Geometrical themes inspired by the N-body problem 9783319714271, 9783319714288

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Lecture Notes in Mathematics  2204

Luis Hernández-Lamoneda Haydeé Herrera Rafael Herrera Editors

Geometrical Themes Inspired by the N-body Problem

Lecture Notes in Mathematics Editors-in-Chief: Jean-Michel Morel, Cachan Bernard Teissier, Paris Advisory Board: Michel Brion, Grenoble Camillo De Lellis, Zurich Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gábor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, New York Anna Wienhard, Heidelberg

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More information about this series at http://www.springer.com/series/304

Luis Hernández-Lamoneda • Haydeé Herrera • Rafael Herrera Editors

Geometrical Themes Inspired by the N-body Problem

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Editors Luis Hernández-Lamoneda Department of Mathematics Mathematics Research Center (CIMAT) Guanajuato, Mexico

Haydeé Herrera Department of Mathematics Rutgers University Camden, NJ USA

Rafael Herrera Department of Mathematics Mathematics Research Center (CIMAT) Guanajuato, Mexico

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-319-71427-1 ISBN 978-3-319-71428-8 (eBook) https://doi.org/10.1007/978-3-319-71428-8 Library of Congress Control Number: 2018931934 Mathematics Subject Classification (2010): 70F07, 37J45, 37J29, 34M99, 53C15, 37D15, 53D12, 57R58 © Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The Seventh Mini-Meeting on Differential Geometry was held from February 17th to 19th, 2015, at the Center for Research in Mathematics (CIMAT), Guanajuato, México. The invited speakers included Adolfo Guillot, Richard Montgomery (Distinguished Visiting Professor for the Mexican Academy of Sciences and the USAMexico Foundation for Science), and Andrés Pedroza. The lectures were organized into three advanced minicourses (three lectures each). This volume consists of the lecture notes of the minicourses whose content we describe briefly: • Guillot’s notes explore some differential equations in the complex domain that can be studied through the understanding of geometric structures (projective, affine) on complex curves. The study of such systems is motivated by the problem of describing the evolution of N particles moving in the plane subject to the influence of a magnetic field. For instance, in the absence of interactions (vanishing interaction constants), the particles move periodically in circles, all of them with the same period (the system is isochronous). The search for the nonzero values of the interaction constants that render the system isochronous is the guiding principle for the theory that is developed. • Montgomery’s notes deal with the solution to a generalization of the wellknown theorem in Riemannian geometry asserting that on a compact Riemannian manifold, every free homotopy class of loops is realized by a periodic geodesic. Inspired by this fundamental geometric fact, Wu-yi-Hsiang posed the question: “Is every free homotopy class realized for the planar Newtonian three-body equation?” In the case of equal or nearly equal masses, and for all nonzero angular momentum sufficiently small, every free homotopy class on the twosphere minus three points is realized by a relatively periodic orbit for the three-body problem. The exposition of the solution to this problem involves not only geometry but also dynamical methods and the McGehee blow-up. The main novelty is the use of energy balance to motivate the mysterious transformation of McGehee. v

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Preface

• Pedroza’s notes present a brief introduction to the recent and important theory of Lagrangian Floer homology and its relation with the solution of Arnol’d conjecture on the minimal number of nondegenerate fixed points of a Hamiltonian diffeomorphism. A sketch of various aspects of Morse theory and an introduction to the basic concepts of symplectic geometry are included, with the aim of understanding the statement of the Arnol’d conjecture and how it relates to Lagrangian submanifolds. We thank all the participants for making the meeting a successful and stimulating mathematical event. We also thank CIMAT’s staff for their help in the organization and smooth running of the event. This meeting was the seventh edition of an annual event intended for researchers and graduate students, with the dual aim of combining a winter school and a research workshop. The meeting was supported by the Mexican Academy of Sciences (AMC), the USA-Mexico Foundation for Science (FUMEC), the Mexican Science and Technology Research Council (CONACyT), and the Center for Research in Mathematics (CIMAT). The organizers were Luis Hernandez-Lamoneda (CIMAT, México), Haydeé Herrera (Rutgers, USA), and Rafael Herrera (CIMAT, México). Guanajuato, Mexico Camden, NJ, USA Guanajuato, Mexico

Luis Hernández-Lamoneda Haydeé Herrera Rafael Herrera

Contents

Complex Differential Equations and Geometric Structures on Curves . . . . Adolfo Guillot Blow-Up, Homotopy and Existence for Periodic Solutions of the Planar Three-Body Problem.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Richard Montgomery A Quick View of Lagrangian Floer Homology .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Andrés Pedroza

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Complex Differential Equations and Geometric Structures on Curves Adolfo Guillot

Abstract These are the notes of a series of lectures on ordinary differential equations in the complex domain delivered at the “Seventh Minimeeting in Differential Geometry” at CIMAT, in Guanajuato, Mexico, in 2015. We use geometric structures on curves as a setting to present some historical results of the theory and as a tool for a better understanding of some classical equations.

These are the notes of a series of lectures delivered at the “Seventh Minimeeting in Differential Geometry” at CIMAT, in Guanajuato, Mexico, in 2015. Some of its material appeared already in the lectures given at the CIMPA School “Singularités des Espaces, des Fonctions et des Feuilletages” that took place in Fez, Morocco, in 2012. The lectures were intended as an invitation to differential equations in the complex domain, focusing on some geometric aspects, and explored some of the complex differential equations that can be studied through the understanding of some geometric structures (projective, affine) on complex curves. The course was addressed to people with some basic knowledge of ordinary differential equations and complex analysis but possibly no previous acquaintance with differential equations in the complex domain. The idea of the lectures was to use the geometric description of some particular differential equations (many of them classical) as a setting to present some historical results of the theory, like Schwarz’s theorem on the uniformization of plane polygons or Fuchs’s theorem on the singularities of linear differential equations. We also use this geometric description as a tool for a better understanding of some of the features of these equations. The author thanks Rafael Herrera for his invitation to lecture at the “Seventh Minimeeting in Differential Geometry”, as well as for his encouragement to write these notes.

A. Guillot () Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, A.P. 273-3 Admon. 3, Cuernavaca 62251, Morelos, Mexico e-mail: [email protected] © Springer International Publishing AG 2018 L. Hernández-Lamoneda et al. (eds.), Geometrical Themes Inspired by the N-Body Problem, Lecture Notes in Mathematics 2204, https://doi.org/10.1007/978-3-319-71428-8_1

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A. Guillot

1 Calogero’s “Goldfish,” Complex Differential Equations and Translation Structures 1.1 The “Goldfish” Many-Body Problem Let us consider the problem of describing the evolution of N particles moving in R3 subject to the equations rR j D !  rP j 

X k¤j

jk

.Prk  rjk /Prj C .Prj  rjk /Prk  rjk .Prj  rP k / ; j D 1; : : : N: jrjk j2

(1)

Here, the position of the jth particle is given by rj 2 R3 and rjk D rj  rk ; the constant jk 2 R measures the influence of the kth particle upon the jth one. There is a magnetic vector field having  D .0; 0; 1/ for direction, and ! 2 R is a natural frequency of the system (! ¤ 0). We denote by fP the derivative of f with respect to the independent variable . If at one instant all the particles are in the plane …  R3 given by z D 0 and all their velocities are tangent to this plane, the particles will remain within this plane throughout their motion. The “goldfish” many-body problem corresponds to the restriction of the system (1) to this plane …, this is, when, for every j, rj .t/ D .xj .t/; yj .t/; 0/. A slightly more general version of this system was introduced by Calogero and Françoise in [7] as a generalization of the original “goldfish,” appearing in [6]. The term “goldfish” was coined by Calogero, borrowing the term from Zakharov, as a metaphor for an extraordinary fish that can only be caught on very rare occasions. In the “goldfish” problem, the particles move within the plane … and are subject to the influence of a magnetic field that is orthogonal to it. In the absence of interactions (when jk D 0 for every j and k), the particles move periodically in circles, and all of them do so with period 2: the period is independent of the initial condition. Systems as these, where for almost every initial condition orbits are periodic with the same period are called isochronous. The following problem will guide our discussion. Problem 1 Determine the values of the interaction constants fjk g that give isochronous settings for system (1) in the “goldfish” setting (the one where the particles evolve within the plane … where the third coordinate vanishes). For example, in the two-body “goldfish” system (1), isochronicity occurs if 12 D 3 and 21 D 3 (this will be proved in Example 5). An orbit is portrayed in Fig. 1a. For 12 D 3:1 and 21 D 3:1, the orbit for the same initial conditions appears in Fig. 1b, where one can see that the orbits do no close like in (a). This system is not isochronous.

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Fig. 1 Two-body problems

In order to address the above problem, we will use Calogero’s “trick.” First, consider complex coordinates uj D xj C iyj . In these, the system reads uRj D i! uP j 

X

jk

k¤j

uP j uP k ; j D 1; : : : ; N: uj  uk

(2)

By introducing the new independent variable t by t./ D

ei!  1 i!

(3)

and defining j .t.// D uj ./, we obtain the system of differential equations j00 D 

X k¤j

jk

j0 k0 j  k

; j D 1; : : : ; N;

(4)

where f 0 stands for the derivative of f (in the complex sense) with respect to the complex parameter t. It is actually this system that we will study, either in the above form as a system of second-order differential equations or in the equivalent form j0 D j ; 0j D 

X k¤j

jk

j k ; j D 1; : : : ; N; j  k

(5)

as a system of first-order ones. Our first task will be to reformulate Problem 1 in the context of the system (5).

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1.2 Differential Equations and Vector Fields More generally, we will consider systems of first-order, autonomous, differential equations in n complex variables of the form y0i D Fi . y1 ; : : : ; yn /; Fi W V ! C; i D 1; : : : n;

(6)

where, for every i, Fi is a holomorphic function defined in a domain V  Cn . As an example, we have system (5) in the domain in C2N where the right-hand side is holomorphic. A local solution to (6) is given by a connected open set U  C and a holomorphic function y W U ! V, y D . y1 ; : : : ; yn / that satisfies Eq. (6). Cauchy’s theorem on the existence and uniqueness of solutions of ordinary differential equations in the complex domain ([29, Ch. XII], [26, Ch. 2]) states that for the differential equation (6), given p 2 V, there exists a neighborhood U  C of 0 and a local solution y W U ! V to (6) such that y.0/ D p (we say that p is an initial condition for the solution y). Moreover, the germ of y at 0 is unique. A point p 2 V is said to be an equilibrium point of (6) if F. p/ D 0. In this case the only solution is the constant one y.t/  p. Since Eq. (6) is autonomous (in the sense that the independent variable does not enter explicitly into the right-hand side), if we have a local solution y W U ! V and b 2 C, then y.t C b/, as a function of t, is also a solution (provided t C b 2 U). It may be the very same solution, if b is one of its periods. More generally, given two b ! V, either the images y.U/ and b b nonconstant solutions y W U ! V and b yWU y.U/ are disjoint (as curves in V) or, by the previous uniqueness, there is some b 2 C such that b y.t/ D y.t C b/, wherever this equality makes sense. In this last case, the two solutions agree (in the intersection of the domains where they are defined) up to a translation. Let us consider the smallest equivalence relation in V such that two points in the image of the same local solution are in the same class. An orbit of the differential equation (6) is a class of this relation. In restriction to the open subset of V where there are no equilibrium points, the orbits are naturally complex curves, locally parameterized by the local solutions. These orbits are also called leaves or integral curves. The associated foliation is the partition of this subset of V (the complement of the equilibrium points) into leaves. A leaf of this foliation is the union of images of local solutions of Eq. (4) that overlap and that is maximal with respect to this property. For the differential equation (5), in the phase space C2N , the set of equilibrium points (the set of initial conditions giving constant solutions) is given by \i fi D 0g. We also have the collision points, those in the set [i¤j fi D j g, where the righthand side of (4) is not holomorphic. We thus have the set V D f.1 ; : : : ; N ; 1 ; : : : ; N / 2 C2N ji ¤ j if i ¤ j and i ¤ 0 for some ig: (7)

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This is the maximal open subset of C2N where the vector field is holomorphic and nonvanishing, and is foliated by one-dimensional orbits. Vector fields give an alternative way to present systems of differential equations as the one we arePconsidering. To the differential equation (6) we may associate n the vector field iD1 Fi . y1 ; : : : ; yn /@=@yi in V. Here, we identify vectors with directional derivatives at points. The local solutions are curves whose tangent vector at every point is the one specified by the vector field. We will talk indistinctly about a vector field in an open subset of Cn or about the associated system of first-order ordinary differential equations, that we will sometimes refer to plainly as a system.

1.3 Translation Structures Each leaf of (6) is an immersed one-dimensional complex manifold. Moreover, it has a translation structure in the sense that follows. Definition 2 Let C be a complex curve (a one-dimensional complex manifold). A translation structure on C is an atlas for its complex structure with charts taking values in C and changes of coordinates lying within the group of translations fz 7! z C bg. There are natural notions of map and isomorphism between curves endowed with translation structures. In a complex curve, there is equivalence between translation structures (compatible with the complex structure) and holomorphic and nowhere-vanishing vector fields. Indeed, in a chart of a translation structure, we may pull back the vector field @=@z from C. Since this vector field is invariant under all the translations of C, vector fields in different charts agree in the curve and give rise to a global nowherevanishing vector field. Reciprocally, for a curve C endowed with a holomorphic and nowhere-vanishing vector field, the inverses of the local solutions of the vector field give an atlas for a translation structure in C. A complex differential equation gives a foliation by complex curves together with a translation structure along each one of its leaves. Complex curves (one-dimensional complex manifolds) are real surfaces (twodimensional real manifolds). We can also say that a translation structure on a real surface is an atlas for its differentiable structure with charts taking values in R2 and changes of coordinates given by translations. We will shift back and forth from the real and complex points of view. A translation structure in a real surface allows one to locally import to it every local notion in R2 that is invariant under translations. For example, in a real surface † with a translation structure, it makes sense to say that a curve  in † is a line segment, the arc of a circle of radius r, or to say that  is traveled at speed one, since all of these are local properties that are invariant under translations. By the same reason, within such a surface †, we can also measure the angle of intersection of two curves, or have, at a point, a vector pointing in the direction .0; 1/ etc.

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Let us interpret the changes of coordinates leading to system (4) in this context. From system (1) we produced system (2). The latter is a system of ordinary differential equations with real time but with complex space. We may also consider it as a system of ordinary differential equations with both complex space and time. When doing so, the solutions uj will be defined in some set U  C and the solutions to the system with real time will be obtained from the solutions to the system with complex time by restricting uj to U \ R. We may now consider the change of independent variables (3) as one between complex differential equations (with both complex time and space). The real trajectories in the complex variable  become circular trajectories of radius ! 1 for the variable t. Thus, in order to recover a solution of the real differential equation (1) from a solution to the complex differential equation (4), it suffices, within the domain where the latter is defined, to travel counterclockwise along a circular arc of radius ! 1 (this makes sense because we have a translation structure in each orbit). From this point of view, isochronicity for the system (1) means that in the integral curves of (4), considered as curves with a translation structure, most of the real curves with radius of curvature r are actually circles of radius r.

1.4 Completeness and Semicompleteness Some of the properties of the solutions of a complex differential equation (like the definitions that follow) can be phrased in terms of the translation structure of its solutions. To some extent, solving a differential equation can be interpreted as understanding the translation structure induced in each and every leaf of its foliation. Definition 3 A translation structure on a complex curve C is said to be • complete if there exists a subgroup  C such that, as curves with a translation structure, C is isomorphic to C= , and • uniformizable if there exists a domain  C and a subgroup  C such that, through its action by translations in C, preserves and, as curves with a translation structure, C is isomorphic to = . (A complete translation structure is uniformizable, and in that case D C.) Definition 4 A holomorphic vector field (in a complex manifold) is said to be • complete if the translation structure of all of its orbits is complete, and • semicomplete if the translation structure of all of its orbits is uniformizable. (A complete vector field is semicomplete.) In other words, a holomorphic vector field is complete if its solutions are defined for all complex time. If a vector field is semicomplete and for one of its leaves C we have that C  = for some  C, a solution of the vector field is defined in , and is maximal as a domain where a solution can be defined. In a semicomplete vector field, its solutions may be globally

Differential Equations and Geometric Structures

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uniformized. In particular, the solutions present no multivaluedness. The notion of semicompleteness was introduced by Rebelo in [38] and makes precise the notion of “absence of multivaluation” in the solutions of complex differential equations. (See also the introduction to [24] for more details.) Example 5 Let us now give examples of the above definitions in the context of the two-body problem (5), when considering the vector field in V: (1) In the case where 12 D 2 and 21 D 2, the vector field is complete, for the most general solution is given by 1 .t/ D

1 1 .at C b/ C cekt C dkt ; 2 .t/ D .at C b/  cekt  d kt 2 2

for a, b, c, d and k in C such that 16cdk2 C a2 D 0. (2) If 12 D 3 and 21 D 3, the vector field is semicomplete (but not complete): the solutions are given by 1 .t/ D

a a 1 1 .at C b/ C .} 0 .t/  ac/; 2 .t/ D .at C b/  .} 0 .t/  ac/; 2 4} 2 2 4} 2

where a; b; c are constants in C and where } is the Weierstrass elliptic function satisfying .} 0 /2 D 4} 3 C a2 c2 . (A fourth parameter may be obtained by replacing }.t/ by 7! }.t C k/ in the above formula.) The function } is a meromorphic one defined in the whole plane having an infinite number of zeros and poles. The vector field is not complete since, for instance, the solution has poles (is not defined) at some of these zeros. (3) In the case 12 D 2, 21 D 2, the vector field is not semicomplete. Solutions are given by z2  .at C b/z C .ct C d/ D Œz  1 .t/Œz  2 .t/: Most solutions will present two ramification points of square root type. If (4) is complete then for every circle in C the corresponding orbit of (1) will be periodic with period 2. If (4) is semicomplete and its solutions are defined in sufficiently big subsets of C (like in sets with discrete complement), then almost every circle will be contained in the domain of definition of the solution for almost every initial condition, and system (1) will be isochronous. In the above examples, we are able to decide upon the completeness, semicompleteness or incompleteness of the equations by explicitly integrating them. The existence of closed forms for the solution of a differential equation occurs only in very exceptional situations. We must resort to some other ways of studying the equations. Some of these are geometric and will be explored in these notes.

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1.5 A Reduction The “goldfish” problem (1) is invariant by translations, rotations and homotheties of the plane in the sense that given a solution (corresponding to a motion of the N bodies), a new one may be obtained by simultaneously applying one of these transformations to the motions of all the bodies. System (4) also has symmetries, inherited from the latter. It is invariant by complex translations and homotheties: if .t/ is a solution and a; b 2 C with a ¤ 0, then a.t/ C b is a solution as well. Equivalently, for the system in the form (5), if ..t/; .t// is a solution and a; b 2 C with a ¤ 0, .a.t/ C b; a.t//

(8)

is also a solution. This gives a group of global transformations of the phase space that maps orbits to orbits, mapping the vector field in one to the vector field in the other, or, equivalently, a mapping between curves endowed with translation structures (a mapping that is a translation in the corresponding charts). We will restrict our study of (5) to the two-body case. In order to simplify this equation, we will begin by factoring out the above-mentioned symmetries. In the set V defined in (7), define the mapping … W V ! C2 , …

.1 ; 2 ; 1 ; 2 / !



2 1 ; 2  1 1   2

 :

The functions 1 .t/ D

1 .t/ 2 .t/ ; 2 D 2 .t/  1 .t/ 1 .t/  2 .t/

are the images of the solutions of (5) under …. The symmetries (8) have been factored out: these expressions are invariant under translations and homotheties of the original solutions, since, by defining .e  i .t/;e i .t// D .ai .t/ C b; ai .t//; a1 .t/ a1 .t/ e 1 .t/ 1 .t/ D D D : e 2 .t/  1 .t/ a2 .t/  a1 .t/ Œa2 .t/ C b  Œa1 .t/ C b  1 .t/  2 .t/  e These new functions satisfy the system of differential equations 10 D 12 C .1  ˛ 1 / 1 2 ; 20 1 for ˛ D 1 12 ,  D 21 .

D

22

C .1  

1

/ 1 2 ;

(9)

Differential Equations and Geometric Structures

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The leaves of Eq. (9) are images of the leaves of (4) under …. The projection … is more than a map between complex curves: it is a map between curves with a translation structure. In particular, if, within the leaf L  V, traveling a distance 2r along a curve of radius of curvature r gives us a closed curve, traveling a distance 2r along a curve of radius of curvature r will give us a closed curve in ….L/ as well. If the two-body original equations (4) induce uniformizable translation structures on the leaves within V, the translation structures induced in their images under … will be uniformizable as well. In other words, if system (9) has multivalued solutions, it is because the original system does too (but the reciprocal is not necessarily true). The new simplified problem is thus the following. Problem 6 Determine the values of ˛ and  such that the vector field corresponding to (9) is semicomplete (such that all the solutions to Eq. (9) are single-valued). More generally, describe the translation structure of the leaves of (9).

1.6 Translation Surfaces and Billiards In order to describe the translation structure of the orbits of (9), we must first know and get acquainted with some translation structures. We will describe a construction that associates real surfaces endowed with translation structures to mathematical billiards in triangular tables. Let T  R2 be a plane triangle. The game of billiards in the table T consists of letting a particle move freely within the interior of T and then reflecting it elastically when it reaches a side of T. (We do not define the future of the particle when it hits a vertex.) The problem is to describe the evolution of most trajectories, or to say something about the periodic orbits, etc. Although a bit off the subject, we will not resist the temptation of quoting the following renowned conjecture. Conjecture 7 Let T  R2 be a plane triangle. The game of billiards in T has a periodic orbit. The conjecture is known to be true for many triangles. For an acute triangle, the pedal triangle (the one having as vertices the feet of the heights) gives a periodic orbit, a result known to Fagnano in 1775. The conjecture also holds for those triangles whose angles are smaller than 100ı (see [41]), and many mechanisms producing periodic orbits exist [45]. We refer the reader to the expository article [44] for an introduction to some aspects of the mathematical theory of billiards in polygonal tables and to the presentation [3] for problems related to periodic orbits on triangular billiards. We will now construct the translation surface associated to a plane triangle. The construction appears in Fox and Kershner [15] although, through their work [46], Katok and Zemlyakov often associate their names to it. The idea for the construction is simple: to reflect the table instead of the trajectories.

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Let Isom.R2 / be the group of isometries of R2 with the Euclidean metric. Let T  R2 be a plane triangle. Let H  Isom.R2 / be the group generated by the reflections 1 , 2 and 3 on the sides `1 , `2 and `3 of T. In the disjoint union th2H h.T/, • glue h.T/ with h i .T/ along their common side h.`i ) and, • if h1 and h2 differ by a translation, identify h1 .T/ with h2 .T/. The resulting surface is the translation surface associated to T. (It is tautologically endowed with a translation structure.) In other words, to construct the translation surface, we start unfolding (or developing) the triangle by successive reflections on its sides and we identify those images differing by a translation. In the translation surface, the trajectories of the billiard have become straight lines: no reflections are involved. The translation surface will have finitely many images of the original triangle if all the internal angles are commensurable to 2, infinitely many otherwise. The construction allows for “degenerate triangles” to be considered as well, like triangles with a vertex at infinity, triangles with a vertex at infinity of angle zero (a triangle having two parallel sides), or triangles with only two sides (sectors); see Fig. 2. The construction can be, more generally, applied to other polygons, including nonconvex, self-overlapping, and degenerate ones. The constructions of the translation surfaces of two different triangles are illustrated in Figs. 3 and 4. In the first, no triangles are actually identified by any translation, and the successive images are only glued along its sides. In the second, corresponding to the triangle with internal angles =2, =3, and =6, the translation surface is a torus, with the points corresponding to the vertices of the triangles removed (a punctured torus). Translation surfaces associated to billiards are exactly the ones we need to understand the system of equations (9). We have the following result from [43]. Theorem 8 (Valdez) A nonradial orbit of the complex differential equation (9) is, as a complex curve endowed with a translation structure, the translation surface of a plane triangle with internal angles ˛, ˇ, and   (where ˇ is defined by the relation ˛ C ˇ C  D 1).

Fig. 2 A true triangle and some degenerate ones, where the construction of a translation surface can still be made

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Fig. 3 Construction of the translation surface for a general triangle

Fig. 4 Construction of the translation surface for the triangle with internal angles =2, =3 and =6

(In this statement, a radial orbit for (9) is one contained in a line through the origin of C2 .) Notice that the statement does not say which specific triangle with the given angles is associated to a particular orbit. We will give a proof of Valdez’s theorem in Sect. 2.4. It will certainly not be the simplest one, but it will guide us through some classical and historical results that will prepare us to study other differential equations.

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Valdez’s theorem gives a solution to Problem 6, which we now detail. Let T be a triangle. Consider the translation surface of T in the neighborhood of one of the vertices of T. By successively reflecting along the two sides defining the vertex, • if the corresponding angle is p=q with p and q relatively prime integers, the sequence of reflected images closes by making p turns after 2q reflections, and • if the corresponding angle is not commensurable with , the sequence of reflected images never closes. Thus, in the translation surface, an arc of a circle of small radius of curvature r centered at the vertex is a closed curve of length 2rp in the first case, and a curve of infinite length (which does not close) in the second one. For the system to be isochronous, the internal angles must be of the form =q, q 2 N. Up to similarities and reflections, there are not many triangles of this kind: the only ones are those with internal angles .=2; =4; =4/, .=3; =3; =3/, .=2; =3; =6/. These triangles are exactly the ones that tessellate the plane by reflections: we can fill the plane with isometric images of these triangles in such a way that the interiors of the triangles do not overlap and such that the reflection along any side of any triangle extends as a symmetry of the tessellation. This is remarkable. In these three cases, the group generated by reflections along the sides is a discrete subgroup of IsomC .R2 /, the group of orientation-preserving isometries of R2 . This group contains two independent translations and the translation surface is a torus, with some punctures corresponding to the vertices. In Fig. 5 we see, for each one of these cases, some reflected images of the original triangle forming a maximal collection not containing two images differing by translations. The translation surface (a punctured torus) is obtained by identifying, by translations, the opposite sides of the corresponding polygon. Most curves will omit the punctures coming from the vertices. There are also some degenerate triangles that tessellate the plane. We have, for example, the degenerate triangle having two right angles and one vertex—of angle zero—at infinity, and the degenerate triangles given by an infinite sector (a vertex of angle =n, a fake vertex with angle  and a vertex at infinity with angle =n). Taking these degenerate triangles into consideration, we may state the result as follows.

