ }. We claim that C k (Σ, M ) is a smooth manifold. Theorem 1.3.4. If Σ is a compact smooth manifold and M is a complete Riemannian manifold, both connected and of finite dimension, then C k (Σ, M ) is a smooth manifold, the various components modeled on the Banach spaces C k (f ∗ T M ) = {C k sections of f ∗ T M }, where f : Σ → M is a smooth map. Here the norm on C k (f ∗ T M ) is XC k = sup{|X(p)| + |∇X| + · · · + |∇k X| : p ∈ Σ}, where ∇ is the Levi-Civita connection and the | · | is length defined by the Riemannian metric on M . To prove this, we first note that since C k (Σ, RN ) is second countable and Hausdorff, so is its subspace C k (Σ, M ). The charts are just restrictions of exponential charts for C 0 (Σ, M ): Suppose that f : Σ → M is C ∞ and (Uf,ε , φf,ε ) is an exponential chart of radius ε centered at f . We then set k = {X ∈ C k (f ∗ T M ) : XC 0 < ε} = Vf,ε ∩ C k (f ∗ T M ), Vf,ε

and k Uf,ε = {g ∈ C k (Σ, M ) : dM (g(p), f (p)) < ε for all p ∈ Σ} = Uf,ε ∩C k (Σ, M ). k : V k → U k by We define ψf,ε f, f, k (X)(p) = expf (p) (X(p)) ψf,ε k → V k by and its inverse φkf,ε : Uf, f,

φf (g)(p) = exp−1 f (p) (g(p)). k , φk ) for C k (Σ, M ). This gives an exponential chart (Uf,ε f,ε

As smooth atlas for C k (Σ, M ), we take   k k , φk ) is an exponential chart on C k (Σ, M ) . , φkf,ε ) : (Uf,ε A = (Uf,ε f,ε

24

1. Infinite-dimensional Manifolds

Since smooth maps f : Σ → M are dense in C k (Σ, M ), the union of the elements of A cover C k (Σ, M ), and to verify that it is a smooth manifold, we need only check that φkf2 ,ε2 ◦ (φkf1 ,ε1 )−1 is smooth where defined, when (Ufk1 ,ε1 , φkf1 ,ε1 ), (Ufk2 ,ε2 , φkf2 ,ε2 ) ∈ A. But this follows from a corresponding ω-lemma: Lemma 1.3.5. Suppose that E1 and E2 are finite-dimensional vector bundles over the compact smooth manifold Σ and that U is a bounded open neighborhood of the zero section of E1 whose restriction to each fiber of E1 is convex. If g : U → (total space of E2 ) is a smooth map which takes the fiber of E1 over p to the fiber of E2 over p (for each p ∈ Σ), and ˜ = {σ ∈ C k (E1 ) : σ(Σ) ⊆ U }, U ˜ → C k (E2 ), defined by ωg (σ) = g(σ), is also smooth. then the map ωg : U The proof is virtually identical to the proof given for Lemma 1.3.2. The proof extends to C k maps because the space C k (Σ, R) has two key properties: (1) it is a Banach algebra, and (2) there is a continuous inclusion from C k (Σ, R) into the Banach algebra C 0 (Σ, R) of continuous functions. Moreover, as before, we have the following: Theorem 1.3.6. If g : M → N is a smooth map, then the map ωg : C k (Σ, M ) → C k (Σ, N ) defined by

ωg (f ) = g ◦ f,

is also smooth. To summarize, for each pair (Σ, M ) of finite-dimensional smooth manifolds, with Σ compact, we have a chain of Banach manifolds, · · · ⊆ C k+1 (Σ, M ) ⊆ C k (Σ, M ) ⊆ C k−1 (Σ, M ) ⊆ · · · ⊆ C 0 (Σ, M ). The intersection of these manifolds is the space C ∞ (Σ, M ) of C ∞ maps from Σ to M , which could be made into a Fr´echet manifold, but we will not describe the details of that construction. We can formulate many problems from the calculus of variations in terms of the infinite-dimensional manifolds that we have constructed. Thus, for example, we can define the action function  1 by J(γ) = |γ  (t)|2 dt, J : C 1 (S 1 , M ) −→ R 2 S1

1.4. Basic mapping spaces

25

and check without much difficulty that J is a smooth map. As we learned in elementary differential geometry courses, the “critical points” for J are the smooth closed geodesics on M . Suppose that Σ is a compact two-dimensional Riemann surface. Thus we can imagine that Σ has a Riemannian metric, but we forget about the metric except for its conformal equivalence class, which we denote by ω. Suppose that {(Uα , (xα , yα )) : α ∈ A} is an atlas of isothermal coordinate charts on Σ, and let {ψα : α ∈ A} be a partition of unity subordinate to {Uα : α ∈ A}. We can then define the Dirichlet integral Eω : C 1 (Σ, M ) −→ R by     ∂f 2 ∂f 2 1 + ψα Eω (f ) = ∂yα dxα dyα . 2 Σ ∂xα α∈A

Once again it is relatively easy to check that Eω is a smooth map on the infinite-dimensional manifold C 1 (Σ, M ). Later we will call “critical points” for Eω harmonic maps. More generally, suppose that Σ is an m-dimensional Riemannian manifold with Riemannian metric expressed in local coordinates (x1 , . . . , xm ) on Σ by m  ηab dxa dxb . a,b=1

If f : Σ → M ⊆ to (ηab ), we set |df | = 2

RN

m  a,b=1

is a smooth map and (η ab ) denotes the matrix inverse

η ab

∂f ∂f · ∂xa ∂xb

and dA =



det(ηab )dx1 · · · dxm .

We can then define the Dirichlet integral E : C 1 (Σ, M ) −→ R

by

E(f ) =

1 2

 |df |2 dA, Σ

which is once again a smooth real-valued function on the infinite-dimensional manifold C 1 (Σ, M ). In the case where the domain is two-dimensional, choice of isothermal parameters leads to exactly the same integrand as before, so this generalizes the previous energy functions to higher dimensional domains.

1.4. The basic mapping spaces For the study of geodesics, harmonic and minimal surfaces and pseudoholomorphic curves, as well as other nonlinear partial differential equations, we need a collection of function spaces with weak enough topologies that we

26

1. Infinite-dimensional Manifolds

can prove convergence of a sequence which is tending toward an infimum of energy on a given component. The infinite-dimensional manifolds that have proven to be most useful in this regard are those modeled on Sobolev spaces. In this section, we describe the simplest of these spaces. If Σ is a compact Riemannian manifold, we can define an inner product (·, ·) on the space C ∞ (Σ, R) of smooth real-valued maps by  (f, g) = (f g + Df, Dg)dA, Σ

where the inner product ·, · on the right is the usual inner product in the cotangent space defined by the Riemannian metric on Σ, and dA denotes the area or volume form on Σ. The inner product (·, ·) makes the space C ∞ (Σ, R) of smooth functions into a pre-Hilbert space. Any pre-Hilbert space has a Hilbert space completion, the set of equivalence classes of Cauchy sequences, as described at the beginning of §1.2. The Hilbert space completion in our case is denoted by L21 (Σ, R), and is called the Sobolev space of L21 -functions on Σ. A second important Sobolev space generalizes the Lp spaces studied in real analysis when 1 ≤ p < ∞. We start by defining a norm  · Lp1 on C ∞ (Σ, R)) by  p  f Lp1 = (|f |p + |Df |p )dA, Σ

where |Df | is the length calculated with respect to the Riemannian metric on Σ. This norm makes C ∞ (Σ, R) into a pre-Banach space and we can construct the Banach space completion as before. This Banach space completion is denoted by Lp1 (Σ, R) and is called the Sobolev space of Lp1 -functions on Σ. Of course, when p = 2 this reduces to the previous example. If we change the Riemannian metric on Σ we change the norm to an equivalent norm, but remember that equivalent norms determine the same open sets and yield the same completion. It is actually “Banachable spaces”, topological vector spaces whose topology comes from a Banach space norm, that are the objects of interest. We can also define versions of these Sobolev spaces for higher order derivatives. Thus we can define the Sobolev norm  · Lp on C ∞ (Σ, R)) by k p   f Lp = (|f |p + |Df |p + · · · + |D k f |p )dA, k

Σ

and construct the completion with respect to this norm, which is denoted Lpk (Σ, R). The resulting space is a Banach space, and a Hilbert space when p = 2. We thus obtain a chain of Banach spaces, · · · ⊆ Lpk (Σ, R) ⊂ · · · ⊆ Lp1 (Σ, R) ⊆ Lp (Σ, R),

1.4. Basic mapping spaces

27

and the intersection of all the spaces in the chain is just the space C ∞ (Σ, R)) of smooth functions. These spaces are essential for the modern theory of partial differential equations and they are compared by means of the Sobolev Lemma: Sobolev Lemma. When p and k are sufficiently large, Lpk (Σ, R) is a subspace of C 0 (Σ, R): (1.12)

Lpk (Σ, R) ⊆ C 0 (Σ, R)

for

pk > dim(Σ).

Banach Algebra Lemma. When pk > dim(Σ), Lpk (Σ, R) is a Banach algebra in an appropriate norm defining the “Banachable” structure. In this section, we will prove two basic cases of these lemmas. More general cases can be found in standard references on partial differential equations, such as Evans [Eva98]. For the case where Σ = S 1 , where S 1 is the unit interval [0, 1] with endpoints identified, the Sobolev Lemma for L21 (S 1 , R) is relatively easy to establish: Lemma 1.4.1. There is a continuous linear injection i : L21 (S 1 , R) → C 0 (S 1 , R) which extends the inclusion C ∞ (S 1 , R) ⊂ C 0 (S 1 , R). Proof. We begin with the sequence of inequalities:   τ |f  (u)|du ≤ |f (τ )| + |f (t)| ≤ |f (τ )| + t

S1

|f  (u)|du,

for t ≤ τ (which we can always arrange by replacing τ by an equivalent parameter τ with τ ∈ [t, t + 1) if necessary). Averaging over τ and using the Cauchy-Schwarz inequality yields     |f (t)| ≤ |f (u)|du + |f (u)|du  ≤

S1

S1

S1

1/2  2 |f (u)| du +



S1

1/2

|f (u)| du 2

≤ (f, f )1/2 ,

where (·, ·) denotes the L21 inner product. Taking the supremum over all t yields (1.13)

f C 0 ≤ f L21 .

Thus a Cauchy sequence with respect to the L21 inner product gets taken under the inclusion C ∞ (S 1 , R) ⊂ C 0 (S 1 , R) to a Cauchy sequence with respect to the C 0 -norm. By definition, an element of L21 is an equivalence class of

28

1. Infinite-dimensional Manifolds

Cauchy sequences, and the map i is defined by sending this equivalence class to the limit of the C 0 Cauchy sequence. It is immediate that i is injective. QED Lemma 1.4.2. After possibly replacing  · L21 with an equivalent norm, L21 (S 1 , R) is a Banach algebra, so multiplication of functions is a continuous bilinear map: L21 (S 1 , R) × L21 (S 1 , R) −→ L21 (S 1 , R). Proof. It follows from the Cauchy-Schwarz inequality that   2 2  2 [(f g) + [(f g) ] ]dt = [(f g)2 + (f  g + f g  )2 ]dt f gL2 = (f g, f g) = 1 1 1 S S  2  2 2   2  2 = [(f g) + (f ) g + 2f gf g + f (g ) ]dt S1     2  2 2 2  2 (f  g  )dt (f g) + (f ) g + f (g ) dt + 2f gC 0 ≤ S1

≤

f 2C 0 g2L2 1

S1

+

g2C 0 f 2L2 1

+

f 2C 0 g2L2 1

+ 2f gC 0 f L21 gL21 .

It then follows from (1.13) that f g2L2 ≤ (constant)f 2L2 g2L2 , 1

1

1

where (constant) denotes a positive constant, finishing the proof of the lemma. QED It follows from this lemma that the multiplication map L21 (S 1 , Hom(Rm , Rn )) × L21 (S 1 , Rm ) −→ L21 (S 1 , Rn ) is continuous. Suppose now that M is a complete connected finite-dimensional Riemannian manifold isometrically imbedded as a proper submanifold of an ambient Euclidean space RN . We let L21 (S 1 , M ) = {γ ∈ L21 (S 1 , RN ) : γ(t) ∈ M for all t ∈ S 1 }, which is a closed subspace of L21 (S 1 , RN ) by Lemma 1.4.1. We claim that L21 (S 1 , M ) is an infinite-dimensional smooth manifold, the proof being just like the proof for C 0 (S 1 , M ). If γ : S 1 → M is a smooth curve, we choose ε > 0 so that for every t ∈ S 1 , expγ(t) : {v ∈ Tf (p) M : |v| < ε} −→ {q ∈ M : dM (q, γ(t)) < ε}

1.4. Basic mapping spaces

29

is a diffeomorphism. We then set Vγ,ε = {X ∈ L21 (S 1 , γ ∗ T M ) : XC 0 < ε}, Uγ,ε = {λ ∈ L21 (S 1 , M ) : dM (λ(t), γ(t)) < ε for all t ∈ S 1 }, and define φγ,ε : Uγ,ε → Vγ,

by

φγ,ε (λ)(t) = exp−1 γ(t) (λ(t)).

Finally, we prove that  A = (Uγ, , φγ, ) : γ : S 1 → M is a C ∞ map and ε > 0 is small enough that φγ, is a homeomorphism } . is a smooth atlas for L21 (S 1 , M ) by the same argument used to establish Lemma 1.3.2. Noting that L21 (γ ∗ T M ) is actually a Hilbert space, we obtain the following: Theorem 1.4.3. If M is a smooth manifold, then L21 (S 1 , M ) is a smooth manifold modeled on the Hilbert spaces L21 (γ ∗ T M ) for γ : S 1 → M a smooth map. Moreover, if g : M → N is a smooth map, then the map ωg : L21 (S 1 , M ) → L21 (S 1 , N ) defined by ωg (γ) = g ◦ γ, is also smooth. We are also interested in Lp1 -maps from a compact oriented surface Σ when p > 2. It turns out that these are H¨older continuous in accordance with the following definition. Definition. If Σ is a metric space, a map f : Σ → R is said to be H¨ older continuous with H¨older exponent γ ∈ (0, 1] if there is a constant C > 0 such that |f (p) − f (q)| ≤ C d(p, q)γ

for all p, q ∈ Σ.

We let C 0,γ (Σ, R) be the space of all functions f : Σ → R which are H¨ older 0,γ continuous. If f ∈ C (Σ, R), we let   f (p) − f (q) : p, q ∈ Σ, p = q . [f ]C 0,γ = sup d(p, q)γ The following lemma states that the space Lp1 (Σ, R) lies within Sobolev range. Lemma 1.4.4. If Σ is a compact oriented surface and p > 2, there is a continuous linear injection i : Lp1 (Σ, R) → C 0,γ (Σ, R), where γ = 1 − 2/p,

30

1. Infinite-dimensional Manifolds

which extends the inclusion C ∞ (Σ, R) ⊂ C 0,γ (Σ, R). Moreover, there is a constant C > 0 such that [f ]C 0,γ ≤ Cf Lp1 . A complete proof of this is given in Evans [Eva98], §5.6.2. We only prove the weaker result that Lp1 (Σ, R) ⊆ C 0,γ (Σ, R). We begin by assuming that Σ is the torus with flat Riemannian metric expressed in terms of suitable conformal coordinates as ds2 = dx2 + dy 2 . We consider a smooth function f (x, y) on the disk D(p, r0 ) of radius r0 about p ∈ Σ defined in terms of Euclidean coordinates centered at p by x2 + y 2 ≤ r02 , then after shifting to polar coordinates (r, θ) defined by x = r cos θ, we see that



s

|f (s, θ) − f (p)| = 0

and hence   2π |f (s, θ) − f (p)|dθ ≤ 0

Thus

2π 0



2π 0

and hence



r0 0



y = r sin θ,

∂f (r, θ)dr ≤ ∂r



s



s

|Df |(r, θ)dr,

0

 |Df |(r, θ)drdθ ≤

0

D(p,r0 )



 

r0

|f (s, θ) − f (p)|sdsdθ ≤

rdr 0

r2 |f (x, y) − f (p)|dxdy ≤ 0 2 D(p,r0 )

D(p,r0 )

 D(p,r0 )

|Df | dxdy. r

|Df | dxdy r



|Df | dxdy. r

It follows from the H¨older inequality that  |Df | r02 dxdy 2 D(p,r0 ) r  (p−1)/p 1/p  dxdy r02 p ≤ |Df | dxdy p/(p−1) 2 D(p,r0 ) D(p,r0 ) r while direct integration yields   dxdy = rp/(1−p) dxdy p/(p−1) D(p,r0 ) r D(p,r0 )  2π  r0 2π(p − 1) (p−2)/(p−1) r0 r1/(1−p) drdθ = . = p−2 0 0

1.4. Basic mapping spaces

31

Thus (1.14) 

r2 |f (x, y) − f (p)|dxdy ≤ 0 2 D(p,r0 )



2π(p − 1) p−2

(p−1)/p

(p−2)/p

r0

Df Lp .

It follows from (1.14) and the H¨older inequality that   2 πr0 |f (0)| ≤ |f (x, y) − f (p)|dxdy + |f (x, y)|dxdy D(p,r0 )

D(p,r0 )

≤ (constant)Df Lp + (constant)f L1 ≤ (constant)Df Lp + (constant)f Lp (area of D(p, r0 ))(p−1)/p ≤ (constant)f Lp1 , which quickly yields the desired result when Σ is the flat torus. If Σ is a more general Riemann surface, we can use the uniformization theorem to give Σ a Riemannian metric of constant curvature and total volume one. Choose r0 > 0 less than the injectivity radius of this metric. A modification of the above argument can then be applied to a normal coordinate ball of radius r0 showing that if p ∈ Σ, then |f (p)| ≤ (constant)f Lp1 ,

and hence

f C 0 ≤ (constant)f Lp1 .

(Note that changing the Riemannian metric on Σ merely replaces the Lp1 norm by an equivalent norm, so adopting the constant curvature metric imposes no restriction.) Thus if a sequence {fi } of smooth functions on Σ converges to a limit in Lp1 -norm, {fi } converges also in C 0 norm to a unique limit function f∞ ∈ C 0 . Thus any element of Lp1 (Σ, R) can be identified with a continuous function and the Lemma is proven. Remark 1.4.5. The previous Sobolev Lemma for Lp1 maps from a surface begins to fail as p approaches 2 from above. The reason is that the highest order term in the L21 -norm is conformally invariant and hence invariant under dilations. In the case where D is the unit disk centered at the origin in R2 this highest order term is  |Df |2 dxdy. D

In contrast, the highest order term in the Lp1 -norm is not invariant under dilations. Given a smooth map f : D → RN which takes the boundary ∂D to a point, we can define a dilated map f : D → RN , where D is the ball of radius  > 0 centered at the origin in R2 , by x y  , . f (x, y) = f  

32

1. Infinite-dimensional Manifolds

Then

 x y  1 Df , ;    and, if x ˜ = x/ and y˜ = y/ denote the coordinates corresponding to x and y on the unit disk D1 , then  (p−2)   1 p |Df | dxdy = |Df |2α d˜ xd˜ y.  D D1 |Df (x, y)| =

Thus, as  → 0, the Lp1 -norm of f approaches infinity as long as p > 2. In particular, a bound on the L21 -norm does not imply a bound on the C 0 -norm. Lemma 1.4.6. If Σ is a compact oriented surface and p > 2, Lp1 (Σ, R) satisfies f gLp1 ≤ 2f Lp1 gLp1 . Thus after passing to an equivalent norm, we can show that Lp1 (Σ, R) is a Banach algebra. Sketch of proof. By the previous lemma, f gLp ≤ f C 0 gLp + gC 0 f Lp ≤ f Lp1 gLp + gLp1 f Lp and D(f g)Lp ≤ f C 0 DgLp +gC 0 Df Lp ≤ f Lp1 DgLp +gLp1 Df Lp . Adding these two inequalities yields the statement of the lemma. QED If Σ is a compact surface and p > 2, we let Lp1 (Σ, M ) = {f ∈ Lp1 (Σ, RN ) : f (p) ∈ M for all p ∈ Σ}, a closed subspace of Lp1 (Σ, RN ) by Lemma 1.4.3. If Σ is a compact surface and p > 2, we claim that Lp1 (Σ, M ) is an infinite-dimensional smooth manifold. In this case, when f : Σ → M is a smooth parametrized surface, we let Vγ, = {X ∈ Lp1 (Σ∗ T M ) : XC 0 < ε}, Uγ, = {g ∈ Lp1 (Σ, M ) : dM (f (p), g(p)) <  for all p ∈ Σ}, and if  > 0 is sufficiently small, we define φγ, : Vγ, → Uγ,

by

φγ, (X)(p) = exp−1 f (p) (X(p)).

We then set A = {(Uγ, , φγ, ) : f : Σ → M is a C ∞ map and  > 0 is small enough that φγ, is a homeomorphism}

1.5. Homotopy type of the space of maps

33

and once again prove that it is a smooth atlas by the same argument used to establish Lemma 1.3.2. Indeed, the same argument holds for Lpk (Σ, M ) so long as pk > 2, and we obtain: Theorem 1.4.7. If Σ is a compact smooth surface and M is a smooth manifold, then for pk > 2, Lpk (Σ, M ) is a smooth manifold modeled on the Banach spaces Lpk (f ∗ T M ) for f : Σ → M a smooth map. Moreover, if g : M → N is a smooth map, then the map ωg : Lpk (Σ, M ) → Lpk (Σ, N ) defined by

ωg (γ) = g ◦ γ,

is also smooth. In exactly the same way, we could show that if Σ is an m-dimensional smooth manifold and pk > m, then Lpk (Σ, M ) is an infinite-dimensional smooth manifold. In the case where Σ is a compact surface, the general Sobolev Lemma (1.12) provide sequences of Banach manifolds and inclusions · · · ⊆ C k (Σ, M ) ⊆ · · · ⊆ C 2 (Σ, M ) ⊆ Lp1 (Σ, M ) ⊆ C 0 (Σ, M ) or · · · ⊆ L2k (Σ, M ) ⊆ · · · ⊆ L22 (Σ, M ) ⊆ Lp1 (Σ, M ) ⊆ C 0 (Σ, M ) when p > 2. It is important to note that the charts defining the atlases for all the manifolds in these sequences are simply the restrictions of the exponential charts we defined on C 0 (Σ, M ). These sequences are ideally suited to a technique commonly used in the theory of elliptic partial differential equations: one starts by proving existence of a solution in a space such as Lp1 (Σ, M ), then uses the method of “elliptic bootstrapping” to show that the solution actually lies in C k (Σ, M ) for higher and higher values of k. The spaces L2k (Σ, M ) in the second sequence have the added advantage of being Hilbert manifolds when k is large, which are sometimes a little easier to work with.

1.5. Homotopy type of the space of maps A continuous map f : X → Y between topological spaces is said to be a homotopy equivalence if there is a continuous map g : Y → X such that f ◦ g and g ◦ f are both homotopic to the identity, and if such a homotopy equivalence exists, we say that X and Y have the same homotopy type. The following theorem on homotopy equivalence was proven quite early in the theory of manifolds of maps; see the survey article of Eells [Eel66] for appropriate references.

34

1. Infinite-dimensional Manifolds

Theorem 1.5.1. Let M be a compact connected Riemannian manifold. Then the inclusions C k (S 1 , M ) ⊆ C 0 (S 1 , M )

and C k (S 1 , M ) ⊆ L21 (S 1 , M )

are homotopy equivalences when k ≥ 1. The point of this theorem is that from the point of view of homotopy theory, L21 (S 1 , M ) is essentially the same as the space of continuous maps C 0 (S 1 , M ) with the compact open topology. This latter space has been extensively studied by topologists and much is known about its homotopy and homology groups, as we will see later. The proof of the theorem is an application of the theory of smoothing operators. In preparation, we suppose that φ : R → R is a smooth map which vanishes outside [−1, 1]. Suppose, moreover, that  φ = 1. φ≥0 and R

For ε > 0, let φε (t) = (1/ε)φ(t/ε), so that supp(φε ) ⊂ [−ε, ε]



and R

φε = 1.

If γ ∈ C 0 (S 1 , RN ), we can regard γ as an element of C 0 (R, RN ) such that γ(t + 1) = γ(t) for all t, and we define φε ∗ γ ∈ C 0 (R, RN ) by   φε (t − τ )γ(τ )dτ = φε (s)γ(t − s)ds. (φε ∗ γ)(t) = R

It is immediately checked that φε ∗ γ is dk dk (φ ∗ γ)(t) = (φ ) ∗ γ = ε dtk dtk

R

C∞

  R

and

 dk φε (t − τ )γ(τ )dτ. dtk

Moreover, (φε ∗ γ)(t + 1) = (φε ∗ γ)(t), and hence φε ∗ γ ∈ C ∞ (S 1 , RN ). We can thus define smoothing operators Sε : C 0 (S 1 , RN ) −→ C k (S 1 , RN ),

Sε : L21 (S 1 , RN ) −→ C k (S 1 , RN )

by Sε (γ) = φ ∗ γ. It is not difficult to show that the maps Sε : C 0 (S 1 , RN ) −→ C k (S 1 , RN ),

Sε : L21 (S 1 , RN ) −→ C k (S 1 , RN )

are continuous. Proof of Theorem 1.5.1. Recall that we regard M as a submanifold of RN . Choose δ > 0 so small that the exponential map exp : N M → RN ,

defined by

exp(v) = p + v,

for v ∈ Np M ,

1.5. Homotopy type of the space of maps

35

maps N Mδ = {v ∈ N M : |v| < δ} diffeomorphically onto M (δ) = {p ∈ RN : d(p, M ) < δ}. Then the “nearest point projection” map r : M (δ) → M , defined by r(p + v) = p for p ∈ M , is a strong deformation retraction from M (δ) to M . To see this, we define h : M (δ) × [0, 1] → M (δ) by

h(p + v, t) = p + (1 − t)v,

and check that (1) h(q, 0) = q, for q ∈ M (δ), (2) h(q, 1) = r(q) ∈ M , for q ∈ M (δ), and (3) h(p, t) = p, for p ∈ M . We have a similar strong deformation retraction on the function space level. The ω-Lemma gives us a smooth map: ωh : L21 (S 1 , M (δ) × [0, 1]) → L21 (S 1 , M (δ)),

defined by

ωh (γ) = h ◦ γ.

We define j : L21 (S 1 , M (δ)) × [0, 1] → L21 (S 1 , M (δ) × [0, 1]) by j(γ, τ )(t) = (γ(t), τ ) and let H = ωh ◦ j. Then (1) H(γ, 0) = γ, for γ ∈ L21 (S 1 , M (δ)), (2) H(γ, 1) = r ◦ γ ∈ L21 (S 1 , M ), for γ ∈ L21 (S 1 , M (δ)), and (3) H(γ, t) = γ, for γ ∈ L21 (S 1 , M ). We can therefore define a strong deformation retraction R : L21 (S 1 , M (δ)) −→ L21 (S 1 , M ) by R(γ) = ωr (γ) = H(γ, 1). In a similar fashion, we can define a strong deformation retraction R : C k (S 1 , M (δ)) −→ C 0 (S 1 , M ), whenever k ≥ 0. Let εk = 2−k and let C 0 (S 1 , M )εk = {γ ∈ C 0 (S 1 , M ) : γ maps the closed interval [(m − 1)2−k , (m + 1)2−k ] into the open ball B(γ(m2−k ); δ) for each integer m such that 0 ≤ m ≤ 2k }, an open set in the compact-open topology. They key point of this set is that when |s − t| < 2−k then the straight line from γ(s) to γ(t) in RN lies entirely

36

1. Infinite-dimensional Manifolds

within M (δ) hence Sε  γ lies within M (δ) when ε ≤ εk . Note that 0

1

C (S , M ) =

∞ 

L21 (S 1 , M )εk+1 ⊂ L21 (S 1 , M )εk ,

C 0 (S 1 , M )εk ,

k=1

and we therefore say that C 0 (S 1 , M ) is a monotone union of the subspaces C 0 (S 1 , M )εk . Similarly, we let L21 (S 1 , M )εk = L21 (S 1 , M ) ∩ C 0 (S 1 , M )εk , C k (S 1 , M )εk = C k (S 1 , M ) ∩ C 0 (S 1 , M )εk ,

when k ≥ 1,

thereby expressing L21 (S 1 , M ) and C k (S 1 , M ) as monotone unions for k ≥ 1. By analogous formulae, we define C 0 (S 1 , M (δ))εk ,

L21 (S 1 , M (δ))εk

and

C k (S 1 , M (δ))εk ,

for k ≥ 1. We can then define smoothing operators Sεk : L21 (S 1 , M )εk −→ C k (S 1 , M (δ))εk , since ⇒

γ ∈ L21 (S 1 , M )εk

Sεk  γ ∈ C k (S 1 , M (δ))εk .

We define s to be the composition of Sεk : L21 (S 1 , M )εk → C k (S 1 , M (δ))εk

and

R : C k (S 1 , M (δ))εk → C k (S 1 , M )εk . We claim that if i : C k (S 1 , M )εk ⊂ L21 (S 1 , M )εk is the inclusion, then s◦i

and

i◦s

are homotopic to the identity. This is easy to verify. To get the homotopy from s ◦ i to the identity, we simply define H1 : C k (S 1 , M )εk × [0, 1] → C k (S 1 , M )εk by

H1 (γ, t) = R ◦ (tSεk + (1 − t)id)) ◦ i(γ).

Similarly, to get the homotopy from i ◦ s to the identity, we define H2 : L21 (S 1 , M )εk × [0, 1] → L21 (S 1 , M )εk by

H2 (γ, t) = i ◦ R ◦ (tSεk + (1 − t)id)(γ),

This shows that for each k ∈ N, the inclusion C k (S 1 , M )εk ⊆ L21 (S 1 , M )εk is a homotopy equivalence.

1.5. Homotopy type of the space of maps

37

To finish the proof, we must take an appropriate limit as k → ∞. Suppose that the metrizable space X is a monotone union of open subsets, by which we mean that we have a sequence of spaces  U1 ⊆ U2 ⊆ U3 ⊆ · · · such that X = {Uk : k ∈ N}. Suppose, moreover, that we let X  = (U1 × [1, 2]) ∪ (U2 × [2, 3]) ∪ (U3 × [3.4]) ∪ · · · topologized as a subset of X × R. We say that X is the homotopy direct limit of the subspaces {Uk : k ∈ N} if the projection p : X  → X on the first factor is a homotopy equivalence. If the subsets Uk are open, then the open cover {Uk : k ∈ N} has a C 0 subordinate partition of unity {ψk : k ∈ N}. In this case, the map  ∞    defined by f (x) = x, kψk (x) f :X→X k=1

is a homotopy inverse to p, showing that X is a homotopy direct limit in this case. Using this argument, one easily verifies that C k (S 1 , M ) is a homotopy direct limit of its subspaces C k (S 1 , M )εk and L21 (S 1 , M ) is a homotopy direct limit of L21 (S 1 , M )εk . Now we apply the following lemma, which is just Theorem A from the Appendix to Milnor’s book on Morse theory [Mil63]: Lemma 1.5.2. Suppose that X is the homotopy direct limit of {Uk : k ∈ N} and that Y is the homotopy direct limit of {Vk : k ∈ N}. If f : X → Y is a continuous map such that f (Uk ) ⊆ Vk and the restriction of f to Uk is a homotopy equivalence from Uk to Vk , then f itself is a homotopy equivalence. We refer the reader to Milnor for the proof of this lemma. It implies that the inclusion C k (S 1 , M ) ⊂ L21 (S 1 , M ) is a homotopy equivalence when k ≥ 1. In a similar manner, one verifies that the inclusion C k (S 1 , M ) ⊂ C 0 (S 1 , M ) is a homotopy equivalence when k ≥ 1, and this completes the proof of Theorem 1.5.1. QED Theorem 1.5.3. Let M be a compact connected Riemannian manifold, Σ a compact connected Riemann surface. Then the inclusions, C k (Σ, M ) ⊆ C 0 (Σ, M ),

C k (Σ, M ) ⊆ Lp1 (Σ, M ) for p > 2, C k (Σ, M ) ⊆ L2k (Σ, M )

for k ≥ 2,

are homotopy equivalences. The proof is essentially the same as for the previous theorem, with 2 replaced by p, except for the definition of smoothing operators defined on Σ. The

38

1. Infinite-dimensional Manifolds

construction of such operators is a standard technique in the theory of partial differential equations. We describe only the case Σ = T 2 here, where T 2 = C/Λ, Λ being a lattice in C; the construction in this case is particularly transparent. (In the general case, the ideas are the same, but one constructs the smoothing operators by piecing together using a partition of unity on Σ.) Note that an element f ∈ Lp1 (T 2 , RN ) can be regarded as a map f : C → RN such that f (z + λ) = f (z) for λ ∈ Λ. Suppose that φ : C → [0, ∞) is a smooth map which vanishes outside D = {z ∈ C : |z| ≤ 1} such that  φ dxdy = 1, C

where (x, y) are the standard coordinates on C. Let φε (z) = (1/ε2 )φ(z/ε), so that  and φ dxdy = 1. supp(φε ) ⊂ {z ∈ C : |z| ≤ } C

If f : C → M comes from an element f ∈ Lp1 (T 2 , RN ), we define φε ∗ f ∈ C ∞ (C, RN ) by  (φε ∗ γ)(z) = φε (z − w)γ(w)dw. C

It is immediately checked that (φε ∗ f )(z + λ) = (φε ∗ f )(z) for λ ∈ Λ, so (φ ∗ f ) can be identified with an element of C ∞ (T 2 , RN ). Thus we can define smoothing operators Sε : C 0 (T 2 , RN ) → C k (T 2 , RN ), Sε : Lp1 (T 2 , RN ) → C k (T 2 , RN ) by Sε (f ) = φε ∗ f . The proof of the theorem for maps from Σ = T 2 now continues in exactly the same way as for maps from S 1 . Remark 1.5.4. It is interesting to consider Sobolev spaces of maps which do not lie in “Sobolev range”. Thus we can define W1p (Σ, M ) = {f ∈ Lp1 (Σ, M ) : f (p) ∈ M for almost all p ∈ Σ }

or

p (Σ, M ) = (closure of C ∞ (Σ, M ) in Lp1 (Σ, M )), H1,S p (Σ, M ) ⊆ W1p (Σ, M ) the inclusion is often for p ≤ dim Σ. Although H1,S strict, and neither space is in general homotopically equivalent to the space C 0 (Σ, M ) of continuous maps. These Sobolev spaces have been studied by Hang and Lin [HL03], among others, and applications to the theory of

1.6. The α- and ω-Lemmas

39

harmonic maps are described in the review article of Brezis [Bre03]. When the dimension of Σ is at least three, harmonic maps from Σ to M can have vastly more complicated singularities than geodesics and harmonic surfaces.

1.6. The α- and ω-Lemmas In this section, we mention a few facts about maps between function spaces which are often useful. For a given choice of compact smooth manifold Σ, we have constructed a covariant functor from finite-dimensional smooth manifolds and smooth maps to infinite-dimensional smooth manifolds and smooth maps, M → C k (Σ, M ),

g:M →N

→

ωg : C k (Σ, M ) → C k (Σ, N ),

where ωg (f ) = g ◦ f . If one studies the proof, one finds that in order to get a C r map ωg one needs to assume that g is C k+r : ω-Lemma 1.6.1. If M is a fixed smooth manifold, a C k+r map g : M → N induces a C r map ωg : C k (Σ, M ) → C k (Σ, N ) for k, r ∈ {0} ∪ N. One might hope that when M is a fixed smooth manifold, there is a similar contravariant functor from compact manifolds Σ to infinite-dimensional smooth manifolds C k (Σ, M ) in which f : Σ1 → Σ2

→

αf : C k (Σ2 , M ) → C k (Σ1 , M ),

where αf (g) = g ◦ f . Toward this end, one finds the following fact proven as Theorem 11.2 of [Abr63]: α-Lemma 1.6.2. If M is a fixed smooth manifold, a C k map f : Σ1 → Σ2 between smooth compact manifolds induces a C k map αf : C k (Σ2 , M ) → C k (Σ1 , M ),

αf (g) = g ◦ f,

for k ∈ N. A combination of these lemmas (as presented in the survey by Eells [Eel66]) is often useful: Theorem 1.6.3. If S, Σ and M are smooth finite-dimensional manifolds with S and Σ compact, then Φ : C k+r (Σ, M ) × C k (S, Σ) −→ C k (S, M ), is C r .

Φ(g, f ) = g ◦ f,

40

1. Infinite-dimensional Manifolds

By taking S to be a point, we can derive the following: Corollary 1.6.4. The evaluation map (1.15)

ev : C r (Σ, M ) × Σ → M

defined by

ev(f, p) = f (p)

is C r . This corollary can can also be proven directly as in [AR67] or [AMR88], page 99. Versions of the α- and ω-Lemmas for Hilbert manifolds were later developed by Ebin [Ebi70], and they were used in [EM70] to study Euler’s equations for fluid flow. One can ask whether the group of L2k diffeomorphisms might be made into a Banach Lie group. Let C 1 D(Σ) denote the topological group of C 1 diffeomomorphisms of a compact manifold Σ, and let Dk (Σ) = C 1 D(Σ) ∩ L2k (Σ, Σ),

for

k ∈ {0} ∪ N,

which can be shown to be a topological group. The following theorem [Ebi70] can be thought of as a combination of the α- and ω-Lemmas: Theorem 1.6.5. If Σ and M are smooth finite-dimensional manifolds with Σ compact, then Φ : L2k+r (Σ, M ) × Dk (Σ) −→ L2k (Σ, M ),

Φ(g, f ) = g ◦ f,

is C r for k, r ∈ {0} ∪ N. The loss of derivatives in the statement of Theorem 1.6.3 prevents Dk (Σ) from being a Banach Lie group under composition. On the other hand, suppose that G is a finite-dimensional Lie subgroup of the group of C ∞ automorphisms which acts on the compact surface Σ. Then Theorem 1.6.2 implies that the map L2k+1 (Σ, M ) × G ⊆ L2k+1 (Σ, M ) × Dk (Σ) −→ L2k (Σ, M ) is C 1 . Thus one gets a C 1 action while losing one derivative in the space of functions.

1.7. The tangent and cotangent bundles Many constructions from the theory of finite-dimensional manifolds are relatively easy to extend to infinite-dimensional Banach or Hilbert manifolds. These include tensors of various ranks, vector fields and differential equations, connections, Riemannian metrics on Hilbert manifolds, Finsler metrics

1.7. Tangent and cotangent bundles

41

on Banach manifolds, differential forms and de Rham cohomology. Many of these constructions are carried out in great detail in the Lang’s book [Lan95] on infinite-dimensional manifolds. We provide a brief summary here. We first extend familiar definitions of tangent and cotangent bundles to the infinite-dimensional context. Let M be an infinite-dimensional smooth manifold modeled on a Banach space E with smooth atlas {(Uα , φα ) : α ∈ A}. Consider the collection of triples (α, p, v), where α ∈ A, p ∈ Uα and v ∈ E. On this collection of triples we define an equivalence relation ∼ by (α, p, v) ∼ (β, q, w)

⇔

p = q and w = D(φβ ◦ φ−1 α )(φα (p))v.

The set of equivalence classes is called the tangent bundle of M and is denoted by T M. Let [α, p, v] denote the equivalence class of (α, p, v) and define π : T M −→ M

by

π([α, p, v]) = p.

˜α = {[α, p, v]; p ∈ Uα , v ∈ E}, and define Let U ˜α −→ E ⊕ E φ˜α : U

by

φ˜α ([α, p, v]) = (φα (p), v).

˜α , φ˜α ) : α ∈ A} is a smooth atlas on T M making T M into a Then {(U smooth manifold modeled on the Banach space E ⊕ E. If p ∈ M, we let Tp M = π −1 (p), the fiber of the tangent bundle over p, and call Tp M the tangent space to M at p. Just as in the finite-dimensional case, elements of Tp M are called tangent vectors. If γ : (a, b) → M is a smooth curve, t ∈ (a, b) and γ(t) ∈ Uα , we define γ  (t) ∈ Tp M

by

γ  (t) = [α, γ(t), D(φα ◦ γ)(t) · 1],

a tangent vector called the velocity vector to γ at t. If F : M → N is a smooth map between manifolds with atlases {(Uα , φα ) : α ∈ A} and {(Vβ , ψβ ) : β ∈ B} and p ∈ M, we can define the differential (F∗ )p : Tp M → TF (p) N by (F∗ )p ([α, p, v]) = [β, F (p), (D(ψβ ◦ F ◦ φ−1 α )(φα (p))(v)], where p ∈ Uα and F (p) ∈ Vβ . Note that if γ : (a, b) → M is a C 1 curve, (F∗ )p (γ  (t)) = (F ◦ γ) (t),

for t ∈ (a, b).

The differentials fit together to form a map of tangent bundles F∗ : T M → TN. In a very similar way, we can also describe the cotangent bundle of M. We consider a similar collection of triples (α, p, v ∗ ), where α ∈ A, p ∈ Uα and

42

1. Infinite-dimensional Manifolds

v ∗ ∈ E ∗ , where E ∗ is the Banach space dual to E, the space of continuous linear maps T : E → R. This time we choose the equivalence relation (α, p, v ∗ ) ∼ (β, q, w∗ )

⇔

∗ ∗ p = q and v ∗ = [D(φβ ◦ φ−1 α )(φα (p))] w ,

where (·)∗ denotes transpose map defined by   ∗ ∗ ∗ −1 [D(φβ ◦ φ−1 α )(φα (p))] w (v) = w (D(φβ ◦ φα )(φα (p))(v), and we let [α, pv ∗ ] denote the equivalence class of (α, p, v ∗ ). The cotangent bundle T ∗ M is the set of these equivalence classes, and it is a smooth manifold modeled on the Banach space E ⊕ E ∗ . Once again, we have a projection π : T M −→ M defined by π([α, p, v]) = p. If p ∈ M, the fiber Tp∗ M of the cotangent bundle over p is called the cotangent space to M at p, and it can be regarded as the dual space to Tp M. A smooth map F : M → N induces a map in the opposite direction (F ∗ )p : TF∗ (p) N → Tp M by (1.16)

∗ ∗ (F ∗ )p ([β, F (p), w∗ ]) = [α, p, (D(ψβ ◦ F ◦ φ−1 α )(φα (p)) (w )].

We can also define various tensor bundles, such as the k-th tensor power and the k-th exterior power of the cotangent bundle. For the first of these, we utilize the Banach space Lk (E, R) of continuous maps k

!" # T : E × E × · · · × E −→ R which are linear in each variable, the so-called space of continuous k-linear maps or continuous k-multlinear maps. For the second, we use its subspace of continuous alternating k-linear maps Lka (E, R). An element T ∈ Lk (E, R) is said to be alternating if T (hσ(1) , . . . , hσ(k) ) = (sgn(σ))T (h1 , . . . , hk ),

for all σ ∈ Sk ,

where Sk denotes the symmetric group on k letters and sgn(σ) denotes the sign of the permutation σ ∈ Sk . A continuous linear map Φ : E → F between Banach spaces induces continuous linear maps Φ∗ : Lk (F, R) → Lk (E, R),

Φ∗ : Lka (F, R) → Lka (E, R)

by means of the formula (Φ∗ T )(v1 , . . . , vk ) = T (Φ(v1 ), . . . , Φ(vk )). To define the k-th tensor power of the cotangent bundle, we start with triples (α, p, T ), where α ∈ A, p ∈ Uα and T ∈ Lk (E, R) and the equivalence relation (α, p, Tα ) ∼ (β, q, Tβ )

⇔

p = q and Tα = Φ∗ Tβ ,

1.7. Tangent and cotangent bundles

43

where Φ = D(φβ ◦ φ−1 α )(φα (p)). The k-th tensor power of the cotangent k ∗ bundle ⊗ T M is the set of equivalence classes. The k-th exterior power is defined the same way, except that T is taken to lie in Lka (E, R). The fibers ⊗k Tp∗ M and Λk Tp∗ M are called the k-th tensor power and the exterior power of the cotangent space to M at p. In the case where M = L21 (S 1 , M ), M being oriented, the tangent bundle has another description, namely T L21 (S 1 , M ) = L21 (S 1 , T M ). To see this, recall how we constructed the atlas on L21 (S 1 , M ). If γ is a smooth element of L21 (S 1 , M ), we let Uγ, = {λ ∈ L21 (S 1 , M ) : dM (λ(t), γ(t)) <  for all t ∈ S 1 }, Vγ, = {X ∈ L21 (S 1 , Rn )) : X(t), X(t) <  for all t ∈ S 1 }, and define ψγ, : Vγ, −→ Uγ,

by

(ψγ, (X))(t) = expγ(t) (X(t)).

For  sufficiently small, ψγ, is a bijection with inverse φγ, : Uγ, → Vγ, , and {(Uγ, , φγ, ) : γ is smooth and  is sufficiently small} is a smooth atlas for L21 (S 1 , M ). Now for each smooth γ, we can construct a lift γ˜ ∈ L21 (S 1 , T M ) by setting γ˜ (t) = 0γ(t) ∈ Tγ(t) M, and construct a corresponding chart on L21 (S 1 , T M ). The remarkable fact is that we can choose the chart to be valid over all of 2 1 ˜γ, = Ω−1 U π (Uγ, ) = {X ∈ L1 (S , T M ) : π ◦ X ∈ Uγ, }. Indeed, we can set V˜γ, = Vγ, × {L21 -sections of γ ∗ T M } ˜γ, by and define ψ˜γ, : V˜γ, → U ψ˜γ, (X, Y ) = (expγ(t) X(t), (d(expγ(t) )X(t) Y (t)). ˜γ, → V˜γ, by φ˜γ, = ψ˜−1 . Then Finally, define φ˜γ, : U γ, −1 −1 (φ˜γ1 , ◦ φ˜−1 γ2 , )(X, Y ) = ((φγ1 , ◦ φγ2 , )(X), D(φγ1 , ◦ φγ2 , )(X)(Y )).

Thus the charts transform exactly the way they should for the tangent bundle. In a quite similar fashion, we can show that if Σ is a compact Riemann surface and p > 2, T Lp1 (Σ, M ) = Lp1 (Σ, T M ).

44

1. Infinite-dimensional Manifolds

It is important to observe that just as the imbedding i : M → RN induces an imbedding ωi : L21 (S 1 , M ) → L21 (S 1 , RN ), so the imbedding i : T M → T RN = R2N induces an imbedding ωi : T L21 (S 1 , T M ) = L21 (S 1 , T M ) −→ L21 (S 1 , T RN ) = L21 (S 1 , R2N ), allowing us to realize T L21 (S 1 , T M ) as a subspace of a Banach space. Similarly, T Lp1 (Σ, M ) can be regarded as a subspace of a Banach space. Note that the Hilbert space inner product allows us to identify the model space L21 (S 1 , Rn ) for M = L21 (S 1 , M ) with its dual. Using this fact, it is not difficult to verify that T ∗ L21 (S 1 , M ) = L21 (S 1 , T ∗ M ),

⊗k T ∗ L21 (S 1 , M ) = L21 (S 1 , ⊗k T ∗ M ),

and Λk T ∗ L21 (S 1 , M ) = L21 (S 1 , Λk T ∗ M ). It is actually the “sections” of the tangent bundle and the exterior powers of the cotangent bundle that will be of most importance for us; these are called vector fields and differential forms, respectively. Definition. A smooth vector field on M is a smooth map X : M −→ T M

such that

π ◦ X = idM .

Note that if X : M → T M is a smooth vector field and f : M → R is a smooth function, we can define the vector field f X : M → T M by (f X )(p) = f (p)X (p). This makes the space of smooth vector fields into a module over the ring of smooth real-valued functions. Example. If M is a finite-dimensional Riemannian manifold and X : M → T M is a smooth vector field on M , then ωX : L21 (S 1 , M ) −→ L21 (S 1 , T M ) = T L21 (S 1 , M )

and

ωX : Lp1 (Σ, M ) −→ Lp1 (Σ, T M ) = T Lp1 (Σ, M ) are smooth vector fields on L21 (S 1 , M ) and Lp1 (Σ, M ) when p > 2.

1.8. Differential forms For calculations on smooth manifolds, differential forms are often more convenient to use than general tensor fields. Definition. A smooth covariant tensor field of rank k on M is a smooth map such that π ◦ φ = idM . φ : M −→ ⊗k T ∗ M

1.8. Differential forms

45

A smooth differential form of degree k on M (or a smooth k-form) is a smooth map φ : M −→ Λk T ∗ M

such that

π ◦ φ = idM .

We let Ωk (M ) denote the space of smooth differential forms of degree k on M. As in the case of vector fields, we can multiply covariant tensor fields or differential forms by functions. An important example of differential one-form occurs when f : M → R is a smooth function. Then for the coordinate chart (Uα , φα ), we have (1.17)

∗ D(f ◦ φ−1 α ) : Uα → L(E, R) = E ,

where E is the model space of M. The differential of f is the smooth one-form df such that df (p) = [α, p, D(f ◦ φ−1 α )(φα (p))],

for p ∈ Uα .

It is readily verified that the local representatives transform as they should under change of coordinates. Moreover, it is easily checked that the differentials of functions at a point p generate the cotangent space. The Leibniz rule for differentiation implies that d(f g) = gdf + f dg. Definition. A point p ∈ M is a critical point for the real-valued function f : M → R if df (p) = 0. An important special case of this construction is the real-valued function  1 2 1 J(γ) = J(γ) = |γ  (t)|2 dt, J : L1 (S , M ) → M, 2 S1 when M is a Riemannian manifold. In this case, regularity theory will show that a critical point in this case is actually a C ∞ map, hence a smooth closed geodesic in M . Using the notion of differential of a function, we can define directional derivatives: Definition. The directional derivative of a function f in the direction of a vector field X , or the Lie derivative of f in the direction of X is the function X (f ) defined by X (f )(p) = df (p)(X (p)), the right-hand side being the dual pairing between cotangent and tangent spaces.

46

1. Infinite-dimensional Manifolds

Lemma 1.8.1. Let X and Y be smooth vector fields on the Banach manifold M. Then there is a unique vector field [X , Y] on M which satisfies the equation ([X , Y](f ))(p) = (X Y(f ))(p) − (YX (f ))(p). Sketch of proof. It suffices to show that the above formula is equivalent to an expression for [X , Y] in terms of a local coordinate chart (Uα , φα ). Suppose that X˜ , Y˜ : Uα → E

are defined by X (p) = [p, α, X˜ (p)],

˜ Y(p) = [p, α, Y(p)].

Using the chain rule, one can check that the vector field [X , Y] is then defined by $ % ˜ X˜ (p) − D X˜ (p)Y(p) ˜ [X , Y](p) = p, α, D Y(p) . QED The vector field [X , Y] is known as the Lie bracket of X and Y; it is easily verified that it satisfies the identity: [f X , gY] = f g[X , Y] + f X (g)Y − gY(f )X . If X1 , . . . , Xk are smooth vector fields on M and φ is a smooth k-form on M, then the smooth function φ(X1 , . . . , Xk ) : M −→ R is defined fiberwise via the continuous (k + 1)-linear map Lka (E, R) × E × · · · × E −→ R, where E is the model space for M. Definition. If φ ∈ Ωk (M ) and ω ∈ Ωl (M ), the wedge product of φ and ω is the (k + l)-form φ ∧ ω defined by (φ ∧ ω)(X1 , . . . , Xk+l )  k!l! = sgn(σ)φ(Xσ(1), . . . , Xσ(k) )ω(Xσ(k+1) , . . . , Xσ(k+l) ). (k + l)! σ∈Sk+l

Here Sk+l is the symmetric group on k + l letters and sgn(σ) is the sign of the permutation σ ∈ Sk+l . We remind the reader that some authors prefer to define the wedge product using the factor k!l! 1 instead of . (k + l)! (k + l)!

1.8. Differential forms

47

With either convention, the wedge product is bilinear and associative, just as in the case of finite-dimensional manifolds, but not commutative. If φ is a k-form and ω an l-form, then φ ∧ ω = (−1)kl ω ∧ φ. We next note that differential forms are “functorial”. If F : M → N is a smooth map and φ is a smooth differential form of degree k on N , we can define a differential form F ∗ φ on M by [F ∗ φ](p) = Fp∗ (φ(F (p)); it follows from (1.16) that F ∗ φ is smooth. Moreover, the wedge product is natural under smooth maps: if F : M → N is a smooth map and φ and ω are smooth differential forms on N , then F ∗ (φ ∧ ω) = F ∗ φ ∧ F ∗ ω. Definition. If φ is a smooth k-form on M and X is a smooth vector field, the interior product ιX φ is the smooth (k − 1)-form on M defined by the formula ιX φ(X2 , . . . , Xk ) = φ(X , X2 , . . . , Xk ), whenever X1 , . . . , Xk are smooth vector fields on M. (It is readily checked that there is a unique such differential form.) It is easily checked that ιX (φ ∧ ψ) = (ιX φ) ∧ ψ + (−1)deg(φ) φ ∧ ιX ψ. Finally, we claim that there is an exterior derivative d, a the collection of R-linear maps from k-forms to (k + 1)-forms which satisfy the following axioms, familiar from finite-dimensional differential topology: (1) If ω is a k-form, the value dω(p) depends only on ω and its derivatives at p. (2) If f is a smooth real-valued function regarded as a differential 0form, d(f ) is the differential of f defined before. (3) d ◦ d = 0. (4) If ω is a k-form and φ is an l-form, then d(ω ∧ φ) = (dω) ∧ φ + (−1)k ω ∧ (dφ). (5) If F : N → M is a smooth map, F ∗ ◦ d = d ◦ F ∗ on differential forms.

48

1. Infinite-dimensional Manifolds

Just as in the finite-dimensional case, one can prove the following: Theorem 1.8.2. There is a unique of linear maps of real vector spaces, d : Ωk (M ) −→ Ωk+1 (M ), which satisfy the five axioms above. Moreover, these linear maps satisfy the explicit formula    (−1)i Xi ω(X0 , . . . , X&i , . . . , Xk ) (1.18) dω(X0 , . . . , Xk ) =  + (−1)i+j ω([Xi , Xj ], X0 , . . . , X&i , . . . , X&j , . . . , Xk ), i 0 such that 1 ([α, p, v], [α, p, v])α < [α, p, v], [α, p, v]p < cp ([α, p, v], [α, p, v])α, cp for all [α, p, v] ∈ Tp M with v = 0. (Thus the topology induced by the Riemannian metric on Tp M agrees with the model space topology.) (2) ·, ·p varies smoothly with p; in other words, p → ·, ·p is a smooth covariant tensor field of rank two. Example 1.9.1. Suppose that we have an isometric imbedding of the smooth compact Riemannian manifold M into RN . Then the Hilbert manifold L21 (S 1 , M ) can be given the Riemannian metric ·, · defined as follows: if X, Y ∈ Tγ L21 (S 1 , M ), we can regard X and Y as maps X, Y : S 1 → RN such that X(t) ∈ Tγ(t) M for each t ∈ S 1 . We set  [X(t) · Y (t) + X  (t) · Y  (t)]dt. X, Y γ = S1

50

1. Infinite-dimensional Manifolds

This can be regarded as the pullback of the “flat” Riemannian metric on the Hilbert space L21 (S 1 , RN ), and it is smooth because the pullback of a smooth covariant tensor field via a smooth map is smooth. Definition. If M is a Hilbert manifold with Riemannian metric p → ·, ·p , the gradient of a C 1 function f : M → R is the vector field grad(f ) defined by for all v ∈ Tp M. grad(f )(p), vp = dfp (v), The idea behind the method of steepest descent is to follow flowlines for the vector field −grad(f ); in favorable cases, these flowlines will converge to a critical point for f . Example 1.9.2. Once again suppose that the smooth compact Riemannian manifold M is isometrically imbedded in RN . If Σ is a compact Riemann surface with a constant curvature metric of total area one in its equivalence class, the Hilbert manifold L22 (Σ, M ) can be given a Riemannian metric ·, · by the following formula: If X, Y ∈ Tf L22 (Σ, M ), we can regard X and Y as maps X, Y : Σ → RN such that X(p) ∈ Tf (p) M for each p ∈ Σ. We set  X, Y f = [X · Y + DX · DY + D 2 X · D 2 Y ]dt, Σ

D2

represent the first and second covariant differentials comwhere D and puted with respect to the Riemannian metric on Σ. This is the pullback of a “flat” Riemannian metric on the Hilbert space L22 (Σ, RN ). If Σ is S 2 or T 2 , the group of isometries of the metric on Σ acts as isometries of this Riemannian metric. Of course, we can do a similar construction for L2k (Σ, M ), when k ≥ 3. In the case of Banach manifolds, the metrics best suited to our applications are not Riemannian, but Finsler. If (Uα , φα ) is a smooth chart on a Banach manifold M, the Banach space norm  ·  pulls back via φα to Tp M: we can define by [α, p, v]α = v.  · α : Tp M → R Definition. A Finsler metric on a Banach manifold M is a function which assigns to each p ∈ M a norm  · p : Tp M −→ R such that: (1) For each p ∈ M, there is some constant cp > 0 such that 1 [α, p, v]α < [α, p, v]p < cp [α, p, v]α , cp

1.9. Riemannian and Finsler metrics

51

for all [α, p, v] ∈ Tp M with v = 0. (Thus the Finsler norm on Tp M is equivalent to the Banach space norm for the model space.) (2)  · p varies continuously with p: Given p ∈ M and ε > 0, there exists an open neighborhood U of p such that q ∈ U implies (1 − ε)[α, q, v]q < [α, p, v]p < (1 + ε)[α, q, v]q , Simplification of notation: We will henceforth write v instead of [α.p, v] for a typical element of Tp M. Note that any Riemannian metric on a Hilbert manifold determines a Finsler metric: simply define ' by vp = v, vp .  · p : Tp M −→ R Even in this special case, however, the norm  · p is only continuous, not smooth as a function on T M. The Riemannian metric on a Hilbert manifold gives a norm-preserving vector bundle isomorphism from Tp M to Tp∗ M. We do not have such an isomorphism in the case of a Finsler metric on a Banach manifold, but the norm  · p on Tp M induces a dual norm (which we also denote by  · p for simplicity) on Tp∗ M: φp = sup{|φ(v)| : v ∈ Tp M and vp = 1}. Example 1.9.3. Once again suppose that the smooth compact Riemannian manifold M is isometrically imbedded in RN . Given a compact Riemann surface Σ and a real number p > 2, the Banach manifold Lp1 (Σ, M ) can be given a Finsler metric as follows: If X ∈ Tf (Lp1 (Σ, M )), we can regard X as a map X : Σ → RN such that X(p) ∈ Tf (p) M for each p ∈ Σ. We then let Xf be the Lp1 -norm of X as a mapping into Euclidean space. Alternatively, this Finsler metric f →  · f can be regarded as the pullback of the “flat” Finsler metric on the Banach space Lp1 (Σ, RN ). If M is a connected Banach manifold with reflexive model space E and Finsler metric  ·  and γ : [0, 1] → M is a C 1 curve, we can define its length L(γ) by  1 γ  (t)dt, L(γ) = 0

where integration along a path can be defined as follows. First we use (1.4) to define path integrals in the Banach space E, then we transport this to paths in the domain of a single coordinate chart and show it is independent of such a chart, then we extend to paths in the Banach manifold itself. We

52

1. Infinite-dimensional Manifolds

can then define a distance function d : M × M → R by (1.19)

(  d(p, q) = inf L(γ) : γ : [0, 1] → M is C 1 , γ(0) = p, γ(1) = q .

Proposition 1.9.4. Given a Finsler metric on a regular Banach manifold, the distance function d defined above defines a metric in the sense of metric spaces, and the metric topology agrees with the manifold topology. For the general proof, we refer to [Pal70] (see the appendix to §2). We merely describe a proof for our Examples 1.9.1, 1.9.2 and 1.9.3. It is quite easily verified that the distance function d satisfies d(p, r) ≤ d(p, q) + d(q, r),

d(p, q) = d(q, p),

and

d(p, p) = 0.

That only leaves the property d(p, q) = 0 ⇒ p = q. Proposition 1.9.5. In each of our examples, L21 (S 1 , M ), L2k (Σ, M ) for k ≥ 2 and Lp1 (Σ, M ) with p > 2, d is a metric and the metric topology agrees with the manifold topology. Moreover, L21 (S 1 , M ), L2k (Σ, M ) for k ≥ 2 and Lp1 (Σ, M ) are complete as metric spaces. Proof. Let us consider Lp1 (Σ, M ). If f, g ∈ Lp1 (Σ, M ) and d(f, g) = 0, then there exist arbitrarily short paths connecting f and g in Lp1 (Σ, M ). But a path connecting f and g in Lp1 (Σ, M ) also connects f and g in the ambient Banach space E = Lp1 (Σ, RN ), so there are arbitrarily short curves connecting f and g in the ambient Banach space. However, by a straightforward modification of the finite-dimensional argument, it is easily verified that if γ : [0, 1] → E is a C 1 -map into a Banach space, then 

1

γ  (t)dt ≥ γ(0) − γ(1),

0

so two distinct points in E cannot be joined by curves of arbitrarily small length. Since the map ωi : Lp1 (Σ, M ) → Lp1 (Σ, RN ) induced by the inclusion i : M → RN is an imbedding, it is now easy to verify that the metric topology agrees with the manifold topology. Finally, since ωi : Lp1 (Σ, M ) → Lp1 (Σ, RN ) is distance-decreasing, a Cauchy sequence {fi } in Lp1 (Σ, M ) is also a Cauchy sequence in Lp1 (Σ, RN ), and must therefore converge. Since Lp1 (Σ, M ) is a closed subset of Lp1 (Σ, RN ), we see that Lp1 (Σ, M ) must be complete as a metric space. The other cases are treated in the same way. QED

1.10. Vector fields and ODEs

53

1.10. Vector fields and ODEs It is well-known that the global qualitative theory of systems of ordinary differential equations is best formulated within the language of vector fields on finite-dimensional manifolds. This theory, including the fundamental existence and uniqueness theorem for systems of ordinary differential equations, can be extended to infinite-dimensional manifolds with almost no change in the proofs. A detailed exposition of this extension is presented in Chapter IV of [Lan95]. Definition. A C 1 curve γ : (a, b) → M is called an integral curve for the vector field X if (1.20)

X (γ(t)) = γ  (t),

for t ∈ (a, b),

where γ  (t) is the velocity vector to γ at t. Just as in the finite-dimensional case, a fundamental Existence and Uniqueness Theorem for Ordinary Differential Equations states that given a smooth vector field X on M, there is a unique integral curve for X which passes through any point of M: Theorem 1.10.1. Suppose that X is a C 2 vector field on M and p ∈ M. Then there is an open neighborhood U of p, an  > 0 and a C 2 map φ : (−ε, ε) × U −→ M such that if φt (q) = φ(t, q) for t ∈ (−, ) and q ∈ U , then (1) each curve t → φt (q) is an integral curve for X , (2) any integral curve for X which passes through U is of the form t → φt (q) for some q ∈ U , (3) φ0 is the inclusion U ⊂ M, and (4) φt ◦ φs = φt+s , whenever, both sides are defined. We will call (−ε, ε) × U a local flow box for X . Idea of proof (following Chapter IV of Lang [Lan95]). The proof is based upon the Contraction Lemma, just like the proof of the Inverse Function Theorem. We can replace the differential equation (1.20) by its local coordinate representation, and consider the initial value problem (1.21)

γ  (t) = f (γ(t)),

γ(0) = q,

54

1. Infinite-dimensional Manifolds

for γ : (a, b) → V and f : V → E, where V is a suitable open subset of a Banach space E. We can assume that f  ≤ K

Df  ≤ L

and

on V . Integrating both sides of (1.21) yields the equivalent integral equation  t f (γ(u))du. (1.22) γ(t) = q + 0

We can assume that the closed ball Bδ (p) of radius δ about p is contained in V and suppose that q ∈ Bδ (p), the open ball of radius δ about p. Let I = [−, ], where  > 0 will be chosen later, and let   X = γ : I → U : γ is continuous, γ(0) = q and γ(I) ⊂ B2δ (p) . We can make X into a complete metric space by defining the distance function d by d(γ1 , γ2 ) = sup{γ1 (t) − γ2 (t) : t ∈ I}. If γ ∈ X, we set



t

f (γ(u))du.

T (γ)(t) = q + 0

We choose ε so that ε < δ/K, and hence T (γ)(t) − q ≤ εK ≤ δ, so T (γ) ∈ X. Finally, we note that d (T (γ1 ), T (γ2 )) = sup {T (γ1 )(t) − T (γ2 )(t) : t ∈ I} ≤ ε sup{f (γ1 (t)) − f (γ2 (t)) : t ∈ I} ≤ εL sup{γ1 (t) − γ2 (t) : t ∈ I} = Ld(γ1 , γ2 ). Thus by choosing ε so that εL < 1, we can ensure that T : X → X will be a contraction. Then, by the Contraction Lemma, T has a unique fixed point γq ∈ X, which must be a solution to the integral equation (1.22). This fixed point γq is C 1 and is the unique solution to the initial value problem (1.21). Thus we can define φ : (−ε, ε) × Bδ (p) → V

by

φ(t, q) = γq (t).

It is relatively easy to check that properties (2), (3) and (4) of the theorem hold and that φ is continuous. It takes a little more work to check that the map φ is C 2 , and for that we refer the reader to the excellent presentation in [Lan95]. QED

1.11. Condition C

55

Once we have the Existence and Uniqueness Theorem, we can piece together the locally defined maps to form a map φ : V −→ M,

where V is an open neighborhood of {0} × M in R × M.

which we call the local flow for X . We say that the maps {φt } defined by φt (q) = φ(t, q) form the one-parameter group of local diffeomorphisms of M which corresponds to the vector field X .

1.11. Condition C We want to apply the existence and uniqueness theorem from the preceding section to find critical points of a C 2 real-valued map f : M → [0, ∞) via the method of steepest descent, where M is an infinite-dimensional manifold. In order to get this method to work, we need to assume that the function f assumes a “compactness condition” introduced by Palais and Smale [PS64]: Definition. Suppose that M is a Banach manifold with a complete Finsler metric. (For example, M might be a Hilbert manifold with a complete Riemannian metric.) Then a C 2 function f : M → [0, ∞) is said to satisfy condition C if whenever {pi } is a sequence in M such that (1) f (pi ) is bounded and (2) df (pi ) is not bounded away from zero, then {pi } possesses a subsequence which converges to a critical point for f . Note that if M is finite-dimensional, f : M → [0, ∞) satisfies Condition C if and only if it is proper. However, Condition C also makes sense and is easy to utilize when M is an infinite-dimensional Hilbert manifold: Theorem 1.11.1. Suppose that M is a Hilbert manifold with a complete Riemannian metric ·, ·. If f : M → [0, ∞) is a C 2 function which satisfies condition C, X = grad(f ) and {φt } is the local one-parameter group of diffeomorphisms corresponding to −X , then (1) for each p ∈ M, φt (p) is defined for all t ≥ 0, and (2) there is a sequence ti → ∞ such that {φti (p)} converges to a critical point for f . The proof will be based upon a collection of inequalities. It follows from the chain rule, that if g : M → R is any smooth function, (1.23)

dg(γ(t))(X (γ(t)) = dg(γ(t))(γ  (t)) =

d (g ◦ γ)(t). dt

56

1. Infinite-dimensional Manifolds

We use this fact, together with the fundamental theorem of calculus, to show that if 0 < t1 < t2 ,  t2 d f (φt (p))dt (1.24) f (φt1 (p)) − f (φt2 (p)) = − t1 dt  t2  t2 df (φt (p))(X (φt (p))dt = grad(f )(φt (p)), X (φt (p)dt = 

t1 t2

=

 grad(f )(φt (p))2 dt =

t1

t1 t2

df (φt (p))2 dt.

t1

On the other hand, using the metric d on the Riemannian manifold M, we have

 t2



d



(1.25) d(φt1 (p), φt2 (p)) ≤

dt (φt (p)) dt t1  t2  t2 X (φt (p))dt = df (φt (p))dt. ≤ t1

t1

Squaring this and using the Cauchy-Schwarz inequality, we obtain   t2 2 2 df (φt (p))dt (1.26) d(φt1 (p), φt2 (p)) ≤  ≤ (t2 − t1 )

t1 t2

df (φt (p))2 dt = (t2 − t1 )(f (φt1 (p)) − f (φt2 (p))).

t1

Let t¯ = sup{t ∈ R : φt (p) is defined}, and if t¯ < ∞, we choose a sequence {ti } of real numbers strictly less than t¯ such that ti → t¯. It follows from (1.26) that {φti (p)} is a Cauchy sequence in M. Since (M, d) is a complete metric space by Proposition 1.9.5, φti (p) → q, for some q ∈ M. But by the Existence and Uniqueness Theorem there is a flow box (−, )×U containing (0, q), and this implies that the curve t → φt (p) can be extended beyond t¯, giving a contradiction. Thus t¯ = ∞ and we see that φt (p) is defined for all t ≥ 0, and the first statement of the theorem is proven. Finally, it follows from (1.24) that  ∞ grad(f )(φt (p))2 dt < ∞, 0

and hence there must exist a sequence ti → ∞ such that df (φti (p)) = grad(f )(φti (p)) → 0. By Condition C, a subsequence of {φti (p)} converges to a critical point for f , finishing the proof of the theorem. Of course, we would like a version of the above theorem for the case of a C 2 function f : M → [0, ∞), where M is only a Banach manifold. However, it

1.11. Condition C

57

is not possible to define Riemannian metrics on M in this case, so we need a replacement for the notion of gradient, such as the following, a definition essentially due to Palais [Pal66]: Definition. Suppose that f : M → R is a C 2 function on a Banach manifold which has a Finsler metric and let U be an open subset of M. A C 1 vector field X : U → T M is called a pseudogradient for f over U if there is a fixed constant ε > 0 such that for each p ∈ U , (1) dfp (X (p)) > εdfp 2 , (2) X (p) < (1/ε)dfp . In both inequalities, dfp  = sup{|dfp (v)| : v ∈ Tp M and v ≤ 1}, which is the dual norm on the cotangent space to M at p. In the case of a Hilbert manifold, X = grad(f ) satisfies both conditions in the definition with  = 1; in other words, a gradient on a Hilbert manifold is also a pseudogradient. The definition was set up so that the following theorem would be true: Theorem 1.11.2. Suppose that M is a Banach manifold with a complete Finsler metric  · . Suppose that f : M → [0, ∞) is a C 2 function which satisfies condition C. Let K = {p ∈ M : df (p) = 0} and let U = M − K. If X is a pseudogradient for f on U , and {φt } is the local one-parameter group of diffeomorphisms corresponding to −X , then (1) for each p ∈ U , φt (p) is defined for all t ≥ 0, and (2) there is a sequence ti → ∞ such that {φti (p)} converges to a critical point for f . The proof is almost identical to the proof of Theorem 1.11.1. Assuming that p is not a critical point for f and that 0 < t1 < t2 , we use the first condition in the definition of pseudogradient to replace (1.24) by the inequality  t2 d (1.27) f (φt1 (p)) − f (φt2 (p)) = − f (φt (p))dt t1 dt  t2  t2 df (φt (p))(X (φt (p))dt ≥ ε df (φt (p))2 dt. = t1

t1

We use the second condition in the definition of pseudogradient to replace (1.25) by   t2 1 t2 X (φt (p))dt ≤ df (φt (p))dt. (1.28) d(φt1 (p), φt2 (p)) ≤ ε t1 t1

58

1. Infinite-dimensional Manifolds

We square as before and use the Cauchy-Schwarz inequality to obtain (1.29) d(φt1 (p), φt2 (p))2 ≤ ≤

t2 − t1 ε2



t2

1 ε2



t2

2 df (φt (p))dt

t1

df (φt (p))2 dt ≤

t1

t2 − t1 (f (φt1 (p)) − f (φt2 (p))). ε3

We can now use (1.29) instead of (1.26) to show that φt (p) is defined for all t ≥ 0. Finally, it follows from (1.27) that  ∞ df (φt (p))2 dt < ∞, 0

which enables us to find a sequence ti → ∞ such that df (φti (p)) → 0, and by condition C, a subsequence of {φti (p)} must converge to a critical point for f . Given a C 2 function f : M → [0, ∞) on a Banach manifold, how can we construct a corresponding pseudogradient on U = M − K, where K is the set of critical points for f ? The standard technique consists of constructing pseudogradients over each set of an open cover of U , and then piecing these together using a partition of unity. Let M be a metrizable infinite-dimensional smooth manifold modeled on a Banach space. According to a well-known theorem of Stone, M must be paracompact. That means that every open cover of M must have an open locally finite refinement. Definition. Let U = {Uα : α ∈ A} be an open cover of M. A partition of unity subordinate to U is a collection {ψα : α ∈ A} of continuous real-valued functions on M such that (1) ψα : M → [0, 1], (2) the support of ψα is a closed subset of Uα , (3) if p ∈ M, there is an open neighborhood V of p which intersects the supports of only finitely many ψα , and ) (4) ψα = 1. It is known (and proven in topology texts) that any open cover of a paracompact Hausdorff space possesses a continuous partition of unity subordinate to a given open cover U = {Uα : α ∈ A}. Moreover, as proven in Lang [Lan95], C ∞ Hilbert manifolds possess partitions of unity. However, for Banach manifolds, we encounter an obstacle: Banach manifolds need not possess C ∞ partitions of unity. As pointed out in the survey of Eells [Eel66], to construct C k partitions of C∞

1.11. Condition C

59

unity one needs to be able to construct nontrivial C k real-valued functions on the model Banach space E with bounded support. Fortunately, in the case where Σ is a Riemann surface and p > 2 the Banach manifold Lp1 (Σ, M ) does possess partitions of unity of class C 2 . To see why, we notice that the function f : Lp1 (Σ, R) −→ R

defined by

f (φ) = φpLp 1

is C 2 , by an argument similar to that given in Example 1.2.6. Let g : R → [0, 1] be a smooth function such that (1) g(s) = 1 when |s| ≤ 1, and (2) g(s) = 0 when |s| ≥ 2. Then the map  fε :

Lp1 (Σ, Rn )

→ [0, 1] defined by

fε (φ) = g

2f (φ) ε



is a C 2 function which equals one on a small neighborhood of the origin and has support contained in the set {φ ∈ Lp1 (Σ, Rn ) : φp ≤ ε}. Using local coordinates, we can transport this function to Lp1 (Σ, M ) thereby obtaining a C 2 function f : Lp1 (Σ, M ) → R which is one in a neighborhood of a given point p and vanishes outside a given open neighborhood of p. From this start we can follow the familiar argument (given in Lang [Lan95], Chapter II, §3) for constructing C 2 partitions of unity subordinate to any open cover on the metrizable smooth manifold Lp1 (Σ, M ). Lemma 1.11.3. If f : Lp1 (Σ, M ) → R is a C 2 function, where p > 2, then f possesses a C 2 pseudogradient X which is tangent to every Lpk (Σ, M ) ⊆ L2k (Σ, M ), for k ∈ N, k ≥ 2. Proof. Suppose that 0 < ε < 1. If p is not a critical point for f , we can √ choose a unit vector u ∈ Tp M such that |dfp (u)| > εdfp ; then √ satisfies v < dfp , dfp (v) > εdfp 2 , v = εdfp u the two conditions in the definition of pseudogradient at the point p. We can extend v to a smooth vector field which is a pseudogradient on some neighborhood of p; for example, we could choose it to be constant in terms of some smooth chart. Thus we can construct a pseudogradient on an open neighborhood about any point p which is not in the set K of critical points of F . If M admits C 2 partitions of unity, one can piece together a C 2 pseudogradient on M − K.

60

1. Infinite-dimensional Manifolds

In the above construction, we can choose v to lie in the dense subspace of Lp1 (Σ, M ) and the C 2 partition of unity on Lp1 (Σ, M ) can be chosen so that it restricts to a C 2 partition of unity on Lpk (Σ, M ) ⊆ L2k (Σ, M )for every k ≥ 2. When this is done, the pseudogradient will be tangent to every Lpk (Σ, M ) for k ≥ 2, finishing the proof of the lemma. QED L2k (Σ, M )

1.12. Birkhoff ’s minimax principle Condition C can often be used to construct critical points for a C 2 function f : M → [0, ∞) corresponding to various topological constraints of M. Indeed, Palais [Pal70] describes how it provides a rigorous derivation of George Birkhoff’s “minimax principle”. Suppose that M is a Banach manifold with a complete Finsler metric and that f : M → [0, ∞) is a C 2 function satisfying Condition C. We let Ma = {p ∈ M : f (p) ≤ a}

so that Ma ⊆ Mb

when

a ≤ b.

We say that c is a critical value for f if there is a critical point p ∈ M for f with f (p) = c. Our goal in this section is to use pseudogradients to show that if there are no critical values between a and b where a ≤ b, then Ma and Mb are homotopy equivalent. This fact sometimes allows us to prove the existence of critical points for f . Definition. For k ∈ N, a C k ambient isotopy of M is a C k map Ψ : [0, 1] × M → M such that each map ψt : M → M, defined by ψt (p) = Ψ(t, p) for t ∈ [0, 1], is a diffeomorphism, with ψ0 = id. We will sometimes denote the ambient isotopy by {ψt : t ∈ [0, 1]}. Theorem 1.12.1 (Deformation Theorem). Suppose that M is a Banach manifold with a complete Finsler metric and C 2 partitions of unity. If f : M → [0, ∞) is a C 2 function satisfying condition C and there are no critical points p for f such that a ≤ f (p) ≤ b where a < b, then (1) there is a C 2 ambient isotopy Ψ = {ψt : t ∈ [0, 1]} of M such that ψ1 (Mb ) ⊂ Ma , and (2) Ma is a strong deformation retract of Mb . Proof. Since f satisfies Condition C, there is a constant k > 0 such that df  ≥ k on {p ∈ M : a ≤ f (p) ≤ b}. Indeed, if not, there would exist a sequence {pi } in {p ∈ M : a ≤ f (p) ≤ b} such that df (pi ) → 0. By condition C, a subsequence of {pi } would converge to a critical point for f in {p ∈ M : a ≤ f (p) ≤ b}, contradicting the hypothesis of the theorem. A

1.12. Birkhoff’s minimax principle

61

similar argument shows that there is an ε > 0 such that there are no critical points p for f such that a − ε < f (p) < b + ε. Using a C 2 partition of unity, we construct a C 2 function η : M → [0, 1] such that (1) η ≡ 1 on {p ∈ M : a ≤ f (p) ≤ b}, (2) η ≡ 0 outside {p ∈ M : a − ε < f (p) < b + ε}. Let Y be a C 2 pseudogradient for f on U = M − K, where K is the critical locus of F and set X = ηY. Theorem 1.11.2 gives us a one-parameter group {φt : t ∈ [0, ∞} of local diffeomorphisms determined by −X with φt (p) defined for all p ∈ M when t ≥ 0 with f (φt (p)) decreasing. We claim that if p ∈ Mb , then φt (p) ∈ Ma for t > (b − a)/(εk). Indeed, if p ∈ Mb − Ma and φt (p) ∈ / Ma , it follows from (1.27) that  t df (φτ (p))dτ ≥ εkt b − a ≥ f (p) − f (φt (p)) ≥ ε 0

or, equivalently, t ≤ (b − a)/(εk). We now set ψt = φct ,

where

c=2

b−a , εk

and define Ψ : [0, 1] × M → M

by

Ψ(t, p) = ψt (p).

Then Ψ is an ambient isotopy such that ψ1 (Mb ) ⊂ Ma . This proves the first assertion of the Theorem. For the second, we let τ (p) be the first time t such that φt (p) ∈ Ma and define Φ : [0, 1]×Mb → Mb by  φt(τ (p)) (p), for p ∈ Mb − Ma , Φ(t, p) = p, for p ∈ Ma . One then checks that Φ is a strong deformation retraction from Mb to Ma , finishing the proof of the theorem. QED Definition. Let F be a family of subsets of M. We say that F is ambient isotopy invariant if A∈F

⇒

ψ1 (A) ∈ F

whenever {ψt : t ∈ [0, 1]} is a C 1 ambient isotopy of M. If F is an ambient isotopy invariant family of sets we set Minimax(f, F ) = inf {sup{f (p) : p ∈ A} : A ∈ F } .

62

1. Infinite-dimensional Manifolds

Theorem 1.12.2 (Birkhoff Minimax Principle). Suppose that M is a smooth Banach manifold with a complete Finsler metric and C 2 partitions of unity. Suppose, moreover, that f : M → [0, ∞) is a C 2 function satisfying condition C and F is a nonempty family of subsets of M which is ambient isotopy invariant. Then Minimax(f, F ) is a critical value for f . Proof. Let c = Minimax(f, F ). If c is not a critical value for f , then Condition C implies that there exists an ε > 0 such that there are no critical points p ∈ M with f (p) ∈ (c − ε, c + ε). But by definition of Minimax(f, F ), there exists an A ∈ F such that A ⊂ Mc+ε . Theorem 1.12.1 then gives a smooth isotopy {ψt : t ∈ [0, 1]} such that ψ1 (Mc+ε ) ⊂ Mc−ε ,

so

ψ1 (A) ⊂ Mc−ε .

Since F is ambient isotopy invariant, ψ1 (A) ∈ F showing that Minimax(f, F ) ≤ c − ε, a contradiction. QED For the simplest application of the minimax principle, we take M0 to be a component of M, and set F = {A ⊆ M : A ⊆ M0 }. Then F is ambient isotopy invariant and the previous theorem implies the following: Corollary 1.12.3. Suppose that M is a smooth Banach manifold with a complete Finsler metric, and that f : M → [0, ∞) is a C 2 function satisfying condition C. Then f assumes its minimum value on each component of M. We can also use invariants from algebraic topology to construct critical points. For example, suppose that M is simply connected and [α] is a nonzero element in πk (M), the k-th homotopy group of M. In this case, elements of πk (M) can be identified with free homotopy classes of maps f : S k → M and we can set F[α] = {h(S k ) such that h : S k → M is a continuous map representing [α]}, a space which is nonempty and ambient isotopy invariant. Hence there is a minimax critical point corresponding to the homotopy class [α]. Alternatively, suppose that x is a nonzero element in Hk (M; G), the singular homology group of M of degree k with coefficients in G, and let Fx = {supp(z) such that z is a singular cycle representing x}, where supp(z) denotes the support of z. Once again Fx is ambient isotopy invariant, and one obtains a minimax critical point corresponding to each homology class x. Of course, there is a dual version for cohomology. In fact, there is a de Rham theory version which makes use of differential forms: If

1.13. de Rham cohomology

63

M has C 2 partitions of unity and a is a differential k-form on M such that da = 0, we can set Fa = { h(A) such that A is a compact oriented manifold of dimension k   ∗ 1 and h : A → M is a C -map such that h a = 0 . A

It follows from the Homotopy Lemma to be proven in the next section that Fa is ambient isotopy invariant, so once again if Fa is nonempty we obtain a minimax critical point corresponding to the closed differential form a. Remark 1.12.4. Minimax critical points for different topological constraints can be the same. For example, a smooth function f : S n → R might have only two critical points even though the homotopy group πk (S n ) is nonvanishing for arbitrarily high k when n ≥ 2 (as explained in §4.1 of [Hat02]). Thus minimax critical points corresponding to nonzero elements of πk (S n ) for many different values of k must agree.

1.13. de Rham cohomology It is relatively straightforward to extend the standard treatments of de Rham cohomology to Banach manifolds which have C 2 partitions of unity. Indeed, if we let Ωk (M) = {C 1 differential forms ω on M of degree k : dω ∈ C 1 }, then whenever ψ is a C 2 smooth real-valued function, ω ∈ Ωk (M)

⇒

ψω ∈ Ωk (M).

We can then make the Ωk (M)’s into a cochain complex in which the differential is the exterior derivative d : Ωk (M) −→ Ωk+1 (M). We say that an element ω ∈ Ωk (M) is closed if dω = 0 and exact if ω = dθ for some θ ∈ Ωk−1 (M). Since d ◦ d = 0, every exact k-form is closed. The quotient space k (M; R) = HdR

closed elements of Ωk (M) exact elements of Ωk (M)

is called the de Rham cohomology of M. If ω ∈ Ωk (M) is closed, we let [ω] k (M; R). In the terminology of algebraic denote its cohomology class in HdR topology, the de Rham cohomology is the cohomology of the cochain complex (1.30)

· · · −→ Ωk−1 (M) −→ Ωk (M) −→ Ωk+1 (M) −→ · · · .

64

1. Infinite-dimensional Manifolds

Note that de Rham cohomology has a cup product defined by [ω] ∪ [φ] = [ω ∧ φ], which makes the direct sum ∗ HdR (M; R) =

∞ 

k HdR (M; R)

k=0

into a graded commutative algebra over R. Moreover, the cup product behaves well under smooth maps: If F : M → N is a smooth map, the linear map F ∗ on differential forms induces a linear map k k (N ; R) −→ HdR (M; R) F ∗ : HdR

such that F ∗ ([ω] ∪ [φ]) = F ∗ [ω] ∪ F ∗ [φ].

Moreover, the identity map on M induces the identity on de Rham cohomology and if F : M → N and G : N → P are smooth maps, then (G ◦ F )∗ = F ∗ ◦ G∗ , so M → H k (M; R),

k k (F : M → N ) → (F ∗ : HdR (N ; R) → HdR (M; R))

is a contravariant functor from the category of smooth manifolds and smooth maps to the category of real vector spaces and linear maps. Lemma 1.13.1 (Poincar´ e Lemma). If U is a convex open subset of a Banach space E or, more generally, any contractible open subset of E, then the de Rham cohomology of U is trivial:  R if k = 0, k HdR (U ; R) ∼ = 0 if k = 0. One can modify the proof that is used in the finite-dimensional case. We only sketch the key ideas—the reader can refer to Lang [Lan95] for details. Since the inclusion from a point into the convex set U is a homotopy equivalence, the Poincar´e Lemma is an immediate consequence of the next lemma. Lemma 1.13.2 (Homotopy Lemma). Smoothly homotopic maps F, G : M → N induce the same map on cohomology, k k (N ; R) −→ HdR (M; R). F ∗ = G∗ : HdR

On the other hand, via functoriality, this follows from the special case of the Homotopy Lemma for the inclusion maps i0 , i1 : M −→ [0, 1] × M,

i0 (p) = (0, p),

i1 (p) = (1, p).

1.13. de Rham cohomology

65

Indeed, if H : [0, 1] × M → N is a smooth homotopy from F to G, then by definition of homotopy, F = H ◦ i0 and G = H ◦ i1 , so i∗0 = i∗1 ⇒ F ∗ = i∗0 ◦ H ∗ = i∗1 ◦ H ∗ = G∗ . This special case, however, can be established by integrating over the fiber of the projection on the second factor [0, 1] × M → M. More precisely, let t be the standard coordinate on [0, 1], T the vector field tangent to the fiber of [0, 1] × M such that dt(T ) = 1. We then define integration over the fiber  1 (ιT ω)(t, p)dt. π∗ : Ωk ([0, 1] × M) → Ωk−1 (M) by π∗ (ω)(p) = 0

∗ ([0, 1]×M) Λk T(t,p)

Here the interior product (ιT ω)(t, p) is an element of and the integration is possible because the exterior power at (t, p) is canonically ∗ ([0, 1] × M). The key to proving that i∗0 = i∗1 in isomorphic to Λk T(0,p) cohomology is the “cochain homotopy” formula i∗1 ω − i∗0 ω = d(π∗ (ω)) + π∗ (dω). This formula can be verified by using naturality to reduce the proof to the finite-dimensional case, and then calculating in local coordinates just as in the familiar finite-dimensional treatment found in [BT82]. Note that dω = 0 ⇒ i∗1 ω − i∗0 ω = d(π∗ (ω)) ⇒ [i∗1 ω] = [i∗0 ω], and hence on the cohomology level i∗0 = i∗1 . This finishes our sketch of the proof of the Homotopy Lemma and the Poincar´e Lemma. Remark 1.13.3. If N is a submanifold of M with inclusion map i : N → M, we let Ωk (M, N ) = ker(i∗ : Ωk (M) → Ωk (N )), and note that the exterior derivative makes this into the k-th cochain group of a cochain complex Ω∗ (M, N ). The cohomology of this complex is called the relative de Rham cohomology of the pair (M, N ) and is denoted by k (M, N ; R). HdR The short exact sequence of cochain complexes (1.31)

0 −→ Ω∗ (M, N ) −→ Ω∗ (M) −→ Ω∗ (N ) → 0

yields a long exact sequence via the “snake lemma” (Theorem 2.16 in [Hat02]) from algebraic topology: k k k (M, N ; R) −→ HdR (M; R) −→ HdR (N ; R) · · · −→ HdR k+1 k+1 (M, N ; R) −→ HdR (M; R) −→ · · · . −→ HdR

This is very useful for calculating de Rham cohomology.

66

1. Infinite-dimensional Manifolds

For us, one of the primary uses of differential forms will be to calculate cohomology of infinite-dimensional manifolds. It is important to realize that ˇ the de Rham cohomology is the same as the singular or Cech cohomology with real coefficients that is studied in algebraic topology: Theorem 1.13.4 (de Rham Theorem). Suppose that either M = Lpk (Σ, M ) where pk > dim(Σ) or M is finite-dimensional, and that M admits C 2 partitions of unity. Then the cohomology of the cochain complex ˇ cohomology of M. Ω∗ (M) is isomorphic to the Cech The proof (due to Andr´e Weil) is via the zig-zag construction described in the excellent text on de Rham theory by Bott and Tu [BT82], which we follow closely. In the case where M is finite-dimensional, we let U = {Uα : α ∈ A} be a locally finite open cover of M by sets which are geodesically convex with respect to a Riemannian metric on M. If M = Lpk (Σ, M ) where pk > dim(Σ), we construct an open cover U = {Uα : α ∈ A} in which each open set Uα is of the form Uα = {g ∈ Lpk (Σ, M ) : g − f C 0 < δ}. The Sobolev Lemma guarantees the existence of such an open cover. We choose δ so small that if p, q ∈ M and d(p, q) < 2δ, then there is a unique minimizing geodesic γp,q : [0, 1] → M

such that γp,q (0) = p,

γp,q (1) = q,

and, moreover, this geodesic depends smoothly on p and q. (Then any two points in a δ-ball about a given point can be connected by a unique such geodesic.) If g1 and g2 are two elements of Uα , we can then define a path Γg1 ,g2 : [0, 1] → Lpk (Σ, M )

by

Γg1 ,g2 (t)(p) = γg1 (p),g2 (p) (t).

It is easily checked in either case that the intersection of any collection of elements from U is contractible. Readers familiar with cohomology theory ˇ will remember that the Cech cohomology of such an open covering is the ˇ same as the Cech cohomology of M (and we do not need to take direct limits). Such an open cover is often called a “good” cover or “Leray” cover. Given a good cover, we construct a double complex K ∗,∗ in which the (p, q)-element is ˇ p (U , Ωq ), K p,q = C which is defined to be the space of functions ω which assign to each distinct ordered (p + 1)-tuple (α0 , . . . , αp ) of indices such that Uα0 ∩ · · · ∩ Uαp = 0 an element ωα0 ···αp ∈ Ωq (Uα0 ∩ · · · ∩ Uαp )

1.13. de Rham cohomology

67

in such a way that if the order of elements in a sequence is permuted, ωα0 ,...,αp changes by the change of the permutation; thus ωα0 α1 = −ωα1 α0 ,

ωαα = 0,

and so forth.

We have two differentials on the double complex, the exterior derivative ˇ p (U , Ωq ) → C ˇ p (U , Ωq+1 ) defined by d:C

(dω)α0 ···αp = dωα0 ···αp ,

ˇ and the Cech differential ˇ p+1 (U , Ωq ) ˇ p (U , Ωq ) → C δ:C defined by (δω)α0 ···αp+1 =

p+1 

(−1)i ωα0 ···αˆ i ···αp+1 ,

i=0

the forms on the right being restricted to the intersection. The first differential is exact except when q = 0 by the Poincar´e Lemma, while in the case q = 0 we find that ˇ p (U , Ω1 )] = C ˇ p (U ; R), ˇ p (U , Ω0 ) → C [Kernel of d : C ˇ ˇ the space of Cech cocycles for the covering U on M. The Cech cohomology of the cover U is by definition the cohomology of the cochain complex (1.32)

ˇ p−1 (U ; R) −→ C ˇ p (U ; R) −→ C ˇ p+1 (U ; R) −→ · · · · · · −→ C

ˇ p (M; R). and is denoted by H The second differential is exact except when p = 0 and it is at this point that the C 2 partition of unity {ψα : α ∈ A} subordinate to U is used. ˇ p (U , Ωq ), we set Indeed, given a δ-cocycle ω ∈ C  ˇ p−1 (U , Ωq ), ψα ωαα0 ···αp−1 ∈ C τα0 ···αp−1 = α

noting that since ψα is C 2 we stay in the class of C 1 forms with C 1 exterior derivatives. Then  (−1)i ψα ωαα0 ···αˆ i ···αp (δτ )α0 ···αp = i,α

and it follows from the fact that δω = 0 that p  (−1)i ωαα0 ···αˆ i ···αp = ωα0 ···αp . i=1

68

1. Infinite-dimensional Manifolds

Hence

 (δτ )α0 ···αp =



 ψα

ωα0 ···αp = ωα0 ···αp ,

α

establishing exactness. When p = 0, we find that 0

1

ˇ (U , Ωq ) = Ωq (M), ˇ (U , Ωq ) → C Kernel of δ : C the space of smooth q-forms on M. We can summarize the previous discussion by stating that the rows and columns in the following commutative diagram are exact: · · · ↑ ↑ ↑ ˇ 0 (U , Ω2 ) → C ˇ 1 (U , Ω2 ) → C ˇ 2 (U , Ω2 ) → · · · 0 → Ω2 (M) → C ↑ ↑ ↑ 0 1 2 1 1 1 ˇ (U , Ω ) → C ˇ (U , Ω ) → C ˇ (U , Ω1 ) → · · · 0 → Ω (M) → C ↑ ↑ ↑ 0 1 2 0 0 0 ˇ ˇ ˇ 0 → Ω (M) → C (U , Ω ) → C (U , Ω ) → C (U , Ω0 ) → · · · ↑ ↑ ↑ 0 1 2 ˇ ˇ ˇ C (U , R) C (U , R) C (U , R) ↑ ↑ ↑ 0 0 0 The remainder of the proof uses this diagram. Given a de Rham cohomolp (M; R) with p-form representative ω we construct a ogy class [ω] ∈ HdR ˇ ˇ p (M; R) corresponding cohomology class s([ω]) in the Cech cohomology H ˇ 0 (U , Ωp ) by simas follows: The differential form defines an element ω 0p ∈ C ply restricting ω to the sets in the cover. It is readily checked that ω 0p is closed with respect to the total differential D = δ + (−1)p d on the double ˇ ∗ (U , Ω∗ ). Using the Poincar´e Lemma, we construct an elcomplex K ∗,∗ = C ˇ 0 (U , Ωp−1 ) such that dω 0,p−1 = ω 0p . Let ω 1,p−1 = δω 0,p−1 ement ω 0,p−1 ∈ C and observe that dω 1,p−1 = 0 and ω 1,p−1 is cohomologous to ω 0p with respect to D. Using the Poincar´e Lemma again, we construct an element ˇ 1 (U , Ωp−2 ) such that dω 1,p−2 = ω 1,p−1 . Let ω 2,p−2 = δω 1,p−2 and ω 1,p−2 ∈ C note that ω 2,p−2 is cohomologous to ω 0p with respect to D. Continue in this ˇ p (U , Ω0 ) which is cohomologous fashion until we reach a D-cocycle ω p0 ∈ C to ω 0p . Since dω p0 = 0, each function ωαp00 ···αp is constant, and thus ω p0 ˇ determines a Cech cocycle s(ω) whose cohomology class is s([ω]). By the usual diagram chasing, the cohomology class obtained is independent of choices made. Moreover, reversing the zig-zag construction described in the preceding paragraph yields an inverse to s. This finishes our sketch of

1.13. de Rham cohomology

69

the proof of de Rham’s theorem; for more details, one can consult [BT82], Chapter 2. Remark 1.13.5. The proof shows that the cohomologies of the two cochain complexes (1.30) and (1.32) are isomorphic. It follows that the cohomology of the cochain complex (1.32) is independent of the choice of good cover. On the other hand, in the case where M has C ∞ partitions of unity the argument can be repeated with C ∞ differential forms to show that the de Rham cohomology is the same whether calculated with C ∞ forms or C 1 forms with C 1 exterior derivatives. Remark 1.13.6. This remark assumes some familiarity with singular cohomology, as treated in Chapter 3 of [Hat02]. One could replace the double complex that occurs in the proof by Ksp,q = Csp (U ), which is defined to be the space of functions ω which assign to each distinct ordered (p + 1)-tuple (α0 , . . . , αp ) of indices such that Uα0 ∩ · · · ∩ Uαp = 0 an element sα0 ···αp in the space of singular cochains within Uα0 ∩ · · · ∩ Uαp with coefficients in Z. The above proof can then be modified to give an ˇ isomorphism from singular cohomology to the Cech cohomology of M.

Chapter 2

Morse Theory of Geodesics

2.1. Geodesics Our next goal is to explain how critical point theory on infinite-dimensional manifolds can be used to produce periodic solutions to a class of important nonlinear ordinary differential equations, the equations for geodesics in Riemannian manifolds. The geodesic equation is a generalization of the simplest second-order linear ordinary differential equation—the equation of a particle moving with zero acceleration in Euclidean space. This equation asks for a vector-valued function γ : (a, b) −→ RN

such that

γ  (t) = 0,

and its solutions are the constant speed straight lines. The simplest way to make this differential equation nonlinear is to consider a proper submanifold M of RN with the induced Riemannian metric, and ask for a function γ : (a, b) −→ M ⊂ RN

such that

(γ  (t)) = 0,

where (·) denotes projection into the tangent space of M . In simple terms, we are asking for the curves which are as straight as possible subject to the constraint that they lie within M . In the terminology of differential geometry, the tangential projection of the ordinary derivative is known as the covariant derivative, and one often writes (γ  (t)) = ∇γ  γ  (t)

or

(γ  (t)) =

Dγ  (t), dt 71

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2. Morse Theory of Geodesics

where ∇ and D are two commonly used notations for the covariant derivative. We can then write the equation (γ  (t)) = 0 in the form ∇γ  γ  (t) = 0. A smooth maps γ which satisfies this equation is called a smooth geodesic in M . According to the well-known Nash Imbedding Theorem, any Riemannian manifold M has a proper isometric imbedding ι : M n → RN into Euclidean space, so the above construction gives a theory of geodesics in any abstract Riemannian manifold. In fact, we can put the geodesic equation into the more general context of simple mechanical systems: Definition. A simple mechanical system is a triple (M, ·, ·, φ), where (M, ·, ·) is a Riemannian manifold and φ : M → R is a smooth function. We call M the configuration space of the simple mechanical system. For a simple mechanical system, “Newton’s equation of motion” is ∇γ  γ  (t) = −(grad φ)(γ  (t)). The left-hand side can be interpreted as the acceleration of a moving particle of unit mass, while the right-hand side is the force (per unit mass) produced by the potential φ. Example 2.1.1. We can apply this to Kepler’s problem of determining the motion of planets around the sun by taking M = R2 − {0, 0} with the standard Euclidean metric and Euclidean coordinates (x, y) and defining φ : M → R by −c , φ=  x2 + y 2 where c is a positive constant. We take c = (Newton gravitational constant)(mass of sun), regard (1/2)γ  (t), γ  (t) as the kinetic energy at time t of a planet of unit mass whose motion is represented by the curve γ : (a, b) → M and φ ◦ γ(t) as the potential energy of the planet in a gravitational field produced by the sun which is located at the origin. In this case, it is shown in books on classical mechanics that the bounded solutions to Newton’s equation of motion, γ  (t) = −(grad φ)(γ  (t)), are ellipses, traversed in accordance with Kepler’s three laws of motion. Example 2.1.2. We can take M = SO(3), the group of real orthogonal 3 × 3 matrices of determinant one, regarded as the space of “configurations”

2.1. Geodesics

73

of a rigid body B in R3 , which has its center of mass located at the origin. We want to describe the motion of the rigid body as a path γ : (a, b) → M . If a point in the rigid body has coordinates (x1 , x2 , x3 ) at time t = 0 , we suppose that the coordinates of the same point at time t will be ⎛ ⎛ ⎞ ⎞ a11 (t) a12 (t) a13 (t) x1 where γ(t) = ⎝a21 (t) a22 (t) a23 (t)⎠ ∈ SO(3), γ(t) ⎝x2 ⎠ , x3 a31 (t) a32 (t) a33 (t) and γ(0) = I, the identity matrix. Then the velocity v(t) of the particle p at time t will be ⎛ ⎞ 3 x1   ⎠ ⎝ x and v(t) · v(t) = aki (t)akj (t)xi xj . v(t) = γ (t) 2 k=1 x3 Suppose now that ρ(x1 , x2 , x3 ) is the density of matter at (x1 , x2 , x3 ) within the rigid body. Then the total kinetic energy within the rigid body at time t is given by the expression   1 ρ(x1 , x2 , x3 )xi xj dx1 dx2 dx3 aki (t)akj (t). K= 2 B i,j,k

We can rewrite this as 1 cij aki (t)akj (t), K= 2

 where

cij =

ρ(x1 , x2 , x3 )xi xj dx1 dx2 dx3 , B

i,j,k

and define a Riemannian metric ·, · on M = SO(3) by  cij aki (t)akj (t). γ  (t), γ  (t) = i,j,k

(1/2)γ  (t), γ  (t)

Then represents the kinetic energy of the rigid body B at time t. A smooth function φ : SO(3) → R can represent a potential energy for the rigid body. We thus obtain a simple mechanical system, which has configuration space SO(3) with a suitable left-invariant metric and potential φ, for which Newton’s equations of motion reduce to Euler’s equations for a rotating rigid body in a gravitational potential. The idea behind the calculus of variations is to regard solutions to ordinary differential equations—such as the geodesic equation or Newton’s equation of motion—as critical points of functions defined on infinite-dimensional manifolds. In the case of geodesics, the infinite-dimensional manifold is L21 ([0, 1], M ) = {γ ∈ L21 ([0, 1], RN ) : γ(t) ∈ M , for all t ∈ [0, 1]}, where M is a proper submanifold of RN , or one of the many useful subspaces of L21 ([0, 1], M ). These include the free loop space L21 (S 1 , M ) = {γ ∈ L21 ([0, 1], M ) : γ(0) = γ(1)},

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2. Morse Theory of Geodesics

which is used to study periodic motion in simple mechanical systems, the space of paths from p to q, Ω(M, p, q) = {γ ∈ L21 ([0, 1], M ) : γ(0) = p, γ(1) = q}, where p and q are points of M , and the space of paths from S0 to S1 , Ω(M, S0 , S1 ) = {γ ∈ L21 ([0, 1], M ) : γ(0) ∈ S0 , γ(1) ∈ S1 }, when S0 and S1 are compact imbedded submanifolds of M . The first step towards formulating the equations of geodesics within critical point theory is to define the Euclidean action  1 1  by JRN (γ) = γ (t) · γ  (t)dt, JRN : L21 ([0, 1], RN ) → R 2 0 the dot denoting the Euclidean dot product. The map JRN is clearly smooth, being the restriction of a continuous bilinear map  1 γ  (t) · λ (t)dt (γ, λ) → 2 S1 to the diagonal. Recall from §1.4 that the isometric imbedding ι : M → RN induces by composition a map ωι : L21 ([0, 1], M ) −→ L21 ([0, 1], RN ) which is also a smooth imbedding. Moreover, the Hilbert space structure on the spaces L21 ([0, 1], RN ) induces Riemannian metrics on L21 ([0, 1], M ) and its subspaces L21 (S 1 , M ) and Ω(M, p, q). We define JM : L21 ([0, 1], M ) → R

JM = JRN ◦ ωι ,

by

a map which is clearly smooth, being the composition of smooth maps. In addition, if φ : M → R is a smooth function, the map  1 2 φ(γ(t))dt γ ∈ L1 ([0, 1], M ) → 0

is also smooth, since it is the composition of ωφ : L21 ([0, 1], M ) −→ L21 ([0, 1], R) with integration, integration being a continuous linear map. Finally, we define the action for the simple mechanical system (M, ·, ·, φ) to be the function   1 1  2  γ (t), γ (t) − φ(γ(t)) dt. JM,φ (γ) = JM,φ : L1 ([0, 1], M ) → R, 2 0 By restriction, we also get smooth maps JM,φ : L21 (S 1 , M ) −→ R,

JM,φ : Ω(M, p, q) −→ R.

2.1. Geodesics

75

With these preparations out of the way, we can now state Hamilton’s principle of least action: the motion of a simple mechanical system is described by a critical point for the action function JM,φ on L21 (S 1 , M ) or Ω(M, p, q). Focusing first on the case of free loops, the case needed for studying periodic motion, we are led to ask: What is the differential (dJM,φ )γ : Tγ (L21 (S 1 , M )) −→ R? We will assume that γ is smooth and that V ∈ Tγ (L21 (S 1 , M )) lies in the space of C ∞ sections of the pullback γ ∗ T M of the tangent bundle T M by γ. To calculate the differential, we suppose that α : S 1 × (−, ) → M is a smooth one-parameter family of curves with α(t, 0) = γ(t) and D2 α(t, 0) = V (t), where D2 denote the partial derivative with respect to the second slot. Then ωα : L21 (S 1 , S 1 × (−, )) −→ L21 (S 1 , M ) is smooth. Define μ ¯ : (−, ) → L21 (S 1 , S 1 × (−, ))

by

μ ¯(τ )(t) = (t, τ ),

¯ is smooth and hence α ¯ = ωα ◦ μ ¯ is a smooth curve in for t ∈ S 1 . Clearly, μ ¯  (0) = V . A straightforward calculation now shows that L21 (S 1 , M ) with α  d ¯ ) = [γ  (t), ∇γ  V  − dφ(γ(t))(V (t))]dt, dJM,φ (γ)(V ) = (JM,φ ◦ α dt 1 S τ =0 where ∇γ  V is the directional derivative of V in the direction of γ  projected into the tangent space, otherwise known as the covariant derivative of V . Since γ is assumed to be smooth, we can integrate by parts to obtain  (2.1) dJM,φ (γ)(V ) = − ∇γ  γ  (t) + (grad φ)(γ  (t)), V (t)dt, S1

the boundary terms cancelling by periodicity. Since V can be an arbitrary smooth vector field along γ, a critical point for JM,φ must be a periodic solution to Newton’s equation of motion, ∇γ  γ  (t) = −(grad φ)(γ  (t)). Thus Hamilton’s principle implies that the motion of a simple mechanical system should be represented by solutions to Newton’s equation of motion. In the case where φ = 0, a critical point for the action is simply a smooth closed geodesic. In a quite similar fashion, one finds that critical points to the function JM,φ : Ω(M ; p, q) → R are solutions γ to Newton’s equation of motion such that γ(0) = p and γ(1) = q. On the other hand, if one calculates the

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2. Morse Theory of Geodesics

derivative on the larger space Ω(M, S0 , S1 ), the integration by parts gives additional boundary terms and we must replace (2.1) by  dJM,φ (γ)(V ) = − ∇γ  γ  (t) + (grad φ)(γ  (t)), V (t)dt S1

+ γ  (1), V (1) − γ  (0), V (0).

This more complicated formula must be used when considering critical points for dJM,φ : Ω(M, S0 , S1 ) → R, and in this case V (0) and V (1) are constrained to be tangent to S0 and S1 . The first variation formula implies that critical points to JM,φ are solutions to Newton’s equation which are perpendicular to S0 and S1 .

2.2. Condition C for the action Our goal is to apply the method of steepest descent to the calculus of the variations problems we just described. For this to work, we need the topology on the space of maps to satisfy two conditions: (1) It must be strong enough so that JM,φ is continuous. (2) It must be weak enough so that suitable sequences {γi } (such as sequences with JM,φ (γi ) bounded and dJM,φ (γi ) not bounded below by zero) will converge in the topology. If the topology is too weak JM,φ will not be continous, while if it is too strong, it will be impossible to establish convergence of minimax sequences {γi }. The two conflicting requirements single out L21 as the appropriate space to use when studying critical points of the action JM,φ . In this section, we show that when M is compact, the action functions JM , JM,φ : L21 (S 1 , M ) −→ R satisfy condition C, thereby making available the minimax principle from §1.11. Moreover, we will show that the critical points of these functions are actually C ∞ curves. Theorem 2.2.1. If (M, ·, ·) is a compact Riemannian manifold, the function JM : L21 (S 1 , M ) → R, defined by  1 γ  (t), γ  (t)dt, JM (γ) = 2 S1 satisfies Condition C: if {γi } is a sequence in L21 (S 1 , M ) such that (2.2)

JM (γi ) is bounded

and

dJM (γi ) → 0,

then it possesses a subsequence which converges to a critical point for JM .

2.2. Condition C for the action

77

Proof. We start by recalling the proof of the Sobolev Lemma from §1.4. Suppose that {γi } is a sequence of elements of L21 (S 1 , M ) satisfying (2.2). We can regard each γi as an element of the space L21 (S 1 , RN ). Then it follows from the Cauchy-Schwarz inequality that for t1 < t2 , 1/2   t2  t2 √   2 |γi (t1 ) − γi (t2 )| ≤ |γi (t)|dt ≤ t2 − t1 |γi (t)| dt t1 t1  √ ≤ t2 − t1 2JRN (γi ). Since JM (γi ) is bounded, we see that {γi } is equicontinuous. Since γi takes values in the compact submanifold M ⊂ RN , {γi } is also uniformly bounded. It therefore follows from Arzela’s Theorem or Ascoli’s Theorem ([Roy88], page 179) that a subsequence of {γi } will converge uniformly to a continuous map γ∞ : S 1 → M . For simplicity, we continue to denote the subsequence by {γi }. To finish the proof, we need to show that dJM (γi ) → 0 ⇒

{γi } is a Cauchy sequence in L21 (S 1 , M ) ⊆ L21 (S 1 , RN ).

To understand dJM (γ), we need to be able to compare an element of Tγ L21 (S 1 , RN ) with its projection into Tγ L21 (S 1 , M ). Recalling that M is a submanifold of RN with inclusion ι : M → RN , we let P denote the vector bundle map P : ι∗ T (RN ) −→ T M which projects onto the tangential component. By the ω-Lemma, we have a smooth map ωP : L21 (S 1 , ι∗ T (RN )) −→ L21 (S 1 , T M ) which is a vector bundle map, both image and range being vector bundles over L21 (S 1 , M ). Lemma 2.2.2. Suppose that ·, · denotes the Riemannian metric on either L21 (S 1 , RN ) or L21 (S 1 , M ) and X is an element of L21 (S 1 , ι∗ T (RN )) with ωπ (X) = γ, where ωπ : L21 (S 1 , ι∗ T (RN )) → L21 (S 1 , M ) is the projection induced by π : ι∗ T (RN ) → M . Then ωP (X), ωP (X) ≤ [1 + CJM (γ)]X, X, where C is some constant which depends on M but is independent of γ. Proof. We can regard P as a section of the bundle i∗ (End(RN )) over M and it possesses a differential dP which is a section of T ∗ M ⊗ [i∗ (End(RN ))].

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2. Morse Theory of Geodesics

Let C1 = sup{dP (v) : v is a unit-length element of T M }, a finite constant since M is compact. Then it follows from the Leibniz rule that (ωP (X))(t) = Pγ(t) X(t)

⇒

Recall that



ωP (X), ωP (X) =

(ωP (X)) (t) = dP (γ  (t))(X(t)) + Pγ(t) X  (t). 

S1

 (ωP (X)) (t)2 + (ωP (X))(t)2 dt,

where  ·  denotes the norm in RN , and hence  [dP (γ  (t))(X(t)) + Pγ(t) X  (t)2 + Pγ(t) (X)(t)2 ]dt ωP (X), ωP (X) = 1 S   [dP (γ (t))(X(t))2 + Pγ(t) X  (t)2 + Pγ(t) (X))(t)2 ]dt ≤ 1 S [C1 γ  (t)2 (X(t)2 + X  (t)2 + X(t)2 ]dt. ≤ S1

Thus,



ωP (X), ωP (X) ≤ C1 sup{(X(t) : t ∈ S } 2

1

S1

γ  (t)2 dt + X, X

≤ (CJM (γ) + 1)X, X, for some positive constant C. This finishes the proof of the lemma. QED We recall that if M is an imbedded submanifold in RN and p ∈ M , then the second fundamental form α : Tp M × Tp M −→ Np M = (normal space to M at p) is the symmetric bilinear map defined by α(γ  (0), γ  (0)) = γ  (0) , for any smooth path γ : (−ε, ε) −→ M such that γ(0) = p. Lemma 2.2.3. If X ∈ L21 (S 1 , ι∗ T (RN )) and ωπ (X) = γ, then  [γ  (t) · X  (t) + α(γ  (t), γ  (t)) · X(t)]dt, (2.3) dJM (γ)(ωP (X)) = S1

where the dot products are taken in the ambient Euclidean space RN . Proof. Note that it suffices to establish (2.3) in the case where γ and X are smooth because both sides are continuous in the L21 -topology. Integration by parts shows that   γ  (t)·ωP (X) (t)(X(t))dt = − γ  (t)·Pγ(t) (X(t))dt. dJM (γ)(ωP (X)) = S1

S1

2.2. Condition C for the action

79

But since α(γ  (t), γ  (t)) is the normal component of γ  (t), γ  (t) · X(t) = Pγ(t) (γ  (t)) · X(t) + α(γ  (t), γ  (t)) · X(t) = γ  (t) · Pγ(t) (X(t)) + α(γ  (t), γ  (t)) · X(t), and hence



dJM (γ)(ωP (X)) = S1

[−γ  (t) · X(t) + α(γ  (t), γ  (t)) · X(t)]dt.

Another integration by parts yields the claim. QED Returning to the proof of the theorem, we let {γi } be a sequence satisfying (2.2) which converges uniformly to a continuous map γ∞ : S 1 → M ⊆ RN . Since JM (γi ) is bounded, γi , γi  = γi 2 is bounded, and hence γi − γj 2 ≤ (γi  + γj )2 is also bounded as i, j → ∞. Lemma 2.2.2 implies that (ωP )γi (γi − γj ) is bounded as well. Since dJM (γi ) → 0, for every ε > 0 there exists N1 ∈ N such that ε (2.4) |dJM (γi )((ωP )γi (γi − γj )) − dJM (γj )((ωP )γj (γi − γj ))| < 3 for i, j > N1 . Here (ωP )γi (γi − γj ) lies in the tangent space to L21 (S 1 , M ) at γi . It follows from (2.4) and the explicit formula (2.3) for dJM that for every  > 0 there exists N ∈ N such that  [γi (t) · (γi (t) − γj (t)) + α(γi (t), γi (t)) · (γi (t) − γj (t))]dt S1  ε [γj (t) · (γi (t) − γj (t)) + α(γj (t), γj (t)) · (γi (t) − γj (t))]dt < , − 3 S1 for i, j > N1 . But |α(γi (t), γi (t))| ≤ (constant)|γi (t)|2 , the constant is a bound for the norm of the second fundamental form of M and hence depends only on the submanifold M . Since for any positive 1 integer m, |γi (t) − γj (t)| < m for i and j sufficiently large,    ≤ (constant)JM (γi ) 1 ≤ ε α(γ (t), γ (t)) · (γ (t) − γ (t))dt i j i i 1 m 3 S as i, j > N2 , a similar implication holding when i is replaced by j with perhaps a different choice of N3 . Hence for every ε > 0 there exists N ∈ N such that  (γ  (t) − γ  (t)) · (γ  (t) − γ  (t))dt < ε, j i j 1 i S

for i, j > N , where N is the maximum of N1 , N2 and N3 . This, together with the C 0 convergence of {γi } implies that {γi } is a Cauchy sequence in

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2. Morse Theory of Geodesics

L21 (S 1 , RN ). Since L21 (S 1 , RN ) is complete, {γi } has a limit in L21 (S 1 , RN ), which is continuous by the Sobolev inequality and therefore must be γ∞ . Thus γi → γ∞ ∈ L21 (S 1 , M ), and by continuity of dJM  on L21 (S 1 , M ), the limit γ∞ must be a critical point for JM . This concludes the proof of Theorem 2.2.1. QED Theorem 2.2.4. Suppose that (M, ·, ·) is a compact Riemannian manifold, φ : M → R is a smooth function and ψ : S 1 → RN is an L21 map. The real-valued functions JM,φ and JM,ψ on L21 (S 1 , M ), defined by     1 and γ (t), γ  (t) − φ(γ(t)) dt JM,φ (γ) = 2 S1 JM,ψ (γ) =

1 2



 S1

 γ  (t), γ  (t) + γ(t) · ψ(t) dt,

satisfy Condition C. Sketch of proof. Suppose that {γi } is a sequence in L21 (S 1 , M ) such that JM,φ (γi ) is bounded and dJM,φ (γi ) → 0. Then JM (γi ) is also bounded and just as before, {γi } is uniformly bounded and equicontinuous and thus by the Arzela-Ascoli Theorem, possesses a subsequence which converges uniformly to a continuous map γ∞ . We can now mimic the preceding proof up to equation (2.4). At this point, we need to account for an extra term in dJM,φ , namely  dφ(γi (t))(Pγi (t) (γi (t) − γj (t)))dt. − S1

However, this term goes to zero, since {γi (t)} is Cauchy in C 0 . Similarly, there is an extra term in JM,ψ which goes to zero. With these minor changes, the argument proceeds to the desired conclusion exactly as before. QED Theorem 2.2.5. Suppose that (M, ·, ·) is a complete Riemannian manifold which has a proper isometric immersion into some Euclidean space RN and that φ : M → R is a smooth function. Then the real-valued functions JM,φ : Ω(M, p, q) → R and defined by JM,φ (γ) =

1 2



JM,φ : Ω(M, S1 , S2 ) → R,

 γ  (t), γ  (t) − φ(γ(t)) dt,

1 0

satisfy condition C. Proof. A straightforward modification of preceding arguments. QED

2.3. Fet-Lusternik Theorem

81

Theorem 2.2.6. If the path γ within L21 (S 1 , M ) or Ω(M, p, q) is a critical point for JM or JM,φ , then γ is C ∞ . Moreover, for k ∈ N ∪ {0}, (1) if ψ ∈ L2k (S 1 , RN ), any critical point for JM,ψ lies in L2k+2 (S 1 , M ), and (2) more generally, if ∇γ  γ  is in L2k , then γ ∈ L2k+2 (S 1 , M ). Proof. We prove only the free loop space cases. If γ is a critical point for JM , it follows from (2.3) that   γ  (t) · X  (t)dt = − α(γ  (t), γ  (t)) · X(t)dt, S1

S1

for all X ∈ Tγ L21 (S 1 , RN ). Thus in the sense of distributions, γ  (t) = α(γ  (t), γ  (t)).

(2.5) Note that γ ∈ L21

⇒

γ  ∈ L2

⇒

α(γ  (t), γ  (t)) ∈ L1 .

Thus it follows from (2.5) and standard theorems of analysis that γ is C 1 . Now we use the technique of “elliptic bootstrapping for ordinary differential equations”: γ ∈ C 1 ⇒ RHS of (2.5) is C 0 ⇒ γ ∈ C 2 ⇒ RHS of (2.5) is C 2 ⇒ · · · . By induction, we see that γ is C ∞ and Theorem 2.2.6 is established for JM . The proofs for JM,φ and JM,ψ are the same except that (2.5) is replaced by γ  (t) = α(γ  (t), γ  (t)) − (grad φ)(γ  (t)) or

γ  (t) = α(γ  (t), γ  (t)) + (ψ(t)) ,

where (·) is the tangential component. QED

2.3. Fibrations and the Fet-Lusternik Theorem To derive conclusions from Condition C for closed geodesics, we need to understand the topology of the Hilbert manifold M = L21 (S 1 , M ). The fact that M is the total space of a “fibration” over M makes it relatively easy to calculate its homotopy groups from the homotopy groups of M . To explain this, we review some of the basic properties of fibrations, referring the reader to Bott and Tu [BT82], §§16 and 17 or Hatcher [Hat02], Theorem 4.41 for more details. A fibration is a continuous map f : E → B between topological spaces which has the homotopy lifting property; this means that if the continuous

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2. Morse Theory of Geodesics

map g˜ : Y → E has a projection into the base f ◦ g˜ : Y → B which can be extended to a homotopy in the base H : Y × [0, 1] → B

with

H(y, 0) = f ◦ g˜(y),

for y ∈ Y ,

then this homotopy H can be lifted to ˜ : Y × [0, 1] → E ˜ H such that H(y, 0) = g˜(y) and

˜ = H. f ◦H

The key facts about fibrations are that if the base B is connected, the fibers Ep = f −1 (p), for p ∈ B, are homotopy equivalent to each other, and the fibration f induces a long exact sequence of homotopy groups · · · −→ πk (Ep , e0 ) −→ πk (E, e0 ) −→ πk (B, p) −→ πk−1 (Ep , e0 ) −→ · · · , where e0 is a choice of base point in Ep . In fact, the only thing needed for the exact sequence is that f be a weak fibration, which means that it has the homotopy lifting property for the case where Y = [0, 1]n , the n-cube, for all choices of n. A key example concerns the path space P = { continuous maps γ : [0, 1] → M such that γ(0) = p }, where p is some choice of base point in M . In this case, the map π : P → M,

π(γ) = γ(1)

is a fibration. Indeed, given g˜ : Y → P

and H : Y × [0, 1] → M with H(y, 0) = π ◦ g˜(y), ˜ : Y × [0, 1] → P by we can define the lift H    t for 0 ≤ t ≤ 1 − (s/2), g ˜ (y) 1−(s/2) , ˜ H(y, s)(t) = H(y, 2t − 2 + s), for 1 − (s/2) ≤ t ≤ 1. ˜ is continuous and has the desired properties. One readily checks that H Indeed, when t = 1 − (s/2),   t = g˜(y)(1) = H(y, 0) = H(y, 2t − 2 + s), g˜(y) 1 − (s/2) so the two pieces of the function fit together continuously, while when we ˜ ˜ is indeed set t = 1, we find that H(y, s)(1) = H(y, 2 − 2 + s) = H(y, s), so H a lift H. The fiber over the base point p of this fibration is Ωp = { continuous maps γ : [0, 1] → M such that γ(0) = p = γ(1) } and is known as the pointed loop space. Its homotopy groups can be computed by the long exact sequence of the fibration, · · · −→ πk (Ωp ) −→ πk (P) −→ πk (M ) −→ πk−1 (Ωp ) −→ · · · .

2.3. Fet-Lusternik Theorem

83

Since P is contractible via the homotopy H : P × [0, 1] → P,

where

H(γ, s)(t) = γ((1 − s)t),

we see that πk (P) = 0 for all k and this long exact sequence collapses to yield πk (Ωp ) ∼ = πk+1 (M ). A second example is the free loop space C 0 (S 1 , M ) = {continuous maps γ : [0, 1] → M such that γ(0) = γ(1)}, which is the total space of a fibration ev : C 0 (S 1 , M ) → M

defined by

ev(γ) = γ(0).

The fiber over p in this case is Ωp once again, and we obtain another long exact sequence · · · −→ πk (Ωp ) −→ πk (C 0 (S 1 , M )) −→ πk (M ) −→ πk−1 (Ωp ) −→ · · · . In this case, however, ev∗ : πk (C 0 (S 1 , M )) → πk (M ) possesses a right inverse i∗ : πk (M ) → πk (C 0 (S 1 , M )) induced by the map i : M → C 0 (S 1 , M ) which takes the point p ∈ M to the constant loop located at p. Hence the exact sequence for ev splits, and we conclude that πk (C 0 (S 1 , M )) ∼ = πk (M ) ⊕ πk (Ωp ) ∼ = πk (M ) ⊕ πk+1 (M ). The consequence is that the homotopy groups of the free loop space C 0 (S 1 , M ) are quite easily determined from the homotopy groups of M . Application to Closed Geodesics: In studying closed geodesics on a Riemannian manifold (M, ·, ·), we let M = L21 (S 1 , M ), an infinite-dimensional Hilbert manifold with a complete Riemannian metric, and define the action  1 γ  (t), γ  (t)dt, J : M −→ R by J(γ) = 2 S1 a function which satisfies condition C by Theorem 2.2.1. We have seen in §1.5 that M = L21 (S 1 , M ) is homotopy equivalent to C 0 (S 1 , M ), the free loop space studied by topologists. Theorem 2.3.1. Let (M, ·, ·) be a compact connected Riemannian manifold and let [S 1 , M ] denote the set of free homotopy classes of continuous maps from S 1 to M . If α ∈ [S 1 , M ] and L21,α (S 1 , M ) = {γ ∈ L21 (S 1 , M ) : the free homotopy class of γ is α}, then J assumes its minimum on L21,α (S 1 , M ). If α = 0, this minimum is achieved at a nonconstant smooth closed geodesic.

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Proof. This is a direct consequence of Theorem 2.2.1 and Corollary 1.12.3. QED The preceding theorem shows a nonsimply connected compact manifold always possesses a smooth closed geodesic. To treat the simply connected case, we need the following theorem due to Lusternik and Fet: Theorem 2.3.2. If (M, ·, ·) is a compact simply connected Riemannian manifold, M contains a nonconstant smooth closed geodesic. Proof of Theorem 2.3.2. Since π1 (M ) = 0, M is orientable and hence Hn (M ; Z) = 0. Let q be the smallest positive integer such that Hq (M, Z) = 0. It follows from the Hurewicz isomorphism theorem that πi (M ) = 0,

for 0 < i < q, and

πq (M ) ∼ = Hq (M, Z) = 0.

It follows from Theorem 1.5.1 that M = L21 (S 1 , M ) is homotopy equivalent to C 0 (S 1 , M ). Hence πk (M) ∼ = πk (M ) ⊕ πk (Mp ) ∼ = πk (M ) ⊕ πk+1 (M ). In particular, πq−1 (M) ∼ = πq (M ) = 0. Since M is simply connected, q ≥ 2 and πq−1 (M) ∼ = πq (M ) is abelian. q−1 Moreover, we can identify πq−1 (M) with [S , M], the space of free homotopy classes of maps from S q−1 to M. Choose a nonzero element α ∈ [S q−1 , M]. Let F = {g(S q−1 ) such that g : S q−1 → M is a continuous map in [α]}. Clearly, F is an ambient isotopy invariant family of sets. Therefore, Minimax(J, F ) is a critical value for J. To show we have a nonconstant smooth closed geodesic in M , we need only verify that Minimax(J, F ) ≥ ε for some ε > 0. Recall that M is isometrically imbedded in an ambient Euclidean space RN . If p ∈ M , let Np M denote the normal space to M in RN and let νM be the disjoint union of all the normal spaces Np M , for p ∈ M , the total space of a smooth vector bundle over M , called the normal bundle. Let N M (δ) denote the union of all the balls of radius δ > 0 in Np M , for p ∈ M , and let M (δ) denote the open δ-neighborhood of M in RN . If δ > 0 is sufficiently small, one can prove that the exponential map exp : N M (δ) −→ M (δ), is a diffeomorphism; this is the content of the tubular neighborhood theorem from differential topology ([Hir97], Chapter 4, §5). Thus if δ is sufficiently

2.4. Second variation

85

small, M is a strong deformation retract of M (δ) and hence M is homotopy equivalent to M (δ). √ For ε > 0 sufficiently small, J −1 ([0, ε]) consists of curves of length ≤ 2ε by the Cauchy-Schwarz inequality, and hence can be contracted to the point γ(0) in RN without leaving M (δ). Thus if g(S q−1 ) ⊂ J −1 ([0, ε]), then g is homotopic to a smooth map g˜ : S q−1 → M0 ,

where

M0 = {γ ∈ M : γ is constant}.

But πq−1 (M0 ) = πq−1 (M ) = 0, and hence g is homotopic to a constant, contradicting the definition of F . Hence Minimax(J, F ) ≥ ε, and M must possess a nonconstant smooth closed geodesic. QED

2.4. Second variation and nondegenerate critical points A defect in the minimax principle is that quite different topological constraints can in fact lead to the same minimax critical points. We need a more refined theory that can analyze the contributions to the topology made by each individual critical point. What Morse discovered was that the contributions of “nondegenerate” critical points are the easiest to analyze, and nondegeneracy is determined by the Hessian at a critical point. Let M be a Banach manifold and let f : M → R be a smooth map. If p ∈ M is a critical point for f , then the Hessian of f at p is the symmetric bilinear map d2 f (p) : Tp M × Tp M → R defined in terms of a chart (Uα , φα ) by d2 f (p)([α, p, v], [α, p, w]) = D 2 (f ◦ φ−1 α )(φα (p))(v, w). It is straightforward to show that this is independent of choice of chart. Indeed, if α : (−, ) → M is a smooth curve with α(0) = p and α (0) = V , then a short calculation shows that d2 2 d f (p)(V, V ) = 2 (f ◦ α) , dt t=0

an expression which is clearly independent of choice of chart. Of course, once one knows d2 f (p)(V, V ), one can determine d2 f (p)(V, W ) for arbitrary V, W ∈ Tp M by the polarization identity: 1 d2 f (p)(V, W ) = [d2 f (p)(V + W, V + W ) − d2 f (p)(V − W, V − W )]. 4 Definition. The Morse index of a critical point p for f is the maximal dimension of a linear subspace of Tp M on which d2 f (p) restricts to a negative definite symmetric bilinear form. The nullity of the critical point p is dim{V ∈ Tp M such that d2 f (p)(V, W ) = 0 for all W ∈ Tp M}.

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We say that the critical point is stable if its Morse index is zero. The Morse index measures the extent to which a critical point fails to be stable. If M is a Hilbert manifold with a Riemannian metric ·, · and f : M → R is a C 2 map with a critical point p, it follows from the Riesz representation theorem that there is a continuous map A : Tp M → Tp M such that (2.6)

d2 f (p)(V, W ) = A(V ), W ,

for

V, W ∈ Tp M.

Definition. Suppose that M is a Hilbert manifold and that f : M → R is a C 2 map. A critical point p for f is Morse nondegenerate if for some choice (and hence every choice) of Riemannian metric ·, · the continuous map A : Tp M → Tp M satisfying (2.6) is an isomorphism; otherwise, it is Morse degenerate. Definition. We say that f : M → R is a Morse function if all of its critical points are Morse nondegenerate. Note that a critical point on a Hilbert manifold is Morse nondegenerate if and only if it has zero nullity. It is harder to give a satisfactory definition of nondegenerate critical point for manifolds modeled on Banach spaces, sufficient for constructions such as adding handles that we describe later. For now, we consider two main examples of Hilbert manifolds from the theory of geodesics: Example A. In the case where M = Ω(M ; p, q), with  1 1  γ (t), γ  (t)dt, J : M −→ R defined by J(γ) = 2 0 there is a standard argument from elementary differential geometry courses which calculates the Hessian at a critical point in terms of the second variation formula or index formula: Proposition 2.4.1. If (M, ·, ·) is a compact Riemannian manifold and J : Ω(M ; p, q) → R is the usual action, then  1 2 [∇γ  (t) V, ∇γ  (t) W  − R(V, γ  (t))γ  (t), W ]dt, (2.7) d J(γ)(V, W ) = 0

for V, W ∈ Tγ Ω(M ; p, q). Let us review how to establish (2.7). We consider a smooth variation of γ given by a smooth family of maps u → γu in Ω(M ; p, q) and let α(t, u) = γu (t),

V (t) =

∂α (t) ∈ Tγ(t) M. ∂u

2.4. Second variation

87

Differentiating twice, we obtain . /  1 d2 ∂ ∂α D ∂α 2 d (J)(γ)(V, V ) = , dt (J(γ )) = u du2 ∂t ∂u ∂t 0 ∂u u=0 u=0 / . /  . 2 ∂α D ∂α D ∂α D ∂α , + , 2 dt = , ∂u ∂t ∂u ∂t ∂t ∂u ∂t S1 u=0 where D denotes the covariant derivative of the Levi-Civita connection on the ambient Riemannian manifold M . Thus / . /  1 . ∂γ D DV DV DV , + , dt. d2 (J)(γ)(V, V ) = ∂t ∂t ∂t ∂u ∂t 0 Using the definition of the Riemann-Christoffel curvature tensor R, we obtain / / .  1 . 0  1 D DV DV DV 2   , + γ (t), R(V, γ (t))V + γ (t), dt. d Jγ (V, V ) = ∂t ∂t ∂t ∂u 0 An integration by parts and use of the fact that γ satisfies the EulerLagrange equation eliminates the last term, thereby yielding (2.7). We can also write (2.8)



1

2

d J(γ)(V, W ) =

Lγ (V ), W dt,

for V, W ∈ Tγ Ω(M ; p, q),

0

where Lγ is the second order differential operator defined by   D DV ◦ + R(V, γ  (t))γ  (t) . Lγ (V ) = − ∂t ∂t When γ is a critical point, we say that V is a Jacobi field along γ if it solves the equation Lγ (V ) = 0, and the nullity of γ in this case is the dimension of the space of Jacobi fields along γ. If the nullity is zero, then γ is a Morse nondegenerate critical point. Theorem 2.4.2. Suppose that (M, ·, ·) is a complete Riemannian manifold and p ∈ M . For almost all choices of q ∈ M , the function J : Ω(M ; p, q) −→ [0, ∞) is a Morse function. Sketch of proof. The proof follows from the fact that there are nonzero Jacobi fields along γ ∈ Ω(M ; p, q) vanishing at the endpoints if and only if q is a critical value for the exponential map expp . Moreover, it follows from Sard’s Theorem (which is proven in [Hir97], Chapter 3) that the set of regular values for expp form a set of full measure in M . Thus for almost all choices of q, all geodesics in Ω(M ; p, q) will be Morse nondegenerate. QED

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The techniques developed in the remainder of this chapter will show that if J : Ω(M ; p, q) −→ [0, ∞) is a Morse function, we have Morse inequalities such as (number of critical points of Morse index λ) ≥ dim Hλ (Ω(M ; p, q); F ), when F is any field. Morse theory for J : Ω(M ; p, q) −→ [0, ∞), together with numerous applications, is worked out via finite-dimensional approximations in Milnor’s classic on Morse theory [Mil63]. Milnor gives an explicit formula (§15) for the Morse index of a critical point γ, which shows that this index is always finite. Example B. We next consider the action integral on the space of closed loops M = L21 (S 1 , M ), with critical points being smooth closed geodesics, the second variation formula in this case being: Proposition 2.4.3. If (M, ·, ·) is a compact Riemannian manifold and J : L21 (S 1 , M ) → R is the usual action, then  2 [∇γ  (t) V, ∇γ  (t) W  − R(V, γ  (t))γ  (t), W ]dt, (2.9) d J(γ)(V, W ) = S1

for V, W ∈ Tγ L21 (S 1 , M ). The proof is almost identical to the proof for Proposition 2.4.1. We can write the formula for the second variation as  2 d J(γ)(V, W ) = Lγ (V ), W dt, Σ

where Lγ is the second-order Jacobi operator defined once again by (2.10)

Lγ (V ) = −

D DV ◦ − R(V, γ  (t))γ  (t). ∂t ∂t

Once again, when γ is a critical point, an element V ∈ Tγ M such that Lγ (V ) = 0 is called a Jacobi field along γ. However, in the closed curve case, the vector field V (t) = γ  (t) is always a nonzero Jacobi field along γ, so closed geodesics can never be Morse nondegenerate in the usual sense. To properly study closed geodesics we need to modify our notion of nondegeneracy. Note that since γ ∈ C ∞ (S 1 , M ), the Jacobi operator extends to a continuous linear operator on completions Lγ : L2k (γ ∗ T M ) → L2k−2 (γ ∗ T M ), for all k ≥ 1. Here L2−1 (γ ∗ T M ) can be regarded as the dual space to L21 (γ ∗ T M ) with respect to the L2 inner product. Symmetry of d2 J implies

2.4. Second variation

89

that the Jacobi operator Lγ satisfies   Lγ (V ), W dt = V, Lγ (W )dt, S1

S1

for all V, W ∈ L2k+2 (γ ∗ T M ),

and hence we say that it is formally self-adjoint. There is a well-developed classical theory for the eigenvalue problem for operators like the Jacobi operator: (1) for each λ ∈ C, the eigenspace Wλ = Ker[Lγ − λι : L2k (γ ∗ T M ) → L2k−2 (γ ∗ T M )], ι being the inclusion, is finite-dimensional, and all the elements of Wλ are C ∞ , (2) if Wλ is empty, then Lγ − λι possesses a continuous inverse, (3) if Wλ is nonempty, in which case we say that λ is an eigenvalue, then λ ∈ R. The eigenvalues can be arranged in a sequence λ1 < λ2 < · · · < λi < · · ·

with λi → ∞

and only finitely many of the eigenvalues are negative. In particular, the Morse index of any critical point γ is always finite. These facts can be proven directly for the ordinary differential operator Lγ , but are also consequences of a basic theorem for second-order elliptic partial differential operators proven in courses on linear PDEs, which is presented in various cases in Chapter 5 of [Tay96], Chapter 3 of [LM89] and in [Eva98]. As Riemannian metric on L21 (S 1 , M ), we can take  [∇γ  (t) V, ∇γ  (t) W  + V, W ]dt, V, W γ = S1

and it also gives rise to a formally self-adjoint operator  D DV + id. Pγ (V ), W dt, where Pγ (V ) = − ◦ V, W γ = ∂t ∂t S1 The map Pγ has zero kernel, so we can define an inverse  2 ∗ 2 ∗ V, W dt, Gγ : L−1 (γ T M ) → L1 (γ T M ) by Gγ (V ), W  = S1

which we call the Green’s operator for Pγ . Then  2 Lγ (V ), W dt = Gγ ◦ Lγ (V ), W  = A(V ), W , d J(γ)(V, W ) = S1

where A = Gγ ◦Lγ . The Green’s operator can be extended to the L2 Sobolev spaces and increases regularity by two, Gγ : L2k−2 (γ ∗ T M ) → L2k (γ ∗ T M ). The operators Lγ and Gγ commute with inclusions L2k ⊆ L2l , when k > l.

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2. Morse Theory of Geodesics

The Euler-Lagrange operator and the gradient: More generally, we can take M = L2k (S 1 , M ) with underlying Riemannian manifold (M, ·, ·), and consider a function J : M → R of the form  J(γ) = L(γ)dt, where L : L2k (S 1 , M ) → L1 (S 1 , M ) S1

is a sufficiently well-behaved function, called the Lagrangian of the variational problem. We assume that we can differentiate to obtain  F (γ), V dA, (2.11) (dJ)(γ)(V ) = S1

for some function F . The equation F (γ) = 0 is called the Euler-Lagrange equation for the variational problem, and F is called the Euler-Lagrange operator . For example, in the case of a simple mechanical system (M, ·, ·, φ), we take 1 L(γ) = γ  (t), γ  (t) − φ(γ(t)), 2 and the corresponding Euler-Lagrange operator is just F (γ) = −∇γ  γ  − (grad φ)(γ  (t)), a nonlinear second-order differential operator. To have an appropriate range for the Euler-Lagrange operator F and k ≥ 1, it is convenient to set (2.12)

˜ 2 (S 1 , T M ) = {(γ, V ) ∈ L2 (S 1 , M ) × L2 (γ ∗ T M )}, L k−2 k k−2

the total space of a smooth vector bundle over L2k (S 1 , M ). Indeed, for k > 2, ˜ 2 (S 1 , T M ) can be thought of as the total space of the pullback of the L k−2 bundle L2k−2 (S 1 , T M ) over L2k−2 (S 1 , M ) via the inclusion ι to L2k (S 1 , M ), but the construction (2.12) also makes sense when k = 2 or k = 1. The Euler-Lagrange map F can then be regarded as a map ˜ 2 (S 1 , T M ). F : L2k (S 1 , M ) → L k−2 If γ is a critical point for J, we can differentiate to obtain a formula for the Hessian,  (2.13) d2 J(γ)(V, W ) = (πV ◦ DF )(γ)(V ), W dA, Σ

where DF denotes the derivative with respect to γ ∈ L2k (Σ, M ) and πV is the vertical projection into the fiber L2k−2 (γ ∗ T M ), and Lγ = πV ◦ (DF )(γ) is just the Jacobi operator at γ. Hence we can regard the Jacobi operator L as the linearization of the Euler-Lagrange operator F at a solution γ to the Euler-Lagrange equation.

2.5. The Sard-Smale Theorem

91

We can define the gradient vector field X = grad(J) on L21 (S 1 , M ) by the equation (2.14)

X (γ), V  = dJ(γ)(V ),

for V ∈ Tγ L21 (S 1 , M ).

We claim that the vector field X = grad(J) is tangent to each of the subpaces L2k (S 1 , M ) ⊆ L21 (S 1 , M ). Indeed, as mentioned before,  D D + id. Pγ (V ), W dt, where Pγ = − ◦ V, W γ = dt dt S1 As γ varies, the differential operators Pγ fit together to form a vector bundle map ˜ 2 (S 1 , T M ), P : L2k (S 1 , T M ) → L k−2 and this vector bundle map P has a “Green’s operator” inverse ˜ 2 (S 1 , T M ) → L2 (S 1 , T M ), G:L k−2

k

a smoothing operator which increases the number of derivatives by two. If ˜ 2 (S 1 , M ) F : L2 (S 1 , M ) → L k

k−2

is the Euler-Lagrange map for Jψ , then (2.14) implies that X = G ◦ F : L2k (S 1 , M ) → L2k (S 1 , T M ). Moreover, if πV ◦ DX (γ) is the linearization of X at a critical point γ, it follows from (2.11) that πV ◦ DX (γ) = G ◦ πV ◦ DF (γ) = G ◦ L = A, where A is a restriction of the Fredholm operator satisfying (2.6).

2.5. The Sard-Smale Theorem Sard’s theorem, which is so useful in understanding transversality theory for finite-dimensional manifolds, possesses an extension to infinite-dimensional manifolds, as developed by Smale. The standard approach to constructing Morse functions on infinite-dimensional manifolds, and understanding transversality of infinite-dimensional manifolds, is based upon the SardSmale Theorem. We say that a point q ∈ M2 is a regular value for the smooth map f : M1 → M2 if p ∈ f −1 (q) ⇒ dfp is onto; otherwise it is called a critical value. Any finite-dimensional Riemannian manifold has a measure which defines volume integrals on open subsets of M ; indeed, if D is an open subset of a coordinate chart (U, (x1 , . . . xn )) on M , we can set  √ gdx1 · · · dxn . Volume of D = D

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2. Morse Theory of Geodesics

Here g is the determinant of the component matrix of the metric tensor, and it is standard to check that the integral for volume is independent of the coordinate chart chosen. Using this notion of integral, one can then define measurable subsets of M and speak of sets of measure zero. The collection of measure zero sets does not depend upon the Riemannian metric. Brown-Sard Theorem 2.5.1. Suppose that f : M1 → M2 is a C k map between finite-dimensional manifolds, where k>0

and

k > dim(M1 ) − dim(M2 ).

Then the subset of M2 consisting of the critical values of f has measure zero. A proof of this theorem can be found in Hirsch’s book [Hir97], Chapter 3. The strange differentiability condition cannot be weakened to k ≥ dim(M1 ) − dim(M2 ). However, this finite-dimensional theorem actually does have a slight improvement that was found by Bates [Bat93]. Let C k,1 denote the class of functions which are C k and, moreover, have k-th order derivatives which are Lipshitz. Bates showed that if f : M1 → M2 is C k,1 , where k > 0 and k ≥ dim(M1 ) − dim(M2 ), then the set of critical values of f has measure zero. We would like a version of Sard’s Theorem for maps f : M1 → M2 between Banach manifolds M1 and M2 modeled on Banach spaces E1 and E2 , respectively. To get a useful theorem, we restrict to the case where the linearization of the map is a Fredholm operator, in accordance with the following definition: Definition. A linear operator L : E → F between Banach spaces is said to be Fredholm if (1) L has finite-dimensional kernel, (2) L has closed range, and (3) L has finite-dimensional cokernel, where the cokernel of L is the quotient space F/(Range(L)). The Fredholm index of a Fredholm operator L is defined by the formula Fredholm index of L = dim(Kernel of L) − dim(Cokernel of L). Examples include linear operators between finite-dimensional Banach spaces; each such operator L : Rn → Rp is Fredholm, and its Fredholm index is just the difference in dimensions n − p.

2.5. The Sard-Smale Theorem

93

The operator A which appeared in the definition of nondegenerate critical point on a Hilbert manifold in the previous section is a Fredholm map of Fredholm index zero. Yet other examples include the Jacobi operators and Green operators Lγ : L2k (γ ∗ T M ) → L2k−2 (γ ∗ T M ),

Gγ : L2k−2 (γ ∗ T M ) → L2k (γ ∗ T M )

encountered in the previous section. In books on functional analysis, it is proven that the Fredholm index is a continuous function from the space of Fredholm operators to the integers, thus constant on any component in the space of Fredholm maps. Definition. A C k map f : M1 → M2 between Banach manifolds M1 and M2 , where k ≥ 1, is said to be a Fredholm map if its linearization (f∗ )p : Tp M1 −→ Tf (p) M2 is a Fredholm operator, for each p ∈ M1 . The Fredholm index of a Fredholm map f : M1 → M2 is the Fredholm index of (f∗ )p for any p ∈ M1 , this being constant if M1 is connected. Of course, in the special case where M1 and M2 are finite-dimensional, any C 1 map f : M1 → M2 is a Fredholm map, and it has Fredholm index dim(M1 ) − dim(M2 ). The Fredholm index can be used to replace one of the hypotheses in the Brown-Sard Theorem. We also need to replace the notion of “measure zero”, since we have no fully adequate notion of measure on infinitedimensional manifolds generalizing the standard measure on submanifolds of finite-dimensional Euclidean space. We say that a subset of M2 is residual or generic if it is a countable intersection of open dense subsets of M2 . According to the Baire Category Theorem, if M2 is a complete metric space (true for most examples we consider), a residual set in M2 is dense. With these preparations out of the way, we can now state Smale’s version of Sard’s Theorem [Sma66]: Sard-Smale Theorem 2.5.2. Suppose that f : M1 → M2 is a C k Fredholm map between Banach manifolds with complete Finsler metrics and countable base, where k>0

and

k > the Fredholm index of f .

Then the set of regular values of f is residual, hence dense. The main idea of Smale’s proof is to reduce to the finite-dimensional case, using a “local representative theorem” for f near p, which is a direct corollary of the inverse function theorem (as described in §1.2). According to this

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theorem, there exist direct sum decompositions of the model spaces for M1 and M2 : ˜ ⊕ F1 , ˜ ⊕ F2 , E1 = E E2 = E where F1 and F2 are both finite-dimensional, and local charts (U, φ) on M1 and (V, ψ) on M2 with φ(p) = 0 and ψ(f (p)) = 0, respectively, such that ψ ◦ f ◦ φ−1 : φ(U ∩ f −1 (V )) −→ E2 is of the special form ψ ◦ f ◦ φ−1 (u, v) = (u, η(u, v)),

for

˜ u ∈ E,

v ∈ F1 ,

where Dη(0, 0) = 0. We denote ψ ◦ f ◦ φ−1 simply by f . Many properties of f itself can be derived from the finite dimensional map v → η(u, v). Using this theorem, we first show that a Fredholm map is locally proper and closed: Lemma 2.5.3. If f : M1 → M2 is a Fredholm map and p ∈ M1 , then there is an open neighborhood U of p such that the restriction of f to the closure U of U is proper and closed. Proof. Suppose that D1 and D2 are closed balls of radius ε > 0 about 0 ˜ and F1 , respectively. To show that f |(D1 × D2 ) is proper, we suppose in E that pi = (ui , vi ) ∈ D1 × D2 and f (pi ) = qi → q ∈ V . We need to show that {pi } has a convergent subsequence. Since D2 is compact, we can arrange that vi → v ∈ D2 after possibly passing to a subsequence. On the other hand, f (ui , vi ) = (ui , η(ui , vi )) → q

⇒

ui → some u ∈ D1 .

Hence pi → (u, v) ∈ D1 × D2 . It follows that f |(D1 × D2 ) is proper. To prove that f |(D1 × D2 ) is closed, we suppose that C is a closed subset of D1 × D2 and that pi = (ui , vi ) is a sequence in C such that f (pi ) converges to q. Then f (pi ) lies in a compact set, so by properness (ui , vi ) lies in a compact set and has a subsequence (still denoted by (ui , vi )) which converges to a point (u∞ , v∞ ) in C, and f (u∞ , v∞ ) = q. It follows that f (C) is closed. QED Returning to the proof of the theorem, we note that since M1 has countable base, it can be covered by a countable collection of open sets Ui such that f |U i has a local representative as described above, with f |U i proper and closed by the lemma. Then for each such Ui the set of critical values for f |U i is closed, and hence the regular values for f |U i form an open set. We need to show that regular values are also dense, but in terms of the local representative, it follows from the finite-dimensional version of Sard’s

2.6. Existence of Morse functions

95

Theorem that for each fixed choice of u0 , there is a regular value v0 for v → ηi (u0 , v), and (u0 , v0 ) is then a regular value. Thus for each Ui , the set of regular values of f |U i is open and dense. Hence the regular values for f itself form an intersection of open dense sets, hence a residual set, proving Theorem 2.5.2. Note that by Corollary 1.2.8, if f : M1 → M2 is a C k Fredholm map between separable Banach manifolds of Fredholm index m, where m < k, then whenever q is a regular value of f , f −1 (q) is a submanifold of of M1 of dimension m.

2.6. Existence of Morse functions We now consider an important application of the Sard-Smale Theorem, the perturbation of a given function to a Morse function. In the case of the action integral, the circle group S 1 = {z ∈ C : |z| = 1} acts on the free loop space L21 (S 1 , M ) via α : L21 (S 1 , M ) × S 1 → L21 (S 1 , M ),

α(γ, e2πis )(t) = γ(t + s),

and this action preserves the function J we have been considering. This ensures that all critical points for J must be Morse degenerate, and indeed all nonconstant multiples of γ  are Jacobi fields along γ. However, there is a simple perturbation of the action integral whose critical points are all Morse nondegenerate, and this perturbation is therefore a Morse function. The perturbation is obtained by making a suitable choice of ψ ∈ L21 (S 1 , RN ), and setting   1 Jψ : L21 (S 1 , M ) → R by Jψ (γ) = γ  (t)2 dt + γ(t) · ψ(t)dt, 2 S1 S1 where the dot denotes the usual dot product in the ambient Euclidean space. In this section, we will show that for most choices of ψ, Jψ is indeed a Morse function. Recall that a subset of a complete metric space is residual if it is the countable intersection of open dense sets. Theorem 2.6.1. For a residual set of ψ ∈ L21 (S 1 , RN ), the function Jψ : L23 (S 1 , M ) → R is a Morse function; that is, all of its critical points are nondegenerate. Thus by the Baire category theorem, Jψ is a Morse function when ψ is chosen in a dense subset of L21 (S 1 , RN ). Note that since critical points of Jψ are automatically elements of L23 (S 1 , M ) when ψ ∈ L21 (S 1 , RN ) (by

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Theorem 2.2.6), it suffices to show that Jψ : L23 (S 1 , M ) → R is a Morse function. The proof illustrates a standard transversality technique in a relatively simple context: Recall that γ is a critical point for Jψ if and only if dJψ (γ)(X) = 0,

for all X ∈ Tγ (L23 (S 1 , M )).

We can write (2.15)

 F (γ, ψ), XdA,

dJψ (γ)(X) = Σ

where F is the Euler-Lagrange operator for Jψ described in §2.4, so that γ is a critical point for Jψ if and only if F (γ, ψ) = 0. A direct calculation (the first variation formula) shows that this Euler-Lagrange map F : L23 (S 1 , M ) × L21 (S 1 , RN ) → L21 (S 1 , T M )

is given by F (γ, ψ) = −∇γ  γ  + ψ ,

where ψ denotes the orthogonal projection of ψ into the tangent space of M . We can regard F as a vector field on L23 (S 1 , M ), depending on a parameter in L21 (S 1 , RN ), which loses two derivatives, and hence takes its values in L21 (S 1 , T M ), F (γ, ψ) = −∇γ  γ  + ψ ∈ {X ∈ L21 (S 1 , T M ) : ωπ ◦ X = ι(γ)}, where ι is the inclusion from L23 to L21 . We adopt the notation that DF denotes the differential of F , (DF )γ,ψ : Tγ,ψ [L23 (S 1 , M ) × L21 (S 1 , RN )] → TF (f,ψ) L21 (S 1 , T M ), while D1 F and D2 F denote restrictions to the two summands in the domain. Note that at a zero (γ, ψ) of F the tangent space to L21 (S 1 , T M ) can be divided into a direct sum T0γ (L21 (S 1 , T M )) = H ⊕ V, where H is horizontal (tangent to the zero section) and V is vertical (tangent to the fiber), and let πV denote the projection onto V along H, so πV ◦DFγ,ψ takes values in Tf L21 (S 1 , M ). Lemma 2.6.2. F is transverse to the zero-section of the vector bundle ωπ : L21 (S 1 , T M ) −→ L21 (S 1 , M ). Proof. Since the dependence on ψ is linear, when we take the partial derivative with respect to the second variable, we obtain πV ◦ (D2 F )(γ, ψ)(η) = η ,

2.6. Existence of Morse functions

97

Figure 1. The overall strategy for proving existence of a Morse function is to construct a submanifold S of the configuration space L23 (S 1 , M ) × L21 (S 1 , RN ), and show that the projection to the space L21 (S 1 , RN ) is Fredholm.

where πV denotes projection into the vertical tangent space at (γ, 0). The formula shows that πV ◦(D2 F )(ψ, γ) maps onto the fiber of ωπ over γ. Hence if γ is a critical point for Jψ , (Range of πV ◦ (D2 F )(γ, ψ)) + (Tangent space to zero-section) = Tangent space at 0γ to L21 (S 1 , T M ), which means that F is transverse to the zero-section. QED Just as in the finite-dimensional case, it follows from Corollary 1.2.8 that the inverse image under a split submersion of a submanifold is itself a submanifold, so F −1 (zero-section) is a submanifold S of L23 (S 1 , M ) × L21 (S 1 , RN ), as illustrated in Figure 1. We can also describe this submanifold as the solution set S = {(γ, ψ) ∈ L23 (S 1 , M ) × L21 (S 1 , RN ) : −∇γ  γ  + ψ = 0}. We would like to determine the tangent space to S, and for this we need to differentiate (2.15) once again to obtain   D1 F (γ, ψ)(X), Y dt = Lγ,ψ (X), Y dt, (2.16) d2 Jψ (γ)(X, Y ) = S1

S1

98

2. Morse Theory of Geodesics

where Lγ,ψ is the Jacobi operator. Let α : T M × T M → N M denote the second fundamental form of M in RN and define for n ∈ Np M , An : T M → T M

by

An (x), y = α(x, y) · n.

Then a calculation (similar to that used to prove Proposition 2.4.1) shows that the Jacobi operator is given by the formula (2.17)

Lγ,ψ (X) = −∇γ  ∇γ  (X) − R(X, γ  )γ  − Aψ(t)⊥ (X).

One readily verifies that for fixed choice of ψ, Lγ,ψ is a Fredholm operator of Fredholm index zero from L23 to L21 as described in §2.4. If (γ, ψ) is a critical point for Jψ , the tangent space to S at (γ, ψ) is obtained by setting πV ◦ DF equal to zero; we obtain T(γ,ψ) S = {(X, η) ∈ Tγ L23 (S 1 , M ) × L21 (S 1 , RN ) : πV ◦ D1 F (γ, ψ)X + πV ◦ D2 F (γ, ψ)η = 0} or, equivalently, (2.18) T(γ,ψ) S = {(X, η) ∈ Tγ L23 (S 1 , M )×L21 (S 1 , RN ) : Lγ,ψ (X)+η = 0}. Lemma 2.6.3. The projection on the second factor π : S → L21 (S 1 , RN ) is a Fredholm map of Fredholm index zero. Proof. We calculate the kernel and image of the map dπ(γ,ψ) : Tγ,ψ S → L21 (S 1 , RN ), obtaining the results: Kernel of dπ(γ,ψ) = {(X, η) ∈ T(γ,ψ) S : η = 0, Lγ (X) = 0}, Range of dπ(γ,ψ) = {Lγ,ψ (X) + χ⊥ : X ∈ Tγ L23 (S 1 , M ), χ ∈ L21 (S 1 , RN )}, where χ⊥ denotes the orthogonal projection of χ into the normal space to M . Note that the kernel of dπ(γ,ψ) is isomorphic to the kernel of Lγ,ψ , while the range of dπ(γ,ψ) is a subspace of L21 (S 1 , RN ) isomorphic to the image within Tγ L21 (S 1 , M ) of Lγ,ψ ; the codimension of this image is just the dimension of the cokernel of Lγ,ψ . Thus dπ(γ,ψ) is a Fredholm operator with the same Fredholm index as Lγ,ψ , namely zero, finishing the proof of Lemma 2.6.3. QED According to the Sard-Smale Theorem 2.5.2, there is a countable intersection of open dense subsets of ψ ∈ L21 (S 1 , RN ) consisting of regular values of π. But if ψ is a regular value for π, then Kernel(Lγ,ψ ) = 0 at each critical point, so all critical points are Morse nondegenerate, and Jψ is a Morse function, establishing Theorem 2.6.1.

2.6. Existence of Morse functions

99

Definition. Suppose that M is a Banach manifold and f : M → R is a C 2 function. A compact finite-dimensional submanifold N of M is said to be a nondegenerate critical submanifold for f (1) if every p ∈ N is a critical point for f ; (2) if p ∈ N , then Tp N is the set of V ∈ Tp M such that d2 f (p)(V, W ) = 0, for all W ∈ Tp M. The Morse index of a connected nondegenerate critical submanifold N is the index of any critical point p ∈ N . Proposition 2.6.4. If M is a compact Riemannian manifold and M0 = {γ ∈ L21 (S 1 , M ) : γ is constant}, then M0 is a nondegenerate critical submanifold of L21 (S 1 , M ) of Morse index zero. Moreover, for some ε > 0, the open neighborhood Uε = {γ ∈ L21 (S 1 , M ) : |γ − M0 |C 0 < ε} has M0 as a strong deformation retract and contains no critical points for the action J other than the elements of M0 . Proof. First note that if γp denotes the constant loop at the point p ∈ M , then Tγp L21 (S 1 , M ) ∼ = L21 (S 1 , Tp M ) and the tangent subspace Tγp M0 consists of the constant maps into Tp M . On the other hand, the normal space to M0 at γp can be identified with    V (t)dt = 0 . Nγp M0 = V ∈ L21 (S 1 , Tp M ) : S1

According to the second variation formula (2.9),  d2 J(γp )(V, W ) = V  (t), W  (t)dt, for V, W ∈ L21 (S 1 , Tp M ), S1

from which we conclude that d2 J(γp )(V, W ) = 0 if V is constant, that is, if V lies in Tγp M0 . On the other hand, it is readily verified that d2 J(γp ) is positive definite on Nγp M0 , so M0 is indeed a nondegenerate critical submanifold of Morse index zero. Let N M0 denote the total space of the normal bundle of M0 and for ε > 0, let Vε = {V ∈ Nγp M0 for some p ∈ M such that |V |C 0 < ε}, and define φ : Vε → U

by

φ(V ) = expp (V ).

100

2. Morse Theory of Geodesics

For ε > 0 sufficiently small, we can define H : Uε × [0, 1] → Uε

by

H(expp (V ), t) = expp (tV ).

Then H is smooth, H(γ, 1) = γ for γ ∈ Uε , H(γ, t) = γ for γ ∈ M0 and all t ∈ [0, 1], so γ → H(γ, 0) is the desired deformation retraction from Uε to M0 . Finally, using a Taylor expansion about points in M0 , one verifies that Uε − M0 contains no critical points for J. QED We can now alter the definition of Jψ slightly. Suppose that η : R → [0, 1] is a smooth function such that  0, for t ≤ ε2 /4, η(t) = 1, for t ≥ ε2 /2. Note that since L(γ)2 ≤ 2J(γ), η(J(γ)) = 1 outside Uε . By condition C, there is a δ > 0 such that dJ(γ) > δ on Vε = {γ ∈ L21 (S 1 , M ) : ε2 /4 < J(γ) < ε2 /2}; otherwise there would be a critical point in the closure of Vε . Given an element ψ ∈ L21 (S 1 , RN ), we consider the function Jψ defined by  (2.19) Jψ (γ) = J(γ) + η(J(γ)) (γ · ψ)dt. S1

It follows from Theorem 2.6.1 that if ψ is generic and sufficiently small, then Jψ ≥ 0 and M0 = Jψ−1 (0) is a critical submanifold of Morse index zero, while all other critical points of Jψ are Morse nondegenerate. So the restriction of Jψ to L21 (S 1 , M ) − M0 is a Morse function.

2.7. Bumpy metrics for smooth closed geodesics With additional work, one can show that if M is a compact manifold, then for generic choice of Riemannian metric on M , all nonconstant smooth closed geodesics have the property that their only Jacobi fields are those generated by the S 1 action on L21 (S 1 , M ). This is the “Bumpy Metric Theorem” of Abraham [Abr70]; a more detailed proof due to Anosov is found in [Ano83]. We will present a proof in this section based upon Bott’s theory of iterated smooth closed geodesics [Bot56]. First, we give some terminology. For k ∈ N, let Met2k (M ) denote the manifold of L2k Riemannian metrics on M , an open subset of the Hilbert space of L2k -sections of the second symmetric power of T ∗ M , for k ∈ N. By a generic Riemannian metric on a smooth manifold M we mean a Riemannian metric that belongs to a countable intersection of open dense subsets of the spaces Met2k (M ) with the L2k -topology, for some choice of k ∈ N.

2.7. Bumpy metrics for geodesics

101

A smooth closed geodesic is prime if it is not a cover of a geodesic of strictly smaller length. We first prove a simpler version of the Bumpy Metric Theorem for the case where the geodesic is prime: Theorem 2.7.1. If M is a compact smooth manifold, then for generic choice of Riemannian metric on M , all nonconstant prime smooth closed geodesics have the property that their only Jacobi fields are those generated by the S 1 action on L21 (S 1 , M ). Thus, for a generic choice of Riemannian metric on M , all nonconstant prime smooth closed geodesics lie on nondegenerate critical submanifolds of dimension one. The techniques for the proof are the same as those used in §2.6. Since critical points for the action function J are smooth, we can work in a Sobolev space of more highly differentiable functions, such as L24 (S 1 , M ), which is also a Hilbert manifold. Proof of Theorem 2.7.1. We need a preliminary step to deal with the fact that J is G-equivariant, where G = S 1 . If N is a compact codimension one submanifold of M with boundary ∂N , we let U (N ) = {nonconstant γ ∈ L22 (S 1 , M ) : γ does not intersect ∂N and has nontrivial transversal intersection with the interior of N }, an open subset of L24 (S 1 , M ). We cover L22 (S 1 , M ) with a countable collection Ui = U (Ni ) of open sets corresponding to a countable collection of codimension two submanifolds Ni . If 0 is a choice of origin in S 1 , we let Fi = {γ ∈ U (Ni ) : γ(0) ∈ Ni }, a C 1 submanifold of L22 (S 1 , M ) by the smoothness theorems in §1.6. Note that Fi meets each nonconstant G-orbit in U (Ni ) in a finite nonempty set of points. We claim that S = {(γ, g) ∈ Fi × Met23 (M ) : γ is a geodesic for the metric g} is a smooth submanifold of Fi ×Met23 (M ). To see this, we let L22 (S 1 , T M ) denote the “pullback” vector bundle over Fi whose fiber at each f ∈ L24 (S 1 , M ) is L22 (S 1 , γ ∗ T M ). We define a C 1 map F : Fi × Met23 (M ) −→ L22 (S 1 , T M )

by F (γ, g) = ∇gγ  γ  ,

where ∇g is the Levi-Civita connection on T M determined by the metric g ∈ Met23 (M ). Let Z denote the image of the zero section of L22 (S 1 , T M ) and note that S = F −1 (Z). Our goal is to show that F is transverse to Z.

102

2. Morse Theory of Geodesics

We can write the first derivative of the action as  dJ(γ)(X) = F (γ, g), Xdt, S1

and differentiating once again gives us the Hessian at a critical point,   2 πV ◦ D1 F (γ, g)(X), Y dt = Lγ,g (X), Y dt, (2.20) d J(γ)(X, Y ) = S1

S1

with D1 F denoting the derivative with respect to the first variable {γ ∈ L22 (S 1 , M ) : γ(0) ∈ Ni }, πV being projection into the vertical space at a zero of F , and (2.21)

Lγ,g : {X ∈ L22 (S 1 , γ ∗ T M ) : X(0) ∈ Tγ(0) Ni } −→ L2 (S 1 , γ ∗ T M )

being the Jacobi operator. Because it is a Jacobi field, the section T (t) = γ  (t) is not in the image of Lγ . However, we claim that (2.22)

{the image of πV ◦ DFγ,g } ⊕ (span of T (t)) = L22 (γ ∗ T M ),

Note that the Jacobi operator πV ◦ (D1 F ) covers all but a finite-dimensional subspace V ⊆ L22 (γ ∗ T M ) and we can take this subspace to consist of Jacobi fields. To establish our claim, we need to show that πV ◦ (D2 F ) covers V. Suppose that γ is a nonconstant geodesic, which will be L24 when the metric is L23 . Choose a point t ∈ S 1 which possesses a neighborhood U such that γ imbeds U into some open set W ⊂ M on which Fermi coordinates (x1 , . . . , xn ) are defined. Such coordinates satisfy the following conditions: (1) γ is described by the equations x2 = · · · = xn = 0, (2) x1 ◦ γ = t, (3) the metric g takes the form δij , for 1 ≤ i, j ≤ n.

)

gij dui duj , such that on γ(U ), gij =

In terms of these local coordinates, the equation for geodesics becomes d2 xk  k dxi dxj = 0. + Γij dt2 dt dt i,j

A perturbation in the metric ) h ∈ Tg M with compact support in W can be written in the form h = hij dxi dxj . Under this perturbation, the only piece of the acceleration that changes is the Christoffel symbol   1  kl ∂gil ∂gjl ∂gij g + − , Γkij = 2 ∂xj ∂xi ∂xl and if Γ˙ kij denotes the derivative of Γkij in the direction of the perturbation,   ˙Γk = 1 ∂hik + ∂hjk − ∂hij , ij 2 ∂xj ∂xi ∂xk

2.7. Bumpy metrics for geodesics

103

and we find that πV ◦ (D2 F )(γ,g) (h) =

(2.23)

n 

 dxi dxj ∂ ∂ = . Γ˙ kij Γ˙ k11 dt dt ∂xk ∂xk n

i,j,k=1

k=1

We can select the perturbation so that h11 = x2 φ,

hij = 0,

for other choices of indices i, j,

where φ has compact support in W and 2 ≤ r ≤ n; we then see that 1 ∂ 1 Γ˙ r11 (t, 0, . . . , 0) = − (h11 )(t, 0, . . . , 0) = − φ(t, 0, . . . , 0). 2 ∂x2 2 Thus the fiber projection of the partial derivative of F with respect to the second variable (in M) is given by the expression 1  ˙r ∂ 1 ∂ =− φ . Γ11 2 ∂xr 2 ∂x2 n

πV ◦ (D2 F )γg (h) = −

r=2

In this manner, we can show that any vector field of the form n  ∂ φr ∂xr r=2

where is an with compact support, lies in the image of πV ◦ D2 Fγ,g , and hence elements of V must be tangent to γ in U . Since elements of V are C 1 and a dense open subset of points of the image of γ can be covered by sets of the form U , we see that elements of V must be tangent to γ over all of S 1 . This enables us to obtain the desired result (2.22). φr

L23 -function

The remainder of the proof is similar to the proof of Theorem 2.6.1. Note that the tangent space to S is given by the formula Tγ,g S = {(X, h) ∈ Tγ Fi × Tg Met23 (M ) : Lγ,g (X) + πV ◦ (D2 F )(h) = 0}, which makes it easy to analyze the kernel and range of the map dπ(γ,g) : T(γ,g) S → Tg M. Indeed, Kernel of dπ(γ,g) = {(X, h) ∈ Tγ Fi × Tg Met23 (M ) : h = 0, Lγ,g (X) = 0}, while Range of dπ(γ,g) = {h ∈ Tg Met23 (M ) : there exists X ∈ Tγ L22 (S 1 , M ) such that X(0) ∈ Tγ(0) Ni and Lγ,g (X) = −πV ◦ (D2 F )(h)}. Thus the kernel of dπ(γ,g) is isomorphic to the kernel of Lγ,g . On the other hand, if h is an element of Tg M such that πV ◦ (D2 F )(h) = 0,

then

(0, h) ∈ T(γ,g) S,

and hence h lies in the range of dπ(γ,g) . It follows that any complement to the range of dπ(γ,g) must inject to a complement to the range of Lγ,g and,

104

2. Morse Theory of Geodesics

in particular, it must be finite-dimensional, so dπ(γ,g) is a Fredholm map. Moreover, the dimension of the cokernel of dπ(γ,g) is no larger than the dimension of the cokernel of Lγ,g . By the earlier transversality argument, dFγ,g maps surjectively onto a complement to the one-dimensional space generated by the Jacobi field T (t), so the dimension of the cokernel of dπ(γ,g) is actually one less than the dimension of the cokernel of Lγ,g . Thus the Fredholm index of dπ is zero, and we can use the Sard-Smale Theorem to show that a countable intersection of open dense subsets of M consist of regular values for π. If g0 is such a “generic metric”, all of the prime geodesics will lie on one-dimensional nondegenerate critical submanifolds, proving Theorem 2.7.1. QED We will next extend the argument from the previous theorem to cover the case in which geodesics are not necessarily prime, thereby obtaining: Bumpy Metric Theorem 2.7.2. If M is a compact smooth manifold, then for generic choice of Riemannian metric on M , all nonconstant smooth closed geodesics have the property that their only Jacobi fields are those generated by the S 1 action on L21 (S 1 , M ). Moreover, the number of S 1 orbits of nonconstant geodesics within L21 (S 1 , M )a = {γ ∈ L21 (S 1 , M ) : J(γ) ≤ a} is finite for any a ∈ (0, ∞). A geodesic which is not prime covers a prime minimal geodesic and our strategy is to study this underlying prime geodesic. We make some definitions following Bott [Bot56]. Suppose that γ : R → M is a smooth geodesic with γ(t + 1) = γ(t). If q ∈ Z and z ∈ S 1 ⊂ C, we let ∞ = {smooth sections X of γ ∗ T M ⊗ C : X(t + q) = zX(t)}, Vq,z

and define an inner product ∞ ∞ × Vq,z −→ C ·, ·q : Vq,z

by



q

X, Y¯ q = 0

.

/  DX D Y¯ ¯ , + X, Y  dt, ∂t ∂t

where D denotes the covariant derivative defined by the Levi-Civita connec∞ with respect to ·, · and tion on M . Let Vq,z denote the completion of Vq,z q define Iq (·, ·) : Vq,z × Vq,z −→ C

2.7. Bumpy metrics for geodesics

by



q

Iq (X, Y¯ ) =

.

0

105

/  DX D Y¯   ¯ , − R(X, γ )γ , Y  dt, ∂t ∂t

which is of course the restriction of d2 J(γ) to Vq,z . Lemma 2.7.3. The inclusion



(2.24)

V1,z ⊂ Vq,1

z q =1

is an isomorphism. Since the inclusion is clearly injective, it suffices to show that it is surjective. If X ∈ Vq,1 and z is a primitive q-th root of unity, we let 1  −j Xz (t) = z X(t + j). q q−1

j=0

Then 1  −j z X(t + (j + 1)) q q−1

Xz (t + 1) =

j=0

=

q−1 z  −(j+1) z X(t + (j + 1)) = zXz (t), q j=0

so Xz ∈ V1,z . Moreover, X=



Xz ,

z q =1

so the inclusion (2.24) is indeed an isomorphism, proving the lemma. Note that if X ∈ V1,z1 and Y ∈ V1,z2 , where z1 and z2 are q-th roots of unity, then   1 . ¯/ D Y DX   , − R(X, γ )γ , Y¯  dt Iq (X, Y¯ ) = ∂t ∂t 0 /  .  2 DX D Y¯   ¯ , − R(X, γ )γ , Y  dt z1 z¯2 + ∂t ∂t 1 /  .  q DX D Y¯ q−1 q−1   ¯ , − R(X, γ )γ , Y  dt z1 z¯2 + ···+ ∂t ∂t q−1 ⎞ ⎛ /   1 . q−1  DX D Y¯ , − R(X, γ  )γ  , Y¯  dt. z1j z¯2j ⎠ =⎝ ∂t ∂t 0 j=0

106

2. Morse Theory of Geodesics

Thus we see that

 qI1 (X, Y¯ ), if z1 = z2 , Iq (X, Y¯ ) = 0, if z1 = z2 , ) and hence the direct sum decomposition z q =1 V1,z of Vq,1 is orthogonal with respect to the index form Iq . Let N (z) denote the nullity of the index form I1 restricted to V1,z , N (z) = dimC {X ∈ V1,z : I1 (X, Y¯ ) = 0 for all Y ∈ V1,z }. The above discussion proves the following lemma due to Bott [Bot56], which plays a key role in his analysis of the relationship between the index and nullity of a prime smooth closed geodesic and the index of nullity of its multiple covers: Lemma 2.7.4. Let γ q denote the q-fold iterate of the smooth closed geodesics γ, so γ q (t) = γ(qt). Then  Nullity of γ q = N (z). z q =1

Proof of Theorem 2.7.2. For the purpose of the proof, we say that a smooth closed geodesic is Morse nondegenerate if it lies on a one-dimensional nondegenerate critical submanifold. For m ∈ N, we let L21 (S 1 , M )m = {γ ∈ L21 (S 1 , M ) : J(γ) < m}. We will prove the following by induction on m: there are only finitely many prime geodesics in L21 (S 1 , M )m and all nonprime geodesics in L21 (S 1 , M )2m are Morse nondegenerate. For the induction step in which m is replaced by 2m, we first use Theorem 2.7.1 and Condition C to show that there are only finitely many new prime geodesics in L21 (S 1 , M )2m , all Morse nondegenerate and isolated. We then perturb the metric in a neighborhood of a given nonprime geodesic in L21 (S 1 , M )4m so that the underlying prime geodesic itself is preserved, but the cover becomes nondegenerate in the perturbed metric. Thus the key part of the inductive step is constructing a perturbation for each nonprime geodesic in L21 (S 1 , M )4m . We do this by constructing a perturbation of the metric on M near the corresponding underlying prime geodesic γ. As in the earlier argument, we choose a point t ∈ S 1 and a neighborhood U containing t such that the underlying prime geodesic γ imbeds U into some open set W ⊂ M on which Fermi coordinates (x1 , . . . , xn ) are defined, coordinates such that: (1) γ is described by the equations x2 = · · · = xn = 0, (2) x1 ◦ γ = t,

2.7. Bumpy metrics for geodesics

107

(3) the metric g takes the form δij , for 1 ≤ i, j ≤ n.

)

gij dui duj , such that on γ(U ), gij =

Following Klingenberg ([Kli78], Proposition 3.3.7), we construct a pertur) bation of the metric g˙ = g˙ ij dxi dxj such that (2.25) n  xr xs αrs (x1 , x2 , . . . , xn ), g˙ ij = 0 if (i, j) = (1, 1). g˙ 11 (x1 , . . . xn ) = r,s=2

Here αrs are smooth functions which vanish outside a small tubular neighborhood of γ, and we make the perturbation sufficiently small that the finitely many nondegenerate S 1 orbits of geodesics in L21 (S 1 , M )2m , perturb to nondegenerate S 1 orbits of geodesics. A straightforward calculation shows that the resulting changes in the Christoffel symbols Γ˙ kij

1 = 2



∂ g˙ ik ∂ g˙ ik ∂ g˙ ik + − ∂uj ∂uj ∂uj



vanish unless at least two of the indices are 1, and if 2 ≤ r, s ≤ n, Γ˙ r11 = −



us αrs ,

s

Γ˙ 1r1 =



us αrs .

s

The corresponding changes in curvature components are given by the formulae ∂ ˙  ∂ ˙  Γljk − Γlik R˙ lijk = ∂ui ∂uj     ˙m − ˙ ljm Γm − Γ˙ lim Γm Γ Γ + + Γ Γljm Γ˙ m lim jk jk ik ik . m

m

m

m

But along γ(U ) all the Γ˙ kij ’s and Γ˙ kij ’s must vanish, so the resulting change in the curvature R1r1s along γ is given by R˙ 1r1s (x1 , 0, . . . , 0) = αrs (x1 , 0, . . . , 0). The Jacobi operator Lγ on normal sections can be expressed in components as follows: If   n n n   ∂ d 2 fr  ∂ fr , then Lγ (X) = − + R1r1s fs . X= 2 ∂xr ∂xr dx1 r=2 r=2 s=2

108

2. Morse Theory of Geodesics

We can consider the family of formally self-adjoint second-order differential operators T of the form   n n  d 2 fr  ∂ T (X) = − + Trs fs . 2 ∂xr dx1 r=2 s=2 For each such operator and each q-th root of unity z ∈ S 1 ⊂ C, we can define the corresponding nullity N (z) = dimC {X ∈ V1,z : T (X) = 0}. For an open dense set of such operators T , N (z) = 0 for every root of unity. The argument described above can be used to perturb the Jacobi operator Lγ for any smooth prime closed geodesic γ so that N (z) = 0 for every root of unity, while leaving the closed geodesic itself fixed. This in turn perturbs the metric near any given nonprime geodesic in L21 (S 1 , M )4m which covers γ so that it is Morse nondegenerate. In particular, this establishes the inductive step described above, allowing us to finish the proof of the Bumpy Metric Theorem 2.7.2. QED

2.8. Adding handles Suppose that M is a smooth manifold modeled on a Hilbert space with a complete Riemannian metric, and that f : M → [0, ∞) is a C 2 Morse function satisfying condition C. From the Deformation Theorem 1.11.1, we know that the topology of Ma = {p ∈ M : f (p) ≤ a} does not change as a increases unless a passes a critical value for f . Our goal now is to understand what happens to the topology of Ma when a passes a critical value c for f such that f −1 (c) contains a finite number of critical points, all Morse nondegenerate. In this section, we carry out that analysis for the case of Hilbert manifolds, and the arguments (which largely follow Smale [Sma64]) are designed so that they can be extended to treat some functions on Banach manifolds. The following theorem states that when a passes a critical value c with f −1 (c) containing a single nondegenerate critical point of Morse index λ, the effect is to add a λ-handle to Ma : Theorem 2.8.1. Suppose that M is a Hilbert manifold with a complete Riemannian metric ·, · and that f : M → [0, ∞) is a smooth function satisfying condition C. If the interval [a, b] contains a single critical value c for f , with c ∈ (a, b), and there is exactly one critical point p for f such

2.8. Adding handles

109

that f (p) = c which is Morse nondegenerate of Morse index λ, then Mb is homotopy equivalent to Ma with a λ-dimensional disk attached along its boundary sphere. Proof. We choose a coordinate chart (U, φ) about the nondegenerate critical point p with φ(p) = 0 ∈ E, where E is the model space for M, and let f¯ = f ◦ φ−1 , a smooth function on the open subset φ(U ) ⊂ E which has a single nondegenerate critical point of index λ at the origin. Expanding the smooth function f¯ in a Taylor series yields (2.26)

1 f¯(x) = c + d2 f¯(0)(x, x) + R1 (x), 2

where R1 (x) ≤ ε1 x2 ,

for x ∈ φ(U ), with ε1 being a positive constant which can be made arbitrarily small by making U small. We can identify Tp M with E and since p is Morse nondegenerate, the self-adjoint bounded linear operator A on E determined by for x, y ∈ E, d2 f¯(0)(x, y) = Ax, y, where ·, · is the Riemannian metric, is invertible. Thus its spectrum is a closed subset of the real axis, bounded away from zero. Corresponding to the restriction of the vector field X = grad(f ) to U is a vector field on φ(U ) with local representative X¯ which has the Taylor series expansion (2.27)

X¯ (x) = Ax + R2 (x),

where R2 (x) ≤ ε2 x,

for x ∈ φ(U ), ε2 being an arbitrarily small positive constant. The vector field X¯ on φ(U ) is closely approximated by its linearization XA which has local representative X¯A (x) = Ax, and one can check that this linearization is a pseudogradient (as defined in §1.11) when ε2 is sufficiently small. If the critical point p has finite index, the negative part of the spectrum is discrete and consists of finitely many eigenvalues. In this case, we let E− be the subspace of E generated by the negative eigenvalues of A and let E+ be the closed orthogonal complement to E− in E. In general, the spectral theorem for bounded self-adjoint operators allows us to divide E into an orthogonal direct sum E = E+ ⊕ E− , each summand being preserved by A, and hence we can think of A as dividing into a “block matrix”   A− 0 A= 0 A+ with A− and A+ self-adjoint invertible operators on the subspaces E− and E+ , respectively. The spectrum of A− lies on the negative real axis, while the spectrum of A+ lies on the positive real axis. Thus if we suppose that x− and x+ are the orthogonal projections of x into E− and E+ , respectively, so x = x− + x+ , the system of differential equations represented by the

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linearization −X¯A is

 dx− /dt = −A− x− , dx+ /dt = −A− x+ .

This is just a linear system with constant coefficients, which has {φt : t ∈ R} as its one-parameter group of diffeomorphisms, where (2.28)

φt (x− , x+ ) = (exp(−tA− )x− , exp(−tA+ )x+ ). The fluid flow for −X¯A described by {φt : t ∈ R} is expanding on the subspace E− , contracting on E+ and closely approximates the fluid flow for X¯ . Let W be an open subset of U with φ(W ) an open ball in E such that ¯ ⊆ U , and let D− (r) and D+ (s) denote closed balls of radius r p∈W ⊆W and s, respectively, about zero in E− ∩ φ(W ) and E+ ∩ φ(W ).