Fig. 5 Part of the construction of the translation surface for the triangles that tessellate the plane

Differential Equations and Geometric Structures

13

Proposition 9 In the cases where 12 ¤ 0 and 21 ¤ 0, for the two-body system (1) to be isochronous, 12 and 21 must be two of the three elements of one of the triples .3; 3; 3/, .2; 3; 6/, .2; 4; 4/, .2; 2; 1/, and .1; n; n/, for n 2 Z . The condition is equivalent to =12 and =12 being two of the internal angles of a (possibly degenerate) triangle that tessellates the plane.1 In principle, this does not prove that system (4) is isochronous, but only that the obstruction provided by its reduction (9) vanishes. It turns out that the necessary conditions in the above proposition are also sufficient, this is, that in these cases the corresponding two-body systems (1) are actually isochronous. We will not prove this here (two instances were already given in Example 5). We refer the reader to [18] for details and references and for an analysis of some sets of interaction constants yielding isochronous settings for Calogero’s “goldfish” (1) when larger sets of bodies are considered. In the case where 12 and 21 vanish, the solutions of (4) are linear functions of t and the corresponding systems (1) are isochronous. The cases where 12 D 0 but 21 ¤ 0, which do not give isochronous settings, will be briefly discussed in Sect. 2.5. From Theorem 8 we also recover the following corollary of [16, Théorème C], implicit in the work of Briot and Bouquet [4]. Corollary 10 When ˛ and  are real, the solution of the differential equation (9) is semicomplete (is free of multivaluedness) if and only if the triple .1=˛; 1=ˇ; 1= /, ˛ C ˇ C  D 1 is, up to reordering, one of .3; 3; 3/, .2; 3; 6/, .2; 4; 4/, .2; 2; 1/, or belongs to the family .1; n; n/, n 2 Z . The condition amounts to  ˛ and  being two of the internal angles of a (possibly degenerate) triangle that tessellates the plane. We refer the reader to [16] and [19, §2] for other proofs of this last result.

2 Affine Geometry We will now consider the group of affine transformations of the complex line, Aff.C/ D fz 7! az C bj a 2 C ; b 2 Cg; which we will refer to as the affine group. As a group of transformations of R2 , it is the group of orientation-preserving similarities of R2 , the one generated by translations, rotations and homotheties. It contains the group IsomC .R2 / of

1 With the case .2; 2; 1/ meaning 12 D 2 and 21 D 2, corresponding to a degenerate triangle with two vertices with right internal angles and a third vertex at infinity (with internal angle zero).

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A. Guillot

orientation-preserving isometries of R2 , which, on its turn, contains the group of translations: .C; C/ ,! IsomC .R2 / ,! Aff.C/: We have the following definition, analogue to Definition 2. Definition 11 A (complex) affine structure on a complex curve is an atlas for its complex structure taking values in C such that the changes of coordinates are in Aff.C/. We say that an affine structure is Euclidean if the changes of coordinates lie within the subgroup IsomC .R2 /. A Euclidean (and thus a translation) structure is a particular kind of affine structure. We may choose to consider the natural translation structure on the orbits of a vector field as a Euclidean or as an affine structure. In an affine structure, traveling at constant speed along a path is a well-defined concept, but the magnitude or the direction of the speed are not. Being a circular arc does make sense, but the radius of this arc is not intrinsically defined. Affine structures appear through some special symmetries of the system (9). If . 1 .t/; 2 .t// is a solution of (9) then, for every z 7! az C b 2 Aff.C/, .a 1 .at C b/; a 2 .at C b//

(10)

is also a solution (provided the expression makes sense). As before, the translational part of this invariance property expresses the fact that the differential equation is autonomous. For b D 0, the formula expresses the fact that system (9) is quadratic and homogeneous. (More generally, if . 1 ; 2 / is a solution of a homogeneous ordinary differential equation of degree dC1, .a 1 .ad t/; a 2 .ad t// is also a solution.) As the element z 7! az C b moves within a neighborhood of the identity in Aff.C/, for most initial conditions the solutions produced by (10) will sweep the initial condition within some open set of the two-dimensional space, and every solution close to a given one will be of the form (10). In the case of the previously studied symmetry (8) of (5), we had a global transformation of the phase space which permuted the orbits while preserving the natural translation structure in each one of them (which was an isomorphism of translation structures when restricted to each of these orbits). The symmetry we have now is a generalization of that one. Here, under a homothety of C2 , the orbits of the vector field are permuted, but the translation structure, the parametrization that makes them solutions of the equation, is no longer respected. However, if we think of the translation structure of the leaves of (9) as an affine structure, then the homotheties map one orbit onto another while preserving this affine structure! The restriction of such a homothety to an orbit induces an isomorphism of the affine structures induced by the natural translation structures. Another way to see this is the

Differential Equations and Geometric Structures

15

following: given a solution .t/, its image under a homothety a .t/ will parametrize another orbit of the same equation, but will not be a solution. In order to make it one we need to modify the parametrization, and this is done by rescaling the independent variable, obtaining the solution a .a/. Let us now factor out this symmetry in order to reduce the system. Consider the mapping   ˛ 1 . 1 ; 2 / 7! :  2 It realizes the quotient under the action of homotheties. (The coefficient ˛= is there to simplify some of the expressions that will follow.) Let .t/ D . 1 .t/; 2 .t// be a solution of (9), and let s.t/ D … ı , s.t/ D

˛ 1 .t/ :  2 .t/

(11)

If we form this expression with the solutions built from via (10), we obtain ˛a 1 .at C b/ D s.at C b/:  a 2 .at C b/ This is, the inverse of s.t/ depends upon the particular solution, but in a controlled way: the inverse of s.t/ is well-defined up to post-composition with an element of the affine group. The fibers of … are transverse to the vector field (9) above 0, 1 and 1. Thus, for every s0 2 C n f0; 1g, there is a local solution 0 W U ! C2 (U  C) such that … ı 0 W U ! C n f0; 1g is a diffeomorphism from U onto a neighborhood of s0 . The inverse of this map is defined in a neighborhood of s0 and takes values in C. If we modify this construction by starting with another solution, will be modified by post-composition with an affine map. All this amounts to the following: the differential equation (9) induces an affine structure in C n f0; 1g whose charts are the inverses of the expressions … ı , for the local solutions  to Eq. (9).

2.1 An Adapted Differential Operator There is a natural differential operator related to the affine group. Let U  C and let f W U ! C. Let A. f ; z/ D

f 00 : f0

(12)

First, A. f ; z/  0 if and only if f is (the restriction of) an affine map: the operator A. f / measures the extent to which f is not affine. Second, A. f ; z/ D A.af C b; z/: the differential operator is invariant under the action of the affine group. Finally, with respect to compositions, A. f ı g; z/ D A. f ; g/g0 C A.g; z/.

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A. Guillot

Let us use this for our problem. Since, in (11), t.s/ is only defined up to postcomposition with an affine transformation, A.t; s/ is well-defined. It can be easily calculated: A.t; s/ D

t.s/00 s00 .t/ ˛1  C˛  : D  D t.s/0 .s0 .t//2 s s1

Thus, the charts of the affine structure induced by (9) on C n f0; 1g through (11) are given by the solutions of the linear differential equation 

f 00 .s/ D

 ˇ1 0 ˛1 C f .s/; s s1

(13)

where ˇ is defined by the equality ˛ C ˇ C  D 1. Schwarz’s theorem on the uniformization of polygons (Theorem 13) will give us a global understanding of the solutions of this equation. Before presenting it, we will analyze the solutions of the above equation around the points s D 0 and s D 1, where it does not satisfy the usual conditions that guarantee the existence of solutions (these points are called singularities). The following result describes the solutions of this equation in the neighborhood of these points. It is a particular instance of Fuchs’s theorem, that will be discussed later (Theorem 18). Proposition 12 Consider the differential equation sf 00 .s/  g.s/f 0 .s/ D 0;

(14)

where g is a holomorphic function in a neighborhood of s D 0. Let  D 1 C g.0/ and suppose that  is not a nonpositive integer. Then, there exists a holomorphic function  defined in a neighborhood of s D 0 with .0/ ¤ 0 such that f .s/ D s .s/ is a solution of the above equation. Notice that the solution s .s/ is in general a multivalued function of s, but that its multivaluedness, being controlled by the factor s , is easily described. Proof Equation (14), considered as a first-order equation for f 0 , can be explicitly R g.s/=s 0 solved, obtaining f .s/ D e . If g.s/ D g.0/ C sG.s/, f 0 .s/ D e

Rs

g.0/d=

e

Rs

G./d

D s1 u.s/

for some holomorphic function u such that u.0/ ¤ 0. If Œs .s/0 D s1 u.s/ then  must be a solution to the equation s 0 C  D u.s/:

(15)

Let us try to find a solution P1 toi this equation via the method of undetermined coefficients. Let u D iD0 ui s be the Taylor development of u and let  D P1 i a s . Equation (15) reads .i C /ai D ui for every i  0. The condition on  i iD0

Differential Equations and Geometric Structures

17

implies that the coefficient of ai in this equation does not vanish and that we may thus set ai D ui =.i C /. We must now prove that the series for  converges. If i is big enough, ji C j  1 and thus jai j jui j if i is big enough. The series for  is convergent, for such is the series for u. Since u.0/ ¤ 0, .0/ ¤ 0. t u If  is a nonpositive integer, logarithmic terms may appear in the solution of (14), like for the equation sf 00 C f 0 D 0, that has log.s/ as solution (a case where  D 0).

2.2 Uniformization of Triangles According to Schwarz, I For his theorem on the uniformization of plane domains, Riemann’s proof was not quite accepted in his time (for a proof, see [1]; for a historical presentation, see [12]). Efforts to formalize Riemann’s proof and efforts to give rigorous proofs for particular cases followed. Schwarz proved Riemann’s theorem for plane polygons bounded by circular arcs. We will now present a simple instance of Schwarz’s result, that of the uniformization of triangles bounded by straight lines. Let H D fz 2 C; =.z/ > 0g be the upper half-plane. Theorem 13 (Schwarz) Let ˛; ˇ;  2 .0; 1/, ˛ C ˇ C  D 1. Let f W H ! C be a nonconstant solution of the differential equation 00

f .s/ D



 ˇ1 0 ˛1 C f .s/: s s1

(16)

Then f is one-to-one, f .H/ is the interior of a triangle of internal angles ˛, ˇ and  , and f extends continuously to the boundary of H (in CP1 ). Let us begin by proving that f has the sought behavior along the real axis. Equation (16) has real coefficients. It can be considered either as a differential equation in the complex domain or as a differential equation in the real one. Moreover, there is compatibility between the solutions of these two viewpoints in the following sense: for every s 2 .0; 1/ there is a solution f W U ! C, defined in a neighborhood U of s in C such that f .U \ R/  R and such that f jU\R is a solution of the real differential equation. There is thus one local solution of (16) that maps the interval .0; 1/ into the real line. The differential equation (16) is a linear one. In particular, if f and g are two linearly independent local solutions of the equation, all other solutions have the form af C bg with a; b 2 C. Since the constant functions are a solutions to (16), all solutions are of the form af C b, for a nonconstant solution f to the equation. (For example, the one mapping the interval .0; 1/ to the reals.) These are compositions of f and of the affine transformation z 7! az C b. Since affine transformations are similarities of R2 and map real lines into real lines, every solution of (16) transforms the interval .0; 1/ into a straight line. The same happens for the other components of R n f0; 1g.

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A. Guillot

We need now to understand the behavior of the solution in the neighborhood of the point 0, that will generate one of the vertices of the triangle. By Proposition 12, there exists a (multivalued) solution of (16) of the form f .s/ D s˛ .s/ with .0/ ¤ 0. p ˛ In the variable z D s .s/, this function is z 7! z˛ . In polar coordinates, if z D rei , z˛ reads .r; / 7! .r˛ ; ˛/. In particular, any determination of s˛ .s/ in a sector has the effect of multiplying by ˛ the angle between the two curves that delimit the sector. Thus, there is a solution to (16) that maps the sector defined by H at 0 to a sector of angle ˛. By our previous discussion, this sector in the image is delimited by two straight lines. This produces a vertex of angle ˛ for the triangle. By the same arguments, in a neighborhood of s D 1, the image of H under f is a sector of angle ˇ delimited by two straight lines. The last vertex of the triangle will be produced by the point s D 1. Let g. y/ D f .1=y/. We have g00 D



 ˇ1 0  1 C g; y y1

which is an equation of the same type. The third vertex of the triangle, of angle  , is formed by the point s D 1. In this way, the image of R [ f1g under f is the contour of a triangle with the specified angles. Let C be this image, as a positively oriented curve. Let T be the interior of the triangle bounded by C. We have yet to prove that f is a biholomorphism from H into the interior of T. Let p 2 C n C. Let E be a curve within the closure of H that follows R [ f1g, the boundary of H, except for a slight modification in order for E to avoid the vertices, as in Fig. 6. We will suppose that the difference between E and R[f1g is contained in the neighborhoods of the vertices covered by Proposition 12. Let C0 D f .E/ be the slight modification of C that remains within T but that avoids the vertices and let T 0 the region bounded by C0 . We will suppose that p is still within T 0 if p was originally in T. The formula for the index of a curve with respect to a point says that 1 2i

0

Fig. 6 The path of integration

I C0

dw D wp

1



1 if p 2 T 0 : 0 if p … T 0

Differential Equations and Geometric Structures

19

Under the change of variables w D f .z/, we have 1 2i

I C0

1 dw D wp 2i

I E

f 0 .z/dz : f .z/  p

Now, the argument principle affirms that if E is a positively oriented simple closed loop in C bounding a region K, then for every meromorphic function h I #.zeros of h in K/  #.poles of h in K/ D E

h0 .z/ dz: h.z/

Since f .z/  p is holomorphic, it has no poles, and  #.zeros of f .z/  p/ D

1 if p 2 T 0 : 0 if p … T 0

Thus, if p … T 0 , there is no z in the region bounded by E such that f .z/ D p, and if p 2 T, there exists a unique z in this region such that f .z/ D p. Thus, f is a biholomorphism onto its image. This finishes the proof of Theorem 13. Remark 14 If in Eq. (16) we have that ˛ < 0, the resulting triangle will be degenerate and will have a vertex at infinity (like some of the triangles appearing in Fig. 2) of angle j˛j.

2.3 The Monodromy of the Linear Equation Consider the linear differential equations of the form g2 .s/f 00 .s/ C g1 .s/f 0 .s/ C g0 .s/f .s/ D 0;

(17)

where, for every i, gi W C ! b C is a meromorphic function which is holomorphic in some set V  C. In general, the solutions of these equations are multivalued. This multivaluedness is described by a group, the monodromy group of the equation. Let z0 2 V. Let f1 and f2 be two linearly independent solutions to the differential equation defined in a neighborhood of z0 , W Œ0; 1 ! V a closed path based at z0 , and e f 1 and e f 2 the analytic continuations of f1 and f2 along . Throughout the analytic continuation, they remain linearly independent solutions of the differential equation. At z0 D .1/, both . f1 ; f2 / and .e f 1 ;e f 2 / are a basis for the space of solutions. Thus, there exists an invertible linear transformation  such that  . fi / D e f i . This linear transformation depends only upon the homotopy class of in V, defining a map  W 1 .V; z0 / ! GL.2; C/, which is a actually a group morphism: the monodromy. Let us calculate this monodromy for the linear equation (16) and describe its action through a polyhedral model. Let f W H ! C be a nonconstant solution of

20

A. Guillot

equation (16), T D f .H/ the triangle uniformized by f . Let V D C n f0; 1g and let W Œ0; 1 ! C n f0; 1g be a path such that .0/ 2 H. We will describe the analytic continuation of f along . If stays forever within H there is nothing to say, so suppose that, eventually, crosses the real axis. We have the following. Proposition 15 (The Schwarz Reflection Principle) If the holomorphic function g is defined in a neighborhood of a real point and is real (maps reals to reals), g.z/ D g.z/. Proof The function g.z/ is holomorphic and agrees with the holomorphic function g.z/ along the real line. t u According to this principle, in order to obtain the solution in the lower half-plane, first reflect onto the upper one (apply complex conjugation), then apply the function, then reflect again. In the context of the linear equation (17), let a; b 2 C be such that the solution g W H ! C of (16) defined by f D ag C b

(18)

maps the first real interval met by the path into R. From the Schwarz reflection principle for g and (18), f .z/ D

a Πf .z/  b C b: a

(19)

On the other hand, the involutive R-linear mapping z 7!

a Œz  b C b a

fixes the real line ` D fax C b; x 2 Rg, image of the real interval under f , and is thus the reflection ` along this line. Equality (19) can be read as f .z/ D ` ı f .z/. Thus, in order to obtain the analytic continuation of f along the path as the latter first crosses the real line, we need to • reflect z onto the upper half-plane z 7! z, • apply f , z 7! f .z/ and • reflect along `, f .z/ 7! ` ı f .z/. The composition of these gives, by (19), f .z/. We can now describe the analytic continuation of f along the path W Œ0; 1 ! C n f0; 1g. We have that .0/ 2 H and that f .H/ is a triangle T. At some point, will cross the real axis along a segment corresponding to a side ` of T. The analytic continuation of f along is a function defined in the lower half-plane H whose image is the reflection of T along `. This goes on. At each time that intersects the real line, the image of the half-plane into which enters under the analytic continuation of f will be the triangle obtained from the immediately preceding one by reflection along the corresponding side.

Differential Equations and Geometric Structures

21

The monodromy of (16) associates to the homotopy class of the closed path W Œ0; 1 ! C n f0; 1g the element of Aff.C/—actually, of IsomC .R/2 —obtained by composing the reflections along the subsequent sides of the triangles met by f ı . (In the group generated by reflections on the sides of T, there is an index-two subgroup consisting of orientation-preserving isometries.) The construction we just described is exactly the first part of the construction of the translation surface, where we started reflecting the triangle along its sides (before identifying copies of the triangle differing by translations).

2.4 Back to the Translation Surface Let us rephrase the construction of the translation surface of a triangle T. The point of view that follows (present, for example, in [27] and [19]) has the advantage of revealing the role that the structure of the group of Euclidean motions plays in our previous definition. As before, let T  R2 be a triangle with sides `1 , `2 and `3 and internal angles ˛, ˇ and  . Let W R2 ! R2 be an orientation-reversing isometry. For each side `i there is a unique orientation-preserving Euclidean motion i of the plane that maps `i to .`i / such that the interiors of T and i .T/ do not intersect. Construct a sphere with three conic points by identifying the side `i of T with the side .`i / of .T/ via i . The three conic points, corresponding to the three vertices of the triangle, have angles 2˛, 2ˇ and 2, in accordance with the GaussBonnet theorem (more precisely, to Descartes’ theorem on the total angular defect of a polyhedron). Let S be the thrice-punctured sphere obtained by removing the conic points. It is naturally endowed with a Euclidean structure. Some other objects come with the Euclidean structure in S. Let … W e S ! S be the universal covering of S. Notice that e S has a natural Euclidean structure that makes … a mapping between curves with Euclidean structures. A chart for this structure from an open subset of e S into C globalizes into a developing map D W e S ! R2 . The group e of deck transformations in S acts by preserving the Euclidean structure, giving rise to a holonomy morphism h W 1 .S/ ! IsomC .R2 / that satisfies, with the developing map D, the following equivariance property: D.  p/ D h. /  D. p/:

(20)

(We refer to [43, §3.4] for details on these). Let us describe these objects. Let p 2 S be a base point lying within the interior of T and let W Œ0; 1 ! S be a closed path. Start developing the surface S by mapping T to itself and then along by taking the reflections along the sides of the images of T as they are crossed by . (This spreads the surface constructed in the first part of our first construction onto the plane.) The resulting map depends only upon the homotopy (with fixed endpoints) class of and is thus well-defined in e S, the universal covering of S. The difference between the identity at p and the map

22

A. Guillot

that results from following is a Euclidean motion h. / 2 IsomC .R2 /, mapping T to the image of T under the composition of (an even number of) reflections on the sides successively met by the image of . This is the holonomy of . By definition, we have relation (20). Let us now rephrase the construction of the translation surface of T. There is a natural map L W IsomC .R2 / ! SO.2; R/ that associates to every Euclidean motion its linear part. There is thus a morphism L ı h W 1 .S/ ! SO.2; R/. This b W morphism has a kernel, and associated to this kernel we have a Galois cover … b S ! S. Tautologically, b S carries a translation structure. It is the translation surface associated to T. It is, by definition, the smallest cover of S that carries a translation structure. Proof of Theorem 8 Let us finally proceed to the proof of Valdez’s theorem. Our first claim is that, as complex curves endowed with affine structures, S—with the structure just defined—and C n f0; 1g—with the structure induced by (9), whose charts are the solutions to (13)—agree. By Fuchs’s theorem, f establishes a biholomorphism between H and the interior of T, which is, by definition, an automorphism of the affine structures. By the Schwarz reflection principle, the analytic continuation of f to H is a biholomorphism to the image of T under a reflection, which proves our first claim. Let C be an integral curve of the vector field (9). Let …jC W C ! S be the projection. It is a Galois cover, whose group of deck transformations is given by the homotheties of C2 that preserve C. It carries, tautologically, a translation structure. We need to prove that it is the smallest cover of C that does. If this cover were not the smallest one having a translation structure, we would have a homothety mapping C to itself while preserving the vector field (the translation structure of C). However, a nontrivial homothety of the form (10) is not an isomorphism of translation structures: if a homothety maps one orbit of X to another and if this homothety is a map of curves with translation structures, it is the identity. We must conclude that …jC is indeed the smallest covering of S carrying a translation structure. This proves Theorem 8. t u

2.5 The Remaining Settings Some settings for the “goldfish” system are out of the scope of Proposition (9). Let us briefly comment on them. If 21 D 0 and 12 ¤ 0, by setting ˛ D 1 12 , Eq. (9) becomes 10 D 12 C .1  ˛ 1 / 1 2 ; 20

D

22

C 1 2 :

(21)

Differential Equations and Geometric Structures

23

As before, by setting s.t/ D 1 .t/= 2 .t/, we have A.t; s/ D

t.s/00 s00 .t/ ˛1 C ˛: D 0 2 D 0 t.s/ .s .t// s

The charts of the affine structure induced by (21) on Cnf0g are given by the solutions of the linear differential equation 00



f .s/ D

 ˛1 C ˛ f 0 .s/: s

There are two singularities of this equation (at s D 0 and s D 1). The one at s D 0 falls within the scope of our previous analysis. In particular, in order to have isochronicity for (1), we must have that ˛ 1 is a nonzero integer. As before, in order to study the singularity at s D 1, let g.z/ D f .1=z/. We have g00 .z/ D 

˛ C .˛ C 1/z 0 g .z/; z2

which has a singular point at z D 0 that is not regular (that is an irregular singular point). This is incompatible with uniformizability (see [24, prop. 6]). For example, for ˛ D 1 we have g.z/ D e1=z , which has an essential singularity at z D 0 and which is very far from having an inverse. In order to have another description of the affine structure induced by f in C in the cases where ˛ D 1=n, n 2 Z , let us consider the affine structure induced in the n-fold ramified cover of C , by setting s D n . Let h. / D f . n /. A straightforward calculation shows that h satisfies the differential equation h00 . / D zn1 h0 . /. This is the affine structure in C induced by the vector field exp.n1 n /@=@ . For n > 0, a detailed description of these translation structures, both in the complex plane and in a neighborhood of 1, has been given, among more general ones, by Álvarez-Parrilla and Muciño-Raymundo in [2]. With this, we close our discussion around the “goldfish” system.

3 Aside: Some Elliptic Functions Some meromorphic functions f W C ! b C have the property of having two independent periods, this is, there exist !1 ; !2 2 C such that f .z C n!1 C m!2 / D f .z/ for every .n; m/ 2 Z2 . These are called elliptic or doubly periodic functions. One of the most famous one is the Weierstrass } function, the solution of the differential equation .} 0 /2 D 4} 3  g2 }  g3

(22)

24

A. Guillot

that has a pole at 0. (Here, .g2 ; g3 / 2 C2 , g23  27g32 ¤ 0.) By using the previously established results, we will prove that for some real values of .g2 ; g3 /, the associated Weierstrass function is indeed elliptic. Let Q.z/ D 4z3  g2 z  g3 , so that Eq. (22) reads .} 0 /2 D Q.}/. We will make the assumption that Q has three real roots, e1 , e2 and e3 . Derivating, } 00 D

1 0 Q .}/: 2

(23)

Let f be the left inverse of }, the function such that f .}.t// D t. Derivating twice and using (22) and (23), 1 0 D f 00 .}/.} 0 /2 C f 0 .}/} 00 D f 00 .}/Q.}/ C f 0 .}/Q0 .}/: 2 Thus, f is a nonconstant solution of the linear differential equation f 00 D 

1 Q0 .t/ 0 f: 2 Q.t/

(24)

We claim that f .H/ is a rectangle and f W H ! f .H/ is a biholomorphism that extends continuously to the boundary. Since 1 1 1 Q0 D C C ; Q z  e1 z  e2 z  e3 then, by the arguments of Sect. 2.2, the image of R under f is a piecewise linear curve with right angles at e1 , e2 , e3 . For the point z D 1, we may argue as before, letting g.w/ D f .1=w/, calculating g00 =g0 , and establishing that g produces a right angle at 0 (that f produces a right angle at 1). By the previous arguments, f jH uniformizes a rectangle. As before, following a solution f along a path consists of reflecting the solution along the sides of the rectangle that are successively met by . Figure 7 illustrates the situation. Let  Isom.R2 / be the group generated by the reflections along the sides of R. Then

Fig. 7 The analytic continuation of the solutions; the two paths give two independent translations (periods)

Differential Equations and Geometric Structures

25

• the images of R under fill the entire plane and • if g1 ; g2 2 are such that the interiors of g1 .R/ and g2 .R/ intersect, then g1 D g2 . These two conditions imply that the inverse of f is a legitimate meromorphic function } W C ! C [ f1g defined in all of the complex plane. Upon following paths like those indicated in Fig. 7, the composition of two reflections along two parallel lines will give a translation. This translation is a period for the inverse function. We have proved that the function } has two linearly independent periods and that it is indeed elliptic. Remark 16 For the equations under consideration, it is not difficult to prove that the aspect ratio of the uniformized rectangle depends only upon the quotient j D 1728g32=.g32  27g23 /. This dependence is transcendental, and it is in general difficult to say the value of j needed to uniformize a rectangle of a specific aspect ratio. However, if g3 D 0, it is easy to see that the rectangle is actually a square. For example, if Q D x.x  1/.x C 1/, the Möbius transformation z 7! .z  1/=.z C 1/ is a symmetry of Eq. (24) that cyclically permutes the four singularities, 1; 1; 0; 1.

4 Projective Geometry There exist systems of ordinary differential equations in three variables having solutions that are not multivalued and that have singularities such as natural boundaries or Cantor sets of essential singularities (this does not happen for systems in two variables [23]). Some examples of systems having this behavior are intimately related to one-dimensional complex projective geometry. As we study these examples, we will find equations that are to (9) what triangles in hyperbolic and spherical geometry are to triangles in the Euclidean plane. In this part we will mainly follow [20].