Lemma 2.8.2. There exist r1 > 0 and s1 > 0 such that (1) D− (r) × D+ (s) ⊆ V and X¯A is transverse to D− (r) × ∂D+ (s) when r ≤ r1 and s ≤ s1 , (2) f¯(∂D− (r1 ) × D+ (s1 )) ≤ c − ε, for some ε > 0, and (3) f¯−1 (c − ) is transverse to D− (r1 ) × x+ for x+ ∈ D+ (s1 ). The first condition follows immediately from the explicit form (2.28) for the explicit fluid flow for X¯A . To prove the second claim, we use the Taylor expansion (2.26) to conclude that 1 1 f¯(x) = c + A− x− , x−  + A+ x+ , x+  + R1 (x) 2 2 1 1 2 2 2 ≤c− −1 x−  + 2 A+ x+  + ε1 x . 2A−  We can choose ε1 smaller than (1/(4A− 2 ). Thus, if we choose r1 small and then s1 much smaller, we can arrange that the second claim holds. The third claim is proven in a similar fashion, and we leave it as an exercise for the reader. In accordance with the terminology of geometric topology, we call ∂D− (0) and D− (0) the descending sphere and descending disk of the critical point p which lie at the core of the λ-handle D− (r1 ) ×D+ (s1 ); ∂D− (0) is a sphere of dimension λ − 1. We also have an ascending sphere ∂D+ (0) and ascending disk D+ (0), which are infinite-dimensional. Returning now to the proof of the theorem, we let η : M → [0, 1] be a smooth function such that η(q) = 0 for q ∈ M − U , η(q) = 1 for q ∈ W , and

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111

let ¯ X = η φ−1 ∗ XA + (1 − η) grad(f ). Note that we can arrange that a and b are very close to c, with their level sets intersecting W . Moreover, we have constructed X so that we can apply Lemma 2.8.2 on the subset of M defined by η ≡ 1. We let {φt : t ∈ R} be the one-parameter group of local diffeomorphisms corresponding to the vector field −X . For q ∈ Mb − Mc− , let τ (q) denote the first time t such that φt (q) ∈ Mc− ∪ φ−1 (D− (r1 ) × ∂D+ (s1 )). Note that τ (q) is finite by the argument for the Theorem 1.10.1 and the transversality conditions of Lemma 2.8.2 show that τ (q) depends continuously on q. Let ht : Mb → Mb by ht (q) = φtτ (x) (q). Then clearly h0 is the identity map and h1 (Mb ) ⊂ Mc− ∪ φ−1 (D− (r1 ) × ∂D+ (s1 )). As in the proof of the Deformation Theorem 1.12.1, ht gives a deformation retraction from Mb to Mc− ∪ φ−1 (D− (r1 ) × ∂D+ (s1 )). We say that Mc−ε ∪ φ−1 (D− (r1 ) × D+ (s1 )) is Mc−ε with a λ-handle attached. This in turn is homotopy equivalent to Ma with a λ-handle attached by the Deformation Theorem. One can check that this is also homotopy equivalent to Ma with a λ-dimensional disk attached along its boundary sphere (an explicit construction for the homotopy equivalence being given in Assertion 4 of §3 of [Mil63]). This proves Theorem 2.8.1. QED Remark 2.8.3. In the proof, the Riemannian metric on M is used only to construct the continuous isomorphism A : Tp M → Tp M which gives the flow near p, and the gradient vector field X is only used away from the critical point p. This suggests the notion of “gradient-like” vector field, which is sufficient for adding handles, but more flexible than an ordinary gradient, a concept utilized by Milnor for finite-dimensional manifolds in [Mil65]: Definition. Suppose that f : M → R is a C 2 Morse function on a Hilbert manifold M with a complete Riemannian metric ·, ·, and let K ⊆ M be the critical locus of f . We say that a C 2 vector field X on M is gradient-like

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for f if M possesses an open cover {U, V } such that (1) the open set U contains the critical set and divides into connected components Up , one for each p ∈ K, such that Up is the domain of a coordinate system φp such that φp (p) = 0, and (2) X |Up = XAp , where XAp has local representative (2.29) X¯Ap (x) = Ap x,

for x ∈ φp (U ), where

d2 (f ◦ φ−1 p )(x, y) = Ap x, y,

the bounded self-adjoint operator Ap : Tp M → Tp M being defined by the Hessian, and (3) the restriction of X to V is a pseudogradient for f . We recall from §1.11 that a pseudogradient is a weakening of the notion of gradient useful for treating critical point theory on Banach manifolds: by definition, the vector field X : V → T M is a pseudogradient for f over U if there is an ε > 0 such that dfq (X (q)) > εdfq 2 ,

X (q) < (1/ε)dfq ,

for q ∈ V .

: p ∈ K} ∪ {ηV } is a C 2 partition of unity with respect to the open If {ηp  cover {Up : p ∈ K} ∪ {V }, and XV is a pseudogradient for f on V , then  ηp XAp + ηV XV X = p∈K

is a C 2 gradient-like vector field for f on M, so long as the open neighborhoods Up are chosen sufficiently small. Even if the Morse function f is only C 2 (so its gradient is only C 1 ), we can construct C 2 gradient-like vector fields for f on M so long as M admits C 2 partitions of unity. Note that it was actually a gradient-like vector field for f that we utilized in the proof of Theorem 2.8.1. Since pseudogradients form an open subset of the space of vector fields, gradient-like vector fields give more flexibility than gradients in constructions involving flows corresponding to Morse functions, because we have considerable freedom to perturb them away from the critical points. We can extend Theorem 2.8.1 to the case of several Morse nondegenerate critical points: Theorem 2.8.4. Suppose that M is a Hilbert manifold with a complete Riemannian metric ·, · and that f : M → [0, ∞) is a smooth function satisfying condition C. If the interval [a, b] contains a single critical value c for f with c ∈ (a, b), and there are finitely many critical point p1 , . . . , pk for f such that f (pi ) = c which are Morse nondegenerate with Morse indices

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113

λ1 , . . . λk , then Mb is homotopy equivalent to Ma with disks of dimension λ1 , . . . , λk attached along their boundary spheres. The proof is a straightforward extension of the proof of Theorem 2.8.1. Remark 2.8.5. Theorems 2.8.1 and 2.8.4 both hold for critical points of infinite Morse index, although attaching handles of infinite index turns out to be invisible from the viewpoint of homotopy theory. In the problems we have been considering from the calculus of variations, the Morse index of critical points is always finite, so the handles constructed via these theorems are finite-dimensional. The Bumpy Metric Theorem 2.7.2 provides motivation for extending Theorems 2.8.1 and 2.8.4 to the case of nondegenerate critical submanifolds. If N ⊂ M is a compact nondegenerate critical submanifold of finite Morse index λ, the normal bundle νN to N is defined via the Riemannian metric ·, · on M; its fiber at p ∈ N is (νN )p = {V ∈ Tp M : V, W  = 0 for all W ∈ Tp N }. The normal bundle has a finite-dimensional subbundle ν− N whose fiber at p is generated by eigenvectors corresponding to the negative eigenvalues of A : Tp M → Tp M, the rank of the bundle ν− N being the Morse index of N . Let D(ν− N ) = {V ∈ ν− N : V  ≤ 1},

S(ν− N ) = {V ∈ ν− N : V  = 1},

the unit disk and unit sphere bundles in the negative normal bundle ν− N . The following lemma states that when a passes a critical value c with containing a single nondegenerate critical submanifold of Morse index λ, the effect is to add a λ-disk bundle to Ma :

f −1 (c)

Theorem 2.8.6. Suppose that M is a Hilbert manifold with a complete Riemannian metric ·, · and that f : M → [0, ∞) is a smooth function satisfying condition C. If [a, b] contains a single critical value c for f with c ∈ (a, b), the set of critical of points with value c forming a nondegenerate compact critical submanifold N of finite Morse index λ, then Mb is homotopy equivalent to Ma with the disk bundle D(ν− N ) attached to Ma along S(ν− N ). The proof is a relatively straightforward modification of the proof for Theorem 2.8.1. It is sometimes convenient to allow N to have more than one component.

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2.9. Morse inequalities Suppose now that M is a Hilbert manifold with a complete Riemannian metric ·, · and that f : M → [0, ∞) is a Morse function satisfying condition C, all critical points having finite Morse index. For a ∈ R, we let Ma = {p ∈ M : f (p) ≤ a}. Condition C implies that each Ma has only finitely many nondegenerate critical points. We would like to investigate how the topology of Ma changes as a increases. There are two ways of tracking the changes in topology, one via the Morse inequalities which we discuss next following Bott’s presentation in [Bot82], the other via the Morse-Witten chain complex for f which will be treated in §2.10. For each nonnegative integer λ, we let μaλ denote the number of critical points for f within Ma of Morse index λ and for a given choice of field F (such as R or Z2 ), we let βλa be the dimension of H λ (Ma ; F ). Then the weak Morse inequalities state that μaλ ≥ βλa .

(2.30)

Instead of proving these directly, we will prove a stronger version of the Morse inequalities in terms of the Morse polynomial for f on Ma and the Poincar´e polynomial for Ma , defined, respectively, by ∞ ∞   a a λ a μλ t , P (t) = βλa tλ , M (t) = λ=0

λ=0

there being only finitely many terms in these series. Then (2.30) is an immediate consequence of: Theorem 2.9.1 (Morse inequalities). M is a Hilbert manifold with a complete Riemannian metric and that f : M → [0, ∞) is a Morse function satisfying condition C. Then there is a polynomial (2.31) Q (t) = a

∞ 

qλa tλ

with qλa ∈ N ∪ {0}

λ=0

such that Ma (t) − P a (t) = (t + 1)Qa (t). We make the simplifying assumption that there is only one critical point at each critical value, so that we can apply Theorem 2.8.1 instead of Theorem 2.8.4. This can be arranged quite easily by simply perturbing f . Our strategy is to apply an induction on a. To start the induction, we note that when we set a < 0, Ma is empty and hence βλa = μaλ = 0. We can therefore set Qa (t) = 0 in this case.

2.9. Morse inequalities

115

For the inductive step, observe first that if the interval [a, b] contains a single critical value c corresponding to a Morse nondegenerate critical point of Morse index λ, then by Theorem 2.9.1,  F, if k = λ, Hk (Mb , Ma ; F ) ∼ = Hk (D λ , S λ−1 ; F ) ∼ = 0, if k = λ. Thus it follows from the exact sequence · · · → Hk (Ma ; F ) → Hk (Mb ; F ) → Hk (Mb , Ma ; F ) → Hk−1 (Ma ; F ) → · · · that either P b (t) = P a (t) + tλ

or

P b (t) = P a (t) − tλ−1 ,

and one can check that the two cases depend on whether the descending sphere (pushed down into Ma ) bounds or not. In the former case, the descending disk can be completed to a cycle representing a new generator of λ-dimensional homology, while in the latter it yields a new relation in (λ − 1)-dimensional homology. Since Mb (t) = Ma (t) + tλ in either case, we see that (Mb (t) − P b (t)) − (Ma (t) − P a (t)) is either 0 or tλ−1 (t + 1). Thus assuming (2.31) for a, we can arrange that Mb (t) = P b (t) + (t + 1)Qb (t) also holds by setting Qb (t) = Qa (t) or Qb (t) = Qa (t) + tλ−1 . This finishes the inductive step, and the theorem follows. We say that a Morse function f : M → [0, ∞) has finite type if for each λ ∈ N ∪ {0}, there are only finitely many critical points for f of Morse index λ. If f : M → [0, ∞) has finite type, we can let μλ denote the number of critical points for f of Morse index λ and it follows from Theorem 2.9.1 that the dimension βλ of Hλ (M; F ) is also finite. If we define the Morse series for f and Poincar´e series for M by M(t) =

∞  λ=0

λ

μλ t ,

P(t) =

∞ 

βλ t λ ,

λ=0

it follows from Theorem 2.9.1 that there is an infinite series ∞  qλ tλ with qλ ≥ 0 such that M(t)−P(t) = (t+1)Q(t), (2.32) Q(t) = λ=0

this being the limiting case of the Morse inequalities as a → ∞. Example 2.9.2. If a smooth proper Morse function f : R2 → [0, ∞) has two local minima at points p1 and p2 in R2 , it must have an additional critical point (say at q ∈ R2 ) of Morse index one (see Figure 2). This consequence

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Figure 2. The graph of a proper Morse function f : R2 → [0, ∞) with three critical points: local minima at p1 and p2 and a saddle point at q.

of the Morse inequalities is known as the “Mountain Pass Lemma”, and it illustrates the “feedback” implicit in the Morse inequalities. Proposition 2.9.3 (Lacunary Principle). If for all choices of λ ∈ N∪{0}, μaλ = 0

⇒

its nearest neighbors μaλ−1 and μaλ+1 are both zero,

then Qa (t) ≡ 0, so μaλ = βλa , for every λ. a a , and hence the = 0 = βλ+1 Indeed, by the weak Morse inequalities, βλ−1 coefficients of tλ−1 and tλ+1 in (t + 1)Qa (t) must be zero. Thus coefficients of tλ−1 , tλ , tλ+1 and tλ+2 in Qa (t) must be zero. From this we conclude that Qa (t) ≡ 0.

Similarly, we can let a → ∞ and obtain the lacunary principle for f on M: If for every λ ∈ N ∪ {0}, μλ = 0 ⇒ μλ−1 = 0 = μλ+1 , then μλ = βλ , for every λ. Example 2.9.4. Suppose that M = Sn , the n-dimensional sphere with metric of constant curvature one, where n ≥ 3, and that p and q are points in M which are not antipodal. Then there is exactly one geodesic from p to q of Morse index k(n − 1), for each k ∈ N ∪ {0}, and hence M(t) = 1 + tn−1 + t2(n−1) + · · · + tk(n−1) + · · · . From the lacunary principle we see that M(t) = P(t), so P(t) = 1 + tn−1 + t2(n−1) + · · · + tk(n−1) + · · · , and we conclude that for any field F ,  n ∼ F, if m = k(n − 1) for k ∈ N ∪ {0}, Hm (Ω(S , p, q); F ) = 0, otherwise.

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117

Thus, if ·, · is any Riemannian metric on Sn , then for generic choice of p and q in Sn , there will be infinitely many geodesics from p to q, at least one being of index k(n − 1) for each k ∈ N ∪ {0}. Example 2.9.5. We would also like Morse inequalities for the case of smooth closed geodesics, in which M = L21 (S 1 , M ), where M is a compact oriented Riemannian manifold. If we give M a “generic” Riemannian metric, Theorem 2.7.2 implies that all nonconstant critical points for the action J : L21 (S 1 , M ) → R lie on one-dimensional nondegenerate critical submanifolds. We can further arrange that the metric is chosen so that only one O(2)-orbit of critical points lies at each critical level. Thus suppose that the interval [a, b] contains a unique critical value c with J −1 (c) containing a unique O(2)-orbit of geodesics which comprises a nondegenerate critical submanifold N . Then Theorem 2.8.6 gives an isomorphism on cohomology Hk (Mb , Ma ; Z) ∼ = Hk (D(ν− N ), S(ν− N ); Z). The fact that M is oriented implies that the normal bundle to any geodesic is trivial, and if we choose the metric so that the Jacobi operators at closed geodesics are generic, we easily verify that for any nondegenerate O(2)-orbit N , ν− N is orientable. It then follows from the Thom isomorphism theorem that if the Morse index of N is λ, then Hk (D(ν− N ), S(ν− N ); Z) ∼ = Hk−λ (N ; Z). Since N is the disjoint union of two circles,  Z ⊕ Z, if k = λ or λ + 1, Hk (D(ν− N ), S(ν− N ); Z) ∼ = 0, otherwise, and hence  Hk (M , M ; Z) ∼ = b

a

Z ⊕ Z, 0,

if k = λ or λ + 1, otherwise.

Thus, if μaλ denotes the number of O(2)-orbits of Morse index λ in and let βλa be the dimension of H λ (Ma ; F ), we obtain as before the weak Morse inequalities L21 (S 1 , M )a

(2.33)

∞  λ=0

μλ tλ (2 + 2t) ≥

∞  λ=0

βλa tλ .

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2.10. The Morse-Witten complex The Morse inequalities do not always completely determine the integer homology H∗ (Mb ; Z). However, implicit in the writings of Thom, Smale, Milnor and others from the 1960s and 1970s is a procedure for associating a chain complex (C∗ (f, X ), ∂) to a Morse function f and an associated gradient-like vector field X , which calculates the homology of M, and one can derive the Morse inequalities from this chain complex. This chain complex is often called the Morse-Witten chain complex , because of the fact that Witten gave an important quantum-mechanical interpretation of the boundary operator [Wit82]. We merely sketch the ideas of the construction here; a much more complete treatment can be found in [Sch93] and Chapter 6 of [Jos08], or derived from Milnor’s treatment of the h-cobordism theorem [Mil65]. For the statement of the next theorem, we assume the reader is familiar with the notion of CW complex; background on this topic can be found in Chapter 0 of [Hat02]. Theorem 2.10.1. If f : M → [0, ∞) is a Morse function on a complete Hilbert manifold that satisfies condition C and all of its critical points have finite index, then for each a ∈ R, Ma has the homotopy type of a finite CW complex with one cell of dimension λ for each critical point of index λ. Sketch of proof. This is a consequence of Theorem 2.8.4 and it is an analog of Theorem 3.5 in [Mil63]. In fact, one can prove that M itself has the homotopy type of a CW complex, but a rigorous proof would require a limiting procedure as a → ∞. Such a procedure is given in Appendix A of [Mil63]. QED Thus the homology of M should be just the cellular homology of the resulting CW complex, as described in books on algebraic topology; see, for example, [Hat02], pages 137–141. This raises the problem of describing the cells and attaching maps, and finding an algorithm for calculating this homology. Such an algorithm can be based upon choice of a gradient-like vector field X for the Morse function f : M → R. Let {φt : t ∈ R} be the one-parameter group of local diffeomorphisms corresponding to −X . Definition. The unstable manifold Wp (f, X ) of a critical point p ∈ M consists of the images of all trajectories t → φt (q) which start at p in the remote past, in other words, such that φt (q) → p as t → −∞. Similarly, the stable manifold Wp∗ (f, X ) of a critical point p consists of the images of

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119

trajectories t → φt (q) such that φt (q) → p as t → ∞. Thus, Wp (f, X ) = {q ∈ M : lim φt (q) = p}, t→−∞

Wp∗ (f, X )

= {q ∈ M : lim φt (q) = p}. t→∞

Lemma 2.10.2. The unstable and stable manifolds Wp (f, X ) and Wp∗ (f, X ) are in fact submanifolds of M. Sketch of proof. Let us consider the case of the unstable manifold Wp (f, X ). First one uses the explicit description (2.29) of the gradient-like vector field X near p to show that for ε > 0 sufficiently small, Wp (f, X )ε = φ−1 p ({x ∈ E− : x < ε}) is part of the unstable manifold. Then one notes that  Wp (f, X ) = {φt (Wp (f, X )ε ) : t ≥ 0}. The properties of smooth flows corresponding to vector fields then show that Wp (f, X ) is a smooth manifold diffeomorphic to an open cell. The proof for Wp∗ (f, X ) is similar, starting with Wp∗ (f, X )ε = φ−1 p ({x ∈ E+ : x < ε}) . It follows from Condition C and the explicit form of X that any orbit q → φt (q) converges to some critical point as t → ∞ (although it may come close to several other critical points first). QED Lemma 2.10.3. We can adjust the gradient-like vector field for f so that if λp is the index of p and λq is the index of q, (1) λp ≤ λq ⇒ Wp (f, X ) ∩ Wq∗ (f, X ) is empty, while (2) λp > λq ⇒ Wp (f, X ) ∩ Wq∗ (f, X ) is a submanifold of dimension λp − λq . Note: One of the advantages of gradient-like vector fields is that they make it easy to achieve the perturbation needed for part 2 of this lemma. Sketch of proof. If Wp (f, X ) ∩ Wq∗ (f, X ) is nonempty, then f (p) > f (q) and we can choose a regular value c for f such that f (p) > c > f (q). Then N = f −1 (c) is a codimension one submanifold of M, and dim (N ∩ Wp (f, X )) = λp − 1,

codim (N ∩ Wq∗ (f, X )) = λq .

Moreover, if we choose c sufficiently close to the Morse nondegenerate critical point p, we see that S = N ∩ Wp (f, X ) is a (λp − 1)-dimensional sphere and

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lies in an open tubular neighborhood U of N with diffeomorphism ψ : U → S × V , where V is an open ball in a Banach space, with ψ(S) = S × {0}. Let N = U ∩ Wq∗ (f, X ), and consider g = π ◦ ψ : N → V, where π is the projection on the second factor. We can check that g is a Fredholm map with Fredholm index λp − 1 − λq , and ψ −1 (S × {x}) ∩ N = {s ∈ N : g(s) = x}. Assuming that X and N are sufficiently smooth, we can use the Sard-Smale Theorem 2.6.2 to choose a regular value x for g. Finally, we can choose x as close as we want to 0, and construct an isotopy from a neighborhood of S to itself which carries S × {0} to S × {x}. Since the conditions defining pseudogradient are open, we can replace X by a new gradient-like vector field so that N ∩ Wp (f, X ) and N ∩ Wq∗ (f, X ) have transverse intersection. If λp ≤ λq , the dimension of N ∩ Wp (f, X ) is less than the codimension of N ∩Wq∗ (f, X ), so Wp (f, X )∩Wq∗ (f, X ) is empty. If λp > λq , then N ∩ Wp (f, X ) ∩ Wq∗ (f, X ) is a submanifold of dimension λp − λq − 1. QED In particular, if λp = λq +1, then N ∩Wp (f, X )∩Wq∗ (f, X ) consists of a finite number of points and Wp (f, X ) ∩ Wq∗ (f, X ) consists of a finite collection of smooth curves from p to q. We can now orient the unstable manifolds and define the Morse-Witten complex of the Morse function f . We let Cλ (f, X ) be the free Z-module (or free abelian group) generated by the critical points pλ,1 , pλ,2 . . . of f of index λ, and define the Morse-Witten boundary ∂ to be the Z-module homomorphism ∂ : Cλ (f, X ) −→ Cλ−1 (f, X ) given by (2.34)

∂(pλ,i ) =



aij pλ−1,j ,

q

where aij ∈ Z is the oriented number of trajectories from pλ,i to pλ−1,j . The sign of a trajectory γ : (−∞, ∞) → M from p to q is determined as follows: First one orients each unstable manifold for f . Then one constructs a trivial vector bundle E along γ which is transverse to T Wq∗ (f, X ) and of complementary dimension such that the fiber Eγ(t) approaches a hyperplane in the negative subspace Tp Wp (f, X ) generated by negative eigenvalues for d2 f (p) as t → −∞, and a subspace of Tq M containing Tq Wq (f, X ) as t → ∞. Using this bundle we translate the oriented negative subspace Tq Wq (f, X ) for

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121

d2 f (q) back to p. If Tq M− denotes the resulting hyperplane in Tp Wp (f, X ), then Tp Wp (f, X ) = Tq M− ⊕ T, where T is the oriented line in the direction of the trajectory leaving p. We assign the positive sign to the trajectory γ if the orientation of Tp Wp (f, X ) agrees with the direct sum orientation, the minus sign otherwise. Example 2.10.4. We reconsider the proper function f : R2 → R, with f representing height as shown in Figure 2 from §2.9. In this case, we have two stable critical points at p1 and p2 , and we can take C1 = free abelian group generated by q, C0 = free abelian group generated by p1 and p2 , with the other groups being zero. If the unstable manifold for q is oriented towards the left, then ∂ : C1 → C0

is defined by

∂(q) = p1 − p2 .

The next theorem says that the Morse-Witten complex (C∗ (f, X ), ∂) is a chain complex which calculates the homology of M: Theorem 2.10.5. Suppose that f : M → R is a nonnegative Morse function on a complete Hilbert manifold that satisfies condition C and all of the critical points of f have finite index. Then ∂ ◦ ∂ = 0 and the homology (with integer coefficients) of the Morse-Witten complex (C∗ (f, X ), ∂) is just the usual homology H∗ (M; Z). Sketch of proof. It suffices to prove this for the restriction of f to Ma which has only finitely many critical points, and we henceforth use the notation M for Ma . After generic choice of X , follows from Lemma 2.10.3 that unstable critical points of index λ intersect only stable manifolds of critical points of index < λ. Thus, if we let M(λ) denote the union of the unstable manifolds in M of index ≤ λ, we obtain an increasing filtration · · · ⊂ M(λ−1) ⊂ M(λ) ⊂ M(λ+1) ⊂ · · · of M by closed subsets. Since the unstable manifolds of index λ are in one-to-one correspondence with generators of H∗ (M(λ) , M(λ−1) ; Z), we can set C∗ (f, X ) equal to this homology group. If we let ∂  denote the boundary homomorphism in the exact sequence of the triple (M(λ) , M(λ−1) , M(λ−2) ),

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one can use the usual argument from the theory of cellular homology to show that ∂  ◦ ∂  = 0 and the cohomology of the Morse complex is the standard homology of M (as in the proof of Theorem 2.35 in [Hat02]). Thus it is relatively easy to see that we do indeed get a chain complex, and we have reduced the proof to the verification that ∂  is the same as the homomorphism ∂ defined by (4.67). To do this, one can follow arguments used by Milnor ([Mil65]) to prove the h-cobordism theorem of Smale. Note that we can adjust the values of the Morse function f while simultaneously multiplying the gradient-like vector field X by a positive function η : Ma → (0, ∞). In this way, we can replace the Morse function f by “self-indexing” function f ∗ which satisfies the conditions f  ≥ −1 and f  (p) = λ, whenever p is a critical point of index λ, by following the proof of Theorem 4.8 in [Mil65], without changing critical points or the boundary maps in the Morse-Witten complex. After this is done, M(λ) is a strong deformation retract of   1 λ+1/2 ∗ . = q ∈ M : f (q) ≤ λ + M 2 To complete the sketch of the proof, we need only verify the following lemma: Lemma 2.10.6. The Morse-Witten boundary ∂ agrees with the boundary ∂  defined by the CW decomposition. The proof can be derived by modifying the proof of Theorem 7.4 in [Mil65], to which we refer for details. Here we give only the roughest description of how the argument goes. The key idea is that if N = (f ∗ )−1 (λ − 1/2), then the CW boundary ∂ can be regarded as a composition Hλ (Mλ+1/2 , Mλ−1/2 ; Z) −→ Hλ−1 (N ; Z) −→ Hλ−1 (Mλ−1/2 , Mλ−3/2 ; Z), where the first map takes the homology class corresponding to the λ-handle for a critical point pλ,i to the homology class of its boundary sphere Siλ−1 ⊆ N , and the second map is induced by inclusion. On the cochain level, we have a corresponding factorization of the coboundary H λ (Mλ−1/2 , Mλ−3/2 ; R) −→ H λ (N ; R) −→ H λ (Mλ+1/2 , Mλ−1/2 ; R), where we use real coefficients so that we can represent cycles by differential forms. Corresponding to a critical point pλ−1,j we can define a “Thom form” θλ−1,j (as described in [BT82]) which represents a cohomology class [θλ−1,j ] ∈ H λ−1 (Mλ−1/2 , Mλ−3/2 ; R)

2.10. Morse-Witten complex

such that

123

 θλ−1,j = δjk Wpλ−1,k

 1 = 0

if j = k, if j = k.

We can think of θλ−1,j as the Poincar´e dual to the class represented by the infinite-dimensional stable manifold for pλ−1,j . One now checks that  θλ−1,j ∈ Z ajq = Siλ−1

is both the integer appearing in the Morse-Witten boundary (2.34) and the integer appearing in the formula for the CW boundary. QED Proposition 2.10.7 (Lacunary Principle). If for all choices of λ ∈ N ∪ {0}, Cλ (f, X ) = 0 ⇒

its nearest neighbors Cλ−1 (f, X ) and Cλ+1 (f, X ) are both zero,

then the boundary ∂ in the Morse-Witten chain complex is zero. The proof is immediate. Note that the Lacunary Principle now yields the integer homology of the loop space of the n-sphere S n when n ≥ 3:  Z, if m = k(n − 1) or k ∈ N ∪ {0}, n ∼ Hm (Ω(S , p, q); Z) = 0, otherwise. In spite of the fact that the Morse-Witten complex gives stronger results, the Morse inequalities are often quite useful, since it is sometimes difficult to calculate the boundary operator in the Morse-Witten complex.

Chapter 3

Topology of Mapping Spaces

3.1. Sullivan’s theory of minimal models We would like to use the Morse theory presented in the preceding chapter to prove the existence of geodesics connecting two given points in a complete Riemannian manifold, or of periodic geodesics in a compact Riemannian manifold. But to do this we need to be able to calculate the basic topological invariants of function spaces such as Ω(M, p, q) and L21 (S 1 , M ). Fortunately, if one is willing to ignore torsion, there is a simple method for calculating the cohomology of Ω(M, p, q) and L21 (S 1 , M ) based upon Sullivan’s theory of minimal models [Sul77]. Indeed, Sullivan’s theory allows us to calculate the de Rham cohomology of the manifolds of maps, homotopy equivalent to C 0 (Σ, M ), where Σ and M are compact manifolds, that arise in studying nonlinear differential equations. We give an introduction to this theory here, and refer to the excellent references [GriM81], [FHT00] and [FOT08] for more detailed presentations. We begin by recalling a few basic definitions. Let F be a field, either the field of rational or real numbers. A F )gradedi commutative algebra over i are is an algebra over F of the form A = ∞ A , in which elements of A i=0 said to be homogeneous of degree i, such that ωi ∈ Ai

and

θj ∈ Aj

⇒

ωi θj = (−1)ij θj ωi ∈ Ai+j .

A differential graded algebra over F (often abbreviated to dga) is a graded commutative algebra A together with a differential d, which is a collection 125

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of linear maps d : Ai → Ai+1 such that d ◦ d = 0 and ωi ∈ Ai

and

θj ∈ Aj

⇒

d(ωi θj ) = (dωi )θj + (−1)i ωi dθj .

The dga is said to be connected if A0 = F , and one-connected if, in addition, A1 = 0. A differential graded algebra map (or dga map) f : (A, d) −→ (B, d) is an algebra homomorphism f : A → B which preserves degrees (f (Ai ) ⊆ B i ) and commutes with the differential (d ◦ f = f ◦ d). We thus obtain a category which has dga’s as its objects and dga maps as its morphisms. Given two graded commutative algebras A=

∞ 

Ai

and

B=

i=0

∞ 

Bi ,

i=0

we can construct their tensor product A⊗B =

∞ 

(A ⊗ B)i ,

(A ⊗ B)i =

i=0

i 

Aj ⊗ B i−j ,

j=0

in which the product is defined by requiring that 

(ω ⊗ φ) · (ω  ⊗ φ ) = (−1)deg φ+deg ω (ωω  ) ⊗ (φφ ), whenever ω, ω  , φ and φ are homogeneous elements. The tensor product of two dga’s is a dga with the differential being defined on the product of homogeneous elements by d(ω ⊗ φ) = dω ⊗ φ + (−1)deg ω ω ⊗ dφ. Among the simplest examples of dga’s is the de Rham complex (Ω∗ (M), d) of a smooth manifold M, for which the field F is the real numbers R and the differential d is the usual exterior derivative. Another important example is the cohomology algebra H ∗ (M, F ), in which the differential is zero. The latter dga is connected if and only if M is connected, one-connected if and only if M is simply connected. Minimal models are defined for smooth manifolds which are simple and of finite type. A manifold M is of finite type if for each k ∈ Z its homology Hk (M ; Z) is finitely generated, and it is simple if it is connected, its fundamental group π1 (M ) is abelian, and this fundamental group acts trivially on the homotopy groups πk (M ) for all k ≥ 2. Connected topological groups are simple, as are the pointed loop spaces of simply connected manifolds. It can be shown that if M is simply connected and of finite type, then the free loop space C 0 (S 1 , M ) is simple and of finite type.

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127

Definition. A real minimal model of a simple smooth manifold M of finite type is a connected dga (A(M ), d) over R, together with a dga map φ : A(M ) → Ω∗ (M ), where Ω∗ (M ) is the de Rham complex of M , which satisfies the following axioms: (1) A(M ) is free as a graded commutative algebra, that is, it is an exterior algebra on generators of positive odd degree tensored with a polynomial algebra on generators of positive even degree. The number of generating elements may be finite or countably infinite, but the number of generators in any given degree is finite. (2) If A+ denotes the ideal in A(M ) consisting of elements of positive degree, then d(A+ ) ⊆ A+ · A+ . (3) We can order the generators x1 , x2 , . . . , xm , . . . so that the degree of xm is ≤ the degree of xm+1 , and if Λ(x1 , . . . , xm ) is the free graded commutative subalgebra generated by the first m generators of A(M ), then dxm+1 ∈ Λ(x1 , . . . , xm ). (This condition is automatically satisfied if the algebra is one-connected.) (4) φ induces an isomorphism on cohomology. Theorem 3.1.1 (Sullivan). Let M be a simple smooth manifold of finite type. Then M possesses a real minimal model, unique up to dga isomorphism. Moreover, the algebra generators in degree k of the minimal model can be taken to be a basis for the real vector space Hom(πk (M ), R). The real vector spaces πk (M )⊗R for k ∈ N are called the de Rham homotopy groups of M . Theorem 3.1.1 states that the minimal model of M is generated by the dual spaces V k = (πk (M ) ⊗ R)∗ = Hom(πk (M ), R) to the de Rham homotopy groups. Although tensoring with R does lose torsion information, the de Rham homotopy groups can often be calculated efficiently by constructing the minimal model. Recall that homotopy groups, although easy to define, are difficult to compute in general if one does not ignore torsion, and indeed the homotopy groups of spheres are not fully known (see [Hat02], Chapter 4). The proof of Theorem 3.1.1 is based upon the theory of “Postnikov towers”, which we discuss later in §3.5. For now, we simply explain the algorithm for constructing the minimal model, following the procedure described by Sullivan’s introductory survey [Sul73].

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We assume first that M is simply connected. It then follows from the Hurewicz isomorphism theorem ([Hat02], Theorem 4.37) that H 2 (M, R) ∼ = V 2 = Hom(π2 (M ), R). We start our construction of the minimal model with a basis x1 , . . . , xm for V 2 , and let A2 (M ) = P [x1 , . . . , xm ], the polynomial algebra generated by x1 , . . . , xm with the trivial differential d. We then define the algebra homomorphism ρ2 : A2 (M ) → Ω∗ (M ) by letting ρ2 (xi ) be a differential form representative for xi , and note that ρ2 is a dga map which induces an isomorphism on cohomology up to degree two and a monomorphism in degree three. We next choose a collection of generators y1 , . . . , yn which correspond generators yn+1 , . . . , yn+l which to a basis for H 3 (M ; R), and additional ) correspond to a basis of relations crij xi ∪ xj = 0 in H 4 (M ; R) among the generators for H 2 , where n + 1 ≤ r ≤ n + l. According to the algorithm these generators y1 , . . . , yn , yn+1 , . . . yn+l should form a basis for V 3 = Hom(π3 (M ), R). We set A3 (M ) = P [x1 , . . . , xm ] ⊗ Λ(y1 , . . . yn+l ), the product of a polynomial algebra with an exterior algebra, with differential d defined by dxi = 0 for 1 ≤ i ≤ m, dyi = 0 for 1 ≤ i ≤ n,  dyr = crij xi ∧ xj for n + 1 ≤ r ≤ n + l. Define the algebra homomorphism ρ3 : A3 (M ) → Ω∗ (M ) to be an extension of ρ2 such that ρ3 (yi ) is a differential form representative for the element of H 3 (M ; R) corresponding to yi for 1 ≤ i ≤ n, and ρ3 (yr ) is a differential form such that  d(ρ3 (yr )) = crij ρ3 (xi ) ∧ ρ3 (xj ), for n + 1 ≤ r ≤ n + l. Then ρ3 induces an isomorphism on cohomology up to degree three and is injective in degree four. In the same manner, we construct a differential graded algebra A4 (M ) = P [x1 , . . . , xm ] ⊗ Λ(y1 , . . . , yn+l ) ⊗ P [z1 , . . . , zq ], with z1 , . . . , zq forming a basis for V 4 = Hom(π4 (M ), R),

3.1. Sullivan’s theory of minimal models

129

together with a dga map ρ4 : A4 (M ) → Ω∗ (M ) which induces an isomorphism on cohomology up to degree four and an injection in degree five. We continue to construct A5 (M ), and so on. As long as the degree of the new generators is no larger than the dimension of the manifold, those generators depend upon the de Rham complex. When the degree of new generators exceeds the dimension of the manifold they are sent to zero under the dga map ρ, but we often need to add new generators to kill off polynomials of lower dimensional generators which must be trivial in cohomology. The union A(M ) of this sequence of differential graded algebras satisfies the axioms for the real minimal model of M . We might next try to extend this algorithm to the case where M is connected but not simply connected. In this case, we would start one degree lower with   π1 (M ) 1 1 ∼ ,R , H (M, R) = V = Hom [π1 (M ), π1 (M )] where [π1 (M ), π1 (M )] is the commutator subgroup of π1 (M ). As the first stage in the minimal model we might try A1 (M ) = Λ(θ1 , . . . , θl ), an exterior algebra on l generators, where l is the rank of H 1 (M, R), with a dga map ρ1 : A1 (M ) → Ω∗ (M ) which takes θ1 to a differential form representing θa . The problem is that this will not induce a monomorphism in degree two cohomology if any linear combination of the two-forms θa ∧θb is exact. To “kill off” the two-forms which are exact, we may need to introduce more generators of degree one, and wedge products of those new generators may need to be “killed”, requiring yet more degree generators in a process that might in the end require an infinite number of generators of degree one. It is here that the hypothesis that M be simple simplifies matters. The theory of Postnikov towers described later shows that under this hypothesis, our choice of A1 (M ) works, and the algorithm proceeds exactly as before. Any smooth map f : M → N between simply connected manifolds of finite type induces a dga map f n : An (N ) → An (M ) for each positive integer n, as well as a dga map in the limit f ∗ : A(N ) → A(M ). We construct these maps f n by induction on n using the corresponding maps f n : Ωn (N ) → Ωn (M ) on differential forms. We thereby obtain a contravariant functor from the category of smooth simply connected manifolds of finite type and smooth maps to the category of differential graded algebras and differential graded algebra maps. As with de Rham cohomology, functoriality turns out to be of great use in calculating minimal models.

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3. Topology of Mapping Spaces

Example 3.1.2. Suppose that M = S 2n+1 , where n ≥ 1. Then A2n+1 (M ) = Λ(x), an exterior algebra generated by x, where x has degree 2n+1 with dx = 0. In this case, ρ2n+1 induces an isomorphism on cohomology so the construction stops at this stage, and the minimal model for M is just Λ(x) with trivial differential. Thus the de Rham homotopy groups for odd-dimensional spheres are  R, if k = 2n + 1, πk (M ) ⊗ R ∼ = 0, otherwise. Example 3.1.3. Suppose that M = S 2n , where n ≥ 1. Then the construction we described above starts with a polynomial algebra A2n (M ) = P [x], where x has degree 2n. But a polynomial algebra on x has nonzero cohomology in all degrees divisible by 2n, so we must “kill” the cohomology in degree 4n by introducing a generator y of degree 4n − 1, yielding A4n−1 (M ) = P [x] ⊗ Λ(y), with the differential d satisfying dx = 0,

dy = x2 .

This time, ρ4n−1 induces an isomorphism on cohomology, so A(M ) = A4n−1 (M ), and we conclude that the de Rham homotopy groups for even-dimensional spheres are  R, if k = 2n or 4n − 1, πk (M ) ⊗ R ∼ = 0, otherwise. Example 3.1.4. If M and N are two simple manifolds of finite type, then A(M × N ) = A(M ) ⊗ A(N ). Example 3.1.5. Suppose that M is a connected sum of m copies of S 3 ×S 3 , M = (S 3 × S 3 )#(S 3 × S 3 )# · · · #(S 3 × S 3 ). In this case, we find that A3 (M ) = Λ(x1 , . . . , x2m ) with trivial differential and A5 (M ) = Λ(x1 , . . . , x2m ) ⊗ Λ(y1 , . . . , yN ), where N = m(2m + 1) − 1. The yj ’s of degree five are needed to kill off all but one of the possible products of the xi ’s. But then we have 2N m products xi yj of degree eight that must be killed off, so A7 (M ) = Λ(x1 , . . . , x2m ) ⊗ Λ(y1 , . . . , yN ) ⊗ Λ(z1 , . . . , zQ ),

3.2. Minimal models for spaces of paths

131

where the zk ’s are of degree seven and Q = 2N m. This introduces yet more products of the form xi zk , which must be killed off by additional generators of degree nine, and so on. If m is large the number of generators of Ak (M ) grows exponentially with k. This implies that the rank of the free part of πk (M ) also grows exponentially with k. The last space is an example of a rationally hyperbolic manifold, in accordance with the following definition: Definition. A compact simply connected smooth manifold is rationally elliptic if the dimension of  πj (M ) ⊗ R π∗ (M ) ⊗ R = j

is finite; otherwise it is called rationally hyperbolic. It is proven in [FHT00] as part of the Dichotomy Theorem that if M is rationally hyperbolic, then k 

dim(πi (M ) ⊗ R)

i=0

grows exponentially with k.

3.2. Minimal models for spaces of paths Once we know the minimal model of M , it is relatively easy to compute the cohomology of Ω(M, p, q) and L21 (S 1 , M ) when M is a simply connected manifold of finite type. Indeed the cohomology of Ω(M, p, q) is particularly easy, because it is an example of an h-space (see [Hat02], page 281 for the definition) and we can apply: Theorem 3.2.1 (Hopf-Leray Structure Theorem). Let M be a connected h-space of finite type. Then the de Rham cohomology of M is the tensor product of an exterior algebra with a polynomial algebra. The proof of this can be found in [Hat02], page 285. Once we know that the cohomology algebra of M is a free graded commutative algebra, the algorithm described in the preceding section collapses, and the minimal model of the h-space M is just the de Rham cohomology itself with trivial differential. As a very special case, we find that for the torus T n = S 1 × · · · × S 1 (n times), the minimal model is A(T n ) = Λ(θ1 , . . . , θn ),

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3. Topology of Mapping Spaces

an exterior algebra on n generators θ1 , . . . , θn of degree one, with trivial differential. On the other hand, since Ω(M, p, q) is an h-space, we see that the minimal model A(Ω(M, p, q)) of the pointed loop space is the free dga generated by a basis for Hom(π∗ (Ω(M, p, q)), R), with trivial differential. Since πk (Ω(M, p, q)) ∼ = πk+1 (M ), this completely determines the minimal model of Ω(M, p, q) in terms of the de Rham homotopy groups πk (M ) ⊗ R of M . It is a little more difficult to determine the minimal model of the free loop space L21 (S 1 , M ), although this can be done quite efficiently, as explained in [VS76]. Recall from the proof of the Fet-Lusternik Theorem in §2.3 that πk (L21 (S 1 , M )) ∼ = πk (M ) ⊕ πk (Ω(M )) ∼ = πk (M ) ⊕ πk+1 (M ). Since the minimal model is generated as an algebra by the dual space of the real homotopy, we see that corresponding to every generator x of A(M ) of degree k we have two generators x and x ˜ of A(L21 (S 1 , M )), of degree k and k − 1, respectively. Thus we see that if the minimal model A(M ) is freely generated as an algebra by elements x1 , x2 , . . . , xk , . . ., then A(L21 (S 1 , M )) must be freely generated by x1 , x2 , . . . , xk , . . . ,

x ˜1 , x ˜2 , . . . , x ˜k , . . . ,

where

deg x ˜i = deg xi − 1.

To finish determining the minimal model of L21 (S 1 , M ), all we need do is determine the differential. One determines the differential in the minimal model via an analysis of the evaluation map ev : L21 (S 1 , M ) × S 1 −→ M

defined by

ev(γ, t) = γ(t).

One starts by considering the case where M = S k , for some integer k, and suppose that h : S k → M = S k is the identity map, a representative for the generator of πk (S k ). ˜ : S k−1 → Ωp (S k ) is the image Lemma 3.2.2. If the homotopy class of h of the homotopy class of h under the boundary map in the exact homotopy sequence of the path space fibration, then (3.1)

˜ k−1 )] × [S 1 ]) = ±[h(S k )], ev∗ ([h(S

where the square brackets denote the corresponding homology classes. Sketch of proof. This reflects the fact that the boundary isomorphism πk−1 (Ω(M )) ∼ = πk (M ) is a special case of the suspension isomorphism from homotopy theory, [S k−1 , Ω(M )] = [ΣS k−1 , M ],

3.2. Minimal models for spaces of paths

133

in which the homotopy classes are pointed and ΣS k−1 is the reduced suspension of S k−1 , which is homeomorphic to S k . More generally, if p is chosen as base point in M , the reduced suspension ΣM of M is defined by ΣM =

M × [0, 1] . (M × {0}) ∪ (M × {1}) ∪ ({p} × [0, 1])

The construction of ΣS k−1 divides S k up into a family of S k−1 ’s with a common base point p. Under the suspension isomorphism, ˜ × id) : S k−1 × S 1 −→ S k ev ◦ (h gets identified with h, from which (3.1) follows. QED It follows from the lemma that if x and x ˜ are the elements of Hom(πk (M ), R) ˜ = 1, then ˜([h]) and Hom(πk−1 (Ωp ), R) such that x([h]) = 1 and x ev∗ (x) = x ⊗ 1 ± x ˜ ⊗ θ, where θ is the generator of A(S 1 ) and we can make the sign positive by replacing x ˜ by −˜ x if necessary. In the case of general M , we suppose that the maps hi : S k → M , for 1 ≤ i ≤ m, represent a basis for πk (M ) ⊗ R and let {x1 , . . . , xm } be a basis for Hom(πk (M ), R), which is dual to {h1 , . . . , hm }. Application of the commutative diagram L21 (S 1 , S k ) × S 1 → S k ↓ ↓ L21 (S 1 , M ) × S 1 → M in which the downward arrows are induced by hi and the horizontal arrows are the evaluation maps ev then shows that ev∗ (xi ) = xi ⊗ 1 + x ˜i ⊗ θ,

(3.2)

for generators x ˜i which go to xi under the boundary in the exact homotopy sequence of the path space fibration. The fact that ev∗ is a dga map now determines the differential d on the generators x1 , . . . , xm . To see this, we first extend the tilde operation to a ˜ on the entire algebra. Then since ev∗ is an algebra right skew-derivation Δ homomorphism, ˜i ⊗ θ) ⊗ (xj ⊗ 1 + x ˜j ⊗ θ) ev∗ (xi xj ) = (xi ⊗ 1 + x ˜j + x ˜i (−1)degxj xj ) ⊗ θ, = (xi xj ) ⊗ 1 + (xi x and, more generally, ˜ (xj , . . . , xj )) ⊗ θ, ev∗ P (xj1 , . . . , xjl ) = P (xj1 , . . . , xjl ) ⊗ 1 + Δ(P 1 l

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3. Topology of Mapping Spaces

for any polynomial P in the generators. The differential d must act on the generators x1 , . . . , xk , . . . for the minimal model of M as before. On the other hand, if dxi = P (xj1 , . . . , xjl ),

where xj1 , . . . , xjl are elements of lower degree,

then

˜ (xj , . . . , xj )) ⊗ θ ev∗ (dxi ) = P (xj1 , . . . , xjl ) ⊗ 1 + Δ(P 1 l while on the other hand, ˜i ⊗ θ) = dxi ⊗ 1 + d˜ xi ⊗ θ. d(ev∗ xi ) = d(xi ⊗ 1 + x ˜ Comparing the two expressions yields the fact that d(˜ xi ) = Δ(dx i ), and ˜ = Δ ˜ ◦ d. Having determined d on the generators x and x hence d ◦ Δ ˜i , we find that d is completely determined by the Leibniz rule. In the above argument, we have also extended (3.2) to arbitrary elements of the minimal model: ˜ ev∗ (x) = x ⊗ 1 + Δ(x) ⊗ θ, for x ∈ A(L2 (S 1 , M )). 1

A(L21 (S 1 , M ))

a bigrading by assigning to each generator xi We give a bidegree (deg xi , 0) and to the corresponding element x ˜i the bidegree (0, deg xi − 1). We say that an element of degree (s, t) has base degree s and fiber degree t, and let E1s,t denote the space of elements of bidegree (s, t). We define a filtration of the minimal model by setting F s = {x ∈ A(L21 (S 1 , M )) : the base degree of x is ≥ s }, and dF s ⊆ F s+1 , because the formula for d shows that the differential increases base degree. This yields a first quadrant spectral sequence (see [McC01]), and one can check that it agrees with the Serre spectral sequence of the fibration L21 (S 1 , M ) → M at the E2 -term and beyond. Remark 3.2.3. There is also a differential form version of the operator ˜ ˜ If ω is an element of Ωk (L2 (S 1 , M )), we let Δ(ω) denote the result of Δ. 1 ∗ integrating ev (ω) over the fiber of π : L21 (S 1 , M ) × S 1 −→ L21 (S 1 , M ). ˜ If ω is a de Rham representative for ρ(x), then Δ(ω) is a de Rham repre˜ sentative for ρ(Δ(x)). A few examples will illustrate how easy it is to calculate the minimal model and the cohomology of the free loop space of a compact simply connected manifold M in simple cases. Example 3.2.4. If M = S 2n+1 , where n ≥ 1, we have seen that the minimal model for M is A(M ) = Λ(x), an exterior algebra generated by an element x of degree 2n + 1 with trivial differential. The minimal model of the free

3.2. Minimal models for spaces of paths

135

loop space in this case is A(Ω1p (M )) = P [˜ x], a polynomial algebra generated by an element x ˜ of degree 2n with trivial differential, while the minimal model of the free loop space is A(L21 (S 1 , M )) = Λ(x) ⊗ P [˜ x], again with trivial differential. Hence we find that x], H ∗ (Ω(M, p, q), R) ∼ = P [˜

H ∗ (L21 (S 1 , M ), R) ∼ x], = Λ(x) ⊗ P [˜

where now x and x ˜ denote cohomology classes. Example 3.2.5. Suppose that M is a compact simply connected Lie group G. According to the Hopf-Leray structure theorem, the cohomology of M over the reals is that of a product of odd-dimensional spheres S n1 × · · · × S nk , the integer k being the rank of the Lie group. Thus it follows from the previous example and Example 3.1.5 that the minimal model for M is A(M ) = Λ(x1 , . . . , xk ), an exterior algebra on elements x1 , . . . , xk of odd degree. The minimal model of the pointed loop space in this case is a polynomial algebra ˜k ], P [˜ x1 , . . . , x

with trivial differential, where

deg(˜ x1 ) = deg(x1 ) − 1,

while the minimal model of the free loop space is x1 , . . . , x ˜k ], A(L21 (S 1 , M )) = Λ(x1 , . . . , xk ) ⊗ P [˜ the tensor product of an exterior algebra and a polynomial algebra, again with trivial differential. Example 3.2.6. If M = S 2n , we have seen that the minimal model for M is A(M ) = P [x] ⊗ Λ(y), a polynomial algebra on an element x of degree 2n tensored with an exterior algebra on an element y of degree 4n − 1, with differential d satisfying dx = 0,

dy = x2 .

The minimal model for the pointed loop space in this case is A(Ω(M, p, q)) = Λ(˜ x) ⊗ P [˜ y ], with trivial differential, while the minimal model for the free loop space is x) ⊗ P [˜ y ], A(L21 (S 1 , M )) = P [x] ⊗ Λ(y) ⊗ Λ(˜ the differential being defined by dx = 0,

dy = x2 ,

d˜ x = 0,

d˜ y = 2x˜ x.

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3. Topology of Mapping Spaces

As a first application of minimal models, we can now establish: Proposition 3.2.7. If M is a compact simple connected Riemannian manifold with finite fundamental group, then H k (Ω(M, p, q); R) and

H k (L21 (S 1 , M ); R)

are nonzero for k lying in an infinite arithmetic sequence, n → k(n) = an+b, where a, b ∈ Z. Proof. The argument comes from Sullivan [Sul73]. Since M has finite fundamental group, it has the same minimal model as its universal cover, which is also compact, and this implies that the minimal model of the universal cover cannot be trivial. Next, one notes that the generators for the minimal model A(M ) for M cannot all be of even degree, because then M itself would have nonvanishing cohomology in arbitrarily high degrees. But this implies that at least one generator for the minimal model of Ω(M, p, q) must have even degree and by the Hopf-Leray structure theorem, the real homology of the loop space Ω(M, p, q) contains a polynomial algebra, which must of course have elements of arbitrarily high degree. To treat the case of the free loop space, we let y be an odd-dimensional generator of minimal degree k in the minimal model for M , and note that the hypothesis on the fundamental group implies that k ≥ 3. Then dy = P (x1 , . . . , xk ), where P is a polynomial (which may be zero) and x1 , . . . , xk are element of even degree < k such that dxi = 0. Moreover, by the formula for the xi = 0 and and since each differential in the minimal model for L21 (S 1 , M ), d˜ term in d˜ y has at least one x ˜i as a factor, ˜k ∧ y˜p ) = 0, d(˜ x1 ∧ · · · ∧ x for an arbitrary choice of the nonnegative integer p. Since d increases base ˜k ∧ y˜p can never be exact, and we see that degree, x ˜1 ∧ · · · ∧ x ˜k ∧ y˜p ] = 0 [˜ x1 ∧ · · · ∧ x

as an element of

H ∗ (L21 (S 1 , M ); R),

and the theorem is proven. QED As a corollary, we obtain the following theorem first presented by Serre [Ser51] in 1951 as an application of his theory of spectral sequences for fibrations: Theorem 3.2.8 (Serre). If M is a compact simply connected Riemannian manifold, then any two points of M can be connected by infinitely many geodesics.

3.2. Minimal models for spaces of paths

137

Proof. If Ω(M, p, q) contained only finitely many geodesics, then Ω(M, p, q) would have the homotopy type of Ω(M, p, q)a for some finite choice of a. After perturbing J to a Morse function, one would be able to use Theorem 2.10.1 to show that Ω(M, p, q)a has the homotopy type of a finite CW complex, and hence Hm (Ω(M, p, q); Z) would have to be zero when m is larger than the dimension of the largest cell in the CW complex. Thus to prove the theorem, it suffices to show that Hm (Ω(M, p, q); Z) = 0 for arbitrarily large m. But this follows from Proposition 3.2.7. QED We claim that a simple mechanical systems (M, ·, ·, φ) in which M is simply connected and compact always have infinitely many periodic solutions to “Newton’s equation of motion” ∇γ  γ  (t) = −(grad φ)(γ  (t)) of any given period. Recall that such solutions are the critical points to the action function JM,φ : L21 (S 1 , M ) −→ R,  [(kinetic energy) − (potential energy)]dt JM,φ (γ) = S1    1   γ (t), γ (t) − φ(γ(t)) dt. = S1 2 Theorem 3.2.9. If (M, ·, ·, φ) is a simple mechanical system, where M is a compact simple Riemannian manifold with finite fundamental group, then the action Jφ : L21 (S 1 , M ) → R has infinitely many critical points of arbitrarily high Morse index. The proof is a straightforward modification of that for Theorem 3.2.8, using the results of Proposition 3.2.7. This theorem applies to the Euler equations for a rigid body, as discussed in Example 2.1.2. We should mention, however, that when φ is nonzero, many other methods have been developed for proving existence of periodic orbits. If φ = 0, Newton’s equations of motion reduce to the geodesic equations on M , but in this case, once we have one critical point for the action J, we can construct infinitely many by simply taking multiple covers of a single underlying smooth closed geodesic: If γ : S 1 → M is a nonconstant smooth closed geodesic, so is γm : S 1 → M,

defined by

γm (t) = γ(mt),

for m ∈ Z,

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which we call a multiple cover of γ when |m| ≥ 2. If m = n, γm and γn are distinct elements of the function space L21 (S 1 , M ), but they represent the same “geometric” geodesic in M . Alternatively, we say that γm and γn are not geometrically distinct when m = n. Unfortunately, Theorem 3.2.8 does not establish the existence of infinitely many geometrically distinct smooth closed geodesics. Recall that a closed geodesic γ : S 1 → M is prime if γ is not a multiple cover of a geodesic of smaller length. By using results of Bott [Bot56] on index and nullity of multiple covers of prime geodesics, Gromoll and Meyer [GroM69a] were able to prove that most compact manifolds do indeed possess infinitely many geometrically distinct geodesics for arbitrary choice of Riemannian metric on M . Gromoll-Meyer Theorem 3.2.10. If the compact manifold M with finite fundamental group satisfies the condition that the Betti numbers of its free loop space L21 (S 1 , M ) are unbounded, then M possesses infinitely many prime closed geodesics, and hence infinitely many geometrically distinct closed geodesics, for any choice of Riemannian metric on M . We will sketch a proof for the special case for generic Riemannian metrics in §3.4 based upon estimates of Gromov.

3.3. Gromov dimension In this section, we will describe how finite-dimensional approximations of spaces of paths yield estimates of Gromov [Gv78], which relate topological complexity to length.1 For this purpose, we need the length functions L : Ω(M, p, q) −→ R

and L : L21 (S 1 , M ) −→ R,

which are defined by  1 γ  (t), γ  (t)dt and L(γ) = 0





L(γ) = S1

γ  (t), γ  (t)dt,

respectively. For each b > 0, we let Ω(M, p, q)[b] = {γ ∈ Ω(M, p, q) : γ has constant speed and L(γ) ≤ b }, L21 (S 1 , M )[b] = {γ ∈ L21 (S 1 , M ) : γ has constant speed and L(γ) ≤ b }. Definition. Let M be a compact connected Riemannian manifold, and let M be either Ω(M, p, q) or L21 (S 1 , M ). For each b > 0, the Gromov dimension 1 We remark that later (§4.9) these estimates will be used to show that no energy is lost in necks during “bubble tree convergence” for certain choices of minimax constraints.

3.3. Gromov dimension

139

of M[b] , denoted by dm(M[b] ), is the maximum of the nonnegative integers k such that every map of a connected polyhedron into M of dimension at most k is homotopic to a map into M[b] . Alternatively, we can say that the Gromov dimension dm(M[b] ) is the largest nonnegative integer such that Hk (M, M[b] ; Z) = 0

for k ≤ dm(M[b] ).

Thus the inclusion induces an isomorphism (3.3)

Hk (M[b] ; Z) ∼ = Hk (M; Z)

for k ≤ dm(M[b] ) − 1,

and we can think of Gromov dimension as giving a rough measure of the extent to which M[b] approximates M in homology. Theorem 3.3.1. If (M, ·, ·) is a compact connected Riemannian manifold with finite fundamental group. Then there are positive constants c1 , c2 , C1 and C2 such that (3.4)

c1 (b − 1) ≤ (dm(Ω(M, p, q)[b] )) ≤ c2 b, C1 (b − 1) ≤ (dm(L21 (S 1 , M )[b] )) ≤ C2 b,

for b > 0. In other words, the Gromov dimension has the same rate of growth as the length b of geodesics in Ω(M, p, q) or L21 (S 1 , M ). We remark that the lower estimates are not true without the hypothesis that M have finite fundamental group. For example, a flat torus has arbitrarily long smooth closed geodesics of index zero which are length minimizing in their homotopy classes. The upper bounds are easier, and follow from the existence of a constant C > 0 such that for any geodesic γ in M , (3.5)

(Morse index of γ) ≤ C(length of γ).

Indeed, an explicit estimate of this type with a constant C that depends upon the curvature of M was worked out by Morse and Schoenberg (see [GKM68], page 176). Here we give an alternate argument for the upper bounds based upon the space of broken geodesics. The space of broken geodesics. We focus on the space of broken geodesics which corresponds to L21 (S 1 , M ), the other case being similar. As usual, we consider S 1 to be the quotient space of the unit interval [0, 1], with endpoints 0 and 1 identified. Choose a large positive integer N and

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define the space of broken geodesics BGN (S 1 , M ) to be the set of continuous maps γ : [0, 1] → M such that γ(0) = γ(1) and the restriction of γ to [i/N, (i + 1)/N ] is a smooth geodesic. Since such a loop γ is continuous and γ  , although not continuous, is square integrable, we see that BGN (S 1 , M ) ⊆ L21 (S 1 , M ). We set L21 (S 1 , M )a = {γ ∈ L21 (S 1 , M ) : J(γ) ≤ a}, BGN (S 1 , M )a = {γ ∈ BGN (S 1 , M ) : J(γ) ≤ a}. It follows from the Cauchy-Schwarz inequality that if γ ∈ L21 (S 1 , M )a , (3.6) 2      (i+1)/N (i+1)/N (i+1)/N a    γ (t)dt ≤ γ (t), γ (t)dt dt ≤ , N i/N i/N i/N and hence if d(p, q) denotes the Riemannian distance between p and q,  d(γ(i/N ), γ((i + 1)/N )) ≤ a/N . If M is a compact Riemannian manifold, there exists δ > 0 such that any two points p and q in the compact Riemannian manifold M with d(p, q) < 2δ are connected by a unique geodesic of length  < 2δ which depends smoothly on p and q. If we choose N so large that a/N < δ, then an element γ of BGN (S 1 , M )a is completely determined by the points γ(i/N ) and we can identify BGN (S 1 , M )a with a subset of M × M × · · · × M (N times). Proposition 3.3.2. Let M be a compact Riemannian manifold. There exists δ > 0 depending on M such that when N > a/δ 2 , BGN (S 1 , M )a is a strong deformation retract of L21 (S 1 , M )a . Recall that by strong deformation retract we mean that there is a continuous map (the deformation retraction) H : L21 (S 1 , M )a × [0, 1] −→ L21 (S 1 , M )a such that (1) H(γ, s) = γ, for all s ∈ [0, 1] if γ ∈ BGN (S 1 , M )a , (2) H(γ, 0) = γ, for all γ ∈ L21 (S 1 , M )a , and (3) H(γ, 1) ∈ BGN (S 1 , M )a , for all γ ∈ L21 (S 1 , M )a . The desired deformation retraction H can be specified by describing the paths γs ∈ L21 (S 1 , M )a such that γs (t) = H(γ, s)(t), and we simply define H by requiring that on each interval [i/N, (i + 1)/N ],  unique geodesic from γ(i/N ) to γ(i + s/N ) on [i/N, (i + s)/N ], γs = γ on [(i + s)/N, (i + 1)/N ].

3.3. Gromov dimension

141

Remark 3.3.3. Bott [Bot82] describes how the space BGN (S 1 , M ) of broken geodesics can be used to develop the Morse theory of geodesics, thereby avoiding infinite-dimensional manifolds, an approach equivalent to that originally adopted by Morse himself. Of course, L21 (S 1 , M )a is closely related to the space L21 (S 1 , M )[b] defined at the beginning of the section. Indeed, the inclusion √ {γ ∈ L21 (S 1 , M )a : γ has constant speed } ⊆ L21 (S 1 , M )[b] , b = 2a is a homotopy equivalence since critical points for L in L21 (S 1 , M )[b] correspond to critical points for J in L21 (S 1 , M )a . Moreover, if we set BGN (S 1 , M )[b] = {γ ∈ BGN (S 1 , M ) : γ has constant speed and L(γ) ≤ b}, the argument for Proposition 3.3.2 can be modified to yield the following: Proposition 3.3.4. Let M be a compact Riemannian manifold. There exists δ > 0 depending on M such that when N > b/δ, BGN (S 1 , M )[b] is a strong deformation retract of L21 (S 1 , M )[b] . Note that L21 (S 1 , M )[b] is homotopy equivalent to a submanifold of the finitedimensional manifold M ×M ×· · ·×M (N times), where N depends linearly on the length b. Since the Morse index is bounded above by N (dim(M )) this gives a linear upper bound (3.5) on Morse index, but it is not as sharp as the one given by Morse and Schoenberg. The lower bounds in (3.4) are more difficult to achieve, and for them we need a second approximation introduced by Gromov [Gv78]. The space of piecewise linear paths for a triangulation. Once again we focus on the space of piecewise linear loops corresponding to L21 (S 1 , M ), the other case being similar. We choose a finite triangulation Δ for the compact manifold M and let P L(S 1 , M ) = {γ ∈ L21 (S 1 , M ) : γ has a constant speed parametrization and is piecewise linear with respect to Δ}. By an argument similar to the one we used for Proposition 3.4.1, we can show that L21 (S 1 , M ) is homotopy equivalent to P L(S 1 , M ) and L21 (S 1 , M )[b] is homotopy equivalent to P L(S 1 , M )[b] = {γ ∈ P L(S 1 , M ) : L(γ) ≤ b}. Let σ1 , . . . , σk be simplices in the triangulation such that any two consecutive elements σi and σi+1 are faces of a simplex in the triangulation (with

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3. Topology of Mapping Spaces

the convention that σk+1 = σ1 . Given pi ∈ σi for 1 ≤ i ≤ k, we let Tσ1 ,...,σk (p1 , . . . , pk ) denote the piecewise linear map which maps [t1 , t2 ] linearly to a segment from p1 ∈ σ1 to p2 ∈ σ2 , then maps [t2 , t3 ] linearly to a segment from p2 ∈ σ2 to p3 ∈ σ3 , and so forth, and finally, maps [tk , t1 ] linearly to a segment from pk ∈ σk to p1 ∈ σ1 . All of this is with respect to some partition 0 ≤ t1 ≤ t2 ≤ t3 ≤ · · · ≤ tk ≤ 1 of [0, 1], which projects to a partition of S 1 . This gives us a map Tσ1 ,...,σk : σ1 × · · · × σk −→ P L(S 1 , M ) which can be thought of as the characteristic map for a cell of dimension ) (dim σi ) in a CW decomposition for P L(S 1 , M ). Now comes Gromov’s key idea: Assume that M is simply connected, and let M (k) denote the k-skeleton of M with respect to the triangulation Δ. Gromov claims that there is a smooth map φ : M → M which is homotopic to the identity and contracts the two-skeleton of Δ to a point. To construct it, one starts by connecting all the vertices to a given point p, thereby obtaining a map H0 : (M (0) × [0, 1]) ∪ (M × 0) −→ M such that H0 |(M × 0) = idM and H0 maps M (0) × 1 to a given point p. Since π1 (M ) = 0 we can extend this to a map H1 : (M (1) × [0, 1]) ∪ (M × 0) −→ M such that H1 |(M × 0) = idM and H1 maps M (1) × 1 to p. Next we extend this to a map H2 : (M (2) × [0, 1]) ∪ (M × 0) −→ M in any way whatsoever, then to (M (3) × [0, 1]) ∪ (M × 0), and so on, finally obtaining a homotopy H : (M × [0, 1]) −→ M whose restriction to M × 1 is a map φ which takes the one-skeleton of M to p and is homotopic to the identity. This induces a map ωφ : P L(S 1 , M ) −→ L21 (S 1 , M ),

ωφ (γ) = φ ◦ γ

which is homotopic to the inclusion. Suppose now that eλ is a cell of dimension λ in the CW decomposition with characteristic map Tσ1 ,...,σk . If γ ∈ eλ , there exist real numbers 0 ≤ t1 ≤ t2 ≤ · · · ≤ tk ≤ 1 such that γ(ti ) ∈ σi and γ is linear on each interval

3.3. Gromov dimension

143

[t1 , t2 ], [t2 , t3 ], . . . , [tk , t1 + 1] as we described before. Then ωφ (γ) takes each interval [ti , ti+1 ] for which dim(σi ) + dim(σi+1 ) ≤ 1 to a point, and hence ) (dim σi ) = λ implies that ωφ (γ) takes all but at most λ of the intervals to a point. Thus γ ∈ eλ

⇒

L(φ ◦ γ) ≤ λ(sup{|Dφ(p)| : p ∈ M })K,

where K is the supremum of the lengths of all images under φ of all linear maps on simplices in the triangulation of M . This implies that L(φ ◦ γ) ≤ C0 λ, where C0 is a positive constant. It follows that if P L(S 1 , M )(λ) is the λ-skeleton of P L(S 1 , M ), then ωφ (P L(S 1 , M )(λ) ) ⊆ L21 (S 1 , M )[C0 λ] . Since any map of a λ-sphere into L21 (S 1 , M ) is homotopic to one into the λ-skeleton, we see that Hλ (L21 (S 1 , M ), L21 (S 1 , M )[b] ; Z) = 0

for b ≥ C0 λ,

and hence (3.7)

λ≤

b C0

⇒

Hλ (L21 (S 1 , M ), L21 (S 1 , M )[b] ; Z) = 0.

This implies that dm(L21 (S 1 , M )[b] ) ≥ C1 b for some positive constant C1 , which proves the lower inequality of Gromov’s theorem when π1 (M ) = 0. If π1 (M ) is finite, we can modify this argument by using a simply connected cover. Corollary 3.3.5. Suppose that (M, ·, ·) is a compact connected Riemannian manifold with finite fundamental group. Then there are positive constants c1 , c2 , C1 , and C2 such that if γ ∈ Ω(M, p, q) is a geodesic, then (3.8)

c1 (length of γ) ≤ (Morse index of γ) + 1 ≤ c2 (length of γ),

while if γ ∈ L21 (S 1 , M ) is a geodesic, then (3.9)

C1 (length of γ) ≤ (Morse index of γ) + 1 ≤ C2 (length of γ).

Proof. Once again we focus on L21 (S 1 , M ) and the upper estimate follows from Proposition 3.3.4. On the other hand, it follows from (3.7) and the exact homology sequence of a triple that a > b ≥ C0 (λ + 1)

⇒

Hλ (L21 (S 1 , M )[a] , L21 (S 1 , M )[b] ; Z) = 0,

and for generic metrics this implies that there are no geodesics of Morse index λ which have length greater than C0 (because according to Morse theory such geodesics would change the homology of L21 (S 1 , M )[a] as a increases). This gives the lower estimate. QED

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One consequence of the Gromov estimates (3.8) is that when the metric a compact Riemannian manifold M is generic, the Morse-Witten complex for the action J : Ω(M, p, q) −→ R has finite type, that is the number of geodesics of each Morse index is finite. Of course, this Morse-Witten complex for J is not of finite type when π1 (M ) is infinite, because in this case there are infinitely many stable geodesics. Gromov pointed out that his estimates for Ω(M, p, q) give information on the rate of growth of the number of geodesics connecting two points p and q with length less than a given bound. For example, suppose that M is a compact simply connected rationally hyperbolic manifold. If M is given a Riemannian metric and p and q are points in general position, then it follows from Corollary 3.3.5 that the number of geodesics of length ≤ b from p to q grows exponentially with b. On the other hand, we can often apply the Gromov estimates for L21 (S 1 , M ) to determine the rate of growth of number of geodesics with length less than a given bound: Theorem 3.3.6. Suppose that M is a compact simply connected manifold and the Betti numbers of the free loop space L21 (S 1 , M ) satisfy the inequality (3.10)

k 

dim Hi (L21 (S 1 , M ); R) ≥ C1 k m ,

i=0

for some positive constant C1 and some positive integer m. Then if M is given a generic Riemannian metric, the number of prime geodesics of length ≤ b is ≥ C2 bm−1 for some positive constant C2 . Sketch of proof. Recall that the Bumpy Metric Theorem 2.7.2 implies that for generic metrics, all geodesics lie on isolated O(2) orbits each a Morse nondegenerate critical submanifold. For such metrics, the Morse inequalities and estimate (3.10) show that if μi denotes the number of such O(2) orbits of Morse index i, then k  4 μ i ≥ C1 k m . i=0

Since the number of geodesics covering a given prime geodesic grows only linearly with length, Gromov’s estimates imply that the number of prime geodesics of length ≤ b must be ≥ C2 bm−1 , for some positive constant C2 . QED Example 3.3.7. If M is a Lie group of rank at least two, we have seen that the minimal model of its free loop space is x1 , . . . , x ˜k ], A(L21 (S 1 , M )) = Λ(x1 , . . . , xk ) ⊗ P [˜

3.4. Infinitely many closed geodesics

145

with trivial differential. Thus, when the rank k is at least two, M will have infinitely many geodesics for any generic metric. A similar argument yields the following: Theorem 3.3.8. Suppose that M is a compact simply connected manifold and the Betti numbers βi of the free loop space L21 (S 1 , M ) satisfy the inequality k 

β i ≥ C3 e C 4 k ,

i=0

for some positive constants C3 and C4 . Then if M is given a generic Riemannian metric, the number of prime geodesics of length ≤ b grows exponentially with b. Although it is straightforward to use the minimal model of a compact simply connected manifold M to construct the minimal model of its free loop space L21 (S 1 , M ), it is not always easy to calculate the cohomology. Thus, although it is easily seen that if M is rationally hyperbolic, the Betti numbers of its pointed loop space satisfy an estimate of the form (3.11)

N 

Hi (Ω(M, p, q); R) ≥ aebN ,

i=0

where a and b are positive constants, it is not known whether a similar conclusion holds for the cohomology of the free loop space. Open Problem 3.3.9. If M is rationally hyperbolic, is it true that N 

Hi (L21 (S 1 , M ); R) ≥ aebN ,

i=0

for some positive constants a and b? In other words, do the Betti numbers of the free loop space of a rationally hyperbolic space have exponential growth? (See [Gv78], [FHT00] and [VS76].)

3.4. Infinitely many closed geodesics The Gromoll-Meyer Theorem is complemented by results of Vigu´e-Poirrier and Sullivan [VS76], who were able use Sullivan’s minimal models to show that the only simply connected manifolds M which have the property that the Betti numbers of L21 (S 1 , M ) are not unbounded are those with the

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rational cohomology ring of a symmetric space of rank one: Theorem 3.4.1 (Vigu´ e-Poirrier and Sullivan). If M is a compact simply connected manifold, then either the cohomology ring H ∗ (M ; R) requires only one generator as an algebra or k 

dim Hi (L21 (S 1 , M ); R) ≥ C1 k 2 ,

i=0

for some positive constant C1 . Corollary 3.4.2. If M is a compact simply connected manifold, then either the cohomology ring H ∗ (M ; R) requires only one generator as an algebra or M has infinitely many prime geodesics for a generic choice of Riemannian metric. The Theorem implies the Corollary by Theorem 3.3.6. Actually, the GromollMeyer Theorem implies the Corollary for an arbitrary choice of metric on M. Proof of Theorem 3.4.1. We follow the elegant argument given in §5.3 of [FOT08]. Our hypothesis is that the cohomology algebra of M requires at least two generators. The first step is to show that the minimal model of M has at least two generators of odd degree. If it does not have at least one odd generator y, then it has two polynomial generators and the differential d must be zero. But this implies that the cohomology is nonvanishing in arbitrarily high degree, not possible for a finite-dimensional manifold M . If it has only one generator y and dy = 0, the cohomology algebra of M still contains a nontrivial polynomial algebra, hence is again nonvanishing in arbitrarily high degree. If it has one generator and dy is nonzero, then the minimal model must contain a polynomial algebra on two generators, and dy can kill only one element in this polynomial algebra, so the cohomology algebra is once again nonvanishing in arbitrarily high degree. So we conclude that the minimal model contains at least two odd generators y1 and y2 . We now order generators by degree, with the xi ’s being the even generators, each xi having degree at least two: x1 , . . . , xn , y1 , xn+1 , . . . , xr , y2 , . . . . Since y1 and y2 have degree at least three, dx1 = 0,

...,

dxn = 0,

dy1 = P1 (x1 , . . . , xn ).

Moreover, dxj = y1 Qj (x1 , . . . , xn ),

for n + 1 ≤ j ≤ r,

dy2 = P1 (x1 , . . . , xr ).

3.4. Infinitely many closed geodesics

147

The minimal model A(L21 (S 1 , M )) of the free loop space has generators x1 , . . . , xn , y1 , xn+1 , . . . , xr , y2 , . . . ,

x ˜1 , . . . , x ˜n , y˜1 , x ˜n+1 , . . . , x ˜r , y˜2 , . . . .

If I is the ideal in A(L21 (S 1 , M )) generated by the even generators x1 , . . . , xr of A(M ), then dI ⊆ I. It follows that we can define a differential on the quotient A(L21 (S 1 , M )) , B= I thereby obtaining a quotient dga (B, d). The elements ˜r · y˜1p · y˜2q , x ˜1 · · · x

p, q ∈ N,

represent linearly independent elements of the cohomology of (B, d), so the cohomology of (B, d) grows at least quadratically with degree. But there is another dga which has the same cohomology as (B, d), namely & 2 (S 1 , M )) = A(L2 (S 1 , M )) ⊗ Λ(z1 , . . . , zr ), A(L 1 1 where z1 , . . . , zr are new generators of odd degree such that dzi = xi for 1 ≤ i ≤ r. Indeed, the dga (B, d) can also be regarded as the quotient A(L21 (S 1 , M )) ⊗ Λ(z1 , . . . , zr ) , J where J is the acyclic differential ideal generated by x1 , . . . , xr and z1 , . . . , zr . & 2 (S 1 , M )) by setting We now filter the dga A(L 1

& 2 (S 1 , M ))) = A≥p (L2 (S 1 , M )) ⊗ Λ(z1 , . . . , zr ). F p (A(L 1 1 This provides a spectral sequence converging to the cohomology of (B, d) which has E2 term E2p,q = H p (A(L21 (S 1 , M ))) ⊗ Λ(z1 , . . . zr )q . The only way this can converge to the cohomology of (B, d) is for the cohomology of A(L21 (S 1 , M )) to grow quadratically with degree. But we have already seen that this implies that M has infinitely many prime geodesics for generic choice of metric. QED Conjecture 3.4.3. It was conjectured by Klingenberg [Kli78] that any compact connected Riemannian manifold of dimension at least two with finite fundamental group has infinitely many geometrically distinct smooth closed geodesics. The Gromoll-Meyer Theorem together with Theorem 3.4.1 might be regarded as a step towards this conjecture. Many additional special cases of this conjecture have been established, and in fact Rademacher has shown that Klingenberg’s conjecture holds for generic metrics (see [Hin84] and [Rad89]).

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3.5. Postnikov towers A rough outline of the proof of Sullivan’s Theorem 3.1.1 goes like this: First, replace the smooth manifold M by its “rationalization” and then construct the “Postnikov tower” of this rationalization. The stages in the Postnikov tower then correspond to stages in the “rational” minimal model of M , and we obtain the real minimal model by tensoring with R. In this section, we describe the ingredients of this construction, but first we need to recall some notions from algebraic topology: rationalization, Eilenberg-MacLane spaces, principal fibrations, and Postnikov towers themselves. Further information on this background with proofs can be found in Chapter 4 of [Hat02] or in [McC01]. One of the important ideas in Sullivan’s theory is that of replacing a connected CW complex of finite type with a “rational” space with the same rational homotopy groups: Given a CW complex of finite type M , its rationalization is a CW complex MQ together with a map i : M → MQ such that (1) πj (MQ ) ∼ = Qrj , where rj ∈ {0} ∪ N, for each j, and (2) i : M → MQ induces an isomorphism on rational homotopy. The rationalization of M is unique up to homotopy equivalence. Moreover, the process of rationalization provides a covariant functor M → MQ ,

(f : M → N ) → (fQ : MQ → NQ ).

For example, suppose that we want to construct the rationalization of the n-dimensional sphere S n , where n ≥ 2. Following Hess [Hes08], we start by taking the wedge sum of an infinite sequence of n-spheres, S1n ∨ S2n ∨ · · · ∨ Skn ∨ · · · , then attaching a sequence of (n + 1)-disks Dkn+1 via attaching maps n φk : ∂Dkn+1 → Skn ∨ Sk+1 n . The which map ∂Dkn+1 with degree one onto Skn and degree k + 1 onto Sk+1 resulting CW complex X(∞) is the direct limit of subcomplexes   n+1 X(r) = (S1n ∨ · · · ∨ Srn ) D1n+1 ∪ · · · ∪ Dr−1 .

Each X(r) has Srn as a strong deformation retract, so is an integral homology sphere, but the inclusion X(r) ⊆ X(r + 1) induces multiplication by r + 1 on H n . Since homology commutes with direct limits, H n (X(∞); Z) = Q and the inclusion i : S n ⊆ X(∞) n of the n-sphere. exhibits X(∞) as the rationalization SQ

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Unfortunately, there seems to be no mechanism for assigning to a CW complex X of finite type a CW complex which has as its homotopy groups the real homotopy groups of X. Given a finitely generated abelian group G, an Eilenberg-MacLane space K(G, n) is a CW complex which has the property that  G, if k = n, πk (K(G, n)) = 0, if k = n. In homotopy theory it is proven that such spaces exist and are unique up to homotopy equivalence, but in devine retribution for making the homotopy groups simple, the cohomology becomes complicated. The exact homotopy sequence for the path fibration of K(G, n), (3.12)

ΩK(G, n) −→ P K(G, n) ↓ ev K(G, n),

shows that the pointed loop space fiber Ω(K(G, n)) is a K(G, n − 1), when n > 1. One might hope that one could use this fibration to calculate the integer cohomology of K(Z, n) by induction on n, using the Serre spectral sequence (as described in [McC01]) for the path space fibration K(Z, n − 1) −→ P K(Z, n) ↓ ev K(Z, n), starting with K(Z, 1) = S 1 and K(Z, 2) = CP ∞ , and such a calculation was carried out for Zp coefficients when p is prime by Serre and Cartan, but the calculations are lengthy. The rational cohomology of K(Q, n) is much simpler: Proposition 3.5.1. The rational cohomology of K(Q, n) is a free dga on a single generator, H ∗ (K(Q, n); Q) ∼ = Λ(x), where Λ(x) is a polynomial algebra when n is even or an exterior algebra when n is odd. This is a standard application of spectral sequences as presented in [McC01], but can also be achieved via a simpler argument based upon a “Hirsch lemma”, as pointed out in [DGMS]. Since the direct product K(G1 , n) × K(G2 , n) is easily seen to be a K(G1 × G2 , n), we see immediately that H ∗ (K(Qr , n); Q) is a polynomial or exterior algebra on r generators according as n is even or odd.

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We can rewrite the fibration (3.12) as K(G, n − 1) −→ (contractible space) ↓ ev K(G, n),

(3.13)

the fiber being an h-group. This suggests regarding the base K(G, n) as analogous to the classifying space for this h-group. A principal fibration over M is a fibration that is pulled back from the fibration (3.13) via some map f : M → K(G, n). In fact, the Hurewicz isomorphism defines a characteristic element c ∈ H n (K(G, n); G) which corresponds to the identity under the isomorphism πn (K(G, n)) −→ G, and the correspondence f → f ∗ c ∈ H n (M ; G) provides a bijection [M, K(G, n)] → H n (M ; G), which classifies principal fibrations. By a Postnikov tower for a connected CW complex M with homology of finite type, we mean a sequence of fibrations p1

p2

p3

pj+1

M0 ←− M1 ←− M2 ←− M3 ← · · · ← Mj ←−−− Mj+1 ← · · · with M0 being a point, together with maps fj : M → Mj

which commute with projections,

pj+1 ◦ fj+1 = fj ,

and satisfy the following axioms: (1) πi (Mj ) = 0, for i > j. (2) fj : M → Mj induces an isomorphism (fj )∗ : πi (M ) → πi (Mj ) for i ≤ j. The idea for constructing the Postnikov tower is to construct Mk for large k by adjoining cells of dimension ≥ k + 1 to M to kill off homotopy of degree ≥ k + 1, thereby obtaining Mk , then adjoin additional k-cells to obtain Mk−1 , and so on, thus obtaining a sequence of inclusions Mk ⊆ Mk−1 ⊆ · · · ⊆ M1 ⊆ M0 . Finally, one transforms theses inclusions into a tower of fibrations which is the Postnikov tower. (See page 251 of [BT82] for more details; more complete treatments are given in [GriM81] or Chapter 4 of [Hat02].) Proposition 3.5.2. If M is simple, each projection pj+1 : Mj+1 → Mj in the Postnikov tower is a principal fibration with fiber K(πj+1 (M ), j + 1),

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which is pulled back via a map kj : Mj → K(πj+1 (M ), j + 2) from the path space fibration K(πj+1 (M ), j + 1) −→ (contractible space) ↓ ev K(πj+1 (M ), j + 2). This is proven via obstruction theory in Chapter VI of [DGMS]. Alternate proofs are found in [Hat02] and [McC01]. Although elegant, the Postnikov tower suffers from a defect. To construct the Postnikov tower of a CW complex M , one needs to calculate all of its homotopy groups, an impossible task. But the problem becomes considerably easier when we follow Quillen and Sullivan into the realm of rational homotopy theory and first replace M with its rationalization MQ . In contrast to M , the Postnikov tower for MQ is easily calculated. It is now possible to understand how the construction we gave at the beginning of the chapter for the minimal model of M is just an algebraic realization of the Postnikov tower of the rationalization MQ . Suppose that M is simply connected for simplicity. Then we can start the rational Postnikov tower at M2 = K(π2 (M ) ⊗ Q, 2) which has as its cohomology the free dga A2 (M ) = P [x1 , . . . , xm ] = P [V 2 ] where

V 2 = Hom(π2 (M ), Q)

and x1 , . . . , xm is a basis for the rational vector space V 2 . At the next stage we consider the principal fibration p3

K(π3 (M ) ⊗ Q, 3) → M3 −→ M2 , which is the pullback of (3.14)

p

→ K(π3 (M ) ⊗ Q, 4) K(π3 (M ) ⊗ Q, 3) → (contractible space) −

via a map k : M2 → K(π3 (M ) ⊗ Q, 4). Recall that the cohomology of K(π3 (M ) ⊗ Q, 4) is an exterior algebra on n + l generators, where n + l is the dimension of the vector space π3 (M ) ⊗ Q. In the universal fibration these are represented by cochains β1 , . . . , βn+l on K(π3 (M ) ⊗ Q, 4) which pull back to the total space, where they are exact, p∗ βi = dαi , with αi restricting to a generator for the cohomology of K(π3 (M ) ⊗ Q, 3). These cochains pull back to cochains β¯i and yi on M2 and M3 , respectively, with p∗ β¯i = dyi . We can choose the representatives in such a way that  crij xi ∧ xj for n + 1 ≤ r ≤ n + l. dy1 = · · · = dyn = 0 and dyr =

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We now set A3 (M ) = P [x1 , . . . , xm ] ⊗ Λ(y1 , . . . yn+l ) = A2 (M ) ⊗ Λ(V 3 ), where V 3 = Hom(π3 (M ), Q). A3 (M ) has as its cohomology the cohomology of M3 and we define a dga map ρ3 : A3 (M ) → H ∗ (M ; Q), which induces an isomorphism on cohomology in degrees ≤ 3 and injection in degree four. Then A3 (M ) is an elementary extension of A2 (M ) in accordance with the following: Definition. Suppose that (A, dA ) is a dga over F , where F = Q or R. We say that a dga (B, dB ) is an elementary extension of (A, dA ) if B = A⊗Λ(V ) where Λ(V ) is a free graded commutative algebra generated by a finitedimensional vector space V over F of a given degree and dB |A = dA ,

dB (V ) ⊆ A.

Having constructed A3 (M ) we next follow the same procedure with the principal fibration p4

K(π4 (M ) ⊗ Q, 4) → M4 −→ M3 , thereby obtaining A4 (M ), an elementary extension of A3 (M ), and so forth. In fact, corresponding to the Postnikov tower, we obtain a sequence of elementary extensions of dga’s. At stage k, the new algebra generators added to form Ak (M ) are in one-to-one correspondence with a vector space basis for Hom(πk (M ), Q). But this process yields a rational minimal model, not a real minimal model. To put that in its proper context, one can give M a simplicial decomposition, at least up to homotopy. In the case of our mapping spaces, this can be done by providing our manifold with a Leray open cover constructed from geodesically convex sets in the range as described in §1.13; the nerve of the covering is then a simplicial complex homotopy equivalent to M . One then constructs a cochain complex Ω∗Q (M ) of rational differential forms. The differential forms in this complex are required to be piecewise linear, linear on each simplex with rational coefficients in terms of barycentric coordinates. Sullivan modifies the proof of de Rham’s theorem to show that Ω∗Q (M ) computes the rational cohomology of M . Modern treatments (as presented in [FHT00] for example) dispense with the triangulation; if M is a CW complex, one can construct a corresponding singular simplicial set and construct the cochain complex of rational differential forms as a functor from simplicial sets instead. A connected CW complex M has a Postnikov tower in which each stage is the pullback of a principal fibration exactly when M is simple (see Theorem 4.69 in Hatcher [Hat02]). But the class of simple CW complexes is not

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closed under the operation of taking function spaces: if M is simple, it is not always the case that Cf0 (Σ, M ) is simple for Σ a compact manifold and f : Σ → M a continuous map. Fortunately, there is a more general class of spaces which is closed under the construction M → (some component of C 0 (Σ, M ), and for which each stage in the Postnikov tower admits a finite refinement into tower or principal fibrations, the class of nilpotent spaces of finite type. Thus for constructions of minimal models, we can weaken the hypothesis that M be simple. Recall that the lower central series of π1 (M ) is defined inductively by G1 = π1 (M ),

G2 = [G1 , π1 (M )],

G3 = [G2 , π1 (M )], . . .

and π1 (M ) is nilpotent exactly when Gn = {1}, where 1 is the identity element, for some n ∈ N. If π1 (M ) is nilpotent, we can set Ni = π1 (M )/Gi , and obtain a tower of quotients · · · → N4 → N3 → N2 =

π1 (M ) , [π1 (M ), π1 (M )]

which stabilizes at π1 (M ) after finitely many stages, yielding a finite refinement of the first stage of the Postnikov tower into a tower of principal fibrations. Similarly, the fundamental group π1 (M ) acts nilpotently on the homotopy group πk (M ) if there exists n ∈ N such that [γ] ∈ π1 (M )

⇒

([γ] − id)n : πk (M ) → πk (M )

is the zero map. We say that a pathwise connected CW complex M of finite type is nilpotent if its fundamental group π1 (M ) is nilpotent, and it acts nilpotently on the homotopy groups πk (M ) for all k ≥ 2. As explained in [McC01] (see page 353), the nilpotent spaces are those spaces which have a Postnikov tower each stage of which has a finite refinement into principal fibrations, and this is enough for the construction of a minimal model. Moreover, if M is a nilpotent space, then according to a theorem of G. W. Whitehead (see [McC01], page 347), Cf0 (Σ, M ) is also nilpotent for any f : Σ → M , when Σ is a compact manifold or a finite CW complex. Definition. A Sullivan model of a nilpotent CW complex M of finite type is a connected dga (A(M ), d) over Q, together with a dga map φ : A(M ) → Ω∗Q (M ), where Ω∗Q (M ) is the de Rham complex of M , which satisfies the

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following axioms: (1) A(M ) is the union of a countable collection {An : n ∈ N} of dga’s, which are arranged in a series Q = A0 ⊆ A1 ⊆ A2 ⊆ · · · ⊆ An ⊆ An+1 ⊆ · · · , such that each An+1 is an elementary extension of An . (2) φ induces an isomorphism on cohomology. The Sullivan model is said to be a (rational) minimal model of M if in addition it satisfied the condition d(A+ ) ⊆ A+ · A+ , where A+ denotes the ideal in A(M ) consisting of elements of positive degree. For example, suppose that (A(M ), dA ) is the minimal model of a nilpotent CW complex of finite type and that x and y are two new generators of degree d ≥ 1 and d + 1. Then B(M ) = A(M ) ⊗ Λ(x, y), with differential dB defined by dB |A = dA , dB x = y, dB y = 0, is not a minimal model, but a Sullivan model of M . Sullivan models are often useful as steps toward constructing minimal models, as we will see later. Theorem 3.5.3 (Sullivan). Every nilpotent CW complex of finite type possesses a rational minimal model, unique up to dga isomorphism, the algebra generators in degree k being a basis for the rational vector space Hom(πk (M ), Q), when k ≥ 2. This is proven in [DGMS] using the Postnikov tower described above. Example 3.5.4. Examples of nilpotent manifolds which are not simple are provided by the nilmanifolds. To construct the simplest of these, we can start with the Heisenberg group H3 (R), the Lie group of matrices ⎛ ⎞ 1 x z ⎝0 1 y ⎠ , with x, y, z ∈ R, 0 0 1 and let M be its quotient by ⎛ 1 m ⎝0 1 0 0

H3 (Z), the discrete group of matrices ⎞ p n⎠ , with m, n, p ∈ Z. 1

This quotient M is compact and its fundamental group π1 (M ) is the discrete group H3 (Z), with lower central series G1 = H3 (Z),

G2 = [π1 (M ), π1 (M )] ∼ = Z,

G3 ∼ = {I}.