4.1 Möbius Transformations The group of projective, Möbius or of fractional linear transformations is the group of self-mappings of the complex projective line CP1 (the extended complex plane C [ f1g) of the form z 7!

az C b ; cz C d

(25)

with a; b; c and d complex numbers (defined up to a common change of sign) such that ad  bc D 1. The group is isomorphic to PSL.2; C/, the quotient of SL.2; C/—the group of 2  2 invertible matrices of determinant 1—by its center,

26

A. Guillot



 1 0 generated by . The identification is given by associating the Möbius 0 1   ab transformation (25) to the matrix . cd This group is also the group of orientation-preserving, angle-preserving transformations of the sphere S2 (with respect to the round metric). The full group of angle-preserving transformations of the sphere is generated by inversions and the geometry of Möbius transformations is also called inversive geometry. In this geometry, being a straight line is not well-defined: despite the fact that lines are preserved by the complex affine group, they are not preserved by inversions, for an inversion may map a line to a circle. Being a circle or a line does make sense (although the radius of a circle, the position of its center or traveling it at constant speed does not). Some well-defined concepts are, for example, the angle between two curves and reflection with respect to a circle or line. Other concepts attached to this geometry are, for instance, the cross-ratio of four points (measuring their relative position in CP1 ), the inversive distance of two nonintersecting circles [10] or the conformal curvature and conformal arc length of a curve [5]. The group of Möbius transformations has the affine group as the subgroup that fixes the point at infinity. Thus, Euclidean geometry is a particular instance of complex one-dimensional projective geometry. Hyperbolic and spherical geometry are other instances of it. If in the fractional linear transformation (25) the coefficients are real, it preserves the upper half-plane H together with its hyperbolic metric. Further, the image in PSL.2; C/ of the group of unitary matrices in SL.2; C/ of the form 

˛ ˇ ˇ ˛



; j˛j2 C jˇj2 D 1

is conjugate to SO.3; R/, the group of rigid rotations of the sphere. In analogy with Definitions 2 and 11, we may say that a complex curve has a projective structure if it has charts for its complex structure taking values in CP1 whose changes of coordinates lie within PSL.2; C/. Particular cases of projective structures are hyperbolic structures—charts in H whose changes of coordinates lie within PSL.2; R/—or spherical ones—charts in CP1 whose changes of coordinates lie within SO.3; R/. In general, we can endow a real or complex manifold M with the geometry of a model. What we mean by model is a simply connected manifold X and a Lie group G acting transitively upon X. By endowing M with the geometry of the model we mean giving an atlas for M with charts taking values in X and changes of coordinates given by restrictions of the transformations that G induces in X. These are called .G; X/-structures by Thurston [42]. The above ones as well as those in Definitions 2 and 11 fall within this scope. The construction of the developing map (20) done in Sect. 2.4 carries on to this more general context.

Differential Equations and Geometric Structures

27

4.2 The Equations of Chazy, Halphen and Ramanujan In [37], Ramanujan considered some holomorphic functions P; Q and R of arithmetical nature defined in the upper half-plane H. Ramanujan’s functions Q and are,  R ab respectively, modular forms of weights 4 and 6. This means that for every 2 cd SL.2; Z/, the group of 2  2 matrices with integer coefficients and determinant 1, 

 at C b Q D .ct C d/4 Q.t/; ct C d   at C b D .ct C d/6 R.t/; R ct C d

(26) (27)

and that Q and R are holomorphic at the cusp (their limits as z ! 1 exists). Ramanujan showed that, together with the auxiliary function P, also defined in H, they satisfy the system of polynomial differential equations dP D . P2 C Œ36 2  1Q/; dt dQ D 4. PQ  R/; dt dR D 6. PR  Q2 /; dt

(28)

for   D i=6,  D 0. The function P is not far from being a modular form: for ab 2 SL.2; Z/, cd 

at C b P ct C d



D .ct C d/2 P.t/ C

6 c.ct C d/: i

(29)

Other classical systems of ordinary differential equations are related to the system (28). If P.t/ is part of a solution to (28) for  D 1=6, P.t/ is also a solution of the third-order equation  000 D 2 00  3. 0 /2 C

4 .6 0   2 /2 ; 36  k2

for k2  2 D 1. This equation was considered by Chazy in [9] and is known as Chazy XII.

28

A. Guillot

System (28) is also closely related to a family of systems introduced by Halphen in [25], given by x0 D ax2 C .1  a/.xy  yz C zx/; y0 D by2 C .1  b/.xy C yz  zx/;

(30)

z0 D cz2 C .1  c/.xy C yz C zx/: (The particular case a D 0, b D 0, c D 0 was considered by Brioschi, Darboux and Halphen; see [19, §3.1].) In the symmetric case b D a, c D a, we can consider the quotient of the system under the action of the group of permutations of three variables. Letting i denote the elementary symmetric functions in x; y; z, .s  x/.s  y/.s  z/ D s3  1 s2 C 2 s  3 , the functions  . P; Q; R/ D

 .3a  2/3 

1 .3a  2/2 2 ; Π1  3 2 ; 9 1 2  2 13  27 3 3 36 432



form a solution to (28) for  D 1 and  D 12 a=.3a  2/: Our aim is to describe the geometry of the solutions of these equations and, through it, to show that there are many semicomplete vector fields within the family (28).

4.3 Halphen’s Invariance Property We will focus on the system of equations (28). In it, the parameter  is inessential and can be modified by reparameterizing the solutions. We will henceforth consider the case  D 1 exclusively. The modularity (26) and (27) of Ramanujan’s functions and relation (29) are particular instances ofa more  general phenomenon: if .P; Q; R/ is a solution to the ab system (28) and A D 2 SL.2; C/, then cd   1 c at C b  ; P 2 .ct C d/ ct C d ct C d   1 at C b b ; Q Q.t/ D .ct C d/4 ct C d   1 at C b b ; R R.t/ D .ct C d/6 ct C d b P.t/ D

(31)

will still form a (not necessarily different) solution, provided the expressions make sense. This is called the invariance property by Halphen [25]. Notice that this

Differential Equations and Geometric Structures

29

expression depends only upon the class of A in PSL.2; C/. We will consider the function s.t/ D Q3 .t/=R2 .t/. It depends upon the chosen solution. The key point, that will be exploited in the paragraphs to come, is that the inverse of s is welldefined up to a fractional linear transformation, independently of the solution considered. Let us, following [20, Section 3], give a Lie-theoretic explanation of the fact that (31) gives a solution of the differential equation (28) whenever .P; Q; R/ does. We will consider SL.2; C/ as a Lie group. Consider the matrices HD

1 2

0



 ; XD

0  12

   01 00 ; YD : 00 1 0

(32)

Under the bracket (commutator) ŒA; B D AB  BA, these matrices satisfy the Lie algebra relations ŒH; X D X; ŒH; Y D Y; Œ Y; X D 2H:

(33)

These matrices form a basis of sl.2; C/, the Lie algebra of 2  2 traceless matrices. Consider the following one-parameter groups in SL.2; C/: 

et=2 0 0 et=2



 ;

1 t 01



 ;

10 t 1

 :

(34)

They are generated, respectively, by the above matrices. (The one-parameter group t 7! eAt is generated by A.) We will consider the three flows (actions of C) in SL.2; C/ given by multiplication to the right by each one of the above one-parameter   ab groups. For example, the orbit of the flow associated to X starting from is cd 

ab cd



1t 01



 D

 a at C b : c ct C d

(35)

To each one of these flows corresponds a vector field in SL.2; C/. Further, since the orbits of one of these vector fields are permuted (while preserving their parametrization) upon multiplication to the left by an element of SL.2; C/, these vector fields are left-invariant. Thus, for each matrix in (32), there is a left-invariant vector field in SL.2; C/, which will be denoted by the same letter. The Lie brackets of these vector fields are, again, given by (33). Let us now consider the vector fields in C3 @ @ @ @ b b D x C 2y C 3z ; YD ; H @x @x @y @z @  @ @ b C 4.xy  z/ C 6.xz  y2 / : X D x2 C y @x @y @z

30

A. Guillot

b and b The third is the one associated to the differential equation (28). The flows of H Y are easily integrated: in time t, they are respectively given by t

.x; y; z/ ! .et x; e2t y; e3t z/;

(36)

t

.x; y; z/ ! .x C t; y; z/:

(37)

The Lie bracket relations of the above vector fields are exactly the ones in (33): b b b b b ŒH; X D b X; ŒH; Y D b Y; Œb Y; b X D 2H:

(38)

The vector fields are linearly independent in the complement of fz2  y3 D 0g. We have sets of vector fields, in both SL.2; C/ and C3 , satisfying the same Lie bracket relations. Lie’s third theorem extends the theorem of existence and uniqueness of solutions from one vector field to a Lie algebra of them. In our setting it states the following. b are Theorem 17 Let p 2 C3 be a point where the vector fields b X, b Y and H linearly independent. There exists a neighborhood U of the identity in SL.2; C/ and holomorphic map ˆ W U ! C3 such that ˆ.I/ D p, that for all q 2 U, and b ˆ.q/ , and Dˆjq Y D b that Dˆjq X D b Xjˆ.q/ , Dˆjq H D Hj Yjˆ.q/ . The germ of ˆ at the identity matrix I is unique. Let us prove that, through this theorem, the bracket relations (38) together with b imply the invariance formula (31). In SL.2; C/, the explicit form of b Y and H consider the equality 

ab cd



1t 01

 D

1 atCb ctCd 0 1

!

1 ctCd

0 0 ct C d

!

1 0 c ctCd 1 

! :

 ab in time t. The cd equality says that, in order to recover this orbit, we may proceed as follows:   1 • start with the orbit of X having the identity as initial condition in time , ; 01 ! 1 atCb ctCd ; • reparametrize the orbit by  D .at C b/=.ct C d/, to obtain 0 1 • for the point with parameter t,!follow the flow of H for time 2 log.ct C d/, this 1 0 to the right; and finally, is, multiply by ctCd 0 ct C d • for the point with parameter t, follow the flow of Y for time c=.ct C d/, this is, ! 1 0 to the right. multiply by c ctCd 1 The left-hand side gives the orbit of X with initial condition

Differential Equations and Geometric Structures

31

Following these steps gives the right-hand side of the equation. Via the correspon3 dence ˆ established   by Lie’s theorem (17), we may translate into C this equality: ab given A D 2 SL.2; C/, in order to obtain the orbit of b X through ˆ.A/, we cd • • • •

start with the orbit of b X through p, .P./; Q./; R.//, reparametrize the orbit by  D .at C b/=.ct C d/; b for time 2 log.ct C d/; for each t, follow the flow (36) of H after this, for each t, follow the flow (37) of b Y for time c=.ct C d/.

This is exactly formula (31). We have here a situation analogue to the one we had in (10). Here, from a solution to the equation and an element A 2 SL.2; C/, we parametrize another orbit of the same equation. This parametrization is not its natural one as a solution to the differential equation, and we must correct it in order to obtain the natural one. The correction is given by applying a fractional lineal transformation to the independent variable. If we consider the projective structure in the orbits of (28) induced by the natural translation structure, relation (31) maps one orbit to another while preserving this projective structure. Let us give a more or less explicit form of ˆ. Let .P.t/; Q.t/; R.t// be a local solution to (28) such that .P.0/; Q.0/; R.0// D p. Let ‰ be the map defined in a neighborhood of the identity of SL.2; C/ and taking values in C3 given by 

ab cd



 7!

      c 1 1 1 b b b  ; ; : P Q R d2 d d d4 d d6 d

By restricting this map to an orbit of the one-parameter groups (34), 

ab cd



1 t 01 



    c 1 1 at C b at C b 7!  ; ; P Q .ct C d/2 ct C d ct C d .ct C d/4 ct C d   1 at C b R .ct C d/6 ct C d

gives the solutions of b X (this is the invariance formula); 

ab cd



et=2 0 0 et=2



 7!

    3t   c e2t e et b b b  ; 4Q ; 6R P 2 d d d d d d d

b in time t applied to ‰.A/; and is the flow (36) of H 

ab cd



1 0 t 1



 7!

      c 1 1 1 b b b  C t; 4 Q ; 6R P 2 d d d d d d d

32

A. Guillot

is the flow (37) of b Y in time t applied to ‰.A/. In this way, ‰ satisfies the conditions in Lie’s theorem (17) and, by uniqueness, ˆ  ‰. By derivating with respect to t in each case, this also shows that the invariance formula (31) implies the bracket relations (38).

4.4 The Schwarzian Derivative In a way analogue to that of Sect. 2, we will use the symmetry (31) to reduce the system of differential equations (28). Let s W C3 Ü CP1 be given by s.t/ D

Q3 .t/ : R2 .t/

(39)

Given a solution  W U ! C3 of the differential equation, we may map by the invariance formula (31) a given solution to the neighboring ones. This is achieved by gliding each point of the orbit within the same fiber of s. As we previously mentioned, the parametrization will not be respected, but its projective class will. In other words, the equation endows the target space of s with a projective structure. The inverse of the function s.t/ defined in (39) depends upon the chosen solution .P.t/; Q.t/; R.t//. However, the inverse of s.t/ is, up to post-composition with a Möbius transformation, independent of the original solution. The differential operator (12) was adapted to affine transformations. We will now introduce a differential operator that plays an analogue role for the group of Möbius transformations, by measuring the degree to which a map fails to be projective. The Schwarzian derivative of the function f with respect to z is   f 000 3 f 00 2 f f ; zg D 0  : f 2 f0 It has the following properties: • f f ; zg  0 if  and only if f .z/ is a Möbius transformation; af C b ; z D f f ; zg; • cf C d • f f ı g; zg D f f ; gg.g0 /2 C fg; zg. In order to prove the first two, we will briefly discuss some facts about the solutions of the Schwarzian differential equation f f ; zg D g.z/:

(40)

A straightforward calculation shows that the ratios of two linearly independent solutions of the associated linear differential equation 1 h00 C g.z/h D 0 2

(41)

Differential Equations and Geometric Structures

33

are solutions to the latter. By the theorem on the existence and uniqueness of solutions, all the solutions of (40) may be obtained from a single one by postcomposing it by fractional linear transformations. (We have three parameters and the Schwarzian differential equation is a third-order one.) This proves the second item. Since z and 1 are linearly independent solutions of h00 D 0 and their ratio is z, this proves the first item as well. As we previously saw, for the solutions of Chazy’s equations (28), the inverse of s.t/ D Q3 .t/=R2 .t/ is defined only up to post-composition with a Möbius transformation. In consequence, its Schwarzian derivative is well-defined. We have  .1  19 /s.s  1/  .1  14 /.s  1/ C 1   2 s 1 ft; sg D  0 2 fs; tg D : .s / 2s2 .s  1/2

(42)

Through s, the differential equation (28) defines a projective structure in C n f0; 1g, whose charts are given by the solutions to the above Schwarzian differential equation. Again, we will resort to Schwarz’ theory of uniformization of plane domains to understand the behavior of t.s/.

4.5 Fuchs’s Theorem We will now state a theorem that describes, locally, the solutions of equations of the form (41) when g has a pole of order at most two at 0. More generally, we will consider the linear differential equations of the form s2 f 00 C g1 sf 0 C g0 f D 0;

(43)

depending upon the holomorphic functions gi . We are interested in the behavior of the solutions in the neighborhood of s D 0, where the equation does not satisfy the usual conditions that guarantee the existence of solutions. For the linear equations of the above type, we can in many cases understand, at least locally, their multivaluedness, as we previously did in Proposition 12. The singularity s D 0 of the differential equation (43) carries the apparently contradictory name of a regular singular point, to say that it is indeed a singular point, but that it is a mild one in the sense that we can give a fairly good description of the behavior of the solutions as we approach the point s D 0. Let us consider the homogeneous equation s2 f 00 C g1 .0/sf 0 C g0 .0/f D 0

(44)

34

A. Guillot

and search for solutions of this equation of the form s for  2 C. If s is a solution,  must be a root the indicial polynomial .  1/ C g1 .0/ C g0 .0/;

(45)

naturally attached to Eq. (43). The roots of this polynomial are called the characteristic exponents. For example, at s D 0, ˛ and 0 are the roots of the indicial polynomial of Eq. (16). If 1 and 2 are the roots of (45) and if they are different, s1 and s2 form a basis for the space of solutions of (44). The following theorem affirms that, in the absence of resonances, the solutions to Eq. (43) are not too far from those of the homogeneous one (44). Theorem 18 (Fuchs) If 1 , 2 are the roots of the polynomial (45) associated to (43) and 1  2 … Z, there exist holomorphic functions 1 and 2 , defined in a neighborhood of s D 0, such that si i .s/ are solutions to Eq. (43). More generally, if 1  2 is not a nonpositive integer, there exists a holomorphic function 1 defined in a neighborhood of s D 0 such that s1 .s/ is a solution to Eq. (43). Again, when the difference of the roots of the indicial equation is an integer, logarithmic terms tend to appear. For a proof of Fuchs’s theorem and the description of the solutions in the presence of resonances (the logarithmic case), we refer the reader to [26, §5.1] and [29, Ch. XV].

4.6 Schwarz and the Uniformization of Triangles, II Schwarz investigated, more generally, the uniformization of polygons in the plane bounded by circular arcs. Theorem 19 (Schwarz) Let ˛; ˇ;  2 .0; 1/ and let f W H ! C [ f1g be a meromorphic solution of the Schwarzian differential equation f f ; sg D

   1  ˛ 2 s.s  1/  1  ˇ 2 .s  1/ C 1   2 s : 2s2 .s  1/2

(46)

The function f is one-to-one, extends continuously to R [ f1g and f .H/ is the interior of a triangle with circular sides and internal angles ˛, ˇ, and  . For the proof we will follow the structure of the proof of Theorem 13. The group of Möbius transformations will now take the role previously played by the group of affine ones. Since Eq. (46) has real coefficients, it has a real solution (one that maps real intervals into the reals). Since all the solutions may be obtained from a single one by post-composing it by Möbius transformations, and since the images of lines under Möbius transformations are circles or lines, all the solutions map real intervals to line segments or circular arcs.

Differential Equations and Geometric Structures

35

For the Schwarzian differential equation (46) at s D 0, the associated linear equation (41) has the form h00 C



 1 2 .1  ˇ / C    h: 4s2

The indicial polynomial is .  1/ C 14 .1  ˇ 2 / and has the roots 12 .1 ˙ ˇ/, whose difference is ˇ. Since ˇ … Z, by Fuchs’s theorem, there are solutions to the linear equation (41) of the form s.1Cˇ/=2 1 .s/, s.1ˇ/=2 2 .s/, with i a holomorphic function for every i. The ratio of these solutions of (41) is a function of the form sˇ . By the discussion around the proof of Theorem 13, any determination of sˇ .x/ in a sector has the effect of multiplying by ˇ the angle between two curves that delimit the sector. The angle that the image of the real axis acquires via f at s D 0 is ˇ. By the same arguments, the angle that the image of the real axis acquires at s D 1 is  . Finally, the fact that the angle that the image of the real axis acquires at s D 1 is ˛ can be established by considering g.z/ D f .1=z/, and calculating the Schwarzian differential equation that g.z/ satisfies. This proves that the image of R [ f1g under f is a triangle bounded by circular arcs with the sought angles. By repeating the arguments of Sect. 2.2, f extends holomorphically and injectively to H. Schwarz’s theorem extends also to the case of null angles (also called horn angles), although this passes inevitably through the understanding of the solutions to Eq. (43) in the presence of resonances. We refer to [26, Ch. 6] and [12, Ch. IV] for details.

4.7 Extending Schwarz’s Function Schwarz’s theorem gives the uniformization of a triangle T as the restriction to H of the solution f to Eq. (46). Let us describe the analytic continuation of f beyond H. The image of monodromy of the linear equation (41) in PSL.2; C/ will control the multivaluedness of f . Let us give a more concrete description of this. As in Sect. 2.3, since Eq. (46) is real, in the neighborhood of every nonsingular real point p there is a solution g to the complex differential equation mapping the reals to the reals. According to the Schwarz reflection principle, extending g beyond the interval J containing p amounts to reflect along g. J/. More generally, extending any solution f W H ! CP1 (which is necessarily the composition of g with a Möbius transformation) amounts to reflecting f along the circular arc f . J/. Let T D f .H/ and recall that it is a triangle limited by three circular arcs `i such that the sum of its interior angles is .˛ C ˇ C  /. Up to a Möbius transformation, we can suppose

36

A. Guillot

that `1 and `2 intersect at 0 and 1, this is, that `1 and `2 are straight lines through the origin, and that the origin is a vertex of T. For the circle `3 , three cases appear: 4.7.1 If ˛ C ˇ C  < 1 (Hyperbolic) The circle `3 does not contain the origin in its interior. There is a unique circle C centered at 0 that intersects the circle `3 orthogonally. Hence, it intersects orthogonally the three sides of T. The interior of C, which we will denote by H, carries a natural hyperbolic metric. With respect to this hyperbolic metric, T is a geodesic triangle. The reflections along its sides preserve H, in restriction to which they are hyperbolic reflections. Let be the group generated by these reflections. The image of f lies within H. If the internal angles of T are not aliquot parts of  (if they are not of the form =k for k 2 Z), T is not a fundamental domain for the action of this group: there are nontrivial elements of that will make the interior of T overlap itself. If the internal angles of T are of the form .=p; =q; =r/, we have Poincaré’s polygon theorem [36], that affirms that • for every point p 2 H there exists  2 such that p 2 .T/, and • for every  2 , if  is not the identity, the interiors of T and .T/ do not intersect. In other words, the images of T under fill H without overlapping: there is a tessellation of the hyperbolic plane by copies of T. (We refer the reader to [33] for a proof and comments around the history of Poincaré’s theorem, including that of its incorrect proofs.) In these cases, the inverse of f , the solution of (46), is welldefined in H as a true function (and not as a multivalued one). It cannot be extended beyond C, which is a natural boundary for this inverse. 4.7.2 If ˛ C ˇ C  D 1 (Euclidean) In this case `3 is necessarily a straight line. The three sides of T are lines and reflections along its sides are actually Euclidean reflections. This is the setting studied in Sect. 2.2. 4.7.3 If ˛ C ˇ C  > 1 (Spherical) The circle `3 contains the origin in its interior. Let W S2 ! C [ f1g be the stereographic projection. Under the inverse of , 0 and 1 are mapped to the south and north pole of the sphere. Up to a homothety (which preserves `1 and `2 ), we may suppose that the three sides of the triangle become great circles (geodesics) of the sphere (this is automatic for `1 and `2 ). Reflections along `i become spherical reflections and the index-two subgroup of of orientation-preserving transformations becomes a group of rotations.

Differential Equations and Geometric Structures

37

The spherical triangles of internal angles =2, =3 and =k, k 2 f2; 3; 4; 5g, tessellate the sphere. For these we get, respectively, the dihedral group D3 and the symmetry groups of the tetrahedron, the cube/octahedron and the icosahedron/dodecahedron. In these cases, the inverse of f , the solution of (46), is a well-defined function.

4.8 The Holonomy Cover In Sect. 1.6, we defined a translation surface for each plane (Euclidean) triangle. More generally, in Sect. 2.4, we defined a translation surface (cover) for any complex curve endowed with an affine structure. An essential fact enabling the construction is that, within the affine group (and within the group of Euclidean transformations), translations form a normal subgroup. In the case of projective geometry, PSL.2; C/ is a simple group (this follows either from the theory of Lie groups or from Dickson’s theorem on the simplicity of PSL.2; F/ for any infinite field F, [14], [30]). For hyperbolic geometry, the group IsomC .H/, isomorphic to PSL.2; R/, is also simple, as is the group SO.3; R/ of isometries of the sphere. There is no fully equivalent analogue notion of translation for these geometries. For a complex curve endowed with a projective structure, one may still define some cover with a simplified geometry. Let S be a complex curve endowed with a projective structure. We have a developing map D W e S ! CP1 and a holonomy morphism h W 1 .S/ ! PSL.2; C/ satisfying relation (20). Let … W S ! S be the Galois covering associated to the kernel of h. The Galois group of … is naturally identified to the image of the holonomy. By condition (20), the developing map induces naturally a map D W S ! CP1 . This is the holonomy cover of S. By definition, it is the smallest cover where one has a globally defined developing map. For a triangle in hyperbolic or spherical geometry, one may construct along these lines a surface that may play the role of the translation one. To fix ideas, let us place ourselves in the hyperbolic setting. Let T  H be a triangle with geodesic sides and let W H ! H be an orientation-reversing isometry. Identify the sides of T with those of .T/ by an orientation-preserving isometry. This produces a thrice-punctured sphere S endowed with a hyperbolic (thus projective) structure, from which one can construct the holonomy cover S. This is the holonomy surface associated to the triangle. In the spirit of our first construction of the translation surface, it goes as follows. Let T  H be a triangle with geodesic sides. Let be the group of isometries generated by the reflections 1 , 2 and 3 on the sides `1 , `2 and `3 of T. In the disjoint union th2 h.T/, identify h.T/ with h i .T/ along their common side h.`i ). This produces a surface, which we may call the holonomy surface associated to the triangle. Notice that acts naturally upon this surface.

38

A. Guillot

4.9 A Replacement for Translation Surfaces In the complement of the point at infinity in CP1 , the vector field @=@z induces a translation structure compatible with the natural projective structure in CP1 (a chart for the translation structure is a chart for the projective one). The holomorphic vector fields in CP1 are those that are, in an affine chart, of the form .az2 C bz C c/@=@z; the ones equivalent to @=@z under a Möbius transformation are the nonzero ones for which b2  4ac D 0. These are called parabolic or unipotent. They are the most general ones that induce, in the complement of their unique zero, translation structures compatible with the projective structure of CP1 . Fix one of such parabolic vector fields Z and let q 2 CP1 be its only zero. Let S be a complex curve endowed with a projective structure. Through the developing map D W e S ! CP1 we may pull back the vector field Z to e S. Let e Z denote this pull-back. In the complement of its zeros, it defines a translation structure compatible with the original projective one, and one may wonder about the existence of an intermediate covering where this vector field is still defined. We may pull it back to the holonomy covering S via D. In 1 the complement of D .q/ the solutions of this vector field Z on S give a translation structure compatible with the projective one. If the holonomy of S has a subgroup P preserving Z, we may consider the quotient of S under the action of this group by deck transformations, in order to obtain a smaller (not necessarily Galois) cover of S that carries a vector field inducing, in the complement of its equilibrium points, a translation structure compatible with the projective structure. Upon considering the maximal subgroup P of the holonomy preserving Z, we get a cover of S admitting a vector field that induces a translation structure compatible with the projective one and that is in correspondence with Z under the associated developing map. This cover is minimal with respect to these properties. In the case where the subgroup of the holonomy preserving Z is trivial, this minimal cover is the holonomy one.

4.10 Back to the Equations We may now describe the translation structures in the orbits of (28) in the case where  is a positive real number. We recall that we have supposed that  D 1 in (28). Proposition 20 For Chazy’s equation (28), for every  2 R,  > 0, the translation structure of a generic solution is, up to the addition of countably many points, projectively equivalent to the holonomy covering of the thrice-punctured sphere associated to a triangle with internal angles =2, =3,  . Proof The differential equation (28) induces a projective structure in the thricepunctured sphere C n f0; 1g. Its charts are given by the solutions to the Schwarzian equation (42). It is induced via (39) by the solutions to Eq. (28). The solution has, as any integral curve of a differential equation, a translation structure (and thus a projective structure). The mapping (39) is one between curves with a projective structure. On the other hand, by gluing a triangle with internal

Differential Equations and Geometric Structures

39

angles .=2; =3; /—hyperbolic if  < 1=6, Euclidean if  D 1=6, spherical if  > 1=6—with its image under a reflection (in the corresponding geometry), we obtain a projective structure in the thrice-punctured sphere S. By Schwarz’s theorem, these projective structures on the thrice-punctured sphere coincide. Let U  C such that 0 2 U and let  W U ! C3 be a local solution such that s ı .0/ 2 C n f0; 1g. Let p D .0/. Consider the solution A obtained from  and A 2 PSL.2; C/ via (31). The solution A is defined in A1 .U/ \ C. If 1 2 A1 .U/, we will add the point 1 to the corresponding orbit (this accounts for the “up to the addition of countably many points” part of the statement). Let C be the orbit of the vector field corresponding to , augmented as we just described and let Y be the associated vector field in C. The restriction sjC W C ! C n f0; 1g is a covering map and Y induces a translation structure (and thus a projective one) in C such that sjC is a mapping of curves endowed with projective structures. The proposition is equivalent the nonexistence of an intermediate covering carrying a compatible vector field. Let us rephrase this condition. Let p0 2 C, p0 ¤ p, be such that s. p0 / D s. p/ and let W and W 0 be neighborhoods of p and p0 that are mapped to each other via W W ! W 0 , D .sjW 0 /1 ı sjW . The proposition is equivalent to the fact that does not map YjW into YjW 0 . Let us prove that this is indeed the case. If Y. p0 / D 0, there is nothing to prove (because in that case the translation structure at p0 is not even induced by Y), so suppose that Y. p0 / ¤ 0. Consider the family of solutions (31) generated by  for b D 0. Evaluating each one of them at t D 0 gives .d2 P.0/  c=d; d4 Q.0/; d 6 R.0//. Thus, these solutions represent in a one-to-one way those that attain the fiber of s containing p at t D 0. One of them corresponds to q0 . The only A 2 PSL.2; C/ with b D 0 that acts, via (31), trivially upon the independent variable (preserving the vector field) is the identity I (this does not hold for the upcoming Eq. (47) and its invariance formula (48), for there the action of I is nontrivial but acts trivially upon the independent variable). Since p0 ¤ p, does not map YjW into YjW 0 . We thus conclude that C is a minimal cover (with respect to Y), as discussed at the end of Sect. 4.9. If p is generic, the holonomy of the projective structure of C will not preserve any compatible vector field, and C will be the holonomy cover of C n f0; 1g. t u A global geometric and dynamical description of the differential equation (28) appears in [20, Section 3]. This description implies that the monodromy group of the projective structure of a given solution is exactly the group that, through (31), preserves the solution. The solutions to these equations enjoy an invariance property similar to modularity. We have the following corollary to Proposition 20. Theorem 21 (Halphen, Chazy) For k 2 Z [ f1g, Eq. (28) is semicomplete (has exclusively single-valued solutions). There are two cases to be considered, the hyperbolic (k > 6, including k D 1) and the spherical one (k < 6). Let us first deal with the hyperbolic one. The hyperbolic triangle of angles =2, =3 and =k tessellates the hyperbolic plane, according to Poincaré’s polygon theorem. See Fig. 8. In this case the developing map D on the holonomy cover is one-to-one, and the solution to (28) is essentially the inverse of D, as a parametrization of the orbit. Notice that it has a natural boundary.

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Fig. 8 The tessellation associated to the hyperbolic triangle with internal angles =2, =3, and =7

In the case k D 1, the modular group PSL.2; Z/ is conjugate to the group of orientation-preserving symmetries of the group of the hyperbolic triangle with an ideal vertex and two vertices with angles =2, =3 (see [32, §III.1]). For Ramanujan’s system, the monodromy of projective structure associated to the solution given by Ramanujan’s functions is PSL.2; Z/. The same reasoning applies in the spherical case. In it, every solution will be a rational one. Chazy [9] proved that particular solutions are given by the logarithmic 1 derivatives (the derivatives of the logarithms) of P 2 k3 for P equal to t2 C 1;

if

k D 2;

t C 6t  3;

if

k D 3;

t5 C t;

if

k D 4;

if

k D 5:

4

11

2

6

t C 11t  t;

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Fig. 9 A rational function invariant under the group of the icosahedron (colored by coloring the domain by pulling back some coloring of the range)

By propagating this solution by (31), one may obtain the general one. For instance, for k D 5, from the above solution, we get s.t/ D

.t30

.t20  228t15 C 494t10 C 228t5 C 1/3 : C 522t25  10005t20  10005t10  522t5 C 1/2

These rational functions are invariant under the action of the monodromy group of the projective structure, which is, in this case, group of orientation-preserving symmetries of the icosahedron. The roots of the numerator of s.t/ are the twenty vertices of the dodecahedron (centers of the faces of the icosahedron). The roots of the numerator of s.t/  1 D

.t11 C 11t6  t/5 .t30 C 522t25  10005t20  10005t10  522t5 C 1/2

are eleven of the twelve vertices of the icosahedron (eleven of the twelve centers of the faces of the dodecahedron), the other one being at t D 1. The roots of the denominators are the thirty middle points of the edges. Figure 9 depicts the function s.

4.11 Equations with Solutions Defined in More Unusual Domains So far, we have seen Euclidean, spherical and hyperbolic geometry at work, but we have not seen any projective structure that does not reduce to one of the above

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types. We will close these notes with the family of differential equations considered in [21]: x0 D x2 C yz. y  z/.Œ˛  y C Œ1  ˛z/; y0 D y.x C yz  z2 /;

(47)

z0 D z.x  yz C y2 /; having parameters ˛;  2 C,  ¤ 0; 1. The principle behind these equations   is ab the same we had before. If .x.t/; y.t/; z.t// form a solution of (47) and 2 cd SL.2; C/, another solution is given by 

      c 1 1 at C b at C b 1 at C b  ; y ; z : x .ct C d/2 ct C d ct C d .ct C d/ ct C d .ct C d/ ct C d (48)

In this case, the vector field associated to (47) forms a Lie algebra isomorphic to sl.2; C/ with the vector fields @=@x, x@=@x C 12 y@=@y C 12 z@=@z. The inverse of s.t/ D y.t/=z.t/ is only defined up to a Möbius transformation, and ft; sg D

s4  4˛s3 C 2.2˛ C 2˛  /s2  4˛s C 2 : 2s2 .s  1/2 .s  /2

(49)

In other words, for each ˛ 2 C we have a projective structure in C n f0; 1; g, whose charts are the solutions to this equation. As before, for Eq. (47) to be semicomplete (for its solutions to be single-valued) it is necessary that this projective structure in C n f0; 1; g is obtained by taking the quotient of an open subset of CP1 under a group of projective transformations. This condition turns out to be also sufficient. The proof goes along the same lines of the one of Proposition 20, except for the fact that (48) does not depend solely upon the class of A in PSL.2; C/ (see [21, thm. 1]). Let us first discuss the case ˛ D 0,  D 1, where, in particular, Eq. (49) is real. The map s 7! .s  1/=.s C 1/ is an order-four symmetry of Eq. (49), cyclically permuting the singularities 0; 1; 1; 1. From Schwarz’s uniformization of polygons with circular sides, the image of H under a solution to the equation is a quadrilateral Q bounded by four circles, whose internal angles are zero (the consecutive sides of Q are tangent). The previously mentioned symmetry implies the existence of an order-four fractional linear transformation cyclically permuting the four vertices. On its turn, this implies the existence of a circle C that intersects orthogonally the four circles and that passes through the vertices. Let H be the disk bounded by C, considered with its hyperbolic metric. The group generated by reflections on the sides of Q has an index-two subgroup  IsomC .H/. We recover explicitly a complete hyperbolic structure of our four-punctured sphere, given by the quotient of H by this group. Our previous arguments show that in this case the function in H that realizes this quotient satisfies the differential equation (49). The

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group can be given explicitly. It is a conjugate of the image in PSL.2; Z/ of the group  3 D

ab cd



 2 SL.2; Z/I a; d  1 mod 3; c; d  0 mod 3 :

A solution to the original system of differential equations with these values of ˛ and  is defined in the upper half-plane; property (48) will make y and z enjoy some modularity properties with respect to 3 (see [35] and [21, §4.3]). For other real values of ˛ and , the solution f to Eq. (49) restricted to H will have for image, according to Schwarz’s theorem, a quadrilateral bounded by four tangent circles. However, in general, there will no longer be a circle that is orthogonal to the four vertices of the quadrilateral. If .˛; / is close enough to .0; 1/; we may still reflect upon the sides of the quadrilateral and fill an open subset of CP1 bounded by a fractal Jordan curve. The function f is one-to-one in the universal covering of C n f0; 1; g and the inverse will still be a well-defined function in . More generally, for all complex values of .˛; / close enough to .0; 1/, the function f will be one-to-one in the universal covering of C n f0; 1; g. This may be seen as a consequence of a theorem of Nehari giving criteria to the injectivity of some functions out of properties of their Schwarzian derivatives [34]. Some deep theorems of Bers tell us even more: that for every  2 C n f0; 1g, there is a maximal connected open set B  C (the Bers slice) such that for all ˛ 2 B , f is one-to-one in the universal covering of C n f0; 1; 1; g. It is a bounded subset of C with fractal boundary. For the parameters at the boundary of the Bers slice, f will be still oneto-one on the universal covering (which is a disk), but the boundary of the image will no longer be a Jordan curve. The theory of Kleinian groups has described many of the things that happen at this boundary. We refer the reader to the discussion and references found in [21]. As for the solutions of (47), the vector field associated to Eq. (47) is semicomplete if ˛ 2 B . The solutions are defined in domains bounded by a round or fractal circle (if ˛ is in the interior of B ) or in other sorts of simply-connected domains (with a wilder behavior at the boundary) if ˛ is in the boundary of B . The above are not the only projective structures in C n f0; 1; g with parabolic monodromy at the ends that can be obtained by taking the quotient of some open subset of CP1 under the action of some group of projective transformations. Consider the kissing Schottky groups ˇ of Möbius transformations generated by A.z/ D z C 2 and B.z/ D z=.1 C z=ˇ/, 12 > ˇ > 0. Let D1 D fjz C ˇj < ˇg, D2 D fjz  ˇj < ˇg, D3 D f=.z/ < 1g, and D4 D f=.z/ > 1g. Notice that the Di are pairwise disjoint and that A.CP1 n D3 / D D4 and B.CP1 n D1 / D D2 (see Fig. 10). By the classical ping-pong lemma (see [11, §II.B]), the group generated by A and B is free, and E D CP1 n[i Di is a fundamental domain for the action of the group (Fig. 10). The discontinuity domain for the action is the union of the images of E, and has an uncountable complement. The curves @Di in the boundary of E get identified by couples under the action of A and B. The gluing gives a four-punctured sphere naturally endowed with a projective structure. This projective structure is

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A

Fig. 10 A fundamental domain for the kissing Schottky group uniformizing a four-punctured sphere

B

also present in the family induced by Eq. (47). In the corresponding case the vector field (47) will be semicomplete: the domains where the solutions are defined are complements of the limit set of the group (which is a Cantor set), at the points of which the solution will present essential singularities. For fixed , the set of values of ˛ such that Eq. (47) is semicomplete has thus nonempty interior (the Bers slice) and may also have isolated points, like the ones we just described.

4.12 A General Construction We finish by giving a unified geometric construction of the equations of Chazy, Halphen, Ramanujan, and   (47). Let U  PSL.2; C/ be the image of the one1T parameter group U D . Let V D PSL.2; C/=U. We may think of it as 01 the space of translation structures in subsets of CP1 that are complements of a point that are compatible with the standard projective structure. We will be interested in it as a homogeneous space under the action to the left of PSL.2; C/. Consider the manifold M D CP1  V. There is a fibration … W V ! CP1 given by the projection onto the first factor, and a horizontal foliation by curves F given by the fibers of the projection onto the second factor. There is a natural action to the left of PSL.2; C/ on M, by fractional linear transformations in the first variable, by the natural action in the second one. The point .0; I/ 2 M has trivial stabilizer. By identifying PSL.2; C/ with the orbit O of .0; I/ 2 M, the horizontal foliation is given by the orbits of the flow (35). The induced vector field is, tautologically, the one inducing the translation structure in each leaf of F . Let C be a complex curve endowed with a projective structure, D W e C ! CP1 its developing map and  W 1 .C/ ! PSL.2; C/ the holonomy morphism. Pull back the fiber bundle … via D in order to get a fibration over C with fibers isomorphic to V. This fibration is endowed with a horizontal foliation. Each one of the leaves of this foliation is naturally endowed with a translation structure. There is an open

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subset of this fibration coming from O where there is a vector field inducing the translation structure in each leaf of F . When C (with its projective structure) is the hyperbolic orbifold with three conic angles, the above fibration with a vector field is equivalent to an open subset in the phase space of Halphen’s equation (30) for a suitable choice of the parameters. When it is the hyperbolic orbifold with three conic points of angles  and 2=3 and 2=k, we get the restriction to a big open subset of Chazy’s equation (28). When C is a four-punctured sphere with a projective structure and the holonomy at each one of the four punctures is parabolic, we obtain the quotient of an open subset of the system (47) under its symmetry .x; y; z/ 7! .x; y; z/.

5 Epilogue We have seen many differential equations with exclusively single-valued solutions. All of these equations had special symmetries and it was because of them that we could establish the absence of multivaluedness of their solutions. Needless to say, most differential equations do not have these symmetries. Nevertheless, for secondorder differential equations, affine structures on curves play a very relevant role in controlling the multivaluedness of their solutions [23, 24]. For third-order equations, we have seen natural boundaries in the domains where their solutions are defined. It is likely that the principle behind the equations of Chazy and Halphen is the only mechanism capable of creating these natural boundaries in the solutions of thirdorder algebraic equations. Translation, affine and projective geometry are the only complex geometries in dimension one. Some higher-dimensional geometries appear in the study of other classical differential equations. For example, the Chazy IX equation [9]  000 D 18. 0 C  2 /. 0 C 3 2 /  6. 0 /2 involves the geometry of the complex two-dimensional affine group (see [22]). We said that integrating a differential equation in the complex domain involves describing the translation structure of its solutions. This is just part of the story. Another important part passes through understanding the behavior of the leaves within the phase space, a problem that is far from being understood. This is a central problem. For a glimpse of some results on the subject, we refer to [8, 13], and [31]. For complete vector fields (or the complete solutions of some vector fields), the orbits have a dynamics of their own (see [40]), which remains to be fully understood. For the general theory (perhaps theories is a better word) of differential equations in the complex domain, we refer the reader to the comprehensive works of Ince [29] and Hille [26], as well as to the more recent one by Ilyashenko and Yakovenko [28]. We recommend the notes by Rebelo and Reis [39] for a more vector-fieldoriented introduction to the theory. For a historical account of the fantastic history of complex differential equations in the nineteenth century, see [17]. Complex

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differential equations played a prominent role in the genesis and development of the uniformization theorem of Poincaré and Kobe. This is the subject of the monograph [12] by Henri Paul de Saint-Gervais. Acknowledgement Partially supported by PAPIIT-UNAM IN108214.

References 1. L.V. Ahlfors, Complex Analysis, 3rd edn. (McGraw-Hill Book, New York, 1978) 2. Á. Álvarez-Parrilla, J. Muciño Raymundo, Dynamics of singular complex analytic vector fields with essential singularities I. Conform. Geom. Dyn. 21, 126–224 (2017) 3. A. Artigue, Órbitas periódicas en billares triangulares. Miscelánea Mat. 59, 19–40 (2014) 4. C. Briot, J.-C. Bouquet, Recherches sur les fonctions doublement périodiques. C. R. Hebdomadaires des Scéances de l’Académie des Sciences 40(7), 342–344 (1855) 5. G. Cairns, R.W. Sharpe, On the inversive differential geometry of plane curves. Enseign. Math. (2) 36(1–2), 175–196 (1990) 6. F. Calogero, The neatest many-body problem amenable to exact treatments (a “goldfish”?). Advances in nonlinear mathematics and science. Physica D 152/153, 78–84 (2001) 7. F. Calogero, J.-P. Françoise, Periodic solutions of a many-rotator problem in the plane. Inverse Probl. 17(4), 871–878 (2001). Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000) 8. C. Camacho, A. Lins Neto, P. Sad, Minimal sets of foliations on complex projective spaces. Inst. Hautes Études Sci. Publ. Math. 68, 187–203 (1988–1989) 9. J. Chazy, Sur les équations différentielles du troisième ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes. Acta Math. 34(1), 317–385 (1911) 10. H.S.M. Coxeter, Inversive distance. Ann. Mat. Pura Appl. (4) 71, 73–83 (1966) 11. P. de la Harpe, Topics in Geometric Group Theory. Chicago Lectures in Mathematics (University of Chicago Press, Chicago, IL, 2000) 12. H.P. de Saint-Gervais, Uniformisation des surfaces de Riemann (ENS Éditions, Lyon, 2010) 13. B. Deroin, C. Dupont, Topology and dynamics of laminations in surfaces of general type. J. Am. Math. Soc. 29(2), 495–535 (2016) 14. L.E. Dickson, Theory of linear groups in an arbitrary field. Trans. Am. Math. Soc. 2(4), 363– 394 (1901) 15. R.H. Fox, R.B. Kershner, Concerning the transitive properties of geodesics on a rational polyhedron. Duke Math. J. 2(1), 147–150 (1936) 16. E. Ghys, J.-C. Rebelo, Singularités des flots holomorphes. II. Ann. Inst. Fourier (Grenoble) 47(4), 1117–1174 (1997) 17. J.J. Gray, Linear Differential Equations and Group Theory from Riemann to Poincaré, 2nd edn. (Birkhäuser Boston, Boston, MA, 2000) 18. A. Guillot, The Painlevé property for quasihomogenous systems and a many-body problem in the plane. Commun. Math. Phys. 256(1), 181–194 (2005) 19. A. Guillot, Semicompleteness of homogeneous quadratic vector fields. Ann. Inst. Fourier (Grenoble) 56(5), 1583–1615 (2006) 20. A. Guillot, Sur les équations d’Halphen et les actions de SL2 .C/. Publ. Math. Inst. Hautes Études Sci. 105, 221–294 (2007) 21. A. Guillot, Some generalizations of Halphen’s equations. Osaka J. Math. 48(4), 1085–1094 (2011) 22. A. Guillot, The geometry of Chazy’s homogeneous third-order differential equations. Funkcial. Ekvac. 55(1), 67–87 (2012)

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23. A. Guillot, Meromorphic vector fields with single-valued solutions on complex surfaces. arXiv:1603.02288. Preprint (2016) 24. A. Guillot, J. Rebelo, Semicomplete meromorphic vector fields on complex surfaces. J. Reine Angew. Math. 667, 27–65 (2012) 25. G.-H. Halphen, Sur certains systèmes d’équations différentielles. Comptes Rendus Hebdomadaires de l’Académie des Sciences XCII 24, 1404–1406 (1881) 26. E. Hille, Ordinary Differential Equations in the Complex Domain (Dover Publications, Mineola, NY, 1997). Reprint of the 1976 original 27. W.P. Hooper, Periodic billiard paths in right triangles are unstable. Geom. Dedicata 125, 39–46 (2007) 28. Y. Ilyashenko, S. Yakovenko, Lectures on Analytic Differential Equations. Graduate Studies in Mathematics, vol. 86 (American Mathematical Society, Providence, RI, 2008) 29. E.L. Ince, Ordinary Differential Equations (Dover Publications, New York, 1944) 30. K. Iwasawa, Über die Einfachheit der speziellen projektiven Gruppen. Proc. Imp. Acad. Tokyo 17, 57–59 (1941) 31. F. Loray, J.C. Rebelo, Minimal, rigid foliations by curves on CPn . J. Eur. Math. Soc. 5(2), 147–201 (2003) 32. W. Magnus, Noneuclidean Tesselations and Their Groups. Pure and Applied Mathematics, vol. 61 (Academic [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, London, 1974) 33. B. Maskit, On Poincaré’s theorem for fundamental polygons. Adv. Math. 7, 219–230 (1971) 34. Z. Nehari, The Schwarzian derivative and schlicht functions. Bull. Am. Math. Soc. 55, 545–551 (1949) 35. Y. Ohyama, Differential equations for modular forms of level three. Funkcial. Ekvac. 44(3), 377–389 (2001) 36. H. Poincaré, Théorie des groupes fuchsiens. Acta Math. 1(1), 1–62 (1882) 37. S. Ramanujan, On certain arithmetical functions. Trans. Camb. Philos. Soc. 22(9), 159–184 (1916) 38. J.C. Rebelo, Singularités des flots holomorphes. Ann. Inst. Fourier (Grenoble) 46(2), 411–428 (1996) 39. J.C. Rebelo, H. Reis, Local theory of holomorphic foliations and vector fields. arXiv:1101.4309. Preprint (2011) 40. J.C. Rebelo, H. Reis, Uniformizing complex ODEs and applications. Rev. Mat. Iberoam. 30(3), 799–874 (2014) 41. R.E. Schwartz, Obtuse triangular billiards. II. One hundred degrees worth of periodic trajectories. Exp. Math. 18(2), 137–171 (2009) 42. W.P. Thurston, in Three-Dimensional Geometry and Topology. Vol. 1, ed. by S. Levy. Princeton Mathematical Series, vol. 35 (Princeton University Press, Princeton, NJ, 1997) 43. F. Valdez, Billiards in polygons and homogeneous foliations on C2 . Ergodic Theory Dynam. Syst. 29(1), 255–271 (2009) 44. F. Valdez, Billares poligonales y curvas de Teichmüller. Universo.math 1(2) (2014). http:// universo.math.org.mx/2014-2/Billares/billares.html 45. Y.B. Vorobets, G.A. Gal’perin, A.M. Stepin, Periodic billiard trajectories in polygons: generation mechanisms. Uspekhi Mat. Nauk 47(3(285)), 9–74, 207 (1992) 46. A.N. Zemljakov, A.B. Katok, Topological transitivity of billiards in polygons. Mat. Zametki 18(2), 291–300 (1975)

Blow-Up, Homotopy and Existence for Periodic Solutions of the Planar Three-Body Problem Richard Montgomery

Abstract Deleting collisions from the configuration space of the planar N-body problem yields a space with a large interesting set of free homotopy classes of loops, classes which are encoded by “syzygy sequences” when N D 3. This expository piece centers on the question “ Is every free homotopy class of loops realized by a periodic solution to the problem?” We report on the recent affirmative answer (Moeckel and Montgomery, Nonlinearity 28:1919–1935, 2015) for the case of non-zero but small angular momentum and three equal or near-equal masses. The key tool is the McGehee blow-up (McGehee, Invent Math 27:191–227, 1974) as implemented by Rick Moeckel in the 1980s. After recounting some history and motivation, about a third of this article exposes the blow-up method. We use an energy balance under scaling transformations to motivate McGehee’s blow-up transformation. We give an explicit description of the blown-up and reduced phase space for the planar N-body problem, N  3 as a complex vector bundle over Œ0; 1/  CPN2 . We end by returning to the angular momentum zero case where we conjecture the answer is ‘no’. We support this conjecture by recent work of Jackman and Rose (The Binary Returns!. arxiv:1512.01852, 2015; Geometric phase and periodic orbits of the equal-mass, planar three-body problem with vanishing angular momentum, Ph.D. Thesis, School of Mathematics and Statistics University of Sydney, 2015).

1 Introduction The following theorem inspired much of my work on the N-body problem. Background Theorem 1 Let .M; ds2 / be a compact Riemannian manifold. Then every free homotopy class of loops on M is realized by a closed geodesic.

R. Montgomery () Mathematics Department, University of California, Santa Cruz, CA, USA e-mail: [email protected] © Springer International Publishing AG 2018 L. Hernández-Lamoneda et al. (eds.), Geometrical Themes Inspired by the N-Body Problem, Lecture Notes in Mathematics 2204, https://doi.org/10.1007/978-3-319-71428-8_2

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If one continuous loop in M can be continuously deformed into another without leaving M then we say that the two loops are “freely homotopic”.1 Free homotopy defines an equivalence relation on loops in M. The resulting equivalence classes are the free homotopy classes of loops. To move from Riemannian geometry to the planar three-body problem we replace the geodesic equations by Newton’s equations, and the Riemannian manifold of the above theorem by the configuration space for the planar three-body problem. This configuration space is the product of three copies of the Euclidean plane. Newton’s equations have singularities along the collision variety where two or more of the bodies collide. Excluding the collision variety from configuration space induces a large space of free homotopy classes where previously there were none. Open Question 1 Is every free homotopy class for the planar Newtonian threebody problem realized by a collision-free periodic solution? We address a reduced version of this question, where “reduced” means modulo the group G of rigid motions of the plane, which is the built-in group of symmetries of Newton’s equations. Consequently we can “reduce” Newton’s equations to obtain equations on the quotient of the three-body configuration space by G. We call this quotient space shape space since its points represent oriented congruence classes of triangles. See [26] for details and a derivation of the structure of shape space. Shape space is diffeomorphic to R3 . Under this diffeomorphism the collision variety (modulo G) becomes three rays issuing forth from the origin. The free homotopy classes of shape space minus collisions are called “reduced free homotopy classes”. Momentarily we describe how to encode reduced free homotopy classes in a simple combinatorial way. We do not require our solutions to be periodic, but rather only that they are “reduced periodic” meaning periodic modulo G. Concretely, if rij denotes the interparticle distances, i.e. the sides of the triangle formed by the three bodies, then we insist that these distances for our solutions satisfy rij .t C T/ D rij .t/ where T is the reduced period.2 The reduced free homotopy classes are conveniently encoded in the astronomical language of “syzygies”. A syzygy is an instant or configuration for which the three bodies lie in a line. Non-collision syzygies are marked 1,2, or 3, depending upon which of the three masses lies in the middle at the syzygy instant. Consider a curve c in the configuration space of the three-body problem which is closed modulo rotation. Write out its syzygy sequence on the circle. We get a periodic list of 1’s 2’s and 3’s. The list is subject to the “non-stuttering” cancelation rule: any time we

1 For those familiar with the fundamental group, we emphasize that the adjective“free” means that there is no fixed base point through which all loops must pass. The space of free homotopy classes typically does not form a group, rather it is isomorphic to the set of conjugacy classes of the fundamental group. 2 Strictly speaking, this T may actually be half the reduced period. For example, if after time T the initial and final triangle are related by a reflection, then after time 2T the two triangles are the same.

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see a 11, a 22 or a 33 we delete it. We call the resulting sequence the “reduced syzygy sequence” of the free homotopy class. For example 32 is the reduced syzygy sequence of 12112321 since 12112321 D 122321 D 1321 D 32 where the last cancellation arises because the word is written on the circle. It can be proved that two reduced-periodic collision-free curves represent the same reduced free homotopy class if and only if their reduced syzygy sequences are equal. Theorem 1 (.RM/2 [24]) For equal or near-equal masses, and angular momenta sufficiently small but nonzero, every reduced syzygy sequence, and thus every reduced free homotopy class for the planar three-body problem is realized by a reduced periodic orbit for the Newtonian planar three-body problem.