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The Lie algebra of the Heisenberg group H3 (R) has basis {X, Y, Z} with bracket [X, Y ] = Z, [X, Z] = 0, [Y, Z] = 0, and the dual one-forms {θX , θY , θZ } satisfy dθX = 0,

dθY = 0,

dθZ = −θX ∧ θY .

Since H1 (M ; Z) is the abelianized fundamental group, H1 (M ; Z) ∼ = Z ⊕ Z and H 1 (M ; R) ∼ = R ⊕ R, the latter group being generated by θX and θY . Thus our algorithm for constructing the minimal model starts with ∼ Λ(θX , θY ), A1,1 = H 1 (M ; R) = and continues with the elementary extension A1,2 = A1,1 ⊗ Λ(θZ ). Since all the homotopy is accounted for, the process now stops and the minimal model of M is A1,2 . Summary. There is a class of CW complexes which contains the simply connected manifolds M of finite type and is closed under the process M → Map(Σ, M ) of taking mapping spaces, when Σ is a given compact smooth manifold. The elements of this class are the nilpotent CW complexes of finite type. Any such space M has a rationalization MQ , which possesses a Postnikov tower, which in turn can be represented via a tower of elementary extensions of dga’s. The inverse limit of the tower of dga’s is the minimal model A(M ) of M , and faithfully reflects the homotopy theory of the rationalization. The generators of the minimal model can be thought of as rational cochains or rational differential forms which live in some finite stage of the Postnikov tower. Of course, some of these forms may go to zero under the map φ : A(M ) → Ω∗Q (M ) to the rational de Rham complex of M itself.

3.6. Maps from surfaces Recall from §1.1, that our ultimate goal is to develop a critical point theory for nonlinear differential equations, not just the ordinary differential equations which occur in classical mechanical systems, but partial differential equations such as the equation for pseudoholomorphic curves, or the equations for harmonic maps or minimal surfaces. To formulate Morse inequalities (or partial Morse inequalities) for partial differential equations, we would often need to study the space Map(Σ, M ), which is one of the completions of the space C ∞ (Σ, M ) of C ∞ maps from a compact manifold Σ to a smooth manifold M , all of these spaces being homotopy equivalent to

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C 0 (Σ, M ) when the completion is suitably strong (see §1.5). For such Morse inequalities to be useful we would like to be able to calculate the homology or cohomology of the space Map(Σ, M ) or the component Mapf (Σ, M ) which contains a given map f : Σ → M , and we claim that the rational cohomology of these spaces is accessible via Sullivan’s method of minimal models. To calculate the homology of Map(Σ, M ), we first construct a CW decomposition of Σ, with Σ(i) denoting the i-skeleton. We can then try to build up the homology of Map(Σ(i) , M )) by induction on i via a sequence of fibrations Ωi+1 (M ) × · · · × Ωi+1 (M ) −→ Map(Σ(i+1) , M ) ↓ ev Map(Σ(i) , M ), where Ωi+1 (M ) denotes the (i + 1)-fold iterated pointed loop space of M and the number of Ωi+1 (M )’s is the number of (i + 1)-cells in the CW decomposition of Σ. The evaluation map ev in this diagram is defined via the attaching maps of the (i + 1)-cells. But it is the homotopy groups, not the homology groups, which lie in an exact sequence for a fibration. This suggests the idea which underlies Thom’s approach to the topology of mapping spaces presented in 1956 (see [Tho56]): Encode the homotopy type of M in a Postnikov tower for M , then use this tower to construct corresponding Postnikov towers for each of the spaces Map(Σ(i) , M ) by induction on i, and finally calculate the cohomology from the tower we obtain for Map(Σ, M ). This is almost always impossible to carry out completely for integer cohomology because the homotopy groups of M are almost never known. But it is relatively easy to calculate homotopy of the spaces in the tower over the rationals or reals, and we have seen that the theory of minimal models describes the rational cohomology of a nilpotent space in terms of a de Rham theory for its Postnikov tower. Based on this idea, as well as later work of Sullivan and Haefliger, Brown and Szczarba [BSz97] give an algorithm which uses the minimal model of a nilpotent manifold M of finite type to construct the minimal model of Mapf (Σ, M ), and this in turn yields the rational cohomology of Mapf (Σ, M ), at least in principal. In this section, we describe how to carry out that algorithm for maps from a compact connected surface Σ into M in a form suitable for applications to the theory of two-dimensional minimal surfaces in an ambient Riemannian manifold, or Sobolev completions for suitable “perturbed” energies, such as the α-energy to be discussed in the next chapter. Further references to a rich literature in the topology of mapping spaces can be found in [Smi10].

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The function spaces needed for harmonic maps and minimal surfaces get further from “Sobolev range” as the dimension of Σ increases. It is when Σ is a surface that we are at least on the border of Sobolev range, and we can hope to see the most definitive results in this case. Indeed, an equivariant Morse theory for the space of connections over a Riemann surface was developed by Atiyah and Bott [AB83] for the YangMills equation over a Riemann surface, in which the corresponding function space has a Riemann surface as domain. (Topological background for the Atiyah-Bott theory will be described briefly in Example 3.6.7.) In this case, it is possible to circumvent much of the analysis because the Yang-Mills function has a large symmetry group, while the theory we develop later focuses on the generic situation in which the symmetry group acting on Map(Σ, M ) is as small as possible. Thus in this section, we focus our attention on the rational homotopy theory of Map(Σ, M ) or Mapf (Σ, M ) when Σ is a surface, even though much of the algebraic theory works when the domain Σ has higher dimension. 3.6.1. The space Map(Σ, M ) when π1 (M ) = π2 (M ) = 0. We first consider first the case where π1 (M ) = π2 (M ) = 0, which implies that Map(Σ, M ) has only one component. If Σ is the torus T 2 , we can construct the minimal model for M = Map(Σ, M ) by first choosing a basis (α, ¯ α ˆ ) for π1 (T 2 ) which singles out a preferred product decomposition T 2 = S 1 × S 1 , and then regard M as an iterated free loop space, Map(T 2 , M ) = Map(S 1 , Map(S 1 , M )). The exact homotopy sequence for the fibration Map(T 2 , M ) −→ Map(S 1 , M ) splits, and since Map(S 1 , M ) is simply connected, we can iterate the construction of the minimal model for the free loop space considered in §3.2. Thus, if x1 , x2 , . . . , xk , . . . are generators for the minimal model A(M ) of M , we get corresponding generators ˆ¯1 , x ˆ¯2 , . . . , x ˆ¯k , . . . ¯1 , x ¯2 , . . . , x ¯k , . . . , x ˆ1 , x ˆ2 , . . . , x ˆk , . . . , x x1 , x2 , . . . , xk , . . . , x for A(Map(T 2 , M )), with ˆi = deg xi − 1, deg x ¯i = deg x

ˆ¯i = deg x1 − 2. deg x

This time we have two operations, bar and hat, which are extended to be ¯ and Δ ˆ which anticommute with each other. The right skew-derivations Δ differential d on the minimal model A(Map(T 2 , M )) is defined by requiring

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that it extend the differential on A(M ), and that it commute with each of these skew-derivations, ¯ =Δ ¯ ◦ d, d◦Δ

ˆ =Δ ˆ ◦ d. d◦Δ

The preferred basis basis (α, ¯ α ˆ ) determines a corresponding coordinate system on T 2 , and two fibrations p¯ : Map(T 2 , M ) −→ Map(S 1 , M ),

where

pˆ : Map(T , M ) −→ Map(S , M ), 2

1

p¯(f )(t) = f (t, 0), where

pˆ(f )(t) = f (0, t),

which possess sections (or right inverses) s¯ : Map(S 1 , M ) −→ Map(T 2 , M ),

where

sˆ : Map(S , M ) −→ Map(T , M ), 1

2

s¯(γ)(t1 , t2 ) = γ(t1 ), where

sˆ(γ)(t1 , t2 ) = γ(t2 ).

This implies that the cohomology of Map(S 1 , M ) pulls back injectively via either p¯∗ or pˆ∗ to a direct summand of the cohomology of Map(T 2 , M ). We arrange that the pullback via p¯ of the minimal model of Map(S 1 , M ) is ¯i ’s, while the pullback via pˆ is generated by the generated by the xi ’s and x ˆi ’s. xi ’s and x Just as in the case of Map(S 1 , M ), the evaluation map ev : Map(T 2 , M ) × T 2 −→ M determines a map of minimal models, ev∗ : A(M ) −→ A(Map(T 2 , M )) ⊗ A(T 2 ). ¯ θ) ˆ on Here the minimal model A(T 2 ) of the torus is the exterior algebra Λ(θ, 1 two elements of degree one. By comparison with the case of Map(S , M ), we find that if x is one of the generators of the minimal model for M , then (3.15)

ˆ ˆ¯ ⊗ θ¯ ∧ θ. ev∗ (x) = x ⊗ 1 + x ¯ ⊗ θ¯ + x ˆ ⊗ θˆ − x

The formula makes it clear how the minimal model must transform under change of basis for π1 (T 2 ), say        α ¯ a b α ¯ a b → , where ∈ SL(2, Z). α ˆ c d α ˆ c d Indeed, we see immediately that the corresponding action on the generators is given by      a b x ¯i x ¯i → , x ˆi x ˆi c d ˆi transform to themselves. This determines the action of ¯ while xi and x SL(2, Z) on the entire minimal model.

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Remark 3.6.1. In analogy with Remark 3.2.3, there are versions of the ¯ Δ ˆ and Δ ˆ ◦Δ ¯ which act on rational cochains or rational difoperators Δ, ferential forms. For example, if ω is an element of Ωk (Map(T 2 , M )), we let ˆ ◦ Δ(ω) ¯ Δ denote the result of integrating ev∗ (ω) over the fiber of π : Map(T 2 , M ) × S 1 × S 1 −→ Map(T 2 , M ). ˆ ◦ Δ(ω) ¯ If ω is a de Rham representative for ρ(x), then Δ is a de Rham ˆ ¯ representative for ρ(Δ ◦ Δ(x)). If Σ is the two-sphere S 2 , the fibration Map(S 2 , M ) → S 2 has a section s : S 2 −→ Map(S 2 , M ) defined by

s(p) = (constant map at p),

and hence the exact homotopy sequence of the fibration splits, and this implies that πk (Map(S 2 , M )) ∼ = πk (M ) ⊕ πk (Ω2 (M )) ∼ = πk (M ) ⊕ πk+2 (M ). The minimal model of Map(S 2 , M ) must therefore be the product of free algebras on elements ˆ ˆ¯2 , . . . , x ˆ¯k , . . . , ¯1 , x x1 , x2 , . . . , xk , . . . , x which are dual to generators of the real homotopy groups of M and Ω2 (M ), respectively. To determine the minimal model for Map(S 2 , M ), we need only determine what the differential d does to these generators. However, we can regard the torus as a sphere with one handle, and there is a degree one collapsing map g : T 2 → S 2 which pinches the handle to a point, and induces a continuous map αg : Map(S 2 , M ) → Map(T 2 , M ). The minimal model for Map(T 2 , M ) must be taken to the minimal model for Map(S 2 , M ) by the dga map αg∗ . This fact completely determines the differential d on generators. ˆ ◦Δ ¯ on A(Map(T 2 , M )) pulls On the other hand, the derivation Δ2 = Δ 2 back via αg to a derivation Δ of degree −2 on A(Map(T 2 , M )) such that ˆ¯, Δ2 (x) = x and the differential d is also completely determined directly from the analog of (3.15), (3.16)

ˆ¯ ⊗ Θ, ev∗ (x) = x ⊗ 1 − x

where Θ is the volume form on the two sphere for the constant curvature metric of total area one. The derivation Δ2 is also determined on rational cochains or rational differential forms, with Δ2 (ω) being minus the integral over the fiber of ev∗ (ω).

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Example 3.6.2. Suppose that M = S 2n+1 where n ≥ 1, an odd-dimensional sphere. Then the minimal model for M is A(M ) = Λ(x), and exterior algebra on an element x of degree 2n + 1, with trivial differential. The minimal model of the free loop space in this case is A(Map(S 1 , M )) = Λ(x) ⊗ P [˜ x],

where

deg(˜ x) = deg(x) − 1,

the tensor product of an exterior algebra and a polynomial algebra, again with trivial differential. The minimal models of the spaces of maps from T 2 and S 2 into M are ˆ¯) A(Map(T 2 , M )) = Λ(x) ⊗ P [¯ x] ⊗ P [ˆ x] ⊗ Λ(x and ˆ¯), A(Map(S 2 , M )) = Λ(x) ⊗ Λ(x with trivial differential in both cases. For this choice of range, the minimal models are isomorphic to the de Rham cohomology. Example 3.6.3. Suppose next that M = S 2n , an even-dimensional sphere, where n ≥ 2. We have seen that the minimal model for M is A(M ) = P [x] ⊗ Λ(y), a polynomial algebra on an element x of degree 2n tensored with an exterior algebra on an element y of degree 4n − 1, with differential d satisfying dx = 0, dy = x2 . Moreover, the algorithm for the minimal model for the free loop space in this case yields x) ⊗ P [˜ y ], A(C 0 (S 1 , M )) = P [x] ⊗ Λ(y) ⊗ Λ(˜ with the differential defined by dx = 0,

dy = x2 ,

d˜ x = 0,

d˜ y = 2x˜ x.

We can iterate the algorithm to obtain the minimal model of the iterated free loop space, ˆ¯] ⊗ Λ(yˆ¯), A(Map(T 2 , M )) = P [x] ⊗ Λ(y) ⊗ Λ(¯ x) ⊗ P [¯ y ] ⊗ Λ(ˆ x) ⊗ P [ˆ y ] ⊗ P [x with differential satisfying the rules dx = 0,

dy = x2 ,

d¯ x = 0,

dˆ x = 0,

d¯ y = 2x¯ x,

dˆ y = 2xˆ x,

ˆ¯ = 0, dx

ˆ¯. dyˆ¯ = 2¯ xx ˆ + 2xx

We conclude that the minimal model for the space of maps from S 2 into M is ˆ¯] ⊗ Λ(yˆ¯), A(Map(S 2 , M )) = P [x] ⊗ Λ(y) ⊗ P [x with differential satisfying dx = 0,

dy = x2 ,

ˆ¯ = 0, dx

ˆ¯. dyˆ¯ = 2xx

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161

More generally, for a compact connected oriented surface Σ of genus g, we can give a systematic treatment based upon a CW decomposition with one zero-cell p0 , 2g one-cells σ1 , . . . , σg and τ1 , . . . , τg , and one two-cell Σ0 , such that the homology classes corresponding to the one-cells satisfy the following conditions on their intersection numbers: [σi ] · [σj ] = 0 = [τi ] · [τj ],

[σi ] · [τj ] = δij ,

1 ≤ i, j ≤ g.

We let Σ(i) denote the i-skeleton of Σ in this CW decomposition and calculate the topological invariants of Map(Σ(i) , M ) by induction on i, starting with Map(Σ(0) , M ) = M . If Ω1 (M ) and Ω2 (M ) denote the h-spaces of pointed maps of the circle and the two-sphere into M , there are two fibrations connecting these spaces, (3.17)

Ω1 (M ) × · · · × Ω1 (M ) −→ Map(Σ(1) , M ) ↓ ev M,

the 2g factors of Ω1 (M ) corresponding to the 2g one-cells, and Ω2 (M ) −→ (3.18)

Map(Σ, M ) ↓ ev Map(Σ(1) , M ).

In both fibrations, the vertical maps are defined by restricting an element f : Σ(i) → M to a lower-dimensional skeleton. Note that Ω1 (M ) and Ω2 (M ) are connected because of our assumption π1 (M ) = π2 (M ) = 0. Moreover, the fibration (3.17) has a section s : Σ −→ Map(Σ(1) , M )

defined by

s(p) = (constant map at p),

while the homotopy exact sequence for the fibration the fibration (3.18) splits, since the boundary map factors through πk−1 (C 0 (S 2 , M )), πk (Ω2 (M )) → πk−1 (Map(S 2 , M )) → πk−1 (Map(Σ, M )), the second map being induced by the map αg : Σg → S 2 which collapses all the handles on Σg to a point. If Σg is a sphere with g handles, then corresponding to each generator x for the minimal model of M , there are 2g + 2 generators for the minimal model of Map(Σg , M ), the space of maps from Σg to M : ˆ¯, ˆ1 , x ¯2 , x ˆ2 , . . . x ¯g , x ˆg , x x, x ¯1 , x ¯g corresponding to the one-cells σ1 , . . . , σg and x ˆ1 , . . . , x ˆg corwith x ¯1 , . . . , x responding to the one-cells τ1 , . . . , τg in the standard CW decomposition of Σg . The differential on the minimal model can be inferred from naturality under maps Σg → T 2 which pinch all but one of the handles to a point.

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Example 3.6.4. If M = S 2n+1 , where n ≥ 1, then ˆ¯), A(Map(Σg , M )) = Λ(x) ⊗ P [¯ x1 , . . . , x ¯g ] ⊗ P [ˆ x1 , . . . , x ˆg ] ⊗ Λ(x with trivial differential. On the other hand, if M = S 2n , where n ≥ 2, then ˆ¯) ¯1 , . . . , x ¯g , x ˆ1 , . . . , x ˆg , x A(Map(Σg , M )) = Λ(x, x ⊗ Λ(y, y¯1 , . . . , y¯g , yˆ1 , . . . , yˆg , yˆ¯), where Λ now denotes a free algebra with some exterior algebra generators and some polynomial generators, and the differential is defined by dx = 0,

dy = x2 , dˆ xi = 0,

d¯ xi = 0, dˆ yi = 2xˆ xi ,

d¯ yi = 2x¯ xi , ˆ¯ = 0, dx

dyˆ¯ = 2



ˆ¯, x ¯i x ˆi + 2xx

where 1 ≤ i ≤ g. 3.6.2. The space Map(Σ, M ) when π1 (M ) = 0 but π2 (M ) = 0. We next relax the assumption that π2 (M ) = 0, but retain the assumption that M be simply connected. In this case the components of the space Map(Σ, M ) are in one-to-one correspondence with π2 (M ) = π0 (Map(Σ, M )), and our goal is to describe the minimal model of Mapf (Σ, M ) = {g ∈ Map(Σ, M ) : g is homotopic to f }, for a given choice of f : Σ → M . Note first that any smooth map f : Σ → M induces a linear function f ∗ : V 2 = Hom(π2 (M ), R) = Hom(H2 (M ; Z)), R) −→ R by

f ∗ (x) = x([f ]),

where [f ] is the homology class of f . As in Remark 3.6.1, integration over the fiber shows that on the component C 0 (Σ, M ) which contains f , ˆ x ¯ = x([f ]) ∈ Q for x ∈ V 2 . We claim that the minimal model for the component Map0 (T 2 , M ) of maps from the torus which contains the constant maps is calculated via ˆ¯ = 0 the same algorithm as before, modified by setting the generators x for x ∈ V 2 . On the other hand, we obtain a Sullivan model (but not a minimal model) for the component Mapf (T 2 , M ) which contains a map f ˆ¯, for each x ∈ V 2 , representing a nontrivial homology class by replacing x with x([f ]) ∈ Q.

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163

The minimal model of Map0 (Σ, M ) is very similar to the case where π2 (M ) = 0, the main difference being that we replace the fibration (3.18) by Ω20 (M ) −→ (3.19)

Map0 (Σ, M ) ↓ ev Map(Σ(1) , M ),

where Ω20 (M ) is the identity component in the second loop space. In the case where Σ = S 2 , (3.19) still has a section provided by the constant maps from S 2 into M which splits the homotopy sequence of the fibration. Use of the map αg : Σg → S 2 as before then shows that the exact homotopy sequence splits for Σg , for any genus g. In particular, a nonzero generator x of degree ˆ¯ for A(Ω20 (M )). On at least three for A(M ) goes to a nonzero generator x ˆ¯ = 0. the other hand, if the generator x has degree two, we must set x However, although Map0 (Σ, M ) is simple when π1 (M ) = π2 (M ) = 0, it is often only nilpotent when π1 (M ) = 0 but π2 (M ) = 0. Example 3.6.5. Suppose that we want to calculate the minimal model of spaces of maps from Σ to a compact simply connected four-manifold M . As a first step, we consider the simpler case of maps to M0 , the noncompact manifold obtained from M by taking out a point. It is well-known (see the proof of Theorem 1.2.1 in [DK90]) that M0 is a wedge sum of two-spheres, m

!" # 2 ∼ M0 = S ∨ S ∨ · · · ∨ S , 2

2

and the two-spheres determine a dual basis x1 , . . . , xm for V 2 . There is no cohomology in dimension three, so we must kill the products xi xj by elements yij ∈ V 3 such that dyij = xi xj ,

for

1 ≤ i ≤ j ≤ m,

and hence V 3 must have dimension (1/2)m(m + 1). Then Map(S 1 , M0 ) has abelian fundamental group, and the degree one generators of the minimal model for Map0 (T 2 , M0 ) are x ¯1 , . . . , x ¯m , x ˆ1 , . . . , x ˆm

and

yˆ¯ij ,

for

1 ≤ i ≤ j ≤ m,

and this is the dual basis to what is called the “de Rham fundamental group” ˆ ¯i = 0, of Map0 (T 2 , M0 ). Since x xj x ˆk , dyˆ ¯ij = 2¯ and we conclude that the de Rham fundamental group of Map0 (T 2 , M0 ) has nonabelian Lie algebra, and hence is nonabelian. We see that Map0 (T 2 , M0 ) is nilpotent of class two in this case.

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3. Topology of Mapping Spaces

To get the minimal model of M instead of M0 , we must impose the relation  Q≡ aij yij = 0 on the generators of V 3 , where Q : H2 (M ; Q) × H2 (M ; Q) −→ Q is the intersection form of the compact four-manifold M . This in turn gives a relation on the generators yˆ ¯ij which can be used to help determine the low degree cohomology of the space Map0 (T 2 , M ). We now turn to the case where f : Σ → M is homotopically nontrivial, and in this case it is no longer true that the long exact homotopy sequence for the fibration Ω20 (M ) −→ Mapf (Σ, M ) ↓ ev (3.20) Map(Σ(1) , M ) necessarily splits. We will construct a Sullivan model of the total space when Σ = S 2 and the above fibration simplifies to (3.21)

Ω20 (M ) −→ Mapf (S 2 , M ) ↓ ev M.

One could determine the Sullivan model for the more general case (3.20) by using the map αg : Σ → S 2 , which pinches all the handles to points, to construct a corresponding map from the fibration (3.21) to (3.20). We can pull the generators x1 , x2 , . . . , xk , . . . for the minimal model A(M ) of M up to Mapf (S 2 , M ) and use the formula (3.16) to determine ˆ ¯i be the result of integrating ev∗ (xi ) over the fiber of ev∗ (xi ). We let x π : Map(S 2 , M ) × S 2 −→ Map(S 2 , M ). When the degree k of xi is at least three, we get a representative of an ˆ element (also denoted by x ¯i ) of the image πk−2 (Ω20 (M )) −→ πk−2 (Mapf (S 2 , M )), which serves as a generator for a Sullivan model of Mapf (S 2 , M ). When the degree of xi is two, we get instead an element x([f ]) of the ground field Q. We claim that the resulting free dga B(Mapf (S 2 , M )) is a Sullivan model for Mapf (S 2 , M ), that is, that the inclusion gives a map ρ : B(Mapf (S 2 , M )) −→ Ω∗Q (Mapf (S 2 , M )) which induces an isomorphism on cohomology. To see this, we give the dga B(Mapf (S 2 , M )) a bigrading by assigning to each generator xi the bidegree

3.6. Maps from surfaces

165

(deg xi , 0) and to the corresponding element x ¯ˆi the bidegree (0, deg xi − 2) when the degree of xi is at least three. We say that an element of bidegree (s, t) has base degree s and fiber degree t, and let E1s,t = A(M )s ⊗ A(Ω20 (M ))t denote the space of elements of bidegree (s, t) in B(Mapf (S 2 , M )). We define a filtration by setting F s = {x ∈ B(Mapf (S 2 , M )) : the base degree of x is ≥ s}, and note that dF s ⊆ F s+1 , so the filtration yields a first quadrant spectral sequence with E2p,0 giving the cohomology of M and E20,q the cohomology of Ω20 (M ). By Zeeman’s Comparison Theorem (Theorem 3.26 in [McC01]), we conclude that ρ gives an isomorphism from the cohomology of B to the cohomology of Mapf (S 2 , M ), as claimed. From the Sullivan model B(Mapf (S 2 , M )) we can then obtain the minimal model by eliminating generators which cancel, and it turns out that this cancellation can be described by the boundary map in the exact homotopy sequence of the fibration (3.21); the following example will explain how this works in practice. Example 3.6.6. As in the preceding example, we consider the case of a simply connected four-manifold M0 which can be represented as m

!" # 2 ∨ S ∨ · · · ∨ S . S M0 ∼ = 2

2

As basis for V 2 we take elements x1 , . . . , xm dual to the two-spheres once again. As basis for V 3 we take yij ,

for 1 ≤ i ≤ j ≤ m,

with dyij = xi xj ,

for V 4 we take zijk ,

for

1 ≤ i ≤ m,

1 ≤ j ≤ k ≤ m,

with dzijk = xi yjk ,

for V 5 we take uijkl ,

for

1 ≤ i, j ≤ m,

1 ≤ k ≤ l ≤ m,

with

duijkl = xi zjkl ,

and so forth. We can suppose that the basis x1 , . . . , xm has been chosen so that ˆ¯1 = x1 ([f ]) = 1, x

ˆ x ¯r = xr ([f ]) = 0,

for

2 ≤ r ≤ m.

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3. Topology of Mapping Spaces

Then with the index convention 2 ≤ r ≤ m, our formula for the differential yields ˆ¯1jkl = zjkl , . . . , ¯1r = xr , dzˆ ¯1jk = yjk , du (3.22) dyˆ¯11 = 2x1 , dyˆ ˆ¯rjkl = 0, . . . . dyˆ¯rs = 0, dzˆ¯rjk = 0, du To obtain the minimal model, one simply eliminates the generators ˆ ¯1jk , u ¯1jkl , . . . , x1 , yjk , zjkl , . . . , yˆ ¯1j , zˆ which cancel in cohomology. When f is homotopically nontrivial, Mapf (S 2 , M0 ) has a relatively simple minimal model, generated by ˆ¯rjkl , . . . , ¯rjk , u yˆ ¯rs , zˆ with trivial differential. We now apply the functor Hom(·, Q) to the exact homotopy sequence of the fibration (3.21), (3.23)

· · · −→ πk (Mapf (S 2 , M )) −→ πk (M )−→πk−1 (Ω20 (M )) −→ · · ·

and obtain another exact sequence in which the dual to the boundary map ∂ is what we call the “transgression”, τ : Hom(πk−1 (Ω20 (M )), Q) −→ Hom(πk (M ), Q). Note that Hom(πk−1 (Ω20 (M )), Q) is the space of degree k − 1 generators for the minimal model of Ω20 (M ) while Hom(πk (M ), Q) is the space of degree k generators for the minimal model of M . Moreover, it follows from (3.22) that         ˆ¯1jkl = zjkl , . . . , τ yˆ¯11 = 2x1 , τ yˆ ¯1r = xr , τ zˆ ¯1jk = yjk , τ u while the transgression vanishes on ˆ¯rjkl , . . . . ¯rjk , u yˆ ¯rs , zˆ The boundary map in (3.23) can also be described in terms of Whitehead products (see §2.1 of [Smi10]). Example 3.6.7. Recall that any principal bundle P over the compact Riemann surface Σ with compact structure group G is the pullback of a universal principal bundle over the classifying space BG for G, and in fact the isomorphism classes of such principal bundles are in one-to-one correspondence with the space [Σ, BG] of homotopy classes of maps from Σ into BG. In §2 of [AB83], Atiyah and Bott show that although the space A of G-connections in a given principal bundle P is contractible, its quotient by the group G of gauge transformations A has the homotopy type of MapP (Σ, BG), G

3.6. Maps from surfaces

167

where MapP (Σ, BG) is the component of Map(Σ, BG) corresponding to P . The rational cohomology of this space is easily calculated by the techniques just described. To illustrate, we can take the case where G = U (2), in which case the minimal model of BG is a polynomial algebra on two generators, A(SU (2)) = P [x, y] = P [c1 , c2 ], where x = c1 and y = c2 are the first and second Chern class of the universal bundle, with trivial differential, where ci has degree 2i. If Σg is a compact Riemann surface of genus g, then the minimal model of the space of connections modulo gauge equivalence is A(MapP (Σg , BSU (2))) = P [x] ⊗ Λ(¯ x1 , . . . , x ¯g ) ⊗ Λ(ˆ x1 , . . . , x ˆg ) y1 , . . . , yˆg ) ⊗ P [yˆ¯], ⊗ P [y] ⊗ Λ(¯ y1 , . . . , y¯g ) ⊗ Λ(ˆ ˆ¯ = 0. with trivial differential, where we have set x

Chapter 4

Harmonic and Minimal Surfaces

4.1. The energy of a smooth map Suppose now that Σ is a compact smooth Riemannian manifold of dimension m. In terms of local coordinates (x1 , . . . , xm ) on Σ we can write the Riemannian metric and area or volume element on Σ as m   ηab dxa ⊗ dxb and dA = det(ηab )dx1 · · · dxm . ds2 = a,b=1

If M is a second Riemannian manifold of dimension n isometrically imbedded in Euclidean space RN and f : Σ → M ⊆ RN is a smooth map, the energy density of f at a given point is given in terms of local coordinates by the formula m  ∂f ∂f 1 η ab · , e(f ) = |df |2 , where |df |2 = 2 ∂xa ∂xb a,b=1

(η ab )

denotes the matrix inverse to (ηab ), and the dot denotes the Euclidean dot product in RN . The energy of a smooth map f : Σ → M ⊆ RN is given by the Dirichlet integral    m 1 ∂f ∂f  e(f )dA = η ab · det(ηab )dx1 · · · dxm , (4.1) E(f ) = 2 Σ ∂xa ∂xb Σ a,b=1

where the integrand is independent of choice of local coordinates. If Σ is one-dimensional, the energy reduces to the action J that we studied in the theory of geodesics. In this chapter we will focus on the case where Σ has dimension at least two. 169

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4. Harmonic and Minimal Surfaces

The energy defines a smooth map E : C 2 (Σ, M ) −→ R. To find the critical points of E, we consider a variation of a given map f which has its support within a coordinate chart (U, (x1 , . . . , xm )) on Σ. Such a variation is simply a smooth family of maps t → f (t) in C 2 (Σ, M ) such that f (0) = f and f (t)(p) = f (p) for all t, when p ∈ Σ − U . In terms of the local coordinates, we set α(x1 , . . . , xm , t) = f (t)(x1 , . . . , xm ), and a straightforward calculation shows that ⎡ ⎤  m  2  ∂α ⎦ ∂ α d ⎣ (E(f (t)) = det(ηab )η ab · dx1 · · · dxm dt ∂t∂xa ∂xb Σ a,b=1 t=0 t=0 ⎡ ⎤    m  ∂α ⎣  ∂ ab ∂α ⎦ · =− det(ηab )η dx1 · · · dxm . ∂xa ∂xb Σ ∂t a,b=1

t=0

We can evaluate at t = 0 and let V (x1 , . . . , xm ) =

∂α (x1 , . . . , xm , 0) ∂t

be the variation field , thereby obtaining the first variation formula, (4.2) ⎤ ⎡    m   1 ∂ ∂f ⎦ dA. dE(f )(V ) = − V · ⎣  det(ηab )η ab ∂xb det(ηab ) a,b=1 ∂xa Σ If f is a critical point for the energy E, then dE(f )(V ) = 0 for all such variation fields V , and f must satisfy the partial differential equation ⎤ ⎡   m   ∂ 1 ∂f ⎦ ⎣  det(ηab )η ab = 0, (4.3) ∂xb det(ηab ) ∂xa a,b=1

where (·) denotes projection into the tangent space to M . Maps f ∈ C 2 (Σ, M ) which satisfy equation (4.3) are called harmonic maps. Just like the equation for geodesics, the equation for harmonic maps is nonlinear because of the projection into the tangent space. In fact, we can rewrite (4.3) in terms of the Levi-Civita connection D on M as   m  D  1 ab ∂f  det(ηab )η = 0. ∂xa ∂xb det(ηab ) a,b=1

4.1. Energy of a smooth map

171

Equivalently, in terms of local coordinates (u1 , . . . , un ) on the ambient manifold M , we can write the equation of harmonic maps as 

1

m 

det(ηab ) a,b=1

∂ ∂xa





det(ηab )η

ab ∂ui

∂xb

 +

m n   a,b=1 i,j,k=1

Γijk η ab

∂uj ∂uk = 0, ∂xa ∂xb

where the Γijk ’s are the Christoffel symbols for the Riemannian manifold M . Harmonic maps between Riemannian manifolds were introduced by Eells and Sampson [ES64] in 1964, who used an approach based upon heat flow to establish the following theorem: Theorem 4.1.1 (Eells and Sampson). If M is a compact connected Riemannian manifold with nonpositive sectional curvatures and Σ is a compact Riemannian manifold, then every component of C 2 (Σ, M ) contains a minimal energy representative, which is a smooth harmonic map. A very nice proof of this theorem using heat flow can be found at the beginning of Chapter 5 of [LW08]. A few years after the work of Eells and Sampson appeared, Hartman [Har67] established a uniqueness result for any component of C 2 (Σ, M ) which does not contain degenerate harmonic maps: Theorem 4.1.2 (Hartman). If Σ and M are compact connected Riemannian manifolds and M has negative sectional curvatures, then any component of C 2 (Σ, M ) which does not contain a harmonic map f : Σ → M with differential of rank ≤ 1 at every point contains a unique harmonic map. Note that if M has nonpositive sectional curvature, the Hadamard-Cartan Theorem asserts that the exponential map expp : Tp M → M is a smooth covering at any point p ∈ M , so M has Euclidean space as its universal cover, and all of its higher homotopy groups are zero, which is the same as saying that M is a K(π, 1), in the language of algebraic topology (see §3.5). Moreover, the Eells-Sampson Theorem does not hold without the strong curvature assumption. Indeed, when M is compact and simply connected, the mapping space C 2 (Σ, M ) is never contractible, and the rational cohomology we calculated in the previous chapter typically provides many topological obstructions to following the heat flow of Eells and Salamon to a map of least energy. When the dimension of Σ is ≥ 3, attempts to minimize energy within a given homotopy class of maps can lead to elements of L21 (Σ, M ) which have quite bad singularities, as discussed for example in [LW08]. However, in the case where Σ has dimension two, we will see that the possible singularities are

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4. Harmonic and Minimal Surfaces

easily analyzed making a rich theory accessible. Two additional phenomena seem to make the case dim Σ = 2 special: (1) The energy from a surface is conformally invariant, and the theory of harmonic maps simplifies considerably due to the existence of isothermal coordinates on Σ. (2) The Sobolev space L21 (Σ, M ) which is most closely adapted to the energy function lies on the border of Sobolev range. It is these phenomena which motivate our restriction to this case dim Σ = 2 for the rest of this chapter. Our goal is to develop the theory of conformal harmonic maps from a two-dimensional domain Σ to an n-dimensional ambient Riemannian manifold M . Let us start by exploring the implications of conformal invariance. It is well-known that any two-dimensional Riemannian manifold possesses isothermal coordinates, a proof based upon Hodge theory being found in Chapter 5, §10 of Taylor’s book on PDE’s [Tay96]. More precisely, any twodimensional Riemannian manifold Σ possesses an atlas {(Uα , (xα , yα )) : α ∈ A} consisting of isothermal charts, charts such that the Riemannian metric on Σ takes the form ds2 = λ2α (dxα ⊗ dxα + dyα ⊗ dyα ), where each λα is a positive smooth real-valued function on Uα . If {ψα : α ∈ A} is a partition of unity subordinate to this atlas, then the energy of a C 2 map f : Σ → M is given by the formula    ∂f ∂f ∂f ∂f 1 ψα · + · dxα dyα E(f ) = 2 α Σ ∂xα ∂xα ∂yα ∂yα . / . / (4.4)  1 ∂f ∂f ∂f ∂f = ψα , , + dxα dyα , 2 α Σ ∂xα ∂xα ∂yα ∂yα where ·, · is the Riemannian metric on M induced by its imbedding into RN , and the equation for harmonic maps simplifies to  (4.5)

∂2f ∂ 2f + ∂x2α ∂yα2

 = 0.

Thus harmonic maps are governed by the tangential projection of the standard Laplace operator when expressed in terms of isothermal coordinates. An atlas of coherently oriented isothermal charts makes an oriented surface Σ into a Riemann surface. To see this, we take two coherently oriented isothermal charts (U, (x, y)) and (V, (u, v)) on Σ with U ∩V nonempty. Then

4.1. Energy of a smooth map

173

on the overlap U ∩ V , (4.6)         du   T   dx dx 2 2 2 = dx dy A A = μ dx dy du + dv = du dv , dv dy dy where

 A=

∂u/∂x ∂u/∂y ∂v/∂x ∂v/∂y



has positive determinant and μ2 is a positive function. But (4.6) says that AT A = μ2 I, where I is the identity matrix, which in turn says that the two columns of A are perpendicular vectors of the same length. This implies that     ∂u/∂x ∂v/∂y =± , ∂v/∂x −∂u/∂y and the fact that A has positive determinant requires that we take the plus sign. This gives the Cauchy-Riemann equations ∂v ∂u = , ∂x ∂y

∂u ∂v =− , ∂y ∂x

which must hold on U ∩ V . Thus for each chart {(Uα , (xα , yα )) : α ∈ A} in the coherently √ oriented isothermal atlas, we can define a complex coordinate zα = xα + −1yα , thereby obtaining an atlas {(Uα , zα ) : α ∈ A} in which complex coordinates are related by holomorphic maps on overlaps. This makes Σ into a one-dimensional complex manifold, otherwise known as a Riemann surface. In this way, Riemannian metrics on Σ are divided into equivalence classes, depending upon which Riemann surface structure is defined by the corresponding isothermal charts. It is often convenient to choose a canonical metric on the surface Σ within the conformal equivalence class determined by a given Riemann surface structure. The uniformization theorem from Riemann surface theory states that a connected Riemann surface Σ has as its universal cover one of three simply connected Riemann surfaces, S 2 = C ∪ {∞},

C

or

D = {z ∈ C : |z| < 1}.

Each of these model spaces possesses a Riemannian metric of constant Gaussian curvature compatible with its conformal structure, and the deck transformations of the universal cover of Σ are isometries with respect to this Riemannian metric. If Σ is closed, it follows from the Gauss-Bonnet formula that the sign of the curvature is positive, zero or negative when Σ has genus zero, one, or at least two, respectively. Once we fix the Riemann surface structure on Σ, the canonical metric is unique (up to diffeomorphism) if we

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impose one of the normalization conditions: (1) K = 1 if Σ = S 2 or, equivalently, the genus g is zero, (2) K = 0 and the metric has total area one if g = 1, (3) and K = −1 if g ≥ 2. If z = x + iy, we can write (4.3) in terms of the complex partial differential operators     √ √ 1 ∂ ∂ 1 ∂ ∂ ∂ ∂ = − −1 , = + −1 ∂z 2 ∂x ∂y ∂ z¯ 2 ∂x ∂y as  2  ∂ f = 0. (4.7) ∂z∂ z¯ Note that we can regard ∂f as a section of E = f ∗ T M ⊗ C, ∂z a complex vector bundle over the Riemann surface Σ. The vector bundle E has a Hermitian metric defined by v, w ∈ Tp M ⊗ C

→

v, w, ¯

where w ¯ is the conjugate of w, and the Levi-Civita connection D on f ∗ T M extends complex linearly to a metric connection on E. In terms of this connection, we can rewrite (4.7) as     √ D ∂f D 1 D D (4.8) = 0, where = + −1 . ∂ z¯ ∂z ∂ z¯ 2 ∂x ∂y The following theorem states that this connection gives E a canonical holomorphic structure, thereby making possible a remarkable relationship between harmonic surfaces and complex analysis: Theorem 4.1.3 (Koszul and Malgrange). If E is a complex vector bundle with Hermitian metric over a Riemann surface Σ and D is a metric connection on E, then there is a unique holomorphic structure on E such that if σ is a section of E, Dσ dz = 0, σ is holomorphic ⇔ D  σ = ∂ z¯ whenever z is a complex coordinate on Σ. In addition to the original argument [KM58], there is a proof in §5 of Atiyah and Bott [AB83] which constructs the local holomorphic charts by showing that the equation D  σ = 0 has sufficiently many solutions. Alternatively, a version of this theorem is proven as Theorem 5.1 in Atiyah, Hitchin and Singer [AHS78] using the Newlander-Nirenberg theorem on integrability of

4.1. Energy of a smooth map

175

almost complex structures, and yet another proof is given in Donaldson and Kronheimer [DK90], Theorem 2.1.53. The upshot is that the equation for harmonic maps from oriented surfaces can be expressed quite simply in terms of Riemann surface theory: Equation (4.7) asserts that a map f : Σ → M is harmonic if and only if in terms of any complex coordinate z on Σ, the section ∂f ∂z

of

E = f ∗T M ⊗ C

is holomorphic with respect to the holomorphic structure on E provided by the Koszul-Malgrange Theorem. The holomorphic structure on E enables us to use techniques from complex analysis, including the Atiyah-Singer theorem for the Cauchy-Riemann operator on Riemann surfaces, also known as the Riemann-Roch Theorem. (A proof of the general Atiyah-Singer theorem is in [LM89], while the more special Riemann-Roch Theorem is presented in books on Riemann surfaces, such as Gunning [Gun66] or Forster [For81], which we recommend for further background.) Let us recall how the Riemann-Roch Theorem goes for line bundles over Riemann surfaces. If L is holomorphic line bundle over the compact Riemann surface Σ, any meromorphic section σ : Σ → L has finitely many zeroes and poles, and the number of zeroes of σ minus the number of poles counted with multiplicity is called the degree of L, this difference being independent of the meromorphic section σ. In particular, a holomorphic line bundle of negative degree must have no holomorphic sections. Once we have the notion of degree, we can state the Riemann-Roch Theorem as (4.9)

dimC O(L) − dimC O(K ⊗ L−1 ) = 1 − g + degree of L,

where O(L) denotes the space of holomorphic sections of L, and K is the canonical line bundle, another name for the cotangent bundle of Σ with its holomorphic structure. The degree of L can also be calculated via the first Chern class or Euler class c1 (L) of L, as studied in the theory of characteristic classes (see Milnor and Stasheff [MS74]), by using the formula degree of L = c1 (L)[Σ] = c1 (L), [Σ] = evaluation of c1 (L) on the fundamental class of Σ. In accordance with the outline of §1.1, we should expect the Atiyah-Singer theorem (4.9) to play a fundamental role in the theory of harmonic maps and the closely related minimal surfaces described in the next section.

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When Σ is the Riemann sphere S 2 , c1 (L)[S 2 ] < 0 ⇒ dim O(L) = 0, and hence the Riemann-Roch Theorem (4.9) gives the dimension of the space of holomorphic sections on the nose in this case, dim O(L) = c1 (L)[S 2 ] + 1.

(4.10)

In particular, if L and L are two holomorphic line bundles of the same degree over S 2 , Riemann-Roch yields dim O(Hom(L, L )) = 1, which implies that L and L are isomorphic as holomorphic bundles. After construction of the basic hyperplane section bundle H of degree one over S 2 , this leads to a classification of the holomorphic line bundles over S 2 : any holomorphic line bundle over S 2 is isomorphic to d

!" # H = H ⊗ H ⊗ · · · ⊗ H, d

for some d ∈ Z.

We will give further striking applications of the Riemann-Roch Theorem to harmonic maps from surfaces in the next section. For now, we calculate the degree of a key holomorphic line bundle which is associated to any harmonic surface in a Riemannian manifold. Definition. A point p ∈ Σ is a branch point for the harmonic map f : Σ → M if (∂f /∂z)(p) = 0, where z is any complex coordinate near p. A nonconstant harmonic map is an immersion except at isolated branch points. If p is a branch point for such a map f , and z is a local complex coordinate defined on a small open neighborhood U of p with z(p) = 0, then we can write (∂f /∂z) = z ν g for some positive integer ν, where g is a section of L over U such that g(p) = 0. We call ν the order or multiplicity of the branch point. The factorization (∂f /∂z) = z ν g shows that even though the locally defined holomorphic section   ∂f ∂f has an isolated zero at p, : Σ → P(E) ∂z ∂z extends to all of U , where P(E) denotes the bundle of projective spaces of fibers of E = f ∗ T M ⊗ C. Moreover, if z and w are two complex coordinates on Σ with overlapping domains, the formula for change of complex coordinates gives ∂f ∂z ∂f ∂f = = (complex-valued function) , ∂w ∂z ∂w ∂z and hence the various locally defined complex derivatives of f determine a holomorphic line subbundle L of E. The line bundle L is isomorphic to the

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177

tangent bundle to Σ exactly when f has no branch points, and in this case c1 (L)([Σ]) = 2 − 2g, in accordance with the Gauss-Bonnet Theorem from the theory of surfaces. If p is a branch point for f and (∂f /∂z) = z ν g in terms of a local coordinate with z(p) = 0, we let w = z ν+1 , then dw = (ν + 1)z ν dz and we can define a section ∂ 1 ∂ ∂ of L which is nonzero near p by = . ν ∂w ∂w (ν + 1)z ∂z Thus we see that the restriction of L to U is obtained from the holomorphic tangent bundle T Σ|(U − {p}) by pasting it to a trivial bundle U × C over U by means of the transition function 1 : U − {p} −→ C − {0}. g0p = (ν + 1)z ν From the trivial bundle over Σ − {p} and this transition function one can construct the point bundle ζpν over Σ. These point bundles are described in more detail in Gunning [Gun66]. The degree of the point bundle ζpν is given by c1 (ζpν )([Σ]) = ν. For a general harmonic map f : Σ → M , the divisor of f is the finite sum ν1 p1 + · · · + νm pm , where p1 , . . . , pm are the branch points of f and ν1 , . . . , νm are their branching orders. The above discussion then shows that (4.11)

m . L∼ = T Σ ⊗ ζpν11 ⊗ · · · ⊗ ζpνm

If ν denotes the total branching order of f , the total number of branch points of f , counted with multiplicity, then the degree c1 (L), [Σ] of the line bundle L is given by the formula (4.12)

c1 (L)([Σ]) = c1 (L), [Σ] = 2 − 2g + ν,

where g is the genus of Σ. Suppose, for example, that h : Σ2 → M is a smooth harmonic map without branch points and that g : Σ1 → Σ2 is a nontrivial holomorphic branched cover. (Thus, if Σ2 = S 2 , the Riemann sphere, we can regard g as a meromorphic function on Σ1 .) Then the composition f = h◦g : Σ1 → M is also harmonic; it is called a branched cover of the harmonic map h : Σ2 → M . Definition. We say that the harmonic map f is prime if it is not a nontrivial branched cover of another harmonic map. If f : Σ → M is a k-fold branched cover of a harmonic two-sphere h : S 2 → M which is free of branch points, then the line bundle Lf for f is related to

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the line bundle Lh for h by c1 (Lf ), [Σ] = kc1 (Lh ), [S 2 ] = 2k, from which one can calculate the total branching order of f in terms of k and the genus g of Σ. Remark 4.1.4. We will sometimes find it convenient to consider critical points for a function closely related to the Dirichlet energy Eω for a given choice of conformal structure ω on Σ, namely the function  θ, (4.13) Eωθ : C 2 (Σ, M ) → R, defined by Eωθ (f ) = Eω (f ) + Σ

where θ is a smooth two-form on M .

4.2. Minimal two-spheres and tori Even more studied than the theory of harmonic maps is the theory of minimal submanifolds of a Riemannian manifold. In this section we will show that minimal submanifolds of dimension two, called minimal surfaces, can be profitably studied in parametric form via harmonic maps. 4.2.1. The area function. Nonsingular parametrized minimal surfaces with the topology of a given compact surface Σ in a Riemannian manifold M can be thought of as critical points of the area function  ∂f ∂f 2 defined by A(f ) = A : C (Σ, M ) −→ R ∂x ∧ ∂y dxdy, Σ

in which (x, y) are arbitrary local coordinates on Σ. In contrast to the energy E, the integrand for area A is independent of the choice of parametrization, and in particular does not depend upon the choice of a metric or conformal structure on Σ, but area and energy are closely related: Proposition 4.2.1. If Σ is given a conformal structure ω and if f ∈ Map(Σ, M ), then E(f ) ≥ A(f ), with equality holding if and only if f satisfies the conditions / . / . / . ∂f ∂f ∂f ∂f ∂f ∂f , = , , , = 0, (4.14) ∂x ∂x ∂y ∂y ∂x ∂y whenever z = x + iy is a choice of complex chart and ·, · denotes the Riemannian metric on the ambient manifold M or, equivalently, / . ∂f ∂f , = 0. (4.15) ∂z ∂z

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179

Proof. We utilize two well-known algebraic identities for vectors v and w in RN :  1 2 |v| + |w|2 , |v||w| ≤ |v ∧ w|2 + |v · w|2 = |v|2 |w|2 , 2 with equality holding only if |v| = |w|. Using these two facts, we find that   ∂f ∂f ∂f ∂f 1 ∂f 2 ∂f 2 (4.16) ∂x ∧ ∂y ≤ ∂x ∂y ≤ 2 ∂x + ∂y , and hence 1 E(f ) = 2

   2 2  ∂f ∂f ∂f + ∂f dxdy ≤ dxdy = A(f ), ∧ ∂x ∂y ∂y Σ Σ ∂x

with equality holding if and only if (4.14) holds at every point of Σ. QED Definition. We say that a map f : Σ → M is weakly conformal with respect to a Riemann surface structure ω on Σ if it satisfies (4.15) when z = x + iy is a complex coordinate for ω. Theorem 4.2.2. A weakly conformal harmonic map f : Σ → M is a critical point for the area function A. Proof. If f is weakly conformal and harmonic, and t → f (t) is a smooth family of maps with f (0) = f , then E(f (t)) ≥ A(f (t)) and

E(f (0)) = A(f (0)) ⇒

d d E(f (t)) = A(f (t)) , dt dt t=0 t=0

so f is critical for the area function. QED Theorem 4.2.2 provides motivation for the following definition of parametrized minimal surface: Definition. A parametrized minimal surface is a weakly conformal harmonic map f : Σ → M . Note that if f is an immersion, we could give Σ the Riemannian metric which makes f an isometric immersion. This gives Σ a Riemann surface structure which makes f weakly conformal, and with this conformal structure, f is harmonic if and only if it is minimal. More generally, if f : Σ → M is a weakly conformal map, then (4.15) shows that the complex line bundle L we described in the preceding section is “isotropic”, that is, L, L = 0. This constrains the possible singularities

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of f ; the differential dfp at any point p ∈ Σ has rank two if f is an immersion in a neighborhood of p, or rank zero if p is a branch point, but never has rank one. We would like to consider minimal surfaces as critical points for a variational problem which involves the Dirichlet integral and the conformal structure, rather than the area function. 4.2.2. Minimal two-spheres. The theory is simpler for Σ = S 2 , which has a unique conformal structure by the uniformization theorem. We can regard Σ = S 2 as C ∪ {∞} and take the atlas defined by the standard coordinate z on C and the coordinate w = 1/z on S 2 − {0}. Then ∂f ∂w 1 ∂f ∂f = =− 2 , ∂z ∂w ∂z z ∂w and (4.17)

∂f bounded near ∞ ∂w

⇒

∂f → 0 like 1/z 2 as z → ∞. ∂z

By the Koszul-Malgrange Theorem 4.1.3, we can regard ∂f /∂z as a holomorphic section on S 2 − {∞} with a removeable singularity at ∞, and by the removeable singularity theorem from complex analysis, ∂f /∂z extends to a holomorphic vector field on S 2 . Thus when calculating energy on the Riemann sphere, we are fully justified in using a single coordinate z = x + iy on C = S 2 − {∞}, and the energy can be expressed by the improper integral / . /  . ∂f ∂f ∂f ∂f 1 , + , dxdy. E(f ) = 2 C ∂x ∂x ∂y ∂y Conversely, if f : C → M is a harmonic map of finite energy, in other words, such that the above integral is finite, a removeable singularity theorem of Sacks and Uhlenbeck (proven later as Theorem 4.6.9) shows that f extends to a harmonic map from S 2 into M . Proposition 4.2.3. A harmonic two-sphere f : S 2 → M is automatically weakly conformal, and hence a parametrized minimal surface. Proof. We let z = x + iy be the standard complex coordinate on C = S 2 − {∞}. Then it follows from (4.7) that . / .   / . / D ∂f ∂f ∂f ∂f ∂ ∂f ∂f , =2 , = 0, so , ∂ z¯ ∂z ∂z ∂ z¯ ∂z ∂z ∂z ∂z is a holomorphic function which extends to a C 2 function on S 2 by (4.17). This function must be constant by the maximum modulus principle, and since it vanishes at ∞, the constant must be zero. QED

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181

For a surface Σ of genus g ≥ 1, it is often convenient to express weak conformality of the harmonic map f : Σ → M by the vanishing of an important tensor field called the Hopf differential : / . ∂f ∂f (4.18) Ωf = , dz ⊗ dz. ∂z ∂z It follows immediately from (4.7) that . / . / D ∂f ∂f ∂ ∂f ∂f , =2 , = 0, ∂ z¯ ∂z ∂z ∂ z¯ ∂z ∂z so the Hopf differential of a harmonic map is indeed a holomorphic quadratic differential, as studied in Riemann surface theory. The argument for Proposition 4.2.3 simply shows that the Hopf differential (4.18) is zero, and one could prove the proposition by simply noting that the the Hopf differential is a section of K2 = K ⊗ K and applying the Riemann-Roch Theorem (4.9) to show that any holomorphic section of this bundle is zero. Another application of holomorphic differentials is provided by the theory of minimal two-spheres in an n-sphere of constant curvature one, M = S n ⊆ Rn+1 . The intersection of this sphere with any three-dimensional plane which passes through the origin in Rn+1 gives a totally geodesic two-sphere within M . In this way we obtain a family of two-dimensional totally geodesic submanifolds of S n , homeomorphic to G3 (Rn+1 ), which can be parametrized as minimal surfaces. When n = 3 these are the only minimal two-spheres, a fact first proven by Almgren (see Lemma 1 of [Alm84]): Theorem 4.2.4. Any nonconstant conformal harmonic map f : S 2 −→ S 3 ⊆ R4 is a parametrization of a totally geodesic two-sphere in S 3 . Proof. Suppose that f : S 2 −→ S 3 ⊆ R4 is any parametrized minimal surface. Then using a dot to denote the usual dot product in R4 , we see that f ·f ≡1

⇒

∂f ∂f ·f =0= · f. ∂z ∂ z¯

Since f is weakly conformal, ∂f ∂f ∂f ∂f 1 ∂f ∂f · =0= · and · = 2, ∂z ∂z ∂ z¯ ∂ z¯ ∂z ∂ z¯ 2λ where λ2 is a positive function on S 2 , and differentiation yields ∂ 2 f ∂f ∂ 2 f ∂f ∂ 2 f ∂f ∂ 2 f ∂f = · = 0 = = · . · · ∂z 2 ∂z ∂z∂ z¯ ∂z ∂ z¯2 ∂ z¯ ∂z∂ z¯ ∂ z¯

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Since f is harmonic, ∂ 2f = μf, ∂z∂ z¯ for some smooth function μ on S 2 . Hence   2   ∂ f ∂ ∂ ∂ 2f ∂ 2f (μf ) · 2 = 0. · = 2 ∂ z¯ ∂z 2 ∂z 2 ∂z ∂z This implies that the quartic differential   2 ∂ f ∂ 2f dz 4 · ∂z 2 ∂z 2 is a holomorphic section of K4 , and the Riemann-Roch Theorem (4.9) implies that it vanishes. But then the space spanned by f , ∂f /∂z, ∂f /∂ z¯ must be preserved by ∂/∂z and ∂/∂ z¯, hence parallel, and this in turn implies that f must map into a three-plane in R4 . QED This argument enables us to classify all of the critical points of the function E : Map(S 2 , S 3 ) −→ [0, ∞). The critical values are 4πd for d ∈ {0} ∪ N. At critical value zero we have the constant maps, at critical value 4π the totally geodesics imbeddings, and at critical value 4πd for d ≥ 2, the critical points are exactly the d-fold branched covers of the totally geodesic imbeddings. Almgren’s Theorem provides a nice application of the Riemann-Roch Theorem, via calculating the dimension of the space of holomorphic sections of the holomorphic line bundle K4 over S 2 . But the Riemann-Roch Theorem also enables us to calculate the dimension of the space O(K2 ) for a Riemann surface of arbitrary genus g. Indeed, the reader can use the Riemann-Roch Theorem (4.9) to show that ⎧ ⎪ if g = 0, ⎨0, 2 (4.19) dimO(K ) = 1, if g = 1, ⎪ ⎩ 3g − 3, if g ≥ 2, and we will see later that this agrees with the complex dimension of the tangent space to the space of complex structures on the connected oriented compact surface of genus g, an even more fundamental application than Almgren’s Theorem. 4.2.3. Minimal tori. In contrast to the Riemann sphere S 2 , the torus T 2 = S 1 × S 1 has a nontrivial space of complex structures, of complex dimension one or real dimension two. To see this, we first note that the uniformization theorem implies that the universal cover of the torus should be C, since the curvature of the canonical metric is zero. To give a global

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183

classification of the complex structures on T 2 , we imagine that we have fixed a basis (α, β) for π1 (T 2 ) ∼ = H1 (T 2 ; Z), representing the two S 1 factors. The universal cover must be holomorphically equivalent to C and the deck transformations corresponding to α and β are invertible holomorphic maps fα , fβ : C −→ C with bounded derivatives fα and fβ . It follows from the maximum modulus principle that these derivatives are constant, so fα (z) = aα z + bα

and

fβ (z) = aβ z + bβ

for complex constants aα , bα , aβ and bβ . But the transformation f (z) = az + b has a fixed point in C unless a = 1 and deck transformations cannot have fixed points, so fα and fβ must be translations. After performing a possible rotation of C and a uniform stretching or contraction, we can arrange that the translations be (4.20)

fα (z) = z + 1,

fβ (z) = z + ω,

where ω = u + iv is a point in the upper half-plane H = {u + iv ∈ C : v > 0}, a parameter space of one complex dimension. Thus the torus must be holomorphically equivalent to the quotient C/Λ, Λ being the lattice in C generated by 1 and ω. Note that the torus inherits a flat Riemannian metric from the universal cover C, and we can rescale it so that it has area one. A holomorphic equivalence between two tori C/Λ1 and C/Λ2 must lift to an invertible holomorpic map f : C → C which takes the generators of Λ1 to corresponding generators of Λ2 , and and it is easily seen that this holomorphic map has doubly periodic derivative, hence constant derivative by the maximum modulus principle, and after normalization to preserve zero, it must be a constant multiple of the identity. The only such lift which takes 1 to 1 is the identity. Hence the conformal structures on T 2 , for a fixed a basis of H1 (T 2 ; Z), are in one-to-one correspondence with points in the upper half-plane H. We say that H is the Teichm¨ uller space of conformal structures on the torus. An orientation-preserving change of basis for the lattice Λ is represented by multiplication by a matrix in SL(2, Z),        α a b α a b (4.21) → for ∈ SL(2, Z). β c d β c d

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The SL(2, Z)-action on the generators ω1 and ω2 of an arbitrary integral lattice in the complex plane is      ω1 a b ω1 → ω2 ω2 c d while the corresponding action on the coordinate ω = u + iv = ω2 /ω1 is (4.22)

ω

→

a + bω . c + dω

Note that while the action of the mapping class group SL(2, Z) on H1 (T 2 ; Z) is faithful, the action on the upper half-plane H has kernel consisting of ±I. By applying a suitable element of SL(2, Z), we can arrange that ω = u + iv lie in the fundamental domain of T1 for this action: D = {ω = u + iv ∈ C : |u| ≤ 1/2, v > 0, |ω| ≥ 1}. The moduli space M1 of conformal structures on the torus is obtained from this fundamental domain by identifying edges. It can be shown that M1 has a Riemann surface structure and is holomorphically equivalent to the complex line C. Thus we have two ways of looking at the Riemann surface structures on uller space H when a choice of basis for H1 (T 2 ; Z) is fixed, T 2 , the Teichm¨ and the moduli space M1 when we ignore choice of basis, M1 being the quotient of H by P SL(2, Z) = SL(2, Z)/{±I}. If T 2 = C/Λ, then any map f : T 2 → M can be lifted to a doubly periodic map f˜ : C → M with periods in the lattice Λ. The standard coordinate z = x + iy on C restricts to local coordinate systems which descend to T 2 , and the standard metric dx2 + dy 2 on C descends to a flat metric on T 2 within the conformal equivalence class corresponding to ω. We can rescale the metric to (1/v)(dx2 + dy 2 ) so that it has total area one. In terms of the coordinates (t1 , t2 ) on C defined by        x = t1 + ut2 , 1 u x + t2 or = t1 0 v y y = vt2 , the unit square 0 ≤ t1 ≤ 1, 0 ≤ t2 ≤ 1 gets taken to the fundamental domain Δ = {t1 + t2 ω : 0 ≤ t1 ≤ 1, 0 ≤ t2 ≤ 1} of the torus. Moreover, in these coordinates, the components of the flat ) metric ηab dta dtb and its inverse are     1 1 1 u2 + v 2 −u u ab , , (η ) = (ηab ) = −u 1 v u u2 + v 2 v

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185

while the area element is given by dA = dt1 dt2 . Hence if f ∈ C 2 (T 2 , M ),  2 2   ∂f 1 ∂f ∂f ∂f ∂f ∂f − 2u |df |2 = (u2 + v 2 ) η ab · = · + ∂ta ∂tb v ∂t1 ∂t1 ∂t2 ∂t2 a,b

or, equivalently, 2 ∂f 2 1 ∂f ∂f + . −u |df | = v ∂t1 v ∂t2 ∂t1 2

From this, it is apparent that the energy     ∂f 2 1 1 1 ∂f 2 1 ∂f + −u v dt1 dt2 (4.23) E(f, ω) = 2 0 0 ∂t1 v ∂t2 ∂t1 is a smooth function of two variables on C 2 (T 2 , M ) × H. This gives an explicit formula for how the energy  1 |df |2 dA, Eω (f ) = E(f, ω) = 2 T2 varies for a fixed choice of f as ω = u + iv varies throughout the Teichm¨ uller space H. (In this formula, |df | and dA are calculated with respect to some metric within the conformal class ω.) Theorem 4.2.5. If (f, ω) is a critical point for the function E : C 2 (T 2 , M ) × H −→ R defined by ( 4.23), then f is a weakly conformal harmonic map, and hence a parametrized minimal surface. Proof. Differentiation with respect to the first variable shows that if (f, ω) is a critical point for E, f must be harmonic. Thus it suffices to show that variation with respect to the variable ω shows that the Hopf differential Ωf vanishes. Note that the differential dz descends from C to the torus C/Λ and hence holomorphic differentials on a torus must be of the form hdz 2 , where h is a holomorphic function on the torus. By the maximum modulus theorem, h must be constant. Hence the only possible Hopf differentials for harmonic tori are cdz 2 , where c is a complex constant. To check that the Hopf differential vanishes, it will be helpful to utilize the Euclidean coordinates x1 = √xv and x2 = √yv , in terms of which the energy of f simplifies to    ∂f 2 ∂f 2 1 + E(f, ω) = ∂x2 dx1 dx2 , 2 Δ ∂x1

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where Δ is a fundamental domain for the lattice Λ. But we can also ask a more general question: Suppose that f0 : T 2 → M is any map, harmonic or not. Can we find conditions on the matrix /  . 1 ∂f ∂f , (4.24) (hab ), defined by hab = dx1 dx2 , 2 Δ ∂xa ∂xb which are necessary and sufficient for the given conformal structure to be critical for the function ω → E(f, ω)? Any other element ω ˜ of Teichm¨ uller space can be represented in terms of our coordinates (x1 , x2 ) for the base conformal structure ω by a metric defined by a constant matrix of determinant one,   σ11 σ12 ∈ SL(2, R) ∩ Sym(2, R), (σab ) = σ21 σ22 where Sym(2, R) denotes the space of 2 × 2 symmetric matrices. The energy of f in terms of the new coordinates is given by . /   ∂f ∂f 1 ab σ , dx1 dx2 , E(f, ω ˜) = 2 Δ ∂xa ∂xb a,b

where

We can represent terms of matrices  1 t → 0



σ 11 σ 12 σ 21 σ 22



 =

 σ22 −σ12 . −σ21 σ11

a curve t → ω(t) in Teichm¨ uller space with ω(0) = ω in by      σ˙ 11 σ˙ 12 0 σ˙ 11 σ˙ 12 , where +t 1 σ˙ 12 −σ˙ 11 σ˙ 12 −σ˙ 11

is symmetric and of trace zero. It is then easily seen that . /   1 ∂f ∂f d E(f, ω(t)) =− σ˙ ab , dx1 dx2 . dt 2 Δ ∂xa ∂xb t=0 a,b

If (f, ω) is critical for E under variations in ω, the above expressions must be zero for all symmetric matrices (σ˙ ab ) of trace zero, which implies that the entries of (4.24) must satisfy h11 = h22 ,

h12 = 0.

But in the case where f is harmonic, the Hopf differential is given by 1 Ωf = (h11 − h22 − 2ih12 )dz 2 , 2 so this does indeed imply that the Hopf differential vanishes, which finishes the proof of the theorem. QED

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187

Our argument shows that if f : T 2 → M is any map, harmonic or not, which is critical for variations in the conformal structure, then the matrix (hab ) defined by (4.24) must be a constant multiple of the identity. As a byproduct of our analysis, we see that there are two types of harmonic tori. Those for which the matrix (hab ) is degenerate are simply parametrizations of smooth closed geodesics, and these can never be conformal with respect to any conformal structure. Those for which the matrix (hab ) is nondegenerate are either conformal or their energy can be decreased by perturbing the conformal structure ω. Note that if the harmonic torus f : T 2 → M is not weakly conformal, its Hopf differential is of the form cdz 2 where c is a nonzero constant, and so it is never zero and f cannot have branch points. However, it may have fold points as the following example shows. Example 4.2.6. If f is an immersion, the fibers of the line bundles L and ¯ are linearly independent at every point. However, a harmonic map f can L ¯ coincide. At such points the rank have points p at which the fibers L and L of f can be at most one. Indeed, there is a degree zero harmonic map f : T 2 → S 2 when S 2 is given the standard Riemannian metric of constant curvature one, which has “fold points” along two circles parallel to the equator. To see how this is constructed, we first note that the metric on S 2 ⊂ R2 with equation x2 + y 2 + z 2 = 1 is expressed in spherical coordinates z = cos φ, θ being the standard angular coordinate in the (x, y)-plane, is ds2 = (cos2 φ)dθ2 + dφ2 = sech2 u(dθ2 + du2 ), where u and φ are related by the equation tanh(u/2) = tan(φ/2). In terms of the standard coordinates (t1 , t2 ) on T 2 which correspond to the factorization T 2 = S 1 × S 1 , the mapping f : T 2 → S 2 can expressed as θ(t1 , t2 ) = t2 ,

φ(t1 , t2 ) = φ(t1 ),

where φ is a (nonconstant speed) parametrization of the geodesic θ = (constant). Note that the circle φ = (constant) has curvature κ = 1/ cos φ and normal curvature κn = 1. From the equation κ2g + κ2n = κ2 , where κg is the geodesic curvature, implies that κg = ± tan φ. Moreover, the curve is traversed with constant speed cos φ. Hence     d2 φ 1 ∂f D ∂f d2 φ D + = 2 + (tan φ)(cos2 φ) = 2 + sin(2φ). 0= ∂t1 ∂t1 ∂t2 ∂t2 dt dt 2 The differential equation we must solve to obtain a harmonic map (the pendulum equation except for constant factors) is equivalent to the first

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order system dφ = ψ, dt

dψ 1 = − sin(2φ). dt 2

Eliminating dt yields dψ (1/2) sin(2φ) =− dφ ψ

which integrates to ψ 2 −

1 cos(2φ) = (constant). 2

Various choices of constant yield harmonic maps for appropriate conformal structures on T 2 . Note that the antipodal map A : S 2 → S 2 for the standard round metric on the two-sphere induces an orientation reversing map A : T 2 → T 2 such that f ◦ A = A ◦ f . We can take the quotient in both domain and range, obtaining thereby a harmonic map from a Klein bottle into the real projective plane RP 2 , which has as its image a M¨obius band.

4.3. Minimal surfaces of arbitrary topology 4.3.1. Teichm¨ uller space. Just like minimal tori, parametrized minimal surfaces of genus g ≥ 2 can be regarded as critical points for a two-variable energy. To define this energy, we need a description for the space of conformal structures (modulo diffeomorphisms homotopic to the identity) on a compact oriented surface Σg of genus g ≥ 2. We will see that the resulting space is an oriented manifold of dimension 6g − 6. Any Riemann surface of genus ≥ 2 has the hyperbolic plane H as its universal cover, and H has a canonical hyperbolic metric of constant Gaussian curvature −1 which induces a hyperbolic metric on Σg . This implies that classifying conformal structures on Σg up to diffeomorphism is the same as classifying hyperbolic metrics up to diffeomorphism. For k ≥ 3, we let Metk−1 (Σg ) denote the space of L2k−1 Riemannian metrics η=

2 

ηab dxa dxb

a,b=1

on Σg , an open subset of the Hilbert space of L2k−1 sections of the bundle of symmetric covariant tensors of rank two, and let Metk−1 0 (Σg ) denote the subspace of metrics of constant Gaussian curvature −1. These spaces of Riemannian metrics are acted upon by groups of diffeomorphisms, Diffk (Σg ) = {φ ∈ L2k (Σg , Σg ) such that φ is a diffeomorphism},

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189

a group under composition, and its subgroups Diffk+ (Σg ) = {φ ∈ Diffk (Σg ) : φ is orientation-preserving}, Diffk0 (Σg ) = {φ ∈ Diffk+ (Σg ) : φ is homotopic to the identity}. As mentioned in §1.6, although Diffk (Σg ) and Diffk0 (Σg ) are smooth Hilbert manifolds, the group multiplications are not smooth, so these are not infinitedimensional Lie groups. However, it is true that the map ψ : Metk−1 (Σg ) × Diffk+ (Σg ) −→ Metk−1 (Σg ),

ψ(η, φ) = φ∗ η,

and its restriction k−1 k ψ : Metk−1 0 (Σg ) × Diff+ (Σg ) −→ Met0 (Σg ),

are smooth. Either of these actions can be further restricted to the subgroup Diffk0 (Σg ). Lemma 4.3.1. The action of Diffk0 (Σg ) on Metk−1 0 (Σg ) is free. Proof. Indeed, if the action of Diffk0 (Σg ) on Metk−1 0 (Σg ) were not free, there would be two distinct isometries id, φ : (Σg , φ∗ η) −→ (Σg , η), both homotopic to the identity. Each would be harmonic, contradicting Hartman’s Theorem 4.1.2 which asserts that the harmonic map between two compact surfaces of negative curvature in a given homotopy class is unique. QED Moreover, one easily checks that the images of the maps ψ˜ : Metk−1 (Σg ) × Diffk (Σg ) −→ Metk−1 (Σg ) × Metk−1 (Σg ) 0

+

0

0

and k−1 k−1 k ψ˜ : Metk−1 + (Σg ) × Diff+ (Σg ) −→ Met+ (Σg ) × Met+ (Σg ),

˜ φ) = (η, ψ(η, φ)), are closed. It therefore follows (see Lemma defined by ψ(η, 2.9.9 in Varadarajan [Var74]) that the orbit spaces, Tg =

Metk−1 0 (Σg ) Diffk0 (Σg )

and

Mg =

Metk−1 0 (Σg ) , Diffk+ (Σg )

are Hausdorff. They are called the Teichm¨ uller space and the Riemann moduli space of conformal structures on Σg , respectively. We would like to make Teichm¨ uller space into a manifold using the techniques of global analysis. The procedure for doing this is found in a landmark article of Earle and Eells [EE69], with an alternate presentation given later in Fischer and Tromba [FT84]. Our arguments will mostly follow Chapter 2 of Schoen and Yau [SY97].

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Theorem 4.3.2. If g ≥ 2, the Teichm¨ uller space Tg is an oriented manifold of dimension 6g − 6. Proof. The idea is simple. We let G = Diffk0 (Σg ), and follow a well-known procedure (see Theorem 2.9.10 in Varadarajan [Var74]) to construct for a k−1 given η ∈ Metk−1 0 (Σg ) a submanifold S ⊂ Met0 (Σg ) such that η ∈ S and ∼ (1) Tη Metk−1 0 (Σg ) = Tη S ⊕ Tη (G · η) and (2) any G-orbit intersects S in only one point. Such a submanifold is called a slice for the action, and it allows us to construct an open neighborhood U of η in S and a G-equivariant diffeomorphism from U × G to an open neighborhood of the orbit through η. To construct this slice at a base metric η, we first suppose that we have chosen isothermal coordinates so that the components of our base metric η are ηab = δab λ2 , for some positive function λ2 , and let ·, · denote the G-invariant L2 inner product on Tη Metk−1 (Σg ) such that  (4.25)

η, ˙ η ˙ = Σ

2 1  η˙ ab η˙ ab dx1 dx2 , λ2

η ˙ =

 η, ˙ η. ˙

a,b=1

Note that if X=

2 

Xa

a=1

∂ ∂xa

is a smooth vector field on Σ with corresponding local one-parameter subgroup {φt : t ∈ R}, it follows from a familiar calculation that 2  d ∗ (φ η) = LX η = · · · = Xa;b dxa dxb , dt t t=0 a,b=1

where the Xa;b ’s are the components of the covariant derivative of X with respect to the metric η. We ask for the conditions under which a oneparameter family of metrics, ηab (t) = ηab + tη˙ ab

for t ∈ (, ),

is perpendicular with respect to ·, · to the orbit through η at t = 0. An integration by parts shows that   2 2 1  1  ˙ = η ˙ X dx dx = − η˙ ab;b Xa dx1 dx2 , 0 = LX η, η 1 2 ab a;b 2 2 Σ λ Σ λ a,b=1

a,b=1

for all ) choices of vector fields Xa , so that η˙ is perpendicular to the G-orbit when η˙ ab;b = 0. This suggests a slice for the action of G on the larger

4.3. Minimal surfaces of arbitrary topology

191

space Metk−1 (Σg ), namely   2  k−1 S˜η = η + η˙ : η˙ ∈ Tη Met (Σg ), η ˙ <  and η˙ ab;b = 0 . b=1

To determine the tangent space to the smaller space Metk−1 0 (Σg ) and obtain the desired section for the action on Metk−1 (Σ ), we utilize the following g 0 lemma: Lemma 4.3.3. Suppose that ηab (t) = ηab + tη˙ ab

for

t ∈ (, )

is a one-parameter family of metrics of constant Gaussian curvature K = −1 such that η˙ ab;b = 0. Then ˙ = 2Tr(η), ˙ Δη(0) (Tr(η))

where

Tr(η) ˙ =

1 (η˙ 11 + η˙ 22 ) λ2

and Δη(0) is the Laplace operator for the base metric η(0). Proof. We use geodesic normal coordinates centered for η at a given point p ∈ Σ. In terms of such coordinates, the curvature tensor of η(t) is given by K det(ηab (t)) = R1212 (t) =

1 (2η12,12 (t) − η11,22 (t) − η22,11 (t)) 2 + (higher order terms),

where the commas denote coordinate derivatives. Differentiating with respect to t and setting t = 0, we obtain Kλ2 (η˙ 11 + η˙ 22 ) =

1 (2η˙ 12;12 − η˙ 11;22 − η˙ 22;11 ) , 2

where we have replaced ordinary derivatives by) covariant derivatives and have evaluated at p. Finally, we use the identity η˙ ab;b = 0 to simplify the right-hand side, ⎛ ⎞ 2  1 η˙ aa;bb ⎠ , Kλ2 (η˙ 11 + η˙ 22 ) = − ⎝ 2 a,b=1

which yields  2  1 1 1  η ˙ = −2K 2 (η˙ 11 + η˙ 22 ), aa λ2 λ2 λ ;bb a,b=1

which finishes the proof. QED

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4. Harmonic and Minimal Surfaces

It follows from the lemma and the maximum principle applied to) the operator Δη(0) − 2 that η˙ 11 + η˙ 22 = 0. This, together with the identity η˙ ab;b = 0 implies that if z = x1 + ix2 , then ∂ (η˙ 11 − iη˙ 12 ) = 0, ∂ z¯ and hence (η˙ 11 − iη˙ 12 )dz 2 is a holomorphic quadratic differential. Thus we set Sη = {η + η˙ : η˙ ∈ Tη Metk−1 ˙ <  and 0 (Σg ), η (η˙ 11 − iη˙ 12 )dz 2 is a holomorphic quadratic differential}, a convex open subset of an affine subspace of a Hilbert space. We have seen that one of the implications of the Riemann-Roch Theorem is that the space of holomorphic quadratic differentials on a Riemann surface of genus g has complex dimension 3g − 3 or real dimension 6g − 6, so the affine space Sη has real dimension 6g − 6. A straightforward calculation shows that the identity map ˙ id : (Σg , η) −→ (Σg , η + η) is harmonic with Hopf differential (1/2)(η˙ 11 − iη˙ 12 )dz 2 . If the G-orbit containing η+η˙ intersected Sη in more than one point, we would have homotopic harmonic maps from (Σg , η) to the same target with two different Hopf differentials, contradicting Theorem 4.1.2 on uniqueness for harmonic maps. We have thus shown existence of the slice, which enables us to construct the local coordinates on Tg . The tangent spaces have continuously varying almost complex structures, showing that Tg is orientable and finishing the proof of Theorem 4.3.2. QED We have actually shown that Metk−1 0 (Σg ) as the total space of a principal bundle with structure group G = Diffk0 (Σg ) over Tg , a point of view emphasized by Earle and Eells. Theorem 4.3.4 (Teichm¨ uller). If g ≥ 2, Tg is diffeomorphic to R6g−6 . We refer to various references for the proof. One argument based upon harmonic maps is presented in §3 of Chapter II of [SY97]. The idea is to choose a base Riemann surface (Σ, η0 ) of genus g, let O(K2 ) denote the space of holomorphic quadratic differentials on (Σ, η0 ), a real vector space of dimension 6g − 6, and define a map (4.26)

Ψ : Tg −→ O(K2 )

by

Ψ(Σ, η) = Ωf ,

4.3. Minimal surfaces of arbitrary topology

193

where Ωf is the Hopf differential of the unique harmonic map f from (Σ, η0 ) to (Σ, η). One then needs to show that Ψ is a diffeomorphism. An alternate argument is based upon the theory of quasiconformal maps. This was Teichm¨ uller’s original approach and it was later simplified in a wellknown paper of Bers [Ber60]. One can find a presentation of this theory in [IT92]. Since Tg is contractible, the principle bundle Metk−1 0 (Σg ) over Tg has a smooth section (4.27)

σ : Tg → Metk0 (Σg ),

so the bundle is trivial. Remark 4.3.5. In Teichm¨ uller theory it is customary to regard O(K2 ) as the holomorphic cotangent space to Teichm¨ uller space Tg at a given point ω. The holomorphic tangent space Tω Tg is then the space of harmonic Beltrami differentials η(d¯ z /dz), where η is an antiholomorphic complexvalued function, and we have a dual pairing    i d¯ z 2 2 O(K ) × Tω Tg → C, → φdz , η φηdz ∧ d¯ z. dz 2 Σ These spaces are also related by an isomorphism (the “raising of an index”) ¯ z2 φ¯ d¯ z ¯ z 2 ) = φd¯ = 2 , (4.28) ι : O(K2 ) −→ Tω Tg defined by ι(φd¯ 2 λ d¯ z dz λ dz where λ2 dzd¯ z is the constant curvature Riemannian metric on Σ. We will sometimes use this isomorphism to identify the space of antiholomorphic quadratic differentials with the holomorphic tangent space. Polarization of (4.25) gives an inner product on either the tangent or cotangent space to Tg which we call the Weil-Petersson metric. In contrast to the the Teichm¨ uller space Tg , the Riemann moduli space Mg is not a manifold, but an orbifold which has singularities at those Riemann surface structures which have larger than normal automorphism groups. It is the quotient of Teichm¨ uller space by the properly discontinuous action of the mapping class group Γg = Diff+ (Σg )/Diff0 (Σg ), an action which preserves the orientation of Tg . There is an alternate approach to the Riemann moduli space Mg based upon algebraic geometry in which Mg is shown to be a quasiprojective variety, as described in Chapter 1 of [ACGH]. We would like to extend Teichm¨ uller theory so that it applies to Riemann surfaces of arbitrary genus, including genus zero and one. According to the uniformization theorem for Riemann surfaces, the universal cover of a connected compact Riemann surface always has a metric of constant curvature, and we let Met0 (Σ) denote the space of Riemannian metrics on the

194

4. Harmonic and Minimal Surfaces

connected oriented surface Σ of genus g which which are normalized1 to have ⎧ ⎪ ⎨constant curvature one if g = 0, constant curvature zero with total area one if g = 1, ⎪ ⎩ and constant curvature −1 if g ≥ 2. However, we encounter a problem when we try to extend Theorem 4.3.2, because the group Diffk0 (Σg ) no longer acts freely on Met0 (Σ) when g = 1 or g = 0. Indeed, in the case of genus one, the group G = S 1 × S 1 ⊆ Diffk0 (T 2 ) preserves the metric of constant curvature zero and total area one, while when the genus is zero, the subgroup SO(3) ⊆ Diffk0 (S 2 ) preserves the metric of constant curvature one (and the larger subgroup P SL(2, C) ⊆ Diffk0 (S 2 ) preserves the conformal structure on S 2 .) To get around this problem in the case of genus one, we simply divide by the action of a smaller group, Diff0,p (T 2 ) = {φ ∈ Diff0 (T 2 ) : φ(p) = p}, the group of diffeomorphims which fix a given point p on the torus T 2 . Any element φ ∈ Diff0 (T 2 ) can be factored uniquely as φ = g ◦ φ0 ,

where

φ0 ∈ Diff0,p (T 2 )

and g ∈ G,

where G = S 1 × S 1 is the group of conformal automorphisms of T 2 which are homotopic to the identity, and this establishes a diffeomorphism (4.29) Diff0 (T 2 ) ∼ = G × Diff0,p (T 2 ) = (S 1 × S 1 ) × Diff0,p (T 2 ), although the group structure is not that of a product. In contrast to Diff0 (T 2 ), the action of Diff0,p (T 2 ) on Met0 (T 2 ) is free, and the action is properly discontinuous and smooth when the spaces are given appropriate Sobolev completions. This enables us to construct a manifold structure on the quotient space T1 =

Met0 (T 2 ) , Diff0,p (T 2 )

by a straightforward modification of the proof of Theorem 4.3.2. As before, we suppose that we have chosen isothermal coordinates so that the components of our base metric η are ηab = δab λ2 , for some positive function λ2 , and ask for the conditions under which a one-parameter family of metrics, ηab (t) = ηab + tη˙ ab

for t ∈ (, ),

is perpendicular to the orbit of the Diff0,p (T 2 )-action with respect to the metric (4.25). The one difference in the argument is that when g = 1, the 1 We will sometimes find it convenient to use a slightly different normalization in which all constant curvature metrics are scaled to have total area one.

4.3. Minimal surfaces of arbitrary topology

195

proof for Lemma 4.3.3 only implies that Δη(0) (η˙ 11 + η˙ 12 ) = 0, which in turn implies that η˙ 11 + η˙ 12 is constant. We then need the condition that elements of Met0 (Σ) have total area one to conclude that η˙ 11 + η˙ 12 = 0, the result parallel to the case g ≥ 2. Once we make this minor modification, we can conclude that T1 is a manifold of dimension two in this case. We need to check that the elements of T1 are in one-to-one correspondence with the space H of conformal structures on T 2 , as presented at the end of §4.2.3. Recall that using the standard flat metric on C, we associated to any element ω ∈ H a corresponding explicit flat metric on T 2 by dividing by the action of the group generated by the deck transformations fα (z) = z + 1,

fβ (z) = z + ω.

This defines a map σ : H → Met0 (T 2 ) and one easily checks that the composition with the projection π : Met0 (T 2 ) → T1 into the space of equivalence classes of constant curvature metrics is a diffeomorphism. Thus we can identify T1 with H, and σ : T1 → Met0 (T 2 ) is a section for the projection π : Met0 (T 2 ) → T1 . Once again we find that Met0 (T 2 ) is a principal fiber bundle over T1 which is trivialized by a section σ : T1 → Met0 (T 2 ).

(4.30)

The earlier argument at the end of §4.2 that Teichm¨ uller space is a two-ball is easier than the new argument we have just given, but we will use the new notation for tangent and cotangent spaces as in the case g ≥ 2. Similarly, when g = 0, we choose three distinct points q, r, s ∈ S 2 and divide by the action of Diff0,q,r,s (S 2 ) = {φ ∈ Diff0 (S 2 ) : φ(q) = q, φ(r) = r, φ(s) = s}. This time any element φ ∈ Diff0 (S 2 ) can be factored uniquely as φ = g ◦ φ0 ,

where

φ0 ∈ Diff0,q,r,s (S 2 )

and g ∈ G = P SL(2, C),

where G is the group of conformal automorphisms of S 2 which are homotopic to the identity, and this establishes a diffeomorphism (4.31) Diff0 (S 2 ) ∼ = P SL(2, C) × Diff0,q,r,s (S 2 ). Once again the action of Diff0,q,r,s (S 2 ) is smooth, free and properly discontinuous, and we can construct the quotient as before, but now the quotient is a point, that is, the corresponding Teichm¨ uller space is trivial. These constructions are carried out by Earle and Eells [EE69], who use the theory of quasiconformal maps to show that Diff0,p (T 2 ) and Diff0,q,r,s (S 2 )

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4. Harmonic and Minimal Surfaces

are contractible. This shows that Diff0 (T 2 ) has the torus S 1 ×S 1 as a strong deformation retract, while Diff0 (S 2 ) has P SL(2, C), and hence its maximal compact subgroup SO(3), as a strong deformation retract. The latter fact was first proven by Smale, using quite different methods. 4.3.2. Energy as a function of two variables. We can now consider the energy as a map of two variables, ˜ : L2 (Σ, M ) × Metk−1 (Σg ) −→ R, (4.32) E k 0 . /   ∂f ∂f  1 ab ˜ E(f, (ηab )) = η , det(ηab )dx1 dx2 , 2 Σ ∂xa ∂xb a,b

where (η ab ) is the matrix inverse of (ηab ), the integrand being independent of choice of local coordinates. If φ ∈ Diffk0 (Σg ), then ˜ η), ˜ ◦ φ, φ∗ η) = E(f, E(f ˜ but this action is only free so Diffk0 (Σg ) acts as a group of symmetries for E, when g ≥ 2. We set ⎧ k ⎪ when g ≥ 2, ⎨Diff0 (Σg ), k 2 G = Diff0,p (T ) when g = 1, ⎪ ⎩ k 2 Diff0,q,r,s (S ) when g = 0, and call G the gauge group for the two-variable energy. The gauge group G acts freely in all cases, and the arguments of §4.3.1 show that the quo˜ tient space of orbits is a smooth infinite-dimensional manifold. Moreover, E descends to a smooth map on this quotient space E:

L2k (Σ, M ) × Metk−1 0 (Σg ) −→ R. k Diff0 (Σg )

Equivalently, we can use the section σ of (4.27) or (4.30) to define a smooth map ˜ σ(ω)). E : L2k (Σ, M ) × Tg −→ R, E(f, ω) = E(f, Since Tg is diffeomorphic to a finite-dimensional Euclidean space, L2k (Σ, M )× Tg is homotopy equivalent to L2k (Σ, M ) for k ≥ 2 or C k (Σ, M ) for k ≥ 2. Theorem 4.3.6. If (f, ω) is a critical point for the map E : C 2 (Σ, M ) × Tg −→ R, then f is harmonic and weakly conformal with respect to ω, and hence a parametrized minimal surface. Proof. We first calculate the derivative of E : Map(Σ, M ) × Met0 (Σ) −→ R,

4.3. Minimal surfaces of arbitrary topology

197

and then restrict to the tangent space to a slice for the action of the symmetry group Diff0 (Σ) which is transversal to the orbits. Note that the first variation, dE(f, η) : Tf,η (Map(Σ, M )) × Met0 (Σ)) −→ R, must vanish when restricted to Tf Map(Σ, M )⊕{0}, so f must be a harmonic map. To calculate the derivative in the other direction, we differentiate (4.32), using a perturbation given by the formula ηab (t) = ηab + tη˙ ab ,

(4.33)

where the variation η˙ ab is trace-free (η˙ 11 + η˙ 22 = 0) and the initial metric is given by the formula  det(ηab ) = λ2 , ηab = δab λ2 , so that for some conformal factor λ2 . Then the formulae d d ηab (t) = η˙ ab , det(ηab )(t) = λ2 (η˙ 11 + η˙ 22 ) = 0 dt dt t=0 imply

 11 12    d η η η˙ 22 −η˙ 12 −2 det(ηab ) 21 22 =λ , η η −η˙ 21 η˙ 11 dt t=0

and thus we find that (4.34) . /    d ∂f d 1 ∂f ab dx1 dx2 E(f, ηab (t)) = det(ηab )η , dt 2 Σ dt ∂xa ∂xb t=0 t=0 a,b . / . / . /   1 ∂f η˙ 11 ∂f ∂f ∂f 2η˙ 12 ∂f ∂f =− , , , − + dx1 dx2 2 Σ λ2 ∂x1 ∂x1 ∂x2 ∂x2 λ2 ∂x1 ∂x2 /  .  1 ∂f ∂f , dx1 dx2 , = −2 Re (η˙ 11 + iη˙ 12 ) 2 ∂z ∂z Σ λ where z = x1 + ix2 . We define a complex bilinear map (4.35)  ¯ z2 = 4 ·, · : Γ(K(2,0) ) × Γ(K(0,2) ) → C by φdz 2 , ψd¯ Σ

1 ¯ φψdA, λ4

L2

inner product on the space of quadratic differenthe real part being an tials. Then we can rewrite (4.34) as 1 d ¯ z 2 , E(f, ηab (t)) = −Re Ωf , (η˙ 11 − iη˙ 12 )d¯ z 2  = −ReΩf , φd¯ (4.36) dt 2 t=0 where Ωf is the Hopf differential, which can be defined even if f is not harmonic. Thus the vanishing of the restriction of dE([f, η]) to {0} ⊕ Tω Tg

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4. Harmonic and Minimal Surfaces

is equivalent to the condition (4.37)

P (Ωf ) = 0,

where P is the L2 -orthogonal projection to holomorphic quadratic differentials. At a critical point for E, the Hopf differential Ωf , which is now holomorphic, must vanish. We conclude that critical points for E are indeed conformal harmonic maps, possibly with branch points. QED Remark 4.3.7. For future reference, we record the first variation formula ¯ z2 we derived to prove Theorem 4.3.6. If X is a section of f ∗ T M and ω˙ = φd¯ is an antiholomorphic quadratic differential regarded as an element of Tω Tg , then (4.38)

/ . / ∂f DX ∂f DX ¯ z2 , + , dxdy −ReΩf , φd¯ dE(f, ω) ((X, ω)) ˙ = ∂x ∂x ∂y ∂y Σ     /  . D ∂f D ∂f ¯ z 2 . + ,X dxdy − ReΩf , φd¯ =− ∂x ∂x ∂y ∂y Σ  .