1.1

My Path Through the Variational Wilderness

Look at every path closely and deliberately. Try it as many times as you think necessary. Then ask yourself, and yourself alone, one question. [. . . ] Does this path have a heart? —from the preface of the book “Keep the River on your Right”, by Tobias Schneebaum; a slight variation on a well-known passage in Carlos Castaneda’s “The Teachings of Don Juan: A Yaqui Way of Knowledge”

The variational proof of the background theorem, Theorem 1, proceeds as follows. Fix a free homotopy class. Define m to the infimum of the lengths of all loops which realize this class. Choose a minimizing sequence: a sequence of loops in the class whose lengths tend to m. Because M is compact, the Arzela-Ascoli theorem guarantees that the sequence has a convergent subsequence. Call c the loop to which the subsequence converges. Standard methods from the calculus of variations now show that c lies in the class and its length m, proving the theorem. The proof just sketched provides an archetypal example of the direct method of the calculus of variations in action. I learned it, and loved it, in grad school. But for me, back in grad school, Celestial Mechanics was the land of old famous long-dead men, a world of very hard problems of no real interest. I was certain I would never work in it. I studiously avoided the entire last part of the book by Abraham-Marsden, since that is the part titled ‘Celestial Mechanics’. Instead, coming out of grad school, I tried my hand in what was in vogue, in gauge theory, symplectic reduction, and eventually I was led into problems in subRiemannian geometry and optimal control where I probably did my first real serious piece of work. I was led into subRiemannian geometry through the work of Wilczek and Shapere who had shown that the problem faced by a falling cat when trying to right herself when dropped from upside down with no angular momentum can be viewed as a kind of optimal control problem mixed with gauge theory. I kept simplifying the cat problem until she consisted of three mass points. At this juncture, I knew I was perilously close to working on the three-body problem but I studiously avoided actually working on it. In 1995 or 1996 Alan Weinstein, a mentor of mine in grad school, told me that Wu-yi Hsiang had been looking

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into the three-body problem from a perspective very similar to mine: variational, combined with equivariant differential geometry. Hsiang and I got together in a cafe on Euclid Avenue in Berkeley one afternoon and spent perhaps 3 hours together. His personality is a force of nature. This force and Hsiang’s optimism and enthusiasm convinced me to begin work on some baby problems within the three body problem. During that visit, Hsiang posed the reduced version of question 1 , and drew pictures on yellow pads of paper which were reminescent of Figs. 4 and 6 of this paper. Very soon after my encounter with Hsiang, I started a seminar at UCSC on the N-body problem. Chris Golé, who had written a wonderful book on Symplectic Twist Maps, was visiting UCSC as an assistant professor and attended regularly. Bill Burke, one of my few real friends on the UCSC faculty, and a physics professor at UCSC also attended. (Bill would die a few years later at 52 when his pickup truck was flipped over in a high wind in Hurricane, Utah, as he was driving home from a Grand Canyon rafting trip.) A month in to our seminar, Chris Golé took me aside and told me “Richard, if you are serious about doing work in the N-body problem then you must go to Paris. You have to visit the Bureau des Longitudes and talk with Alain Albouy, Alain Chenciner and Jacques Laskar. They do phenomenal work in mathematical celestial mechanics.” The next year I had a sabbatical 1997–1998, and my family and I took that sabbatical year at CIMAT, in Guanajuato, Mexico. While there I began my book on SubRiemannian Geometry and also invited myself to Paris in the Spring. I stayed in Paris for 6 weeks that first time and became lifelong friends with the two Alains and with Jacques. My conversations with Albouy and Chenciner and eventual collaboration with Chenciner began slowly, and evolved out of their incredibly careful, thorough, and exacting referee work of some of my papers for the journal “Nonlinearity”. I will not repeat the story of how, at the end of the millennium, in December of 1999, Chenciner and I rediscovered the remarkable figure eight orbit of Cris Moore. (You can find a version of that story on my web page, for example.) Upon seeing an early draft of our paper, Phil Holmes told us that his student C Moore had done similar work 6 years before us [28]. And Robert MacKay pointed Chnenciner to an amazing paper of Poincaré, over a century earlier, in which Poincaré [30] used the direct method to answer a variant of question 1. He proved the existence of a reduced periodic solution in almost every reduced homology class, provided we replace the Newtonian 1=r2 force law by a 1=r3 “strong-force” law.3 In addition to his beautiful 3 page paper, I recommend [7] or [4] for history and details of successes and failures of the variational method applied to the N-body problem. The main challenge in getting the direct method of the calculus of variations to work in a three-body problem is the non-compactness of its configuration space. Minimizing sequences of loops may leave the configuration space, escaping ‘to infinity’, in their attempts to minimize. See Fig. 1. The most challenging escape

3 For a 1=ra potential, so 1=raC1 force law, the action of an orbit segment with an isolated collision is finite if and only if a < 2.

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Fig. 1 Slip the circles off the ends. . .

to infinity to prevent is not the ultraviolet escape (distances between bodies tending to infinity) but rather the infrared escape: distances between bodies going to zero, which is to say the possibility that a minimizing sequence of paths converges to a path in which two or more of the bodies suffer a collision. After the figure eight work, I continued trying to use variational methods and geometric perspectives to establish new results for the Newtonian N-body problem. Our “choreographic” success, my new French, Catalan, and New Jersey friends [9], and the surprising simplicity and beauty of the variational proof of Theorem 1 kept me on a 17 year long path through the underbrush of the calculus of variations in trying to establish some version of Theorem 1. Throughout that time, I worked almost exclusively in the case of zero angular momentum. The reason I insisted on restricting to zero angular momentum is that the restriction arises in an extremely natural way out of the map from configuration space through shape space. One realizes this projection to be a Riemannian submersion and that being orthogonal to the fibers (the rotational orbits of a single triangle) is equivalent to having zero angular momentum, and minimizers between orbits must have zero angular momentum. In physical terms, if you fix a curve on shape space, and minimize the length of all realizations of this shape curve by curves in configuration space, then the resulting minimizers are exactly those curves which project onto the given curve and having angular momentum zero. In retrospect, the long variationally based path I took was a path with a heart. But it did not bring me closer to a resolution of Question 1.

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1.2 Breakthrough I had a sabbatical in Portland in the Spring of 2014. I had largely given up on making progress on Question 1 and decided that it was time that I performed some numerical experiments to get some idea of whether the answer was yes or no. To do this, I set myself the task of sorting through orbit segments for the zero-angular-momentumequal-mass-three-body problem, by numerically integrating a large variety of initial conditions up until they generated a syzygy sequence of some fixed length N. I planned to take this as “raw data” and look for gaps—certain nonoccurrences of subwords—in the resulting length N words. I figured I could get up to words of length N=10, by shooting from the collinear plane in shape space. I rediscovered for the fourth or fifth time that programming and numerical analysis are not paths with a heart for me. Within a month I could see that I would need more than a year to complete my appointed task. I would need help! At first I tried to enlist the aid of my old kayak friend and life-long Fortran programmer Michael Schlax who was living nearby in Corvalis. But it soon became clear that this would be too slow—Michael was a geophysicist and statistician by training, not a mathematician. And I was not enjoying learning Fortran. So I decided to seek out my old friend Carles Simó and get one of his students, or even better, him, to aid me in my numerical searches. Carles Simó has a reputation as one of the most inventive, careful numerical analysts working in celestial mechanics. He is also a mathematician and a friend. So we can talk. My subRiemannian book had gotten me invited to a trimester at IHP in Paris for the Fall quarter of 2014 and I took advantage of that trip to invite myself to spend time with Carles in Barcelona. The first afternoon I met with Carles and explained Question 1. He looked at me with his piercing eyes and asked “Richard, why do you care?” I had been working on this problem 17 years. Carles’ question was a stab to my heart! It knocked the breath out of me. But I knew Carles did not mean to hurt— he is simply a direct man who does not waste time or mince words. So I tried to explain why I cared. Carles listened. The next morning when we met again, he began “Richard, if what you think is true about this Question, (that all free homotopy classes are realized) then there has to be a dynamical mechanism.” With those few words I switched paths! I abandoned the variational path, and asked myself what “dynamical mechanisms” do I know? What mechanisms which work for general Hamiltonian systems? I realized I knew only two “dynamical mechanisms”: that of KAM torii and that hyperbolic tangles. KAM would be of no help. But Moser, in his famous book [29], had shown clearly how tangles yield symbolic dynamics in a celestial mechanics problem, the Sitnikov problem. I wracked my brain. Who had done similar work, but for the full three-body problem? Rick Moeckel! And Rick had been my main collaborator these last three years! I reread some of Rick’s papers [17, 20, 22] from the 1980s. I discovered that back then he had essentially solved my problem! A few small gaps remained to fill, but Moeckel had done the huge bulk of the work nearly 30 years earlier.

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The goal of the remainder of this paper is to explain what Rick did, how we used it, and try to motivate his work and in particular the McGehee blow-up.

2 Background: Equations and Solutions 2.1 Equations The classical three-body problem asks that we solve the system of non-linear ODEs: m1 qR 1 D

F21 C F31

m2 qR 2 D

F12 C F32

m3 qR 3 D

F23 C F13 :

(1)

where Fab D Gma mb

qa  qb 3 rab

(2)

is the force exerted by mass ma on mass mb and rab D jqa  qb j; qa 2 Rd ; and ma ; G > 0: Here a; b D 1; 2; 3 label the bodies. The dimension d for us will be 2. The standard value is d D 3. The ma represent the values of point masses whose instantaneous 2 positions are qa .t/. The double dots indicate two time derivatives: qR D ddt2q . The constant G is Newton’s gravitational constant and is physically needed to make dimensions match up. Being mathematicians, we can and do set G D 1.

2.2

Solutions of Euler and Lagrange

The only solutions to the three-body problem for which we have explicit formulae were found by Euler [10] and Lagrange [14] in the last half of the eighteenth century. See Figs. 2 and 3. Their solutions are central to our story. For Lagrange’s solution, place the three masses at the vertices of an equilateral triangle and drop them: let them go from rest. They shrink homothetically towards their common center of mass, remaining equilateral at each instant. The solution ends in finite time in triple collision. This motion forms half of Lagrange’s triple collision solution. To obtain the other half of Lagrange’s solution use time-reversal invariance to continue this solution backwards in time. In the full solution the three masses explode out of triple collision, reach a maximum size at the instant at

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Fig. 2 A Lagrange solution

Fig. 3 An Euler solution

which we dropped the three masses, and then shrink back to triple collision, staying equilateral throughout. A surprise is that the Lagrange solution works regardless of the mass ratios m1 Wm2 Wm3 . For Euler’s solutions, place the masses on the line in a certain order: qi < qj < qk so as to form a special ratio qk  qj W qj  qi . (This special ratio depends on the mass ratios and also the choice of mass mj on the middle and is the root of a fifth degree polynomial whose coefficients depend on the masses.) Again drop them. They stay on the line as they evolve and again the similarity class of the (degenerate) triangle stays constant: this ratio of side lengths stays constant. (In case the two masses at the ends are equal then the special ratio is 1W1: place mj at the midpoint of mi and mk .) The solutions described are part of a family of explicit solutions. For every one of the solutions in these families the similarity class formed by the three masses stays constant in time during the evolution. Each mass moves on its own Keplerian conic with the center of mass of the triple as focus, the solutions described above being the special case of degenerate (colinear) ellipses. We derive these families analytically in Sect. 4.3.1 below.

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All together these solutions form five families. The corresponding shapes are called “central configurations”. The Lagrange solutions count as two, one shape for each orientation of a labelled equilateral triangle. The Euler solutions count as three, one for each choice of mass in the middle. For almost all (Newtonian) time the solutions of Theorem 1 are very close to one of the three Euler solutions. The Lagrange solutions act as bridges between various Eulers.

3 Shape Sphere: Blow-Up and Reduction, First Pass A basic aid to understanding the planar three-body problem is the shape sphere, a two-sphere whose points represent oriented similarity classes of triangles. At each instant of time the three bodies form the vertices of a triangle. Call two triangles “oriented similar” if one can be brought to the other by a composition of translations, rotations, and scalings. The resulting space of equivalence classes forms the shape sphere. See Fig. 4. This sphere has 8 marked points, the 5 central configurations just described LC ; L ; E1 ; E2 ; E3 and the 3 binary collision points labelled B12 ; B23 ; B31 . The sphere’s equator represents the space of collinear triangles. The 3 binary collision points, and 3 Euler central configurations lie on this equator, interleaved so as to be alternating. The earliest occurring picture of the shape sphere in the context of celestial mechanics with which I am familiar is [18]. You will find a detailed exposition of the shape sphere and its relation to the three-body problem in [26]. L

Fig. 4 The shape sphere. Lagrange points, Euler points, and collision points marked. The equator consists of collinear triangles. Figure courtesy of Rick Moeckel

E2

B23 E3 B13

L

B12 E1

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We summarize how the shape sphere arises out of the three-body problem. The configuration space for the three-body problem, with collisions allowed, is C3 with a point q D .q1 ; q2 ; q3 / 2 C3 representing the three vertices of the triangle—the positions of the three bodies. We have identified C with R2 in the standard way: x C iy 2 C corresponds .x; y/ 2 R2 . A standard trick from Freshman physics allows us to restrict the problem to the center-of-mass zero subspace: Ecm D fq 2 C3 W m1 q1 C m2 q2 C m3 q3 D 0g Š C2  C3 : (See the beginning of Sect. 5 below.) In Ecm the binary collision locus become three complex lines which intersect at the origin 0. The origin represents triple collision. The masses endow C3 with a canonical metric called the “mass-metric” (Eq. (4)) and relative to that metric the distance from triple collision is given by r where r2 D m1 jq1 j2 C m2 jq2 j2 C m3 jq3 j2 : (See Eq. (8).) Take the sphere fr D 1g WD S3  C2 Š Ecm : Because the three-body equations are invariant under rotations they descend to ODEs on the quotient of C2 D Ecm by the group S1 of rotations. This quotient space is topologically an R3 . We call this R3 “shape space”. To understand this quotient note that the rotation action leaves r unchanged but moves points on S3 around according to .Z1 ; Z2 / 7! .uZ1 ; uZ2 /, u 2 S1  C. (Here Z1 ; Z2 are any complex linear coordinates for Ecm .) This is the circle action used to form the Hopf fibration: Hopf W S3 ! S3 =S1 D S2 D shape sphere : Points of the quotient R3 represent oriented congruence classes of triangles: planar triangles modulo translation and rotation, but not scaling. Express R3 in spherical coordinates .r; s/; s 2 S2 . Then the origin r D 0 corresponds to triple collision. A point s on the sphere represents a ray rs; r  0 of triangles all having the same shape. The collision locus C D fr12 D 0 or r23 D 0 or r31 D 0g is represented by the three rays corresponding to the three binary collision points B12 ; B23 ; B31 2 S2 . Newton’s equations break down at triple collision r D 0. McGehee blow-up is a change of variables (Eq. (12)) which converts Newton’s equations to a system of ODEs which is well-defined when r D 0. The locus r D 0 in the new variables is called “the collision manifold” and forms a bundle over the shape sphere. The blown-up system of ODEs has exactly 10 fixed points, all on the collision manifold, a pair of fixed points lying over each of the five central configurations. For a chosen central configuration, one element of the pair corresponds to the homothetic arc incoming to triple collision, as in our original description of the Lagrange solution, while the other element of the pair corresponds to the initial segment of that solution which explodes out from triple collision.

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The 10 fixed points on the collision manifold have stable and unstable manifolds, parts of which stick out of the collision manifold, and which intersect in complicated ways, as per the Smale Horseshoe and heteroclinic tangles. See Fig. 5. Moeckel investigated these manifolds and their relations in seminal works [17–21], and [22] where he proved existence of “topological heteroclinic tangles” between them. Simó and Suslin had also proved existence of connections between the various collision manifolds with careful numerical evidence in [36]. One finds the following abstract graph

in several of these papers ([20], p. 53, Theorem 10 . Figure 2 of [22] becomes our graph after deleting the vertices labelled with B’s and edges incident to these B’s.)

Fig. 5 The equilibria arising upon blow-up and relations between their stable and unstable manifolds. The purple and green arrows are the rest cycles described in Fig. 6

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Fig. 6 The concrete graph, embedded in the shape sphere

Moeckel’s theorem in [20], based on the intersections between stable and unstable manifolds of the 10 fixed points, asserts that all paths in this graph are “realized” by solutions to the three-body problem provided the angular momentum, energy and masses are as per Theorem 1. Embed this graph in the shape sphere as indicated by Fig. 6. Call the embedded graph the “concrete connection graph”. The dynamical relevance of the concrete connection graph has to do with the Isosceles three-body problem. When two of the masses are equal, say m1 D m2 , then the isosceles triangles r13 D r23 form an invariant submanifold of the threebody problem whose dynamics is called the “Isosceles three-body problem”. These 2–3 Isosceles triangles form a great circle in shape space which passes through both Lagrange points, the binary point B23 , and the Euler point E1 . If all three masses are equal we have have three Isosceles subproblems represented by three great circles on the shape sphere. Take one-half of each great circle, namely that half whose endpoints are the two Lagrange points and which contains the Euler point. In this way we form the concrete connection graph whose edges are Isosceles semi-circles. Observe that the shape sphere minus the three binary collision points retracts onto the concrete connection graph. Theorem 1 follows immediately from this observation and Moeckel’s theorem referred to above, once we know that the realizing solutions of Moeckel’s theorem, projected onto the shape sphere, stay C0 -close to corresponding edges in the concrete connection graph. For details see Sect. 6 of this article or [24].

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4 Metric Set-Up: McGehee Blow-Up The method of blow-up in celestial mechanics was invented by McGehee in [15], originally for the 4-body problem on the line. It is no more work to perform the blow-up for the N-body problem in d-dimensional Euclidean space, rather than our special case of the three-body problem in the plane. The d-dimensional N-body equations are: ma qR a D †b¤a Fba ;

q a 2 Rd

(3)

with the forces Fba as above.

4.1 Metric Reformulation Let E D .Rd /N denote the N-body configuration space. Write points of E as q D .q1 ; : : : ; qN / and think of the points as the N-gons in d-space. The masses endow E with an inner product, hq; vi D †ma qa  va

(4)

called the mass inner product. Here  denotes the standard inner product on Rd . Then the standard kinetic energy is KD

1 hPq; qP i: 2

(5)

Let r be the gradient associated to this metric: dfq .v/ D hrf .q/; vi, so that .rf /a D 1 @f ma @Eq . Then the N-body equations take the simple form a

qR D rU.q/

(6)

where U is the negative of the standard potential V: U D V D †a 0g and M int .h; J0 / D fH D h; J D J0 ; r > 0g

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Energy and angular momentum are not defined at r D 0 so we have excluded r D 0. Set M.h/ D Closure.M int .h//;

M.h; J0 / D Closure.M int .h; J0 //;

(18)

the closure being within the blown-up phase space. (The superscript “int” is for “interior”.) We will need to understand the boundaries of these spaces, which is their intersection with the extended collision manifold r D 0; in other words we must understand how these invariant submanifolds approach the extended collision manifold fr D 0g as r ! 0. The following notation will be useful in this endeavor. Definition 3 (Notation) For F D F.q; v/ a homogeneous function on EE write FQ for the scale-invariant version of F achieved by multiplying F by r˛ where ˛ is the degree of homogeneity of F with respect to our weighted scaling. Thus: F.q; v/ D Q y/. r˛ F.s; According to “energy balance” both the potential energy, kinetic energy, and total energy are homogeneous of degree 1. Thus Q U.s/ D rU.q/ Q is homogeneous of degree 0 and can be viewed as a function on the sphere where U Scm . And Q D rH H

(19)

where Q y/  U.s/ Q Q y/ D 1 hy; yi  U.s/ D K. H.s; 2 Q U Q are homogeneous of degree 0. The angular momentum is homogeneous of and K; degree 1=2 so that Q y/ J D r1=2 J.s; where JQ is scale invariant and equals †ma sa ^ ya . If follows immediately from Eq. (19) that Q D 0; r D 0g: @.M.h// D fH while using in addition Eq. (20) we see that Q D 0; JQ D 0; r D 0g: @.M.h; 0// D fH We give these submanifolds separate names.

(20)

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Q D 0; r D 0g. Definition 4 The full collision manifold is M0 D fH Definition 5 The “standard collision manifold” is the locus Q D JQ D 0g: C WD fr D H Exercise 8 Show that M0 and C are invariant submanifolds of the blown-up flow by using Eq. (13) to show that d Q Q H D  H: d 1 d Q J D   JQ d 2 hold everywhere on the blown-up phase space. Thus the extended collision manifold contains the full collision manifold M0 which in turn contains the standard collision manifold C and these are invariant submanifolds. The equilibria all lie on C. The following theorem is fundamental. Theorem 2 (Sundman) If r ! 0 along an honest solution, then J D 0 for that solution and hence that solution tends to C as r ! 0. Moreover, the solution tends to the subset of equilibria within C. Here we are using the hopefully obvious Definition 6 An “honest solution” to the blown-up equations is a solution such that r > 0. The honest solutions are just the reparameterizations of solutions to our original Newton’s equations according to the blown-up time. Remark The standard collision manifold C is the space most authors refer to when they speak of the “collision manifold” for the N-body problem. Chenciner (see also [6]) argues that the standard collision manifold is the dilation quotient of the N-body phase space.

4.7 Aside: Parabolic Infinity Set u D 1=r and view u D 0 as a neighborhood of infinity.

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Exercise 9 Show that under the change of variables r 7! u D 1=r, with s; y unchanged the r0 equation becomes u0 D u with  as before. Now u D 0 becomes an invariant submanifold for the flow. We have there a kind of dual to the theorem of Sundman above. First we need a definition. Definition 7 A solution escapes parabolically to infinity as the Newtonian time t ! 1 if its energy H D 0 and if in the limit the unrescaled kinetic energy K.v/ tends to zero as t ! 1. Theorem 3 (Parabolic) Every solution which escapes parabolically to infinity tends to the subset of equilibria in the blown up variables .s; y/. There is one important difference to keep in mind now. Solutions with nonzero angular momentum can and do escape parabolically to infinity, while no solutions with angular momentum zero limit to the collision manifold.

5 Quotient by Rotations Newton’s equations and their McGehee blow-ups (Eq. (13)) are invariant under the group G of rigid motions and so descend to ODEs on the quotient space of their phase spaces by G. Working on this quotient instead of the original helps our intuition enormously in the case N D 3 and d D 2. We describe the quotient and some aspects of the quotient flow for general N and d D 2. The group G of rigid motions is the product of two subgroups, the translation group and the rotation group. We have already formed the quotient of phase space by translations when we went to center-of-mass frame, i.e. by restricting to s; y 2 Ecm . To form the remaining quotient by rotations it is much cleaner to restrict to the planar case d D 2. Henceforth we assume that we are working with the planar N-body problem, d D 2. We identify R2 with C as before. Thus E Š CN and Ecm Š CN1 . Represent rotations as unit complex scalars u 2 S1  C acting on .q; v/ 2 Ecm Ecm by .q; v/ 7! .uq; uv/ and on McGehee coordinates by .r; s; y/ 7! .r; us; uy/. Definition 8 The blown up reduced phase space in the planar case is the quotient of the blown-up center of mass phase space Œ0; 1/Scm Ecm Š Œ0; 1/S2N3 CN1 by the group of rotations. Upon deleting the collision locus C we denote this quotient by PN D .Œ0; 1/  .S2N3 n C/  CN1 /=S1 : Let us begin to try to understand this quotient by momentarily forgetting the velocities (v or y) and the fact that we deleted the collision locus C. The circle action

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sends a blown-up configuration .r; s/ to .r; us/, s 2 Scm . So we need to understand the quotient of the sphere Scm D S2N3 by this action of S1 . It is well known that this quotient Scm =S1 is isomorphic to the complex projective space CPN2 WD P.Ecm / with the projection map Scm ! Scm =S1 being the Hopf fibration. Hence the quotient of the .r; s/ by S1 yields Œ0; 1/  CPN2 . To better understand the meaning of points of CPN2 , work with a general q 2 Ecm , not necessarily a unit length vector. We insist only that q ¤ 0 and allow the scalar u to vary over the larger group C S1 of all nonzero complex numbers. The resulting quotient is well-known to be .CN1 n f0g/=C D CPN2 . The action of u 2 C on q 2 CN1 n f0g is precisely the action of rotating and scaling the (centered) N-gon q. Definition 9 The projective space CPN2 D P.Ecm / just constructed is called shape space. Its points represent oriented similarity classes of planar N-gons. We have realized the configuration part of the quotient after blow-up as Œ0; 1/  CPN2 where CPN2 is the shape space. When N D 3 the shape space is the shape sphere described above.

5.1 Collision Locus The condition that a configuration q D .q1 ; : : : ; qN / represent a collision is that qa D qb for some a ¤ b; 1 a; b N. This condition is complex linear when viewed in homogeneous coordinates Œq  1 ; q1 ; : : : ; qN  and so defines a complex hyperplane, a CPN3  CPN2 . There are N2 pairs .a; b/ and so we have to delete N 2 hyperplanes from our shape space. The union of these hyperplanes, viewed projectively, is the collision locus: C D fŒq D Œq1 ; q2 ; : : : ; qN  2 CPN2 W qa D qb some a ¤ bg: We use the same symbol for the collision locus before or after quotient.

5.2 Accounting for Velocities In the last few paragraphs above we dropped the velocity y. The quotient map .r; s/ 7! .r; Œs/ from Œ0; 1/  Scm ! Œ0; 1/  CPN2 expresses Œ0; 1/  Scm as a principal S1 bundle over Œ0; 1/  CPN2 . Now include the velocity y. The quotient procedure with y included is precisely the procedure used to construct an associated vector bundle to a principal bundle. (See for example [13] or [37].) Realizing this, we see that the quotient PN is a complex vector bundle over Œ0; 1/  .CPN2 n C/ whose rank is N  1, its fiber being parameterized by y 2 Ecm Š CN1 . What is this vector bundle?

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Proposition 2 PN D Œ0; 1/  T.CPN2 n C/  R2 as a vector bundle over Œ0; 1/  .CPN2 n C/. The final R2 factor can be globally Q where  D hs; yi represents the time rate of change of size coordinatized by .; J/ Q and where J D his; yi is also equal to r1=2 J off of r D 0 where J is the usual total angular momentum of the system. The fiber variable tangent to shape space CPN2 represents “shape” velocity. In the case of N D 3 we have CPN2 D CP1 D S2 , the shape sphere previously discussed in Sect. 3. Then P3 D Œ0; 1/  T.S2 n C/  R2 D Œ0; 1/  R  .S2 n C/  R3 ; where C D fB12 ; B23 ; B31 g is the set of three binary collision points.

5.2.1 Velocity (Saari) Decomposition Passing through a configuration q 2 Ecm we have two group-defined curves: the scalings q;  2 R of q and the rotations uq; u 2 S1 of q. The tangent spaces to these curves are orthogonal, and together with the orthogonal complement of their span they define a geometric splitting of Tq Ecm D Ecm Tq Ecm D (scale) C (rotation) C (horizontal) D Rq

˚

iRq

˚

fv W J.q; v/ D 0; .q; v/ D 0g

(21) (22)

where Definition 10 The horizontal space at q is the orthogonal complement (rel. the mass metric) of the sum of first two subspaces Rq and iRq, i.e. it is the orthogonal complement to the C-span of q. Refer to Exercise 7 and the definition of  to see why the horizontal space at q is the zero locus of J.q; v/ and .q; v/. Unit vectors spanning the scale and rotation spaces are s and is. Consequently, if we take a v 2 Tq Ecm and decompose it accordingly we get v D hs; vis C his; viis C vhor

(23)

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and the scale invariant version: Q C yhor I y D s C Jis

 D hs; yi; JQ D his; yi

(24)

where the subscript “hor” on v and y denote their orthogonal projections onto the horizontal subspace. Remark Saari [34] pointed out the importance of the horizontal-vertical splitting of Eq. (23 ) in celestial mechanics. This splitting is thus often called the “Saari decomposition” in the context of the N-body problem.

5.2.2 Proof of Proposition 2 The decomposition (Eq. (24)) of y is S1 -equivariant. The coefficients of the first two terms  and JQ D his; yi are S1 -invariant functions and so are well defined functions on the quotient PN . The horizontal term yhor , as y varies at fixed s, sweeps out the horizontal subspace at s and these subspaces, as s varies, forms the horizontal distribution associated to a connection on the principal S1 -bundle Scm ! CPN2 . It is a basic fact about principal G-bundles with connection that the union of the horizontal spaces for the connection forms a G-equivariant vector bundle over the total space, and the quotient of this vector bundle by G is canonically isomorphic to the tangent space to the base space. Writing Œs; y to denote the S1 equivalence class of the pair .s; y/ we see that the set of all Œs; yhor ’s forms TCPN2 . Q determine y uniquely. It follows that the map Now s, together with .yhor ; ; J/ Q is a vector bundle isomorphism between the vector bundles Œs; y 7! .Œs; yhor ; .; J// .Scm  Ecm /=S1 and TCPN2  R2 over CPN2 . The radial scaling coordinate r goes along for the ride, without any change. t u Because the decompositions of Eqs. (23) and (24) are orthogonal and the second decomposition is scale invariant it follows that total kinetic energy decomposes as 1 2 1 J2 Kshape .Œs; yhor / C C 2 2 r 2r r 2 2 Q 1  J C Kshape /: D . C r 2 2

K.q; v/ D

(25)

The final term Kshape is formed by computing the squared length of the horizontal factor yhor and is canonically identified with the kinetic energy of the standard Fubini-Study metric on the shape space CPN2 . Remark The kinetic energy decomposition (25) shows that for J ¤ 0 the manifolds M int .H0 ; J/ is already closed in PN so that M.h; J/ D M int .h; J/

(26)

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Q shows that U Q  1 J 2 =r C O.r/ holds on Indeed, the energy equation rh D H 2 int M .h; J/ which shows that if for a sequence pi 2 M int .H0 ; J/ we have that r.pi / ! 0 then U.si / ! 1 so that the shape si of these points pi are converging to the collision locus C  CPN2 on the shape space. But we deleted C in forming PN .

5.3 Euler-Lagrange Family in Reduced Coordinates What does a planar central configuration family (Sect. 4.3.1) look like in the blownup reduced coordinates of PN ? We largely follow the exposition of Moeckel [21, section 2]. Let scc be a planar central configuration and Œscc  2 CPN2 the corresponding point in shape space. For the associated central configuration family, the shape does not change. In particular the shape velocity yhor D 0. Thus among our full set of Q of PN D Œ0; 1/T.CPN2 nC/R2 we have Œs; yhor  D variables .r; Œs; yhor ; .; J// Q change along the solution Œscc ; 0 being constant, and only the variables .r; ; J/ curves of the family. This size r and angle  of the curves in the family specify the homography factor  D .t/ D rei where .t/ solves the Kepler problem as per Exercise 6. Since the shape does not change, the shape velocity yhor is identically zero along each of these solutions and so Kshape D 0. Thus along such a solution 1 1 1 J2 1 KQ D  2 C JQ2 D  2 C 2 2 2 2 r (see Eq. (25)). But JQ D r1=2 J and J is constant along solutions so the change of r Q So we can think of the only variables and choice of J determines the change of J. for the family as being ; r. Fix the energy h. We can then view the central configuration family as a oneparameter family of curves in the .; r/ plane, the parameter being the angular momentum J. Indeed the energy equation reads: rh D

1 2 1 2  C J =r  U.scc /: 2 2

and since U.scc / is constant, this defines a one-parameter family of curves. We plot these curves in the ; r plane for various values of the angular momentum J below in Fig. 7. Observe the rest point cycle in this picture: the closed curve passing through the two equilibria. This curve is the union of two solution curves, a top arch which is an honest solution, and a bottom return curve lying in M0 . The top arch lies on

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Fig. 7 A central configuration family in ; r coordinates. The arch and ‘floor’ r D 0 comprise the rest cycle

M.h; 0/4 and is the ejection-collision orbit described when we introduced central configurations in Sect. 2.2 by describing Lagrange’s solution: the orbit explodes out of total collision along the shape scc achieves a maximum size and shrinks back to p triple collision. It connects the rest point R 2 C having shape p scc and  D 2U.scc / > 0 with the rest point R having shape scc and  D  2U.scc / < 0. Q D 0g D M0 The bottom ‘return road’ lies on the full collision manifold fr D 0; H and yields a return route from R to R. This rest point cycle is the limit of the family of the periodic central configuration solutions with J ¤ 0 as J ! 0. It is important for later on that the return road DOES NOT lie on the standard collision manifold C, and that it does not consist of rest points.

Notational Convenience. We have just used the symbol M.h; 0/  PN for what used to be a submanifold of the phase space before quotient. We will continue to use the same notation for any G-invariant submanifold or function on phase space before or after the quotient procedure. Thus we have:

4

C; M0 ; M.h/; M int .h/; M.h; J0 /; etc.  PN :

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6 A Gradient-Like Flow! A flow is called “gradient-like” if it admits a continuous function f , which we call a ‘Liapanov function’ which is strictly monotone decreasing along all solution curves except for the equilibria. (See Robinson [32, p. 357.].) The dominant aspect of the flow on the full collision manifold M0 is that it is gradient-like with  as Liapanov function. See Proposition 3 below. Exercise 10 Use Eqs. (13) to derive the identity 1 Q  0 D KQ   2 C H 2

(27)

(See for example Moeckel [18, eq. (1.6)].) Exercise 11 Use the “Saari decomposition” of kinetic energy (Eq. (25)) to show that 1 1 KQ   2 D Kshape C JQ2 : 2 2 Conclude, using the previous exercise, that 1 Q D 0g:  0 D Kshape C JQ2  0 on M0 D fr D 0; H 2

(28)

You have proved much of Proposition 3  0  0 everywhere on the full collision manifold M0 . Moreover  is constant along a solution lying in M0 if and only if that solution is one of the equilibria. Remark Proof of Proposition 3. In Exercise 11, Eq. (28) you proved that  0  0. It remains to show that any solution which lies on the locus where  0 D 0 is in fact an equilibrium. Equation (28) implies that  0 D 0 if and only if Kshape D 0 and JQ D 0. Now Kshape .Œs; y/ D 0 if and only if yhor D 0. So,  0 D 0 if and only if both yhor and JQ D 0. But then the only nonzero term in the Saari decomposition of Eq. (24) is the real term so that y D s with  2 R. Take inner products with s to find that  D , or y  s D 0 for any such point. Assume now that we have a solution curve .s./; y.// of such points in M0 lying on the locus  0 D 0. Differentiating the equation y./ D ./s./ using the blow-up equations we see that y0 D  0 s C s0 . But  0 D 0 by assumption and s0 D y  s D 0 by the blow-up equations, so y0 D 0 along the solution: our curve is an equilibrium. t u

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Example 1 Return the central configuration family s D scc described in the earlier Sect. 5.3 and its limiting “return path” on M0 indicated in Fig. 7. From rh D KQ  U, U D U.scc /, and Kshape D 0 we have that 0 D 12  2 C 12 JQ2  U.scc / or 12  2 C 12 JQ2 D U.scc / for this return path. On the other hand  0 D Kshape C 12 JQ2 D 12 JQ2 . These equations yield two important conclusions : (1) the only rest points on the family are at C where JQ D 0 and (2) as we approach points along the return path from the interior M int .h/ we have that J ! 0 and r ! 0 in such a way that JQ D r1=2 J tends to a finite nonzero limit.

6.1 Making Moeckel’s Manifold with Corner into a Manifold with a T In [21], at the beginning of section 2, Moeckel constructs a certain manifold with corners in preparation for perturbing the heteroclinic tangles lying on M.h; 0/ into the realms of M.h; /. (He denotes his manifold with a corner by M0C and later simply M.) Dynamics on this manifold-with-corners is essential to our proof of Theorem 1. I had a hard time making sense of this manifold so I rederived what Moeckel did in a slightly different way. I get a “manifold with a T” instead of Moeckel’s manifold with a corner. A “T ” is made out of two corners. One corner is Moeckel’s manifold with a corner and the other is a reflection of it. The corner itself is our good friend C, the standard collision manifold (Fig. 8). Recall that M.h/ is a hypersurface in PN , and as such is a manifold with boundary, whose boundary is our friend the full collision manifold M0 D fr D Q D 0g. 0; H O Definition 11 M.h/ D M.h; 0/ [ M0  M.h/. O M.h/ is a codimension 1 subvariety of the smooth manifold with boundary M.h/. It is the zero locus of the function rJQ restricted to M.h/ and as such has two algebraic components: r D 0 which is our full collision manifold M0 , and J D 0 which forms O M.h; 0/. The singular locus of M.h/ is the intersection C D fr D 0; JQ D 0g of these two components. All the rest point cycles described above associated to the O O central configurations lie on this M.h/. M.h/ is comprised of two “manifolds with corners”, namely frJQ D 0; JQ  0g and frJQ D 0; JQ 0g. The first of these is Moeckel’s manifold with a corner. O M.h/ is to be viewed as the limit as J ! 0 of the manifolds M.h; J/. Proposition 4 For S  R a subset of the line of angular momentum values, set O M int .h; S/ D [J2S M int .h; J/. Then M.h/ D \>0 M int .h; .; //. The proof of the proposition follows in a routine way from our expressions Q J D r1=2 JQ and the kinetic for scaled energy and angular momentum rh D H, energy decomposition of Eq. (25). It is useful to recall, Eq. (26) that the M.h; J/ D M int .h; J/ are closed for J ¤ 0.

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As an alternative to the description of the proposition, we can either let J ! 0 from above or below. Set O C .h/ D lim M.h; J/ M J!0C

and O  .h/ D lim M.h; J/: M  J!0

Then one can show without difficulty that O M.h/ D MC .h/ [ M .h/; O O  .h/ D f p 2 M.h/; O O C .h/ D f p 2 M.h/; JQ  0g and M JQ 0g being the two with M manifolds with corners described earlier, Moeckel’s manifold with a corner being O C. M What is a Manifold with a ‘T’? Suppose we have two real-valued functions x; y on an n-dimensional manifold Q such that 0 is a regular value for both functions and .0; 0/ is a regular value of the map .x; y/ W Q ! R2 . Then the locus fxy D 0; y  0g is a manifold with a T. Its singular locus is fx D y D 0g. A manifold with a T is locally diffeomorphic to the product of the “upside down T” xy D 0; y  0 in the xy plane, by an Rn2 . See Fig. 8.

O Fig. 8 M.h/ inside M.h/ is the zero level set of rJQ

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6.2 Finishing Up the Proof of Theorem 1 The idea of Moeckel is that hyperbolic structures persist on perturbation, and that the various stable-unstable connections between Euler and Lagrange central O configuration points on M.h/ are sufficiently “hyperbolic” that they persist into M.h; / for  ¤ 0 small. Nonzero angular momentum is needed to get orbits connecting from R’s to R in finite time since the rest cycle of Fig. 7 takes infinite blown-up time. Moeckel cannot carry out the “perturbation of hyperbolic” idea literally because he cannot establish the needed hyperbolicity or transversality. Instead, following an earlier idea of Easton, he replaces hyperbolicity by a weaker notion of “topologically transverse” between collections of “windows” transverse to the flow. This notion is sufficiently flexible and stable to allow Moeckel to perturb the various formal connections to get actual orbits realizing walks in the abstract graph introduced in Sect. 3. By following the details of his proof, three decades later, we were able to verify that his realizing solutions when projected onto the shape space do indeed stay C0 -close to the concrete connection graph as described in Sect. 3. The hypothesis of equal or near equal masses is needed to insure that (some of) the eigenvalues for the linearization at the Euler equilibria are complex. This complexity implies a “spiralling” of the Lagrange stable/unstable manifolds around the Euler unstable/stable manifolds and is needed to insure that all connections in the abstract connection graph are realized.

7 A Conjecture: Non-existence Theorem 1 asserts the existence of a family of small-angular momentum solutions which realize any given free homotopy class. What about our original problem, described in Sect. 1.1, of realizing classes for the angular momentum zero threebody problem? The simplest classes of all are those which wind once around a binary collision. They are represented by a curve in which two of the masses rotate once around their common center of mass while the third body remains motionless, far away. We call these classes “tight binary classes”. Their syzygy sequence is ij where i; j are the two moving masses. Conjecture 1 There is no reduced-periodic solution to the equal-mass zero angular momentum three-body problem which realizes a tight binary class.

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We present four pieces of evidence supporting the conjecture. 1. 2. 3. 4.

Hyperbolic Pants. (Not) Hanging out at Infinity. Danya Rose’s Bestiary. Failure of limits.

7.1 Hyperbolic Pants Continue to take the masses all equal, but change the potential from Newton’s 1=r potential to the ‘strong-force’ 1=r2 —the same one investigated by Poincaré [30]. I proved in [27] that the tight binary class is not realized by a reduced periodic solution of this modified problem.

7.2 Hanging Out at Infinity I tried to establish existence of tight-binary type zero angular momentum periodic solutions of “Earth-Moon-sun type” using perturbation theory, following Meyer [16] as a guidebook. If they existed these would be solutions in which 1 and 2 are far away, moving counterclockwise in an approximate circle about their common center of mass while that center of mass moves clockwise in a slow circle about mass 3. Such a motion could not realize a single tight binary as a periodic solution in the inertial frame since there are many months (1–2 circles) in a single slow year. But we only care about relative motion, so it looks conceivable that such a motion could be reduced periodic, executing 1  2 in a single reduced period. Connor Jackman, a UCSC graduate student, recently proven that this approach cannot succeed [12]. To explain his result, we begin by investigating the relevant Hill region. Fix the angular momentum to be zero and the energy to be some negative constant. Project this codimension two energy-momentum level set onto shape space to achieve the Hill region. If U is the negative of the potential so that H D K  U with K the kinetic energy and if we set H D h < 0 then the Hill region is the region for which U  h. The result is depicted in Fig. 9. The radial coordinate R in that picture is a measure of the overall size of the triangle. From the picture we see that imposing a constraint of the form R > R0 for R0 large enough, breaks the Hill region into three components. Each component is associated to a partition of the three bodies into a tight binary configuration and a far mass. Insisting that R >> 1 is equivalent to insisting that in each component one of the two distances is much smaller than the other two. That is, insisting R >> 1 is the same as saying that we are working within the realm of perturbation theory.

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Fig. 9 Cutting a ball from a pair of pants

We can now state Jackman’s theorem Theorem 4 Fix angular momentum zero, total energy to some negative constant. Then there is an R0 > 0 (large) such that any solution (periodic or not!) which begins in the region R > R0 must enter the region R < R0 . Jackman’s theorem holds for any angular momentum, for any mass distribution as long as the masses are all positive and any negative energy. The F0 will depend on the masses, the angular momentum, and the energy. As a corollary to the theorem, we see that there is no orbit having the pattern described at the beginning of this section.

7.3 The Bestiary of Danya Rose Danya Rose is a recent PhD [33] from Sydney, Australia, who worked under the direction of Holger Dullin. The heart of his thesis is a systematic, very detailed numerical study of the equal-mass zero angular momentum three-body problem which contains over 300 non-collinear, non-isosceles solutions. These solutions are meticulously laid out in over 700 pages of Appendix F of his thesis. I reproduce two sample pages here, below. He titles this appendix “A bestiary of periodic orbits”. The Bestiary contains no solutions which even come close to representing a simple tight binary. The pages come in pairs, one of which contains an array of statistics of the solution, and the other consists of four pictures, one being that solution drawn as three curves in inertial space, another begin the corresponding curve on the shape sphere, and a third being the curve viewed on the regularized shape sphere.

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I describe some details of his search strategy so you can decide for yourself how convincing the data is. At the end of this subsection, I reproduce two pages from the Bestiary. The search proceeds in two basic steps. The first I call “gravitational billiards” and relies crucially on the fundamental domain defined by the group generated by the discrete symmetries available when the masses are equal. The second step is a careful numerical integration on the regularized shape space.

7.3.1 Coding Gravitational Billiards When the masses are equal the problem admits mass interchange qi $ qj as a discrete symmetry. These symmetries are involutions and generate the symmetric group on three letters acting on configuration space by permuting the position coordinates. On the shape sphere the mass interchange involution qi $ qj acts as a half-twist about the binary collision ray rij D 0. We add to these involutions the reflection about any line in the plane which generates reflection about the equator on the shape sphere. Together these involutions generate a 12 element group, the dihedral group D6 associated to a regular hexagon, which acts on configuration space, mapping solutions to solutions. We used this group to great advantage in [8]. The D6 action on configuration space induces an action on the shape space and the shape sphere. A fundamental domain for the action on the shape sphere consists of a spherical triangle whose vertices are a binary collision, say B12 , a neighboring Euler point, say E1 , and the ‘upper’ Lagrange point, LC . See Fig. 10 where, following Rose, we relabel these points B; M and E. The three edges are labelled A, O and C. The edges A and O correspond to Acute and Obtuse Isosceles triangles. C represents for collinear triangles. Fig. 10 A fundamental domain on the shape sphere, with labels

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Fig. 11 A fundamental domain on the shape space is a curvilinear tetrahedron

The corresponding fundamental region on the shape space R3 is the inverse image of this spherical triangle under central projection, intersected with the Hill region. This shape space fundamental domain is an curvilinear (ideal) tetrahedron. See Fig. 11. The outer face of this tetrahedron lies on the Hill boundary, labelled “F” by Rose, F being for Freefall. It corresponds to the single face of the spherical triangle under central projection. Three of the four vertices of the tetrahedron lie on this face, one vertex being ideal, at infinity along the binary ray. The other two vertices on this Hill boundary face correspond to the isosceles and Lagrange brake initial conditions. The tetrahedron’s fourth vertex is triple collision, at the origin of R3 . The other three faces of the tetrahedron correspond to the three edges of the spherical triangle. The three edges incident to triple collision continue to carry the labels of their corresponding vertices in the spherical triangle.

84 Table 1 Faces of the fundamental tetrahedron = edges plus face of fundamental spherical triangle Table 2 Relevant edges of fundamental tetrahedron = vertices of fundamental spherical triangle

R. Montgomery A O C F B E M

Acute (Isosceles) Obtuse (Isosceles) Collinear Freefall (Hill boundary) Binary Equilateral (= Lagrange) Midpoint (= Midpoint)

We summarize Rose’s coding of the tetrahedral elements in Tables 1 and 2. A theorem of mine [25] asserts that any negative energy zero angular momentum solution must repeatedly intersect “C”—collinearity, and hence cannot stay inside the interior of a Fundamental domain for all time. Since such a solution must hit ‘C’ we may as well start on C. Inspired by this theorem, Rose starts off with initial condition on the collinear face, and ‘shoots’ in to the domain by choosing velocities pointing in. He numerically integrates. That solution enters into the three-dimensional interior table and leaves it through some other face. Instead of leaving, he can reflect that solution back in, using the associated reflection of that face, and continue. Equivalently, each time we hit a face, we apply Snell’s law to the associated vector, reflect it back in to the region and continue by applying Newton’s equations to this new initial condition. In this way, we get a billiard problem, on the tetrahedral “table”. The trajectories bounce off the walls by the standard billiard reflection principle, and move inside the interior according to the zero-angular momentum reduced Newton’s equations. For any such solution, we simply list the tetrahedral boundary elements hit, in the order of hitting, as per the standard practice of symbolic dynamics. This list is a word in the seven letters A,O,C, F, B, M, E. Rose calls this the orbit’s “sequence type” The simplest realization of a tight binary would have sequence type COCO. Remark on Vertices Following Rose, we ignore the vertices of the tetrahedron in our listing. Here are some good reasons for ignoring the vertices. Two of the vertices lying on the Hill boundary, correspond to the Lagrange and Euler homothety solutions. They represent single known solutions The remaining vertex on F is the ideal one of a binary collision in “free fall” . The corresponding “solution” is a “hard binary”: an ideal motion in which two of the masses are eternally stuck together in collision, falling in to the third mass. The final vertex is triple collision, perhaps the most interesting, about which volumes have been written, including the bulk of the present paper. But a periodic solution cannot hit triple collision. Rose cuts off his integrations when they get too close to triple collision, so this vertex is not relevant for Rose’s investigations.

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We now copy two pages from the Bestiary. Following that we will explain a bit about the regularized shape sphere and the integration method. From Danya Rose’s Bestiary, pp. 163 , 164.

7.3.2 B-Mode, Unstable: t0 (8, 5) Isotropy subgroup: {.I; 0/, . 2 ; 12 /, .s1 ; 14 /, . 2 s1 ; 14 /, .s2 ; 12 /, . 2 s2 ; 0/, .s3 ; 34 /, . 2 s3 ; 34 /} Sequence type: .A C 0 /4

W OCACOACAOACACOA D 67:01921804 D 14:96879087 D 12:36682876 D 8:57712227 D 0:00000000 1 1 0 0 0:94542673 1:41582468 BC1:08224817C BC0:96447581C C C B B C C B B BC0:94542673C B 0:00000000C z2 D B z1 D B C C BC1:17170924C BC0:00000000C C C B B @ 0:00000000A @ 0:00000000A C1:17170924 C1:72244433 R1 D 0  2 s2 R D s 2 1 0 3 1 C25:42672460 C25:42672460 C 0:00000000i B C1:00000000 C B C1:00000000 C 0:00000000i C C B C B C B C B B C1:00000000 C B C0:25229380 C 0:96765068i C DB C C jj D B B C1:00000000 C B C0:25229380  0:96765068i C C B C B @ C1:00000000 A @ C1:00000000 C 0:00000000i A C0:03932870 C0:03932870 C 0:00000000i

Tp Tr G  W

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αj

1 0

−1

τ

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7.3.3 Regularized Shape Sphere and Numerics Binary collisions are a singularity of the ODEs defining the three-body problem and create havoc with numerics. Levi-Civita regularized a single isolated binary collision in the planar problem by using a kind of branched cover over the collision point followed by a time change. Lemaître extended what Levi-Civita did so as to apply to all three binary collisions in a democratic manner. His work was followed up by Murnaghan, Waldvogel who wrote several sets of explicit polynomial regularized equations. See also [23] for a geometric perspective on the regularization maps with many pictures. Rose integrates the reduced regularized equations as written down by Waldvogel, using a symplectic integrator developed along with Dullin. Rose keeps track of when and how solutions cross the various faces using an elegant idea due to Henon for accurately computing Poincaré sections. An added bonus is that in a natural system of coordinates, denoted ˛1 ; ˛2 ; ˛3 , the faces of the regularized fundamental tetrahedron have a very simple description.

7.4 Failure of Limits What happens if we take the solutions guaranteed by our Theorem 1 and let the angular momentum J ! 0? All of them limit to various concatenations of Euler collision-ejection central configuration solutions joined at triple collision. In other words: they all die in total collision. Moeckel and I spend a few weeks trying to establish other hyperbolic-based ‘return mechanisms” in the spirit of Fig. 7 above which would work for J D 0. Such a mechanism would have allowed us to construct the requisite symbolic dynamics and thereby get existence of near-but-not collision periodic solutions having J D 0 and the desired reduced syzygy sequences. Our efforts repeatedly failed. Acknowledgements I am grateful for critique and feedback in the preparing this article from Alain Albouy, Alain Chenciner, Carles Simó, Rick Moeckel, Connor Jackman and Gabriel Martins. Wu-yi Hsiang asked the central question which inspired this research. As described, a conversation with Carles Simó completely redirected my methods to the ones that were ultimately successful. The main result owes its existence to Rick Moeckel. I am thankful to the participants of the CIMAT school in Guanajuato, Mexico, to the organizer of that school Rafael Herrera, to Gil Bor for good Israeli salads, company, and piano playing and to Patricia Carral and Eyal Bor for their hospitality and use of a comfortable bed during this school. After much of this writing was completed I learned from Holger Dullin about Danya Rose’s work. The heart of this exposition, the sections on McGehee blow-up were first TeXed up for a seminar I ran with Rafe Mazzeo at Stanford over a decade ago and I would like to express gratitude to Rafe and all the seminar attendees. I would like to thank both Dullin and Rose for a number of email correspondences. Finally I wish to thank NSF grant DMS-1305844 for essential support.

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References 1. A. Albouy, Symétrie des configurations centrales de quatre corps. C. R. Acad. Sci. Paris 320(2), 217–220 (1995) 2. A. Albouy, V. Kaloshin, Finiteness of central configurations of five bodies in the plane. Ann. Math. (2) 176(1), 535–588 (2012) 3. A. Albouy, H. Cabral, A. Santos, Some problems on the classical n-body problem. Celest. Mech. Dyn. Astron. 113, 369–375 (2012). http://arxiv.org/abs/1305.3191 4. G. Bor, R. Montgomery, Poincare y el problem de N-cuerpos. Miscelanea matematica de la Sociedad Matematica Mexicana, May issue, no. 57 ‘extraordinario’ ( 2013) 5. J. Chazy, Sur certaines trajectoires du problème des n corps. Bull. Astron. 35, 321–389 (1918) 6. A. Chenciner, Collisions totales, mouvements complètement paraboliques et réduction des homothéties dans le Problème des N corps. Reg. Chaotic Dyn. 3, 93–106 (1998) 7. A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry, in ICM 2002, vol. 3 (2002) 8. A. Chenciner, R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses. Ann. Math. 152, 881–901 (2000) 9. A. Chenciner, J. Gerver, R. Montgomery, C. Simó, Simple choreographies of N bodies: a preliminary study, in Geometry, Mechanics, and Dynamics, ed. by P. Newton, P. Holmes, A. Weinstein, Volume in honor of the 60th birthday of J.E. Marsden (Springer, Berlin, 2002) 10. L. Euler, Considérations sur le problème de trois corps. Mémoires de l’Académie de Sciences de Berlin 19, 194–220 (1770) 11. M. Hampton, R. Moeckel, Finiteness of relative equilibria of the four-body problem. Invent. Math. 163(2), 289–312 (2006) 12. C. Jackman, The Binary Returns! arxiv:1512.01852 (2015) 13. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. 1 (Interscience Pub., New York, 1969) 14. J.-L. Lagrange, Essai sur le Probleme des Trois Corps (Prix de l’Academie Royale des Sciences de Paris, tome IX, 1772) ( in vol. 6 of Oeuvres (p. 292)) 15. R. McGehee, Triple collision in the collinear three-body problem. Invent. Math. 27, 191–227 (1974) 16. K. Meyer, Periodic Solutions of the N-Body Problem. Lecture Notes in Mathematics, vol. 1719 (Springer, Berlin, 1999) 17. R. Moeckel, Orbits of the three-body problem which pass infinitely close to triple collision. Am. J. Math. 103(6), 1323–1341 (1981) 18. R. Moeckel, Orbits near triple collision in the three-body problem. Indiana Univ. Math. J. 32(2), 221–239 (1983) 19. R. Moeckel, Heteroclinic phenomena in the isosceles three-body problem. SIAM J. Math. Anal. 15, 857–876 (1984) 20. R. Moeckel, Chaotic orbits in the three-body problem, in Periodic Solutions of Hamiltonian Systems and Related Topics, ed. by P.H. Rabinowitz (D. Reidel Pub., Dordrecht, 1987) 21. R. Moeckel, Chaotic dynamics near triple collision. Arch. Rat. Mech. 107(1), 37–69 (1989) 22. R. Moeckel, Symbolic dynamics in the planar three-body problem. Reg. Chaotic Dyn. 12(5), 449–475 (2007) 23. R. Moeckel, R. Montgomery, Symmetric regularization, reduction, and blow-up of the planar three-body problem. Pac. J. Math. 262(1), 129–189 (2013) 24. R. Moeckel, R. Montgomery, Realizing all reduced syzygy sequences in the planar three-body problem. Nonlinearity 28, 1919–1935 (2015). Also arXiv:1412.2263 25. R. Montgomery, Infinitely many syzygies. Arch. Ration. Mech. Anal. 164(4), 311–340 (2002) 26. R. Montgomery, The three-body problem and the shape sphere. Am. Math. Monthly 122(4), 299–321 (2015). Also: arXiv:1402.0841 27. R. Montgomery, Fitting hyperbolic pants to a three-body problem. Ergodic Theory Dynam. Syst. 25(3), 921–947 (2005)

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28. C. Moore, Braids in classical gravity. Phys. Rev. Lett. 70, 3675–3679 (1993) 29. J. Moser, Stable and Random Motion in Dynamical Systems (Princeton University Press, Princeton, 1973) 30. H. Poincaré, Sur les solutions périodiques et le principe de moindre action. C.R.A.S. Paris 123, 915–918 (1896) 31. G. E. Roberts, A continuum of relative equilibria in the five-body problem. Physica D 127, 141–145 (1999) 32. C. Robinsion, Dynamical Systems, 2nd edn. (CRC Press, West Palm Beach, 1999) 33. D. Rose, Geometric phase and periodic orbits of the equal-mass, planar three-body problem with vanishing angular momentum. Ph.D. thesis, School of Mathematics and Statistics University of Sydney, 2015 34. D. Saari, From rotation and inclination to zero configurational velocity surfaces, I, a natural rotating coordinate system. Celest. Mech. 33, 299–318 (1984) 35. C. Simó, Relative equilibrium solutions in the four-body problem. Celest. Mech. 18, 165–184 (1978) 36. C. Simó, A. Susín, Connections between invariant manifolds in the collision manifold of the planar three-body problem, in The Geometry of Hamiltonian Systems, ed. by T. Ratiu, Mathematical Sciences Research Institute Series (Springer, Berlin, 1991), pp. 497–518 37. N. Steenrod, The Topology of Fiber Bundles (Princeton University Press, Princeton, 1951)

A Quick View of Lagrangian Floer Homology Andrés Pedroza

Abstract In this note we present a brief introduction to Lagrangian Floer homology and its relation to the solution to the Arnol’d Conjecture, on the minimal number of non-degenerate fixed points of a Hamiltonian diffeomorphism. We start with the basic definition of a critical point on smooth manifolds, in order to sketch some aspects of Morse theory. Introduction to the basics concepts of symplectic geometry are also included with the idea of understanding the statement of the Arnol’d Conjecture and how it is related to the intersection of Lagrangian submanifolds.