Remark 4.3.8. No matter what the genus of the orientable surface Σ, we get a well-behaved energy on the Teichm¨ uller space level, E : Map(Σ, M ) × T −→ R, but it is invariant under the full symmetry group for E, which depends upon the genus g: (1) When g = 0, E is invariant under the action of the group G = P SL(2, C) of holomorphic automorphisms of S 2 . (2) When g = 1, E is invariant under an action of the semi-direct product G ψ Γ1 where G = S 1 × S 1 is the group of conformal automorphisms homotopic to the identity and Γ1 = SL(2, Z) with ψ : SL(2, Z) → Aut(S 1 × S 1 ) inducing the usual homomorphism to H1 (S 1 × S 1 ; Z) ∼ = Z ⊕ Z. (3) When g ≥ 2, E is invariant under an action of the mapping class group Γg =

Diff+ (Σg ) . Diff0 (Σg )

(Note that in the case g = 1, the group SL(2, Z) acts with kernel ±I on the Teichm¨ uller space T1 .) Of course, we are really interested in classifying points in the quotients obtained by dividing out by the remaining actions.

4.3. Minimal surfaces of arbitrary topology

199

Indeed, we might regard distinct points of the quotient space (4.39)

M(Σg , M ) =

Map(Σg , M ) × Met0 (Σg ) Diff+ (Σg )

as “geometrically distinct” parametrized minimal surfaces, but this quotient is not a manifold because the group actions have different isotropy groups at different points. A similar problem occurred in the theory of closed geodesics as presented in Chapter 2, because the action  1 2 1 (S , M ) −→ R, defined by J(γ) = γ  (t), γ  (t)dt, J : L1 2 S1 is invariant under an action of the symmetry group G = S 1 . One might be tempted to develop a Morse theory on the quotient space of G-orbits, but the quotient is not a manifold, and Bott pointed out in [Bot82] that efforts to use the quotient led to mistakes in early work on closed geodesics. A safer approach for closed geodesics is based upon equivariant Morse inequalities involving the G-equivariant cohomology of L21 (S 1 , M ) (as discussed, for example, in Hingston’s article [Hin84]). Similarly, in the case of minimal surfaces, we prefer to do analysis on the well-behaved manifold Map(Σ, M ) × T . Instead of using the actual quotient (4.39), we utilize the homotopy quotient for constructing cohomology constraints for constructing minimax minimal surfaces. Our strategy is to use G  Γ-equivariant cohomology of Map(Σ, M ) × T to construct partial equivariant Morse inequalities for E. Remark 4.3.9. If θ is a smooth two-form on M , we can define the θ-energy  k−1 θ 2 θ ˜ ˜ ˜ f ∗ θ, E : Lk (Σ, M ) × Met0 (Σg ) −→ R by E (f, ω) = E(f, ω) + Σ

˜ is the energy defined by (4.32). This energy is invariant under where E action of Diffk0 (Σg ), and hence it descends to a well-defined function E θ : L2k (Σ, M ) × Tg −→ R. If θ is closed, the integral of θ over Σ depends only on the de Rham cohomology class of θ and is constant over each component of Map(Σ, M ), and in this case, E θ has exactly the same critical points as E. In general, however,  θ ˙ η) ˙ = dE(f, ω)(X, ˙ η) ˙ + f ∗ LX (θ), dE (f, ω)(X, Σ

where LX denotes the Lie derivative of θ in the direction of X. This modified energy has a theory quite similar to that for E itself, and it is useful in Gromov’s theory of J-holomorphic curves, including those which are not compatible or tame with respect to a symplectic structure on M ; see [Gv78].

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The argument for Theorem 4.3.6 shows that critical points for E θ are weakly conformal. We will see that much of the analysis of the energy E can be extended to E θ . 4.3.3. Nonorientable minimal surfaces. We now describe how nonorientable minimal surfaces can be studied via their oriented double covers. (Readers interested only in the oriented case can skip this topic on a first reading.) If f0 : Σ0 → M is a nonorientable parametrized surface in M , the nonorientable domain Σ0 has an oriented double cover Σ with projection π : Σ → Σ0 , and there is an orientation-reversing sheet interchange map A : Σ → Σ with A2 = id such that Σ0 is the quotient of Σ under the action of A. We study f0 via its oriented lift f = f0 ◦ π : Σ → M . Consider first the function space needed to study minimal projective planes. A parametrized projective plane in M can be regarded as a smooth map f : S 2 −→ M such that f ◦ A = f, where A : S 2 → S 2 is an “antipodal map”. If we want to preserve the P SL(2, C)-symmetry, we need to consider a family of such antipodal maps Ax parametrized by a point x the Poincar´e disk H3 = {x ∈ R3 : x · x ≤ 1}. We define the antipodal map A0 which corresponds to the origin 0 ∈ H3 as the restriction of the endomorphism x → −x of R3 to the unit sphere S 2 ⊆ R3 , an isometry with respect to the standard constant curvature metric η0 on S 2 which has Gaussian curvature one. Any linear fractional transformation g ∈ P SL(2; C) defines an isometry gˆ of H3 with its hyperbolic metric which restricts to a conformal map of S 2 , and we set x = x(g) = gˆ(0),

ηx = g ∗ η0 ∈ Met0 (S 2 ) and

Ax = g ◦ A0 ◦ g −1 .

We thus obtain a family of antipodal maps x → Ax , each such map being an isometry for the constant curvature metric ηx ∈ Met0 (S 2 ), and for each antipodal map we can define a projection πx : S 2 → RPx2 , the image being a copy of the projective plane. We then let : {Mapx (RP 2 , M ) : x ∈ H3 }, (4.40) M(RP 2 , M ) = the total space of a fiber bundle over H3 with fiber being Mapx (RP 2 , M ). Over the family M(RP 2 , M ) we have a corresponding family x → A of antipodal maps on Map(S 2 , M ). The full symmetry group G = P SL(2, C) acts on M(RP 2 , M ), while only its maximal compact subgroup G0 = SO(3) acts on any fiber.

4.3. Minimal surfaces of arbitrary topology

201

Any parametrized surface f : S 2 → M of nonzero area has a well-defined center of mass x(f ) defined using the energy density e(f ) = (1/2)|df |2 of f by ; (1/4π) S 2 Xe(f )dAS 2 x(f ) = , E(f ) where dAS 2 is the area element for the constant curvature one metric on S 2 and X : S 2 → R3 is the standard inclusion onto the unit sphere in R3 . If f : S 2 → M is the lift of a nonconstant parametrized projective plane f0 : RPx2 → M defined by means of the antipodal map Ax , then it has center of mass x. We will usually regard Map(RP 2 , M ) as the fixed point set of the involution A0 on Map(S 2 , M ), Map(RP 2 , M ) = {f ∈ Map(S 2 , M ) : f ◦ A0 = f } ⊆ Map(S 2 , M ), but sometimes it will be convenient to use the larger space M(RP 2 , M ), which represents parametrized projective planes whose oriented double covers have varying centers of mass. We can construct a similar function space for studying minimal Klein bottles. A parametrized Klein bottle can be regarded as being covered by a parametrized torus, the domain having a flat metric of area one that is invariant under an orientation-reversing deck transformation As , for some s ∈ S 1 , which consists of a translation composed with a reflection, expressed in terms of standard coordinates (t1 , t2 ) on the torus as As (t1 , t2 ) = (t1 + 1/2, −t2 + s). Then As preserves the circle t2 = s/2, and it satisfies the identity A2s (t1 , t2 ) = (t1 + 1, t2 ). If we fix s, we once again have a one-to-one correspondence Maps (K 2 , M ) ∼ = {f ∈ Map(T 2 , M ) : f ◦ As = f }, although this destroys part of the S 1 × S 1 -symmetry of the torus. We let : M(K 2 , M ) = {Maps (K 2 , M ) : s ∈ S 1 }, the total space of a fiber bundle over S 1 with fiber being Maps (K 2 , M ). The symmetry group G = S 1 × S 1 acts on M(K 2 , M ), while only the subgroup G0 = S 1 acts on any individual fiber. We will use the notation Map(K 2 , M ) to denote the space in which A0 is the distinguished choice of orientationreversing deck transformation. Finally, if f0 : Σ0 → M is a nonorientable parametrized surface with oriented double cover f : Σ → M of genus g at least two, we let A : Σ → Σ

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4. Harmonic and Minimal Surfaces

be a sheet interchange map for the double cover. The appropriate function space for the nonorientable surface is then Map(Σ0 , M ) ∼ = {f ∈ Map(Σ, M ) : f ◦ A = f }. When the genus of the double cover of Σ0 is at least one, we can define the Teichm¨ uller space for Σ0 as the quotient TΣ0 =

Met0 (Σ0 ) , Diff0 (Σ0 )

with Met0 (Σ0 ) being the space of metrics of constant curvature zero and total area one-half when g = 1 and constant curvature −1 when g ≥ 2, and Diff0 (Σ0 ) is the space of diffeomorphisms of Σ0 which are homotopic to the identity. Each element of Met0 (Σ0 ) pulls back to an element of Met0 (Σ), and once we have fixed A, each element of Diff0 (Σ0 ) pulls back to a unique element of Diff0 (Σ) which commutes with A. Thus TΣ0 ∼ =

{η ∈ Met0 (Σ) : A∗ η = η} . {φ ∈ Diff0 (Σ) : A ◦ φ = φ ◦ A}

In the case of a double cover of a Klein bottle, we arrange that the differential of A = As fixes the generator corresponding to 1 in the fundamental parallelogram, and the orthogonal line in the complex plane must then be preserved under the reflection u + iv → u − iv, so the Teichm¨ uller space TK of flat Klein bottles with total area one-half consists of the positive real numbers ω = iv with v > 0, the fixed point set of the involution on the upper half-plane, A : H −→ H,

A (u + iv) = −u + iv.

When the oriented double cover Σ has genus g ≥ 2, the proof of Theorem 4.3.2 carries over to show that the nonorientable Teichm¨ uller space TΣ0 is a smooth manifold. The sheet interchange map A induces a conjugate linear map A on quadratic differentials on Σ, and the argument for Theorem 4.3.2 shows that the cotangent space to TΣ0 is the space of holo¯ z 2 . Thus morphic quadratic differentials hdz 2 on Σ such that A (hdz 2 ) = hd¯ the dimension of TΣ0 is exactly half the dimension of the Teichm¨ uller space of its double cover. For the record, we mention that the harmonic map proof of Teichm¨ uller’s Theorem 4.3.4 can be extended to show that TΣ0 is diffeomorphic to a finite-dimensional Euclidean space, and therefore Map(Σ0 , M ) × TΣ0

is homotopy equivalent to

Map(Σ0 , M ),

when Σ0 is nonorientable, just like in the oriented case. Indeed, the diffeomorphism Ψ of (4.26) can be used to transport the involution A on

4.3. Minimal surfaces of arbitrary topology

203

holomorphic quadratic differentials to an involution A on the Teichm¨ uller space T of the oriented double cover, and one finds that TΣ0 = {ω ∈ T : A (ω) = ω}. In all cases, each nonorientable parametrized surface f0 : Σ0 → M can be lifted to an orientable double cover f : Σ → M , establishing a bijection Map(Σ0 , M ) ∼ = {f ∈ Map(Σ, M ) : f ◦ A = f }, with A : Σ → Σ being a suitable sheet interchange map on the oriented surface Σ which double covers Σ0 . The map A : Map(Σ, M ) → Map(Σ, M ) induces a real-linear involution A on f ∗ T M , as well as on the space of sections of f ∗ T M . Moreover, the involutions preserve both the metric and the pullback of the Levi-Civita connection. We can construct a direct sum decomposition Γ(f ∗ T M ) = Γ+ (f ∗ T M ) ⊕ Γ− (f ∗ T M ), where Γ+ (f ∗ T M ) = {X ∈ Γ(f ∗ T M ) : A (X) = X}, Γ− (f ∗ T M ) = {X ∈ Γ(f ∗ T M ) : A (X) = −X}. It is the space of sections of Γ+ (f ∗ T M ) which can be regarded as deformations of the nonorientable minimal surface f0 : Σ → M itself, as opposed to deformations which would separate the two sheets of the orientable double cover. In summary, when Σ0 is nonorientable, up to a factor of two, the twovariable energy E : Map(Σ0 , M ) × TΣ0 −→ R, can be thought of as the energy of the oriented double cover restricted to the fixed-point set of an involution, E : {f ∈ Map(Σ, M ) : f ◦ A = f } × {ω ∈ T : A (ω) = ω} −→ R. We can apply the first variation formula (4.38) to this lift, obtaining (4.41)

/ . / ∂f DX ∂f DX ¯ z2 , + , dxdy −ReΩf , φd¯ dE(f, ω) ((X, ω)) ˙ = ∂x ∂x ∂y ∂y Σ     /  . D ∂f D ∂f ¯ z 2 , =− + ,X dxdy − ReΩf , φd¯ ∂x ∂x ∂y ∂y Σ ¯ z 2 being an antiholomorphic quadratic differential ¯ z 2 ), φd¯ with ω˙ = Re(φd¯  .

tangent to TΣ0 , X an element of Γ+ (f ∗ T M ). This yields the following consequence: Theorem 4.3.10. If Σ0 is a nonorientable compact connected surface and (f, ω) is a critical point for the map (4.42)

E : Map(Σ0 , M ) × TΣ0 −→ R,

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then f is harmonic and weakly conformal with respect to ω, and hence a nonorientable parametrized minimal surface. Remark 4.3.11. If Σ0 = RP 2 the symmetry group of (4.42) is SO(3) instead of P SL(2, C). Sometimes it is convenient to extend (4.42) to E : M(RP 2 , M ) −→ R. Note that if M0 (RP 2 , M ) = {(f, x) ∈ M(RP 2 , M ) : E(f ) = 0}, then

M0 (RP 2 , M ) ∼ = {f ∈ Map(S 2 , M ) : f ◦ Ax(f ) = f },

where x(f ) is the center of mass of f . Thus SO(3)-orbits of nonconstant minimal projective planes in M can be regarded as P SL(2, C)-orbits of nonconstant minimal two-spheres invariant under the family of antipodal maps {Ax : x ∈ H3 }. Similarly, if Σ = K 2 , we can restore the symmetry group to S 1 × S 1 by extending (4.42) to E : M(K 2 , M ) × TK −→ R, and S 1 -orbits of nonconstant minimal Klein bottles can be regarded as certain S 1 × S 1 -orbits of minimal tori.

4.4. The α-energy 4.4.1. Perturbations of the energy. For the case where Σ is a compact oriented surface of arbitrary genus g ≥ 0 with conformal structure ω and M is a smooth Riemannian manifold, we would like to develop a critical point theory for  1 |df |2 dA, Eω : Map(Σ, M ) −→ R, Eω (f ) = 2 Σ where Map(Σ, M ) denotes the space of C ∞ maps from Σ to M . Indeed, we would like a similar theory for the two-variable energy E : Map(Σ, M )(Σ, M ) × T −→ R,

E(f, ω) = Eω (f ),

where T is the Teichm¨ uller space for Σ, in analogy with the Morse theory we developed earlier for  1 |γ  (t)|2 dt. J : L21 (S 1 , M ) −→ R, J(γ) = 2 S1 To carry this out, we would need to complete the space Map(Σ, M ) with respect to an appropriate Sobolev norm, chosen so that Eω would satisfy Condition C. For the energy Eω , however, this seems to require that we complete the space Map(Σ, M ) with respect to the L21 -topology, a topology

4.4. The α-energy

205

which is too weak for the usual techniques of global analysis, since the Sobolev inequality barely fails to show that L21 -functions are continuous. Thus Sacks and Uhlenbeck [SU81], [SU82] were led to consider perturbations of the energy that could be defined on L2α 1 (Σ, M ), for α > 1, which does lie within C 0 (Σ, M ). Among the simplest such perturbations are functions  1 defined by Fφ,η (f ) = φ(|df |2 )dA, Fφ,η : Map(Σ, M ) → R 2 Σ where η is a Riemannian metric on Σ and φ : [0, ∞) → [0, ∞) is a C 1 function, smooth on (0, ∞), which has suitable growth properties. The differential of this function is just the first variation as calculated in §4.1, and if we carry this out in terms of isothermal coordinates (x, y) on Σ, we obtain / . / .  ∂f DV ∂f DV  2 dFφ,η (f )(V ) = , + , dxdvy. φ (|df | ) ∂x ∂x ∂y ∂y Σ An integration by parts yields     /  . D D  2 ∂f  2 ∂f φ (|df | ) + φ (|df | ) , V dxdy, dFφ,η (f )(V ) = − ∂x ∂y ∂y Σ ∂x which vanishes for all variation fields V if and only if f satisfies the EulerLagrange equation     D D  2 ∂f  2 ∂f φ (|df | ) + φ (|df | ) = 0. (4.43) ∂x ∂x ∂y ∂y As a first attempt, we might take φ(t) = tα , where α > 1. Thus for a given Riemannian metric η on Σ, we could define  1 2α & & |df |2αdA. (4.44) Eα,η : L1 (Σ, M ) → R by Eα,η (f ) = 2 Σ &α,η This perturbation has been much studied, but the critical points of E are not necessarily smooth. One can see the crux of the problem if we set M = R, in which case the Euler-Lagrange equation     ∂ ∂ 2α−2 ∂f 2α−2 ∂f |df | + |df | =0 ∂x ∂x ∂y ∂y can be rewritten in terms of polar coordinates (r, θ) on the plane as     1 ∂ 1 ∂ 2α−2 ∂f 2α−2 ∂f r|df | + 2 |df | = 0. r ∂r ∂r r ∂θ ∂θ If D is the unit disk in the (x, y)-plane, the function f : D → R defined in polar coordinates by f (r, θ) = r(2α−2)/(2α−1) satisfies this equation. Although this function is not C 1 , it can be checked that it does lie in L2α 1 (D, R). Thus 2α & we see that critical points for Eα,η in L1 are not necessarily C ∞ .

206

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However, the slight modification in which φ(t) = (1 + t)α − 1 provides a function for which all critical points are C ∞ , and that is the primary choice for the theory that we develop. (In PDE terminology, we would say that this choice renders the variational equation uniformly elliptic.) Definition. Suppose that M is a smooth manifold with Riemannian metric g. Given a Riemannian metric η on Σ, the α-energy is the function Eα,η : L2α 1 (Σ, M ) → R defined by  1 [(1 + |df |2 )α − 1]dA, (4.45) Eα,η (f ) = 2 Σ for α > 1, where dA is calculated with respect to η and |df | is calculated with respect to η and g. For ω ∈ T , the (α, ω)-energy Eα,ω is just the function Eα,η , where η is chosen to be the constant curvature metric within the conformal class ω which has constant curvature and total area one, which we have seen is unique up to the action of Diff0 (Σ). The normalization of the metric on Σ used in this definition differs by a scale factor from the normalization we used in Teichm¨ uller theory (see §4.3.1). Note that Eα,η is nonnegative and assumes the value zero exactly at the constant maps. Moreover, the α-energy Eα,ω approaches the usual energy Eω as α → 1. The α-energy on M is a composition of the map ωi , induced via Theorem 1.4.7 by an isometric imbedding i : M → RN , with the α-energy on RN . N The latter function, Eα,ω : L2α 1 (Σ, R ) → R, is clearly continuous, and it is 2 also C with derivatives  (4.46) dEα,ω (f )(V ) = α (1 + |df |2 )α−1 df · dV dA Σ

and

 2

(1 + |df |2 )α−1 dV1 · dV2 dA  + 2α(α − 1) (1 + |df |2 )α−2 (df · dV1 )(df · dV2 )dA,

(4.47) d Eα,ω (f )(V1 , V2 ) = α

Σ

Σ

by the same arguments we used in Example 1.2.6. Since the composition of 2 C 2 maps is C 2 we see that the map Eα,ω on L2α 1 (Σ, M ) is also C . If we let ω ∈ T vary and set Eα (f, ω) = Eα,ω (f ), we get a C 2 -function Eα : L2α 1 (Σ, M ) × Tg → R. Definition. A critical point f ∈ L2α 1 (Σ, M ) for Eα,ω is called an α-harmonic map for the conformal structure ω. If (f, ω) ∈ L2α 1 (Σ, M ) × T is a critical point for Eα , f is called an α-minimal surface.

4.4. The α-energy

207

In complete analogy with harmonic maps, we can calculate the first variation formula for α-harmonic maps from a compact Riemann surface Σ. Indeed, it follows from (4.43) that   / .   /  . D D 2 ∂f 2 ∂f dEα,ω (f )(V ) = −α γ ,V + γ ,V dxdy, ∂x ∂x ∂y ∂y Σ where γ 2 = (1 + |df |2 )α−1 , and (x, y) is a choice of conformal coordinates. This leads to the Euler-Lagrange equation     D D ∂f ∂f (4.48) (1 + |df |2 )α−1 + (1 + |df |2 )α−1 = 0. ∂x ∂x ∂y ∂y This equation can also be written in terms of the standard Laplace operator acting on f :     D ∂f D ∂f + (4.49) ∂x ∂x ∂y ∂y   ∂ ∂ 2 ∂f 2 ∂f (log(1 + |df | )) + (log(1 + |df | )) . = −(α − 1) ∂x ∂x ∂y ∂y 4.4.2. Condition C for the α-energy. Our next goal is to show that the function Eα,ω : L2α 1 (Σ, M ) → R satisfies Condition C: Theorem 4.4.1. If α > 1, the function Eα,ω : L2α 1 (Σ, M ) → R satisfies Condition C. In other words, if {fi } is a sequence of points in L2α 1 (Σ, M ) such that Eα,ω (fi ) ≤ E0 ,

dEα,ω (fi ) → 0,

where E0 is some constant, and

then {fi } possesses a subsequence which converges in L2α 1 (Σ, M ) to a critical point for Eα,ω . The proof of Theorem 4.4.1 is very similar to the proof of Condition C for the action integral in the theory of smooth closed geodesics. If V is tangent α to L2α 1 (Σ, M ) and φ(t) = (1 + t) − 1, then / . / .  DV ∂f DV ∂f  2 , + , dxdy dEα,ω (f )(V ) = 2 φ (|df | ) ∂x ∂x ∂y ∂y Σ / . / .  DV ∂f DV ∂f , + , dxdy = 2α (1 + |df |2 )α−1 ∂x ∂x ∂y ∂y Σ or, equivalently,

 .

dEα,ω (f )(V ) = −2α Σ

D ∂x

 γ

2 ∂f

∂x



D + ∂y

 γ

2 ∂f

∂y



/ ,V

dxdy,

2 2 α−1 and D denotes covariant for V ∈ Tf (L2α 1 (Σ, M )), where γ = (1 + |df | ) N derivative. More generally, we can consider an element V ∈ Tf (L2α 1 (Σ, R ))

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and project it into the tangent space to L2α 1 (Σ, M ). Starting with the or∗ N thogonal projection P : i T R → T M into the tangent space, we use the ω-Lemma to construct a smooth map ∗ N 2α ωP : L2α 1 (Σ, i T R ) −→ L1 (Σ, T M ). ∗ N Lemma 4.4.2. If V ∈ Tf (L2α 1 (Σ, i T R ),   ∂V ∂f ∂V ∂f · + · dEα,ω (f )(ωP (V )) = 2 φ (|df |2 ) ∂x ∂x ∂y ∂y Σ      ∂f ∂f ∂f ∂f −V · α , +α , dxdy, ∂x ∂x ∂y ∂y

where α : T M × T M → N M is the second fundamental form of M in RN . Proof. In the proof, we can suppose that f and V are C ∞ , since the C ∞ ⊥ ⊥ maps are dense in L2α 1 . We write V = ωP (V ) + V , where ωP (V ) and V are the components of V which are tangent and normal to M . Then          ∂ ∂ ∂f ∂f ∂f ∂ γ2 ·V = γ2 · ωP (V ) + γ2 ·V⊥ ∂x ∂x ∂x ∂x ∂x ∂x      D ∂f ∂f 2 ∂f 2 γ · ωP (V ) + γ α , ·V⊥ = ∂x ∂x ∂x ∂x and, similarly,         D ∂f ∂f ∂f ∂f ∂ γ2 ·V = γ2 · ωP (V ) + γ 2 α , · V ⊥, ∂y ∂y ∂y ∂y ∂y ∂y where the dot denotes the dot product in an ambient Euclidean space RN which contains M as an isometrically imbedded submanifold. It follows that dEα,ω (f )(ωP (V ))       ∂ ∂ ∂f ∂f γ2 + γ2 ·V = −2α ∂x ∂x ∂y ∂y Σ       ∂f ∂f ∂f ∂f 2 ⊥ +γ α , +α , ·V dxdy. ∂x ∂x ∂y ∂y An integration by parts now yields the statement of the lemma. QED Lemma 4.4.3. There is a constant C depending only on M such that ≤ C(1 + Eα,ω (f ))V L2α ωP (V )L2α 1 1

∗ N for V ∈ Tf (L2α 1 (Σ, i T R )).

Proof. To prove this, we apply the Leibniz rule to the equation (ωP (V ))(p) = Pf (p) V (p),

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209

where Pf (p) is the projection from RN into the tangent space Tf (p) M , obtaining     ∂P ◦ f ∂V ∂ (ωP (V ))(p) = (V (p)) + Pf (p) (p) , ∂x ∂x ∂x     ∂ P ◦ ∂f ∂V (ωP (V ))(p) = (V (p)) + Pf (p) (p) . ∂y ∂y ∂y Thus that after possibly replacing the usual L2α 1 norm with an equivalent norm, we can write = ωP (V )L2α + dωP (V )L2α , ωP (V )L2α 1 but

  



∂P ◦ f ∂V (V (p))dx + P (p) dx

dωP (V )L2α =

f (p)

2α

∂x ∂x

   L



∂P ◦ f ∂V (V (p))dy + Pf (p) (p) dy

+

2α

∂y ∂y L ≤ C1 sup{(V (p)) : p ∈ Σ} df L2α + C2 DV L2α , where C1 and C2 are positive constants. Thus we see that $ % ≤ C V  ωP (V )L2α 2α Eα,ω (f ) + C4 DV  2α + C5 V L2α , 3 L L 1 1 where C3 , C4 and C5 are yet other positive constants and we have used the fact that the C 0 -norm is less than some constant times the L2α 1 -norm. The last estimate yields the statement of the lemma. QED Proof of Theorem 4.4.1. Let {fi } be a sequence from L2α 1 (Σ, M ) ⊂ N ) such that E (Σ, R (f ) is bounded and dE (f ) → 0. By the L2α α,ω i α,ω i 1 Sobolev Lemma 1.4.4 each fi ∈ C 0 (Σ, M ). Moreover, since M is comolder estimate pact, {fi } is uniformly bounded, and it follows from the H¨ in Lemma 1.4.4 that {fi } is equicontinuous. Therefore, the Arzela-Ascoli Theorem from real analysis implies that a subsequence of {fi } converges uniformly to a continuous map f∞ : Σ → M . Since Eα,ω (fi ) is bounded and fi is bounded in C 0 , it follows that fi is N bounded in L2α 1 (Σ, R ). It then follows from Lemma 4.4.3 that ωP (fi − fj ) N is bounded in L2α 1 (Σ, R ). Since dEα,ω (fi ) → 0, |dEα,ω (fi )(ωP (fi − fj )) − dEα,ω (fj )(ωP (fi − fj ))| → 0

as

i, j → ∞.

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Using Lemma 4.4.2, we can rewrite this as       ∂f ∂f ∂f ∂f ∂f ∂f j j i i i i  2 2 φ (|dfi | ) · − + · − ∂x ∂x ∂x ∂y ∂y ∂y Σ       ∂fi ∂fi ∂fi ∂fi , +α , · (fi − fj ) dxdy − α ∂x ∂x ∂y ∂y       ∂fj ∂fj ∂fi ∂fj ∂fi ∂fj  2 · − + · − − 2 φ (|dfj | ) ∂x ∂x ∂x ∂y ∂y ∂y Σ       ∂fj ∂fj ∂fj ∂fj , +α , · (fi − fj ) dxdy → 0 − α ∂x ∂x ∂y ∂y as i and j approach infinity. Note that since energy is bounded,    ∂f ∂f i i α , · (fi − fj )dxdy ≤ (constant) sup |fi − fj | → 0 and ∂x ∂x Σ    α ∂fi , ∂fi · (fi − fj )dxdy ≤ (constant) sup |fi − fj | → 0 ∂y ∂y Σ as i and j approach infinity, and hence    2 φ (|dfi |2 )dfi , dfi − dfj dA − φ (|dfj | )dfj , dfi − dfj dA → 0 Σ

Σ

or, equivalently,  (4.50) 2α (1 + |dfi |2 )α−1 dfi , dfi − dfj dA Σ  2 α−1 dfj , dfi − dfj dA → 0, −2α (1 + |dfj | ) Σ

as i and j approach infinity. We can rewrite this as |dEα,ω (fi )(fi − fj ) − dEα,ω (fj )(fi − fj )| → 0. To finish the proof, we need a lemma concerning the function ψ : R2N +1 −→ R defined by

ψ(v) = |v|2α .

Lemma 4.4.4. For a given choice of α > 1, there is a constant c ≥ (1/16) such that (Dψ(v) − Dψ(w))(v − w) = 2α(|v|2α−2 v − |w|2α−2 w) · (v − w) ≥ c|v − w|2α for all v, w ∈ R2N +1 − {0}. Proof. A direct calculation shows that Dψ(v)(w) = 2α|v|2α−2 v · w, D 2 ψ(v)(w, w) = 4α(α − 1)|v|2α−4 (v · w)2 + 2α|v|2α−2 w · w ≥ 2α|v|2α−2 w · w.

4.4. The α-energy

211

Hence  1 Dψ(v)(v − w) − Dψ(w)(v − w) = D 2 ψ(w + ψ(v − w))(v − w, v − w)dt 0  1 |w + t(v − w)|2α−2 |v − w|2 dt ≥ c|v − w|2α , ≥ 2α 0

for some constant c > 0. To see that c can be chosen to be larger than 1/16, note that by the triangle inequality, either 1 1 1 |w| ≥ |v − w| ⇒ |w + t(v − w)| ≥ |w| ≥ |v − w| for t ∈ [0, 1/4] 2 2 4 or 1 1 1 |v| ≥ |v − w| ⇒ |v + (1 − t)(w − v)| ≥ |v| ≥ |v − w| for t ∈ [3/4, 1]. 2 2 4 This proves the lemma. QED We now set

 v=

∂fi ∂fi , 1, ∂x ∂y



 ,

w=

∂fj ∂fj 1, , ∂x ∂y

 .

Then it follows from (4.50) and Lemma 4.4.4 that  |dfi − dfj |2α dA → 0 as i, j → ∞. Σ N Since fi → f∞ in C 0 , it follows that {fi } is a Cauchy sequence in L2α 1 (Σ, R ). N By completeness of the Banach space L2α 1 (Σ, R ), there exists an eleN 2α N ˜ ˜ ment f˜∞ ∈ L2α 1 (Σ, R ) such that fi → f∞ in L1 (Σ, R ). Clearly, f∞ = 2α f∞ ∈ L1 (Σ, M ), and Theorem 4.4.1 is proven. QED

The following integral formula obtainable from Lemma 4.4.4 will be useful later: N Lemma 4.4.5. If f, V ∈ L2α 1 (Σ, R ), then



|dEα,ω (f + V )(V ) − dEα,ω (f )(V )| ≥ c

|dV |2α dA, Σ

for some c > 0. Proof. Simply set   ∂ ∂ (f + V ), (f + V ) , v = 1, ∂x ∂y in Lemma 4.4.4 and integrate over Σ. QED

 w=

∂f ∂f 1, , ∂x ∂y



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4. Harmonic and Minimal Surfaces

Remark 4.4.6. One can check that Condition C also holds for the function &α,η of (4.44) when α > 1, although the critical points of this function are E not necessarily smooth. Condition C also holds for various lower-order perturbations of the αenergy. For example, if α > 1 and ψ ∈ L2k (Σ, RN ) for k ≥ 2, the perturbed function  2α f · ψdA Eα,ψ,ω : L1 (Σ, M ) → R, defined by Eα,ψ,ω (f ) = Eα,ω (f ) + Σ

satisfies Condition C, because the additional linear term involving ψ does not affect the argument of Theorem 4.4.1. The reader can check that another example is the φ1 -energy, where φ1 : [0, ∞) → [0, ∞) is a C 1 function such that φ1 (t) = (1 + t)α − 1 for t ∈ (0, t0 ], φ1 (t) ≥ α(1 + t)α−1

for t ∈ (0, ∞).

In other words, we can allow a higher growth rate near infinity without destroying Condition C. 4.4.3. Regularity. Finally, we consider regularity of critical points for the α-energy: Regularity Theorem 4.4.7. If 1 < α < 3/2, any critical point f ∈ ∞ L2α 1 (Σ, M ) for Eα,ω is C . Sketch of proof. The argument uses techniques from nonlinear partial differential equations, and following [SU81], we divide it into three steps: First we show that the critical point f is in L22 , then in a H¨older space C 1,β , and finally, we use the Schauder theory [GT83] together with elliptic bootstrapping to prove that f is C ∞ . Step 1. We show that f lies in L22 (Σ, M ), using the theory of difference quotients presented in Evans [Eva98], §8.3. Regularity is a local condition, and we need only show that a critical point f is regular near a given point p ∈ Σ. Let U and V be small open neighborhoods of p with V¯ ⊂ U . Our strategy is to work in local coordinates (u1 , . . . , un ) defined on a neighborhood of f (U ) in M and a local isothermal coordinates (x1 , x2 ) on U ⊂ Σ. The coordinate system on M allows us to regard f |U as an Rn -valued map. We abbreviate the composition ui ◦ f to ui , and let η=

2  a,b=1

ηab dxa dxb

and

g=

n  i,j=1

gij dui duj

4.4. The α-energy

213

denote the Riemannian metrics on Σ and M , respectively. Let pia

∂ui = ∂xa

Then 1 Eα,ω (f |U ) = 2

|df | = 2

so

n 2  

gij (u1 , . . . un )η ab (x1 , x2 )pia pjb .

a,b=1 i,j=1

 L(p, x, u)dx1 dx2 , U

where

L(p, x, u) = (1 + |df |2 )α



det(ηab ).

The fact that f |V is (α, ω)-harmonic is expressed by the Euler-Lagrange condition ⎡ ⎤    ∂L ∂L ∂ ⎣ (ζ 2 vi ) + ζ 2 vi ⎦ dx1 dx2 = 0, (4.51) i ∂x ∂p ∂u a i U a a,i

i

for every smooth Rn -valued test function v = (v1 , . . . , vn ) on U , ζ : Σ → [0, 1] being a suitable smooth cutoff which is one on V and zero outside U . Note that the first derivatives of the Lagrangian L are given by  ∂ ∂L = α(1 + |df |2 )α−1 i (|df |2 ) det(ηab ), i ∂pa ∂pa where

 ∂ 2 (|df | ) = gij (u1 , . . . un )η ab pjb . ∂pia 2

n

b=1 j=1

Similarly, the second derivatives of L are given by ∂2L ∂pia ∂pjb

 2 (|df | ) det(ηab ) ∂pia ∂pjb  ∂ ∂ + α(α − 1)(1 + |df |2 )α−2 i (|df |2 ) j (|df |2 ) det(ηab ), ∂pa ∂p ∂2

= α(1 + |df |2 )α−1

b

where ∂2 ∂pia ∂pjb

(|df |2 ) = gij (u1 , . . . , un )η ab .

Note that by scaling the coordinates appropriately, we can make |∂gij /∂uk | and |∂ 2 gij /∂uk ∂xl | less than  for any given  > 0. As a consequence of these estimates, we find that the Euler-Lagrange operator for the (α, ω)-energy is uniformly elliptic: n 2  

∂ 2L

ξi ξj j a b i ∂pa ∂pb a,b=1 i,j=1

≥ αγ

2

n 2   a,b=1 i,j=1

gij (u1 , . . . un )η ab

 det(ηab )ξai ξbj ,

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4. Harmonic and Minimal Surfaces

where γ 2 = (1+|df |2 )α−1 . The estimates also allow us to apply the difference quotient approach described in Evans [Eva98] to show that the map u = (u1 , . . . un ) : V → Rn is in L22 . Since Σ can be covered by finitely many neighborhoods V , it follows that f ∈ L22 (Σ, M ), finishing Step 1. &α and We remark that it is uniform ellipticity that fails for the function E this explains why critical points of this function are not necessariy smooth. Step 2. For the second step, we recall that equation (4.49) for α-harmonic maps can be written in the form (Δf ) = −(α − 1)d(log(1 + |df |2 )) · df, where

Df (Δf ) = ∂x1



∂f ∂x1



Df + ∂x2



∂f ∂x2

 ,

D denoting the Levi-Civita connection on M . Since we know by Step 1 that f is L22 , we are justified in differentiating on the right-hand side, thereby obtaining the following result after a short calculation: (4.52)

∂2f B(d2 f, df ) ∂2f + 2 = A(f )(df, df ) − (α − 1) df, 2 1 + |df |2 ∂x1 ∂x2

where A(f ) and B are bilinear maps, A(f ) being the familiar second fundamental form of M in RN . We can put the last term on the left-hand side obtaining L(f ) = A(f )(df, df ), where L(u) =

B(d2 u, df ) ∂ 2u ∂ 2u + 2 + (α − 1) df, 2 1 + |df |2 ∂x1 ∂x2

L being a second-order differential operator with coefficients in L∞ . Moreover, since α − 1 < 1/2, the operator L is uniformly elliptic. The linear operator L defines a bounded linear map from L42 to L4 , which can be regarded as a small perturbation of the scalar Laplace operator when α is close to one. If we restrict L to a small disk D about a given point p and u ∈ Ker(L), each component of u will assume its maximum value on the boundary ∂D of D. Thus if we impose Dirichlet boundary conditions on u − f so that u and f agree on ∂D, the map L : L42 → L4 will be injective. On the other hand, Δ : L42 → L4 has a continuous inverse G0 : L4 → L42 , and we can think of G0 as an approximate inverse to L when α − 1 is small. Given h ∈ L4 , then for appropriate choices of norms on L42 and L4 , the map u → G0 (Lu − h) + u is a contraction, and it must therefore possess a unique fixed point. This implies that L is surjective, and the open mapping theorem implies that the inverse G to L is continuous.

4.4. The α-energy

215

Since the critical point f is continuous and df ∈ L2 , A(f )(df, df ) is in L4 , and there is a unique u ∈ L42 (D, M ) satisfying the Dirichlet boundary conditions such that L(u) = A(f )(df, df ). This u must of course be f and hence f ∈ L42 . We thus conclude that the restriction of f to a small neighborhood of any point lies in L42 . Since L42 is contained in the H¨older space C 1,β for suitable β, we conclude that f ⊂ C 1,β (Σ, M ). Step 3. The final step is via the technique of “elliptic bootstrapping” which is applied to the partial differential equation (4.53)

L(f ) = A(f )(df, df ),

using the Schauder theory. The necessary Schauder estimates are described in [GT83], Chapter 8 for scalar operators, the general case being a standard extension within PDE theory. The result is that f ∈ C 1,β ⇒ right side of (4.53) ∈ C 0,β ⇒ f ∈ C 2,β ⇒ right side of (4.53) ∈ C 1,β ⇒ f ∈ C 3,β ⇒ · · · . This shows that f is C ∞ and completes our sketch of the argument for the theorem. QED Remark 4.4.8. One can prove Theorem 4.4.7 for various perturbations of the α-energy. For example, if ψ ∈ L2k (Σ, RN ), where k ≥ 2, we can define the perturbed function  2 f · ψdA, Eα,ψ,ω : Lk (Σ, M ) → R, by Eα,ψ,ω (f ) = Eα,ω (f ) + Σ

and the elliptic bootstrapping argument for regularity given above works until the solution is in L2k+2 . Another example of a function which satisfies Condition C and has critical points in L2k+2 is (4.54)  n  2α ηi (f ) f · ψi dA, F : L1 (Σ, M ) → R, defined by F (f ) = Eα,ω (f ) + i=1

Σ

where each ψi ∈ L2k (Σ, RN ) is subject to a uniform C 2 bound and each ηi : L2k (Σ, M ) → R is a C 2 -function. Yet another example of a function satisfying Condition C with smooth critical points is  θ θ : L2α (Σ, M ) → R, defined by E (f ) = E (f ) + θ, (4.55) Eα,ω α,ω 1 α,ω Σ

where θ is a smooth two-form on M , or the function obtained from it by adding additional perturbation terms such as those in (4.54). This is a useful perturbation of the conformally invariant function E θ described in Remark 4.3.9.

216

4. Harmonic and Minimal Surfaces

4.5. Morse theory for a perturbed energy Once we have condition C and regularity, we can apply Liusternik-Schnirelmann theory to the functions  1 2α Eα,ω : L1 (Σ, M ) → R, defined by Eα,ω (f ) = [(1 + |df |2 )α − 1]dA 2 Σ where the metric on Σ has area one, or θ : L2α Eα,ω 1 (Σ, M ) → R,

defined by

 θ Eα,ω (f ) = Eα,ω (f ) +

θ, Σ

where θ is a smooth two-form on M , and establish the existence of critical points for these functions corresponding to various topological constraints via the minimax construction described in §1.12. We can conclude for example that any component of L2α 1 (Σ, M ) contains an element f which minimizes θ Eα,ω or Eα,ω . But we would like a more refined theory; in fact, we would like to establish Morse inequalities and define a Morse-Witten complex when the critical points are Morse nondegenerate. Such Morse inequalities should relate critical points of these perturbed energies to the cohomology of Map(Σ, M ) which we studied in §3.6. Of course, when the canonical metric on Σ (of constant curvature 1, −1 or zero with total area one) has a nontrivial group G of isometries homotopic to the identity, that is, when Σ is a sphere or a torus, we have an action of G on the space of solutions, all critical points will lie on G-orbits and there are no critical points which are Morse nondegenerate in the usual sense. We then have the choice of developing equivariant θ to Morse inequalities or perturbing the functions Eα,ω Eα,ω  f · ψdA, Eα,ψ,ω (f ) = Eα,ω (f ) + Σ (4.56) θ θ Eα,ψ,ω (f ) = Eα,ω (f ) + f · ψdA, Σ

functions which have isolated critical points, satisfy Condition C and have critical points at least as smooth as ψ. Indeed, the notion of nondegenerate critical point can be extended to these functions, and we will prove in this section that the perturbed functions defined by (4.56) are Morse functions for generic ψ, that is, all of their critical points are Morse nondegenerate. In this section, we describe the Morse theory developed by Uhlenbeck [U72], which applies to such functions. This leads us to a strategy for proving the existence of critical points to the functions Eω , or the related function Eωθ defined in Remark 4.1.4: Use Morse inequalities or a Morse-Witten complex to prove existence of critical points for the perturbed functions Eα,ψ,ω

4.5. Morse theory for a perturbed energy

217

θ and Eα,ψ,ω and then investigate the limits as the perturbations are turned off, that is, as α → 1 and ψ → 0.

4.5.1. Second variation of energy and α-energy. There is a Morse θ theory for Eα,ψ,ω and Eα,ψ,ω which is quite parallel to the Morse theory for J developed in Chapter 2. (We will focus on the function Eα,ψ,ω , leaving the θ mostly to the reader.) To start the theory, extension to the case of Eα,ψ,ω we need formulae for the first and second variations. If f is a critical point for Eα,ψ,ω , then a calculation similar to that which yields (4.46) shows that   2 α−1 df, DV dA + ψ · V, dEα,ψ,ω (f )(V ) = α (1 + |df | ) Σ

Σ

for all V ∈ Tf L2α 1 (Σ, M ), where DV denotes the covariant differential of V with respect to the Levi-Civita connection on M . We would also like a formula for the second derivative such as that given in Proposition 2.4.1. In order to state this formula, we define an operator K : Tf L2α 1 (Σ, M ) → (Σ, M ) in terms of the Riemann curvature R of M by the formula Tf L2α 1    / .  ∂f ∂f ∂f ∂f + R V, , W dxdy, K(V ), W dA = R V, ∂x ∂x ∂y ∂y where (x, y) are isothermal coordinates on Σ. Proposition 4.5.1. If f is a critical point for Eα,ψ,ω , then (4.57) 2

 (1 + |df |2 )α−1 [DV, DW  − K(V ), W ]dA  + 2α(α − 1) (1 + |df |2 )α−2 df, DV df, DW dA Σ  + ψ · α(V, W )dA,

d Eα,ψ,ω (f )(V, W ) = α

Σ

Σ

for all V, W ∈ in RN .

Tf L2α 1 (Σ, M ),

where α is the second fundamental form of M

Although (4.57) looks complicated at first, it simplifies when α → 1 to give the second variation formula for harmonic maps: Corollary 4.5.2. If f is ω-harmonic, then  2 (4.58) d Eω (f )(V, W ) = [DV, DW  − K(V ), W ]dA, Σ

for all V, W ∈ Tf L2α 1 (Σ, M ).

218

4. Harmonic and Minimal Surfaces

Just as in the theory of geodesics, a parametrized harmonic surface f : Σ → M is said to be stable if the index form defined by (4.58) is positive semidefinite. Proof of Proposition 4.5.1. This is quite similar to the proof of Proposition 2.4.1. For simplicity, we assume that ψ = 0, leaving the discussion of the extra term as an exercise for the reader. We consider a variation of f which has its support within a given coordinate chart (U, (x, y)) on Σ. Recall that such a variation is a smooth family of maps t → f (t) in L2k (Σ, M ) with f (0) = f , and let F (x, y, t) = f (t)(x, y),

V (x, y) =

∂F (x, y, 0) ∈ Tf (x,y) M. ∂t

Setting γ(t)2 = (1 + |df (t)|2 )α−1 , we obtain d2 d2 Eα,ω (f )(V, V ) = 2 (Eω (f (t)) dt t=0 .  . / /  ∂ ∂F D ∂F 2 2 ∂F D ∂F γ , +γ , dxdy , =α ∂x ∂t ∂x ∂y ∂t ∂y Σ ∂t t=0 where D as usual denotes the covariant derivative of the Levi-Civita connection on the ambient Riemannian manifold M . We can rewrite this as . / . /  D ∂F D ∂F D ∂F D ∂F 2 2 , + , γ (4.59) d Eα,ω (f )(V, V ) = α ∂t ∂x ∂t ∂x ∂t ∂y ∂t ∂y Σ / . / . ∂F D 2 ∂F ∂F D 2 ∂F , 2 + , 2 dxdy + ∂x ∂t ∂x ∂y ∂t ∂y t=0 . / . /  2 ∂F D ∂F ∂F D ∂F dγ , + , dxdy . +2α ∂x ∂t ∂x ∂y ∂t ∂y Σ dt t=0 When we evaluate at t = 0, we find that df, DV  dγ 2 = (α − 1)γ 2 , dt 1 + |df |2 where V is the variation field, so we can rewrite (4.59) as / . / DV DV DV DV , + , d Eα,ω (f )V, V ) = α (1 + |df | ) ∂x ∂x ∂y ∂y Σ / . / . ∂f D DV ∂f D DV , + , dxdy + ∂x ∂t ∂x ∂y ∂t ∂y  + 2α(α − 1) (1 + |df |2 )α−2 df, DV 2 dA. .



2

2 α−1

Σ

4.5. Morse theory for a perturbed energy

219

Finally, applying the definition of the Riemann-Christoffel curvature tensor R, we obtain / . / .  DV DV DV DV 2 2 α−1 , + , d Eα,ω (f )(V, V ) = (1 + |df | ) ∂x ∂x ∂y ∂y Σ .   / .   / ∂f ∂f ∂f ∂f + , R V, V + , R V, V ∂x ∂x ∂y ∂y / . / . ∂f D DV ∂f D DV , + , dxdy + ∂x ∂x ∂t ∂y ∂y ∂t  + 2α(α − 1) (1 + |df |2 )α−2 df, DV 2 dA. Σ

An integration by parts and use of the Euler-Lagrange equation eliminates the third term, thereby yielding (4.57) in the case where ψ = 0. QED 4.5.2. The Jacobi operator for α-energy. If f is a critical point for the function Eα,ψ,ω , then for every k ∈ N with k ≥ 2, we can define a second-order linear partial-differential operator (4.60)  2 ∗ 2 ∗ 2 L : Lk (f T M ) → Lk−2 (f T M ) by d Eα,ψ,ω (f )(V, W ) = L(V ), W dA, Σ

which we call the Jacobi operator at f . Note that the Jacobi operator L extends to a continuous linear map L : L21 (f ∗ T M ) → L2−1 (f ∗ T M ), if we let L2−1 denote the dual space to L21 with respect to the L2 inner product. If ι : L2k (f ∗ T M ) → L2k−2 (f ∗ T M ) is the inclusion and λ ∈ C, we can also define associated operators Lλ = L − λι. Of course, there is a similar definition of the Jacobi operator for the θ function Eα,ψ,ω at a critical point f . If L is a Jacobi operator at f , we call Wλ = Ker(Lλ ) = {V ∈ L2k (f ∗ T M ) : L(V ) = λV } the eigenspace for λ ∈ C. We say that λ is an eigenvalue if Wλ is nonzero, and nonzero elements of Wλ are called eigenvectors for eigenvalue λ, We can apply the theory of formally self-adjoint elliptic operators to the Jacobi operator, as explained in [Eva98], [LM89] and [Tay96], and find that the operator L defined by (4.60) is a Fredholm operator, which satisfies the additional conditions: (1) each eigenspace Wλ = Ker(Lλ ) is finite-dimensional, (2) all the elements of Wλ are C ∞ ,

220

4. Harmonic and Minimal Surfaces

(3) if Wλ is zero, then Lλ possesses an inverse, Gλ : L2k−2 (f ∗ T M ) → L2k (f ∗ T M ), which is called a Green’s operator , (4) all eigenvlaues are real, and (5) the eigenvalues can arranged in a sequence λ1 < λ2 < · · · < λi < · · ·

with λi → ∞,

with only finitely many of the eigenvalues being negative. If f is a critical point for Eα,ψ,ω , we can also define an inner product 2α ·, · : Tf L2α 1 (Σ, M ) × Tf L1 (Σ, M ) −→ R

by the somewhat complicated formula,  V, W  = α (1 + |df |2 )α−1 [DV, DW  + V, W ]dA Σ  + 2α(α − 1) (1 + |df |2 )α−2 df, DV df, DW dA Σ  ψ · α(V, W )dA, + Σ

which is useful because it so close to (4.57). We can assume that ψ is chosen sufficiently small that ·, · is positive definite, and then, since the critical point f is a smooth function, ·, · is actually equivalent to the simpler L21 inner product ·, ·0 defined by  V, W 0 = [DV, DW  + V, W ]dA. Σ

We can define a second-order linear partial-differential operator  J : L21 (f ∗ T M ) −→ L2−1 (f ∗ T M ) by V, W  = J(V ), W dA, Σ 2 ∗ Lk (f T M )

→ L2k−2 (f ∗ T M ) which again restricts to a Fredholm operator J : for all k ≥ 2. However, since ·, · is positive-definite, the eigenvalues of J are all positive and J itself has a Green’s operator inverse. It is here that the complicated formula for the L21 inner product on L21 (f ∗ T M ) pays off, because L − J is of zero order, and hence L − J : L2k (f ∗ T M ) −→ L2k−2 (f ∗ T M ) is a compact operator, for all k ≥ 1. In parallel with the theory of smooth closed geodesics, we can define a continuous linear map (4.61) A : L21 (f ∗ T M ) −→ L21 (f ∗ T M ) by AV, W  = d2 Eα,ψ,ω (f )(V, W ).

4.5. Morse theory for a perturbed energy

221

In fact, the operator A = J −1 ◦ L is of the form A = id + K : L21 (f ∗ T M ) −→ L21 (f ∗ T M ), where K is compact, and restricts to an operator A = id + K : Lpk (f ∗ T M ) −→ Lpk (f ∗ T M ), where K is compact, for p ≥ 2 and k ≥ 2, by the Lp theory of elliptic operators. If p = 2α, we can choose q so that (1/p) + (1/q) = 1 and define the operator A directly on Lp1 (f ∗ T M ) by (4.61). Indeed, any V ∈ Lp1 (f ∗ T M ) defines a continuous linear functional W → d2 Eα,ψ,ω (f )(V, W ),

for

W ∈ Lq1 (f ∗ T M ),

and by the duality between Lp1 and Lq1 this linear functional defines an element AV ∈ Lp1 (f ∗ T M ). This defines a continuous linear map (4.62)

2α A : Tf L2α 1 (Σ, M ) −→ Tf L1 (Σ, M )

such that 2α K = A − id : Tf L2α 1 (Σ, M ) −→ Tf L1 (Σ, M )

is a compact operator, as before. 4.5.3. Morse inequalities for a perturbed energy. We recall some θ is Morse nondefinitions from §2.4: A critical point f for Eα,ψ,ω or Eα,ψ,ω degenerate if its Jacobi operator L has trivial kernel. The Morse index of f is the sum of the dimensions of the eigenspaces of L for all the negative θ is a Morse function if all of its critical eigenvalues. Finally, Eα,ψ,ω or Eα,ψ,ω points are Morse nondegenerate. Theorem 4.5.3. For k ≥ 2 and a residual set of ψ ∈ L2k (Σ, RN ), the functions θ : L2k+2 (Σ, M ) −→ R Eα,ψ,ω , Eα,ψ,ω defined by (4.56) are Morse functions, that is, all of their critical points are nondegenerate. Note that since all critical points are smooth, we can work within the spaces L2k for k large which enables us to deal with continuous functions throughout the argument. We now just follow the proof of Theorem 2.6.1, with minor changes. In particular, we construct an Euler-Lagrange map F : L2k+2 (Σ, M ) × L2k (Σ, RN ) −→ L2k (Σ, T M ) and show that it is transversal to the zero section. As before, we use this transversality to show that S = {(f, ψ) ∈ L2k+2 (Σ, M ) × L2k (Σ, RN ) : F (f, ψ) = 0}

222

4. Harmonic and Minimal Surfaces

is a smooth submanifold of the product. Finally, we show that the projection of π : S → L2k (Σ, RN ) onto the last factor of the product is a Fredholm map of Fredholm index zero. It then follows from the Sard-Smale Theorem that a residual set of ψ consists of regular values for the projection, and hence all critical points in π −1 (ψ) are Morse functions, proving the theorem. For the function Eα,ψ,ω , we now let M = Lp1 (Σ, M ), where p = 2α, with its standard Finsler metric, and let Ma = {f ∈ Lp1 (Σ, M ) : Eα,ψ,ω (f ) ≤ a}. a subspace which is empty when a ≤ −min{|ψ(p)| : p ∈ M }. Morse theory should tell us how the topology of Ma changes as a increases. According to Theorem 1.12.1, if [a, b] contains no critical values Ma is a strong deformation retract of Mb , so Ma and Mb are homeomorphic. The Handle Addition Theorem 2.8.1 can now be modified so that it applies to the Morse function Eα,ψ,ω , and hence the Morse inequalities of §2.9 can be extended to Eα,ψ,ω : Theorem 4.5.4. If the interval [a, b] contains a single critical value c for Eα,ψ,ω or Eα,ψ,ω , there is exactly one critical point f0 which has c and this critical point is Morse nondegenerate of Morse index λ, then Mb is homotopy equivalent to Ma with a handle of index λ attached. This is proven more generally for functions on Banach manifolds with “weakly nondegenerate” critical points in [U72]. The argument is easily extended to finitely many critical points on the same critical level. Proof in our case. We can modify the proof of Theorem 2.8.1 using the definition of gradient-like vector field presented in §2.8 applied now to the Banach manifold Lp1 (Σ, M ). Suppose that (U, φ) is a smooth chart on Lp1 (Σ, M ) with values in the model space E = Lp1 (f0∗ T M ) such that φ(f0 ) = ¯ ⊆ U with φ(W ) 0. Let W be an open subset of U such that f0 ∈ W ⊆ W an open ball centered at 0 = φ(f0 ). Let η : M → [0, 1] be a C 2 -function such that η(g) = 0 for g ∈ M − U , η(g) = 1 for g ∈ W , and let (4.63)

X = η φ−1 ∗ XA + (1 − η) X0 ,

where X0 is a pseudogradient and XA is the vector field defined by the Fredholm map A : Lp1 (f0∗ T M ) → Lp1 (f0∗ T M ) of (4.62). As in §2.8, we can divide the model space E into a direct sum, E = E− ⊕ E+ , with E− and E+ being the subspaces spanned by eigenvectors for

4.5. Morse theory for a perturbed energy

223

negative and positive eigenvalues, respectively, with E− being finite dimensional. As before, we can write A in “block matrix” form,   A− 0 , A= 0 A+ with A− and A+ being the restrictions to the subspaces E− and E+ , respectively. Within W , the vector field −X is then represented in the coordinate system (U, φ) by the dynamical system  dx− /dt = −A− x− , (4.64) dx+ /dt = −A+ x+ , which expands along E− and contracts along E+ . Lemma 4.5.5. If F = Eα,ψ,ω ◦ φ−1 is Morse nondegenerate at 0 = φ(f0 ), then dF (X)(X A (X)) > εX2Lp > 0,

(4.65)

1

for some ε > 0, when X is a nonzero element of the model space Tf0 Lp1 (Σ, M ) and XLp1 < δ for some δ > 0. Proof. Recall that A = id + K, where K is compact. We express the model space as a direct sum decomposition Tf0 Lp1 (Σ, M ) = F ⊕ G, which is invariant under A, with F being finite dimensional and (K|G)Lp1 < ε1 , where K|G is the restriction of K to G. We can replace the Lp1 - and L21 norms by equivalent norms which respect this decomposition. It then suffices to prove (4.65) for the two cases X ∈ F and X ∈ G. If X ∈ F and XLp1 < δ,

(4.66)

dF (X)(X A (X)) = dF (X)(AX)  1 d2 F (tX)(AX, X)dt ≥ d2 F (0)(AX, X) − ε1 X, X = 0

≥ AX, AX − ε2 X, X, the positive constant ε2 depending on the continuity of the second derivative of F . We can make ε2 small by choosing δ small, and thus arrange that dF (X)(X A (X)) = dF (X)(AX) ≥ ε3 X2Lp , 1

where ε3 is a small positive constant, since the alent on the finite-dimensional space F .

Lp1 -

and

L21 -norms

are equiv-

If X ∈ G and XLp1 < δ, (4.65) follows from Lemma 4.4.5 and the fact that the restriction of A to G is a small perturbation of the identity. QED

224

4. Harmonic and Minimal Surfaces

It follows from Lemma 4.5.5, together with the definition of pseudogradient, that dEα,ψ,ω (X ) is bounded below by a positive constant on the annular region on which η is neither zero nor one. Thus it follows from (4.64) that orbits for −X must either flow down to Ma within a finite time interval or converge to f0 . From this point, the argument for adding handles is exactly the same as in §2.8. One can refer to [U72] for an argument treating a more general case of “weakly nondegenerate” critical points. We remark that the gradient-like vector field X can be chosen to be tangent to each of the submanifolds L2k (S 2 , M ) ⊆ L2α 1 (Σ, M ), for k ∈ N, k ≥ 2. The reason is that we can choose local pseudogradients and representatives XA in local coordinates near each critical submanifold to have this property. Elliptic bootstrapping shows that the critical point is in fact smooth, and convergence of the flow holds in all the Sobolev norms. It is only in the convergence of an orbit to a critical point as t → ∞ that the L2α 1 -norm is essential. We can define stable and unstable manifolds of nondegenerate critical points just as we did before and construct the Morse-Witten complex. If X is a gradient-like vector field for Eα,ψ,ω , we let Ck (Eα,ψ,ω , X ) denote the free Z-module generated by the critical points fk,1 , fk,2 . . . for Eα,ψ,ω of index k and let ∂ be the Z-module homomorphism ∂ : Ck (Eα,ψ,ω , X ) −→ Ck−1 (Eα,ψ,ω , X ) defined by (4.67)

∂(fk,j ) =



ajq fk−1,q ,

q

where ajq ∈ Z is the oriented number of trajectories from fk,j to fk−1,q , the orientation being determined as in §2.10. Theorem 4.5.6. The Z-module homomorphisms thus defined satisfy the identity ∂ ◦ ∂ = 0 and the resulting Morse-Witten complex · · · → Ck+1 (Eα,ψ,ω , X ) → Ck (Eα,ψ,ω , X ) → Ck−1 (Eα,ψ,ω , X ) → · · · calculates the homology of C 0 (Σ, M ). θ . Once we have the Handle Of course, there is a similar theorem for Eα,ψ,ω Addition Theorem 4.5.4, the proof of Theorem 4.5.6 is a straightforward modification of the argument presented in §2.10. Let Map0 (Σ, M ) denote the component within the space Map(Σ, M ) of maps containing a given map f : Σ → M . Then the Morse-Witten complex provides Morse inequalities just like in the Morse theory of geodesics, which relate critical points of θ on Map0 (Σ, M ) to the homology of Map0 (Σ, M ). Eα,ψ,ω or Eα,ψ,ω

4.6. Bubbles

225

When Σ is an oriented compact surface, the Morse inequalities together with the cohomology of Map(Σ, M ) as worked out in §3.6 imply that the number of such critical points will often grow exponentially with Morse index. On the other hand, Condition C implies that there are only finitely many critical points for Eα,ψ,ω below a given energy level a. The preceding discussion has the following consequence: Corollary 4.5.7. Let M = Lp1 (Σ, M ) and Ma = {f ∈ M : Eα,ψ,ω ≤ a}. Then each Ma is smoothly homotopy equivalent to a finite-dimensional manifold with boundary and M has the homotopy type of a countable CW complex. The smooth homotopy equivalence in the first statement of the corollary can be achieved inductively as we add handles. The last statement is an analog of Theorem 17.3 in Milnor’s book on Morse theory [Mil63], and can be proven by the direct limit argument given in the appendix of that reference.

4.6. Bubbles 4.6.1. The Bochner Lemma. We have seen how to construct critical points for the α-energy Eα,ω , but what we are really interested in are the harmonic maps, the critical maps for the usual energy Eω that we hope to obtain in the limit as α → 1. We need some tools to study this limit and one of the key tools is the Bochner Lemma, which we discuss in this section. The Bochner Lemma gives an estimate for the Laplacian of the energy density in terms of curvature, and it has become a useful tool which appears in many other global analysis problems. Our treatment of the Bochner Lemma follows Lin and Wang [LW08], and is based upon classical tensor analysis calculations. Let Σ and M be compact Riemannian manifolds with metrics (ηab ) and (gij ), respectively, with M as usual being isometrically imbedded in Euclidean space RN . We suppose that f : Σ −→ M ⊆ RN is a smooth map and for a given choice of local coordinates (x1 , x2 ) on Σ, we consider the vector-valued maps fa =

∂f ∈ RN ∂xa

which have components fai =

∂fi , ∂xa

with respect to local coordinates (u1 , . . . , un ) on M . Here we use the conventions on ranges of indices: 1 ≤ a, b ≤ 2,

1 ≤ i, j, k, l ≤ n.

226

4. Harmonic and Minimal Surfaces

Recall that the energy density is given by 1 1  ab η fa · fb = gij η ab fai fbj , e(f ) = 2 2 where (η ab ) is the matrix inverse to (ηab ). Finally, we let ˜ abcd ) and RM = (Rijkl ) R Σ = (R ˜ ab ) be the Ricci be the curvature tensors of Σ and M , and let RicΣ = (R curvature of Σ. Recall that if we divide the Laplacian of the vector-valued map f : Σ → RN into tangential and normal components, Δη f = (Δη f ) + (Δη f )⊥ , the condition that f be harmonic is just (Δη f ) = 0. On the other hand, the normal component of the Laplacian is expressed in terms of the second fundamental form α(f )(p) : Tp M × Tp M −→ Np M of M within RN by (Δη f )⊥ (p) =



η ab α(f )(p)(fa , fb ).

Theorem 4.6.1 (Bochner Lemma). If f : Σ → M is a smooth map, then (4.68) Δη (e(f )) = |∇df |2 + d[(Δη f ) ], df    ˜ ab fc · fd − η ac η bd Rijkl f i f j f k f l . + η ac η bd R a b c

d

in T ∗ Σ⊗

where ∇ denotes covariant derivative with respect to the connection f ∗ T M . Hence, if f is harmonic, then   ˜ ab fc · fd − η ac η bd R η ac η bd Rijkl fai fbj fck fdl . (4.69) Δη (e(f )) = |∇df |2 +

To prove this, we use normal coordinates centered a point p ∈ Σ and similar coordinates at f (p) in M , and calculate Δη (e(f ))(p). To keep the notation simple, we leave the point p out of our expressions. Then, using the fact that ηab = δab , we find that   |fa;b |2 + fa , fa;bb  Δη (e(f )) =    ˜ bcab fc  (4.70) fa , fb;ba  − fa , R = |fa;b |2 +    ˜ ab fa , fb , fa , (Δη f )a  + R = |fa;b |2 + where we have used the local coordinate formula for covariant derivatives on Σ,  ˜ dcab fc . R fd;ba − fd;ab =

4.6. Bubbles

227

On the other hand, fa , (Δη (f ))a  = fa , [(Δη f ) ]a  + fa , [(Δη f )⊥ ]a   fa , α(f )(fb , fb )a . = fa , [(Δη f ) ]a  + Now we use the fact that fa , α(f )(fb , fb ) = 0 to conclude that  Δη (f ), (α(f )(fb, fb ) (4.71) fa , (Δη (f ))a  = fa , [(Δη f ) ]a  −  (α(f )(fa , fa ), (α(f )(fb , fb ). = fa , [(Δη f ) ]a  − Finally, we note that   α(f )(fa , fb ), α(f )(fa , fb ) (4.72) |fa;b |2 = |∇df |2 + and we substitute (4.71) and (4.72) into (4.71) and obtain  ˜ ab fa · fb R Δη (e(f )) = |∇df |2 + d[(Δη f ) ], df  + $ %  − α(f )(fa , fa ), (α(f )(fb , fb ) − α(f )(fa , fb ), α(f )(fa , fb ) . This last equation implies (4.68) by the Gauss equation. ˜ then (4.69) simplifies If Σ is a compact surface with Gaussian curvature K, to  ˜ |2 − (4.73) Δη (e(f )) = |∇df |2 + K|df Rijkl fai fbj fak fbl . In terms of isothermal coordinates (x, y) = (x1 , x2 ) on Σ, the Riemannian metric and energy density take the form  2  2  ∂f 1 ∂f ηab dxa dxb = λ2 (dx2 + dy 2 ), e(f ) = 2 + . 2λ ∂x ∂y Closely related is the area density 1 a(f ) = 2 λ

∂f ∂f ∂x ∧ ∂y .

It follows from inequality (4.16) that a(f ) ≤ e(f ) with equality if and only if f is conformal. We can rewrite (4.73) in terms of a(f ) as (4.74)

˜ ) − 2K(σ)a(f )2 , Δη (e(f )) = |∇df |2 + 2Ke(f

228

4. Harmonic and Minimal Surfaces

where K(σ) is the sectional curvature of the two-plane generated by ∂f /∂u and ∂f /∂v, and if K(σ) ≤ 1, our inequality between a(f ) and e(f ) implies that ˜ ˜ (4.75) Δη (e(f )) ≥ |∇df |2 + 2Ke(f ) − 2e(f )2 ≥ 2Ke(f ) − 2e(f )2 . In particular, we get a quadratic estimate on the Laplacian of energy density ˜ = 0 and K(σ) ≤ 1: when K (4.76)

Δη (e(f )) ≥ −2e(f )2 .

This is the estimate which is the key to understanding the limits of αharmonic maps. Corollary 4.6.2. If M has nonpositive sectional curvatures, there are no harmonic maps from the two-sphere S 2 into M and any harmonic map from the torus T 2 into M must be totally geodesic. If M has negative sectional curvatures, there are no harmonic maps from S 2 or T 2 into M . ˜ ≡ 1. Proof. If Σ = S 2 , we can choose the Riemannian metric on Σ so that K The continuous function e(f ) must assume a maximum at some point. At this point the Bochner inequality shows that Δ(e(f )) > 0, a contradiction. ˜ ≡ 0. In If Σ = T 2 , we can choose the Riemannian metric on Σ so that K this case, the inequality must be an equality. Hence ∇df = 0 which implies that ∂f ∂f and ∂x1 ∂x2 are parallel sections of the pullback of the bundle f ∗ T M . This implies of course that f is totally geodesic. The case where M has negative sectional curvatures is proven in a similar fashion. QED 4.6.2. Local control of energy density. Our next goal is to use the Bochner Lemma to investigate the limits of α-energy critical points as α → 1. We will see that the Bochner Lemma gives a uniform C 1 estimate on harmonic and α-harmonic maps whose total energy is small. This in turn is crucial in understanding a key feature of harmonic and α-harmonic maps: the phenomenon of bubbling. Throughout this section, we use the notation  2  2 ∂f 1 1 + ∂f dudv, e(f ) = |df |2 , so that e(f )dA = ∂y 2 2 ∂x for isothermal parameters (x, y) on Σ. Suppose that Dr is a disk of radius r in the plane R2 with the standard flat metric, and that M is a compact Riemannian manifold, the metric on M being normalized so that its sectional

4.6. Bubbles

229

curvatures satisfy the inequality K(σ) ≤ 1. Then the Bochner Lemma yields the quadratic estimate (4.76): Δη (e(f )) ≥ −2e(f )2 . We can exploit this estimate as in [Sch84] to prove the following: ε-Regularity Theorem 4.6.3. Let Σ be a compact Riemann surface and M a compact Riemannian manifold. Then there are constants r0 ≥ 0, ε0 > 0, C > 0 and α0 ∈ [1, ∞) which depend upon Σ and M such that if Dr is a disk of radius r ≤ r0 in Σ and f : Dr → M is an α-harmonic map for α ∈ [1, α0 ], then ;  e(f )dA (4.77) e(f )dA < ε0 ⇒ sup e(f ) < C Dr 2 . πr Dr/2 Dr In other words, if the energy on a ball of small radius is sufficiently small, the maximum of the energy density on a ball of half the radius is bounded by a constant multiple of its average over the original ball. We start by proving this for ordinary harmonic maps on disks in the complex plane when the metric on the plane is flat: Lemma 4.6.4. Suppose that f : D1 → M is a harmonic map, where D1 is the unit disk in the complex plane, with the standard Euclidean metric ds2 and M is a compact Riemannian manifold with sectional curvatures satisfying the inequality K(σ) ≤ 1. Then for any r0 ∈ (0, 1/2),  (4.78) e(f )dA < π(r02 − 4r04 ) ⇒ max σ 2 sup e(f ) < 4r02 . σ∈(0,1]

D1

D1−σ

Proof. Choose σ0 ∈ (0, 1] so that σ02 sup e(f ) ≥ σ 2 sup e(f ), D1−σ0

D1−σ

for all σ ∈ (0, 1],

and choose p0 ∈ D1−σ0 so that e0 = e(f )(p0 ) = sup{e(f )(p) : p ∈ D1−σ0 }. It suffices to show that if σ02 e0 ≥ 4r02 , then the hypothesis of (4.78) cannot hold. But  σ 2 0 sup e(f ) ≤ σ02 sup e(f ) = σ02 e0 ⇒ sup e(f ) ≤ 4e0 , 2 D1−σ /2 D1−σ0 D1−σ /2 0

and the inequality σ02 e0 ≥ 4r02 implies that σ0 r0 √ ≤ , 2 e0

0

230

4. Harmonic and Minimal Surfaces

√ so that the disk of radius r0 / e0 about p0 is contained in D1 . If we rescale this disk to radius r0 , and define a new harmonic map g : Dr0 → M by   q g(q) = f p0 + √ we have e(g)(0) = 1 and e(g) ≤ 4 on Dr0 . e0 It now follows from the quadratic Bochner estimate (4.76) that g : Dr0 → M satisfies Δ(e(g)) ≥ −2(e(g))2 ≥ −32. Our idea now is to exploit the maximum principle for the Laplace operator: If we define a smooth function h : Dr0 → R in terms of the standard Euclidean coordinates (u, v) on the disk by h = e(g) − [1 − 8(u2 + v 2 )], then Δh = Δ(e(g)) + 32 ≥ 0, and hence by the maximum principle,   2 hdA ≤ [e(g) − (1 − 8(u2 + v 2 ))]dA 0 = πr0 h(0) ≤ Dr0



Dr0

≤



2π

e(g)dA −



0

Dr0

r0

(1 − 8r2 )rdrdθ,

0

and an integration yields the inequality   e(f )dA ≥ e(g)dA ≥ π(r02 − 4r04 ). D1

Dr0

Thus we have proven the contrapositive of the assertion (4.78). QED Proof of Theorem 4.6.3 for harmonic maps and flat metrics on Dr . Note first that π 2 1 r02 ≤ ⇒ r ≤ π(r02 − 4r04 ). 8 2 0 Thus, if   π π e(f )dA < , we can set e(f )dA = r02 , 16 2 D1 D1 and (4.78) implies  π e(f )dA = r02 ≤ π(r02 − 8r04 ) (4.79) 2 D1 ⇒

2

max σ sup e(f )
0, then there is a finite collection of points {p1 , . . . , pl } ⊆ Σ, and a subsequence of {fm } (which we still denote by {fm }), such that {fm } converges uniformly in C k on compact subsets of Σ − {p1 , . . . , pl }, for any nonnegative integer k, to an ω∞ -harmonic map f∞ : Σ − {p1 , . . . , pl } −→ M. Proof. We begin by choosing ε0 > 0 in accordance with (4.77) so that when r ≤ r0 ,  Cε0 e(f )dA < ε0 ⇒ sup e(f ) < . (4.81) πr2 Dr/2 Dr

4.6. Bubbles

233

Let rm = (1/2)m , for each positive integer m and let Om be an open cover of Σ by disks Drm (pi ) of radius rm about the points pi , so that each point of Σ is covered by at most h disks and the disks Drm /2 (pi ) still cover M . (We can choose h to be independent of m.) Then  e(fm ) ≤ hE0 . i

Drm (pi )

Hence there are at most hE0 /ε0 of the disks in the cover Om on which  e(fm ) ≥ ε0 . Drm (pi )

For each m ∈ N we let {p1m , . . . , plm } be the center points of these disks, where l ≤ hE0 /m. After possibly passing to a subsequence, we can arrange that p1m → p1 , . . . , plm → pl . Since M is compact, the {fm }’s are uniformly bounded and since each |dfm | is bounded on any compact subset of Σ−{p1 , . . . , pl }, the {fm }’s are equicontinuous on compact subsets of Σ − {p1 , . . . , pl }. By Arzela’s Theorem the {fm }’s converge to a continuous function f∞ on Σ − {p1 , . . . , pl }, and the convergence is uniform on compact subsets. To finish the proof, we need to show that the convergence is uniform on compact subsets of Σ − {p1 , . . . , pl } in each C k , and for this we use “elliptic bootstrapping”, as in the proof of the Regularity Theorem 4.4.7 for α-harmonic maps. Indeed, if M is isometrically imbedded in RN , it follows from (4.52) that the equation for which f must satisfy to be an α-harmonic map is (4.82)

∂2f B(d2 f, df ) ∂ 2f + + (α − 1) df = α(f )(df, df ), ∂x2 ∂y 2 1 + |df |2

where α(f ) and B are bilinear maps, α(f ) being the familiar second fundamental form. It follows from (4.81) that {dfm } is bounded in Lp for all p on any disk within Σ − {p1 , . . . , pl }. It then follows from multiplication theorems for Sobolev spaces that the right-hand side of (4.82) is bounded in Lp for all p on any such disk, and hence elliptic estimates give that f is bounded in Lp2 for all p. But now one of the Sobolev imbedding theorems implies that a subsequence of {fm } converges in C 1,β for some β > 0 on any compact subset of Σ−{p1 , . . . , pl }. Next we can apply Schauder estimates to see that a subsequence of {fm } converges in C 2,β , then in C 3,β , and so forth. Finally, we obtain a subsequence of {fm } which converges uniformly in every C k on compact subsets of Σ − {p1 , . . . , pl }, and the theorem is proven. QED

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4. Harmonic and Minimal Surfaces

Theorem 4.6.7. Suppose that M is a compact Riemannian manifold with nonpositive sectional curvatures. If ω is a conformal structure on the compact Riemann surface Σ, there is an ω-energy minimizing harmonic map in any free homotopy class [Σ, M ]. The assumption that the sectional curvatures are nonpositive, allows us to eliminate the pesky quadratic term from the ε-Regularity Theorem. In this case, Theorem 4.6.3 implies that for some α0 ∈ (1, ∞) and some r0 ≥ 0, whenever Dr is a disk of radius r ≤ r0 in Σ and f : Dr → M is an αharmonic map with α ∈ (1, α0 ],  e(f )dA. (4.83) sup e(f ) < 4CL Dr/2

Dr

Since the right-hand side of (4.83) is bounded, the energy density is uniformly bounded, making blowup impossible. We do not need to assume that the integral of energy over D4 is small in this case, and hence we do not need the complicated covering argument utilized in the proof of Theorem 4.6.6. Instead, we simply take a sequence αm → 1 and for each αm choose an αm -harmonic map fm : Σ → M which lies in a given component of [Σ, M ] and minimizes the αm -energy on that component. Then (4.83) implies that e(fm ) is bounded on any ball Dr/2 , which implies that {fm } converges uniformly on compact subsets. Elliptic bootstrapping then implies that the sequence {fm } converges uniformly in all C k , proving the theorem. Theorem 4.6.7 is just the Eells-Sampson Theorem 4.1.1 in the special case where Σ has dimension two. Similarly, we can prove Hartman’s Theorem 4.1.2 in the special case where Σ is a surface: Theorem 4.6.8. Suppose that M is a compact Riemannian manifold with negative sectional curvatures. If (Σ, ω) is any Riemann surface, there is at most one ω-harmonic map in any free homotopy class [Σ, M ] which does not contain a degenerate harmonic map of rank zero or one. Indeed, the corresponding statement for (α, ω)-harmonic maps follows from the fact that since    / .  ∂f ∂f ∂f ∂f + R V, , V < 0, R V, ∂x ∂x ∂y ∂y where (x, y) are isothermal coordinates on Σ, any (α, ω)-harmonic map is nondegenerate and strictly stable by (4.57), so long as f has rank two on an open set. Moreover, if there were two (α, ω)-harmonic maps in the same component of C 0 (Σ, M ), there would have to be an unstable critical point by the Morse inequalities, which cannot happen.

4.6. Bubbles

235

To prove the corresponding statement for the limiting case of ω-harmonic maps, we can use the implicit function theorem. For this, we define a smooth map Eω : L2k (Σ, M ) × [1, α0 ) −→ R

by Eω (f, α) = Eα,ω (f ),

and observe that at α = 1 we obtain the usual energy, Eω (f, 1) = Eω (f ). We can differentiate to obtain the Euler-Lagrange map Fω : L2k (Σ, M ) × [1, α0 ) −→ L2k−2 (Σ, T M ). If f is an (α, ω)-harmonic map, we can compose the differential DFω of Fω∗ at (f, α) with the projection onto the vertical obtaining the corresponding Jacobi operator, L = πV ◦ DFω (f, α) : Tf L2k (Σ, M ) ⊕ R −→ Tf L2k−2 (Σ, M ). Note that L is a Fredholm operator with Fredholm index one, and that f is a Morse nondegenerate critical point for Eα,ω if and only if L is surjective. Suppose now that f : Σ → M is an ω-conformal harmonic map, and note that f must be Morse nondegenerate by Corollary 4.6.2. It follows from the implicit function theorem that if U is a sufficiently small C 2 neighborhood of (f, 1) in L2k (Σ, M ) × [1, α0 ), then (Fω )−1 (zero section of T (Map(Σ, M ))) is a smooth submanifold of U of dimension one. This submanifold consists of critical points of Eα,ω for α varying in some interval [1, α0 ) for which the corresponding Jacobi operators L are surjective. All the points of this submanifold are Morse nondegenerate critical points for Eα,ω . Thus two distinct ω-harmonic maps f, g : Σ → M in the same component of C 0 (Σ, M ) would give rise to two distinct one-parameter families fα , gα of (α, ω)-harmonic maps in the same component of C 0 (Σ, M ), contradicting the uniqueness of (α, ω)-harmonic maps. This finishes the proof of Theorem 4.6.8. 4.6.4. Removable singularities. In §4.5 we have shown existence of many α-energy critical points, corresponding to numerous topological constraints, while in §4.6.2 we have shown that given a sequence {fm : Σ → M } of αm -harmonic maps with αm > 1 such that αm → 1 and E(fm ) ≤ E0 for some constant E0 > 0, there is a collection of points {p1 , . . . , pl } and a subsequence of {fm } (which we still denote by {fm }) which converges uniformly in C k on compact subsets of Σ − {p1 , . . . , pl }, for every k ∈ N, to a map (4.84)

f∞ : Σ − {p1 , . . . , pl } −→ M,

236

4. Harmonic and Minimal Surfaces

which is harmonic on Σ − {p1 , . . . , pl }. The following theorem implies that f∞ extends to a harmonic map defined on the entire Riemann surface Σ: Removeable Singularity Theorem 4.6.9. Let D be the unit disk in the complex plane C. If f : D − {0} −→ M is a harmonic map of finite energy, then f extends to a smooth harmonic map on the entire disk D. This is proven as Theorem 3.6 in [SU81], and [Par96] provides a more conceptual proof which we follow. We sketch the proof, focusing on the case where the metric on the disk D is the standard flat metric. We can write this flat metric in terms of polar coordinates centered at the origin as ds2 = dr2 + r2 dθ2 = e−2u (du2 + dθ2 ),

(4.85)

where r = e−u . If we use the (r, θ) coordinates and p is the point with coordinates (r, θ), then the disk of radius r about p is completely contained within the punctured disk. Thus it follows from the ε-Regularity Theorem 4.7.1 that there is a constant ε0 > 0 such that whenever the energy E(f ) on the punctured disk satisfies E(f ) < ε0 , then ˜ 0. r2 (supremum of e(f ) on the disk of radius r/2 about p) < Cε This implies that when E(f ) < ε0 , ' ∂f ∂f ˜ 0, (supremum of r , on the disk of radius r/2 about p) ≤ 2Cε ∂r ∂θ and these estimates become

' ∂f ∂f ˜ 0 supremum of , ≤ 2Cε ∂u ∂θ

(4.86)

in terms of the (u, θ) coordinates. By choosing ε0 > 0 sufficiently small, we can make these suprema as small as desired. Thus when the energy is finite, we can arrange that |(∂f /∂u)| and |(∂f /∂u)| be bounded by an arbitrarily small constant by restricting to a ball of small enough radius, then rescaling to unit radius. The (u, θ) coordinates establish an isomorphism from the punctured disk to a half-infinite cylinder [0, ∞) × S 1 and the energy of f : D → R is simply     1 ∞ 2π ∂f 2 ∂f 2 E(f ) = ∂u + ∂θ dudθ. 2 0 0 We set 1 K(u) = 2



2π 0

2 ∂f (u, θ)dθ, ∂u

1 P (u) = 2



2π 0

2 ∂f (u, θ)dθ, ∂θ

4.6. Bubbles

237

and use the Euler-Lagrange equation, D2 f D2 f + = 0, 2 ∂u ∂θ2 to show that K  (u) − P  (u) = 0, so K(u) = P (u) + c, where c is a constant. But then boundedness of energy implies that K(u) ≡ P (u). Our next goal is to calculate P  (u), and we find that   /  2π . ∂f D ∂f  , dθ, P (u) = ∂u ∂θ ∂θ 0 while

 / . 2   / D ∂f D ∂f ∂f , + , dθ P (u) = ∂u ∂θ ∂u2 ∂θ ∂θ 0 / .   /  2π .  2  ∂f ∂f ∂f ∂f D D f ∂f + R , , dθ. , ≥ ∂θ ∂u2 ∂θ ∂u ∂θ ∂u ∂θ 0 But the Euler-Lagrange equation and integration by parts yields / /  2π .  2   2π .  2  D D f ∂f D D f ∂f dθ = − dθ , , ∂θ ∂u2 ∂θ ∂θ ∂θ2 ∂θ 0 0  2π 2 2 D f = ∂θ2 dθ. 0 



2π

.

D ∂u



∂f ∂θ



Recall our standing assumption that M ⊆ RN and note that by a Fourier expansion, we can prove that  2π 2 2  2π 2 ∂f ∂ f 2P (u) = ∂θ (u, θ)dθ ≤ ∂θ2 (u, θ)dθ 0 0 and 2 2   2π  2 2 ∂ f (u, θ) − D f dθ ≤ C2 sup ∂f P (u), ∂θ2 ∂θ ∂θ2 0

where the constant depends on the second fundamental form of M . Similarly,  /  2π .  ∂f ∂f ∂f ∂f ∂f , , dθ ≤ C3 sup P (u), R ∂u ∂θ ∂u ∂θ ∂θ 0 where the constant C3 now depends on the curvature of M . Putting it all together, we get the estimate, ∂f  P (u) ≥ 2P (u) − C4 sup P (u). ∂θ But the supremum of ∂f /∂θ can be made arbitrarily small by the estimate (4.86) allowing us to make the error terms arbitrarily small and get an estimate (4.87)

P  (u) ≥ P (u).

238

4. Harmonic and Minimal Surfaces

This estimate together with the maximum principle implies that P (u) must decay exponentially: P (u) = K(u) ≤ C5 e−u . In the (r, θ) coordinates this gives e(f ) ≤ C6 r. The last estimate is enough to establish H¨ older continuity of the extension of f to D1 . Then the usual bootstrapping argument shows that the limit is C ∞ , completing the proof. Returning now to our sequence {fm } of αm -harmonic maps with αm → 1 and E(fm ) ≤ E0 , we note that as measures, the sequence of Radon measures {e(fm )dA} converges weakly to  e(f∞ )dA + mi δpi dA, i=1l

where mi is a nonnegative constant and δpi is the Dirac delta-function located at pi . If mi is zero, we can throw the point pi away, since in this case Theorem 4.6.9 implies that after possibly passing to a subsequence {fm } will converge on a neighborhood of pi . If mi > 0 is nonzero, we call pi a bubble point of the sequence and mi measures the amount of energy lost at the bubble point. We next investigate what happens at the bubble points p1 , . . . , pl . For each m, choose qim ∈ Drm (pim ) so that e(fm )(qim ) = sup{e(fm )(q) : q ∈ Drm (pim )}. Note that qim → xi as j approaches ∞, and let  bim = e(fm )(qim ). If no subsequence of the fm ’s converge near pi , then bim → ∞. Assume that the latter alternative holds, and define   p ˜ ˜ fm : Drm bim (0) → M by fm (q) = fm qim + . bim One readily verifies that e(fim ) ≤ 1 and e(fim )(0) = 1, so bubbling cannot occur at 0 but it may occur at finitely many other points {q1 , . . . , qm }. In this case, a subsequence of the fm ’s converges to a nonconstant harmonic map f˜∞ : C − {q1 , . . . , qm } −→ M, the convergence being uniform in C k for all k on compact subsets of C − {q1 , . . . , qm }.

4.7. Existence of minimal two-spheres

239

Recall that as a Riemann surface the complex plane is just the Riemann sphere S 2 minus a point. Therefore the removable singularity theorem implies that f˜∞ extends to a smooth harmonic map fˆ∞ : S 2 → M . Moreover, this must be a nonconstant minimal two-sphere, since we normalized the energy density to be one at the origin. Remark 4.6.10. Each harmonic two-sphere which bubbles off carries with it a certain minimal amount of energy. Indeed, a harmonic two-sphere h : S 2 → M is automatically conformal and hence a minimal surface, and from the Gauss equation, its Gaussian curvature K : S 2 → R is bounded above by K0 , the maximum value of all sectional curvatures on M . Hence by the Gauss-Bonnet theorem,  4π KdA ≤ K0 (Area of h(S 2 )) ⇒ E(h) ≥ . 4π = K0 S2 In particular, if the energy of a sequence of αm -harmonic maps is bounded by E0 , the number of harmonic two-spheres that can bubble off in the limit is ≤ (4πE0 )/K0 .

4.7. Existence of minimal two-spheres 4.7.1. Minimal two-spheres in simply connected manifolds. With the analysis of bubbling at our disposal, we can now prove a theorem due to Sacks and Uhlenbeck [SU81], which can be thought of as the analog within minimal surface theory of the Fet-Lusternik Theorem 2.3.2 on existence of smooth closed geodesics: Theorem 4.7.1 (Sacks and Uhlenbeck). If M is a compact simply connected Riemannian manifold, then M contains at least one nonconstant minimal two-sphere. Proof. The argument is very similar to that given in §2.3 for the FetLusternik Theorem. There is a least integer q ≥ 2 such that Hq (M ; Z) = 0, and it follows from the Hurewicz theorem that πi (M ) = 0,

for 0 < i < q, and πq (M ) ∼ = Hq (M, Z) = 0.