1 Introduction Many elegant results in mathematics have to deal with the fixed-point-set of a function. For example: Brouwer fixed-point theorem, Lefschetz fixed-point theorem, Banach fixed-point theorem and Poincaré-Birkhoff theorem, just to name a few. Furthermore, these results are fundamental in their own area of mathematics and have interesting consequences in diverse areas of mathematics; differential equations, topology and game theory among others. Symplectic geometry has its own fixed-point theorem, which was conjectured by Arnol’d [1] in 1965. The Arnol’d Conjecture was motivated by Poincaré-Birkhoff theorem: An areapreserving diffeomorphism of the annulus which maps the boundary circles to themselves in opposite direction, must have at least two fixed points. The generalization of Poincaré-Birkhoff theorem fits in symplectic geometry and not in volume-preserving geometry. The Arnol’d Conjecture establishes a lower bound on the number of fixed points of a Hamiltonian diffeomorphism in terms of the topology of the manifold. The fixed points of a Hamiltonian diffeomorphism, (in fact any diffeomorphisms) can be seen as the intersection of its graph and

A. Pedroza () Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo No. 340, 28045 Colima, Colima, Mexico e-mail: [email protected] © Springer International Publishing AG 2018 L. Hernández-Lamoneda et al. (eds.), Geometrical Themes Inspired by the N-Body Problem, Lecture Notes in Mathematics 2204, https://doi.org/10.1007/978-3-319-71428-8_3

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the diagonal. In the context of symplectic geometry, is the intersection of two Lagrangian submanifolds. In 1987, Floer [10] developed a homological theory that focused on the intersection of Lagrangian submanifolds. In particular, under some hypotheses, he proved the Arnol’d Conjecture for a particular class of closed symplectic manifolds. This theory is called Lagrangian Floer homology. In these notes we sketch how Lagrangian Floer homology is defined. In fact we review some aspects of Morse theory from its basics; like non-degenerate critical points, the Hessian, flow lines of the gradient vector field up to Morse homology. The reason being, that Lagrangian Floer homology emulates in many aspects Morse homology. Also we cover the basics of symplectic manifolds and Hamiltonian diffeomorphisms. The last section deals with Lagrangian Floer homology and how it is used to prove the Arnol’d Conjecture. For the basic notions of differential geometry the reader can look at [41]; for the aspects of symplectic geometry [7] and [23]; and also [24] where the analytical aspect of holomorphic curves is covered. For details and proofs on the construction of Lagrangian Floer homology see [3, 32] and [33]. For an excellent introduction to Fukaya categories see [4]; and [40] for a detail treatment of the subject.

2 Morse-Smale Functions Let M be a smooth manifold of dimension n and f W M ! R a smooth function. A point p 2 M is called a critical point of f if the differential dfp W Tp M ! R at p is the zero map. Denote by Crit. f / the set of critical points of f . Notice that Crit. f / can be the empty set, however if M is compact then it is not empty, since a smooth function on M has a maximum and a minimum. Let p 2 M be a critical point of f and .x1 ; : : : ; xn / a coordinate chart about p. The Hessian matrix of f at p relative to the chart .x1 ; : : : ; xn /, is the n  n matrix  Hess. f ; p/ D

 @2 f . p/ : @xi @xj

A critical point p is said to be non-degenerate if the matrix Hess. f ; p/ is nonsingular. Note that the Hessian matrix is symmetric, hence if it is non-singular its eigenvalues are real and non-zero. The index of f at a non-degenerate critical point p, which is denoted by ind. f I p/, is defined as the number of negative eigenvalues of the Hessian matrix at p. The definition of the index at a non-degenerate critical point given above depends on the coordinate system; however it can be shown that it is independent of the coordinate system about the critical point. There is an alternative definition of the index of a function at a non-degenerate critical point, that does not needs a

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coordinate system. For a critical point p 2 M of f define the bilinear form dfp2 W Tp M  Tp M ! R Yf /, where e Y is any vector field on M whose value at p is Y. as dfp2 .X; Y/ WD X.e Notice that since p is a critical point of f , the bilinear form dfp2 is symmetric, X; e Y D Œe X; e Yp . f / D e X p .e Yf /  e Y p .e Xf /: 0 D dfp Œe In this context, p is called non-degenerate if the bilinear symmetric form dfp2 is nondegenerate. The index of f at p is defined as the number of negative eigenvalues of the symmetric bilinear form dfp2 . The two definitions given of non-degenerate critical point agree. The same applies for the two definitions of the index of a nondegenerate critical point. For further details, see [3, Ch. 1] and [29]. Definition 1 A smooth function f W M ! R for which all of its critical points are non-degenerate is called a Morse function. Now we consider some examples in the case when M D R2 . The origin is the only critical point of the function f .x; y/ D x2 C y2 . Moreover is a non-degenerate critical point and its index is zero. The origin is also the only non-degenerate critical point of the functions g.x; y/ D x2  y2 and h.x; y/ D x2  y2 . In these cases the index at the origin is 1 and 2 respectively. These three examples describe the general behavior of a function on R2 near the origin when it is a non-degenerate critical point. The precise statement on the behavior of a function near a non-degenerate critical point is given by Morse lemma. Theorem 1 (Morse Lemma) Let f W Rn ! R be a smooth function such that the origin is a non-degenerate critical point of index . Then there exists a coordinate chart .u1 ; : : : ; un / about the origin such that f .u1 ; : : : ; un / D f .0/  u21      u2 C u2C1 C    C u2n : It goes without saying that Morse lemma also holds for smooth functions defined on arbitrary manifolds. A consequence of Morse lemma, as stated above, is that there exists a neighborhood about the origin in Rn so that it is the only critical point in such neighborhood. Corollary 1 Non-degenerate critical points of a smooth function are isolated. Note that a Morse function defined on a compact manifold has finitely many critical points. The main reason behind the study of Morse functions is to understand the topology of the manifold. Thus for a smooth function f W M ! R and a 2 R define the level set Ma WD fx 2 Mj f .x/ ag  M:

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Notice that when a0 is the absolute minimum of f , then Ma is empty for every a < a0 . And in the case when a1 is the absolute maximum of f , then Ma D M for every a1 a. Now we explain what we mean by understanding the topology of the manifold; one aspect is that the manifold can be constructed from information from a fixed Morse function on it. Consider a compact manifold M, a smooth Morse function f W M ! R and for simplicity assume that p0 ; : : : ; pk are all the critical points, with i D ind. f I pi / and i < iC1 for i 2 f0; : : : ; k  1g. Thus f achieves its minimum at p0 and ind. f I p0 / D 0; and it achieves its maximum at pk and ind. f I pk / D n. In order to build the manifold M from the critical points of f , one starts with the point Mf . p0 / D f p0 g. Then from Theorem 2 below, it follows that Ma has the same homotopy type has Mf . p0 / for a 2 . f . p0 /; f . p1 //. By an -cell we mean a space homeomorphic to the closed ball of dimension . Hence, Ma is homeomorphic to the n-cell for a 2 . f . p0 /; f . p1 //. The next step is to analyze the next non-degenerate critical point p1 2 M. In this case for a 2 . f . p1 /; f . p2 //, it follows that Ma has the same homotopy type as Mf . p0 / with a 1 -cell attached. That is Ma ' Mf . p0 / [g e1 , where g W @.e1 / ! Mf . p0 / is a gluing function. This process continues at every critical point. That is for a 2 . f . pi /; f . piC1 // the space Ma as the same homotopy type has to Mf . pi1 / with an attached i -cell. The last step asserts that Ma ' Mf . pk2 / [ ek1 for a 2 . f . pk1 /; f . pk //; that is Ma is homeomorphic to M minus an open ball. Therefore M is homeomorphic to Ma with a n-ball attached. Note that the change of topology between the level sets occurs precisely at the critical points of f : Below, we carry out the same process described above for RP1 : Therefore when f W M ! R is a Morse function, is possible to describe the topology of the level sets Ma as a increases; in particular the topology of M. Furthermore, there is an alternative approach to understand the topology of M using a Morse function. This is called Morse homology and it will be describe in Sect. 3. Theorem 2 Let f W M ! R be a Morse function. • If f has no critical value in Œa; b, then Ma is diffeomorphic to Mb . • If f has only one critical value in Œa; b of index , then Mb has the same homotopy type as that of Ma [g e , for some gluing function g. As above, Ma [ e means that e is attached to Ma by some gluing function g W @.e / ! Ma . Note that @.e / is diffeomorphic to S1 . In the next example, we show how Theorem 2 is used to obtain the whole manifold M, by attaching one -cell at a time. Example 1 Consider the real projective space RPn , the set of lines through the origin in RnC1 . A point in RPn is represented in homogeneous coordinates as Œx0 W : : : W xn . Let a0 ; : : : ; an be distinct real numbers, define f W RPn ! R by f .Œx0 W    W xn / D

a0 x20 C    C an x2n : x20 C    C x2n

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So defined f is smooth and since the aj 0 s are distinct it has n C 1 non-degenerate critical points, that are p0 WD Œ1 W 0 W    W 0; p1 WD Œ0 W 1 W    W 0; : : : ; pn WD Œ0 W    0 W 1. Thus f is a Morse function; moreover the critical point pj has index j. The reader is encouraged to verify the statements made above. And also to get the same conclusions for the case of the complex projective space CPn with the function f .Œz0 W    W zn / D

a0 jz0 j2 C    C an jzn j2 : jz0 j2 C    C jzn j2

Now we look at the particular case of RP1 ; recall that RP1 is diffeomorphic to the circle. In this particular case take a0 D 0 and a1 D 1, so f takes the form f .Œx0 W x1 / D

x20

x21 : C x21

In this case f has only one critical point of index 0, namely at Œ1 W 0. It also has only one critical point of index 1 at Œ0 W 1. These points correspond to the maximum and minimum of f . In terms of Theorem 2 the circle is obtained as follows. We start with the 0-cell that is just a point, that is M0 D fŒ1 W 0g. Since f has no critical values in the interval Œ0; 1=2 other than 0, then from Theorem 2 if follows that M0 has the same homotopy type has M1=2 . Notice that M1=2 is a semicircle, the south hemisphere. Next comes the other critical point Œ0 W 1. It has index 1; thus a 1-cell is attached to M1=2 . That is, the two points of @.e1 / get glued to M1=2 to obtain the circle. See Fig. 1. An important aspect to consider is the existence of Morse functions on a given manifold. It turns out that there are plenty of Morse functions. More precisely, the set of Morse functions on a closed manifold is C2 -dense in the space of smooth functions. The reason that the C2 -topology is needed is because the concept of nondegenerate critical points involves derivatives up to second-order. In theory, is not

RP1

1-cell

0-cell (a)

M1/2 (b)

(c)

Fig. 1 Morse decomposition of RP1 with respect to f . In (a) the 0-cell that corresponds to p0 of index 0. In (b) the submanifold M1=2 , diffeomorphic to a point, is attached a 1-cell that corresponds to the point p1 of index 1. Finally, (c) the result after attaching the 1-cell

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to difficult to understand the topology of M via a Morse function as above. Next we take this idea a step further to recover the homology of M. Fix a Riemannian metric g on M and let h; i be the induced inner product on its tangent bundle. The gradient vector field, grad. f /, of the function f W M ! R is defined by the equation hgrad. f /; Xi D X. f / for every vector field X on M. Notice that if p is a critical point of f , then grad. f /p D 0. And conversely, if grad. f /p D 0 then p is a critical point of f : Therefore Crit. f / equals the zero set of grad. f /. In order to simplify the exposition, from now on we assume that M is compact. Denote by  W R  M ! M the flow of the negative gradient vector field of f . Thus for x 2 M ˇ @.t; x/ ˇˇ D grad. f /x : @t ˇtD0 The reason to consider the negative gradient vector field is only a matter of convention. Note that grad. f / D grad.f / and if p is a non-degenerate critical point of f , then ind. f I p/ D n  ind.f I p/ where n is the dimension of M. Also notice that grad. f /. f / < 0 outside the set of critical points of f , hence grad. f / points in the direction in which f is decreasing. The way to think about the index of a non-degenerate critical point is the number of linearly independent directions in which the grad. f / decreases. Let p be a point where grad. f / vanishes, then consider all points of M that under the flow  converge to p as t goes to infinity; ˇ   ˇ W s . f ; p/ WD x 2 M ˇˇ lim .t; x/ D p : t!C1

Similarly, ˇ n o ˇ W u . f ; p/ WD x 2 M ˇ lim .t; x/ D p ; t!1

the set of all points in M that have p has a source. Since grad. f / vanishes at p, then the critical point p is a fixed under the flow, hence p 2 W s . f ; p/ and p 2 W u . f ; p/. The submanifolds W u . f ; p/ and W s . f ; p/ are called the unstable manifold and stable submanifold of f at p, respectively. Theorem 3 If p is a non-degenerate critical point of f , then W u . f I p/ is a smooth submanifold of M of dimension ind. f I p/. Instead, if we consider the function f the set critical non-degenerate points of f and f agree. Moreover W u . f I p/ D W s .f I p/ and W s . f I p/ D W u .f I p/. Hence W s . f I p/ is also a smooth submanifold of M of dimension n  ind. f I p/.

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N

Fig. 2 The flow lines of the gradient vector field of f .x; y; z/ D z on S2 with respect to the standard Riemannian structure

S

Example 2 Let M D S2 be the unit sphere in R3 centered at the origin and f W S2 ! R defined as f .x; y; z/ D z. Then the poles N D .0; 0; 1/ and S D .0; 0; 1/ are the critical points of f . Furthermore they are non-degenerate, N has index 2 and S has index 0. Consider the Riemannian structure on S2 induced from the standard Riemannian structure on R3 . Then grad. f / is the vector field that points downwards, as shown in Fig. 2, and W u . f ; N/ D S2 n fSg; W s. f ; N/ D fNg; W u . f ; S/ D fSg and W s . f ; S/ D S2 n fNg: Example 3 Consider the function f W R2 ! R given by f .x; y/ D cos.2x/ C cos.2y/: So defined f induces a smooth function on the flat two-dimensional torus T2 D R2 =Z2 , which we still denote by f . There are 4 non-degenerate critical points on the torus, p1 D Œ0; 0; p2 D Œ0; 1=2; p3 D Œ1=2; 0 and p4 D Œ1=2; 1=2; of index 2; 1; 1 and 0 respectively. Consider the Riemannian structure on T2 induced from the canonical Riemannian structure on R2 . Then the flow of grad. f / can be seen in Fig. 3. Notice that there are only two lines that connect p2 to p4 . And a 1-dimensional family of flow lines that connect p1 to p4 , whose points determine four open connected components of the torus. Also Fig. 3 gives a description of the stable and unstable submanifolds. Observe that every interior point of Œ0; 1  Œ0; 1, lies in a flow line that ends at p4 . That is, W s . f I p4 / D T2 n f@.Œ0; 1  Œ0; 1/g: Similarly we have that W u . f I p1 / equals T2 n f p2 ; p3 ; p4 g [ f.x; 1=2/jx 2 Œ0; 1 n f1=2gg [ f.1=2; y/j y 2 Œ0; 1 n f1=2gg;

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1.0

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Fig. 3 The flow lines of the gradient vector field of cos.2x/ C cos.2y/ on T2 with respect to the flat Riemannian structure

and W u . f I p2 / D f.x; 1=2/jx 2 Œ0; 1 n f1=2gg; W u . f I p3 / D f.1=2; y/j y 2 Œ0; 1 n f1=2gg: Let p and q be non-degenerate critical points of a smooth function f W M ! R. Then the set W u . f I p/ \ W s . f I q/ consists of points of M that belong to a flow line u W R ! M of grad. f /u.t/ that connects p to q; that is du .t/ D grad. f /u.t/ ; dt

lim u.t/ D p and

t!1

lim u.t/ D q:

t!C1

(1)

We know from Theorem 3 that W u . f I p/ and W s . f I q/ are submanifolds of M, but their intersection might not be a smooth manifold. Hence a smooth function f W M ! R is said to satisfy the Smale condition if for any pair of critical points p and q, W u . f I p/ and W s . f I q/ intersect transversally. In particular W u . f I p/ \ W s . f I q/ is a submanifold of M. The function that appears in Example 2 satisfies the Smale condition. In this example the intersection of any pair of stable and unstable submanifolds is either empty, a point, the sphere minus a point or the sphere minus two points. Also the Moorse function in Example 3 satisfies the Smale condition. In particular, notice that W u . f I p2 / \ W s . f I p4 / consists of two disjoint open intervals.

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The type of functions that are of interest in this note are the Morse-Smale functions. For an arbitrary compact manifold and Riemannian metric, there always exists a Morse-Smale function. Furthermore, in some sense there are plenty of such functions. Then if .M; g/ is a Riemannian manifold and f W M ! R a MorseSmale function we write M . f I p; q/ for the set of points of M that belong to a flow trajectory of grad. f / that goes from p to q as in Eq. (1). Notice that in this case M . f I p; q/ is a smooth submanifold of M of dimension ind. f I p/  ind. f I q/. Note that the submanifold M . f I p; q/ admits a natural action of R defined as .s:u/.t/ WD u.t C s/ for s 2 R. The action is in fact free and the orbit space of this action is denoted by MO. f I p; q/. Hence MO. f I p; q/ is identified as the space of trajectories that joint p to q: So defined, the space of points that belong to a flow line of grad. f / that connect p to q, M . f I p; q/, is not necessarily compact. For example, in the case of the twosphere in Example 2 we have that M . f I N; S/ is S2 nfN; Sg. In this example if we add the critical points we obtain a compact space, namely the whole manifold S2 . Note that in this example MO. f I N; S/ is diffeomorphic to S1 : But it is not always the case that by adding the critical points p and q to M . f I p; q/ that it becomes a compact space. For instance, in the torus case of Example 3 the space M . f I p1 ; p4 /[f p1 ; p4 g is not compact. In general, the way to compactify the space of trajectories MO. f I p; q/ is by adding broken trajectories. A broken trajectory from p to q is a collection of flow lines fu1 ; : : : ; ur g of grad. f / such that uj connects the critical points xj to xjC1 for j 2 f1; : : : ; rg where p D x1 and q D xrC1 : Consider the bigger set of flow lines that connect p to q, namely the ordinary flow trajectories plus the broken trajectories, M . f I p; q/ WD MO. f I p; q/ [ fbroken trajectories from p to qg: Recall that the index of critical points of f decreases along flow lines. Hence the number of flow lines that compose a broken trajectory from p to q is less than ind. f I p/  ind. f I q/. Hence if ind. f I p/  ind. f I q/ D 1 there are no broken trajectories connecting p to q and M . f I p; q/ D MO. f I p; q/. That is, MO. f I p; q/ is compact in this case and it consists of finitely many points. The proof of the next result can consulted in [3, Chp. 3] and [38]. Proposition 1 Let .M; g/ be a closed Riemannian manifold, f W M ! R a Morse-Smale function and p; q critical points of f . Then the natural action of R on M . f I p; q/ is free. Moreover M . f I p; q/ is smooth and compact of dimension ind. f I p/  ind. f I q/  1: The important case that would be relevant later on is the case when ind. f I p/  ind. f I q/ D 2. Usually in this case the space MO. f I p; q/ is not compact, so we must add broken trajectories. Hence M . f I p; q/ is a finite collection of closed intervals and circles. In Example 3 consider u1 .t/ WD Œ0; t and u2 .t/ WD Œt; 1=2, for t 2 .0; 1=2/; two flow lines of grad. f /. The flow line u1 connects p1 to p2 , and u2 connects p2 to p4 . Hence fu1 ; u2 g is a broken trajectory that connects p1 to p4 . Note that MO. f I p1 ; p4 / is diffeomorphic to four copies of .0; 1/; and there are eight broken

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trajectories that must be added to obtain M . f I p; q/. For instance, fu1 ; u2 g is one of them. Henceforth M . f I p; q/ is diffeomorphic to four copies of Œ0; 1:

3 Morse Homology We are going to define the Morse-Witten complex of .MI f ; g/; the Riemannian manifold and the Morse-Smale function. For simplicity we will use Z2 coefficients, keep in mind that it is possible to use integer coefficients. In order to define Morse homology with integer coefficients, one must prove that is possible to have a coherent system of orientation on the compact moduli spaces. In the case of Z2 coefficients the orientation of the moduli spaces is also irrelevant, only the boundary components of the moduli spaces of dimension two are important. See for example [38], where they use integer coefficients. Denote by Crit . f / the set of critical points of index  and by C . f / the Z2 -vector space generated by the elements of Crit . f /. For  … f0; 1; : : : ng, define C . f / to be the trivial vector space. If p and q are critical points of f such that ind. f I p/ D ind. f I q/ C 1, then by Proposition 1 MO. f I p; q/ is a finite set of points. Denote by #Z2 MO. f I p; q/ the number of points of MO. f I p; q/ module 2. The boundary operator, @ W C . f / ! C1 . f /, is the linear map defined on generators p 2 C . f / as @ . p/ WD

X

#Z2 MO. f I p; q/ q:

q2Crit1 . f /

Notice that if MO. f I p; q/ is zero-dimensional, then M . f I p; q/ consists of finitely many lines that connect p to q. This geometric description of M . f I p; q/ is useful when computing the boundary operator @; this will be seen for instance below in Example 4. The reason why @ is called the boundary operator is given by the next result. In order to compute @1 ı @ one must consider the moduli spaces M . f I p; r/ where ind. f I p/  ind. f I r/ D 2: For p 2 Crit2 . f /, @1 @ . p/ WD

X

X

#Z2 .#MO. f I p; q/  #MO. f I q; r// r

r2Crit2 . f / q2Crit1 . f /

where #MO. f I p; q/ stands for the number of points of MO. f I p; q/. Notice that M . f I p; r/ is a one-dimensional compact manifold; hence it is the union of a finite collection of closed intervals and circles. Hence its boundary consists of a even number of points which are [q2Crit1 . f / #MO. f I p; q/  #MO. f I q; r/ and correspond to the broken trajectories from p to r that go through q.

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Theorem 4 The operator satisfies @1 ı @ D 0. The complex .˚ C . f /; @/ is called the Morse-Witten complex of .MI f ; g/. Its homology Ker @ Im @C1

MH .MI f ; g/ WD

is called the Morse homology of .MI f ; g/ with Z2 -coefficients. Note that the relevant moduli spaces M . f I p; q/ for the definition of Morse homology are those whose dimension is at most two. Remark 1 As mentioned above is possible to define Morse homology with Z coefficients. For, W u .f ; p/ is an orientable submanifold of M for any critical point p. Hence, one fixes an orientation on W u .f ; p/ for every critical point. This yields an orientation on W u .f ; p/ \ W s .f ; q/ and hence on M . f I p; q/ and MO. f I p; q/. Thus if ind. f I p/ D ind. f I q/ C 1, then M . f I p; q/ D MO. f I p; q/ is a finite set of points each of which has a sign. Set n. f I p; q/ to be the sum of these signs; then the boundary operator over Z-coefficients is defined as @ . p/ WD

X

n. f I p; q/ q:

q2Crit1 . f /

However the statement @1 ı @ D 0 is delicate in this case. One must take into consideration that the orientation of the moduli spaces of dimension two induced the right orientation on its boundary; the one-dimensional moduli spaces. For example see [38]. Recall from Example 2, that on S2 we defined a Morse function with the poles N and S as critical points of index 2 and 0 respectively. The Riemannian structure on the sphere was induced from the canonical Riemannian structure on R3 . Further, we calculated the stable and unstable submanifolds of N and S. From this calculation, it follows that f is a Morse-Smale function. Therefore C0 . f / D Z2 hSi, C2 . f / D Z2 hNi and the boundary operator is the zero map. Hence ( 2

MH .S I f ; g/ D

Z2

if  D 0; 2

0

if  ¤ 0; 2:

Example 4 In this example we consider the function on the two-dimensional torus T2 defined in Example 3. Notice that the function is Morse-Smale. Hence C0 . f / D Z2 hp4 i; C1 . f / D Z2 hp2 ; p3 i; and C2 . f / D Z2 hp1 i. Counting trajectory flow lines, we get @p1 D 2p2 C 2p3 D 0, @p2 D 0, @p3 D 0, and @p4 D 0. Therefore,

2

MH .T I f ; g/ D

8 ˆ ˆ 0, and still S1 ./ is a Lagrangian submanifold. Taking n copies of this example, it follows that the n-dimensional -torus S1 ./      S1 ./ is a Lagrangian submanifold of .R2 ; !0 /      .R2 ; !0 / D .R2n ; !0 /. Thus for a given symplectic manifold .M; !/ and  small enough, by Darboux’s Theorem we have that the n-dimensional -torus is a Lagrangian submanifold of .M; !/.

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In the particular case of .R2 ; !0 /, we have that the two-dimensional torus S1  S1 is a Lagrangian submanifold. Furthermore, the torus is the only oriented surface that can be embedded as a Lagrangian submanifold in .R2 ; !0 /.

5 Symplectic and Hamiltonian Diffeomorphisms There are two types of symmetries associated to a symplectic manifold. Recall that we assumed that symplectic manifold is closed. In the non-compact case, one has to consider diffeomorphisms with compact support. A diffeomorphism  W .M; !/ ! .M; !/ is said to be a symplectic diffeomorphism if   .!/ D !. The set of symplectic diffeomorphisms of .M; !/ forms a group under composition and is denoted by Symp.M; !/. In fact the group of symplectic diffeomorphisms is an infinite dimensional space, its Lie algebra consists of vector fields X such that the 1-form !.X; / is closed. Among the group of symplectic diffeomorphisms we have the second type of symmetries, called Hamiltonian diffeomorphisms. A symplectic diffeomorphism  is called Hamiltonian diffeomorphism if there exists a path of symplectic diffeomorphisms ft g0t1 and a smooth function H W Œ0; 1  M ! R, such that 0 D 1M , 1 D , and if Xt is the time-dependent vector field induced by the equation d  t D X t ı t ; dt then !.Xt ; / D dHt . The set of Hamiltonian diffeomorphisms is a group under composition and is denoted by Ham.M; !/. As in the symplectic case, Ham.M; !/ is an infinite dimensional space and its Lie algebra consists of vector fields X such that the 1-form !.X; / is exact. A Hamiltonian diffeomorphism  is called autonomous, if there exists a path ft g, as in the definition of Hamiltonian diffeomorphism, such that Xt is independent of t. In other words autonomous Hamiltonian diffeomorphisms are the image of the exponential map of Hamiltonian vector fields. Alternatively, the group Ham.M; !/ can be described as the group generated by autonomous Hamiltonian diffeomorphisms [5]. Not only is Ham.M; !/ a subset Symp.M; !/, as the definition suggest; the group of Hamiltonian diffeomorphisms is a normal subgroup of the group of symplectic diffeomorphisms. As we explain below in most cases is a proper subgroup. Among other properties of Ham.M; !/ is that it is connected with respect to the C1 topology; Symp.M; !/ does not have to be connected. Further if Symp0 .M; !/ is the connected component of the group of symplectic diffeomorphisms that contains the identity map and H 1 .M; R/ D 0, then Ham.M; !/ D Symp0 .M; !/:

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For example Ham.CPn ; !FS / D Symp0 .CPn ; !FS / for n  1: However if H 1 .M; R/ ¤ 0, Ham.M; !/ is properly contained in Symp0 .M; !/: Below we will see an example where Ham.M; !/ is a proper subgroup of Symp0 .M; !/. An important remark about a Hamiltonian diffeomorphism is that its fixed point set is non-empty. Recall that we are assuming that .M; !/ is closed; in the noncompact case the assertion is false. For instance, on .R2n ; !0 / a translation map is a Hamiltonian diffeomorphism that is fixed-point free. As for the case of compact symplectic manifolds it is straightforward to justify that the fixed point set is non-empty in the case of autonomous Hamiltonians. For if  is an autonomous Hamiltonian, then there exists ft g such that d t D X ı t ; dt

and

!.X; / D dH:

(2)

Since the manifold is assumed to be compact, then the set of critical points of H W M ! R is non-empty. But ! is non-degenerate, hence by Eq. (2) the set of critical points of H coincides with the zero set of X. If X vanishes at p it follows by Eq. (2) that p is a fixed point of the flow ft g; in particular p is a fixed point of 1 D : The fact that Hamiltonian diffeomorphisms on compact manifold always have fixed points is not shared by symplectic diffeomorphisms that are not Hamiltonian. Example 8 Consider the flat two-dimensional torus .T2 D R2 =Z2 ; ! D dx ^ dy/. For a fix ˛ 2 .0; 1/, the translation map  ˛ Œx; y WD Œx C ˛; y preserves the area and hence is symplectic diffeomorphism. Notice that since ˛ ¤ 0; 1, the map  ˛ has no fixed points. Hence  ˛ is not a Hamiltonian diffeomorphism for any ˛ 2 .0; 1/. Moreover  ˛ lies in the identity component of the group of symplectic diffeomorphism. Therefore Ham.T2 ; !/ is a proper subgroup of Symp0 .T2 ; !/. As mentioned above, symplectic diffeomorphisms give rise to Lagrangian submanifolds. In the next example we show how this is done and highlight the importance of this example in the study of fixed points of Hamiltonian diffeomorphisms. Example 9 Let  W .M; !/ ! .M; !/ be a symplectic diffeomorphism, thus   .!/ D !. Then the graph of  is an embedded submanifold of dimension 2n in M  M, that is j W M ! M  M is given by j.x/ D .x; .x// and its image is the graph of , graph./ WD f.x; .x//j x 2 Mg: Furthermore the graph of  is a Lagrangian submanifold of .M  M; 1 .!/  2 .!//; for j .1 .!/  2 .!// D !    .!/ D 0:

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The above computation shows the relevance of the minus sign that appears in the symplectic form of .M  M; 1 .!/  2 .!// in Example 7. In the case when  W M ! M is a Hamiltonian diffeomorphism, when know that the fixed point set is non empty; further this set is in one-to-one correspondence with the intersection points of graph./ with the diagonal . As pointed out above, there are symplectic diffeomorphisms , such that graph./ and  have no points in common. For instance  ˛ of Example 8. Notice that for any Hamiltonian diffeomorphism  W .M; !/ ! .M; !/ and Lagrangian submanifold L  .M; !/, .L/ is again a Lagrangian submanifold. An important fact that will be useful in the context of Lagrangian Floer homology is the following. A Lagrangian submanifold L  .M; !/ is called non-displaceable if for every Hamiltonian diffeomorphisms  W .M; !/ ! .M; !/, the Lagrangian submanifolds L and .L/ have points in common. Otherwise, L is called displaceable. Hence we are considering the intersection of two particular Lagrangian submanifold, L and .L/. This is part of the phenomenon that Lagrangian Floer homology attempts to answer, intersection or non-intersection of Lagrangian submanifolds. Consider the two-dimensional sphere .S2 ; !/ with any area form, let us try to understand the intersection of a particular pair of Lagrangian submanifolds. Consider the Lagrangian submanifold L to be any circle that lies entirely in a hemisphere. Thus there always exists a rotation  W .S2 ; !/ ! .S2 ; !/, which is in fact a Hamiltonian diffeomorphism of the 2-sphere, such that L and .L/ have no points in common. That is L is displaceable. Now consider the case when L is such that both components U and V of S2 nL have equal area. Recall that any Hamiltonian diffeomorphism preserves area; hence for any Hamiltonian diffeomorphism , .U/ D U or .U/ \ V is not empty. Then for any Hamiltonian diffeomorphism , we have that L \ .L/ is non-empty if the Lagrangian submanifold is such that the two components of S2 n L have equal area. Hence the non-displaceable Lagrangian are precisely the embedded circles that split the sphere in two pieces of equal area. In fact one of the current problems in symplectic geometry is to determine which Lagrangian submanifolds are non-displaceable or displaceable. The higher dimensional analog of the above example, for the case when L splits the sphere in two parts of equal area, is the Lagrangian submanifold RPn in .CPn ; !FS /. One of the triumphs of Lagrangian Floer homology is the proof that RPn is non-displaceable. This result was proved by Y.-G. Oh in [30]; where he defined Lagrangian Floer homology for monotone Lagrangian submanifolds. Now we go back to the case of Lagrangian submanifolds induced by symplectic diffeomorphisms as in Example 9. Hence let  W .M; !/ ! .M; !/ be a symplectic diffeomorphism and graph./  .M  M; 1 .!/  2 .!// which is a Lagrangian submanifold. Note that in this example the Lagrangian submanifold graph./ it is actually the image of the Lagrangian  under the symplectic diffeomorphisms 1  of .M  M; 1 .!/  2 .!//. That is graph./ D .1  /./:

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In fact when  W .M; !/ ! .M; !/ is a Hamiltonian diffeomorphism we know that .1  /./ \  is non empty. The intersection points are in one-to-one correspondence with the fixed points of . In this case .1  / is also a Hamiltonian diffeomorphism of .M  M; 1 .!/  2 .!//. Lagrangian Floer homology gives a stronger result, it shows that  is non displaceable, that is ˚./ \  ¤ ; for any Hamiltonian diffeomorphisms ˚ of .M  M; 1 .!/  2 .!//, not necessarily those induced from Hamiltonians of .M; !/: Moreover it gives a lower bound on the cardinality of ˚./ \  under some non degeneracy conditions of ˚. That is it solves the Arnol’d Conjecture. The problem of estimating the number of fixed points of a Hamiltonian diffeomorphism, is a particular case of the wider problem of estimating the number of intersection points of two Lagrangian submanifolds. In broad terms, that is the objective of Lagrangian Floer homology. As seen in the definition, Hamiltonian diffeomorphisms have a strong connection with smooth functions. A manifestation of this connection was the nice link between the fact that a smooth function on a closed manifold admits critical points; and the fact the on a closed symplectic manifold the fixed point set of a Hamiltonian diffeomorphism is non-empty. In 1965, Arnol’d [1] conjectured an analog result of Theorem 6, but for the case of Hamiltonian diffeomorphisms on closed symplectic manifolds instead of Morse functions on arbitrary manifolds. See also [2, Appendix 9]. His motivation was the Poincaré-Birkhoff annulus theorem: An area preserving diffeomorphism of the annulus such that the boundary circles are turned in opposite directions must have at least two fixed points. A fixed point p 2 M of a Hamiltonian diffeomorphism  is said to be non-degenerate if 1 is not an eigenvalue of the linear map ;p W Tp M ! Tp M. Note that non-degenerate fixed points are isolated, and in the case of a closed symplectic manifold there are a finite number of them. Conjecture 1 (Arnol’d) Let .M; !/ be a closed symplectic manifold and  a Hamiltonian diffeomorphism such that all of its fixed points are non-degenerate. Then #f p 2 Mj. p/ D pg 

2n X

Rank Hj .M; R/:

jD0

For two-dimensional symplectic manifolds, the conjecture was proved by Eliashberg [9]; in [8] C.C. Conley and E. Zehnder proved the conjecture for the symplectic torus manifold with the standard symplectic form; and for the complex projective space with the Fubini-Study symplectic form the conjecture was proved by B. Fortune and A. Weinstein in [12]. The real break through in solving the Arnol’d Conjecture was made by A. Floer in [10]. In a series of papers A. Floer developed a homological theory based on holomorphic techniques, which were introduced by Gromov [18], and the new approach to Morse theory developed by Witten [42]. Under some assumption on the symplectic manifold A. Floer developed Hamiltonian Floer homology, using

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holomorphic cylinders, in order to find a lower bound to the number of fixed points of a Hamiltonian diffeomorphism. Then he generalized this approach to develop Lagrangian Floer homology, now using holomorphic stripes, in order to determine the minimum number of intersection points of a particular pair of Lagrangian submanifolds. The Arnol’d Conjecture has been proved for arbitrary symplectic manifolds. Some reference for the proof of the conjecture, sometimes under some restrictions and others in full generality are: Fukaya and Ono [14], Hofer and Salamon [19], Liu and Tian [20], Ono [34], Oh [30], Ruan [37]. Some of the techniques introduced in [14] on the proof of the Arnol’d Conjecture have been re-evaluated. For instance the Kuranishi structure on the moduli space of holomorphic strips MJ . p; q; L0 ; L1 /, that will be defined in the next section, as well as its virtual fundamental class. Recently Pardon [35] has given an alternative approach to this problem using techniques from homological algebra. There is also a series of articles by McDuff and Wehrheim, [22, 25, 26] and [27], where they treat this problem using tools from analysis.

6 Lagrangian Floer Homology The construction of Lagrangian Floer homology emulates to a certain extent the construction of Morse homology described above. The manifold in consideration to define it is a certain space of trajectories which is infinite dimensional; and the function defined on it is a certain action functional. The critical points turn out to be constant trajectories, that give rise to the differential complex used to define Lagrangian Floer homology. Is important to point out while the construction of Lagrangian Floer homology follows the spirit of the construction of Morse homology, new complications emerge that were not present before. Just to have an idea of this, it suffices to say that Lagrangian Floer homology is not always defined do to the fact that the square of the differential map is not always equal to zero. Let L0 and L1 be two compact Lagrangian submanifolds in .M; !/ that intersect transversally. Consider the space of smooth trajectories from L0 to L1 , P.L0 ; L1 / WD f W Œ0; 1 ! Mj is smooth, .0/ 2 L0 and .1/ 2 L1 g: endowed with the C1 -topology. Notice that the constant paths in P.L0 ; L1 / are the ones that correspond to the intersection points L0 \ L1 . From now on we write P for P.L0 ; L1 /. The space P is not necessarily connected, thus we fix O in P and consider the component that contains , O which we denoted by P./. O Relative to O consider the f O / of P.O /. Elements of P. f O / are denoted by Œ; w universal covering space P. where w is a smooth path in P.O / from O to . That is w W Œ0; 1  Œ0; 1 ! M is a smooth map such that w.s; / 2 P.O / for all s 2 Œ0; 1, w.0; / D O and w.1; / D .

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f O / is not the right space to define the action functional. The The space P. right space is the Novikov covering of P.O /, which is defined by an equivalence f O /. For the sake of making the exposition less technical we are not relation on P. going to define the Novikov covering of P./, O instead we are going to impose strong assumptions on the symplectic manifold .M; !/ and the pair of Lagrangian submanifolds L0 and L1 in order to define the action functional on f P./ O and carry out a similar procedure as in Morse theory. Thus from now on we assume that the symplectic manifold .M; !/ is such that Z S2

f ! D 0

for every Œ f  2 2 .M/, where f is a smooth representative. A symplectic manifold that satisfies this condition is said to be symplectically aspherical. The symplectic tori .T2n ; !/, for n  1, are examples symplectically aspherical since 2 .T2n / is trivial. However there are plenty of symplectically aspherical manifolds with non trivial 2 , even in dimension four [17]. As we will see in the next paragraph, if .M; !/ is symplectically aspherical then the action functional is well defined on the f 0 ; L1 I O /. This hypothesis on .M; !/ is also useful since it rules out the covering P.L appearance of bubbles; that is holomorphic spheres attached to holomorphic strips. See for example [10]. As mentioned above, Lagrangian Floer homology is defined on more generally symplectic manifolds, even in the presence of bubbles. Further we also assume that L0 and L1 are simply connected. Then the action functional A W f P.L0 ; L1 I O / ! R is defined as Z A .Œ; w/ D

Œ0;1Œ0;1

w !;

that is, the symplectic area of w.Œ0; 1  Œ0; 1/  .M; !/. To see that A is well defined, let .; w/ and .; w0 / represent the same class. Then we have a map defined on the cylinder, w#w0 W S1  Œ0; 1 ! M where w#w0 .s; 0/ is a loop in L0 and w#w0 .s; 1/ is a loop in L1 . Since the Lagrangian submanifolds are assumed to be simply connected, there exists a 2-disk contained in L0 whose boundary is the loop w#w0 .s; 0/. This observation also applies to L1 . That is, we added the caps to the cylinder to obtain topological a 2-sphere. Since .M; !/ is symplectically aspherical, the symplectic area of the 2-sphere is zero. Notice also that the symplectic area of each cap is zero, since the symplectic form is identically zero on Lagrangian submanifolds. Thus the symplectic area of the cylinder w#w0 W S1  Œ0; 1 ! M is equal to zero. Then we have that Z 0D



S1 Œ0;1

Z

w#w ! D

Z

D Œ0;1Œ0;1

w ! C



Œ0;1Œ0;1

Z

Œ0;1Œ0;1

w ! C w !

Z Œ0;1Œ0;1

w !

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Hence it follows that the action functional is well defined on the covering f 0 ; L1 I O /. In local coordinates the action functional takes the form P.L Z A .Œ; w/ D

1

Z

0

1 0

 !

 @w @w ; ds dt: @s @t

Lagrangian Floer homology is defined emulating the way Morse homology is defined. The finite dimensional manifold in Morse homology is replaced by the f 0 ; L1 I O /. And the function into consideration is infinite dimensional space P.L the action functional. However the analytical difficulties in this setting are more complex than in the Morse scenario. To follow the path of Morse homology we need to define a Riemannian structure f 0 ; L1 I O /. Denote by J the space of almost complex structures on .M; !/. on P.L Recall that an almost complex structure J 2 J on .M; !/ is said to be !-compatible if for every nonzero vector v, !.v; Jv/ > 0

!. J; J/ D !.; /:

and

For an !-compatible almost complex structure J, we have that gJ .; / WD !.; J/ defines a Riemannian metric on .M; !/. Is important to note that compatible almost complex structures exist in abundance on any symplectic manifold. Let J D fJt g0t1 be a smooth family of !-compatible almost complex structures on .M; !/; hence we have fgt g0t1 a smooth family of Riemannian metrics. Then on f P.O / we define a Riemannian metric associated to fgt g0t1 as Z hh 1 ; 2 ii WD

1

0

gt . 1 .t/; 2 .t//dt

f O /. As in the Morse theory case, we compute the gradient of A W for 1 ; 2 in T P. f O / ! R with respect to hh; ii, P. Z dA.Œ;w/ . / D

1 0

Z

1

D 0

Z 0

 @ ; .t/ dt @t



 @ ; Jt .Jt .t// dt @t



 @ ; Jt .t/ dt @t

! !

1

D



gt

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 @ ; Jt D @t 

@ D Jt ; : @t That is, gradA .Œ; w/ D Jt

@ : @t

Since Jt is an automorphism of TM for each t, the gradient of A vanishes at Œ; w if and only if  W Œ0; 1 ! .M; !/ is a constant path. Thus the critical points of the action functional A are of the form Œ; w where  is a constant path, corresponding to an intersection point of L0 with L1 . If p is in L0 \ L1 , the constant path at p is denoted by `p . As in the Morse theory case, a flow line of gradA connecting p to q is a smooth f O / such that function u W R ! P. du D gradA ; ds

lim u.s/ D `p ;

s!1

and

lim u.s/ D `q :

s!C1

(3)

Unwrapping this, is better to write u W R  Œ0; 1 ! .M; !/ and the first equation of Eq. (3) as @u @u C Jt D 0: @s @t

(4)

Note that Eq. (4) is the Cauchy-Riemann equation, @J .u/ D 0, with respect to the !-compatible almost complex, fJt g, structure on .M; !/. A smooth map u W R  Œ0; 1 ! .M; !/ is said to be a J-holomorphic strip in .M; !/ if @J .u/ D 0. Then the space of connecting flow lines (actually J-holomorphic strips in .M; !; J/) that connect p with q is defined as as MOJ . p; q; L0 ; L1 / WD fu W R  Œ0; 1 ! .M; !/ j u is smooth, satisfies Eq. (3) and u.s; / 2 P.L0 ; L1 /g: In the case when .M; !/ is non compact but exact, and the Lagrangian submanifolds are still compact, one imposes an additional condition on the flow lines. That is to say, in addition to Eq. (3) the map u W R  Œ0; 1 ! .M; !/ is required to have finite energy, Z RŒ0;1

u ! < 1:

(5)

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In the case of a compact symplectic manifold, a holomorphic strip u with u.s; / 2 P.L0 ; L1 / has finite energy, Eq. (5), if and only if satisfies the limit conditions lim u.s/ D `p ;

s!1

and

lim u.s/ D `q

s!C1

for some p; q 2 L0 \ L1 . For the details see Robbin and Salamon [36]. Note that the strip R  Œ0; 1i  C is conformally equivalent with the closed unit disk D2  C minus two points on the boundary. Thus sometimes u is also referred as a holomorphic disk. For p; q 2 L0 \ L1 , let C1 .R  Œ0; 1; MI L0 ; L1 / be the set of smooth maps u W R  Œ0; 1 ! M with the limit behavior as in (3). Thus we have a bundle map [u C1 .u .TM// ! C1 .R  Œ0; 1; MI L0 ; L1 /; where the space C1 .u .TM// is the space of vector fields along u.R  Œ0; 1/, that is sections of u .TM/ ! R  Œ0; 1. Notice that the Cauchy-Riemann equation (4) defines a section, u 7! @J .u/, of this bundle. Moreover the moduli space MOJ . p; q; L0 ; L1 / is precisely the zero locus of this section. In order to show that the moduli space is a finite dimensional manifold, the section @J must intersect transversally the zero-section. Transversality is one of the problems in defining Lagrangian Floer homology, it is a delicate issue of the subject. The issue of transversality of the section @J is in fact relaxed, from the one stated above in the sense that the smooth condition on the map u is relaxed. The smooth condition is weakened to the Sobolev space Wpk .R  Œ0; 1; MI L0 ; L1 / for k > p=2 and p > 1. However, the new zero locus obtained in this setting coincides with the previous one due to elliptic regularity; that is if u 2 Wpk .R  Œ0; 1; MI L0 ; L1 / is such that @J .u/ D 0, then u is in fact smooth. The main result in this direction is that there exists a dense subset Jreg .L0 ; L1 / of C1 .Œ0; 1; J / of !-compatible almost complex structures such that for J D fJt g 2 Jreg .L0 ; L1 / and every u 2 MOJ . p; q; L0 ; L1 / the linearized operator D.@J /u W f 2 Wpk .u .TM//j .s; 0/ 2 L0 and .s; 1/ 2 L1 g ! Wpk1 .u .TM// is a surjective Fredholm operator. Furthermore the index of the operator D.@J /u is the Maslov index of the map u W RŒ0; 1 ! .M; !/. Below we give the definition of the Maslov index of u. It then follows that the kernel of D.@J /u is finite dimensional and is identified with the tangent space of MOJ . p; q; L0 ; L1 / at u. For the details of these assertions see [10]. As in the finite dimensional Morse theory case, the space of flow lines MOJ . p; q; L0 ; L1 / admits an action of R on the s coordinate. The quotient space by this action is denoted by MJ . p; q; L0 ; L1 /. The space MJ . p; q; L0 ; L1 / still needs to be taken further apart; namely the homotopy class of an element needs to be taken into consideration. Let ˇ 2 2 .M; L0 [ L1 /, and define MJ . p; q; L0 ; L1 I ˇ/ as the elements u 2 MJ . p; q; L0 ; L1 / such that Œu D ˇ.

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In order to address the dimension of MJ . p; q; L0 ; L1 /, for the moment consider only one Lagrangian submanifold L of .M; !/. Then to each smooth map u W .D2 ; @D2 / ! .M; L/, one gets a trivial fibration u .TM/ ! D2 that is symplectic. Furthermore, the fibration is trivial as symplectic bundles, u .TM/ ' R2n D2 . Then when the fibration is restricted to @D2 , it defines a loop of Lagrangian subspaces of .R2n ; !0 /. That is, if .R2n / represents the Grassmannian of Lagrangian subspaces then the trivialized fibration induces a map uL W S1 ! .R2n /. The Maslov index L .u/, of u W .D2 ; @D2 / ! .M; L/ is defined to be the integer .uL / .1/ 2 1 ..R2n // ' Z: The Maslov index of u is well defined, it does not depend on the symplectic trivialization; furthermore it only depends on the homotopy type of u relative to L. Hence the Maslov index induces a group morphism L W 2 .M; L/ ! Z. The above concept extends to the case when two Lagrangian submanifolds L0 and L1 are involved. For the definition of the Maslov index see [2, 23] or [36]. Theorem 9 Let L0 and L1 be compact Lagrangian submanifolds of .M; !/ that intersect transversally. Then there exists a dense subset Jreg .L0 ; L1 / in C1 .Œ0; 1; J / of !-compatible almost complex structures, such that for J D fJt g 2 Jreg .L0 ; L1 /, p and q in L0 \ L1 , and ˇ 2 2 .M; L0 [ L1 /, the space MOJ . p; q; L0 ; L1 I ˇ/ is a smooth manifold. Moreover its dimension is given by the Maslov index .ˇ/. The proof of this result appears in [10] and [30]. So far we imposed conditions on L0 ; L1 and .M; !/, in order to have a more transparent exposition of the subject. All the statements made so far hold for arbitrary closed symplectic manifolds and compact Lagrangian submanifolds, with the corresponding adaptations. However the next results that we are going to state does not hold in general. In fact it is well understood that Lagrangian Floer homology can not be defined on arbitrary symplectic manifolds for arbitrary Lagrangian submanifolds. See [15] and [30] for more information on this peculiarity. For instance one can impose the condition that the pair of Lagrangians submanifolds of .M; !/ must be monotone and that the Maslov index of the Lagrangians has to be greater than 2. A Lagrangian submanifold L  .M; !/ is called monotone if there exists  > 0, such that ! D L . One important fact that follows by considering monotone Lagrangian submanifold, is that the Maslov index of a non constant holomorphic disk with boundary in the Lagrangian is positive. Under these conditions, Oh [30] defined Lagrangian Floer homology. There are less restrictive conditions for which Lagrangian Floer homology is well defined; see [15] and [40]. The advantage of the monotone assumptions is that the complex is just the Z2 -vector space generated by the intersection points of the Lagrangian submanifolds; and the sum in the definition of the boundary operator (6) is a finite sum. Basically the same picture as in the Morse theory case. From now we are going to assume that the symplectic area of any 2-disk with boundary in the Lagrangian submanifold is zero. Thus from now on we assume that .M; !/ is a closed symplectic manifold and L0 ; L1 are closed Lagrangian submanifolds such that Œ!  2 .M; Lj / D 0 for j D 0; 1. Under these conditions we will define the Floer complex of the Lagrangian submanifolds L0 and L1 .

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Then under these hypothesis, when . p; q; ˇ/ D 1 the space MJ . p; q; L0 ; L1 I ˇ/ is compact and hence is a finite collection of points. The reason why the space is compact is because Œ!  2 .M; Lj / D 0 rules out the existence of holomorphic spheres and disks with boundary in the Lagrangian submanifolds. Since a convergent sequence of holomorphic strips, under Gromov’s topology, converges to a holomorphic strip with the possible union of holomorphic spheres and disks, the moduli space is compact. See for example [30]. In some sense, this is most straightforward way scenario to define Lagrangian homology; avoid holomorphic spheres and disks. Now the advantage of using the field Z2 , is that we don’t have to worry about orientations of the moduli spaces, that is assigning .C/ or ./ to each component of MJ . p; q; L0 ; L1 I ˇ/ Recall that J D fJt g is given as in Theorem 9. Denote by #Z2 MJ . p; q; L0 ; L1 I ˇ/ the number of points of MJ . p; q; L0 ; L1 I ˇ/ module 2. Before defining the boundary operator as before, we note that the solutions of (4) might determine an infinite number of homotopy classes of 2 .M; L0 [L1 /. Thus we introduce the Novikov field over Z2 to give meaning to the possible infinite number of homotopy classes of connecting orbits. The Novikov field is defined as  WD

8 1