Let M be the space of smooth maps from S 2 to M and let π : M → M by π(f ) = f (0), the evaluation of f at the basepoint 0 ∈ S 2 = C ∪ {∞}. It is a well-known fact from homotopy theory that π is a fibration with fiber Mp = π −1 (p), the set of basepoint preserving maps from S 2 to M . Moreover, πk (Mp ) ∼ = πk+2 (M )

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and the fibration π induces a long exact sequence · · · → πk (Mp ) → πk (M) → πk (M ) → πk−1 (Mp ) → · · · . We note, moreover, that π∗ : πk (M) → πk (M ) possesses a right inverse i∗ : πk (M ) → πk (M)

induced by the inclusion i : M → M

which takes a point to the constant map at that point. Hence the long exact sequence splits and we conclude that πk (M) ∼ = πk (M ) ⊕ πk (Mp ) ∼ = πk (M ) ⊕ πk+2 (M ). Thus the homotopy groups of M are completely determined by the homotopy groups of M . In particular, πq−2 (M) ∼ = πq (M ) = 0. Since M is simply connected, q ≥ 2 and πq−1 (M) ∼ = πq+1 (M ) is abelian. Moreover, we can identify πq−2 (M) with [S q−2 , M], the space of free homotopy classes of maps from S q−2 into M. Choose a nonzero element α ∈ [S q−2 , M]. Let F = {g(S q−2 ) such that g : S q−2 → M is a continuous map in [α]}. Then F is an ambient isotopy invariant family of sets, so Minimax(Eα , F ) is a critical value for Eα . Recall that M is isometrically imbedded in an ambient Euclidean space and let M (δ) denote the open δ-neighborhood of M in RN for δ > 0. By the tubular neighborhood theorem, if δ is sufficiently small, M is a strong deformation retract of M (δ). Moreover, for  > 0 sufficiently small, any map f : S 2 → M of energy <  can be contracted to its center of mass in RN without leaving M (δ). RN

Suppose now that g(S q−2 ) ⊂ Eα−1 ([0, ]). Then g is homotopic to a smooth map g˜ : S q−1 → M0 ,

where

M0 = {γ ∈ M : γ is constant }.

Hence Minimax(Eα , F ) ≥ . We now take a decreasing sequence of real numbers {αm } with αm → 1. The Minimax Theorem 1.11.2 gives a corresponding sequence {fm } of critical points for Eαm which have energy ≥  > 0. Either a subsequence converges to a nonconstant harmonic map or a bubble forms in the limit in accordance with the theory presented in the previous section. In the latter case, the bubble provides a nonconstant harmonic map. In either case we get a nonconstant harmonic map f∞ : S 2 → M which is automatically conformal, since there is only one conformal structure on S 2 . The image is a nonconstant minimal two-sphere, which proves the theorem. QED

4.7. Existence of minimal two-spheres

241

4.7.2. Realizing generators for π2 (M ) by minimal spheres. Let M be a compact manifold and suppose that f : D1 → M is a map with f (∂D1 ) = p which represents an element [f ] ∈ π2 (M, p). Suppose, moreover, that γ : [0, 1] → M is a smooth map with γ(0) = p = γ(1) which represents an element [γ] ∈ π1 (M, p). We can then define γ  f as follows: Let (r, θ) be polar coordinates on D1 and set  f (2r, θ), for 0 ≤ r ≤ 1/2, g(r, θ) = (γ  f )(r, θ) = γ(2r − 1), for 1/2 ≤ r ≤ 1. Then [γ], [f ] → [g] = [γ  f ] gives an action of π1 (M, p) on π2 (M, p). The orbits of this action comprise the space [S 2 , M ] of free homotopy classes of maps from S 2 into M . Alternatively, this action makes π2 (M, p) into a Z[π1 (M, p)]-module, where Z[π1 (M, p)] is the group ring of π1 (M, p). This action is discussed in more detail in Chapter 4 of [Hat02]. There are at most countably many generators for π2 (M, p) as a Z[π1 (M, p)]-module. Theorem 4.7.2 (Sacks and Uhlenbeck). Suppose that M is a compact Riemannian manifold. Then one can represent a generating set for π2 (M, p) as a Z[π1 (M, p)]-module by area-minimizing two-spheres, which may contain self-intersections and branch points. Note that the minimal two-spheres need not pass through the point p. If M is simply connected, it follows from the Hurewicz Theorem that π2 (M, p) ∼ = H2 (M ; Z), and hence we obtain the following: Corollary 4.7.3. If M is a compact simply connected Riemannian manifold, then a set of generators for H2 (M ; Z) can be represented by minimal two-spheres. Before proving Theorem 4.7.2, we make a few remarks regarding the symmetry of the functions E, Eα : C 2 (S 2 , M ) → R. We regard S 2 as the one-point compactification C ∪ {∞} of the complex plane C with the standard coordinate z = x + iy = reiθ = eu+iθ , and note that E is invariant under all the linear fractional transformations: If φ : S 2 → S 2 is the diffeomorphism defined by   az + b a b , for ∈ SL(2, C), φ(z) = c d cz + d then E(f ◦ φ) = E(f ). Note that     a b −1 0 = c d 0 −1

⇒

f ◦ φ = f,

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so the group of linear fractional transformations is actually G = P SL(2, C) = SL(2, C)/{±I}. In contrast, when α > 1, the maps Eα0 , Eα : C 2 (S 2 , M ) → R defined by   1 1 Eα0 (f ) = |df |2α dA and Eα (f ) = [(1 + |df |2 )α − 1]dA 2 Σ 2 Σ are invariant only under the smaller group of isometries SO(3) ⊆ P SL(2, C). Indeed, critical points of Eα0 must be parametrized so that the “center of mass” is zero: Lemma 4.7.4. Let X : S 2 → R3 denote the standard inclusion, and let 0 denote the origin in R3 . If f ∈ C 2 (S 2 , M ) is a critical point for Eα0 with α ∈ (1, ∞), then  X|df |2α dA = 0. S2

Moreover, there is a smooth function ψα : [0, ∞) → R such that ψα (0) = 0, ψα (t), ψα (t) > 0 for t > 0, and if f is a critical point for Eα , then  Xψα (|df |2 )dA = 0. (4.88) S2

In the argument for this lemma, we utilize the standard metric on S 2 of constant curvature one, expressed in terms of our standard coordinates by ds2 =

4 1 (dr2 + r2 dθ2 ) = (du2 + dθ2 ), (1 + r2 )2 cosh2 u

where r = e−u . (This metric is related by scaling to the constant curvature metric of total area one.) For t ∈ R, we define a linear fractional transformation φt : S 2 → S 2 by u ◦ φt = u + t, θ ◦ φt = θ, so that t → φt is a one-parameter subgroup. The energy density is then given by the formula  2 1 ∂f 2 ∂f 2 + cosh2 (u + t), e(f ◦ φt ) = 2 ∂u ∂θ and it is straightforward to calculate its derivative at t = 0: 1 d d 2 e(f ◦ φt ) |d(f ◦ φt )| = (4.89) dt 2 dt t=0 t=0  2  2 ∂f ∂f = + sinh u cosh u = |df |2 tanh u. ∂u ∂θ

4.7. Existence of minimal two-spheres

243

Using this fact, we can perform a straightforward calculation to obtain  d 0 E (f ◦ φt ) = (α − 1) |df |2α(tanh u)dA. dt α 2 t=0

S

In terms of the standard Euclidean coordinates (x, y, z) on R3 , S 2 is represented by the equation x2 + y 2 + z 2 = 1, and a straightforward computation using stereographic projection from the north pole to the (x, y)-plane shows that sinh u r2 − 1 = = tanh u, z= 2 r +1 cosh u so if f is a critical point for Eα0 , then  d 0 = Eα0 (f ◦ φt ) = (α − 1) |df |2αzdA. dt 2 S t=0 But we can take the z-axis to be any line passing through the origin, and hence  X|df |2α dA = 0, S2

for the metric of constant curvature one, and hence for the constant curvature metric of total area one. More generally, we can use (4.89) to calculate the derivative of Eα to obtain  1 d d 2 α 2 Eα (f ◦ φt ) = (1 + |d(f ◦ φt )| ) sech (u + t)dudθ dt 2 dt §2 t=0  t=0  =α (1 + |df |2 )α−1 tanh u|df |2 dA − (1 + |df |2 )α tanh udA. S2

S2

Thus, if f is a critical point for Eα ,  [α(1 + |df |2 )α−1 |df |2 − (1 + |df |2 )α ]zdA, 0= S2  = [α(1 + |df |2 )α−1 |df |2 − (1 + |df |2 )α − 1]zdA, S2

where we have used the fact that the average value of z on S 2 is zero. We obtain (4.88) by setting  t α(1 + t)α−1 t − (1 + t)α + 1 τ = α dτ. (4.90) ψα (t) = α−1 (1 + τ )2−α 0 One easily verifies that this function has the desired properties, and has a smooth limit as α → 1, namely  t τ dτ = t − log(1 + t). (4.91) ψ1 (t) = 0 (1 + τ )

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Remark 4.7.5. It is sometimes convenient to demand that harmonic maps f : S 2 → M satisfy the side condition  Xψ1 (|df |2 )dA = 0, S2

thereby reducing the symmetry group of the problem from P SL(2, C) to a maximal compact subgroup SO(3). Lemma 4.7.6. Suppose that M is a compact Riemannian manifold. Then there is an 0 > 0 depending on an upper bound K0 > 0 for the sectional curvatures for M such that every nonconstant α-harmonic map f : S 2 → M for α ≤ α0 ∈ (1, ∞) is sufficiently close to one, satisfies Eα (f ) ≥ E(f ) ≥ 0 . Since M is compact, we can rescale so that the sectional curvatures are bounded above by one. We give S 2 the constant curvature metric η of total ˜ = 2π. Then it follows from (4.75) area one, which has Gaussian curvature K that Δη (e(f )) ≥ 2(2π − e(f ))e(f ). It then follows from the ε-regularity Theorem 4.6.3 that there is an 0 > 0 such that α ∈ (1, α0 ] and E(f ) < 0 ⇒ e(f ) < 2π. This yields a contradiction to the maximum principle for Δη unless e(f ) is identically zero. Lemma 4.7.7. Suppose that {fm } is a sequence of nonconstant critical points for Eαm with αm → 1 such that fm converges to a harmonic map f∞ : S 2 → M on S 2 − {0}, uniformly on compact subset in C k for all k. Then f∞ cannot be a map to a point. Indeed, it follows from the previous lemma that Eα (fm ) ≥ 0 and hence if f∞ were constant a nontrivial bubble would form near the point 0 ∈ S 2 . Thus energy density would have to concentrate at the point 0 and this would contradict (4.88). Proof of Theorem 4.7.2. The proof is an induction which makes use of a succession of variational problems. Let F1 = {f ∈ C 2 (S 2 , M ) : the free homotopy class of f is nontrivial}, and let μ1 = inf{E(f ) : f ∈ F1 }.

4.7. Existence of minimal two-spheres

245

Suppose that αm → 1 and let fm be an element of F1 which achieves a minimum for Eαm on F1 . Either a subsequence of {fm } converges without bubbling to a minimal two-sphere f∞ : S 2 → M which lies within F1 , in which case we have a nonconstant minimal two-sphere f ∈ F1 such that E(f1 ) = μ1 , or at least one nonconstant minimal two-sphere h : S 2 → M bubbles off at a point p1 ∈ S 2 . We need to show that the second possibility leads to a contradiction. Recall that a subsequence of {fm } will converge to a harmonic map f∞ : S 2 → M , uniformly on compact subsets of S 2 − {p1 , . . . , pl }, where p1 , . . . , pl are the bubble points. By Lemma 4.7.7, either f∞ is nonconstant or there is at least one other bubble point. Let us suppose that f∞ is nonconstant. (The other case can be treated in a similar fashion.) We will perform surgeries on small circles about p1 to divide each αm -harmonic map ˆm : S2 → M . into a base map fˆm : S 2 → M and a bubble h In more detail, let Drm (p1 ) be a disk of radius rm chosen so that rm → 0 as m → ∞,  e(fm )dA ≥ 0 and fm (Drm (p1 ) − D(1/3)rm (p1 )) ⊆ Nδ (q), Drm (p1 )

where q = f∞ (p1 ) and Nδ (q) is the domain of a geodesic coordinate system of radius δ in M . Then define a map fˆm : S 2 → M by  fm (p), for p ∈ S 2 − Drm (p1 ), fˆm (p) = expq (η(p)(expq )−1 fm (p)), for p ∈ Drm (p1 ), where η : S 2 → [0, 1] is a smooth function such that  1 on S 2 − D(2/3)rm (p1 ), η≡ 0 on D(1/3)rm (p1 ). Let Rm : S 2 → S 2 be the reflection through the circle ∂D(1/2)rm (p1 ), and ˆ m : S 2 → M by define h  −1 2 ˆ m (p) = expq (η ◦ Rm (p)(expq ) Rm ◦ fm (p)), for p ∈ S − Drm (p1 ), h for p ∈ Drm (p1 ), fm (p), ˆ m agrees with fm Thus fˆm agrees with fm outside Drm (p1 ) while Rm ◦ h inside Drm (p1 ). We can clearly arrange that ˆ m ) < Eα (fm ) + 0 Eαm (fˆm ) + Eαm (h m 2 if m is sufficiently large, where 0 is the constant appearing in Lemma 4.7.6.

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ˆ m must be homotopically nontrivial with At least one of the maps fˆm or h energy less than the infimum over F1 , and this provides the desired contradiction. Thus we obtain a homotopically nontrivial minimal two-sphere f1 : S 2 → M and we let Γ1 denote the Z[π1 (M, p)]-submodule of π2 (M, p) generated by [f1 ]. If Γ1 = π2 (M, p), we let F2 = {f ∈ C 2 (S 2 , M ) : the free homotopy class of f is not in Γ1 }, set μ2 = inf{E(f ) : f ∈ F2 }, and proceed exactly as before. One thereby obtains a minimal two-sphere f2 : S 2 → M which does not lie in Γ1 . We then let Γ2 be the Z[π1 (M, p)]-submodule of π2 (M, p) generated by [f1 ] and [f2 ], and so forth. For the inductive step, we suppose we have already constructed a Z[π1 (M, p)]-submodule Γk−1 of π2 (M, p) generated by minimal two-spheres [f1 ], . . . , [fk−1 ]. If Γk−1 = π2 (M, p), we let Fk = {f ∈ C 2 (S 2 , M ) : the free homotopy class of f is not in Γk−1 }, we let μk = inf{E(f ) : f ∈ Fk }, and we verify that there is a nonconstant minimal two-sphere fk ∈ Fk such that E(fk ) = μk . Theorem 4.7.2 follows by induction on k. The above theorem can be applied to compact three-dimensional Riemannian manifolds, in which case the surfaces which minimize area are free of branch points by a theorem of Osserman and Gulliver (see [O70] and [Gul73]).2 In fact, Meeks and Yau [MY80] carry out the tower construction of Papakyriakopoulos with minimal surfaces to show that one can choose the generators of π2 (M, p) as a Z[π1 (M, p)]-module to be either imbedded minimal two-spheres or double coverings of imbedded minimal projective planes: Theorem 4.7.8 (Meeks and Yau). Suppose that M is an oriented compact three-dimensional Riemannian manifold. Then there is a collection {f1 , . . . , fk , . . .} of generators for π2 (M, p) as a Z[π1 (M, p)]-module, each of which is either an embedded minimal two-sphere or a doubly covered imbedded projective plane. We refer to [MY80] for the proof. This is one of three topological theorems for three-manifolds proven via minimal surfaces by Meeks and Yau, the others being the Loop Theorem and Dehn’s Lemma. 2 If we make a generic choice of Riemannian metric, we can instead apply the Bumpy Metric Theorem 5.1.1 of the next chapter.

4.7. Existence of minimal two-spheres

247

We say that a compact oriented three-dimensional manifold M is irreducible if every two-sphere smoothly imbedded within it bounds a ball. It is prime if it is either irreducible or diffeomorphic to S 2 × S 1 . We claim that it follows from Theorem 4.7.8 that when M is a compact oriented three-dimensional manifold with nontrivial π2 , one can divide M along finitely many homotopically nontrivial minimal two-spheres which separate M and obtain a connected sum decomposition (4.92)

M = M1 M2  · · · Mk ,

with each summand Mi being prime. We call (4.92) the prime decomposition of M . To prove the claim, we suppose that π2 (M ) = 0. Then according to Theorem 4.7.8, there is either an imbedded two-sphere in M or an imbedded projective plane RP 2 ⊆ M . In the case of an imbedded sphere, it may either separate M into two components or not. In the former case, the imbedded two-sphere divides M into a nontrivial direct sum M = M1 M2 , each summand having nontrivial π2 . (If either summand had trivial π1 it would be a homotopy sphere and the two-sphere would be homotopic to a constant.) If the minimal two-sphere is nonseparating, a tubular neighborhood of the two-sphere is a trivial bundle E = S 2 × [−1, 1] over S 2 and we can connect S 2 × {1} to S 2 × {−1} by a smooth imbedded curve C in M which intersects E only at its endpoints. Let D be a tubular neighborhood of C. Then E ∪ D is diffeomorphic to S 2 × S 1 minus a closed three-ball after smoothing corners. It follows that either M is diffeomorphic to S 2 × S 1 , or we can once again construct a connected sum decomposition M = M1 M2 in which both summands have nontrivial π2 . In the case of an imbedded projective plane RP 2 ⊆ M , since M is oriented a closed tubular neighborhood of RP 2 has an imbedded two-sphere as boundary, and either M is diffeomorphic to the oriented manifold RP 3 , or this two-sphere once again divides M into a nontrivial direct sum, each summand having nontrivial π2 . Using the process just described, we can continue constructing direct sum decompositions (4.92) with larger and larger values of k. This process either continues forever or stops when further decomposition would lead to summands with trivial π1 . At each stage the van Kampen Theorem from algebraic topology provides a decomposition of π1 (M ) into a free product π1 (M ) ∼ = π1 (M1 ) ∗ π1 (M2 ) ∗ · · · ∗ π1 (Mk ), the fundamental group of each summand being nontrivial. A theorem of Grushko [Sta65] states that the minimum number of generators needed for

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a free product G1 ∗ G2 equals the sum of the minimum number of generators needed for G1 and for G2 . Thus, since π1 (M ) is finitely generated, the number of summands is bounded and the process of dividing into connected sums must stop at a finite stage. (This also gives a proof that the number of generators needed for the Meeks-Yau Theorem is finite.) One might wonder whether a summand with trivial fundamental group might be further divided into a nontrivial direct sum. But the resolution of the Poincar´e conjecture implies that this cannot happen, and that each summand which is not diffeomorphic to S 2 × S 1 is irrreducible. Thus one obtains the prime decomposition (4.92) as claimed. Existence of this prime decomposition was established by Kneser without using the solution to the Poincar´e conjecture, and uniqueness was proven by Milnor. The remarkable consequence of the Meeks-Yau argument is that the decomposition occurs along imbedded two-spheres which can be taken to be minimal with respect to a given preassigned Riemannian metric. Example 4.7.9 (Failure of Morse inequalities for E). The connected sum decomposition provides many examples of free homotopy classes within [S 2 , M ] which cannot be represented by area-minimizing minimal two-spheres, showing that full Morse inequalities cannot hold for E. Here is an explicit example illustrating how much of the critical point theory for α-harmonic maps gets lost as α → 1. Starting with a lens space L(3, 1) of constant curvature one, we consider the connected sum M = L(3, 1)L(3, 1)L(3, 1) along isolated imbedded minimal two-spheres N1 and N2 of very small radius of curvature which minimize within their free homotopy classes. From van Kampen’s Theorem it follows that the fundamental group is generated by elements a, b and c, with the relations a3 = b3 = c3 = 1. The second homotopy group π2 (M ; p) is generated as a π1 (M )-module by the two imbedded minimal spheres f1 : S 2 → N1 ,

f2 : S 2 → N2 .

We can construct an imbedded two-sphere N = N1 N2 by connecting N1 with N2 by a very thin tube and a corresponding imbedding f : S 2 → N which represents a free homotopy class in [S 2 , M ] which is not freely homotopic to any multiple of [f1 ] or [f2 ]. If α0 is sufficiently close to one, then for each α ∈ (1, α0 ] there is a minimizing α-minimal two-sphere fα in the component of Map(S 2 , M ) representing this free homotopy class. As α → 1 a subsequence of these α-minimal two-spheres should approach a configuration consisting of N1 , N2 and a minimal geodesic connecting N1 and N2 of some length bounded away from zero. The free homotopy class of f is not represented by a minimal two-sphere in M .

4.8. Existence of higher genus minimal surfaces

249

4.8. Existence of higher genus minimal surfaces 4.8.1. Tori of least area. Just as we proved existence of minimal spheres in §4.7, we would like to prove existence of minimal surfaces of least area of higher genus using the two-variable energy E : M(Σ, M ) −→ R, where M(Σ, M ) =

Map(Σ, M ) × Met0 (Σg ) Map(Σ, M ) × Tg = , Diff+ (Σg ) Γg

as described at the end of §4.3. Here Met0 (Σg ) is the space of Riemannian metrics on Σg of constant curvature and total area one, Diff+ (Σg ) is the space uller space and of orientation-preserving diffeomorphisms of Σg , Tg is Teichm¨ Γg is the mapping class group. We can try to approach this problem by first developing a Morse theory for the α-energy Eα : M(Σ, M ) −→ R and then let α → 1. But there are three problems which must be controlled: (1) The most essential problem is bubbling, as described in §4.6. (2) A second problem is that a minimal surface might be a nontrivial cover (possibly branched) of a minimal surface of lower area. (3) Finally, as α → 1 the conformal structure of a sequence of critical points might approach infinity within the moduli space. We start by considering the case where g = 1 and Σg is a torus T 2 . We say that a smooth map f : T 2 → M has rank k if the image of f : π1 (T 2 ) −→ π1 (M ) is an abelian group with k generators. If f0 : T 2 → M has rank two, then so do all the elements in the component Mapf0 (T 2 , M ) = {f ∈ Map(T 2 , M ) : f is homotopic to f0 }. For k ∈ {0} ∪ N, we let Map(k) (T 2 , M ) = {f ∈ C 2 (T 2 , M ) : f has rank k }. For example, if f (π1 (T 2 )) ∼ = Z ⊕ Z2 , then f ∈ Map(2) (T 2 , M ). Note that the mapping class group Γ = SL(2, Z) preserves Map(k) (T 2 , M ), so Eα induces a map Eα : M(k) (T 2 , M ) −→ R,

where

M(k) (T 2 , M ) =

Map(k) (T 2 , M ) × T . Γ

Moreover, if f ∈ Map(2) (T 2 , M ), f ◦φ=f

for some

φ∈Γ

⇒

φ = identity,

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so the mapping class group SL(2, Z) acts freely on Map(2) (T 2 , M ) × T , and (k) if Mα (T 2 , M ) denotes the completion with respect to the L2α 1 -norm, then (k) 2 Mα (T , M ) will be a smooth Banach manifold. Recall that the α-energy descends to a C 2 map on the quotient 2 Eα : M(k) α (T , M ) −→ R. (1)

We expect that sequences tending to a minimum for α-energy in Mα (T 2 , M ) would degenerate to a closed geodesic. But two of our problems are con(k) trolled at least partially in Mα (T 2 , M ): this space does not contain any branched covers of spheres and the following lemma shows that critical points cannot have conformal structure approaching infinity: (2)

Lemma 4.8.1. The map Eα : Mα (T 2 , M ) → R satisfies Condition C. Let us recall that Condition C asserts is that if [fi , ωi ] is a sequence of points in M(2) (Σ, M ) on which Eα is bounded and for which dEα ([fi , ωi ]) → 0, and if for each i, (fi , ωi ) ∈ Map(2) (T 2 , M ) × T is a representative for [fi , ωi ], then there are elements φi ∈ Γ such that a subsequence of (fi ◦ φi , φ∗i ωi ) converges to a critical point for Eα on Map(2) (T 2 , M ) × T . To prove this, we recall that for the torus, the Teichm¨ uller space T is the upper half-plane, and after a change of basis we can arrange that an element ω ∈ T lies in the fundamental domain (4.93)

D = {u + iv ∈ C : v > 0, −(1/2) ≤ u ≤ (1/2), u2 + v 2 ≥ 1}

for the action of the mapping class group Γ = SL(2, Z). The moduli space R is obtained from D by identifying points on the boundary. The complex torus corresponding to ω ∈ T can be regarded as the quotient of C by the abelian subgroup generated by d and ωd, where d is any positive real number, or alternatively, this torus is obtained from a fundamental parallelogram spanned by d and ωd by identifying opposite sides. The fundamental parallelogram of area one can be regarded as the image of the unit square {(t1 , t2 ) ∈ R2 : 0 ≤ ti ≤ 1} under the linear transformation    1    1 1 1 u t x t → =√ 2 , 0 v t2 t y v where z = x + iy is the usual complex coordinate on C. A straightforward calculation gives a formula for the usual energy   2 2  ∂f 1 + ∂f dxdy E(f, ω) = ∂y 2 P ∂x 2    ∂f 2 1 ∂f ∂f 1 v 1 + 2 − u 1 dt1 dt2 , = 2 P ∂t v ∂t ∂t

4.8. Existence of higher genus minimal surfaces

251

P denoting the image of the unit square. The only way that ω can approach the boundary of Teichm¨ uller space while remaining in the fundamental domain D is for v → ∞. The rank two condition implies that the maps t1 → f (t1 , b) must be homotopically nontrivial for each choice of t2 = b, and hence the length in M of t1 → f (t1 , b) is bounded below by a positive constant c. This implies that   1 1 1 ∂f 2 1 2 v dt dt E(f, ω) ≥ 2 0 0 ∂t1 (4.94)  v 1 c2 v ≥ (length of t1 → f (t1 , b))2 db ≥ 2 0 2 by the Cauchy-Schwarz inequality, and hence Eα (f, ω) (which is ≥ E(f, ω)) must approach infinity. Suppose now that [fi , ωi ] is a sequence of points in M(2) (T 2 , M ) on which Eα is bounded and for which dEα ([fi , ωi ]) → 0, and for each i, (fi , ωi ) ∈ Map(T 2 , M )×T is a representative for [fi , ωi ]. Then the projection [ωi ] ∈ R of ωi ∈ T is bounded, and must therefore have a subsequence which converges to an element [ω∞ ] ∈ R. Hence there are elements φi ∈ Γ such that a subsequence of φ∗i ωi converges to an element ω∞ ∈ T . Then Eα,ω∞ (fi ◦ φi ) is bounded and dEα,ω∞ (fi ◦ φi ) → 0, so by Condition C for Eα,ω∞ , a subsequence of {(fi ◦ φi , φ∗i ωi )} converges to a critical point for Eα on Map(T 2 , M ) × T . This establishes Condition C for the function Eα . An application is given by the following theorem, proven by Schoen and Yau [SY79] and Sacks and Uhlenbeck [SU82]: Theorem 4.8.2. Every component of M(2) (Σ, M ) contains an element [f, ω] which minimizes the function E : M(2) (T 2 , M ) −→ R. If (f, ω) is a representative, then f : T 2 → M is conformal and harmonic with respect to ω, and hence a minimal surface. Proof. Let C be a component of M(2) (T 2 , M ). Choose a decreasing sequence αm → 1 and for each αm , a corresponding critical point [fm , ωm ] ∈ M(2) (T 2 , M ) which minimizes Eαm on C. We can assume that (4.95)

Eαm ([fm , ωm ]) → inf{E(f, ω) : [f, ω] ∈ C}.

Since [ωm ] must lie in a bounded region of the Riemann moduli space R1 , we can arrange that [ωm ] converges after passing to a subsequence, and hence after choosing suitable representatives, arrange that ωm converges to

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4. Harmonic and Minimal Surfaces

an element ω ∈ T1 . By Theorem 4.6.6, we can pass to a further subsequence so that either {fm } converges uniformly on compact subsets of T 2 to an ω-harmonic map f : T 2 → M or nonconstant minimal two-spheres bubble off as α → 1. But in the latter case, we can perform a surgery just like we described in the proof of Theorem 4.7.2 and obtain a new parametrized torus fˆm : T 2 → M which lies in the same component C and has smaller energy 0 Eαm (fˆm , ωm ) < Eαm (fm , ωm ) − , 2 contradicting (4.95). So no bubbling can occur, and we obtain a critical point (f, ω) for E which minimizes E within the component C. QED Remark 4.8.3. When the dimension of the ambient manifold M is three, the local regularity result of Osserman and Gulliver again implies that the minimal tori found by Theorem 4.8.2 are immersed. Suppose now that we want to investigate the topology of a compact oriented three-dimensional manifold. We start by using π2 (M ) to express M as a direct sum decomposition (4.92) in which each of the summands is prime. One can show that each prime summand is one of three types: it is diffeomorphic to S 1 × S 2 , it is finitely covered by S 3 , or all of its homotopy groups are trivial except for π1 = π, so it is a K(π, 1). The next step in exploring the topology is to try to divide up the K(π, 1) summands along imbedded tori. An imbedded torus or Klein bottle in M is incompressible if the inclusion induces an injection on π1 . The positive resolution of Thurston’s geometrization conjecture (as described in [MF10]) implies that any prime simply connected compact threemanifold M can be divided along imbedded tori into manifolds with boundary consisting of tori, the interior of each having one of eight possible geometries with finite volume. According to Theorem 5.1 of Hass and Scott [HS88], for a given choice of Riemannian metric on M , each decomposing torus is either isotopic to an imbedded minimal torus or to the boundary of a nonorientable [−1, 1]-bundle over an imbedded minimal Klein bottle. Many of these tori are incompressible, so in this way we get many examples of geometrically interesting incompressible minimal tori in compact oriented three-manifolds with arbitrary Riemannian metric. 4.8.2. Surfaces of least area of genus ≥ 2. The existence theory for incompressible minimal tori of least area presented in the previous section can be extended to surfaces of higher genus. Suppose that Σ is a compact oriented surface of genus g ≥ 2. We set Map (Σ, M ) = {f ∈ C 2 (Σ, M ) : f : π1 (Σ) → π1 (M ) is injective},

4.8. Existence of higher genus minimal surfaces

253

and call it the space of incompressible maps from Σ to M . (We use this terminology even if f is not an imbedding.) We define M (Σ, M ) to be the corresponding quotient space M (Σ, M ) =

Map (Σ, M ) × T , Γ

with Γ being the mapping class group, and let Mα (Σ, M ) be the completion of M (Σ, M ) with respect to the L2α 1 -norm. Lemma 4.8.4. If Σ is a compact oriented surface of genus g ≥ 2, then the map Eα : Mα (Σ, M ) → R satisfies Condition C. Proof. The proof makes use of two ingredients, a collar theorem of Keen [Kee74], and the structure of the Bers compactification of the moduli space Mg of conformal structures on Σ. We suppose that Σ is given the hyperbolic metric of Gaussian curvature −1 corresponding to the conformal structure ω ∈ T . The collar theorem is a fundamental result of Riemann surface theory, and states that if γ is a closed geodesic of this metric of length ≤ k1 , where k1 is a positive constant, then there is a collar region C ⊆ Σ of fixed area k2 > 0 about γ. Indeed, we can arrange that γ lifts to the map 

γ˜ : [0, l] → H

2

defined by

iπ γ˜ (t) = exp t + 2

˜ where and the collar region is of the form C = π(C), (4.96) C˜ = {reiθ ∈ H2 : 1 ≤ r < el , π − θ0 < θ < θ0 }, where

 ,

cot θ0 =

k2 . 2l

Lemma 4.8.5. Suppose that Σ has a closed geodesic γ whose length with respect to the hyperbolic metric corresponding to ω is l ≤ k1 , and that C is the collar region about γ described above. If f : Σ → M is any smooth map, then the ω-energy of f |C is at least k4 /l, for some positive constant k4 . Proof. We consider the lift f˜ : C˜ → M , which has energy density  2 ˜ ˜ = 1 |df |2 ≥ 1 |(f ◦ γ˜θ ) (t)| , e(f) 2 2 |˜ γθ (t)|2

where

γ˜θ (t) = exp (t + iθ) .

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4. Harmonic and Minimal Surfaces

Straightforward calculation shows that 1 |˜ γθ (t)|2 = sin2 θ

Thus we find that  ˜ = E(f˜|C)  ≥

π−θ0 θ0



π−θ0 

θ0

el 1 el 1

2 ∂ f˜ and |(f˜ ◦ γ˜θ ) (t)|2 = r2 , ∂r 2 r2 sin2 θ ∂ f˜ so e(f ) ≥ . ∂r 2

drdθ r sin2 θ 2   2 1 π−θ0 l ∂ f˜ r ∂ f˜ drdθ = dtdθ, 2 ∂r 2 θ0 0 ∂t

e(f˜)

where we have set r = et in the inequality,  2 ˜ L(f ◦ γ˜θ ) = 0

last integral. But by the Cauchy-Schwarz l

 l ˜ 2 ∂ f˜ ∂f dt ≤ l dt, ∂t 0 ∂t

and since f˜◦ γ˜θ is a closed curve in M which is not homotopic to a constant, L(γθ ) ≥ k3 , for some positive constant, k3 . Hence 2 ˜ ≥ k3 (π − 2θ0 ) , E(f, ω) ≥ E(f˜|C) 2l which yields the assertion of the lemma, since θ0 → 0 as l → 0. QED

To finish the proof of Lemma 4.8.4, we suppose that {[fm , ωm ]} is a sequence of points in Mα (Σ, M ) such that Eα ([fm , ωm ]) is bounded. We claim that the sequence {[ωm ]} must stay in a compact region of the Riemann moduli space Mg . To see this, we make use of the Bers compactification of the moduli space Mg , as described in Appendix B of [IT92]. After passing to a subsequence, we can assume that {[ωm ]} converges to a point in the compactification. But points in the compactification are Riemann surfaces with nodes and and as one approaches a Riemann surface with nodes from inside the moduli space, some homotopically nontrivial loop must have its length go to zero, and then the energy will go to infinity by Lemma 4.8.5. Thus {[ωm ]} must remain bounded, and after passing to a subsequence we can assume that [ωm ] → [ω∞ ] ∈ Mg . Now we use the fact that Eα,ω∞ satisfies Condition C to conclude that Eα : Mα (Σ, M ) → R also satisfies Condition C. QED Theorem 4.8.6. If Σ is a compact oriented Riemann surface of genus g ≥ 2, then every component of the space M (Σ, M ) of incompressible maps from

4.8. Existence of higher genus minimal surfaces

255

Σ to M contains an element [f, ω] which minimizes the function E : M (Σ, M ) −→ R. If (f, ω) is a representative, then f : Σ → M is conformal and harmonic with respect to ω, and hence a minimal surface. To prove this, we use essentially the same argument as we used in the proof of Theorem 4.8.2. Example 4.8.7. Let M be a compact three-dimensional manifold with constant negative sectional curvature. We say that a component Map0 (Σ, M ) of Map(Σ, M ) is incompressible if whenever f ∈ Map0 (Σ, M ), f induces a monomorphism on fundamental groups. Since Σ and M are both EilenbergMacLane spaces, it follows from Theorem 8.1.11 of [Spa66] that incompressible components correspond to conjugacy classes of monomorphisms π1 (Σ, p0 ) → π1 (M, f (p0 )). The mapping class group acts as outer automorphisms on π1 (Σ, p0 ), and incompressible components of Map(Σ, M ) × T project to components of the quotient M (Σ, M ). We further call the incompressible component prime if there is no unbranched cover p : Σ → Σ0 , with the genus of Σ0 strictly less than the genus g of Σ, together with a map f0 : Σ0 → M , injective on fundamental groups, such that f = f0 ◦ p. When this happens, we also say that the corresponding component of M (Σ, M ) is prime. We can state these conditions in terms of surface subgroups, a surface subgroup of genus g ≥ 2 within π1 (M ) being a subgroup with presentation −1 −1 −1 a1 , . . . , ag , b1 , . . . , bg |a1 b1 a−1 1 b1 · · · ag bg ag bg = 1,

and hence isomorphic to the fundamental group of a compact oriented surface of genus g. Then components of M (Σ, M ) correspond to surface subgroups of π1 (M ) of genus g, while prime components of M (Σ, M ) correspond to surface subgroups of genus g which are not contained in any surface subgroup of genus strictly less than g. Kahn and Markovic [KM12a, KM12b] show that when M is any compact three-manifold with constant negative curvature metric, there are many such components. Indeed, they show that there is a constant c > 0 such that there are at least (cg)2g components of M (Σ, M ) in which the domain has genus g, and the rate of growth with genus implies that most of these must be prime. Theorem 4.8.6 implies that each prime incompressible component of Map(Σ, M ) contains a prime area minimizing minimal surface, and we thus obtain prime minimal surfaces of arbitrarily high genus in any compact hyperbolic three-manifold M . In particular, any hyperbolic three-manifold has

256

4. Harmonic and Minimal Surfaces

infinitely many “geometrically distinct” closed minimal surfaces. Since they are minimizing they must be immersions by the theorem of Osserman and Gulliver cited earlier, but we expect that they are only rarely imbeddings.

4.9. Unstable minimal surfaces We have seen in the preceding sections that it is sometimes possible to use the α-energy to prove the existence of minimal surfaces of least area in Riemannian manifolds subject to appropriate constraints, in spite of the obstructions provided by bubbling. We now consider the question of when it is possible to prove the existence of minimal surfaces corresponding to minimax constraints, in particular constraints that come from homology or cohomology classes of positive degree for the space Map(Σ, M ). 4.9.1. The minimax construction. If Σ is a compact connected surface with conformal structure ω, the α-energy Eα,ω : L2α 1 (Σ, M ) −→ R satisfies Condition C, and we can use the minimax construction from §1.12 to construct minimax critical points corresponding to various topological constraints. For example, given an element x ∈ Hλ (Map(Σ, M ); Q), we let Fx denote the space of subsets h(A) ⊆ Map(T 2 , M ), where A is a compact oriented manifold of dimension λ, and h : A → Map(Σ, M ) is a smooth map such that h∗ takes the fundamental class of A to a nonzero multiple of x. It follows from Corollary 4.6.7 and Thom’s resolution of the Steenrod representability problem for rational homology (see §15 of [CF64] for example) that for each choice of x, the collection Fx is nonempty. Since Fx is also ambient isotopy invariant, it follows from the minimax principle (Theorem 1.12.2) that μα,ω (x) = inf{sup{Eα,ω (h(x)) : x ∈ A} : (A, h) ∈ Fx } is a critical value for Eα,ω , represented by a minimax (α, ω)-harmonic map fα : Σ → M . Similarly, we can use cohomology with arbitrary coefficients, in particular, with coefficients in R or Q, as studied in Chapter 3. Thus, for example, given an element a ∈ H λ (Map(Σ, M ); Q),

4.9. Unstable minimal surfaces

257

we let Fa denote the space of subsets h(A) ⊆ Map(T 2 , M ), where A is a compact oriented manifold of dimension λ, and h : A → Map(Σ, M ) is a smooth map such that



h∗ a = 0. A

Once again, for each choice of a, the collection Fa is nonempty and ambient isotopy invariant, so μα,ω (x) = inf{sup{Eα,ω (h(x)) : x ∈ A} : (A, h) ∈ Fa } is a critical value for Eα,ω , represented by a minimax (α, ω)-harmonic map fα : Σ → M . The Birkhoff Minimax Principle therefore provides various nonconstant α-harmonic maps into M . But to obtain genuine harmonic maps or minimal surfaces, we need to take the limit of minimax critical points fα constructed as above as α → 1. 4.9.2. Bubble trees. Suppose therefore that {αm : m ∈ N} is a sequence of real numbers decreasing monotonically to one, and that for each n ∈ N, fm is a minimax critical point for  1 2 [(1+|df |2 )αm −1]dA, Eαm ,ω : C (Σ, M ) −→ M defined by Eαm ,ω (f ) = 2 Σ corresponding to one of the minimax constructions described in the preceding section. Then the sequence Eαm ,ω (fm ) decreases monotonically to some nonnegative limit. Moreover, the Morse index of the elements in the sequence is ≤ λ. Can we find a subsequence of {fm : m ∈ N} which has a limit consisting of a collection of harmonic surfaces as m → ∞? In what sense is the limit achieved? We claim that if π1 (M ) is finite, the α-energy critical points generated by the minimax construction for a given homology or cohomology class converge in a sense to be made precise to a “bubble tree.” Definition. A bubble tree based upon a compact Riemann surface Σ = Σg of genus g ≥ 0 with conformal structure ω is a collection of maps indexed by the vertices and edges of a rooted tree T , with root denoted by 0. The root of the tree corresponds to an ω-harmonic base map f0 : Σ → M,

when v = 0,

and the other vertices correspond to harmonic two-spheres or bubble maps fv : S 2 → M,

when v = 0,

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4. Harmonic and Minimal Surfaces

while the edges correspond to smooth maps called necks, γv : [0, 1] → M, which join the parent p(v) of a vertex to a vertex v (with γv only defined when v = 0), so that γv (0) ∈ Image(fp(v) ), We the We M.

γv (1) ∈ Image(fv ).

allow a vertex v to correspond to a constant map, but only when v is parent of another vertex. In this case we say that fv is a ghost bubble. also allow some of the necks γv to be constant maps to a single point of The energy of the bubble tree is  E(fv ). Eω (f0 ) + v =0

Figure 1 gives an illustration of a bubble tree. If M is compact, we can assume without loss of generality that we have rescaled the Riemannian metric on M so that it satisfies the condition Kr (σ) ≤ 1. If we assume we have an upper bound on energy of a bubble tree, we get a limit on the number of possible bubbles from the following elementary lemma: Lemma 4.9.1. Suppose that the real sectional curvatures of the Riemannian metric on M satisfy the inequality Kr (σ) ≤ 1. Then any nonconstant minimal two-sphere f : S 2 → M must have area at least 4π. Proof. It follows from the Gauss equation that the induced Gaussian curvature Kf on S 2 ≤ Kr ((fp )∗ Tp S 2 ) and hence from the Gauss-Bonnet formula that if f : S 2 → M is any minimal two-sphere,  E(f ) ≥

Kf dAf ≥ 4π,

the last inequality being strict if f has nontrivial branch locus. QED We can now state the following: Theorem 4.9.2. Suppose that M is a compact manifold with a finite fundamental group, and that {fm : m ∈ N} is a sequence of critical points for Eαm ,ω with αm → 1, which have uniformly bounded energy and uniformly bounded Morse index. Then a subsequence of {fm : m ∈ N} converges (in the sense described below) to a bubble tree, with the necks converging to geodesics of finite length and of energy zero. Moreover, the sequence Eαm ,ω (fm ) converges to the energy of the bubble tree.

4.9. Unstable minimal surfaces

259

Figure 1. As α → 1 a subsequence of α-energy critical points will converge to a bubble tree.

In particular, there is no energy loss in the necks. Our proof that there is no energy loss uses the Gromov estimates of §3.3. Indeed, we will show that a subsequence of {fm } converges in the sense of varifolds (as described in [Mor08] or [Sim84]) to the sum of the images of the harmonic maps at the vertices, but we will actually show a stronger conergence in terms of suitable reparametrizations of domains. Note that Theorem 4.9.2 applies to a sequence {fm : m ∈ N} of critical points for Eαm ,ω with αm → 1 which are constructed via a minimax construction from a homology class in H∗ (Map(Σ, M ); Q), or a cohomology class in H ∗ (Map(Σ, M ); Q). Proof of the theorem. According to Theorem 4.6.6, {fm } has a subsequence (denoted again by {fm } for simplicity) which converges in C k for all k on compact subsets of the complement in Σ of a finite subset {p1 , . . . , pl } of “bubble points” to a smooth ω-harmonic map f0 : Σ → M, with |dfm | → ∞ at each bubble point. We assume that Σ has its canonical metric of constant curvature and total area one. For each m, we divide the Riemann surface Σ into a collection of metric disks D1;m , . . . , Dl;m of small radius about the bubble points {p1 , . . . , pl } and a base Σ0;m = Σ − (D1;m ∪ · · · ∪ Dl;m ).

260

4. Harmonic and Minimal Surfaces

 , Each disk Di;m is further decomposed into a union Di;m = Ai;m ∪ Bi;m  where Ai;m is an annular neck region and Bi;m is a bubble region, a smaller disk which is centered at pi . Following Parker [Par96], we arrange that the  is ≤ (constant)/m3 , radius of Di;m is ≤ (constant)/m, the radius of Bi;m and  1 |dfm |2 dA = CR , 2 Ai;m

where CR is a small positive renormalization constant, too small to allow bubbling in the annulus Ai;m . Let Bi;m be a disk with the same center  as Bi;m but with m times the radius and let Ai;m = Di;m − Bi;m . If the renormalization constant is sufficiently small, the supremum of |dfm | on Ai;m must go to zero because of the Sacks-Uhlenbeck ε-Regularity Theorem 4.6.3. We expand the disk Bi;m to a disk of radius m by means of a conformal contraction Ti;m : Dm (0) → Bi;m , Dm (0) being the disk of radius m in C. An argument based upon Theorem 4.6.6 shows that a subsequence of gi;m = fm ◦ Ti;m : Dm (0) → M,

m = 1, 2, . . .

converges uniformly in C k on compact subsets of C, or on compact subsets of C − {pi,1 , . . . , pi,li }, where {pi,1 , . . . , pi,li } is a finite set of new bubble points, to a harmonic map gi : S 2 → M . If there are new bubble points {pi,1 , . . . , pi,li } in C = S 2 − {∞}, the process can be repeated. Around each new bubble point pi,j , we construct a small disk Di,j;m which is further subdivided into an annular region Ai,j;m and a smaller disk Bi,j;m on which bubbling will occur. Once again, we construct conformal contractions Ti,j;m : Dm (0) → Bi,j;m and a subsequence of the maps gi,j;m = fm ◦ Ti,j;m will converge to a harmonic two-sphere gi,j : S 2 → M on the complement of finitely many points. The process can be repeated; for each fm in the sequence, we may have several level-one bubble regions Bi;m , each of which may contain several level-two bubble regions Bi,j;m , each of which may contain several level-three bubble regions, and so forth. For each sufficiently large m ∈ N, Σ is a disjoint union of a base Σ0;m , neck regions Ai1 ,...,ik ;m = Di1 ,...,ik ;m − Bi1 ,...,ik ;m ,

(4.97) and bubble regions (4.98)

Δi1 ,...,ik ;m = Bi1 ,...,ik ;m −



Di1 ,...,ik ,j;m ,

j

from which disks around bubble points of higher level have been deleted. The restriction gi1 ,...,ik ;m of fm to the bubble regions Δi1 ,...,ik ;m corresponding to a given multi-index (i1 , . . . ik ) converge to a harmonic two-sphere gi1 ,...,ik which may have zero energy (in which case it is a ghost bubble), but ghost

4.9. Unstable minimal surfaces

261

bubbles always have bubble points in addition to ∞. Since each bubble with no children is nonconstant and requires energy ≥ 4π, the process must terminate after finitely many steps, yielding harmonic maps corresponding to the vertices of a finite tree T (a base ω-harmonic surface f0 : Σ → M and several minimal two-spheres corresponding to the nonzero vertices). It remains to discuss convergence of suitable reparametrizations of the restrictions fm |Ai;m ,

fm |Ai,j;m ,

...

fm |Ai1 ,...,ik ;m ,

...

of the αm -harmonic maps to the neck regions (4.97). We conformally expand each Di1 ,...,ik ;m in (4.97) to a disk of unit radius, and reparametrize each fm |Ai1 ,...,ik ;m on part of a punctured disk |z| ≤ 1, |z| = 0 in the complex plane C. In terms of the coordinates (u, θ) defined by z = x + iy = reiθ = e−u+iθ , we can regard the punctured disk as an infinitely long cylinder, with the usual Euclidean metric ds2 = dx2 + dy 2 becoming ds2 = dr2 + r2 dθ2 = e−2u (du2 + dθ2 ). which we then restrict to the finite cylinder 0 ≤ u ≤ b, with ∂Bi1 ,...,ik ;m corresponding to r = e−b . Thus the annuli Ai1 ,...,ik ;m are conformally equivalent to cylinders [0, bm ] × S 1 with bm → ∞, and the maps fm |Ai1 ,...,ik ;m are reparametrized as maps gm : [0, bm ] × S 1 −→ M, on these cylinders. These in turn can be reparametrized as maps   1 1 hm : [0, 1] × S −→ M, hm (t, ϕ) = gm (bm t, bm ϕ) , bm on the conformally equivalent cylinders     1 1 , where S 1 [0, 1] × S 1 bm bm

has radius 1/bm ,

which we claim collapse to a parametrized curve in the limit; we claim that a subsequence of the sequence {hm } either converges to a points or a geodesic with parametrization independent of ϕ under our hypotheses. It is useful to compare these maps with maps contructed from a given constant speed geodesic or more generally, a constant speed parametrized curve γ : [0, 1] → M , that can serve as models for the limit. For a given choice of b > 0, we define   1 1 −→ M, by hγ,b (t, ϕ) = γ(t), hγ,b : [0, 1] × S bm

262

4. Harmonic and Minimal Surfaces

a map which degenerates to γ as t → ∞, and define gγ,b : [0, b] × S −→ M, 1

  t . by gγ,b (t, θ) = γ b

For any choice of b > 0, gγ,b is harmonic if and only if γ is a constant speed geodesic. Moreover, for any constant speed curve 1 γ : [0, 1] → M we have |γ  | ≡ L, J(γ) = L2 , 2 where L is the length of γ, and hence 2πJ(γ) L2 =π . b b Thus if we have a bound on the length L, the energy of the geodesic must go to zero as b → ∞. E(hγ,b ) =

(4.99)

We now apply a variation of the argument we gave earlier for the Removeable Singularity Theorem 4.6.9. It follows from the ε-Regularity Theorem 4.6.3 for α-harmonic maps that a sufficiently small bound on the energy of the reparametrizations gm of fm |Ai1 ,...,ik ;m gives a bound of the form (4.86) on the derivatives of gm , ∂gm ∂gm 1 , ∂u ∂θ ≤ C on [1, bm − 1] × S . Once one has this estimate, one has a C 0 bound on energy density, so the bootstrapping argument of Theorem 4.6.6 shows that the restrictions of gm to any cylinder [a, b] × S 1 ,

which is contained in [1, bm − 1] × S 1 for m ≥ M ,

must converge in C k for all k to a harmonic map g∞,a,b : [a, b] × S 1 −→ M in the limit. Now we refer once again to the argument for the Removeable Singularity Theorem 4.6.9. This gives the estimate (4.87):  1 2π ∂g∞,a,b 2  (u, θ)dθ. P (u) ≥ P (u), where P (u) = 2 0 ∂θ This differential inequality implies an exponential decay estimate for P (u). For simplicity, we state this estimate for the interval [−c, c]; one can always reduce to this case by replacing u by u − u0 for an appropriate choice of u0 . Lemma 4.9.3. If P : [−c, c] → R satisfies the estimate P  (u) ≥ P (u), then (4.100)

P (u) ≤

sinh(u) cosh(u) (P (c) + P (−c)) + (P (c) − P (−c)). 2 cosh(c) 2 sinh(c)

4.9. Unstable minimal surfaces

263

Proof of the lemma. The function φ : [−b, b] → R defined by φ(u) = P (u) −

cosh(u) sinh(u) (P (c) + P (−c)) − (P (c) − P (−c)) 2 cosh(c) 2 sinh(c)

satisfies the inequality (4.101)

φ (u) ≥ φ(u),

for u ∈ (−c, c),

and is zero at the two endpoints. The function φ must assume its maximum value at some point u0 ∈ [−c, c]. If φ(u0 ) > 0, then u0 cannot be an endpoint and hence φ (u0 ) ≤ 0, contradicting (4.101). Thus φ is nonpositive on the entire interval [−c, c], which immediately implies (4.100). QED This lemma implies that the function P (u) decays exponentially on the interior of any closed interval contained in each [0, bm ] for m large. Moreover, as in the proof of Theorem 4.6.9, we can show that  1 2π ∂g∞,a,b 2 (u, θ)dθ = P (u) + c, K(u) = 2 0 ∂u which shows that g∞,a,b is close to a constant speed geodesic on any closed interval contained in each [0, bm ] for m large. Finally, we use the fact that π1 (M ) is finite. Recall that by assumption each fm |Ai1 ,...,ik ;m must have finite index, and hence any geodesics obtained in a limit of any sequence of subintervals must also have finite index. Now the Gromov estimates of Corollary 3.3.5 show that the lengths of the geodesics approximations satisfy an a priori bound L(γ) ≤ L0 for some L0 > 0 which depends only on the bound on the Morse index. Since bm → ∞, it follows that the energy in the necks goes to zero as m → ∞ as claimed. In particular, we get convergence in the sense of varifolds to the sum of the images of the harmonic maps at the vertices and, after passing to a subsequence, we see that the reparametrizations hm of the αm -harmonic cylinders fm |Ai1 ,...,ik ;m must converge to geodesics of finite length, proving Theorem 4.9.2. QED Remark 4.9.4. The notion of bubble tree is due to Parker and Wolfson [PW93]. Parker [Par96] used the notion of bubble tree to describe limits of sequences of harmonic or α-harmonic maps, and in the case of harmonic map sequences, he showed that the necks have zero length in the limit, and hence the geodesics in the bubble tree must reduce to points. Examples include limits of sequences of J-holomorphic curves in a symplectic manifold with compatible Riemannian metric, so Parker’s results apply to the Gromov compactification for the space of J-holomorphic curves that is part of the foundation for symplectic topology [MS04]. Our proof of Theorem 4.9.2 has been based to a large extent on Parker’s article.

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Remark 4.9.5. Theorem 4.9.2 is due to Chen and Tian [CT99] for the case of minimizing sequences; they used the fact that the lengths of minimizing geodesics are bounded to show that there was no energy loss in the necks for minima. Remark 4.9.6. Suppose that M is the connected sum, M = CP 2 (S 1 × S 3 )CP 2 , for which the fundamental group is Z, with a metric which makes the necks connecting the summands very small. Then infinitely many distinct components of Map(S 2 , M ) can be constructed by taking least energy representatives for Map(S 2 , CP 2 ) based at points a and b in the extreme summands and connecting them by a path in some component of Ω(M, a, b). If one takes a minimax construction for each of these components one obtains the same minimax energy, say E0 , so Map(S 2 , M )E0 +ε = {f ∈ Map(S 2 , M ) : E(f ) ≤ E0 + ε} has infinitely many components each of which could be represented by a distinct bubble tree, which is the limit of a sequence of minimizing α-energy critical points with α → 1. One can take sequences of α-energy critical points in Map(S 2 , M )E0 +ε for which the lengths of the corresponding curves in Ω(M, a, b) grow so rapidly that energy is actually lost in the limit, so some assumption on finiteness appears to be necessary for Theorem 4.9.2 to be valid as stated. (See §4 of [Par96] for similar constructions of Palais-Smale sequences which fail to converge.) This example illustrates one source of noncompactness for bubble trees that can be ruled out by assuming that π1 (M ) is finite. On the other hand, with no assumption on π1 (M ), it follows from Theorem 4.2.1 of Jost [Jos91] that sequences constructed from minimax constraints do have subsequences which have no energy loss in the necks; the proof is based upon the technique of substitution which is presented there. Theorem 4.9.2 has the following immediate consequence: Theorem 4.9.7. Suppose that Σ is a Riemann surface of genus g ≥ 0, that M is a compact manifold with a finite fundamental group, and that {(fm , ωm ) : m ∈ N} is a sequence of critical points for Eαm : L2α 1 (Σ, M ) × Tg −→ R with αm → 1 with energy and Morse index subject to a universal bound. Assume, moreover, that the elements ωm ∈ Tg are bounded. Then a subsequence of {(fm , ωm ) : m ∈ N} will converges to a bubble tree, with the base converging to a conformal harmonic map, and the necks converging

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265

to geodesics of finite length and of energy zero. Moreover, the sequence Eαm (fm , ωm ) converges to the energy of the bubble tree. There are cases in which the convergence of ωm ∈ Tg can be assured, at least after passing to a subsequence. For example, in the case where Σ = T 2 , the torus, Lemma 4.8.1 implies that Map(2) (T 2 , M ) × T1 −→ R Γ1 satisfies Condition C, and this allows us to determine a sequence of minimax solutions with bounded ωm for arbitrary constraint from the homology of (2) Mα (T 2 , M ). 2 Eα : M(2) α (T , M ) =

4.9.3. Sources of noncompactness for the two-variable energy. In §4.5.3, we established Morse inequalities, and a Morse-Witten complex, for generic perturbations of the α-energy Eα,ω : Map(Σ, M ) −→ M when α > 1, and of course there is a direct limit complex as α → 1 which also calculates the homology of Map(Σ, M ). But for the theory of minimal surfaces, we need to allow the conformal structure ω ∈ T to vary, and replace the energy function Eα,ω by Eα : Map(Σ, M ) × T −→ R,

where

Eα,ω (f ) = Eα (f, ω).

We then try to construct critical points for the two-variable energy E : Map(Σ, M ) × T −→ R. by taking the limit as α → 1 while keeping ω ∈ T bounded.3 Which elements of the cohomology of Map(Σ, M ) does this limiting process represent by minimax critical points for the two-variable energy E? The process we have outlined encounters three potential problems, which we call sources of noncompactness: (1) As α → 1, we must allow for bubbling, with the limit being a collection of minimal two-spheres or a base minimal surface of genus g ≥ 1 together with a collection of minimal two-spheres. (2) For a minimizing sequence, the conformal structure on Σ may approach the boundary of moduli space, implying a possible degeneration to a surface with lower genus, or a division of the base minimal surface into two or more components. 3 Actually, the two-variable energy is invariant under the group of symmetries G  Γ, where G is the group of complex automorphisms of Σ which are homotopic to the identity, and Γ is the mapping class group, so we expect G  Γ-orbits of minimal surfaces in the limit. For the best results we should use G  Γ-equivariant homology or cohomology in the construction of our constraints.

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(3) The sequence may approach a nontrivial branched covers of a prime minimal surface of lower energy. Branched covers count as critical points within the space of functions, although they are not geometrically distinct from the covered surface.

The first two problems have already been encountered several times in this chapter when constructing area-minimizing representatives of homotopy or homology classes, and we have seen how to deal with them in several contexts. Resolving the third problem, multiple covers of a given prime minimal surface, is crucial to understanding unstable minimal surfaces. It arose also in the theory of closed geodesics, but is more complicated for minimal surfaces because they may be branched and we will see in §5.8 that branched covers of a prime minimal surface do not always lie on nondegenerate critical submanifolds, even when the metric on the ambient manifold M is generic. Handling these sources of noncompactness seems particularly daunting in the case in which the domain has genus zero (Σ = S 2 ). In this case, the symmetry group P SL(2, C) is noncompact, there is no real distinction between base and bubbles in the bubble tree, and in some cases the bubbles in the bubble tree might all be parametrizations of the same underlying geometric minimal two-sphere. Moreover, although there is only one imbedded minimal two-sphere within S 2 , namely id : S 2 → S 2 , the topology of Map(S 2 , S 2 ) is already rather complicated (see [Seg79]), suggesting that partial Morse inequalities are difficult to formulate in this case. A few results via perturbation are possible for minimal two-spheres, as long as we prevent bubbling altogether, so that the theory of bubble trees is unnecessary. For example, if the sectional curvatures of the Riemannian metric on the ambient manifold M satisfy the inequality Kr (σ) ≤ 1 and we have a lower bound E0 on the energy of nonconstant minimal two-spheres in Map(S 2 , M ), it follows Lemma 4.9.1 that no bubbling is possible below energy E0 + 4π. Thus under the restriction Kr (σ) ≤ 1, we can analyze minimal two-spheres of energy < 8π. Using this idea we were able to establish equivariant Morse inequalities for minimal two-spheres of low Morse index in S n with a suitably pinched Riemannian metric (see [Moo90]), chosen to be generic in accordance with the Bumpy Metric Theorem of the next chapter. The remaining case in which Σ is a surface of genus at least one seems to be more accessible. In this case, we can distinguish between the base minimal surface and the bubbles, and we have found evidence that certain cohomology classes prevent degeneration and bubbling when the dimension of the ambient manifold is sufficiently large.

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267

4.10. An application to curvature and topology The Morse theory of the α-energy can sometimes establish relationships between curvature and topology of Riemannian manifolds. Just as the theory of geodesics uncovers relations between curvature and topology of Riemannian manifolds (through classical theorems such as Synge’s theorem, Myers’ theorem and the theorem of Hadamard and Cartan), we might hope to apply the stability theory of minimal surfaces to find relationships between curvature and topology of Riemannian manifolds. The problem we consider is that of finding the weakest curvature condition which implies that a compact simply connected n-manifold is homeomorphic to a sphere. When n = 3, the solution of the Poincar´e conjecture [MF10], [MT14] implies that no curvature condition is needed, so we might as well assume that n ≥ 4. When n ≥ 4, the normal bundle to an immersion f : S 2 → M is almost never parallelizable, so parallel sections are usually unavailable, and the rough idea behind our argument is to use holomorphic sections of the normal bundle to estimate the Morse index of a minimal or harmonic two-sphere, and then use Morse theory to show that many of the homotopy groups of M must vanish.

4.10.1. Complex form of second variation. By Corollary 4.5.2, the second variation of energy at a harmonic map f : Σ → M is given by the formula  d2 Eω (f )(V, W ) = Lf (V ), W dA, Σ

where Lf is the Jacobi operator , defined by (4.102)       D D ∂f ∂f ∂f ∂f 1 D D Lf (V ) = − 2 ◦ + ◦ + R V, + R V, . λ ∂x ∂x ∂y ∂y ∂x ∂x ∂y ∂y We say that an element V ∈ Tf M is a Jacobi field along f if Lf (V ) = 0. Theorem 4.10.1. The Jacobi operator can be written in the following complex form:     ∂f ∂f 4 D DV ◦ + R V, , Lf (V ) = − 2 λ ∂z ∂ z¯ ∂z ∂ z¯ where the Riemann-Christoffel curvature tensor R has been extended to be complex linear.

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Proof. Straightforward calculations show that, on the one hand,   D D D D D D √ D D D D 4 ◦ = ◦ + ◦ + −1 ◦ − ◦ ∂z ∂ z¯ ∂x ∂x ∂y ∂y ∂x ∂y ∂y ∂ x ¯   D D √ ∂f ∂f D D ◦ + ◦ + −1R , , = ∂x ∂x ∂y ∂y ∂x ∂y while on the other,       ∂f ∂f ∂f ∂f ∂f ∂f 4R ·, = R ·, + R ·, ∂z ∂ z¯ ∂x ∂x ∂y ∂y       √ ∂f ∂f ∂f ∂f − R ·, + −1 R ·, ∂x ∂y ∂y ∂x       ∂f ∂f ∂f ∂f ∂f ∂f √ = R ·, + R ·, − −1R , , ∂x ∂x ∂y ∂y ∂x ∂y the last step following from the Bianchi symmetry. Substitution into (4.102) now yields the theorem. QED Corollary 4.10.2. The second variation of ω-energy is given by (4.103)  /  . ¯ / .  ∂f ∂f DV D W 2 ¯ ¯ , − R V ∧ ,W ∧ dxdy, d Eω (f )(V, W ) = 4 ∂ z¯ ∂z ∂z ∂ z¯ Σ where V and W are sections of the complex vector bundle E = f ∗ T M ⊗ C and R(x ∧ y), v ∧ w = R(x, y)w, v. Proof. Simply substitute into (4.102) and integrate by parts. QED When V = W , we can also write (4.103) as  $ %  2 D V − K(σ) |V |2 dA, (4.104) d2 Eω (f )(V, V¯ ) = 2 Σ

where K(σ) is the sectional curvature of the complex two-plane spanned by ∂f /∂z and V , and D  : Γ(E) −→ Γ(E ⊗ K) ¯ is the ∂-operator defined by the Levi-Civita connection of M , in accordance with the Koszul-Malgrange Theorem 4.1.3. The first term in this formula vanishes when V is holomorphic, that is, when D  V = 0. The idea of using holomorphic sections to simplify second variation mirrors a standard technique for proving theorems on curvature and topology via geodesics: one constructs a parallel orthonormal frame along the geodesic and applies second variation. For minimal surfaces, it is usually not possible to construct parallel sections, but the Riemann-Roch Theorem often provides a large supply of holomorphic sections. Recall that if f : Σ → M

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269

is conformal and harmonic, the local sections ∂f /∂z : U → E = f ∗ T M ⊗ C generate a holomorphic isotropic line bundle L within E. We let L⊥ = {V ∈ E : V, L = 0}, the orthogonal complement of L, and since L is isotropic, L ⊆ L⊥ . Then L⊥ is a holomorphic subbundle of E and we can regard the quotient space N = L⊥ /L

(4.105)

as a “holomorphic normal bundle” to f . Corollary 4.10.3. When V is a section of L, the second variation formula simplifies to  . ¯ / DV D W 2 ¯ , dxdy, (4.106) d Eω (f )(V, W ) = 4 ∂ z¯ ∂z Σ and hence V is a Jacobi field if and only if it is holomorphic. The proof is achieved by noting that in this case, R(V ∧ (∂f /∂z)) = 0. For minimal two-spheres, the dimension of the space of holomorphic sections of L is ν + 3, where ν is the branching order. 4.10.2. Isotropic curvature. Just as studying the stability of geodesics leads to sectional curvature, studying stability for minimal surfaces leads to the notion of isotropic curvature. Recall that in terms of normal coordinates (u1 , . . . , un ) centered at p on a Riemannian manifold M , the Riemannian metric can be expressed by a Taylor series 1 Rikjl (p)uk ul + higher order terms, gij = δij − 3 k,l

where the Rikjl ’s are components of the Riemann-Christoffel curvature tensor . Moreover, the curvature operator is defined in terms of these components to be the linear map R : Λ2 Tp M −→ Λ2 Tp M such that

 R

  ∂ ∂ ∂ ∂ ∧ Rijkl (p) ∧ . = ∂ui p ∂uj p ∂uk p ∂ul p k,l

If z and w are linearly independent elements of Tp M ⊗ C, we say that the sectional curvature of the complex two-plane σ spanned by z and w is R(z ∧ w), z¯ ∧ w ¯ , z ∧ w, z¯ ∧ w ¯

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where the bar denotes complex conjugation. The complex two-plane σ is said to be isotropic if z, z = w, w = z, w = 0. Definition. The Riemannian manifold M is said to have positive isotropic curvature if K(σ) > 0, whenever σ is an isotropic complex two-plane. Note that for isotropic two-planes to exist the dimension of the ambient manifold M must be at least four. Isotropic curvature is related to stability properties of a minimal two-sphere f : S 2 → M because the first term in the integrand of the second variation formula  / .    DV 2 ∂f ∂f 2 ¯ dxdy, d Eω (f )(V, V¯ ) = 4 ∂ z¯ − R V ∧ ∂z , V ∧ ∂ z¯ S2 vanishes when V is a holomorphic section, and as we will see, there are always holomorphic sections which together with ∂f /∂z spans an isotropic two-plane when M has dimension at least four. Our analysis of the holomorphic sections rests on the following theorem, which is special to two-spheres, as opposed to other Riemann surfaces: Grothendieck Theorem 4.10.4. Any holomorphic line bundle over the Riemann sphere S 2 = CP 1 divides into a holomorphic direct sum of holomorphic line bundles. This theorem, proven in [Gk57], allows us to write the normal bundle N of (4.105) as a direct sum of line bundles, N = L1 ⊕ L2 ⊕ · · · ⊕ Ln−2 ,

c1 (L1 )[S 2 ] ≥ · · · ≥ c1 (Ln−2 )[S 2 ],

where

where c1 (Li )[S 2 ] is the first Chern class of Li evaluated on the fundamental class of S 2 . Since the Riemannian metric is invariant under the Levi-Civita connection, it extends to a holomorphic complex bilinear form ·, · : N × N −→ C, and in particular, N is isomorphic to its dual. Thus the line bundles in the above sequence can be arranged so that c1 (Li ) = −c1 (Ln−i−1 ) for each i. If V is a holomorphic section of one of the line bundles Li in the direct sum decomposition of E, then / . ∂f 2 : S2 → C and V, V, V  : S → C ∂z are holomorphic, and hence V, V  = (constant),

.

∂f V, ∂z

/ = (constant).

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271

In particular, if V is a holomorphic section of Li where Li has positive first Chern class, or more generally if V is any holomorphic section such that V, V  = 0 at least one point, then / . / . ∂f ∂f ∂f = , = 0, V, V  = V, ∂z ∂z ∂z and ∂f span an isotropic two-plane ∂z at every point where they are linearly independent, which happens at an open dense set of points. If M has positive isotropic curvatures, it then follows from the index formula (4.103) that d2 Eω (f )(V, V¯ ) < 0. V

and

Let m be the number of line bundle summands Li of E with positive first Chern number, c1 (L1 )[S 2 ] ≥ · · · ≥ c1 (Lm )[S 2 ] ≥ 1,

c1 (Lm+1 )[S 2 ] ≤ 0

and let E0 be the direct sum of all of the line bundle summands with zero Chern class; the dimension of the space O(E0 ) of holomorphic sections of E0 is n−2m−2. If V1 , . . . , Vm are nonzero holomorophic sections of L1 , . . . , Lm , respectively, then Vi , E0  = 0 for 1 ≤ i ≤ m. If {W1 , . . . , Wl } is a basis for O(E0 ), then Wi , Wj  is constant, and therefore there is a maximal isotropic holomorphic subbundle I0 ⊂ E0 ) which has rank ≥ (1/2)(rank of E0 )-1). If V is the space of holomorphic sections of L1 ⊕ · · · ⊕ Lm ⊕ I0 , then dim V ≥ (1/2)(dim M − 3) and ∂f linearly independent ⇒ d2 Eω (f )(V, V¯ ) < 0. ∂z This yields a lemma which will be needed for the proof in the next theorem: V ∈V

and

Lemma 4.10.5. Suppose that M is a Riemannian manifold which has positive isotropic curvature. Then the Morse index of any harmonic twosphere f : S 2 → M is at least (1/2)(dim M ) − (3/2). 4.10.3. Manifolds of positive isotropic curvature. The following theorem [MM88] generalizes a well-known “sphere theorem” due to Berger, Klingenberg and Toponogov: Sphere Theorem 4.10.6. Suppose that M is a compact smooth simply connected Riemannian manifold of dimension at least four which has positive isotropic curvature. Then M is homeomorphic to a sphere. The idea behind the proof is to show that M is a homotopy sphere and apply the solutions to the generalized Poincar´e conjecture in dimensions greater than four to conclude that M is homeomorphic to a sphere.

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By definition, a compact connected manifold M of dimension n is a homotopy sphere if πq (M ) = 0, for all integers q such that 0 < q < n. Thus if M is not a homotopy sphere there must be an integer q with 0 < q < n such that πq (M ) = 0, and we choose the smallest such integer. By the Hurewicz isomorphism theorem, q is also the smallest positive integer such that Hq (M ; Z) = 0. By Poincar´e duality, we can assume that q ≤ n/2. Just as in §5.4, we let M be the space of smooth maps from S 2 to M and let π : M → M be the evaluations map defined by π(f ) = f (0) where 0 ∈ S 2 = C ∪ {∞} is the basepoint. Since the fibration π possesses a section, the long exact homotopy sequence of π splits, and if Mp denotes the subset of M consisting of maps such that f (0) = p, we conclude that ∼ πk (M ) ⊕ πk (Mp ) = ∼ πk (M ) ⊕ πk+2 (M ), πk (M) = which implies that

πq−2 (M) ∼ = πq (M ) = 0.

We now apply Morse theory to a perturbation Eα,ψ of the energy E, where ψ is chosen so that all nonconstant critical points of Eα,ψ are Morse nondegenerate, and let α → 1 and ψ → 0. We thereby obtain a sequence {fm } of critical points for Eαm ,ψm , each having Morse index no more than q− 2. By the bubbling argument presented in §4.6.3, we find that a subsequence (still denoted by {fm }) converges uniformly in every C k on every compact subset of S 2 − {p1 , . . . , pl }, where p1 , . . . , pl are a finite number of bubble points, to a harmonic map on Σ − {p1 , . . . , pl }, which can be extended to a smooth harmonic map f∞ : Σ → M by the Sacks-Uhlenbeck removeable singularity theorem. Suppose first that f∞ is nonconstant. In that case, we claim that the Morse index of f∞ is no larger than q − 2. For this, we need the following lemma: Lemma 4.10.7. Suppose that fm : S 2 → M is a sequence of αm -harmonic maps in M which converge in C k on compact subsets of S 2 − {x1 , . . . , xl } to a smooth harmonic map f∞ : S 2 → M . Then Morse index of f∞ ≤ lim inf Morse index of fm . Proof. To prove the lemma, we first recall the second variation formula (4.57) for α-harmonic two-spheres:  2 (1 + |df |2 )α−1 [∇V, ∇W  − K(V ), W ]dA d Eα (f )(V, W ) = α S2  (1 + |df |2 )α−2 df, ∇V df, ∇W dA. + 2α(α − 1) S2

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273

Note that the second term is dominated by the first and goes to zero when α is close to one, and the first term approaches d2 E(f )(V, W ) when the support of V and W does not contain any bubble points. We claim that if d2 E(f∞ ) is negative definite on a fixed linear space V of dimension d, so is d2 Eα (fm ) when m is sufficiently large. To see this, we make use of a radial cutoff function near the bubble points which was first introduced by Choi and Schoen [CS75] in a different context. First, define a piecewise smooth function η : [0, ∞) → [0, ∞) by ⎧ ⎪ if r ≤ ε2 , ⎨0, (4.107) η(r) = 2 − (log r)/(log ε), if 2 ≤ r ≤ ε, ⎪ ⎩ 1, if ε ≤ r, so that

and

⎧ ⎪ if r ≤ ε2 , ⎨0, dη (r) = (−1)/(r log ε), if ε2 ≤ r ≤ ε, ⎪ dr ⎩ 0, if ε ≤ r,

2  ε 2π 2π dη (r) rdrdθ = . dr = − 2 dr log ε 0 0 ε2 r(log ε) Thus, if we define ψi : S 2 → R so that it is one outside an ε-neighborhood of the bubble point xi and in terms of polar coordinates (ri , θi ) about the bubble point xi satisfies the condition ψi = η ◦ ri , then  −C , where C is a positive constant, |dψi |2 dA ≤ log ε S2 

2π



ε

and a similar estimate holds for ψ = ψ1 · · · ψl , a cutoff function which vanishes at every bubble point. If ε > 0 is chosen sufficiently small, then V ∈V

⇒

d2 E(f )(ψV, ψV ) < 0

⇒

d2 Eα (f )(ψV, ψV ) < 0.

This shows that d2 Eα (f ) is negative definite on a space of dimension d and proves the lemma. QED It follows from Lemma 4.10.7 that if f∞ : S 2 → M is nonconstant, it must have index ≤ q − 2 ≤ (1/2)(dim M ) − 2 which contradicts Lemma 4.10.5. But if f∞ is constant, a nonconstant sphere must bubble off; that is, there must be a family of conformal reparametrizations gm : S 2 → S 2 , such that fm ◦ gm converges to a nonconstant harmonic sphere fˆ∞ in C k on compact subsets of S 2 − {p1 , . . . , pl }, where p1 , . . . , pl are a finite number of bubble points. We can choose pm ∈ S 2 such that dfm (pm ) = sup{dfp  : p ∈ S 2 },

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and after rotations we can arrange that all the pm ’s are equal and in fact that z(pm ) = 0, where z is the standard coordinate on S 2 − {∞} = C. Let rm = dfm (pm ), and let gm : S 2 → S 2 be the conformal map expressed in terms of the standard coordinate as hm (z) = rm z. Finally, let hm = fm ◦ gm , a sequence of maps which converge to fˆ∞ on compact subsets of S 2 − {p1 , . . . , pl }. We must now replace d2 Eα (fm ) by d2 Eα (fm ◦gm ) in the above argument. Once again, one obtains a nonconstant two-sphere fˆ∞ in the limit. Moreover, a calculation similar to that for f∞ leads to the conclusion that fˆ∞ has index ≤ q − 2 ≤ (1/2)(dim M ) − 2, which once again contradicts Lemma 4.10.5. Thus M must be a homotopy sphere, and it follows from positive resolutions of the generalized Poincar´e conjecture ([Mil65] and [FQ90]) in dimensions ≥ 4 that M is homeomorphic to a sphere, proving Theorem 4.10.6. Remark 4.10.8. It can be shown that if the real sectional curvatures K(σ) of a Riemannian manifold satisfy the inequalities

(4.108)

1 < K(σ) ≤ 1, 4

then the manifold has positive isotropic curvatures, but not conversely. The sphere theorem proven by Berger, Klingenberg and Toponogov made the pinching hypothesis (4.108) on real sectional curvatures and is therefore weaker than the sphere theorem we have proven. The complex projective space with the standard Fubini-Study metric is simply connected, has real sectional curvatures lying in the range [1/4, 1] and has nonnegative isotropic curvature, but is not homeomorphic to a sphere. There are other techniques one can apply to the problem of finding relationships between curvature and topology, including the Ricci flow. Using Ricci flow, Brendle and Schoen [BS09a] were able to prove that a compact simply connected smooth manifold which satisfies (4.108) must actually be diffeomorphic to a sphere. Remark 4.10.9. Conformally flat four-manifolds of positive scalar curvature automatically have positive isotropic curvature, and it is easy to construct a conformally flat metric of positive scalar curvature on the connected sum of a finite number of S 3 × S 1 ’s. Alternatively, since the connected sum of manifolds of isotropic curvature also admits a metric of positive isotropic curvature by Micallef and Wang [MWa92], it follows from the fact that S n−1 × S 1 has a metric of positive isotropic curvature when n ≥ 4 that the

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275

connected sum of a finite number of S n−1 × S 1 ’s also admits such a metric, thereby showing that the fundamental group of a compact manifold of positive isotropic curvature can be a free group of arbitrary rank. On the other hand, Fraser [Fra03] used a second variation argument to show that the fundamental group of a compact manifold of dimension at least five with positive isotropic curvatures cannot contain a free abelian group of rank two, and the proof was extended to the case of dimension four in [BS09b].

Chapter 5

Generic Metrics

5.1. Bumpy metrics for minimal surfaces Two-dimensional minimal surfaces in a general Riemannian manifold can exhibit bewildering complexity. For example, Calabi [Cal67] discovered that minimal two-spheres within an n-dimensional sphere S n (n ≥ 4) of constant curvature one must have area 2πd where d is an integer, but the moduli space of such minimal two-spheres grows in complexity with n and d. Another class of examples is provided by holomorphic curves in K¨ahler manifolds. Intersections of algebraic K3 surfaces with complex hyperplanes give moduli spaces of holomorphic area-minimizing surfaces of high dimension which disappear suddenly when the K¨ahler structure is varied to be nonalgebraic. However, we might hope that the full space of minimal surfaces in a smooth manifold M is considerably simpler when we make a suitable choice of generic Riemannian metric on M . Just as a generic choice of proper function f : M → R on a smooth manifold M is necessary for Morse theory, so a generic choice of Riemannian metric on a compact smooth manifold M should simplify the relationships between the topology of M and the types of minimal surfaces within M , making analysis of the critical locus for the energy more tractable. Definition. By a generic Riemannian metric on a smooth manifold M we mean a Riemannian metric that belongs to a countable intersection of open dense subsets of the spaces Met2k (M ) of L2k Riemannian metrics on M with the L2k topology, for some choice of k ∈ N, k ≥ 2.

277

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In other words, we put a countable number of conditions on the first k derivatives of the metric, each satisfied by on open dense set of metrics. The Sard-Smale Theorem 2.5.2 implies that “most” Riemannian metrics will be generic. The question we consider in this chapter is: What properties can we assume for the nonconstant parametrized minimal surfaces in Riemannian manifolds with suitable generic metrics? We claim that for generic metrics, we can arrange that all parametrized minimal surfaces are branched covers of prime minimal surfaces which lie on nondegenerate critical submanifolds, each of which is an orbit for the identity component of the group of conformal automorphisms of the compact conformal surface which serves as domain. Definition. A parametrized minimal surface f : Σ → M is a branched cover of a (nonconstant) parametrized minimal surface f0 : Σ0 → M , if f = f0 ◦ g, where g : Σ → Σ0 is a holomorphic branched cover of oriented domains when Σ0 is orientable, or the composition of such a branched cover with an unbranched double cover when Σ0 is nonorientable. A nonconstant parametrized minimal surface f : Σ → M is prime if it is not a branched cover of a parametrized minimal surface of strictly lower energy. Of course, oriented minimal surfaces form the core of the theory, but we also want to be able to consider minimal projective planes and other nonorientable surfaces, so we allow the domain Σ0 of a prime parametrized minimal surface to be nonorientable. We can study such nonorientable prime parametrized minimal surfaces by means of their oriented double covers. To study transversality of oriented parametrized minimal surfaces, we must account for the fact that if (f, ω) is a critical point for the two-variable energy E : Map(Σ, M ) × Tg −→ R, the entire G-orbit of (f, ω) consists of critical points, where G is the identity component of the group of symmetries of E for the domain Σ. When Σ is oriented, this is the group of holomorphic automorphisms of Σ, and there are three cases: (1) If Σ is the two-sphere, G = P SL(2, C) acts on Map(Σ, M ) via composition on the right. (2) Similarly, if Σ is the torus, G = S 1 × S 1 acts on Map(Σ, M ). (3) If Σ has genus at least two, G is trivial. When Σ is nonorientable, we follow the discussion in §4.3.3 and take the identity component G of the group of symmetries to be (1) SO(3) when Σ is the projective plane,

5.1. Bumpy metrics for minimal surfaces

279

(2) S 1 when Σ is the Klein bottle, and (3) the trivial group, when the oriented double cover of Σ has genus ≥ 2. Even for generic metrics, the critical set of minimal spheres and minimal tori is never Morse nondegenerate in the usual sense; if (f, ω) is a critical point for the two-variable energy E, the entire G-orbit of (f, ω) consists of critical points. The best we can hope for is that they lie on nondegenerate critical submanifolds as defined at the end of §2.6, each such submanifold having the same dimension as G. Our main goal in this chapter will be to prove the following: Bumpy Metric Theorem 5.1.1. If M is a compact smooth manifold of dimension at least three, then for a generic choice of Riemannian metric on M , all prime compact parametrized minimal surfaces f : Σ → M are free of branch points and lie on nondegenerate critical submanifolds, each such submanifold being an orbit of the group G of conformal automorphisms of Σ which are homotopic to the identity. This is analogous to the Bumpy Metric Theorem 2.7.2 of Abraham [Abr70] for closed geodesics. Our proof follows [Moo06] with a few improvements, except that we do not make the unnecessary assumption that M has dimension at least four. (Regrettably, the author published an unnecessary erratum to [Moo06] which wrongly claimed that a more awkward organization of the argument was necessary.) A theorem of Brian White [Whi91] gives a similar bumpy metric theorem for imbedded minimal surfaces, while B¨ohme and Tromba [BT81] argue that minimal disks in Euclidean space Rn with boundary a Jordan curve are Morse nondegenerate for generic choice of boundary when n ≥ 4. Discussion of boundary value problems lies outside the scope of this book, so we refer to [DHT10] for background for the B¨ohme-Tromba theorem, as well as a discussion of its extension by Tomi and Tromba [TT95] to the Plateau problem for minimal surfaces of more general topological type. A key point of Theorem 5.1.1 is that for generic choice of metric, branch points do not exist on prime closed parametrized minimal surfaces. This is not true for minimal surfaces which are not prime; for example, nontrivial branched covers of immersed minimal two-spheres in generic metrics do indeed have branch points, and the branch points of such covers will persist under deformations of the metric. The Bumpy Metric Theorem allows us to develop a transversality theory for the space of solutions to the minimal surface equation when the

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Riemannian metric on the ambient space is generic. We consider the subset Σ(s) = {(p1 , . . . , ps ) ∈ Σs : pi = pj when i = j} of the s-fold cartesian product Σs , for s a positive integer, as well as the multidiagonal in the s-fold cartesian product M s , Δs = {(q1 , . . . , qs ) ∈ M s : q1 = q2 = · · · = qs }. In accordance with [GG73] (see Chapter III, §3), we then say that an immersion f : Σ → M has transversal crossings if for every s > 1, the restriction of f s = f × · · · × f : Σs −→ M s to Σ(s) is transversal to Δs . Thus, if Σ is a compact surface and M has dimension at least five, an immersion with transversal crossings is a oneto-one immersion and hence an imbedding, while if M has dimension four, such an immersion has only double points and the intersections at double points are transverse. If M has dimension three, transverse triple points are possible. Transversal Crossing Theorem 5.1.2. Suppose that M is a compact smooth manifold of dimension at least three. Then for a generic choice of Riemannian metric on M , (1) every prime compact parametrized minimal surface f : Σ → M is an immersion with transversal crossings, and (2) any two distinct prime compact parametrized minimal surfaces have transverse intersections. We emphasize that the minimal surfaces considered in Theorems 5.1.1 and 5.1.2 are not required to be area-minimizing or even stable. The message of Theorem 5.1.2 is that the self-intersection set of minimal surfaces in the generic case becomes more complicated as the dimension of the ambient space decreases, a phenomenon quite familiar to lowdimensional topologists. Recall that it is the failure of the “Whitney trick” to eliminate self-intersection of opposite parity on surfaces in four-manifolds which explains the failure of the h-cobordism theorem to hold for manifolds of dimension four (as explained in Chapter 1 of [DK90]). Of course, we would like to understand the full critical locus for the energy function, including branched covers of prime minimal surfaces. As we will explain in §5.8, the Bumpy Metric Theorem implies that branched covers of spheres lie on integral submanifolds, but these submanifolds are not usually nondegenerate. On the other hand, part of Bott’s argument on iterated closed geodesics can be extended to minimal tori, showing that

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unbranched covers of prime minimal tori by tori lie on nondegenerate critical submanifolds, each an orbit for the action of G = S 1 × S 1 (see [Moo07]). We can give an explicit example of an isolated G-orbit of minimal twospheres which appears as the limit of a family of minimal tori in a wellknown K¨ahler surface, namely an elliptic fibration over S 2 with fibers of varying topological type, the generic member of which is a torus. (See the discussion of elliptic K¨ahler surfaces in Chapter 4 of Griffiths and Harris [GH78], for example.) There are typically several singular fibers; these may include fibers which have a single transverse self-intersection (called a “fishtail” fiber) or a simple branch point singularity (called a cusp). A fiber with a single transverse self-intersection or branch point singularity is a holomorphic curve with virtual genus one (see [GH78], page 471) which is realized as a parametrized surface via its normalization, but the resulting parametrized curve has domain of genus zero. In the sense of geometric measure theory, the nonsingular fibers provide a sequence of integral currents which converge to the singular fiber, while in the mapping context, the nonsingular fibers are represented by maps with different domain than the singular fiber, indeed in a different mapping space. Although they are close as integral currents, we cannot provide the singular fishtail fiber and the nonsingular fibers with a well-behaved common description in the mapping setting because the normalizations have different genus as parametrized minimal surfaces. For an explicit example, we consider the rational map from P 2 C to P 1 C defined by two generic cubics as in §3.1 of [GS99]. To get a genuine smooth map we blow up P 2 C at the nine points of intersection obtaining a genuine smooth map on a new domain, 9

!" # 2 2 2 ∼ π : E(1) = P C P C · · · P C −→ P 1 C, which is one of the basic examples of elliptic fibrations. In the generic case there are twelve fishtail fibers, which are topologically immersed twospheres with transverse self-intersections. Cusps can occur for such elliptic fibrations but not generically, that is, cusps have higher codimension than fishtails even within the realm of smooth algebraic surfaces, consistent with Theorem 5.1.1.

5.2. Local behavior of minimal surfaces We start the proof of the Bumpy Metric Theorem by adapting the definition of injective point from the theory of J-holomorphic curves (see §2.5 of McDuff and Salamon [MS04]), and showing that injective points are open and dense for prime parametrized minimal surfaces.

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5.2.1. Injective points. Recall that if Σ is a compact connected oriented surface of genus g, a critical point (f, ω) ∈ L2k (Σ, M ) × Tg

for the function E : L2k (Σ, M ) × Tg −→ R

is called an oriented parametrized minimal surface in M . If Σ0 is a compact connected nonorientable surface, then Σ0 has an orientable double cover π : Σ → Σ0 with a sheet interchange map A : Σ → Σ such that π ◦ A = π. A critical point (f˜, ω) for the function E : L2k (Σ, M ) × Tg −→ R such that f˜ ◦ A = f˜ induces a map f : Σ0 → M which is called a nonorientable parametrized minimal surface in M , and is studied via its orientable double cover f˜. Definition. If f : Σ → M is a parametrized minimal surface, either oriented or nonorientable, we say that p ∈ Σ is an injective point for f if df (p) = 0

and

f −1 (f (p)) = p.

If f : Σ → M is connected and has injective points, we say it is somewhere injective. Lemma 5.2.1. If a conformal harmonic map f : Σ → M is prime, its injective points form an open dense subset of Σ. Suppose that U1 and U2 are open neighborhoods of p1 and p2 within Σ with f (p1 ) = f (p2 ) = q and suppose that each tangent plane f∗ Tpi Σ has nonsingular projection on the two-plane Π ⊆ Tq M . Choose normal coordinates (u1 , . . . , un ) on an open neighborhood of q in M with ui (q) = 0, so that the plane spanned by the coordinate vectors (∂/∂u1 ) and (∂/∂u2 ) at q is Π. Let Bε be a ball of small radius ε about 0 in the (u1 , u2 )-plane. Near the intersection point, we can consider the two surfaces f |U1 and f |U2 as graphs of functions hi : Bε → Rn−2 ,

i = 1, 2,

which satisfy an explicit nonlinear partial differential equation. Our convention is to denote the Riemannian metric on M and the metric on Bε induced by the immersion (u1 , u2 ) → (u1 , u2 , hi (u1 , u2 )) by n  i,j=1

gij dui duj

and

2 

ηab dua dub ,

a,b=1

respectively. With this convention, it follows from a relatively straightforward calculation (presented in the Appendix to [MWh95]) that each hi is

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283

an Rn−2 -valued solution to an elliptic system in “nonparametric form,” 2 

η ab (u1 , u2 , h, Dh)

a,b=1

∂ 2h + φ(u1 , u2 , h, Dh) = 0, ∂ua ∂ub

where (η ab ) is the matrix inverse of (ηab ), the operator on the left having scalar symbol. It is readily checked that the difference h = h2 − h1 satisfies an equation of the same form, 2 

η ab (u1 , u2 , h, Dh)

a,b=1

∂ 2h + φ(u1 , u2 , h, Dh) = 0, ∂ua ∂ub

in which the coefficients depend upon h1 , which is now regarded as a known function of u1 and u2 , with h or equivalently h2 being regarded as unknown. Thus by Corollary 1 of Theorem 1.1 in [MWh95] (a corollary of an earlier theorem of Hartman and Wintner), we see that either h ≡ 0 or (5.1) |h(u1 , u2 )| = O(|(u1 , u2 )|m ) ⇒

for some positive integer m, and h(u1 , u2 ) = p(u1 , u2 ) + o1 (|(u1 , u2 )|m ),

where p(u1 , u2 ) is a nonzero homogeneous vector-valued harmonic polynomial of degree m, where h − p is o1 (|z|m ) means that h − p is o(|z|m ) and D(h − p) is o(|z|m−1 ). If h ≡ 0, then f |U1 and f |U2 agree on a small neighborhood, and we can use the unique continuation theorem of Aronszajn [Aro57] or an argument of Sampson [Sam78] to show that f |U1 and f |U2 extend to the same map on Σ.1 If f is prime this implies that p1 = p2 . Since the domain is only two-dimensional, the harmonic polynomials in (5.1) are easily described; indeed, p(u1 , u2 ) = Re(a(u1 + iu2 )m ), where a is a nonzero element of Cn−2 , and we conclude that the set of points in U1 × U2 at which f |U1 and f |U2 do not intersect is open and dense. The lemma follows from the fact that we can cover Σ × Σ by finitely many pairs of such neighborhoods. Remark 5.2.2. Lemma 5.2.1 is not always true for harmonic maps which are not conformal. The degree zero map f : T 2 → S 2 of Example 4.2.6 has fold points and is not somewhere injective. Moreover, if g : S 2 → S n is a totally geodesic two-sphere when S 2 has the round metric, then g ◦ f : T 2 → S n is also harmonic but not somewhere injective. 1 Use of the Aronszajn theorem should not be surprising; unique continuation is part of the foundation for Donaldson theory and Seiberg-Witten theory as well.

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Wood [Woo77] constructs additional examples of harmonic maps f : Σ1 → Σ2 from one surface with Riemannian metric to another with a complicated singularity locus which fail to be somewhere injective; compositions of his example, g ◦ f : Σ1 → M , where g : Σ2 → M is totally geodesic, also fail to be somewhere injective. 5.2.2. Self-intersection sets. It is sometimes useful to have a more explicit description of the set of self-intersections. A description of the selfintersection set of a minimal surface is a key ingredient of the proof of Theorem 4.7.8, the geometric sphere theorem for three-manifolds developed by Meeks and Yau [MY82]. It is also used in the study by Freedman, Hass and Scott [FHS83] of decomposition of three-manifolds along tori. It is not surprising that a similar description of the self-intersection set can be established in the case in which the ambient space M has arbitrary dimension at least three. We describe such an extension to higher codimension in this subsection. An explicit normal form near a branch point is given by Theorem 1.4 of Micallef and White [MWh95] which states that if p is a branch point of a weakly conformal harmonic map f : Σ → M , then there are normal coordinates (u1 , . . . , un ) centered at f (p) on the ambient Riemannian manifold M and a C 2,α diffeomorphism z → ξ(z) near p on Σ, z being a holomorphic coordinate centered at p, such that ξ(z) = cz + O(|z|2 ) for some nonzero complex constant c, and if we abbreviate the composition ui ◦ f ◦ ξ to ui , then for some ν ∈ N, (5.2)

u1 (z) + iu2 (z) = ξ(z)ν+1 ,

ur (z) = fr (ξ(z)) for 3 ≤ r ≤ n,

where each fr (ξ) is o1 (|ξ|ν+1 ). Here o1 (|z|k+1 ) stands for a term which is o(|z|k+1 ), and which has a derivative that is o(|z|k ). Thus in terms of the nonconformal coordinate ξ, the error term in u1 + iu2 vanishes. If we use ξ as the local (nonconformal) parameter on Σ, we can write (5.2) as (5.3)

ξ → (ξ ν+1 , f˜(ξ)),

where f˜ takes values in Rn−2 .

Theorem 1.4 of [MWh95] goes on to state that if μν+1 = 1, then either f˜(μξ) − f˜(ξ) is identically zero, or (5.4)

¯ξ¯ρ + o1 (|ξ|ρ ), f˜(μξ) − f˜(ξ) = aξ ρ + a

where a is a nonzero element of Cn−2 and ρ > ν + 1. We say that a branch point p for a harmonic map f is ramified of order σ if σ + 1 is the largest integer such that there is an open neighborhood U of p and a smooth map ψ : U → D, where D is the unit disk centered at 0

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285

in the complex plane, such that (1) ψ(p) = 0, (2) ψ|(U − {p}) is a (σ + 1)-sheeted cover of D − {0}, and (3) if q, r ∈ U − {p}, then f (q) = f (r) ⇔ ψ(q) = ψ(r). If σ = ν ≥ 1, the branching order of p, we say that p is a false branch point. At the other extreme, we say that the branch point is primitive or unramified if the largest such integer σ is zero. Note that a branch point is primitive precisely when f˜(μξ) − f˜(ξ) is not identically zero for any nontrivial (ν + 1)-fold root of unity. Lemma 5.2.3. If f : Σ → M is a prime parametrized minimal surface, all its branch points are primitive. Proof. The idea is to use the Aronszajn unique continuation theorem and a theory of branched immersions, like that presented by Gulliver, Osserman and Royden in §3 of [GOR73]. We begin by defining an equivalence relation ∼ on points of Σ by setting p ∼ q if there are open neighborhoods Up and Uq of p and q, respectively, and a conformal or anti-conformal diffeomorphism ψ : Up → Uq such that f ◦ ψ = f . (We allow ψ to be orientation-reversing so that we can allow for branched covers of nonorientable surfaces.) Using Aronsjazn’s unique continuation theorem, one shows that ∼ is indeed an equivalence relation and that the quotient space Σ0 is a topological manifold (possibly nonorientable) which is smooth except at the branch points. We can define f0 : Σ0 → M by f0 ([p]) = f (p), where [p] denotes the equivalence class of p, so that if π : Σ → Σ0 is the quotient map, f0 ◦ π = f . We note that any point equivalent to a branch point is itself a branch point. The restriction of f0 to Σ0 minus the equivalence classes of the branch points is a harmonic map of finite energy. It therefore follows from the removeable singularity theorem of Sacks and Uhlenbeck (Theorem 4.6.9) that the restriction of f0 can be extended to the equivalence classes of the branch points so as to be a harmonic map, which is conformal so long as f is conformal. Thus f is a branched cover of f0 , with allowances made for nonorientability. The hypothesis that f is prime implies that Σ0 must equal Σ, π is the identity, and every branch point must be primitive, finishing our argument. QED The following is a special case of Lemma 2.4 of Cheng [Che76]. Lemma 5.2.4. Suppose that f is a smooth real-valued function on an open neighborhood of the origin in R2 and p is a nonzero homogeneous harmonic

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polynomial of degree ρ ≥ 2 on R2 such that f (ξ) − p(ξ) = o1 (|ξ|ρ ), Then there is a local f (ξ) = p(Φ(ξ)).

C1

for some positive integer ρ.

diffeomorphism Φ of R2 , fixing the origin, such that

We refer to [Che76], pages 48 and 49, for an elegant proof of this lemma. With Lemmas 5.2.3 and 5.2.4 at our disposal, we are ready to present the main result of this section. Lemma 5.2.5. Let f : Σ → M be a prime compact connected parametrized minimal surface in a Riemannian manifold of dimension at least three, with self-intersection set K = {p ∈ Σ : f −1 (f (p)) contains more than one point}, a compact subset of Σ. Then each point within K belongs to one of two classes: (1) an isolated point of self-intersection (of which there can be only finitely many), (2) a point which lies in a neighborhood U ⊂ Σ such that K ∩ U is a subset of a finite number of C 1 curve segments of finite length in U intersecting at the point. We remark that since Σ is compact, Lemma 5.2.5 implies that the selfintersection set K is contained in a finite number of points and a finite number of C 1 curves of finite length. If M has dimension three, Lemma 5.2.4 mostly follows from Lemma 2 in [MY82] or Lemma 1.4 in [FHS83], and we can use these sources as models for the proof. To begin the proof, we note that if p ∈ K, there are only finitely many points in f −1 (f (p)). Indeed, if there were infinitely many points in f −1 (f (p)), there would be a convergent sequence {pi } of such points since Σ is compact. This would contradict the local form of the parametrized minimal surface in a neighborhood of the limit p0 of the sequence. By an easy induction on the number of points in the preimage of a self-intersection point, it suffices to consider the self-intersection set of two harmonic maps f1 : D1 → M and f2 : D2 → M from disks centered at p1 and p2 , respectively, such that f1 (p1 ) = f2 (p2 ) = q, as well as the selfintersection of a single such map near a branch point. We start with the first case under the assumption that neither p1 nor p2 is a branch point. If K is the intersection set of f1 and f2 , we need to show that any point in K lies in one of the two classes of points described in

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287

the conclusion of the lemma. The tangent planes Π1 and Π2 at q to f1 (D1 ) and f2 (D2 ), respectively, can intersect in a point or a line, or the two planes may coincide. If the planes intersect in a point, then p1 and p2 belong to the first class in the conclusion of the lemma. If Π1 and Π2 intersect in a line, the planes make an angle θ with each other, where 0 < θ < π. In this case there is a plane Π which makes an acute angle with both Π1 and Π2 . On the other hand, if Π1 = Π2 , we simply set Π = Π1 = Π2 . In either case, each of the harmonic surfaces can be regarded as a graph over Π near q, in terms of normal coordinates on M centered at q. In more detail, we choose normal coordinates (u1 , . . . , un ) on an open neighborhood of q in M with ui (q) = 0, so that the plane spanned by the coordinate vectors (∂/∂u1 ) and (∂/∂u2 ) at q is Π. Let Bε be a ball of radius ε about 0 in the (u1 , u2 )-plane. We can consider the two surfaces as graphs of functions hi : Bε → Rn−2 ,

i = 1, 2,

which satisfy an explicit nonlinear partial differential equation. As in the preceding section, we denote the Riemannian metric on M and the metric on Bε induced by the immersion (u1 , u2 ) → (u1 , u2 , hi (u1 , u2 )) by n 

gij dui duj

i,j=1

and

2 

ηab dua dua ,

a,b=1

respectively, and it follows as in the preceding section that each hi is an Rn−2 -valued solution to an elliptic system in “nonparametric form”, 2 

η ab (u1 , u2 , h, Dh)

a,b=1

∂ 2h + φ(u1 , u2 , h, Dh) = 0, ∂ua ∂ub

where (η ab ) is the matrix inverse of (ηab ), the operator on the left once again having scalar symbol. It is readily checked that the difference h = h2 − h1 satisfies an equation of the same form, 2  a,b=1

η ab (u1 , u2 , h, Dh)

∂ 2h + φ(u1 , u2 , h, Dh) = 0, ∂ua ∂ub

in which the coefficients depend upon h1 , which is now regarded as a known function of u1 and u2 , with h or equivalently h2 being regarded as unknown. Thus by Corollary 1 of Theorem 1.1 in [MWh95] (a corollary of a theorem

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5. Generic Metrics

of Hartman and Wintner), we see that either h ≡ 0 or |h(u1 , u2 )| = O(|(u1 , u2 )|m ) for some positive integer m, and ⇒

h(u1 , u2 ) = p(u1 , u2 ) + o1 (|(u1 , u2 )|m ),

where p(u1 , u2 ) is a nonzero homogeneous vector-valued harmonic polynomial of degree m. We recall that stating that h − p is o1 (|z|m ) means that h − p is o(|z|m ) and D(h − p) is o(|z|m−1 ). It cannot be the case that h ≡ 0 by Lemma 5.2.3, since f is assumed to be prime. Since the domain is only two-dimensional, such harmonic polynomials are easily described; indeed, p(u1 , u2 ) = Re(a(u1 + iu2 )m ),

where a is a nonzero element of Cn−2 .

If m = 1, the two surfaces intersect in a smooth curve near q. If m > 1, this vector-valued harmonic polynomial vanishes at a single point or on a C 1 collection of curves. In the former case, the two surfaces intersect in an isolated point and we are done. In the latter case, we can choose a one-dimensional linear subspace V of Rn−2 such that if πV denotes the orthogonal projection into this subspace, πV ◦ p is not identically zero. The zero set of the Rn−2 -valued function h is contained in the zero set of the scalar function πV ◦ h, so it suffices to show that the zero set of πV ◦ h is a finite collection of C 1 curves passing through the origin. But now, following [MY82], we can apply Lemma 5.2.4 to conclude that there is a C 1 local diffeomorphism Φ near the origin in R2 such that πV ◦ h = πV ◦ p ◦ Φ. Thus πV ◦ h vanishes on the intersection of a finite number of C 1 curves in terms of the parameter ξ, and by the canonical form described at the beginning of the section, also in terms of the conformal parameter z. This establishes the desired conclusion in this case. To finish the proof in the case that branch points are present, recall that we need to consider: (1) the self-intersection set of a single small minimal disk f : D → M in which D is centered at a branch point p, and (2) the intersection of two minimal disks f1 : D1 → M and f2 : D2 → M centered at p1 and p2 , with f (p1 ) = f (p2 ) = q, in which one or both of the points p1 and p2 are branch points. If we were to make the assumption that f is not only minimal, but also locally area minimizing, we could apply Theorem 4.2 of [MWh95] to show that the self-intersection set of the restriction of f to a neighborhood of the branch point p is empty when the neighborhood is sufficiently small, completing the first of these steps.

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289

In general, the self-intersections near a branch point p on a single minimal disk f : D → M can be treated by the canonical form (5.2). We note that for particular points ξ1 and ξ2 of D, f (ξ1 ) can equal f (ξ2 ) only if ξ2 = μξ1 , where μ is a (ν + 1)-fold root of unity, and f˜(μξ1 ) = f˜(ξ2 ), where f˜ is defined by (5.3). According to Lemma 5.2.3, since f is prime, it is primitive, and hence f˜(μξ) − f˜(ξ) is not identically zero for any nontrivial (ν + 1)-root of unity. Thus we can apply (5.4) to conclude that ¯ξ¯ρ ) + o1 (|ξ|ρ ), πV ◦ f˜(μξ) − πV ◦ f˜(ξ) = πV (aξ ρ + a where πV is the projection into a three-dimensional subspace of Rn contain¯) = 0. Applying Lemma 5.2.4 ing the tangent plane at p such that πV (a + a once again shows that πV ◦ f˜(μξ) − πV ◦ f˜(ξ) vanishes on a finite number of C 1 curves (in terms of the nonconformal parameter ξ) passing through p. Once again, since the transformation from the conformal parameter z to ξ is C 2,α , the self-intersection set of f |D lies within a finite collection of C 1 curves of finite length, finishing the argument in this case. We next consider the case in which two minimal disks f1 : D1 → M and f2 : D2 → M intersect at the centers p1 and p2 of the disks. If the tangent planes Π1 and Π2 to f1 (D1 ) and f2 (D2 ) at q = f (p1 ) = f (p2 ) intersect at a point, then the two minimal disks also intersect at a point, and p1 and p2 are isolated points of self-intersection. Suppose next that the two planes Π1 and Π2 coincide and p1 is a branch point of order ν, while p2 is not a branch point. In this case, we replace f2 : D2 → M by a new disk f2 : D2 → M centered at p2 such that f2 (z) = f2 (z ν+1 ). Thus f2 is a branched cover of f2 , and f2 has a false branch point at its center p2 with exactly the same order as the branch point at p1 . Both f1 and f2 have nonconformal parametrizations ξ → (ξ ν+1 , f˜1 (ξ)),

ξ → (ξ ν+1 , f˜2 (ξ)),

in which ξ is related to conformal parameters z1 on D1 and z2 on D2 by C 2,α diffeomorphisms. It cannot be the case that f˜1 − f˜2 is identically zero, because this would contradict unique continuation, since f is prime. Therefore it follows from Remark 1.6 of [MWh95] that if μ is any nontrivial (ν + 1)-fold root of unity, ¯ξ¯ρ + o1 (|ξ|ρ ), f˜1 (μξ) − f˜2 (ξ) = aξ ρ + a where a is a nonzero element of Cn−2 and ρ > ν + 1. An application of Lemma 5.2.4 again shows that f1 and f2 intersect in a finite number of C 1 curves of finite length intersecting at p1 or p2 , in terms of the parameter ξ. This holds also in terms of conformal parameters z1 for f1 and z2 for f2 , just as before. The images of these curves in the disk D2 are C 1 curves intersecting at p2 .

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5. Generic Metrics

If Π1 and Π2 coincide and both p1 and p2 are branch points of branching order ν1 and ν2 , respectively, we let ν + 1 be the lowest common multiple of ν1 + 1 and ν2 + 1. We choose positive integers σ1 and σ2 so that ν + 1 = σ1 (ν1 + 1) and ν + 1 = σ2 (ν2 + 1). For i = 1, 2, we then replace fi : Di → M by a new disk fi : Di → M such that fi (z) = fi (z σi ). The two disks f1 and f2 are no longer necessarily primitive, but we can still apply Remark 1.6 of [MWh95] to show that if μ is any (ν + 1)-root of unity, either f˜1 (μξ) − f˜2 (ξ) ≡ 0

or

f˜1 (μξ) − f˜2 (ξ) = aξ ρ + a ¯ξ¯ρ + o1 (|ξ|ρ ),

where a is a nonzero element of Cn−2 and ρ > ν + 1. Once again the first case would contradict unique continuation, since f is prime, so we can apply Lemma 5.2.4 yet again to conclude that f1 and f2 intersect in a finite collection of C 1 curves of finite length, and once again we argue that f1 and f2 themselves intersect in a finite number of C 1 curves of finite length. Finally, we need to consider the case in which the two planes Π1 and Π2 intersect in a line. As before, if p1 is a branch point of order ν, while p2 is not a branch point, we replace f2 by a branched cover f2 so that f1 and f2 both have branch points of order ν. If p1 and p2 are branch points of order ν1 and ν2 we let ν + 1 be the lowest common multiple of ν1 + 1 and ν2 + 1, and replace both f1 and f2 by branched covers f1 and f2 so that f1 and f2 both have branching order ν. Each of the maps f1 and f2 has a canonical form (5.61), but since the tangent planes make an angle with each other, the normal coordinates in M that are used to realize these forms are also at an angle. Thus we can choose two normal coordinate systems (u1 , u2 , u3 , u4 , . . . , un )

and

(u1 , v2 , v3 , u4 , . . . , un )

related to each other by a rotation, (5.5)

u2 = av2 − bv3 ,

u3 = bv2 + av3 ,

a2 + b2 = 1,

where b = 0, so that by (5.2), u1 (ξ1 ) + iu2 (ξi ) = ξ1ν+1 , u1 (ξ2 ) + iv2 (ξ2 ) =

ξ2ν+1 ,

ur (ξ1 ) = o1 (|ξ1 |ν+1 ) v3 (ξ2 ) = o1 (|ξ2 |

ν+1

for 3 ≤ r ≤ n, ),

us (ξ2 ) = o1 (|ξ2 |ν+1 ) for 4 ≤ s ≤ n,

where ξ1 and ξ2 are nonconformal parameters on D1 and D2 , respectively. Note that f1 and f2 can intersect only where Re(ξ1 ) = Re(ξ2 ). Since both ξ1 and ξ2 give Euclidean parametrizations of Π1 and Π2 , respectively, we can arrange that ξ1 and ξ2 are related by a local diffeomorphism from a neighborhood of 0 in D1 to a neighborhood of 0 in D2 under which ξ1 → ξ2 with u1 (ξ1 ) = u1 (ξ2 ).

5.2. Local behavior of minimal surfaces

291

It follows from (5.5) that u3 (ξ2 ) = b Im(ξ1ν+1 ) + o1 (|ξ1 |ν+1 ) = b Im(ξ2ν+1 ) + o1 (|ξ2 |ν+1 ), v3 (ξ1 ) = −b Im(ξ2ν+1 ) + o1 (|ξ2 |ν+1 ) = −b Im(ξ1ν+1 ) + o1 (|ξ1 |ν+1 ). According to Lemma 5.2.4, we can find local C 1 diffeomorphisms Φ1 and Φ2 of D1 and D2 so that if Φi (ηi ) = ξi , v3 ◦ Φ1 (η1 ) = −b Im(η1ν+1 )

u3 ◦ Φ2 (η2 ) = b Im(η2ν+1 ).

The set of points in D1 taken by f1 : D1 → M to points of f2 (D2 ) are points such that Im(η1ν+1 ) = 0, while the points of D2 taken by f2 : D2 → M to points of f1 (D1 ) are points such that Im(η2ν+1 ) = 0. Thus we find that the points of K contributed in a neighborhood of a self-intersection of two disks whose tangent planes intersect in a line lie on finitely many C 1 curves of finite length. Thus we have covered all cases and Lemma 5.2.5 is proven. Example 5.2.6. Whitney’s theorems imply that imbeddings are dense in Map(Σ, M ) when M has dimension at least five, while immersions are dense when M has dimension at least four. But immersions are no longer dense in Map(Σ, M ) when M has dimension three. For an explicit example, we first consider the map g : S 2 → S 2 , where S 2 is the one-point compactification of the complex plane C, defined by z → z 2 , a branched cover with branch points at zero and infinity. We compose this with the standard inclusion S 2 ⊆ R3 , obtaining a map f0 : S 2 → R3 . The resulting map is easily perturbed to a map f : S 2 → R3 which is an immersion in the complement of the north and south poles and has intersection set a single meridian stretching between poles, and f cannot be perturbed to a nearby immersion, close in the C k topology for k large. Indeed, the projection of f to the plane R2 orthogonal to the polar axis maps a small circle around the north pole to a closed curve with winding number two in the sense of the Whitney-Graustein theorem (see [Sma58]), and a nearby immersion f˜ would also map this circle with winding number two. But the immersion f˜ would then give a homotopy of this curve to a curve with winding number one, contradicting the Whitney-Graustein theorem. An adaptation of this argument shows that any minimal surface f : Σ → M in a three-manifold with nontrivial branch locus cannot be approximated by an immersion. This means that Theorem 5.1.1 cannot perturb minimal surfaces with branch points to minimal surfaces without them. As we will see, the proof instead shows that any minimal surface with branch points is a degenerate critical point for E, which vanishes entirely under appropriate metric variations.

292

5. Generic Metrics

5.3. The two-variable energy revisited At the beginning of Chapter 4, we explained how oriented parametrized minimal surfaces f : Σ → M are critical points for a two-variable version of the Dirichlet energy  1 |df |2η dAη , (5.6) E : Map(Σ, M ) × Met0 (Σ) −→ R, E(f, η) = 2 Σ where |df |η and dAη are calculated with respect to the Riemannian metric η ∈ Met0 (Σ). Here Met0 (Σ) is the space of Riemannian metrics on the oriented surface Σ of genus g which have ⎧ ⎪ ⎨constant curvature one if g = 0, constant curvature zero with total area one if g = 1, ⎪ ⎩ and constant curvature −1 if g ≥ 2. This energy (5.6) is invariant under an action of the group Diff0 (Σ) of diffeomorphisms homotopic to the identity, such that φ ∈ Diff0 (Σ) acts via the map (f, η) ∈ Map(Σ, M ) × Met0 (Σ)

→

(f ◦ φ, φ∗ η).

It is therefore possible to divide out by the action of the gauge group G: (1) Diff0 (Σ) when g ≥ 2, (2) Diff0,p (T 2 ) where p is a point in T 2 when g = 1, (3) or Diff0,q,r,s (S 2 ) where q, r and s are distinct points in S 2 when g = 0, the restricted action in each case being free, thus obtaining a simpler energy (5.7)

E : Map(Σ, M ) × T −→ R,

where T is the Teichm¨ uller space of Σ. The energy for a nonorientable parametrized surface f0 : Σ0 → M with domain Σ0 is one-half the energy on the oriented double cover f ◦ π, where π : Σ → Σ0 is the oriented double cover of domains (see §4.3.3). We restrict attention to orientable surfaces until Extension 5.3.11, where we describe the extension needed to adapt the oriented theory to nonorientable surfaces. In this section, we calculate the second variation of the two-variable energy E and use that to determine the tangential Jacobi fields for E. We are forced to study second variation for the two-variable energy E rather than the more customary second variation of area A because it is the twovariable energy E which has a family of perturbations Eα : Map(Σ, M ) × Met0 (Σ) −→ R,

α > 1,

5.3. Two-variable energy revisited

293

which are well-behaved from viewpoint of global analysis.2 Although each immersion f : Σ → M inherits a canonical conformal structure which renders f conformal, Example 5.2.6 shows that immersions are not even dense in the space of mappings when the ambient space has dimension three. To study maps which have singularities, it is useful to have a formula which allows the metric on Σ to be independent from the (perhaps singular) metric induced on Σ by the Riemannian metric on M . 5.3.1. Tensor analysis on a Riemann surface. To fully understand the second variation formula for the two-variable energy, we need a simple description of the tangent space to Teichm¨ uller space as the cokernel of a first-order elliptic operator. It is convenient to describe the tangent space to Teichm¨ uller space using 2 the tensor analysis over the Riemann surface S described in a seminal and highly recommended article of Calabi (§2 of [Cal67]), easily extended to an arbitrary compact Riemann surface Σ. Calabi used this tensor analysis to extend the proof of Almgren’s Theorem 4.2.4 that all minimal two-spheres in the round S 3 are totally geodesic to an analysis of minimal two-spheres in the sphere S n of constant curvature one for n ≥ 4. He showed (among other things) that all minimal two-spheres in the round S n have area and energy an integer multiple of 4π; but when n ≥ 4 the moduli space of such minimal two-spheres is far more complicated than in Almgren’s case, and a rich theory was subsequently developed by many authors. For a given Riemann surface Σ, we let K(1,0) = (T ∗ Σ)1,0 and K(0,1) = (T ∗ Σ)0,1 denote the holomorphic cotangent bundle and antiholomorphic cotangent bundle of Σ, respectively. More generally, we set  p  q (5.8) K(p,q) = T p,q Σ = (T ∗ Σ)1,0 ⊗ (T ∗ Σ)0,1 , a complex line bundle whose sections can be represented in terms of a local complex parameter z in the form f dz p d¯ zq , where f is a complex-valued function, and the direct sum becomes a graded commutative algebra ∞ ∞   ∗∗ T p,q Σ. T Σ= p=0 q=0

The real inner product on T Σ extends to a complex bilinear inner product ·, · on the sum of all K(p,q) ’s, which pairs K(p,q) with K(q,p) , or to a Hermitian inner product with respect to which all the K(p,q) ’s are orthogonal, 2 Recall that by the discussion in §4.5, fixing the second variable in E α and applying an additional small perturbation gives a function which satisfies the full Morse inequalities on the mapping space Map(Σ, M ).

294

5. Generic Metrics

and the real structure on the complexification determines a conjugate linear conjugation map C : K(p,q) → K(q,p) . z on Σ determines a collection of complex A Riemannian metric λ2 dzd¯ linear operators D  : Γ(T p,q Σ) −→ Γ(T p+1,q Σ),

D  : Γ(T p,q Σ) −→ Γ(T p,q+1 Σ),

the covariant differentials with respect to the Levi-Civita connection, which are characterized by the following conditions: (1) the Leibniz rule holds, that is, D  (στ ) = D  (σ)τ + σD  (τ ),

D  (στ ) = D  (σ)τ + σD  (τ ),

¯ on T p,0 , (2) the operator D  is the usual ∂-operator ¯ )dz p = ∂f dz p ⊗ d¯ z, D  (f dz p ) = (∂f ∂ z¯ (3) conjugation transforms ∇ to ∇ , ¯, D  σ = D  σ z is preserved, (4) the metric λ2 dzd¯ D  (λ2 dzd¯ z ) = 0 = D  (λ2 dzd¯ z ). Indeed, these conditions imply that D  and D  must be given by the explicit formulae   ∂(log λ) ∂f  p q − 2pf dz p+1 d¯ z )= zq , D (f dz d¯ ∂z ∂z   (5.9) ∂(log λ) ∂f − 2qf dz p d¯ zq ) = z q+1 . D  (f dz p d¯ ∂ z¯ ∂ z¯ It is easily verified that on Γ(T p,q ), 1 ∂ 2 (log λ) = (q − p)Kλ2 dzd¯ z, ∂z∂ z¯ 2 where K is the Gaussian curvature of the Riemannian metric on Σ, which we will usually take to be constant. (5.10)

D  ◦ D  − D  ◦ D  = 2(p − q)

The Riemannian metric also determines an isomorphism Λ : Γ(T p,q ) −→ Γ(T p+1,q+1 ),

Λ (f dz p d¯ z q ) = f λ2 dz p+1 d¯ z q+1 .

Using this isomorphism, we can express everything in terms of the bundles T p,0 , which have holomorphic transition functions. For example, we can replace the operator D  : Γ(T p,0 ) −→ Γ(T p,1 )

with Λ−1 ◦ D  : Γ(T p,0 ) −→ Γ(T p−1,0 ).

5.3. Two-variable energy revisited

295

For simplicity of notation, we will denote this operator simply by D  , with the hope that no confusion results. Thus     ∂(log λ) ∂f  p p+1  p −2 ∂f D (f dz ) = − 2pf dz , D (f dz ) = λ dz p−1 , ∂z ∂z ∂ z¯ and

p D  ◦ D  − D  ◦ D  = K. 2 The key observation is that all tensors on Σ can be expressed in terms of special case of differentials f dz p , and this allows use of the geometry of holomorphic bundles over S 2 : Calabi repeatedly uses the fact that S 2 has no nonzero holomorphic differentials f dz p with p > 0 as a main step in his proofs. Once we fix a base metric η, we can write an arbitrary variation of metrics in terms of the bundles (5.8), and complexification immediately yields (5.11)

Tη (Met(Σ)) ⊗ C = Γ(K(2,0) ) ⊕ Γ(K(1,1) ) ⊕ Γ(K(0,2) ).

Indeed, this description has been adopted by some physicists in presentations of string theory (see [DHo99]). Sections of K(1,1) are exactly the metric deformations which correspond to Weyl rescalings, while (5.12)

Tη (Met0 (Σ)) ⊗ C = Γ(K(2,0) ) ⊕ Γ(K(0,2) ).

Moreover, one checks that the operator (5.13)

D  : Γ(K(0,1) ) → Γ(K(0,2) )

represents the infinitesimal action of vector fields on quadratic differentials induced by the action of Diff0 (Σ) on Met0 (Σ). Lemma 5.3.1. The holomorphic tangent space to the Teichm¨ uller space Tg is naturally isomorphic to the cokernel of (5.13), while the cotangent space is isomorphic to the space of holomorphic quadratic differentials {φdz 2 ∈ Γ(K(2,0) ) : D  (φdz 2 ) = 0}. This is just a restatement of the conclusion of arguments presented in §4.3. Recall that the tangent space to Teichm¨ uller space is isomorphic to the tangent space to a slice of the Diff0 (Σ)-action on Met0 (Σ). We choose the slice so that it is tangent to the one-parameter families of metrics ηij (t) = ηij (0) + tη˙ ij ,

with

ηij (0) = λ2 δij ,

for some conformal factor λ2 , the (η˙ ij ) being required to be L2 -perpendicular to the orbits of Diff0 (Σ) at (f, η). When the genus is at least two, the

296

5. Generic Metrics

argument in §4.3.1 shows that η˙ satisfies the zero-trace condition η˙ 11 + η˙ 22 = 0, as well as the condition that φdz 2 = (η˙ 11 − iη˙ 12 )dz 2

(5.14)

be a holomorphic quadratic differential, so that 2    2 η˙ ab dxa dxb . Re φdz = a,b=1

Our formula for second variation in tangential directions will use the operator  : Γ(L) −→ Γ(L ⊗ K(0,1) ), DL in which the holomorphic tangent bundle K(0,1) of (5.13) is replaced by the ramified tangent bundle L, ramified at the branch points of the parametrized minimal surface. 5.3.2. Second variation of energy. Our next goal is to extend the calculations of §4.3.2 to provide a formula for the second variation of E : Map(Σ, M ) × Met0 (Σ) −→ R at a critical point (f, η). We then restrict to the tangent space to a slice for the action of the diffeomorphism group to obtain the corresponding formula on the Teichm¨ uller space level. At a critical point (f, η), the second derivative of energy E : Map(Σ, M ) × Met0 (Σ) −→ R is a sum of three terms. If X is a section of f ∗ T M and η˙ is a trace-free variation of the metric on Σ, then ˙ (X, η)) ˙ = d2 Eη (f )(X, X) (5.15) d2 E(f, η)((X, η),    . DX ∂f / d  ab  η det(ηab ) , dx1 dx2 +2 dt ∂xa ∂x Σ a,b

t=0

b

  2    . ∂f ∂f / 1 d ab + det(ηab ) , η dx1 dx2 , 2 Σ dt2 t=0 ∂xa ∂xb a,b

with Eη denoting energy for fixed choice of metric η on Σ. We can think of these three terms as second-order partial derivatives in the three directions: Map(Σ, M ) × Map(Σ, M ),

Map(Σ, M ) × Met0 (Σ),

Met0 (Σ) × Met0 (Σ).

The first term in (5.15) is the familiar second variation formula for harmonic maps,    DX2 − R(X ∧ df ), X ∧ df  dA, (5.16) d2 Eη (f )(X, X) = Σ

5.3. Two-variable energy revisited

297

where in terms of the complex parameter z = x1 + ix2 on Σ,   1 DX 2 DX 2 2 DX = 2 + , λ ∂x1 ∂x2 where D denotes covariant differentiation, and R(X ∧ df ), X ∧ df  / .  / .    ∂f ∂f ∂f ∂f 1 , X + R X, ,X , R X, = 2 λ ∂x1 ∂x1 ∂x2 ∂x2 R being the Riemann-Christoffel curvature tensor of M . To evaluate the other two terms, we take a metric variation ηab (t) = λ2 δab + tη˙ ab , with η˙ 11 + η˙ 22 = 0 as in §4.3.2. Differentiation yields d det(ηab (t)) =0 (5.17) dt t=0

and (5.18)

d2 2 2 det(ηab (t)) = 2η˙ 11 η˙ 22 − 2η˙ 12 η˙ 12 = −2(η˙ 11 + η˙ 12 ). 2 dt t=0

Using (5.17) and the fact that   11 12    1 η22 −η12 η η det(ηab ) 21 22 =  , η η det(ηab ) −η21 η11 we find that

   11 12  d 1 η˙ 22 −η˙ 12 η η , det(ηab ) 21 22 = 2 η η dt λ −η˙ 21 η˙ 11 t=0

which can be rewritten as  d  ab  η det(ηab ) = −λ−2 η˙ ab . (5.19) dt t=0 Similarly, we can use (5.17) and the chain rule to conclude that    11 12    2 d2  d 1 η η η −η 22 12  det(ηab ) 21 22 = 2 η η −η η dt2 dt 21 11 ) det(η ab t=0 t=0     2 1 d2 0 λ =− det(ηab ) , 0 λ2 2[det(ηab )]3/2 dt2 t=0 which by (5.18) can be rewritten as  d2  ab  2 2 det(ηab ) = λ−4 (η˙ 11 + η˙ 12 )δab . η (5.20) dt2 t=0

298

5. Generic Metrics

It now follows from (5.19) that the second term in (5.15) is . /   η˙ ab DX ∂f −2 , dx1 dx2 , λ2 ∂xa ∂xb Σ a,b

while from (5.20) and the trace-free condition we conclude that the third term is   η˙ ab 2 ∂f 2 1 λ2 ∂x dx1 dx2 . 2 Σ b a,b

Adding the three terms together yields our formula for the second variation: ˙ (X, η)) ˙ d2 E(f, η)((X, η),

. /   η˙ ab DX ∂f dx1 dx2 , = d Eη (f )(X, X) − 2 2 ∂xa ∂xb Σ a,b λ  1 2 2 + (η˙ + η˙ 12 )σ 2 dx1 dx2 , 4 11 λ Σ 2

(5.21)

where

2 ∂f 2 ∂f 2 ∂f = = 2 . σ = ∂x1 ∂x2 ∂z The polarization identity then yields the following: 2

Proposition 5.3.2. The second variation of the two-variable energy E at a critical point (f, ω) is given by ˙ = d2 Eη (f )(X, Y ) ˙ (Y, ζ)) d2 E(f, η)((X, η),  . / . /   ζ˙ab DX ∂f η˙ ab DY ∂f , , + 2 dx1 dx2 − (5.22) λ2 ∂xa ∂xb λ ∂xa ∂xb Σ a,b  1 + (η˙ ζ˙ + η˙ 12 ζ˙12 )σ 2 dx1 dx2 , 4 11 11 λ Σ where X and Y are sections of f ∗ T M and η˙ and ζ˙ are trace-free variations in the metric on Σ. The second variation (5.21) depends only on the projections of 2  a,b=1

η˙ ab dxa dxb ,

2 

ζ˙ab dxa dxb ∈ Tη Met0 (Σ)

a,b=1

into Teichm¨ uller space, that is it is unchanged when the metric is subjected to a variation tangent to the infinite-dimensional symmetry group Diff0 (Σ). From a conceptual point of view, it is important to maintain this infinitedimensional symmetry, because a fortuitous change of the metric on Σ might simplify the formula in a given context. We can regard reparametrization

5.3. Two-variable energy revisited

299

via a diffeomorphism as analogous to a change of gauge in Donaldson or Seiberg-Witten theory. Remark 5.3.3. We can specialize to obtain the classical second variation formula for area of immersions (found in [Sim68], Theorem 3.2.2 or [Law80], Theorem 32). We let Imm(Σ, M ) = {smooth immersions f : Σ → M }, and define A : Imm(Σ, M ) −→ R by

A(f ) = E(f, η(f )),

where η(f ) is the Riemannian metric induced by the immersion f on Σ. When calculating the second variation of A we can restrict to variations which are sections of the normal bundle to f , since a tangential variation does not change the area of the immersed surface. To carry this out in more detail, we suppose that X ⊥ is a section of the normal bundle (f ∗ T M )⊥ to f . A variation in the direction of X ⊥ will produce a corresponding variation σ(X ⊥ ) = (η˙ ab ) in the induced metric on Σ, which is given by the formula (5.23) / . / .  / . ∂f DX ⊥ ∂f DX ⊥ ∂f ⊥ D + = −2 X , , η˙ ab = , , ∂xa ∂xb ∂xa ∂xb ∂xa ∂xb the last equality using the fact that X ⊥ is perpendicular to ∂f /∂x1 and ∂f /∂x2 . Note that since f is harmonic, it follows from the right-hand expression that η˙ 11 + η˙ 22 = 0, so the variation is infinitesimally area preserving. It follows from (5.23) that / . 1 DX ⊥ ∂f , = η˙ ab , (5.24) ∂xa ∂xb 2 and hence the second and third terms in the second variation formula (5.21) are     η˙ ab 2 η˙ ab 2 1 − λ dx1 dx2 and 2 λ dx1 dx2 , Σ Σ a,b

a,b

respectively, and the second variation itself is given by  

2



2 ⊥ ⊥ ˙ (X , η)) ˙ = d E(f, η)((X , η),

DX ⊥ dA Σ a

   η˙ ab 2 1 ⊥ ⊥ − λ2 dA − R(X ∧ df ), X ∧ df dA. 2 Σ Σ a,b

300

5. Generic Metrics

A direct calculation shows that

2  1 . DX ⊥ ∂f / 2  η˙ ab 2 ⊥

= 4

=4 , (DX )

, λ2 λ2 ∂xa ∂xb a,b

a,b

from which we obtain the classical formula for second variation of area under normal variations:   (5.25) d2 E(f, ω) (X ⊥ , σ(X ⊥ )), (X ⊥ , σ(X ⊥ ))  $ % DX ⊥ 2 − 2(DX ⊥ ) 2 − R(X ⊥ ∧ df ), X ⊥ ∧ df  dA =  Σ$ % (DX ⊥ )⊥ 2 − (DX ⊥ ) 2 − R(X ⊥ ∧ df ), X ⊥ ∧ df  dA. = Σ

In this formula, (DX ⊥ ) 2 , depends only on the second fundamental form of the immersion f , as one verifies from (5.23). Polarization once again yields the following: Proposition 5.3.4. The second variation of area A is given by  $< = < = 2 ⊥ ⊥ (DX ⊥ )⊥ , (DY ⊥ )⊥ − (DX ⊥ ) , (DY ⊥ ) d A(f )(X , Y ) = 2 Σ % −R(X ⊥ ∧ df ), Y ⊥ ∧ df  dA, where X ⊥ and Y ⊥ are sections of normal bundle (f ∗ T M )⊥ , and (·) and (·)⊥ denote projections into tangent and normal bundles, respectively. This is particularly useful when Σ and M are orientable, and M has dimension three, because in this case the normal bundle is a trivial line bundle. It is also useful to specialize the second variation formula of Proposition 5.3.2 to the case of tangential sections X and Y of f ∗ T M . If f : Σ → M is an arbitrary parametrized minimal surface with possible branch points and (f ∗ T M ) is the tangential projection of f ∗ T M , we have an isomorphism τ : (f ∗ T M ) → Γ(L), where L is the holomorphic subbundle of f ∗ T M ⊗ C generated by ∂f /∂z as described in §4.1, so that     1 ∂f ∂f ∂f ∂f ∂f ∂f = −i + φ2 ⇒ τ φ1 = 2(φ1 + iφ2 ) . ∂z 2 ∂x1 ∂x2 ∂x1 ∂x2 ∂z √ We can use this isomorphism to transport multiplication by −1 on L to an almost complex structure J on (f ∗ T M )⊥ such that     ∂f ∂f ∂f ∂f , J . =− = J ∂x1 ∂x2 ∂x2 ∂x1

5.3. Two-variable energy revisited

301

In terms of this almost complex structure J, we then have Lemma 5.3.5. If X and Y are tangential sections of f ∗ T M , that is, if X = X and Y = Y and η˙ and ζ˙ are variations in the metric on Σ, then (5.26) /  . DX DY DY DX 2 ˙ ˙ (Y, ζ)) = +J , +J dx1 dx2 d E(f, η)((X, η), ∂x1 ∂x2 ∂x1 ∂x2 Σ . / . /   ζ˙ab DX ∂f η˙ ab DY ∂f − , , + 2 dx1 dx2 λ2 ∂xa ∂xb λ ∂xa ∂xb Σ a,b  1 + (η˙ ζ˙ + η˙ 12 ζ˙12 )σ 2 dx1 dx2 . 4 11 11 λ Σ Proof. This is an immediate consequence of Proposition 5.3.2 and the fact that   1 DX DX Dτ (X) . = +J ∂ z¯ 2 ∂x1 ∂x2 5.3.3. Tangential Jacobi fields. Parallel to our earlier treatment of Jacobi operators for the action on Map(S 1 , M ) and the ω-energy on Map(Σ, M ), we can define a Jacobi operator L for the two-variable energy E : Map(Σ, M ) × T −→ R at any critical point (f, ω). This operator is defined by  2 L(X1 , ω˙ 1 ), (X1 , ω˙ 1 ) dA, d E(f, ω)((X1 , ω˙ 1 ), (X2 , ω˙ 2 )) = Σ

for X1 , X2 ∈ Γ(f ∗ T M ) and ω˙ 1 , ω˙ 2 ∈ Tω T , the inner product in the integral on the right being the direct sum of the pullback Riemannian metric on f ∗ T M and the Weil-Peterson metric on Teichm¨ uller space. We say that (X, ω) ˙ is a Jacobi field if L(X, ω) ˙ = 0. Our next goal is to calculate the Jacobi fields which are tangential , the Jacobi fields (X, ω) ˙ such that X is tangent to f (Σ) at almost all points. These Jacobi fields can be obtained by projection under the quotient map Met0 (Σ) → T from the second variation formula for E : Map(Σ, M ) × Met0 (Σ) −→ R discussed in preceding pages. We say that (X, η) ˙ ∈ Tf Map(Σ, M ) × Tη Met0 (Σ) is a Jacobi field if ˙ = 0 for all (Y, ζ) ˙ ∈ Tf Map(Σ, M ) × Tη Met0 (Σ). ˙ (Y, ζ)) d2 E(f, η)((X, η), Here η˙ and ζ˙ are trace-free variations in the metric.

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5. Generic Metrics

Proposition 5.3.6. The tangential Jacobi fields (X, η) ˙ ∈ Tf Map(Σ, M ) × Tη Met0 (Σ) for the energy E are solutions to the equation (5.27)

DX η˙ 11 ∂f η˙ 12 ∂f DX +J − 2 − 2 = 0. ∂x1 ∂x2 λ ∂x1 λ ∂x2

Proof. We start by showing that d2 E(f, η)((X, η), ˙ (X, η)) ˙ =0

⇔

(X, η) ˙ satisfies (5.27).

˙ the second term in (5.26) takes the form When (X, η) ˙ = (Y, ζ), . / . /   η˙ 11 DX ∂f DX ∂f − , , −2 λ2 ∂x1 ∂x1 ∂x2 ∂x2 Σ . / . / η˙ 12 DX ∂f DX ∂f + 2 , , + dx1 dx2 , λ ∂x2 ∂x1 ∂x1 ∂x2 which can be rewritten as . / . /  1 DX DX ∂f ∂f − dx1 dx2 , (η˙ 11 + J η˙ 12 ) , (J η˙ 11 − η˙ 12 ) −2 2 ∂x1 ∂x1 ∂x2 ∂x1 Σ λ or . /  DX ∂f 1 DX + J , ( η ˙ + J η ˙ ) −2 dx1 dx2 . 11 12 2 ∂x1 ∂x2 ∂x1 Σ λ Moreover, the third term is . /  2 2 η˙ 11 + η˙ 12 ∂f ∂f , dx1 dx2 . λ4 ∂x1 ∂x1 Σ Adding the various terms together yields (5.28)

d2 E(f, ω)((X, η), ˙ (X, η)) ˙ 2  DX DX η ˙ η ˙ ∂f ∂f 11 12 = ∂x1 + J ∂x2 − λ2 ∂x1 − λ2 J ∂x1 dx1 dx2 . Σ

Suppose now that we consider d2 E(f, ω)((X, η), ˙ (Y, 0)), where X is a ∗ section of (f T M ) and Y is an arbitrary section of f ∗ T M . In this case, the second term in (5.26) takes the form . / . /   η˙ 11 DY ∂f DY ∂f , , − − λ2 ∂x1 ∂x1 ∂x2 ∂x2 Σ . / . / DY ∂f DY ∂f η˙ 12 , , + dx1 dx2 , + 2 λ ∂x2 ∂x1 ∂x1 ∂x2

5.3. Two-variable energy revisited

303

while the third term vanishes. Thus in this case, we obtain (5.29) d2 E(f, ω)((X, η), ˙ (Y, 0)) /  . DX η˙ 11 ∂f η˙ 12 ∂f DY DY DX +J − 2 − 2 , +J dx1 dx2 . = ∂x1 ∂x2 λ ∂x1 λ ∂x2 ∂x1 ∂x2 Σ Since any variation is the sum of a variation (X, η˙ ab ), where X is tangent and a variation (Y, 0), it follows that the Jacobi fields (X, η˙ ab ) for which X is tangent are exactly the solutions to the equation (5.27). QED We can express the equation (5.27) in terms of our holomorphic line bundle L ⊆ f ∗ T M ⊗ C and its distinguished locally defined holomorphic sections ∂f /∂z as φ¯ ∂f DZ − 2 = 0, where φdz 2 = (η˙ 11 − iη˙ 12 )dz 2 (5.30) ∂ z¯ λ ∂z is a holomorphic quadratic differential. Corollary 5.3.7. The tangential Jacobi fields (X, ω), ˙ with X being a section of (f ∗ T M ) , correspond to pairs (Z, φdz 2 ) where Z is a section of L and φdz 2 is a holomorphic quadratic differential which together satisfy (5.30). We can write the above equation as  ¯   φ¯ ∂f φ d¯ ∂f DZ z  d¯ z= 2 d¯ z= dz , (5.31) DL (Z) = ∂ z¯ λ ∂z λ2 dz ∂z  is the ∂-operator ¯ on L, analogous to the operator D  we studied where DL in §5.3.1, but with the holomorphic tangent bundle K(0,1) of Σ replaced by the ramified holomorphic tangent bundle L of f . Note that the first factor in large parentheses on the right-hand side of (5.31) is just the extremal Beltrami differential which corresponds to the holomorphic quadratic differential φdz 2 . If ιL : Tω T −→ Γ(L ⊗ K(0,1) ) is the operator defined by multiplying an antiholomorphic quadratic differential by the section (∂f /∂z)dz and then contracting with the metric, ¯ z 2 ∂f φ¯ ∂f ¯ z 2 ) = φd¯ dz = 2 d¯ z, (5.32) ιL (φd¯ 2 λ dzd¯ z ∂z λ ∂z and we can write the equation (5.31) for tangential Jacobi fields as ¯ z 2 ). (5.33) D  (Z) = ιL (φd¯ L

Definition. We let J (L) denote the space of complex tangential Jacobi fields, ¯ z 2 ) ∈ Γ(L) × Tω T which satisfy (5.33)}. J (L) = {((Z, φd¯

304

5. Generic Metrics

¯ z 2 ) ∈ J (L), we say that If (Z, φd¯ ¯ z 2 ) ∈ Re (J (L)) (X, ω) ˙ = Re(Z, φd¯ is the corresponding real tangential Jacobi field . Note that if Σ is oriented of genus zero or one, J (L) contains a subspace g of elements of the form (Z, 0), where Z is a holomorphic section generated by the action of the symmetry group G, the identity component of the group of conformal automorphisms of Σ, which is P SL(2, C) when g = 0, S 1 × S 1 when g = 1, and trivial when g ≥ 2. We claim that whenever a parametrized minimal surface (f, η) has a nontrivial branch locus, the branch points give rise to additional tangential Jacobi fields. Recall from (4.11) that the first Chern class of the line bundle L depends on the genus g of Σ and the total branching order ν of f : c1 (L)[Σ] = degree of L = 2 − 2g + ν. Proposition 5.3.8. When Σ is oriented, the complex dimension of the space of tangential Jacobi fields for a critical point (f, η) for the two-variable energy E is given by the formula: ⎧ ⎪ ⎨3 + ν if g = 0, dimC J (L) = 1 + ν if g = 1, ⎪ ⎩ ν if g ≥ 2. Proof. This proposition is proven by induction on the total branching order ν. When the total branching order ν is zero, L is just the holomorphic tangent bundle to Σ, and the only solutions to (5.33) are just the holomorphic sections of this bundle, which is just the tangent space to the identity component G of the group of conformal automorphisms. Thus when ν = 0, we see that dimC J (L) is three, one or zero when the genus g is zero, one or ≥ 2, respectively, and the proposition holds in this case. For the inductive step, we will apply the Riemann-Roch Theorem for  , which we can write in the form the operator DL  ) − dim(cokernel of D  ) = 1 − g + degree of L, dim(kernel of DL L  is the space O(L) of holomorphic sections of L, where the kernel of DL  is isomorphic to orthogonal complement of the while the cokernel of DL  in Γ(L ⊗ K(0,1) ). This orthogonal complement is the space image of DL of antiholomorphic sections of L ⊗ K(0,1) which is conjugate isomorphic to O(K ⊗ L−1 ). Recall from (4.11) that the ramified tangent bundle L over Σ can be written as (5.34) L∼ = T Σ ⊗ ζ ν1 ⊗ · · · ⊗ ζ νm , p1

pm

5.3. Two-variable energy revisited

305

where T Σ is the holomorphic tangent bundle to Σ, ζpi is a point bundle at the branch point pi and νi is the branching order at pi , while m 1 K ⊗ L−1 ∼ ⊗ · · · ⊗ ζp−ν . = K ⊗ K ⊗ ζp−ν m 1

(5.35)

We now assume the proposition is true for any line bundle L of the form (5.34) with “total branching order” ν, and prove it is true for L ⊗ ζp . Note that on the one hand, we have a monomorphism of holomorphic sections O(L) → O(L ⊗ ζp ), on the other hand, an epimorphism O(K ⊗ L−1 ) → O(K ⊗ L−1 ⊗ ζp−1 ). Applying the Riemann-Roch Theorem, we find that either  (1) the dimension of the kernel of DL⊗ζ is one larger than the dimenp  sion of the kernel of DL , but the dimensions of the cokernels are the same, or

(2) the dimensions of the kernels are the same, but the dimension of  is one less than the dimension of the cokernel the cokernel of DL⊗ζ p  of DL . In either case, dim J (L ⊗ ζp ) is one larger than dim J (L). In the first case  has increased by one, in the this is because the kernel of the operator DL⊗ζ p latter case it is because an additional one-dimensional subspace of the space of antiholomorphic quadratic differentials has been covered by the operator  . QED DL⊗ζ p If f : Σ → M is a weakly conformal harmonic map with divisor of branch points D = ν1 p1 + · · · + νm pm , we let z1 , . . . , zm be local complex coordinates on Σ centered at p1 , . . . , pm , and define a map (5.36) Ψ : J (L) −→ by



νi

m 

!" # L pi ⊕ · · · ⊕ L pi

i=1



 ¯ z2 = Ψ Z, φd¯

  DZ D νi −1 Z (pi ), . . . , (pi ) , . . . . . . . , Z(pi ), ∂zi ∂ziνi −1 

Proposition 5.3.9. The linear map Ψ defined by (5.36) is surjective, and its kernel is the space g of holomorphic sections generated by the holomorphic automorphisms of Σ.

306

5. Generic Metrics

Proof. Any element in the kernel of Ψ must be a holomorphic section of T Σ, hence an element of the space g. Thus Ψ induces a monomorphism νi

m !" #  J (L) ˜ Ψ: −→ L pi ⊕ · · · ⊕ L pi ∼ = Rν , g i=1

and this induced map must then be an isomorphism by Proposition 5.3.8. QED From of view of the Riemann surface Σ, the section Z appearing   the point ¯ z 2 has poles at the branch points. Since Ψ ˜ is an isomorphism, in Z, φd¯ any combination of principal parts at branch points can be realized by an element of J (L). Corollary 5.3.10. If f : Σ → M is a weakly conformal harmonic map with divisor of branch points D = ν1 p1 + · · · + νm pm , then there exist linearly independent elements z 2 ) ∈ J (L) for 1 ≤ i ≤ m, such that Zi (pj ) = δij ei , (Zi , φ¯i d¯ where ei is a unit-length element chosen from Lpi , the fiber of L at pi . Definition. By J0 (L) we will denote the kernel of the map (5.37) Ψ0 : J (L) −→

n 

L pi

i=1

  ¯ z 2 = (Z(p1 ), Z(p2 ), . . . , Z(pn )) . defined by Ψ Z, φd¯ It is a vector space of real dimension 2(ν −n)+dim G. We will sometimes let J1 (L) denote a 2n-dimensional complement of J0 (L) within J (L). Roughly speaking, the idea behind the proof of the Bumpy Metric Theorem is that the space J1 (L) can be “covered” by perturbations in the metric on M . Extension 5.3.11. Tangential Jacobi fields for nonorientable minimal surfaces. In accordance with the discussion of §4.3.3, we can extend the results just presented to nonorientable parametrized minimal surfaces. Suppose that f0 : Σ0 → M is a nonorientable parametrized minimal surface with oriented double cover f : Σ → M , and that A : Σ → Σ is the orientation-reversing sheet interchange map such that f ◦ A = f , which we choose in accordance with the discussion in §4.3.3. If E = f ∗ T M ⊗ C, the sheet interchange map A induces a conjugate linear endomorphism A of Γ(E). Recall that the tangential component of the complexified tangent

5.3. Two-variable energy revisited

307

bundle of M pulled back via f has a direct sum decomposition (f ∗ T M ⊗ C) = L ⊕ L, the two summands being interchanged by conjugation, and A also interchanges the line bundle L with its conjugate L. We can therefore construct a direct sum decomposition Γ(E) = Γ+ (E) ⊕ Γ− (E), with ¯ ¯ Γ− (E) = {Z ∈ Γ(E) : A (Z) = −Z}, Γ+ (E) = {Z ∈ Γ(E) : A (Z) = Z}, the sections of Γ+ (E) being regarded as complex deformations of the nonorientable minimal surface f0 : Σ → M . We remark that the conjugate of any ¯ z 2 ) ∈ J (L) is a solution tangent Jacobi field (Z, φd¯ ¯ ¯ ¯ φdz 2 ) to D Z dz = φ ∂f dz, (Z, ∂z λ2 ∂ z¯ for some holomorphic quadratic differential φdz 2 . We now let ¯ z 2 ) ∈ J (L) : Z ∈ Γ+ (E)}, J+ (L) = {(Z, φd¯ the space of complex Jacobi fields of the oriented double cover which correspond to deformations of the nonorientable minimal surface f0 . Note that when Σ is a projective plane or a Klein bottle, J+ (L), contains the subspace ¯ g = {Z ∈ O(L) : A (Z) = Z}, a real subspace of the space of holomorphic sections of L invariant under the conjugate linear endomorphism A , the real dimension of g being three or one in the two cases. Thus not all Jacobi fields for f descend to Jacobi fields for f0 , only those which are invariant under A . If f0 has no branch points, J+ (L) = g. On the other hand, any branch point for a nonorientable parametrized minimal surface f0 : Σ0 → M corresponds to two branch points on the oriented double cover f which are interchanged by A, and sections of J+ (L) take conjugate values at branch points pi and A(pi ). Thus the inductive argument for Proposition 5.3.8 yields a corresponding statement for the nonorientable parametrized minimal surface f0 : Σ0 → M if J (L) is replaced by J+ (L), ⎧ 2 ⎪ ⎨3 + 2ν if Σ0 = RP , dimR J+ (L) = 1 + 2ν if Σ0 = K 2 , ⎪ ⎩ 2ν if the oriented double cover of Σ0 has g ≥ 2, with ν denoting the total branching order of f0 , the real dimension contributed by each branch point on the nonorientable surface being two. Similarly, versions of Proposition 5.3.9 and Corollary 5.3.10 continue to hold. The real parts of elements of J+ (L) are invariant under the real involution A∗ , and hence are well-defined real tangential Jacobi fields on the

308

5. Generic Metrics

Figure 1. The overall strategy of the proof of the Bumpy Metric Theorem is to construct submanifolds like P ∅ of configuration space, and show that the projections from each of these submanifolds to the space of metrics is Fredholm.

nonorientable base Σ0 . The preceding discussion enables us to use the Riemann-Roch formula from Riemann surface theory on the oriented double cover to calculate the dimension of the space of real tangential Jacobi fields for nonorientable minimal surfaces with branch points.

5.4. Minimal surfaces without branch points 5.4.1. Overview of proof. In this section we prove the following proposition, which makes the provisional assumption that there are no branch points, and which we use as a model for the proof of the Bumpy Metric Theorem itself. Proposition 5.4.1. Suppose that M is a compact connected smooth manifold of dimension at least three. For a generic choice of Riemannian metric on M , every prime compact parametrized minimal surface f : Σ → M with no branch points is as nondegenerate as allowed by its connected group G of symmetries. If G is trivial, it is Morse nondegenerate for the two-variable energy E in the usual sense. If G is nontrivial, then all such minimal surfaces lie on nondegenerate critical submanifolds for E which are orbits for the G-action. Following the transversality arguments we presented earlier in the context of geodesics (see §2.6 and §2.7), there are two steps to the proof of this

5.4. Minimal surfaces without branch points

309

proposition. Let Met(M ) denote the space of Riemannian metrics on M . The first step is to show that the space of prime minimal surfaces without branch points, (5.38) P ∅ = {(f, ω, g) ∈ Map(Σ, M ) × T × Met(M ) : f is a prime conformal ω-harmonic immersion for g with no branch points} is a submanifold of the configuration space Map(Σ, M ) × T × Met(M ). Here the superscript ∅ denotes that the branch locus is empty. The second step is to show that the projection π : P ∅ → Met(M ) to the space of metrics is a Fredholm map of Fredholm index dΣ , where dΣ is the real dimension of the connected group G of symmetries: (1) dΣ = 6, when Σ = S 2 and G = P SL(2, C), (2) dΣ = 3, when Σ = RP 2 and G = SO(3), (3) dΣ = 2, when Σ = T 2 and G = S 1 × S 1 , (4) dΣ = 1, when Σ is the Klein bottle K 2 and G = S 1 , and (5) dΣ = 0, when Σ (or its oriented double cover) has genus ≥ 2. One then uses the Sard-Smale Theorem 2.5.2 to show that almost all metrics are regular values for the projection, and that for metrics which are regular values, the minimal surfaces satisfy the conclusion of the proposition. Later we will extend this argument to show that branch points can be eliminated by a suitable additional metric perturbation. 5.4.2. The Euler-Lagrange operator. The first step in proving Proposition 5.4.1 or the Bumpy Metric Theorem itself uses the first variation formula (4.38), which we now know holds for parametrized surfaces of arbitrary genus, to construct an Euler-Lagrange operator (5.39) F : (Map(Σ, M ) × Tg ) × Met(M ) −→ T (Map(Σ, M ) × Tg ) = Map(Σ, T M ) × T Tg with F (f, ω, g) ∈ T(f,ω) (Map(Σ, M ) × Tg ). This is defined by requiring that  ˙ Tg , (5.40) dE(f, ω) ((X, ω)) ˙ = F0 (f, ω, g), XdA + F1 (f, ω, g), ω Σ

where F0 : (Map(Σ, M ) × Tg ) × Met(M ) −→ Map(Σ, T M ), F1 : (Map(Σ, M ) × Tg ) × Met(M ) −→ T Tg ,

310

5. Generic Metrics

are the two components of F , and ·, ·Tg is the inner product on the tangent space to Teichm¨ uller space. It follows from (4.38) that the first component of F is      ∂f Dg ∂f 1 Dg + , F0 (f, ω, g) = − 2 λ ∂x1 ∂x1 ∂x2 ∂x2 where (x1 , x2 ) are isothermal coordinates with respect to ω, and D g denotes the covariant derivative with respect to the Levi-Civita connection determined by the Riemannian metric g on M , while the second component is given by / . ∂f ∂f , dz 2 ∈ Tω T . F1 (f, ω, g) = −Ωf = − ∂z ∂z For fixed choice of metric g ∈ Met(M ), the critical points for the energy E are the points (f, ω, g) such that F (f, ω, g) lies in the zero-section of the tangent bundle to Map(Σ, M ) × Tg . We can regard F as defining a section of the bundle E −→ Map(Σ, M ) × T × Met(M )

(5.41)

which has total space E = {(f, ω, g, X, ω) ˙ ∈ Map(Σ, M ) × T × Met(M ) × Map(Σ, T M ) × T T such that π(X) = f and π(ω) ˙ = ω}. If (f, ω, g) is a zero for F , we set ˙ is L2 -perpendicular (5.42) V = {(X, ω) ˙ ∈ Tf Map(Σ, M ) ⊕ Tω T : (X, ω) to the image of πV ◦ DF(f,ω,g) }, where πV is the vertical projection on the fiber, well-defined at zeros of F . We let D1 F denote the derivative of F at a critical point (f, ω, g) in the direction of Map(Σ, M ) × T ; this partial derivative is obtained by differentiating (5.40) to obtain (5.43) d2 E(f, ω) ((X1 , ω˙ 1 ), (X2 , ω˙ 2 ))  = L0 (f, ω, g)(X1 , ω˙ 1 ), X2 dA + L1 (f, ω, g)(X1 , ω˙ 1 ), ω˙ 2 Tg . Σ

Here L0 and L1 are the two components of the Jacobi operator Lf,ω,g = (πV ◦ D1 F )(f,ω,g) : Γ(f ∗ T M ) ⊕ Tω T −→ Γ(f ∗ T M ) ⊕ Tω T , which is defined implicitly by the second variation formula of Proposition 5.3.1, and extends to a Fredholm map Lf,ω,g = (πV ◦ D1 F )(f,ω,g) : L2k (f ∗ T M ) ⊕ Tω T −→ L2k−2 (f ∗ T M ) ⊕ Tω T , when k is large. The operator L covers all but the finite-dimensional space of Jacobi fields.

5.4. Minimal surfaces without branch points

311

Recall that when Σ (or its oriented cover) has genus zero or one, the space of Jacobi fields includes g(f, ω, g) ⊆ Tf Map(Σ, M ) ⊕ Tω T = E(f,ω,g) the space generated by the G-action, G being the group of symmetries of Σ. The idea behind the first step in the proof of Proposition 5.4.1 is to show that F is as transversal to the zero section as allowed by the G-action, and to do this it suffices to check that V = g. The following lemma, together with the discussion in §5.3.3, will accomplish this goal. But we have written the proof of the Lemma so that it aplies with no change to prime minimal surfaces with nontrivial branch locus, so that we can use it in the proof of the general case of the Bumpy Metric Theorem. Key Lemma 5.4.2. If (f, ω) is a prime minimal surface with possibly nontrivial branch locus, the only elements of T(f,ω) (Map(Σ, M ) × T ) that can be perpendicular to the image of every variation in the metric are the tangential Jacobi fields (X, ω), ˙ which correspond to elements of J (L) if Σ is oriented or J+ (L) if Σ is nonorientable. Proof. The idea behind the proof is to calculate the metric deformation operator , Pf,ω,g = −(πV ◦ D2 F )(f,ω,g) : L2k−1 (Met(M )) −→ L2k−2 (f ∗ T M ) ⊕ Tω T , the partial derivative with respect to the metric g ∈ Met(M ). We will divide D2 F into components D2 F0 and D2 F1 , and focus first on D2 F0 . Recall from Lemma 5.2.1 that prime minimal surfaces are somewhere injective. Thus the set W ⊂ Σ consisting of points p such that p is not a branch point and f −1 (f (p)) = p is open and dense by hypothesis. Let us choose an open neighborhood U of a point p ∈ W such that U ⊂ W and f imbeds U onto f (Σ) ∩ V for some open set V ⊂ M . (If Σ is nonorientable, we choose U to be an open subset of the oriented double cover of Σ such that U ∩ A(U ) is empty, and carry out the following calculations on the double cover.) Arrange, moreover, that V is the domain of local coordinates (u1 , . . . , un ) such that ui (f (p)) = 0 and (1) f (U ) is described by the equations u3 = · · · = un = 0, (2) ua ◦ f = xa on f (U ), for a = 1, 2, where x1 + ix2 is a holomorphic coordinate on U ,

312

5. Generic Metrics

(3) the canonical metric on Σ having constant curvature and total area one is expressed as  ·, ·0 = ηab dxa dxb , ηab = λ2 δab , for 1 ≤ a, b ≤ 2, where λ2 is a positive function, and (4) the Riemannian metric g on the ambient space takes the form   δir , for 1 ≤ i ≤ n and 3 ≤ r ≤ n, ·, · = gij dui duj , gij = 2 σ δij , for 1 ≤ i, j ≤ 2, when restricted to f (Σ) ∩ V , where σ 2 is a positive function. Such coordinates can be constructed using the exponential map restricted to the normal bundle of the surface f (Σ) ∩ V in M . With respect to a general Riemannian metric on V with metric components gij , the equation for harmonic maps can be written as 1 F0 (f, ω, g) = − 2 λ



∂ 2 uk ∂ 2 uk + ∂x21 ∂x22



   1 ∂ui ∂uj ∂ui ∂uj k Γ + − = 0, λ2 ij ∂x1 ∂x1 ∂x2 ∂x2 i,j

where the Γkij ’s are the corresponding Christoffel symbols. We call −F0 the tension of the harmonic map f in agreement with the terminology of Eells and Sampson [ES64]. If we specialize to our given choice of coordinates, the equation for harmonic maps simplifies to − Γk11 − Γk22 = 0, and we want to calculate the change in the expression for tension as we vary the metric. A perturbation in the metric g˙)∈ Tg Met(M ) with compact support in V can be written in the form g˙ = g˙ ij dui duj , where the g˙ ij ’s are smooth functions on V . Under such a perturbation, the only part of the tension that changes is the Christoffel symbol    1 ∂gil ∂gjl ∂gij k kl g Γl,ij , where Γl,ij = + − . Γij = 2 ∂uj ∂ui ∂ul If Γ˙ k,ij denotes the derivative of Γk,ij in the direction of the perturbation, 1 Γ˙ k,ij = 2



∂ g˙ jk ∂ g˙ ij ∂ g˙ ik + − ∂uj ∂ui ∂uk

 .

5.4. Minimal surfaces without branch points

313

From this formula, we can calculate that   n 1  ˙ ∂ui ∂uj ∂ui ∂uj (5.44) D2 F0 (f, ω, g), X (g) ˙ =− 2 + Γk,ij fk λ ∂x1 ∂x1 ∂x2 ∂x2 i,j,k=1

=−

1 λ2

n 

(Γ˙ k,11 + Γ˙ k,22 )f k ,

when

X=

n  i=1

k=1

fi

∂ ∂ui

is a fixed vector field along f . Thus according to definition (5.42) of V,   2 n  f k Γ˙ k,aa dx1 dx2 + D2 F1 (f, ω, g), ω ˙ (g) ˙ = 0, (X, ω) ˙ ∈V ⇒ − Σ k=1 a=1

for all perturbations g˙ in the metric. If we set g˙ 11 (u1 , . . . , un ) = g˙ 22 (u1 , . . . , un ) =

n 

ur φr (u1 , u2 ),

r=3

where φr is a smooth function on U with compact support, and let g˙ ij = 0 for the other choices of indices i, j, a straightforward calculation shows that the fiber projection of the partial derivative of F0 with respect to g is given by the expression ˙ = πV ◦ (D2 F0 )(f,ω,g) (g)

2 n n 1   ∂ g˙ aa ∂ 1  ∂ g˙ 11 ∂ = , 2λ2 ∂ur ∂ur λ2 ∂ur ∂ur r=3 a=1

r=3

˙ = 0 since the variation of the induced metric on while πV ◦ (D2 F1 )(f,ω,g) (g) Σ is via Weyl rescalings that do not change conformal structure. Thus we see that any vector field of the form n  ∂ φr (u1 , u2 ) , ∂ur r=3

where φr is a smooth function on U with compact support, lies in the image of πV ◦ (D2 F )(f,ω,g) . From this we can show that elements of V ⊗ C must restrict to sections of (f ∗ T M ⊗ C) over U . Since a dense open subset of points of Σ can be covered by sets of the form U and elements of V belong to the kernel of an elliptic operator and are hence smooth, (X, ω) ˙ ∈V

⇒

X ∈ Γ((f ∗ T M ⊗ C) ).

Using the above calculation, we could easily finish the proof if Map(Σ, M ) is replaced by the space of imbeddings, by replacing the Teichm¨ uller space in the domain of F by the space Met(Σ) and letting any tangential metric variation g˙ ∈ Met(M ) be tracked by a corresponding variation η˙ ∈ Met(Σ). But when f is only somewhere injective, η˙ can be an arbitrary variation, while g˙ must have support in U , an open dense subset of Σ.

314

5. Generic Metrics

It will be useful to have an explicit formula for the effect of a tangential variation in the metric. To determine the conditions satisfied by the tangential components of elements of X when (X, ω) ˙ ∈ V, we set (5.45) g˙ 11 = −g˙ 22 = φ1 , g˙ 12 = −φ2 , g˙ ij = 0 for other choices of indices i, j, where φ1 and φ2 have compact support within V , and a short calculation shows that 1 ∂ g˙ 11 ∂ g˙ 12 1 ∂ g˙ 22 ∂φ1 ∂φ2 + − = − , Γ˙ 1,11 + Γ˙ 1,22 = 2 ∂u1 ∂u2 2 ∂u1 ∂u1 ∂u2 ∂ g˙ 12 1 ∂ g˙ 11 1 ∂ g˙ 22 ∂φ2 ∂φ1 − + =− − . Γ˙ 2,11 + Γ˙ 2,22 = ∂u1 2 ∂u2 2 ∂u2 ∂u1 ∂u2 Hence in this case,  ∂ ∂ = fa ⇒ ∂ua ∂xa a=1 a=1      1 ∂φ2 ∂φ1 ∂φ1 ∂φ2 1 ˙ =− 2 − − f + − f2 D2 F0 (f, ω, g), X(g) λ ∂x1 ∂x2 ∂x1 ∂x2

(5.46) X =

and



0

(5.47) Σ

2 

2

fa

1 πV ◦ (D2 F0 )(f,ω,g) (g), ˙ X dA       ∂φ2 ∂φ1 ∂φ1 ∂φ2 1 − − f + − f 2 dx1 dx2 . =− ∂x1 ∂x2 ∂x1 ∂x2 Σ

In particular, f 1 (∂/∂x1 ) + f 2 (∂/∂x2 ) is perpendicular to the range of the map πV ◦ (D2 F0 )(f,g) if and only if       ∂φ2 ∂φ2 ∂φ1 1 ∂φ1 2 − − f +f − dx1 dx2 = 0 ∂x1 ∂x2 ∂x1 ∂x2 Σ or, equivalently, if and only if   1     1 ∂f 2 ∂f 2 ∂f ∂f − + φ1 − φ2 dx1 dx2 = 0, ∂x1 ∂x2 ∂x2 ∂x1 Σ for all choices of functions φ1 and φ2 with compact support in U , and this implies that f 1 (∂/∂x1 ) + f 2 (∂/∂x2 ) must be the restriction to U of a superposition of real or imaginary parts of a holomorphic section of L. Since U is dense in Σ, (X, 0) ∈ V ⇒ X ∈ O(L) ⊕ O(L), where O(L) is the space of holomorphic sections of L. When Σ is nonorientable, X must also be invariant under the conjugate linear map A , so in all cases the variations (X, 0) within V must lie in the space g of infinitesimal symmetries.

5.4. Minimal surfaces without branch points

315

In terms of the complex coordinate z = x1 + ix2 on Σ (or its oriented double cover), the tangential variation g˙ constructed above can be regarded as the real part of a quadratic differential φdz 2 = (φ1 + iφ2 )dz 2 . If   ∂ ∂ ∂ 1 2 −i Z = (f + if ) = 2(f 1 + if 2 ) , ∂x1 ∂x2 ∂z so that the real part of Z is the vector field X considered above, then (5.46) yields (5.48) 2 D2 F0 (f, ω, g), Z(g) ˙ =− 2 λ



∂ (f + if ) (φ1 + iφ2 ) ∂ z¯ 1

2

 =−

1 ∂φ dz(Z), λ2 ∂ z¯

where we have extended F0 (f, ω, g), · to be complex linear, and the integration by parts of the previous paragraph yields     ∂φ dz(Z) dx1 dx2 πV ◦ (D2 F0 )(f,ω,g) (g), ˙ ZdA = − (5.49) ∂ z¯ Σ Σ    DZ dx1 dx2 . = φdz ∂ z¯ Σ The fact that f is conformal, say f ∗ ·, · = σ 2 (dx21 + dx22 ), allows us to write /   / . . σ2 DZ 2 ∂f DZ ∂f ∂f , = which implies that dz = 2 , , ∂ z¯ ∂z 2 ∂ z¯ σ ∂ z¯ ∂ z¯ and comparison with (5.49) yields   πV ◦ (D2 F0 )(f,ω,g) (g), ˙ ZdA = (5.50) Σ

Σ

2 λ2

.

DZ φ ∂f , ∂ z¯ σ 2 ∂ z¯

/ dA.

To calculate D2 F1 , we let ¯ z 2 = (η˙ 11 + iη˙ 12 )d¯ z2 ψd¯ be an antiholomorphic quadratic differential and set η˙ 22 = −η˙ 11 . We then consider a smooth family of metrics t → g(t) = ·, ·t , and the corresponding family of Hopf differentials . . . / / / ∂f ∂f 1 ∂f ∂f ∂f ∂f , − , − 2i , dz 2 Ωf (t) = 4 ∂x1 ∂x1 t ∂x2 ∂x2 t ∂x1 ∂x2 t 1 = [(g11 (t) − g22 (t)) − 2ig12 (t)] dz 2 . 4 Using the real part of (4.35) as inner product, we find that ¯ z 2  = −ReΩf (t), ω ˙ ReF1 (f, ω, g(t)), ψd¯  1 [η˙ (g (t) − g22 (t)) + 2η˙ 12 g12 (t)]dx1 dx2 . =− 2 11 11 Σ λ

316

5. Generic Metrics

We differentiate this with respect to t, letting g˙ denote the derivative of g(t) at t = 0, obtaining the result (5.51)



1 [η˙ (g˙ − g˙ 22 ) + 2η˙ 12 g˙ 12 ]dx1 dx2 2 11 11 Σ λ  2 =− [η˙ g˙ + η˙ 12 g˙ 12 ] dA. 4 11 11 Σ λ

¯ z2 = − ˙ ψd¯ ReD2 F1 (f, ω, g)(g),

Since

.

∂f ∂f , ∂z ∂ z¯

/ =

σ2 , 2

we conclude that

/ . ¯ ψ ∂f φ ∂f , dA. (5.52) λ2 ∂z λ2 ∂ z¯ Σ ¯ z 2 is From (5.50) and (5.52) we conclude that if Z is a section of L and ψd¯ an antiholomorphic quadratic differential, then /  . ψ¯ ∂f φ ∂f DZ 2 ¯ − 2 , dA. ˙ (Z, ψd¯ z ) = 2 D2 F (f, ω, g)(g), ∂ z¯ λ ∂z λ2 σ 2 ∂ z¯ Σ ¯ z 2 ), then (X, ω) Thus, if (X, ω) ˙ = Re(Z, ψd¯ ˙ ∈ V; that is, (X, ω) ˙ is perpendicular to all variations in the metric, if and only if ψ¯ ∂f DZ = 2 , (5.53) ∂ z¯ λ ∂z on the open dense set U , but this implies that (X, ω) ˙ must be an element of the space of tangential Jacobi fields. QED ¯ z 2  = −Re ˙ ψd¯ ReD2 F1 (f, ω, g)(g),



2 σ2

Remark 5.4.3. In the process of the proof, we have obtained expressions (5.48) and (5.52) for the image of the tangential variation g˙ under the metric deformation operator , (5.54)

˙ = (−πV ◦ D2 F0 (f, ω, g)(g), ˙ −πV ◦ D2 F1 (f, ω, g)(g)) ˙ Pf,ω,g (g)   2 ∂φ ∂f , g˙ , = Re λ2 ∂ z¯ ∂ z¯

in which φdz 2 denotes the quadratic differential corresponding to the metric variation g. ˙ 5.4.3. The Fredholm projection. We now return to the proof of Proposition 5.4.1, which assumes that f is an immersion. Under this assumption, it follows from §5.3.3 that the only tangential Jacobi fields are those generated by the G-action, where G is the group of symmetries. It therefore follows from Lemma 5.4.2 that V is just the space g of vector fields generated

5.4. Minimal surfaces without branch points

317

by the G-action, and it follows easily that the space P ∅ defined by (5.38) is indeed a submanifold. Our next step consists of verifying: Lemma 5.4.4. If P ∅ is the manifold of triples (f, ω, g) where g ∈ Met(M ) and f : Σ → M is a conformal ω-harmonic immersion with no branch points, then the projection π : P ∅ → Met(M ) is a Fredholm map of Fredholm index dΣ , where dΣ is the real dimension of G. Proof. First we need to determine the tangent space to P ∅ at (f, ω, g), and this is a straightforward calculation which ends with the result: ˙ g) ˙ ∈ Tf Map(Σ, M ) ⊕ Tω T ⊕ Tg Met(M ) : (5.55) T(f,ω,g) P ∅ = {(X, ω, ˙ + (πV ◦ D2 F )(f,ω,g) (g) ˙ = 0}. (πV ◦ D1 F )(f,ω,g) (X, ω) We can also write the equation for the tangent space as (5.56)

L(X, ω) ˙ = P (g), ˙

where L = L(f,ω,g) is the Jacobi operator at (f, ω, g) and P = −(πV ◦ D2 F )(f,ω,g) is the deformation operator defined on variations g˙ ∈ Tg Met(M ), which specializes to the operator (5.54) for tangential variations g˙ described in the preceding section. We can calculate the kernel and the range of the map dπ(f,ω,g) : T(f,ω,g) P ∅ → Tg Met(M ), and we find that ˙ g) ˙ ∈ Tf Map(Σ, M ) ⊕ Tω T ⊕ Tg Met(M ) : Kernel of dπ(f,ω,g) = {(X, ω, g˙ = 0, L(X, ω) ˙ = 0}, while ˙ = L(X, ω), ˙ Range of dπ(f,ω,g) = {g˙ ∈ Tg Met(M ) : P (g) for some (X, ω) ˙ ∈ Tf Map(Σ, M ) ⊕ Tω T }. Recall that the Jacobi operator L = L(f,ω,g) : (L2k sections of f ∗ T M ) −→ (L2k−2 sections of f ∗ T M ) is Fredholm of Fredholm index zero, when k ≥ 2. Moreover, both the kernel of L and the L2 orthogonal complement to its range contain a dΣ -dimensional subspace generated by the G-action. The kernel of dπ(f,ω,g) is isomorphic to the kernel of L, while P (g) ˙ =0

⇒

L(0, 0) = P (g) ˙

⇒

g˙ ∈ Image(dπ(γ,g) ),

318

5. Generic Metrics

and hence the complement to the range of dπ(γ,g) must inject to a complement to the range of Lγ,g . In particular, the complement to the range of dπ(f,ω,g) is finite-dimensional and the projection dπ(f,ω,g) is a Fredholm map when restricted to P ∅ . Moreover, the L2 orthogonal complement to the range is isomorphic to the space of metric deformations h such that P (h) = −πV ◦ (D2 F )(f,ω,g) (h) is a Jacobi field. It therefore follows from Lemma 5.4.2 that that the codimension of the range of dπ(f,ω,g) is the dimension of the space of Jacobi fields which are orthogonal to g. Since the dimension of the kernel of dπ(f,ω,g) is the dimension of the space of Jacobi fields, we find that the Fredholm index of dπ(f,ω,g) is the dimension of g or dΣ . QED Finally, we use the Sard-Smale Theorem 2.5.2 to conclude that a generic subset of Met(M ) consists of regular values for the projection π on the second factor. If g0 is a regular value of the projection, π −1 (g0 ) consists of immersed minimal surfaces for g0 whose only Jacobi fields are the automatic ones generated by the action of G. Thus π −1 (g0 ) divides into dΣ -dimensional nondegenerate critical submanifolds of exactly the dimension forced by the symmetry, each such submanifold being a G-orbit of critical points. This finishes the proof of Proposition 5.4.1.

5.5. Minimal surfaces with simple branch points The crux of the proof is showing that branch points do not occur on prime minimal surfaces for generic metrics. To do this, we make use of the branch point stratification, which we now describe, focusing first on orientable surfaces. By a branch type for total branching order ν, we mean a finite sequence of positive integers Λ = (ν1 , . . . , νm ), such that ν1 ≥ ν2 ≥ · · · ≥ νm

m 

and

νi = ν.

i=1

We say that a conformal harmonic map f : Σ → M has ) branch type Λ if its divisor of branch points can be written as D = νi pi where Λ = (ν1 , . . . , νm ) has branch type Λ. We say that a branch type is simple if νi = 1 for all i. Note that there are only finitely many branch types Λ with total branching order |Λ| = ν, for fixed choice of ν ∈ Z. Define a partial order on branch types by demanding that  Λ = (ν1 , . . . , νm ) ≤ Λ = (ν1 , . . . , νm )

⇔

m ≤ m

and

νi ≤ νi

for 1 ≤ i ≤ m,

5.5. Minimal surfaces with simple branch points

319

and we write Λ < Λ if Λ ≤ Λ and Λ = Λ . There is a unique branch type ∅ of total branching order 0, and ∅ ≤ Λ for every branch type Λ. We would like to extend the argument for Proposition 5.4.1 from the space P ∅ of prime minimal surfaces without branch points to the space P of all prime minimal surfaces. The natural strategy for achieving this would be to show that the Euler-Lagrange map F : Map(Σ, M ) × T × Met(M ) −→ T (Map(Σ, M ) × T ) is as transversal to the zero section of the tangent bundle as allowed by the G-action at every point (f, ω, g) of P. Thus it would suffice to find variations g˙ in the metric such that the image of the map (5.57)

g˙

→

˙ P (g) ˙ = −πV ◦ D2 F (f, ω, g)(g)

covers the tangential Jacobi fields provided by the branch points, a complement to the space g of tangential fields generated by the G-action. In this case the intersection of the image of F with the zero section would contain the submanifold P, and the differential of the projection from P to the space Met(M ) of metrics would fail to be surjective at prime minimal surfaces with branch points, implying that such minimal surfaces do not exist for generic metrics. In a very rough sense, we might think of these metric variations as “killing” the Jacobi fields corresponding to branch points on a minimal surface. In this section, we construct a family of metric variations whose image under (5.57) covers the space of tangential Jacobi fields for simple branch points (but not the entire space of such Jacobi fields when a parametrized minimal surface f has branch points of branching order ≥ 2). This will extend the argument for Proposition 5.4.1 from the space P ∅ to the space (5.58) P s = {(f, ω, g) ∈ Map(Σ, M ) × T × Met(M ) : f is a prime conformal ω-harmonic immersion for g with simple branch points}, and give a proof of the Bumpy Metric Theorem under the additional assuption that all branch points are simple. Where might the variations which kill the tangential Jacobi fields come from? Since the Jacobi fields we want to eliminate are tangential, we expect that the variations in the metric should also be tangent to Σ. Moreover, we expect the variations to have a holomorphic description, so we should look for holomorphic or meromorphic sections (suitably modified) for some holomorphic line bundle over Σ. We mention that the stratification by branch type extends immediately to nonorientable surfaces. Each branch point pi of the nonorientable minimal surface f0 : Σ0 → M lifts to two branch points p˜i and pˆi of the oriented

320

5. Generic Metrics

double cover f which are interchanged by A, so if ν is the total branching order of f0 , the total branching order of f will be 2ν. 5.5.1. Conformally finite Riemann surfaces. A Riemann surface is conformally finite if it is conformally equivalent to a compact Riemann surface with finitely many points (called punctures) removed. Such a surface Σ0 is constructed from a compact oriented surface Σ of genus g by deleting m distinct points {p1 , . . . , pm }, where m > 0 and we will arrange that m ≥ 3 if g = 0. For a given simple divisor D = p1 + · · · + pm , we let Diff0,D (Σ) = {φ ∈ Diff0 (Σ) : φ(pi ) = pi for 1 ≤ i ≤ m}, where Diff0 (Σ) is the group of diffeomorphisms homotopic to the identity, and define the Teichm¨ uller space Tg,m for Riemann surfaces of genus g with m punctures by (5.59)

Tg,m =

Met0 (Σ) , Diff0,D (Σ)

where as usual Met0 (Σ) is the space of constant curvature metrics suitably normalized: we can take the curvature to be 1, 0 or −1, with area being one when the curvature is zero. This is a straightforward extension of the definition given in §4.3.1, according to which the usual Teichm¨ uller space with no punctures is T = Met0 (Σ)/G, where G is the gauge group, ⎧ ⎪ when g ≥ 2, ⎨Diff0 (Σg ), 2 G = Diff0 (T , p) when g = 1, ⎪ ⎩ 2 Diff0 (S , q, r, s) when g = 0. Here, p is a point on T 2 and q, r and s are distinct points on S 2 , providing examples of simple divisors D = p or D = q + r + s. The definition (5.59) of Teichm¨ uller space for Riemann surfaces with punctures is essentially that used by Robbin and Salamon [RoS06], except that these authors use the space of conformal structures Conf(Σ) in place of our space of constant curvature metrics Met0 (Σ). To determine the tangent space to Tg,m , we simply construct a slice S as in §4.3.1, the tangent space to the slice being the L2 orthgonal complement to the orbit of the Diff0,D (Σ)-action in the tangent space to Diff0 (Σ) at a given point. We find that elements of this orthogonal complement must have restrictions to Σ − {p1 , . . . , pm } which are holomorphic quadratic differentials with singularities allowed at the branch points. But the singularities are restricted to be at most simple poles by the requirement that we remain within the space of L2 -sections. Thus the argument for Theorem 4.3.2 extends to show that Tg,m is a smooth orientable manifold of real dimension

5.5. Minimal surfaces with simple branch points

321

6g − 6 + 2m. In fact, Teichm¨ uller’s theorem extends to conformally finite Riemann surfaces (see [Abi80] or [Ber81]), and shows that Tg,m is diffeomorphic to Euclidean space of dimension 6g − 6 + 2m so long as this number is positive. (There is a single conformal structure on the sphere minus three points.) As the reader might suspect, Teichm¨ uller theory for the theory of nonorientable surfaces f0 : Σ0 → M with punctures reduces to Teichm¨ uller theory for the oriented double cover. Each branch point pi of the nonorientable minimal surface f0 : Σ0 → M lifts to two branch points p˜i and pˆi of the oriented double cover f which are interchanged by the sheet interchange map A, and a diffeomorphism of Σ0 which fixes the divisor D0 = p1 + · · · + pm on Σ0 corresponds to a diffeomorphism of Σ which commutes with A and fixes the divisor D = p˜1 + · · · + p˜m + pˆ1 + · · · + pˆm . Therefore as Teichm¨ uller space for Σ0 with m punctures (where m > 0 and 2 m ≥ 3 if Σ0 = RP ) we can take {η ∈ Met0 (Σ) : A η = η} Met0 (Σ0 ) ∼ , = Diff0,D0 (Σ0 ) {φ ∈ Diff0,D (Σ) : φ ◦ A = A ◦ φ} which can be thought of as the fixed-point set of an action of the sheet interchange map on Tg,2m where g is the genus of the oriented double cover. It is convenient to have a description of Teichm¨ uller space which allows the points to vary. Such a description is ⎛ ⎞ m !" # ⎝Σ × · · · × Σ −Δ(m) ⎠ × Met0 (Σ) (5.60)

Tg,m ∼ =

Diff0 (Σ)

,

where Δ(m) is the “fat diagonal” consisting of m-tuples (p1 , . . . , pm ) for which any two of the pi ’s are equal. The extra tangential variations in Tg,m (2m real variations when Σ has genus at least two) can be thought of as follows: If we consider a fixed conformal structure on the compact surface Σ and allow the punctures {p1 , . . . , pm } to vary, we obtain a family of varying conformal structures on the punctured surface. This family can be realized by a family of diffeomorphisms of Σ, isotopic to the identity, which do not leave the given punctures fixed. (There is of course a description for the Teichm¨ uller space of a compact connected orientable surface with m punctures similar to (5.60).) Suppose now that f : Σ → M is a prime parametrized minimal surface, and suppose for simplicity that it has no singularities except branch points, and in particular has no self-intersections. Suppose, moreover, that there

322

5. Generic Metrics

are diffeomorphisms φ of Σ and φ˜ of M such that φ˜−1 ◦ f = f ◦ φ−1 . Then by invariance of the map E : Map(Σ, M ) × Met(Σ) × Met(M ) −→ R under diffeomorphims on both domain and range, we find that E(f, φ∗ η, g) = E(f ◦ φ−1 , η, g) = E(φ˜−1 ◦ f, η, g) = E(f, η, φ˜∗ g). Thus, metric deformations induced by diffeomorphisms of the punctured domain which extend to diffeomorphisms on M preserve energy. But what about a family of diffeomorphisms t → φt with φ0 = id of Σ which does not / Diff0,D (Σ) for t = 0? This gives a deformapreserve the divisor D, φt ∈ tion of the metric on Σ, and the corresponding deformation of the metric on the range can result in a change in the induced conformal structure on the punctured Riemann surface Σ − {p1 , . . . , pm }, where {p1 , . . . , pm } are the branch points, and these must perturb away the minimal surface, because they move away from the unique conformal structure which makes the original map conformal. Recall that the cotangent space to Tg,m consists of meromorphic quadratic differentials on the compact surface Σ which have the possibility of simple poles only at the punctures. This suggests that the metric deformations that kill the tangential Jacobi fields of f : Σ → M when the branch points are simple should be constructed from cutoffs of meromorphic differentials with simple poles. The argument carrying out this strategy is easiest when the minimal surface f : Σ → M has no self-intersections, but we will see that we can make it work when the self-intersection set is a smooth one-complex, as described in §5.2.2, by multiplying by suitable cutoff functions. 5.5.2. Eliminating simple branch points. For our construction of metric variations which kill branch points, we utilize a canonical form for a minimal surface in the neighborhood of a branch point. Recall that in terms of a local holomorphic coordinate z centered at a branch point p of order ν, we can write (∂f /∂z)(z) = z ν g(z) where g is a holomorphic section of f ∗ T M ⊗ C with g(0) = 0. Moreover, it follows directly from Taylor’s theorem that in terms of normal coordinates (u1 , . . . , un ) centered at f (p), we can express f as (5.61) (u1 + iu2 )(z) = cz ν+1 + o2 (|z|ν+1 ), ur (z) = o2 (|z|ν+1 ) for 3 ≤ r ≤ n, c being a nonzero complex constant, and by changing the complex coordinate we can arrange that c = 1. Here we use a commonly used extension of the

5.5. Minimal surfaces with simple branch points

323

“little o” notation, according to which a term v is o(|z|ν ) if v/|z|ν → 0 as |z| → 0 and ok (|z|ν ) is defined inductively for k ∈ N by v is ok (|z|ν )

⇔

v is o(|z|ν ) and its derivative is ok−1 (|z|ν−1 ).

Stronger results than (5.61) are found in the article of Micallef and White [MWh95]. The canonical form (5.61) with c = 1 implies that if we regard w = ¯ then w is closely u1 + iu2 as a complex coordinate on Σ with conjugate w, approximated by z ν+1 and, in fact, z, dw = [(ν + 1)z ν + o1 (|z|ν )] dz + o1 (|z|ν )d¯ dw ¯ = [(ν + 1)¯ z ν + o1 (|z|ν )] d¯ z + o1 (|z|ν )dz. ¯ by We can define sections ∂/∂w and ∂/∂ w ¯ of L ⊕ L       ∂ ∂ ∂ = 1, d w ¯ = 0, dw = 0, dw ∂w ∂w ∂w ¯

 dw ¯

∂ ∂w ¯

 = 1,

and derive asymptotic formulae for these sections near the branch point,   1 ∂ ∂ ∂ −ν = + o1 (|z|−ν ) , + o1 (|z| ) ν ∂w (ν + 1)z ∂z ∂ z¯   (5.62) 1 ∂ ∂ ∂ = + o1 (|z|−ν ) . + o1 (|z|−ν ) ∂w ¯ (ν + 1)¯ zν ∂ z¯ ∂z Recall that the Euler-Lagrange operator F of §5.4.2 divides into two components, F0 and F1 , with values in T Map(Σ, M ) and T T , respectively, F0 being minus the tension. To differentiate F0 , we utilize the formula   n ∂ui ∂uj ∂ 1  kl ˙ ∂ui ∂uj πV ◦ (D2 F0 )(f,ω,g) (g) ˙ =− 2 g Γl,ij + , λ ∂x1 ∂x1 ∂x2 ∂x2 ∂xk i,j,k=1

which follows directly from (5.44). The canonical form near the branch point implies that    (ν + 1)2 r2ν + o(r2ν ) if i = j = 1 or i = j = 2, ∂ui ∂uj ∂ui ∂uj + = ∂x1 ∂x1 ∂x2 ∂x2 otherwise, o(r2ν ) where r = |z|, and hence / .  ∂ ˙ fi dA (5.63) πV ◦ (D2 F0 )(f,ω,g) (g), ∂xi =−

2    (ν + 1)2 r2ν + o(r2ν ) (Γ˙ k,11 + Γ˙ k,22 )f k dx1 dx2 . k=1

324

5. Generic Metrics

We wish to restrict attention to trace-free tangential variations represented by quadratic differentials, φdz 2 = (φ1 + iφ2 )dz 2 = (g˙ 11 − ig˙ 12 )dz 2 ,

(5.64)

with

g˙ 22 = −g˙ 11 ,

where we set g˙ ij = 0 for other choices of the indices i, j. Recalling our earlier calculation of the variation in Christoffel symbols, we determine the analog of (5.47) for a variation g˙ supported near the branch point p to be 

0

(5.65) Σ

1 πV ◦ (D2 F0 )(f,g) (g), ˙ X dA      ∂φ1 ∂φ2 2 2ν 2ν (ν + 1) r + o(r ) f1 − =− ∂u ∂u 1 2 Σ    ∂φ2 ∂φ1 f 2 dx1 dx2 , + − − ∂u1 ∂u2

while to differentiate F1 , we use the inner product (5.51). After complexification, the derivative D2 F with respect to change in the metric can be represented by a map B : (trace-free tangential variations in Tg Met(M )) × (Γ(L) × Q(Σ)) −→ C, ¯ z2, where Q(Σ) is the space of antiholomorphic quadratic differentials ψd¯ defined in accordance with (5.51) by    1  0 2 ¯ 2 2 ¯ z ) = ˙ Z dA − φψdA, (5.66) B φdz , (Z, ψd¯ πV ◦ (D2 F )(f,g) (g), 4 Σ Σ λ and φdz 2 and ψdz 2 are expressed as metric variations g˙ and η˙ via (5.64). We recall that when f : Σ → M is a weakly conformal harmonic map with divisor of branch points D = ν1 p1 + · · · + νm pm , Proposition 5.3.9 provides us with a collection of tangential Jacobi fields, z 2 ) ∈ J (L) for 1 ≤ i ≤ n, such that (Zi , ψ¯i d¯

Zi (pj ) = δij ei ,

where ei is a fixed unit-length element in the fiber over pi . If we choose the complex coordinate zi on Σ near pi so that (∂f /∂zi ) = ziνi g, where g(pi ) has unit length, we can take ei = g(pi ). Then Zk has the expansion (5.67)

Zk = hki (zi )

∂f , ∂zi

where

hki (zi ) =

δki (1 + o1 (|zi |)). ziν

The following lemma constructs metric variations g˙ which kill one of these fields, hence a subspace of complex dimension one of the tangential Jacobi

5.5. Minimal surfaces with simple branch points

325

fields, for each branch point, in accordance with (5.66): Lemma 5.5.1. Suppose that f : Σ → M is a weakly conformal harmonic map with oriented domain Σ and divisor of branch points D = ν1 p1 + · · · + νm pm , and that f is an imbedding except for the singularities at the branch points. Then there is a collection φ1 dz 2 , . . . , φm dz 2 of smooth quadratic differentials on Σ with compact support lying within Σ − {branch points}, such that    1, if j = k, z 2 ) = δjk = (5.68) B φj dz 2 , (Zk , ψ¯k d¯ 0, if j = k, where B is defined by (5.66). Remark 5.5.2 on radial cutoff functions. Here is the idea behind the proof of Lemma 5.5.1 in the case where all branch points are simple. In this case, the quadratic differentials φ1 dz 2 , . . . , φm dz 2 are metric deformations which approximate meromorphic quadratic differentials with simple poles at the branch points, and these meromorphic quadratic differentials lift after raising an index to the tangent space to the Teichm¨ uller space of the punctured surface. But we want the branch points themselves to be fixed, so we multiply by a radial cutoff function ζ near each branch point. We have already seen the construction for a suitable cutoff function in the proof of Lemma 4.10.7 (see also [CS75]), a cutoff which sometimes allows us to ignore the contribution of a single point to a double integral over a surface. We first define a piecewise smooth map η : [0, ∞) → [0, ∞) by ⎧ ⎪ if r ≤ ε2 , ⎨0, (5.69) η(r) = 2 − (log r)/(log ε), if ε2 ≤ r ≤ ε, ⎪ ⎩ 1, if ε ≤ r, so that (5.70)



2π



lim

ε→0 0

0

while (5.71)

  2π 2π(ε − ε2 ) dη (r) rdrdθ = lim dr = lim = 0, dr ε→0 ε2 (log ε) ε→0 log ε

ε



2π



ε

lim

ε→0 0

0

b dη (r)rdrdθ = lim ε→0 r dr



2π



ε

b 0

0

dη (r)drdθ = 2πb. dr

If p ∈ Σ, we can  choose a local conformal coordinate z = x1 + ix2 centered at p and let r = x21 + x22 so that we can define the ε-ball Bε (p) about p as

326

5. Generic Metrics

the set of points satisfying r ≤ ε, for a small ε, 0 < ε < 1. The radial cutoff function we will use in our constructions is then the map ζ : B2ε (p) −→ [0, 1] defined by

ζ = η ◦ r.

Proof of Lemma 5.5.1. Let us assume first that Σ has genus at least two, leaving the modifications for genus zero and one to the end of the argument. Moreover, we first prove a slightly simpler provisional assertion regarding the map B0 : (trace-free tangential variations in Tg Met(M )) × Γ(L) −→ C,   1  0 2 defined by B0 φdz , Z = ˙ Z dA. πV ◦ (D2 F )(f,g) (g), Σ

We claim that: There is a collection φ1 dz 2 , . . . , φm dz 2 of smooth quadratic differentials on Σ with compact support lying within Σ − {branch points}, such that    1, if j = k, 2 (5.72) B0 φj dz , Zk = δjk = 0, if j = k, We choose one of the branch points pj and construct a meromorphic quadratic differential φj dz 2 with a pole of order one at pj and no other poles. (The existence of such a meromorphic quadratic differential, under the assumption that Σ has genus at least two, follows from the Riemann-Roch theorem.) We need to multiply this quadratic differential by a cutoff function near the poles so that it determines a well-defined deformation of the metric, and we do this in accordance with Remark 5.5.2. We thus construct a smooth cutoff function ζi : Σ → [0, 1] which is identically one outside the ε-ball about pi , identically zero on an ε2 -ball about pi , and depends only on the radial coordinate r centered at pi . We then let ζ = ζ1 · · · ζm , and set φj = ζφj , thereby obtaining a new quadratic differential φj dz 2 which is holomorphic except in small neighborhoods of the branch points, where it vanishes. Since we are are making the temporary assumption that f is an imbedding on the support of ζ, the quadratic differential φj dz 2 can indeed be realized by a deformation of the metric on the ambient manifold M . In accordance with (5.45), the components in the metric in local coordinates adapted to the conformal harmonic map vary by g˙ 11 = −g˙ 22 = φ1 ,

g˙ 12 = −φ2 ,

when

φj dz 2 = (φj1 + iφj2 )dz 2

with g˙ ij = 0, for other choices of the indices i, j.

5.5. Minimal surfaces with simple branch points

327

Indeed, if U is an open ball in Σ disjoint from the ε-balls about the branch points, we can apply the argument for the Key Lemma 5.4.2 over U , and since φj is holomorphic on U it follows from (5.48) that  πV ◦ (D2 F0 )(f,ω,g) (φj ), Zk dA = 0. U

  Thus the only contributions to the integral B0 φj dz 2 , Zk come from the small neighborhoods of the branch points on which the derivative of the cutoff function ζ is nonzero. We focus on a small neighborhood of pj in Σ on which we have a holomorphic coordinate z centered at pj , and on which we corresponding coordinate w = u1 + iu2 on M which approximates z ν+1 to which we can apply (5.62). Since φ dz 2 is a meromorphic differential, we find that ∂ (φ + iφ ) ∂w ¯ j1  j2   1 ∂ −ν −ν ∂ + o(|z| (φj1 + iφj2 ) = + o(|z| ) ) (ν + 1)¯ zν ∂ z¯ ∂z = o(|z|−ν−1 ), where the extra factor |z|−1 comes from the fact that φj1 + iφj2 = b/z + o(|z|−1 ) for some complex constant b, while its derivative with respect to z is o(|z|−1 ). Hence on a punctured neighborhood of the origin, we have      ∂φj1 ∂φj2 ∂φj2 ∂φj1 1 − − −i − = o(|z|−ν−1 ). 2 ∂u1 ∂u2 ∂u1 ∂u2 But we are interested in the variation defined by φ = (φj1 + iφj2 )dz 2 = ζφ = (ζφj1 + iζφj2 )dz 2 , where ζ is the radial cutoff, and in this case the calculation yields 1 ∂ ∂ζ (φj1 + iφj2 ) = (φj1 + iφj2 ) + ζ(r)o(|z|−ν−1 ) ν ∂w ¯ (ν + 1)¯ z ∂ z¯ 1 ∂r + ζ(r)o(|z|−ν−1 ). = (φj1 + iφj2 )ζ  (r) ν (ν + 1)¯ z ∂ z¯ In terms of polar coordinates z = reiθ ,   '  1 ∂ ∂ ∂r 2 2 = +i x1 + x2 ∂ z¯ 2 ∂x1 ∂x2 y 1 1 1 x +i = (cos θ + i sin θ) = eiθ , = 2 r r 2 2

328

5. Generic Metrics

while

1 be−iθ ζφj1 + iζφj2 = b + o(|z|−1 ) = + o(|z|−1 ), z r where b is a nonzero complex constant. Hence     ∂φj2 ∂φj1 ∂φj1 ∂φj2 1 −i − − − 2 ∂u1 ∂u2 ∂u1 ∂u2  bζ (r) 1 = + ζ(r)o(|z|−ν−1 ) 2r (ν + 1)¯ zν or, equivalently, (Γ˙ 1,11 + Γ˙ 1,22 ) − i(Γ˙ 2,11 + Γ˙ 2,22 ) =

1 bζ  (r) + ζ(r)o(|z|−ν−1 ). r (ν + 1)¯ zν

Using the expression (5.67) for Zk , we obtain  2   (Γ˙ 1,aa − iΓ˙ 2,aa )(hki (z)) a=1

1 bζ  (r) hki (z)) + ζ(r)o(|z|−2ν−1 ) r (ν + 1)¯ zν bζ  (r) 1 = hki (z)) + ζ(r)o(|z|−2ν−1 ). r (ν + 1)¯ zν =

The real part of the last expression is the integrand in (5.63), so substitution into the complex form of (5.63) yields  0 1 πV ◦ (D2 F0 )(f,ω,g) (φj ), Zk dA Bε (pj )





= −(ν + 1) 

Bε (pj )

 ζ  (r) r + ζ(r)o(1) drdθ (δjk b) r

(δjk b)ζ  (r)drdθ + Error(ε),

= −(ν + 1) B (pj )

an expression linear in both φ and Z, where Error(ε) → 0 as ε → 0. Thus  ε ζ  (r)dr = ζ(ε) − ζ(0) = 1 0  0 1 πV ◦ (D2 F0 )(f,ω,g) (φj ), Zk dA = −2πδjk (ν + 1)b. ⇒ lim ε→0 Bε (p ) j

Finally, we choose b so that 2π(ν + 1)b = −1. A similar calculation shows that  0 1 πV ◦ (D2 F0 )(f,ω,g) (φj ), Zk dA = 0, when i = j. lim ε→0 Bε (p ) i

5.5. Minimal surfaces with simple branch points

329

These calculations show that when we insert our collection Z1 , . . . , Zm of holomorphic sections of L and our collection φ1 dz 2 , . . . , φm dz 2 of quadratic differentials into B0 , the results converge to (5.72) as ε → 0. Choosing ε > 0 sufficiently small and replacing Z1 , . . . , Zm by suitable linear combinations of Z1 , . . . , Zm enables us to establish (5.72) exactly. Next, we replace the provisional assertion (5.72) by the assertion (5.68) in the statement of the lemma. To do this, we note that there was some freedom in constructing the quadratic differential φj dz 2 . Recall that it was constructed from a meromorphic differential φj dz 2 with a simple pole at pj and no other poles. To φj we could have added an arbitrary holomorphic quadratic differential on Σ without affecting B0 in the limit, because the contribution of a bounded holomorphic differential goes to zero. Indeed, suppose that Teichm¨ uller space T has dimension N and that ψ1 dz 2 , ψ2 dz 2 , . . . , ψN dz 2 is a basis for the holomorphic quadratic differentials at a point ω ∈ T . If φj dz 2 is obtained from a meromorphic differential with a single simple pole at pj and no other singularities by the cutoff process we described, there exist unique constants cji ∈ C such that ? > N  2 2 ¯ 2 = 0, for 1 ≤ k ≤ N , φj dz − cji ηi dz , ψk d¯ z i=1

where ·, · is the complex bilinear form defined by (4.35), which also appears in the second term of (5.66). This enables us to cancel the second term in B and achieve the assertion (5.68) of the lemma when Σ has genus at least two. In the case that Σ has genus zero or one, the preceding proof does not quite work because we may not be able to construct holomorphic differentials with poles at prescribed points. To remedy this problem in the case where Σ = S 2 , we introduce three additional points {q, r, s} not in the branch locus and allow the meromorphic differential φj dz 2 to have simple poles at these points; it is then possible to prescribe a meromorphic quadratic differential which has a single additional simple pole at any particular choice of the branch point of f . To compensate for this, we replace each Zk by Zk = Zk + Wk , where Wk is an element of the space g of infinitesimal symmetries corresponding to the G-action, chosen so that Zk (q), Zk (r) and Zk (s) all vanish. We multiply by cutoff functions near each of the points q, r and s and carry through the calculation we presented above. Assuming that the meromorphic differential φj dz 2 is not regular, but has a simple pole at q, the above calculation yields a zero contribution at q since Zk (q) = 0.

330

5. Generic Metrics

Since Zk (r) and Zk (s) also vanish, we find that the contributions at r and s are also zero. When Σ = T 2 the argument is similar, except we need only let the meromorphic differential φj dz 2 have an additional simple pole at a single predetermined point p ∈ T 2 . Once again, we replace each Zk by Zk = Zk + Wk , where Wk is an infinitesimal symmetry so that Zk (p) = 0, and find that this forces the contribution at p to be zero. This finishes the proof of Lemma 5.5.1. QED Finally, we show that the hypothesis of Lemma 5.5.1 can be weakened to allow for a self-intersection set as described in Lemma 5.2.5. There are two steps: We first show that we can replace each φi dz 2 by a metric variation which vanishes at finitely many points of self-intersection. We then show that we can also modify the variation at finitely many curves of selfintersection. Lemma 5.5.3. Given a finite subset F = {q1 , . . . , qN } ⊆ Σ, disjoint from the branch locus, and a trace-free tangential variation φdz 2 in Tg Met(M ) which is bounded at each point of F , we can construct a family of smooth functions ζε : M → [0, 1] such that each ζε φdz 2 vanishes in a neighborhood of every point of F , and     lim B0 ζε φdz 2 , Z = B0 φdz 2 , Z , ε→0

for all Z ∈ Γ(L). Proof. We can again make use of the radial cutoff functions described in Remark 5.5.2 (although the estimate is actually far more elementary this time). Thus we define ζi : Σ → R so that it is one outside an ε-neightborhood of the point qi ∈ F , and in terms of polar coordinates (ri , θi ) centered at qi , it is expressed by ζi = φ ◦ ri , where φ is defined by (5.69). We then let ζε = ζ1 · · · ζN , a cutoff function which vanishes at every element of F . The calculation (5.70) then shows that family {ζε : 0 < ε ≤ ε0 } has the properties necessary to establish the above limit. QED Lemma 5.5.4. Given a finite collection of disjoint C 1 curves of selfintersection F = {C1 , . . . , CN } ⊆ Σ, disjoint from the branch locus, and a trace-free tangential variation φdz 2 in Tg Met(M ) which is bounded at each point of F , we can construct a family of smooth functions ψε : M → [0, 1] such that each ψε φdz 2 vanishes in a neighborhood of every point of F , and     lim B0 ψε φdz 2 , Z = B0 φdz 2 , Z , ε→0

for all Z ∈ Γ(L).

5.5. Minimal surfaces with simple branch points

331

Proof. The proof is similar to that of the previous lemma, but now we multiply by a smooth cutoff function ψε : Σ → [0, 1] which vanishes along a given C 1 curve in the collection, which we denote by C. In a tubular neighborhod U of C we can choose a local coordinate system (s, t) such that s is arc length along C and |t| measures distance from C in the background metric on Σ. We then choose a C 1 function ψε (t) with values in [0, 1], depending only on t, such that  1, for |t| ≥ ε, ψε (t) = 0, for |t| ≤ ε/2, and consider the quadratic differential ψε φdz 2 near C. We need to show that the limit      lim B0 φdz 2 , Z − B0 ψε φdz 2 , Z ε→0  1 0 πV ◦ (D2 F )(f,g) ((1 − ψε )g˙ j ), Z dA = lim ε→0 Σ

is zero. But the integrand on the right is bounded with support in an ε-neighborhood of C. Thus as in the preceding lemma, we see that the perturbation once again does not alter the result. QED We can summarize the results of this section as follows: Proposition 5.5.5. If (f, ω) is a critical point for E such that f is prime and has divisor of branch points D = ν1 p1 + · · · + νm pm , then each branch point pi contributes a two-dimensional real space of tangential Jacobi fields which are killed by metric variations. Proof. This is an immediate consequence of Lemmas 5.5.1, 5.5.3 and 5.5.4 when Σ is oriented. When Σ is nonorientable, replace the statement of Lemma 5.5.1 with a slightly different assertion: Suppose that f0 : Σ0 → M is a weakly conformal harmonic map with nonorientable domain Σ and divisor of branch points D0 = ν1 p1 + · · · + νm pm , and that f is an imbedding except for the singularities at the branch points. Then there is a collection φ1 dz 2 , . . . , φm dz 2 of smooth quadratic differentials on the oriented double cover Σ of Σ0 with compact support lying within Σ − {preimages of branch points},

332

5. Generic Metrics

such that

 1, if j = k, 0, if j = k, z 2 ) are elements of where B is defined by (5.66) and the elements (Zk , ψ¯k d¯ J+ (L), tangential Jacobi fields invariant under the sheet interchange map A , as described in Extension 5.3.11. The proof of the new statement is exactly the same as the proof of Lemma 5.5.1 except that all deformations are required to be invariant under the sheet interchange map A. Thus, for example, the divisor of branch points for the double cover f : Σ → M is   B φj dz 2 , (Zk , ψ¯k d¯ z 2 ) = δjk =

p1 + pˆ1 ) + · · · + νm (¯ pm + pˆm ), D = ν1 (¯ where p¯i and pˆi are the two preimages of pi . Once one has the new version of Lemma 5.5.1, the proofs of Lemmas 5.5.3 and 5.5.4 apply for nonorientable surfaces with no change. QED It is clear from the construction that the metric variations constructed are exactly the variations which move a given branch point pi on Σ through a two-dimensional space, moving the puncture in a two-parameter family of directions on Σ and hence changing the conformal structure. 5.5.3. Bumpy metrics for simple branch points. We now have the means to prove the Bumpy Metric Theorem for prime minimal surfaces with only simple branch points, which is the key case: Proposition 5.5.6. Suppose that M is a compact connected manifold of dimension at least three. For a generic choice of Riemannian metric on M , every prime compact parametrized minimal surface f : Σ → M with all branch points of order ≤ 1 has in fact no branch points at all, and is as nondegenerate as allowed by its connected group G of symmetries. If G is trivial, all such minimal surfaces are Morse nondegenerate, while if G is nontrivial, then all such minimal surfaces lie on nondegenerate critical submanifolds for E which are orbits for the G-action. In fact, the proof is just a generalization of the proof of Proposition 5.4.1, in which we replace the space P ∅ defined by (5.38) by the subspace P s ⊆ Map(Σ, M ) × T × Met(M ) of (5.58). To establish Proposition 5.5.6, we need only check that the conclusion holds for elements of P s . We will show that because of Proposition 5.5.5, the argument for Proposition 5.4.1 now extends directly from P ∅ to P s . Once again, we study P s by means of the Euler-Lagrange map F : Map(Σ, M ) × T × Met(M ) −→ Map(Σ, T M ) × T T

5.5. Minimal surfaces with simple branch points

333

as in §5.4.2, which we again regard as a section of the vector bundle E = {(f, ω, g, X, ω) ˙ ∈ Map(Σ, M ) × T × Met(M ) × Map(Σ, T M ) × T T such that π(X) = f and π(ω) ˙ = ω}. The space P s consists of the points (f, ω, g) such that F (f, ω, g) = 0. If (f, ω, g) is a zero for F , we let ˙ is perpendicular (5.73) V = {(X, ω) ˙ ∈ Tf Map(Σ, M ) ⊕ Tω T : (X, ω) to the image of πV ◦ DF(f,ω,g) }, where πV is projection on the fiber, and we need to show that V = g, where g is the space of infinitesimal complex automorphisms generated by the G-action. But the Key Lemma 5.4.2 shows that all variations excepts for the tangential Jacobi variations are killed by metric variations, while Proposition 5.5.5 allows us to kill all Jacobi fields for prime minimal surfaces with simple branch points except for those coming from g. In other words, Image(DF(f,ω,g)) + g(f,ω,g) + T(f,ω,g) Z = T(f,ω,g,0,0) E, where Z is the zero section, and it follows from the implicit function theorem that P s is a submanifold as in the case of immersions. Thus it remains only to check that the argument of §5.4.3 continues to hold for parametrized minimal surfaces with simple branch points. We need to check that the projection π : P s → Met(M ) is a Fredholm map with Fredholm index being equal to dim G. But ˙ g) ˙ ∈ Tf Map(Σ, M ) ⊕ Tω T ⊕ Tg Met(M ) : T(f,ω,g) P s = {(X, ω, L(X, ω) ˙ = P (g)}, ˙ where L = L(f,ω,g) is the Jacobi operator at (f, ω, g) ∈ P s and P is the deformation operator which restricts to the operator (5.54) for tangential variations. As before we find that the kernel of dπ(f,ω,g) : T(f,ω,g) P s → Tg Met(M ) is isomorphic to the kernel of the Jacobi operator L and as before we find that P (g) ˙ =0

⇒

L(0, 0) = P (g) ˙

⇒

g˙ ∈ Image(dπ(γ,g) ),

so any complement to the range of dπ(γ,g) must inject to a complement to the range of L(f,ω,g) . Thus dπ(γ,g) is a Fredholm map, and a complement to the range is isomorphic to the space of metric deformations h such that P (h) is a Jacobi field. It follows that the range of dπ(f,ω,g) has finite-dimensional complement, while Lemma 5.4.2 and Proposition 5.5.5 imply that the complement of the range of dπ(f,ω,g) has the same dimension as the space of

334

5. Generic Metrics

Jacobi fields orthogonal to g. Since the dimension of the kernel of dπ(f,ω,g) is the dimension of the space of Jacobi fields, we find that the Fredholm index of dπ(f,ω,g) is the dimension of g, as in the case of immersions. Finally, we apply the Sard-Smale Theorem to conclude that the set of regular values of π : P s → Met(M ) forms a residual set. If g admits parametrized minimal surfaces with a nontrivial divisor of simple branch points, then those minimal surfaces have nontrivial Jacobi fields, which implies that g cannot be a regular value of the projection. Therefore such minimal surfaces cannot occur for generic choice of metric, and Proposition 5.5.6 is established. We remark that the nullity of any minimal surface with at least one simple branch point must be at least two, because any nonzero tangential ¯ z 2 ) within J (L) must have linearly independent real and Jacobi field (Z, ψd¯ imaginary parts. We will return to discuss implications of this point in Remark 5.6.6.

5.6. Higher order branch points To finish the proof of the Bumpy Metric Theorem, we need to extend our argument to branch points of order higher than one, and in this case the argument for Lemma 5.5.1 kills only a proper subspace of the space of extra tangent Jacobi fields, namely those arising from the tangent space to the punctured conformal surface Σ − {p1 , . . . , pm }, where p1 , . . . , pm are the branch points. Our goal is to show that the closed parametrized minimal surfaces of a branch type Λ for which some branch point has branching order at least two lie within a suitable submanifold of a configuration space which has a Fredholm projection with negative Fredholm index to the space of metrics on M . To construct this submanifold we employ the branch point stratification described at the beginning of §5.5. Definition. For a given branch type Λ, the configuration space on which ˜ m (Σ, M ) × Met(M ), where we will work is Am (Σ, M ) = M ⎛

⎞ m !" # Map(Σ, M ) × ⎝Σ × · · · × Σ −Δ(m) ⎠ × Met0 (Σ) (5.74)

˜ m (Σ, M ) = M

Diff0 (Σ)

with m being the number of branch points within Λ. Here Δ(m) denotes the fat diagonal consisting of m-tuples (p1 , . . . , pm ) for which no two of the elements are equal. The space of minimal surfaces of branch type Λ for

5.6. Higher order branch points

335

varying metrics is (5.75) P Λ = {([f, p, η], g) ∈ Am (Σ, M ) such that f is a prime conformal harmonic map with respect to η and g of branch type Λ with branch locus p = (p1 , . . . , pm )}. The map f determines the branch locus, so in fact we have a bijection from ˜ m (Σ, M ) to Map(Σ, M )×Tg,m , but we use (5.74) because we need to keep M track of the branch points. Our goal is to show that P Λ lies in the union of a countable collection of submanifolds, each of which has a Fredholm projection to Met(M ) with Fredholm index ≤ −2. The constructions in this section will not require Σ to be orientable; when Σ is nonorientable, local isothermal coordinates on Σ lift to a holomorphic coordinate on the oriented double cover. 5.6.1. Submanifolds of Map(Σ, M ). We start by describing for each branch type Λ, a corresponding space of maps within Map(Σ, M ), or more properly within ⎞ ⎛ m !" # Map(Σ, M ) × ⎝Σ × · · · × Σ −Δ(m) ⎠ . We define MapΛ (Σ, M ) to be the set of

  (f, (p1 , . . . , pm )) ∈ Map(Σ, M ) × Σ × · · · × Σ − Δ(m)

such that f is an immersion except at each point pi where the νi -jet of f is zero, while the (νi + 1)-jet is nonzero. (Saying that the ν-jet of f at p is zero means that all covariant derivatives of f of order between one and ν vanish at p.) We observe that it suffices to prove the Bumpy Metric Theorem for the part of the space Map(Σ, M ) for which α-energy is bounded by some positive constant, and it follows via Condition C from Morse theory for the α-energy that energy bounds yield bounds on index plus nullity. Thus we need only consider minimal surfaces with a fixed bound on the total branching order, which in turn gives a bound ν0 on the branching order of any branch point. Using the Sobolev imbedding theorem and the smoothness of the νi -jet evaluation map for manifolds of C k maps, as described in [AR67], Theorem 10.4, one checks that the L2k completion of MapΛ (Σ, M ) is a C k−ν0 −2 submanifold of finite codimension in   L2k (Σ, M ) × Σ × · · · × Σ − Δ(m) ,

336

5. Generic Metrics

and we can choose k large enough to make the evaluation map C  for any choice of  ∈ N. We next determine the tangent space to MapΛ (Σ, M ) in the case where Λ = (1, 1, . . . , 1) at a point (f, (p1 , . . . , pm )). Suppose that, for t ∈ (−, ), t → (f (t), (γ1 (t), . . . , γm (t))) is a variation in MapΛ (Σ, M ) with (f (0), (γ1 (0), . . . , γm (0))) = (f, (p1 , . . . , pm )) . About a given point pi we choose local isothermal coordinates (x1 , x2 ) on Σ such that x1 (pi ) = x2 (pi ) = 0. If we differentiate the defining relation df (t)(γi (t)) = 0 with respect to t and set t = 0, we obtain  ∂ 2f ∂ 2f (pi ) + (p)(xb ◦ γi ) (0) = 0. ∂xa ∂t ∂xa ∂xb 2

b=1

The tangent vector corresponding to this variation is (X, v1 , . . . , vm ) ∈ Tf Map(Σ, M ) ⊕ Tp1 Σ ⊕ · · · ⊕ Tpm Σ, where the components satisfy the conditions ∇X(pi ) = −d2 f (pi )(vi , ·), and we regard the two sides of this equation as linear maps from Tpi Σ to Tf (pi ) M . These conditions can also be obtained from the explicit derivatives of the one-jet evaluation map as described in Abraham and Robbin [AR67], page 27. In the general case where Λ = (ν1 , . . . , νm ), one can show by a similar argument that the tangent space consists of the elements (X, v1 , . . . , vm ) ∈ Tf Map(Σ, M ) ⊕ Tp1 Σ ⊕ · · · ⊕ Tpm Σ which satisfy the conditions ∇j X(pi ) = 0, for 1 ≤ j ≤ νi − 1 and ∇νi X(pi ) = −(dνi+1 f )(pi )(vi , ·, . . . , ·), where (dνi+1 f )(pi ) is a well-defined symmetric map since the lower derivatives vanish. The formula (f, (p1 , . . . , pm )) → (p1 , . . . , pm ) gives a map m

!" # Map (Σ, M ) −→ Σ × · · · × Σ −Δ(m) , Λ

in which the fiber over (p1 , . . . , pm ) is a submanifold MapΛ (Σ, M )(p1 , . . . , pm ) with tangent space consisting of X ∈ Tf Map(Σ, M ) such that ∇j X(pi ) = 0,

for 1 ≤ j ≤ νi .

5.6. Higher order branch points

337

Indeed, the manifold MapΛ (Σ, M ) is foliated by these fibers, submanifolds of finite codimension. We now construct a configuration space within Am (Σ, M ) adapted to the branch type Λ by imposing some further conditions on a point ((f, (p1 , . . . , pm ), η), g) ∈ MapΛ (Σ, M ) × Met0 (Σ) × Met(M ). Suppose that Λ = ν1 p1 + · · · + νm pm , focus on one of the points pi , and simplify notation by abbreviating (pi , νi ) to (p, ν). We use the metric g = ·, · on the ambient manifold M in formulating the next lemma: Lemma 5.6.1. There exists a finite collection of linear conditions on the (ν + 1)-jet of f at p such that the (ν + 1)-jet of f satisfies these conditions if and only if whenever z is a complex coordinate on Σ centered at p, then  (5.76)

jν

∂f ∂z

 = jν (z ν h(z)) ,

h(p), h(p) = 0,

h(p) = 0,

where jν denotes ν-jet at p, and h is a section of f ∗ T M ⊗ C defined over some open neighborhood U of p. Note: Condition (5.76) just says that if f is conformal on Σ0 , then we can extend ∂f /∂z to an isotropic section over Σ so that at each pi , ∂f = z νi h, ∂z where z is a complex coordinate on some neighborhood of pi which is centered at pi , and h is isotropic and nonzero at p. Proof of lemma. It is convenient to express these conditions in terms of canonical isothermal coordinates (x1 , x2 ) centered at p ∈ Σ for the standard constant curvature metric on Σ, and normal coordinates (u1 , . . . , um ) on M determined by the Riemannian metric on M and centered at f (p) ∈ M . The jet conditions (5.76) are then expressible in terms of the coefficients of the Taylor expansions. We can consider the (ν + 1)-order differential dν+1 f : Tp Σ × · · · × Tp Σ −→ Tf (p) M as a polynomial approximation to f near p, which we can write as u = (u1 (x1 , x2 ), u2 (x1 , x2 ), . . . , um (x1 , x2 )).

338

5. Generic Metrics

If x1 + ix2 = r cos θ, we can also write this as ⎡



ui (x1 , x2 ) = rν+1 ⎣ =r

ν+1

 

⎤ aijk cosj θ sink θ⎦

j,k≥0,j+k=ν+1

√

cijk exp

−1(j − k)θ 2



 : j + k = ν + 1, j ≥ 0, k ≥ 0

,

where the cijk ’s are complex constants satisfying the conditions cijk = c¯ikj . The jet condition (5.76) can now be expressed in terms of the vectors cν+1−j = (c1j,ν+1−j , . . . , cmj,ν+1−j ), cj = ¯

0 ≤ j ≤ ν + 1,

as (5.77)

c0 · c0 = 0,

c0 = ¯ cν+1 = 0,

cj = 0

for j = 0, ν + 1.

Note that these conditions are invariant under change of normal coordinates on M , and they imply that the Fourier expansion of the rν+1 -term in ui (r, θ) is a linear combination of cos((ν + 1)θ) and sin((ν + 1)θ). QED The conditions we have described are invariant under the action of the group Diff0 (Σ) of diffeomorphisms isotopic to the identity, so we can pass to the quotient in (5.74) obtaining a submanifold AΛ (Σ, M ) = {([f, (p1 , . . . , pm ), η], g) ∈ An (Σ, M ) : (f, (p1 , . . . , pm )) ∈ MapΛ (Σ, M ) and conditions (5.77) hold at every pi }. One easily checks that AΛ (Σ, M ) is a submanifold of finite codimension in Am (Σ, M ) which contains all conformal harmonic maps f : Σ → M of branch type Λ. Moreover, at each point of AΛ (Σ, M ) we have a well-defined family (5.78)

([f, (p1 , . . . , pm ), η], g)

→

(E1 (p1 ), . . . , Em (pm ))

of two-planes (each Ei (pi ) varying smoothly because the (νi + 1)-jet evaluation map is smooth), such that Ei (pi ) is spanned by the image of the (νi + 1)-jet of f at pi , for 1 ≤ i ≤ m. The key feature of the submanifold AΛ (Σ, M ) is that it comes equipped with the family of two-planes (5.78), one at each point pi , the elements of which are automatically the images of Re(L)(pi ) (on the oriented double cover) whenever the element of AΛ (Σ, M ) is a conformal harmonic map.

5.6. Higher order branch points

339

5.6.2. Submanifolds of configuration space containing P Λ . Given a branch type Λ = (ν1 , . . . , νm ) which is not simple, we want to cover the space P Λ of prime minimal surfaces of branch type Λ by submanifolds lying in a countable open cover of Am (Σ, M ), submanifolds which have Fredholm projection with negative Fredholm index to the space of metrics on M . If the genus of Σ (or its oriented two-fold cover) is at least two, we do not have to deal with a positive-dimensional group G of symmetries, so we treat that simpler case first. Then we describe the minor modifications that are necessary for the remaining cases. Proposition 5.6.2. Suppose that Σ (or its oriented double cover) has genus ≥ 2. If Λ = (ν1 , . . . , νm ) is a given branch type of total branching order ν, then there is a countable open covering {Wi : i ∈ N} of P Λ by open sets in Am (Σ, M ), and for each i ∈ N a submanifold m QΛ i ⊆ Wi ⊆ A (Σ, M ) Λ such that P Λ ∩ Wi ⊆ QΛ i and the projection π : Qi → Met(M ) is a proper Fredholm map of Fredholm index zero.

Proof. To prove this we start with a given conformal harmonic map ˜ m (Σ, M ) × Met(M ) ([f0 , p0 , η0 ], g0 ) ∈ P Λ ⊆ Am (Σ, M ) = M of branch type Λ = (ν1 , . . . , νm ). It suffices to construct an open neighborhood U of this element of Am (Σ, M ) and a submanifold QΛ ⊆ U ⊆ Am (Σ, M ) so that all conformal harmonic maps of branch type Λ within U lie in QΛ and the projection from QΛ to Met(M ) is Fredholm of Fredholm index zero. We can assume that U is the domain for a submanifold chart for the submanifold AΛ (Σ, M ) constructed in the previous section. Thus we can assume that the model Banach space for Am (Σ, M ) is a direct sum E ⊕ F , where F is finite-dimensional and that we are given a submanifold chart φ : U −→ E ⊕ F,

with φ(AΛ (Σ, M ) ∩ U ) = φ(U ) ∩ (E ⊕ {0}),

the dimension of F being the codimension of AΛ (Σ, M ) in Am (Σ, M ). Moreover, we can assume that φ ([f0 , p0 , ω0 ], g0 ) = 0. Within the open set U , we want to obtain a submanifold of lower codimension than AΛ (Σ, M ) ∩ U . To do this, we first use the submanifold chart described above to extend the two-planes (5.78) from the submanifold AΛ (Σ, M ) ∩ U to all of U . We then utilize complex coordinates zi on Σ (or its double cover) in a neighborhood of each pi , and the Levi-Civita

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connection D for the Riemannian metric g on M . If ΠE(pi ) denotes the orthogonal projection onto the two-plane E(pi ), we can then impose the conditions that (5.79) ΠE(pi )

∂f D2 f D νi f (pi ) = ΠE(pi ) 2 (pi ) = · · · = ΠE(pi ) νi (pi ) = 0, ∂zi ∂zi ∂zi ΠE(pi )

D νi +1 f (pi ) = 0, ∂ziνi +1

where νi is the branching order at pi . (The inequality in (5.79) is included to eliminate branch types which are > Λ.) Thus we impose only 2νi real linear conditions at the branch point pi . Imposing the conditions at all of the branch points yields a new configuration space B Λ (Σ, M ), a submanifold of codimension 2ν in Am (Σ, M ) ∩ U where ν is the total branching order, such that AΛ (Σ, M ) ∩ U ⊆ B Λ (Σ, M ) ⊆ Am (Σ, M ) ∩ U . A conformal harmonic maps of branch type Λ within U lies within B Λ (Σ, M ) because the differentials of f at each branch point pi must vanish up to order νi . Moreover, the codimension of B Λ (Σ, M ) exactly equals the dimension of the space of tangential Jacobi fields that are introduced by the branch points. By Proposition 5.3.9, the conditions (5.79) prevent any elements ¯ z 2 ) described from the space of tangential Jacobi fields (X, ω) ˙ = Re(Z, ψd¯ Λ in §5.3.3 from lying in the tangent space to B (Σ, M ). This implies that any conformal harmonic map within B Λ (Σ, M ) ⊆ Am (Σ, M ) ∩ U must have branch type Λ. To apply the technology of §5.4.2, we replace the space U ⊆ Am (Σ, M ) by its preimage   U˜ ⊆ Map(Σ, M ) × Σm − Δ(m) × Tg × Met(M ), under the quotient map which divides by the Diff0 (Σ)-action, which contains the point (f0 , p0 , ω0 , g0 ). As in §5.4.2, the Euler-Lagrange map (5.80)

F : Map(Σ, M ) × Tg × Met(M ) −→ Map(Σ, T M ) ⊕ T Tg

defines a section of the smooth bundle E defined in (5.41) over Map(Σ, M ) × Tg × Met(M ) in such a way that conformal harmonic maps are just the elements of F −1 (Z), where Z is the image of the zero section in E, and we can pull the bundle E back to U˜ . Differentiation of the Euler-Lagrange map (5.80) yields the Jacobi operator L(f,ω,g) = πV ◦ (D1 F )(f,ω,g) , which is Fredholm of index zero (since we are in the special case where G is trivial). Since (f0 , ω0 , g0 ) has branch type Λ of total branching order ν, (5.81)

L(f0 ,ω0 ,g0 ) : Tf0 Map(Σ, M ) ⊕ Tω0 Tg −→ Tf0 Map(Σ, M ) ⊕ Tω0 Tg

5.6. Higher order branch points

341

has a (2ν)-dimensional space of tangential Jacobi fields, (5.82)

Re (J (L)) = Span{(X1 , ω˙ 1 ), . . . , (X2ν , ω˙ 2ν )},

and Jacobi fields in this space are smooth by elliptic regularity. The (2ν)dimensional space spanned by these Jacobi fields can be translated about the neighborhood U˜ thereby generating a subbundle F ⊆ E of fiber dimension 2ν. We can then define an L2 -orthogonal projection p : E −→ F ⊥ to the L2 -orthogonal complement F ⊥ to F in E. The bundles are invariant under the action of Diff0 (Σ), hence project to bundles over AΛ (Σ, M ) ∩ U . Let Z be the image of the zero-section in the restriction of F ⊥ to B Λ (Σ, M ). We claim that p ◦ F Λ is transverse to Z at the point (f0 , ω0 , g0 ), and hence transverse to Z in some neighborhood B Λ (Σ, M )∩W of (f0 , ω0 , g0 ), with W ⊆ U . From this we will conclude that Q = (p ◦ F Λ )−1 (Z) ∩ W is a submanifold of W which contains all of the conformal harmonic maps of branch type Λ within W. To prove transversality at (f0 , ω0 , g0 ), we note that we can use our original Jacobi operator L to define a corresponding operator ˜ m (Σ, M ) −→ T(f,(p ,...,p ),ω) M ˜ m (Σ, M ) & : T(f,(p ,...,p ),ω) M L m m 1 1 & = L⊕I, which is also Fredholm of index zero and divides into a direct sum L where I : Tp1 Σ ⊕ · · · ⊕ Tpm Σ −→ Tp1 Σ ⊕ · · · ⊕ Tpm Σ & to our new configis just the identity map. We next restrict the operator L uration space to obtain ˜ n (Σ, M ), LΛ : T(f,p,ω) B Λ (Σ, M ) −→ T(f,p,ω) M a Fredholm operator of Fredholm index −2ν, because we have cut down the dimension of the domain with the 2ν real linear conditions described in (5.79). Finally, we compose with the projection into F ⊥ obtaining a map (5.83)

⊥ p ◦ LΛ : T(f,p,ω) B Λ (Σ, M ) −→ F(f,p,ω) ,

which has Fredholm index zero at (f0 , ω0 , g0 ) since we have eliminated spaces of equal dimension from the kernel and cokernel. Since (f0 , ω0 , g0 ) is prime, it is somewhere injective, hence injective on an open dense subset of Σ, so we can apply Key Lemma 5.4.2 to show that the derivative with respect to the metric πV ◦ D2 F (f0 , ω0 , g0 ) : Tg0 Met(M ) −→ Tf0 Map(Σ, M ) ⊕ Tω Tg0

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5. Generic Metrics

covers the complement of the space spanned by the tangential Jacobi fields (5.82) in the space of all Jacobi fields. On the other hand, the tangential Jacobi fields (5.82) get taken to zero under the projection p, so ⊥ p ◦ πV ◦ D2 F Λ (f0 , ω0 , g0 ) : Tg0 Met(M ) −→ F(f 0 ,ω0 ,g0 )

covers a complement to the range of the projected Jacobi operator of (5.83). This implies transversality at the point (f0 , ω0 , g0 ). Transversality is an open condition, so F Λ |(B Λ (Σ, M ) ∩ W) must be transversal to Z when W is a sufficiently small open neighborhood of (f0 , ω0 , g0 ) within U ⊆ An (Σ, M ). Thus   Q = (p ◦ F Λ )−1 (Z) ∩ B Λ (Σ, M ) ∩ W is indeed a submanifold containing all points of P Λ ∩ W. It remains only to show that Q has Fredholm projection to the space of metrics with Fredholm index zero. Recall that Fredholm maps are locally proper by Theorem 1.6 of [Sma66]. Thus, since the Sobolev completions have countable bases, it will then follow that we can cover P Λ by a countable collection of such open sets Wi satisfying all the conditions of Proposition 5.6.2, thereby finishing its proof. The proof that the projection is Fredholm of Fredholm index zero is a straightforward modification of the argument for Lemma 5.4.4. Indeed, ˙ g) ˙ ∈ T(f,ω,g) B Λ (Σ, M ) : (5.84) T(f,ω,g) Q = {(X, ω, ˙ + p ◦ πV ◦ (D2 F )(f,ω,g) (g) ˙ = 0}, p ◦ LΛ (X, ω) where the Jacobi operator p ◦ LΛ is of Fredholm index zero, with Kernel of (p ◦ LΛ ) = J ∼ = (complement to Range of (p ◦ LΛ )), where J is a finite-dimensional space of nontangential Jacobi fields. For the differential of the projection, dπ(f,ω,g) : T(f,ω,g) Q → Tg Met(M ), we find that

Ker(dπ(f,ω,g) ) ∼ = Ker(p ◦ LΛ ),

while the range of dπ(f,ω,g) consists of the elements ˙ ∈ Range(p ◦ LΛ ), g˙ ∈ Tg Met(M ) such that P (g) where

˙ P (g) ˙ = −πV ◦ (D2 F )(g)

is the metric deformation operator. Thus the range of dπ(f,ω,g) is the preimage of a closed space of finite codimension, hence a closed subspace of finite codimension itself. It follows that dπ(f,ω,g) is indeed a Fredholm map, and the dimension of the cokernel of dπ(f,ω,g) is no larger than the dimension

5.6. Higher order branch points

343

of the cokernel of L. But tranversality shows that d(p ◦ F Λ )(f,ω,g) is surjective, so anything not in the image of (p ◦ LΛ ) must be covered by T and the cokernels of (p ◦ LΛ ) and dπ must have equal dimension. Hence dπ(f,ω,g) is a Fredholm map of Fredholm index zero, finishing the proof of Proposition 5.6.2. QED It remains to discuss the cases in which Σ has genus zero or one, and so has a positive-dimensional group G of symmetries. In these cases, we need to modify the previous statement a little and utilize a construction that is also useful in other contexts: Proposition 5.6.3. Suppose that Σ has genus zero or one, and Λ = (ν1 , . . . , νm ) is a given branch type. Then there is a countable collection {Wi : i ∈ N} of open sets in Am (Σ, M ), and for each i ∈ N a submanifold ˜ m (Σ, M ) × Met(M ) QΛ ⊆ Wi ⊆ Am (Σ, M ) = M i

such that every element of P Λ lies on the G-orbit of some element in QΛ i , → Met(M ) is a proper Fredholm map for some i, and the projection π : QΛ i of Fredholm index zero. Proof. To prove this, we modify the argument of Proposition 5.6.2 to allow for the G-action. Suppose first that Σ is the two-sphere S 2 . If N is a compact codimension two submanifold of M with boundary ∂N , we let (5.85) U (N ) = {f ∈ L2k (S 2 , M ) : f does not intersect ∂N and has nonempty transversal intersection with the interior of N }, which is an open subset of L2k (S 2 , M ). Given three disjoint compact codimension two submanifolds with boundary, say Q, R and S, we let (5.86)

U (Q, R, S) = U (Q) ∩ U (R) ∩ U (S),

also an open subset of L2k (S 2 , M ). We cover L2k (S 2 , M ) with a countable collection of sets U (Qj , Rj , Sj ) defined by a sequence j → (Qj , Rj , Sj ) of triples of such codimension two submanifolds, where j ∈ N. We choose three points q, r and s in S 2 and let (5.87) Fj (S 2 , M ) = {f ∈ U (Qj , Rj , Sj ) : f (q) ∈ Qj , f (r) ∈ Rj , f (s) ∈ Sj }. When Σ = S 2 , Fj (Σ, M ) meets each P SL(2, C)-orbit in U (Qj , Rj , Sj ) in a finite number of points. It follows from the Sobolev imbedding theorem and smoothness of the evaluation map on the space of C k−2 maps (for example, by Proposition 2.4.17 of [AMR88]) that the evaluation map ev : C k (S 2 , M ) × Σ −→ M,

defined by

ev(f, p) = f (p),

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5. Generic Metrics

is C k−2 . Thus when k is large, we can regard Fj (Σ, M ) as a submanifold of Map(S 2 , M ) of codimension six with tangent space Tf Fj (Σ, M ) = {sections X of f ∗ T M : X(q) ∈ Tf (q) Qj , X(r) ∈ Tf (r) Rj , X(s) ∈ Tf (s) Sj }. Similarly, in the case where Σ is the torus T 2 , we define U (N ) by (5.85) and choose a sequence j → Nj of smooth compact codimension two submanifolds of M such that U (Nj ) cover Map(T 2 , M ). Finally, we choose a base point p ∈ Σ to break the symmetry, and let (5.88)

Fj (T 2 , M ) = {f ∈ U (Nj ) : f (p) ∈ Nj }.

We can perform similar constructions to break the SO(3) symmetry of L2k (RP 2 , M ) and the S 1 symmetry of L2k (K 2 , M ), where K 2 is the Klein bottle. Alternatively, we can use the extended functions E : M(RP 2 , M ) −→ R and

E : M(K 2 , M ) × TK −→ R

of Remark 4.3.11 to restore the larger symmetry groups P SL(2, C) and S 1 × S 1 . Thus for real projective planes, we choose N as in (5.85) and let U (N ) = {(f0 , x) ∈ M2k (RP 2 , M ) : f0 does not intersect ∂N and has nonempty transversal intersection with the interior of N }, where M2k (RP 2 , M ) is the L2k completion of M(RP 2 , M ), choose three distinct points q0 , r0 and s0 in RP 2 , and let Fj (RP 2 , M ) = {(f0 , x) ∈ U (Qj ) ∩ U (Rj ) ∩ U (Sj ) : f0 (q0 ) ∈ Qj , f0 (r0 ) ∈ Rj , f0 (s0 ) ∈ Sj }. If q, r and s are preimages of q0 , r0 and s0 in S 2 , {(f0 , x) ∈ Fj (RP 2 , M ) : E(f0 ) = 0} ∼ = {f ∈ Fj (S 2 , M ) : f ◦ Ax(f ) = f, E(f ) = 0}, where Fj (S 2 , M ) is defined by (5.87). A similar construction yields open sets Fj (K 2 , M ) for the Klein bottle. We next construct local configuration spaces   m (m) (Σ, M ) = F (Σ, M ) × Σ − Δ × Tg × Met(M ) Am j j such that every conformal harmonic maps f : Σ → M of branch type Λ has a G-orbit which intersects one of the spaces Am j (Σ, M ). (Of course, if Σ is nonorientable, we must replace Map(Σ, M ) by M(Σ, M ) in the definition of the configuration space.) The constructions in the proof of Proposition 5.6.2 Λ can be applied to each Am j (Σ, M ) yielding a submanifold Aj (Σ, M ) of finite codimension in Am j (Σ, M ) as before.

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345

Finally, we run through the proof of Proposition 5.6.2 starting with a conformal harmonic map m ([f0 , p0 , η0 ], g0 ) ∈ P Λ ∩ AΛ j (Σ, M ) ⊆ Aj (Σ, M ).

Exactly the same proof now leads to a countable covering {Wi : i ∈ N} of P Λ ∩ Anj (Σ, M ) by open sets in Am j (Σ, M ), together with a countable collection of submanifolds, m QΛ i ⊆ Wi ⊆ Aj (Σ, M ),

such that Λ P Λ ∩ Wi ∩ Am j (Σ, M ) ⊆ Qi

and each projection QΛ i → Met(M ) is a proper Fredholm map of Fredholm index zero. This is exactly what is needed to establish Proposition 5.6.3. QED Corollary 5.6.4. Suppose that Σ is a compact connected Riemann surface and Λ is a given branch type. For generic choice of metric on M , there are only countably many G-orbits of prime minimal surfaces (f, ω) from Σ into M of branch type Λ. This is a comforting intermediate step, but of course will be replaced soon by a much stronger result—there are no minimal surfaces with nontrivial branch type for generic choice of metric. 5.6.3. Bumpy metrics for higher order branch points. To finish the proof of the Bumpy Metric Theorem, we need only treat the remaining case of higher order branch points. Proposition 5.6.5. Suppose that Λ = (ν1 , . . . , νm ) is a given branch type, Λ = ∅. Then there is a countable open cover {Wi : i ∈ N} of P0Λ ⊆ Am (Σ, M ), and for each i ∈ N a submanifold n RΛ i ⊆ Wi ⊆ A (Σ, M )

such that every element of P0Λ lies on the G-orbit of some element in RΛ i , → Met(M ) is a proper Fredholm map for some i, and the projection π : RΛ i of Fredholm index −2m, where m is the number of branch points. Proof. The proof of Proposition 5.6.5 is a very slight modification of the proofs for Propositions 5.6.2 and 5.6.3. Assume first Σ has genus at least two. As before, we start with a given conformal harmonic map ˜ m (Σ, M ) × Met(M ) ([f0 , p0 , η0 ], g0 ) ∈ P0Λ ⊆ Am (Σ, M ) = M

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5. Generic Metrics

of branch type Λ, and construct an open neighborhood U of this element in Am (Σ, M ) and a corresponding submanifold RΛ ⊆ QΛ ⊆ U ⊆ Am (Σ, M ), where QΛ is the submanifold constructed in Proposition 5.5.2, so that all conformal harmonic maps of branch type Λ within U lie in RΛ , and the projection from RΛ to Met(M ) is Fredholm of Fredholm index −2m. The construction proceeds at first exactly like the proof of Proposition 5.5.2, the first change being that we replace the space Re (J (L)) in (5.82) by the smaller space Re (J0 (L)), Re (J0 (L)) being the kernel of the map Ψ0 defined by (5.37). As before, this 2(ν − m)-dimensional space can be translated about the coordinate neighborhood U , thereby generating a subbundle F0 ⊆ E of fiber dimension 2(ν − m). We can then then define an L2 -orthogonal projection p0 : E −→ F0⊥ to the L2 -orthogonal complement F0⊥ to F0 in E. Let Z be the image of the zero-section in the restriction of F0⊥ to the submanifold B Λ (Σ, M ) constructed in the earlier proof. The hypothesis that ([f0 , p0 , η0 ], g0 ) has weakly transversal crossings implies that the hypotheses of Lemmas 5.1.1 and 5.1.3 are satisfied. Therefore the metric variations constructed in the proof of Lemmas 5.1.1 and 5.1.3 exist and cover a complement to the smaller space Re (J0 (L)) within Re (J (L)). This allows us to show that p0 ◦F Λ is transverse to Z at the point (f0 , ω0 , g0 ), and hence transverse to Z in some neighborhood B Λ (Σ, M ) ∩ W of (f0 , ω0 , g0 ), with W ⊆ U . Then RΛ = (p0 ◦F Λ )−1 (Z)∩W is a submanifold of W which contains all of the conformal harmonic maps of branch type Λ within W. But F0 has codimension 2m in F , and instead of (5.83), we now obtain a Jacobi operator   Λ Λ p0 ◦ L : T(f,p,ω) B (Σ, M ) −→ F0⊥ (f,p,ω)

which has Fredholm index −2m. Carrying through the rest of the argument, we find that the submanifold RΛ we obtain as the inverse image of the zero section now has Fredholm projection to Met(M ) with Fredholm index −2m instead of zero. This finishes the proof when Σ (or its double cover) has genus at least two. The case where Σ has genus zero or one is now treated exactly as in the proof of Proposition 5.6.3. We replace Am (Σ, M ) by a sequence of submanifolds   m (m) (Σ, M ) = F (Σ, M ) × Σ − Δ × Tg × Met(M ) Am j j

5.7. Proof of the Transversal Crossing Theorem

347

of codimension dim G in Am (Σ, M ), and once again the previous argument yields a sequence of submanifolds of the configuration space which contain representatives of every G-orbit of minimal surfaces of branch type Λ which satisfy the conditions in the definition of P0Λ , the submanifolds projecting to the space of metrics with Fredholm index −2m. This finishes the proof of Proposition 5.6.5. QED 5.6.4. Proof of the Bumpy Metric Theorem 5.1.1. We simply gather together the preceding results. It follows from Proposition 5.6.5 that for g belonging to a residual subspace of Met(Σ), P Λ ∩ π −1 (g) is empty whenever Λ = ∅. In other words, we have proven that for a residual set of Riemannian metrics on M , no prime parametrized minimal surfaces have branch points, and we can work with the much simpler submanifold of minimal immersions with no branch points, P ∅ = {(f, ω, g) ∈ Map(Σ, M ) × T × Met(M ) : f is a prime minimal immersion, conformal with respect to ω and g}. Then the argument of §5.4 for the space P ∅ of minimal immersions for varying metrics establishes Theorem 5.1.1. QED Remark 5.6.6. We saw at the end of §5.5 that nullity of any parametrized minimal surface with at least one simple branch point must be at least two, and the arguments of this section show that the same is true for branch points which are not simple. Thus we can say that the space of metrics which contain minimal surfaces with nontrivial branch locus “has codimension two” within the space of all metrics, and it follows from Theorems 3.1 and 3.3 of Smale [Sma66] that along a generic path within Met(M ) parametrized minimal surfaces cannot have branch points.

5.7. Proof of the Transversal Crossing Theorem Proof of Theorem 5.1.2. As usual, we replace the spaces Map(Σ, M ) and Met(M ) of smooth maps and smooth Riemannian metrics on M by their Sobolev completions L2k (Σ, M ) and Metk−1 (M ), for a large integer k (where Metk−1 (M ) denotes the L2k−1 completion of the space of smooth Riemannian metrics on M ), thereby obtaining Hilbert manifolds. However, to keep the notation simple, we continue to denote the completions by Map(Σ, M ) and Met(M ). As we saw in §5.4.2, (5.89) P ∅ = {(f, ω, g) ∈ Map(Σ, M ) × T × Met(M ) : f is a prime immersed conformal ω-harmonic map}

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5. Generic Metrics

is a smooth submanifold. The Main Theorem of [Moo07] implies that if g0 is generic metric on M , then all prime conformal harmonic maps for g0 are immersed and hence lie in P ∅ . Moreover, for any such metric, each element of Ng0 = π2−1 (g0 ) ∩ P ∅ is either a nondegenerate critical point for the energy when the genus g of Σ (or its oriented double cover) is ≥ 2, or lies in a nondegenerate critical submanifold which has the same dimension as the group G of symmetries for Σ when Σ is a sphere, torus, projective plane or Klein bottle. Here π2 : Map(Σ, M ) × T × Met(M ) −→ Met(M ) is the projection on the last factor. Recall that first variation of energy E gives rise to an Euler-Lagrange map F : Map(Σ, M ) × T × Met(M ) −→ T (Map(Σ, M ) × T ) from which we can calculate the tangent space to P ∅ , the result being ˙ h) ∈ Tf Map(Σ, M ) ⊕ Tω T ⊕ Tg Met(M ) : T(f,ω,g) P ∅ = {(X, ω, L(X, ω) ˙ + πV ◦ (D2 F )(f,ω,g) (h) = 0}, where L is the Jacobi operator of E, D2 F is the derivative with respect to Met(M ) and πV denotes projection into the vertical tangent space at a zero of F . It follows from this expression and from Lemma 5.4.2 that if (f, ω, g0 ) is any element of Ng0 , then the projection on the first factor, (5.90) π1 : P ∅ −→ Map(Σ, M ) × T has surjective differential at (f, ω, g0 ). (Recall that if Σ has a positive-dimensional group G of conformal automorphisms, the orbits of the G-action generate the tangential Jacobi fields.) Thus all pairs (f  , ω  ) sufficiently close to (f, ω) lie in the image of P ∅ , and can be realized by parametrized minimal surfaces for metrics which are near g0 . To prove the first statement of Theorem 5.2.1, we construct a countable cover of Map(Σ, M ) × T × Met(M ) by product open balls Ui × Vi , Ui ⊂ Map(Σ, M ) × T ,

Vi ⊂ Met(M ),

such that if Ui × Vi intersects P ∅ , (1) it is the domain for a submanifold chart for P ∅ , (2) the restriction of π2 : Ui × Vi → Vi to P ∅ ∩ (Ui × Vi ) is proper, and (3) the restriction of π1 : Ui ×Vi → Ui to P ∅ ∩(Ui ×Vi ) is a submersion.

5.8. Branched covers

349

The second condition can be arranged by Theorem 1.6 of [Sma66] and the last condition follows from (5.90). Let Imm(Σ, M ) denote the space of C 1 immersions of Σ into M . It follows from standard transversality theory for finite-dimensional manifolds (see Corollary 3.3 of Chapter III, §3 of [GG73]) that U = {f ∈ L2k (Σ, M ) ∩ Imm(Σ, M ) : f has transversal crossings} is an open dense subset of L2k (Σ, M ) ∩Imm(Σ, M ). Since π0 is a submersion, π0−1 (U ) ∩ P ∅ ∩ (Ui × Vi ) is an open dense subset of P ∅ ∩ (Ui × Vi ). It follows that for g in an open dense subset Vi of Vi , the immersions in Ng ∩ (Ui × Vi ),

where

Ng = π2−1 (g) ∩ P ∅

have transversal crossings. Note that Wi = Vi ∪ (Met(M ) − V i ) is open and dense. Metrics g which lie in the intersections of the Wi ’s, a countable intersection of open dense subsets of Met(M ), have the property that Ng contains only immersions with transversal crossings, establishing the first assertion of Theorem 5.2.1. The second statement is proven by the same argument, modified to the case where Σ is a compact surface with two components instead of one. QED

5.8. Branched covers The Bumpy Metric Theorem makes possible a description of the branched covers of prime minimal surfaces in a compact Riemannian manifold of dimension at least three with generic Riemannian metric. These branched covers lie on smooth critical submanifolds for the energy function E which have dimension determined by the branching order. However, we will see later (Example 5.8.2) that these critical submanifolds are not always nondegenerate in the sense of Bott for generic choice of Riemannian metric. This implies that the theory of branched covers of minimal two-spheres, for example, is somewhat different from the theory of iterated geodesics as studied by Bott [Bot56] and others. Suppose that h : S 2 → M is a prime minimal two-sphere, which we take to be immersed since we assume throughout this section that the Riemannian metric on M is generic. If Σ is a compact Riemann surface and g : Σ → S 2 is a holomorphic map (also known as a meromorphic function on Σ), then the composition f = h ◦ g : Σ → M is a conformal harmonic map, hence a minimal surface in its own right. For each integer d ≥ 2, we set Hold (Σ, S 2 ) = {g : Σ → S 2 : g is holomorphic and has degree d},

350

5. Generic Metrics

and we call this the space of holomorphic branched covers of S 2 by Σ of degree d. For an explicit construction of elements of Hold (S 2 , S 2 ), we take two effective divisors D0 and D∞ on S 2 of degree d, which have no points in common. Then there will be a unique meromorphic function g on S 2 with divisor (g) = D0 − D∞ , up to multiplication by a nonzero complex scalar. In fact, if we regard S 2 as C ∪ {∞}, and arrange after a holomorphic automorphism of S 2 that neither divisor includes infinity, say D0 =

d  i=1

pi ,

D∞ =

d 

qi ,

then g = c

i=1

(z − p1 ) · · · (z − pd ) (z − q1 ) · · · (z − qd )

is the corresponding meromorphic function, the complex number c being an arbitrary element of C − {0}. (Note that we allow repetition among the pi ’s and qi ’s.) From this description, it is clear that Hold (S 2 , S 2 ) is a complex manifold of complex dimension 2d + 1; indeed, ⎡ ⎤ d d !" # !" # ⎢ ˆ⎥ Hold (S 2 , S 2 ) = (C − {0}) × ⎣S 2 × · · · × S 2 × S 2 × · · · × S 2 −Δ ⎦, ˆ is the multidiagonal, where Δ ˆ = {(p1 , . . . , pd , q1 , . . . , qd ) : some pi equals some qj } . Δ Note that Hold (S 2 , S 2 ) is noncompact and, in fact, when one of the pi ’s moves into coincidence with one of the qi ’s the L2k -norm of g must go to infinity, when k ≥ 2. We let Imm(S 2 , M ) = {f ∈ Map(S 2 , M ) : f is an immersion}, and for each integer d ≥ 1, we define a map Φd : Imm(S 2 , M ) × Hold (S 2 , S 2 ) → Map(S 2 , M ) by

Φd (h, g) = h ◦ g,

a map which will be C ∞ when the mapping spaces are completed with respect to the L2k -topology when k is large. For a fixed choice of immersed prime minimal two-sphere h : S 2 → M , Φd,h = Φd (h, ·) : Hold (S 2 , S 2 ) → Map(Σ, M ) is an embedding onto a submanifold Md,h ⊆ Map(S 2 , M ), the submanifold of d-fold covers of h. This construction can be extended to branched covers of a minimal twosphere by an oriented surface Σ of genus g ≥ 1. Recall that Abel’s Theorem

5.8. Branched covers

351

([GH78], Chapter 2, §2) gives g conditions which the difference D0 − D∞ of divisors d d   pi and D∞ = qi D0 = i=1

i=1

of degree d must satisfy in order to be the divisor of a meromorphic function. Indeed, Abel’s Theorem states that if {ω1 , . . . , ωg } is a basis for the space of holomorphic one-forms on Σ, then D0 − D∞ is the divisor of a meromorphic function if and only if μ(D0 , D∞ ) = 0, where   g  g  qi  qi  μ(D0 , D∞ ) = ω1 , . . . , ωg ∈ Cg . i=1

pi

i=1

pi

From this description, we might expect that for a generic choice of conformal structure ω ∈ T , the (noncompact) space Hold (Σ, S 2 ) of holomorphic maps of degree d from Σ to S 2 is a complex manifold of complex dimension 2d + 1 − g. But for the critical locus for the two-variable energy we need to allow the complex structure on Σ to vary, so we let T denote the Teichm¨ uller space of Σ and consider the space (5.91) Hd (Σ, S 2 ) = {(g, ω) ∈ Map(Σ, S 2 ) × T : g is holomorphic and has degree d when Σ has the conformal structure ω}, which we expect to be a manifold of complex dimension 2d + 2g − 2, when g ≥ 2 (since we have a 3g − 3-dimensional space of deformations in the Teichm¨ uller direction), and a complex manifold of dimension 2d + 1 when g = 1. According to the Riemann-Hurwitz formula, if g : Σ → S 2 is a branched cover of degree d, then ν = (total branching order) = 2d + 2g − 2, so the expected complex dimension of Hd (Σ, S 2 ) is just the total branching order ν when g ≥ 2, and ν + 1 when g = 1, the extra complex deformation coming from the S 1 × S 1 -action. We can now extend our earlier construction and define for each integer d ≥ 1 a map Φd : Imm(S 2 , M ) × Hd (Σ, S 2 ) → Map(Σ, M ) × T by

Φd (h, (g, ω)) = (h ◦ g, ω),

a map which we expect to be C ∞ when the mapping spaces are completed with respect to the L2k topology for k large. For a fixed choice of immersed prime minimal two-sphere h : S 2 → M , Ψd,h = Φd (h, ·) : Hd (Σ, S 2 ) → Map(Σ, M ) × T

352

5. Generic Metrics

is an embedding onto a submanifold Md,h ⊆ Map(Σ, M ) × T , the submanifold of d-fold covers of h with the topology of Σ. These constructions can be extended further to yield the following: Proposition 5.8.1. Suppose that Σ0 is a compact oriented surface of genus g0 with a fixed conformal structure and that h : Σ0 → M is an immersed minimal surface, where M has dimension at least three. Then (5.92) Hd (Σ, Σ0 ) = {(g, ω) ∈ Map(Σ, Σ0 ) × T : g is holomorphic and has degree d when Σ has the conformal structure ω}, is a smooth manifold of dimension 2ν+dimR G, where ν is the total branching d (Σ, Σ0 ). Moreover, the map order of any element (g, ω) ∈ H Ψd,h : Hd (Σ, Σ0 ) → Map(Σ, M ) × T ,

Ψd (g, ω) = (h ◦ g, ω)

is an embedding onto a submanifold Md,h of Map(Σ, M ) × T . Proof. Note first that by the Riemann-Hurwitz formula, ν = 2g − 2 − d(2g0 − 2), which depends only on the genera of Σ and Σ0 . The only statement in d (Σ, Σ0 ) is a smooth the theorem which is not immediate is the fact that H manifold. To see this, we let M = Σ0 and consider the two variable energy E : Map(Σ, Σ0 ) × T → R. As before, the first derivative of E gives rise to an Euler-Lagrange map F : Map(Σ, Σ0 ) × T → T (Map(Σ, Σ0 ) × T ), and zeros of this map are just the conformal harmonic maps, which in this case are either holomorphic or antiholomorphic. At a critical points (f, ω) for E, the second derivative of E is given by the second variation formula for E. If f is nonconstant, all the Jacobi fields will be tangential Jacobi fields as described in §5.3.3, and the real dimension of the space of such Jacobi fields can be calculated in terms of the total branching order ν of f . According to Proposition 5.3.5, this dimension is (1) 2ν if Σ has genus at least two, (2) 2ν + 2 if Σ has genus one, and (3) 2ν + 6 if Σ has genus zero.

5.8. Branched covers

353

Thus it follows from the implicit function theorem that the zero set of F is a smooth submanifold with tangent space consisting of the Jacobi fields, and d (Σ, Σ0 ) = {(g, ω) ∈ Map(Σ, Σ0 ) × T : g is holomorphic and H has degree d when Σ has the conformal structure ω} is indeed a smooth manifold of real dimension 2ν when Σ has genus at least two, real dimension 2ν +2, when Σ has genus one, and real dimension 2ν +6, when Σ has genus zero as claimed. QED Example 5.8.2. Let us consider the case in which the ambient manifold M = S 3 , the three-sphere of constant curvature one. According to Almgren’s Theorem 4.2.4, the nonconstant minimal two-spheres of least energy are totally geodesic of area 4π, and lie on a nondegenerate critical submanifold N ⊆ Map(S 2 , S 3 ) which is diffeomorphic to SO(3), the Grassmannian or oriented three-planes in an R4 which contains S 3 as a round sphere. In this case, it is easy to write out the formula for second variation of area (Proposition 5.3.4) because the normal bundle is trivial and all sections are function multiples of a canonical unit length normal section E3 . Since S 2 is a totally geodesic submanifold of S 3 the second fundamental form vanishes, and we obtain the second variation formula  2 (5.93) d A(f )(gE3 , hE3 ) = − (Δg + 2g)hdA where g, h : S 2 → R S2

are smooth functions and Δ is the standard Laplace operator on S 2 . The spectrum of Δ is calculated in basic calculus courses by separation of variables. The Riemannian metric on S 2 takes the form ·, · = dφ2 + sin2 φdθ2

on

V1 = {(φ, θ) : 0 < φ < π, 0 ≤ θ ≤ 2π},

where we identify θ = 0 with θ = 2π. The standard formula for the Laplace operator in these coordinates gives      ∂ ∂ ∂ 1 ∂ 1 sin φ + . Δ=− sin φ ∂φ ∂φ ∂θ sin φ ∂θ Numerous books on applied mathematics detemine that the nontrivial solutions to the eigenvalue equation −Δg = λg occur when λk = k(k + 1) for k ∈ {0} ∪ N, the multiplicity of the eigenvalue λk being 2k + 1. This implies that the index form (5.93) at a totally geodesic two-sphere has Morse index one and nullity three, the nullity agreeing with the dimension of critical submanifold diffeomorphic to SO(3).

354

5. Generic Metrics

Consider now the spectrum for the two-fold branched cover in which the branch points are antipodal. We can treat this with the Riemannian metric (5.94) ·, ·2 = dφ2 + sin2 φdθ2 on V2 = {(φ, θ) : 0 < φ < π, 0 ≤ θ ≤ 2dπ}, where we identify θ = 0 with θ = 4π. The same technique of separation of variables shows that if we set x = cos φ, eigenfunctions should be of the form u(x) cos((m/2)θ) and u(x) sin((m/2)θ) where m ∈ {0} ∪ N and u : [−1, 1] → R is a solution to the associated Legendre DE     μ2 d m 2 du (1 − x ) + λ− (5.95) u = 0, μ = . 2 dx dx 1−x 2 This equation has regular singular points at x = ±1. When m = 1 we find that the solutions which are bounded at x = ±1 are linear combinations of (1 − x2 )1/4 cos((1/2)θ) and (1 − x2 )1/4 sin((1/2)θ). By elliptic regularity we can remove the singularities of these solutions to the eigenvalue equation at the north and south poles, and we thus find that the lowest nonzero eigenvalue is λ = 3/4 with multiplicity two. This is consistent with the “trombone principle” which states that the lowest nonzero eigenvalue of the larger domain V2 should be smaller than that of V1 . For the record, we mention that with more work one could verify that the nontrivial solutions to the eigenvalue equation −Δg = λg for this double cover occur when λk = (k/2)((k/2) + 1) for k ∈ {0} ∪ N, the multiplicity of λk being k + 1. The above calculation shows that any branched cover of order two with branch points antipodal with respect to the constant curvature metric must have Morse index at least three, and this estimate continues to hold for minimal surfaces near the totally geodesic two-spheres in generic metrics near the constant curvature one metric on S 3 . But we get a different result when the two branch points are close in terms of the constant curvature metric on S 2 . Start with a perturbed generic metric with an imbedded two-sphere f : S 2 → S 3 of Morse index one (known to exist by Lemma 4.10.7) and consider a one-parameter family in the space of double branched covers of f , f ◦ gt : S 2 → M,

t ∈ [0, 1],

such that g0 has two antipodal branch points and g1 has two branch points which almost coincide in the constant curvature metric. The endpoint f ◦ g1 gives two approximate copies of the base surface f connected by a thin neck (another instance of bubbling), and the Morse index of this configuration is

5.8. Branched covers

355

only two, one for each copy. It follows that for some t ∈ (0, 1), the second variation of f ◦ gt must have nonzero nullity. This simple calculation shows that the space of two-fold branched covers of the imbedded two-sphere f : S 2 → S 3 does not form a nondegenerate critical submanifold. On the other hand, unbranched covers, at least unbranched covers of tori lie on nondegenerate critical submanifolds for generic metrics. Suppose that f0 : Σ0 → M is a fixed somewhere injective prime minimal surface in the Riemannian manifold M , and we consider perturbations of the metric on M which have one-jet fixed along the image of f0 . More precisely, we let Met(M, f0 ) denote the space of Riemannian metrics g on M such that the one-jet j1 (g) of g agrees with the one-jet j1 (g0 ) of g0 at points of f0 (Σ0 ). Theorem 5.8.3. Suppose that M is a compact smooth manifold of dimension at least three with Riemannian metric g0 . (1) If f0 : Σ0 → M is a Morse nondegenerate prime minimal torus or Klein bottle with no branch points, then for a generic choice of Riemannian metric in Met(M, f0 ), all minimal tori and Klein bottles which cover f0 are also Morse nondegenerate. (2) If f0 : Σ → M is a nonoriented Morse nondegenerate prime minimal surface of any genus with no branch points, then for a generic choice of Riemannian metric in Met(M, f0 ), the oriented double cover of f0 is also Morse nondegenerate. This is proven in [Moo07] with an additional assumption that the dimension of M be at least four, but the proof works when the dimension of M is three. In the case of minimal tori covering minimal tori the argument is similar to that for Theorem 2.7.2, and like that argument is based upon Bott’s theory of iterated closed geodesics.

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PS64. R. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964), 165-171. Par96. T. Parker, Bubble tree convergence for harmonic maps curvature flow , J. Differential Geometry 44 (1996), 595-633. PW93. T. Parker and J. Wolfson, Pseudo-holomorphic maps and bubble trees, J. Geometric Analysis 3 (1993), 63-98. Rad89. H. Rademacher, On the average indices of closed geodesics, J. Differential Geometry 29 (1989), 65-83. RS80. M. Reed and B. Simon, Methods of mathematical physics I: Functional analysis, Academic Press, New York, 1980. RoS06. J. Robbin and D. Salamon, A construction of the Deligne-Mumford orbifold , J. Eur. Math. Soc. 8 (2006), 611-699. Roy88. H. L. Royden, Real analysis, Third edition, Prentice-Hall, New York, 1988. Rud86. W. Rudin, Real and complex analysis, Third Edition, McGraw-Hill, New York, 1986. SU81. J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Annals of Math. 113 (1981), 1-24. SU82. J. Sacks and K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271 (1982), 639-652. Sam78. J. H. Sampson, Some properties and applications of harmonic mappings, Annales ´ Scientifiques de l’Ecole normale Sup´erieure 11 (1978), 211-228. Sch84. R. Schoen, Analytic aspects of the harmonic map problem, in Seminar on nonlinear partial differential equations, Springer, New York, 1984, pp. 321-358. SY79. R. Schoen and S. T. Yau, Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature, Annals of Math. 110 (1979), 127-142. SY97. R. Schoen and S. T. Yau, Lectures on harmonic maps, International Press, Boston, 1997. Sch93. M. Schwarz, Morse homology, Birkh¨ auser Verlag, Basel, 1993. Seg79. G. Segal, The topology of spaces of rational functions, Acta Mathematica 143 (1979), 39-72. Ser51. J. P. Serre, Homologie singuli`ere des espaces fibr´ees, Annals of Math. 54 (1951), 425-505. Sim84. L. Simon, Geometric measure theory, Australian National University, Canberra, Australia, 1984. Sim68. J. Simons, Minimal varieties in riemannian manifolds, Annals of Math. 88 (1968), 62-105. Sma58. S. Smale, Regular curves on Riemannian manifolds, Trans. Amer. Math. Soc. 87 (1958), 492-512. Sma64. S. Smale, Morse theory and a nonlinear generalization of the Dirichlet problem, Annals of Math. 80 (1964), 382-396. Sma65. S. Smale, On the Morse index theorem, J. Math. Mech. 14 (1965), 1049-1055; Corrigendum 16 (1967), 1069-1070. Sma66. S. Smale, An infinite-dimensional version of Sard’s theorem, Amer. J. Math. 87 (1966), 861-866.

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Index

α-energy, 206 α-harmonic map, 206 α-Lemma, 39 α-minimal surface, 206 ε-Regularity Theorem, 229 λ-handle, 110 ω-Lemma, 20, 39 action integral, 24, 74 Almgren’s Theorem, 181 ambient isotopy, 60 ambient isotopy invariant, 61 area function, 178 ascending disk, 110 ascending sphere, 110 atlas, 18 Baire Category Theorem, 93 Banach algebra, 6 Banach Algebra Lemma, 27 Banach manifold, 17 Banach space, 4 Banachable space, 6 Birkhoff Minimax Principle, 62, 257 Bochner Lemma, 226 bootstrapping, 15, 81, 215 branch point, 176 false, 285 primitive, 285 branch point stratification, 318 branch type, 318 branched cover of a harmonic map, 177

broken geodesics, 139 Brown-Sard Theorem, 92 bubble point, 238 bubble tree, 257, 259 bubble tree convergence for harmonic surfaces, 258 for minimal surfaces, 265 Bumpy Metric Theorem for closed geodesics, 104 for minimal surfaces, 279 for prime closed geodesics, 101 ˇ Cech cohomology, 67 center of mass, 201 for α-energy, 242 chain rule, 8 closed differential form, 63 Condition C, 55 for α-energy, 250, 253 for geodesics, 76 for the α-energy, 207 configuration space for higher order branch points, 334 for simple mechanical systems, 72 connected dga, 126 cotangent bundle to a Banach manifold, 41 critical point, 45 Morse nondegenerate, 86 stable, 86 critical value, 91 cup product, 64 curvature operator, 269

365

366

cusp singularity, 281 de Rham cohomology, 63 de Rham homotopy group, 127 de Rham Theorem, 66 degree of a holomorphic line bundle, 175 descending disk, 110 descending sphere, 110 dga, 125 diffeomorphism group of Σ Diff+ (Σ), 189 Diff0 (Σ), 189 Diff0,D (Σ), 320 Diff0,p (T 2 ), 194 Diff0,q,r,s (S 2 ), 195 diffeomorphism of manifolds, 18 differential form, 45 differential graded algebra, 125 differential of a function, 45 directional derivative, 45 Dirichlet energy integral, 25, 169 disk bundle addition, 113 divisor of a harmonic surface, 177 Eells-Sampson Theorem, 171 Eilenberg-MacLane space, 149 elementary extension of a dga, 152 elliptic fibration, 281 energy, 169 energy density, 169 energy loss in necks, 259 equivariant cohomology, 199 Euler class, 175 Euler equations for a rigid body, 73 Euler-Lagrange map, 96 Euler-Lagrange operator for closed curves, 90 for the two-variable energy, 309 evaluation map, 40 exact differential form, 63 Existence and Uniqueness Theorem for ODE’s, 53 exponential chart, 19 exterior derivative, 47 Fet-Lusternik Theorem, 84 fibration, 81 finite type, 126 Finsler metric, 50 first Chern class, 175 fishtail singularity, 281 flow box, 53

Index

Fr´echet derivative, 7 Fr´echet space, 16 Fredholm index, 92 Fredholm map, 93 Fredholm operator, 92 Fredholm projection, 317 free homotopy class, 241 free loop space, 73 fundamental domain of Teichmuller space T1 , 184 gauge group G for two-variable energy, 196, 292, 320 generic Riemannian metric, 100, 277 geodesic, 72 good cover, 66 graded commutative algebra, 125 Green’s operator, 220 Gromoll-Meyer Theorem, 138 Gromov dimension, 139 Gromov estimates, 143, 263 Grothendieck Theorem, 270 Grushko Theorem, 247 H¨ older continuous function, 29 Hahn-Banach Theorem, 6 Hamilton’s principle of least action, 75 handle addition, 108 harmonic map, 170 Hartman Theorem, 171 Hessian of f , 85 Hilbert manifold, 17 Hilbert space, 3 homotopy direct limit, 37 homotopy equivalence, 33 Homotopy Lemma, 64 homotopy lifting property, 81 homotopy type, 33 Hopf differential, 181 Hopf-Leray Theorem, 131 incompressible Klein bottle, 252 torus, 252 incompressible component of mapping space, 255 incompressible map from Σ to M , 253 injective point, 282 integral curve, 53 interior product, 47 Inverse Function Theorem, 13

Index

irreducible three-manifold, 247 isothermal coordinates, 172 isotropic two-plane, 270 Jacobi field for closed geodesics, 88 for fixed end-point geodesics, 87 tangential, 301 Jacobi operator for the α-energy, 219 Kepler problem, 72 kinetic energy, 72 Koszul-Malgrange Theorem, 174 Lacunary Principle, 116, 123 Leibniz rule, 7 Leray cover, 66 Lie bracket, 46 Lie derivative, 45 local flow for a vector field, 55 lower central series, 153 Lp norm, 12 Meeks-Yau Theorem, 246 method of steepest descent, 49 metric deformation operator, 311, 316 minimal Klein bottle, 201 minimal model, 127, 154 for Map(S 1 , M ), 132 of Map(Σ, M ), 155 minimal projective plane, 200 minimal surface branched cover, 278 prime, 278 somewhere injective, 282 minimal torus, 182 minimal two-sphere, 180 minimax critical point, 61 minimax principle, 60 Morse function, 86, 87, 95, 221 of finite type, 115 Morse index of a critical point, 85 of a critical submanifold, 99 Morse inequalities, 114 equivariant, 199, 266 failure for the usual energy E, 248 for the perturbed α-energy, 224 Morse polynomial, 114 Morse series, 115 Morse-Witten boundary, 120

367

Morse-Witten complex, 120 for the perturbed α-energy, 224 Mountain Pass Lemma, 116 multiplicity of a branch point, 176 Newton’s equation of motion, 72 nilpotent CW complex, 153 nilpotent fundamental group, 153 nondegenerate critical submanifold, 99, 279 nonorientable minimal surface, 200 norm, 4 nullity, 85 one-connected dga, 126 one-parameter group, 55 Open Mapping Theorem, 6 order of a branch point, 176 Palais-Smale compactness condition, 55 parametrized minimal surface, 179 partition of unity, 58 path space fibration, 82 periodic motion, 74 piecewise linear paths for a triangulation, 141 Poincar´e Lemma, 64 Poincar´e polynomial, 114 Poincar´e series, 115 point bundle, 177 pointed loop space, 82 positive isotropic curvature, 270 Postnikov tower, 150 pre-Banach space, 4 pre-Hilbert space, 3 prime decomposition of a three-manifold, 247 prime harmonic map, 177 prime incompressible component of mapping space, 255 prime smooth closed geodesic, 101 prime three-manifold, 247 principal fibration, 150 pseudogradient, 57 pullback bundle, 18 radial cutoff function, 273, 325 rank k parametrized torus, 249 rationalization of a CW complex, 148 rationally elliptic manifold, 131 rationally hyperbolic manifold, 131 regular value, 91 Regularity for the α-energy, 212

368

Removeable Singularity Theorem, 236 residual subset, 93 Riemann moduli space, 184, 189 Riemann surface, 173 conformally finite, 320 Riemann-Christoffel curvature tensor, 269 Riemann-Roch Theorem, 175 Riemannian metric, 49 Sacks-Uhlenbeck energy, 206 Sacks-Uhlenbeck Theorem, 239, 241 Sard-Smale Theorem, 93 second variation for closed geodesics, 88 for fixed end-point geodesics, 86 for harmonic surfaces, 217 of area, 300 of energy, complex form, 268 of the α-energy, 217 of the two-variable energy, 296 sectional curvature complex, 269 self-intersection set of a minimal surface, 284 seminorm, 4 sheet interchange map, 200 simple mechanical system, 72 smooth function on a Banach space, 7 smooth manifold, 17 smooth map between manifolds, 18 smoothing operators, 34 Sobolev Lemma, 27, 77 Sobolev norm, 26 Sobolev range, 29 sources of noncompactness, 265 Sphere Theorem, 271 split subspace of a Banach space, 15 stable harmonic surface, 218 stable manifold, 118 Sullivan model, 153 surface subgroup, 255 symmetry group for E full, 198 identity component G, 278 tangent bundle to a Banach manifold, 41 tangent vector

Index

to a Banach manifold, 41 tangential Jacobi field complex form, 304 for the two-variable energy, 301 tangential Jacobi fields for nonorientable minimal surfaces, 306 Taylor’s Theorem, 11 Teichm¨ uller space for punctured surfaces, 320 for surfaces of genus ≥ 2, 189 for tori, 183 Teichm¨ uller Theorem, 192 tensor bundle, 42 tensor field, 44 Transversal Crossing Theorem, 280 transversal crossings, 280 two-variable energy E, 196, 292 Uniform Boundedness Theorem, 6 uniformization theorem, 173 unstable manifold, 118 vector field, 44 gradient, 50, 91 gradient-like, 111 weak Morse inequalities, 114 weakly conformal map, 179 wedge product, 46 Weil-Petersson metric, 193

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For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/gsmseries/.

During the last century, global analysis was one of the main sources of interaction between geometry and topology. One might argue that the core of this subject is Morse theory, according to which the critical points of a generic smooth proper function on a manifold M determine the homology of the manifold. Morse envisioned applying this idea to the calculus of variations, including the theory of periodic motion in classical mechanics, by approximating the space of loops on M  F] E ½RMXIHMQIRWMSREP QERMJSPH SJ LMKL dimension. Palais and Smale reformulated Morse’s calculus of variations in terms of MR½RMXIHMQIRWMSREP QERMJSPHW ERH XLIWI MR½RMXIHMQIRWMSREP QERMJSPHW [IVI JSYRH useful for studying a wide variety of nonlinear PDEs. 8LMWFSSOETTPMIWMR½RMXIHMQIRWMSREPQERMJSPHXLISV]XSXLI1SVWIXLISV]SJGPSWIH geodesics in a Riemannian manifold. It then describes the problems encountered when extending this theory to maps from surfaces instead of curves. It treats critical point theory for closed parametrized minimal surfaces in a compact Riemannian manifold, establishing Morse inequalities for perturbed versions of the energy function on the mapping space. It studies the bubbling which occurs when the perturbation is turned SJJXSKIXLIV[MXLETTPMGEXMSRWXSXLII\MWXIRGISJGPSWIHQMRMQEPWYVJEGIW8LI1SVWI Sard theorem is used to develop transversality theory for both closed geodesics and closed minimal surfaces. This book is based on lecture notes for graduate courses on “Topics in Differential Geometry”, taught by the author over several years. The reader is assumed to have taken basic graduate courses in differential geometry and algebraic topology.

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