Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. (MN-27) [Course Book ed.] 9781400856459

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EXISTENCE AND REGULARITY OF MINIMAL SURFACES ON RIEMANNIAN MANIFOLDS

by Jon T. Pitts

Princeton University Press and University of Tokyo Press

Princeton, New Jersey 1981

Copyright © 1981 by Princeton University Press All Rights Reserved

Published in Japan Exclusively by University of Tokyo Press in other parts of the world by Princeton University Press

Printed in the United States of America by Princeton University Press, Princeton, New Jersey

Library of Congress Cataloging in Publication Data will be found on the last printed page of this book

Research supported in part by a grant from the National Science Foundation

For my parents, Bishop and Mabel Pitts, and my wife, Karen, whose love and patience have been my inspiration

CONTENTS

INTRODUCTION

1_

CHAPTER 1.

N O N T E C H N I C A L SYSTEMIC D E S C R I P T I O N A N D ILLUSTRATIVE EXAMPLES

12

1.1 1.2 1.3 1.4

The v a r i a t i o n a l c a l c u l u s Almost minimizing varifolds Stable m a n i f o l d s T o p o l o g y of r e g u l a r c r i t i c a l surfaces

13 28 41 46

CHAPTER 2.

PRELIMINARIES

48

2.1

48

2.2 2.3 2.4 2.5-2.7

D e f i n i t i o n s , n o t a t i o n , and terminology Curvature formulas Variations Useful varifold operations C o n s e q u e n c e s of first v a r i a t i o n

CHAPTER 3.

ALMOST MINIMIZING VARIFOLDS

91

3.1-3.2 3.3

Almost minimizing varifolds Stability of a l m o s t m i n i m i z i n g varifolds E q u i v a l e n t f o r m u l a t i o n s of almost minimizing The c l a s s o f c o m p a r i s o n surfaces V a r i f o l d tangents I n t e g r a l i t y theorem f o r a l m o s t minimizing varifolds

94 96

3.4-3.9 3.10-3.11 3.12 3.13

68 73 75 84

g7 127 131 133

CHAPTER 4.

EXISTENCE OF ALMOST MINIMIZING VARIFOLDS

138

4.1-4.2 4.3

Homotopy relations Stationary varifolds and critical sequences Nontrivial homotopy classes Covering lemmas Existence theorems for almost minimizing varifolds

139 147

CHAPTER 5.

POINTWISE CURVATURE ESTIMATES FOR STABLE HYPERSURFACES

178

5.1-5.2 5.3-5.7

Preliminary notation and estimates Curvature integral estimates for

180 186

4.4-4.7 4.8-4.9 4.10-4.13

5.8-5.13 5.14-5.15

stable hypersurfaces Pointwise estimates for subsolutions of elliptic equations on varifolds Pointwise curvature estimates on stable hypersurfaces

153 159 162

202 214

CHAPTER 6.

DECOMPOSITION THEOREM FOR STABLE HYPERSURFACES

224

6.1 6.2 6.3-6.5 6.6

Decomposition of a varifold

226 226 227 229

6.7-6.11 6,12 6.13-6.17

Useful constants Decomposition theorem Tangent cones of limits of stable manifolds Miscellaneous estimates "Little" decomposition theorem Proof of decomposition theorem

232 248 263

CHAPTER 7.

REGULARITY

288

7.1 7.2-7.3

Preliminary remarks Regularity of area minimizing hypersurfaces Regularity of limits of stable manifolds Compactness theorem for stable hypersurfaces Disk theorem for stable cones Regularity of comparison surfaces in the class β Tangent cones of almost minimizing varifolds Regularity of almost minimizing varifolds in a disk Regularity of almost minimizing varifolds in an annulus Existence of regular minimal hypersurfaces Existence of manifolds of general critical type

290 291

7.4 7.5 7.6 7.7 7.8 7.9-7.11 7.12 7.13 7.14

REFERENCES

293 296 296 298 299 301 319 324 325

327

INTRODUCTION

In this monograph, we develop a comprehensive variational calculus with which we explore the existence and regularity of minimal surfaces on riemannian manifolds. Our principal conclusion is the following theorem. THEOREM A. EXISTENCE THEOREM FOR REGULAR MINIMAL HYPERSURFACES ON RIEMANNIAN MANIFOLDS (7.13). If

2 < k < 5, max{k,4} < ν < ® , and

M

is^ a_

(k+1)-dimensional compact riemannian manifold of class (f+1) , then k

M

supports a_ nonempty, compact.

dimensional. imbedded. minimal submanifold

(without boundary) of class

v

.

In these dimensions this theorem answers completely a more general question; namely, for what positive integers k

and

η

does a smooth, compact,

η

dimensional,

riemannian manifold support a regular closed minimal submanifold of dimension

k ?

Classically the only case

in which there were satisfactory answers of great generality was when

k = 1

and

(existence of closed geodesies).

η

was arbitrary

The first breakthrough

to higher dimensions without severe restrictions on the ambient manifold came in 1974 when we established a precursor of theorem A, valid when

k = 2

and

η = 3.

(This was announced in [PJ2], later revised and distributed in [PJ3, PJ4, PJ5].) (There have been further developments since then in the case historical remarks below.)

k = 2 ; see the

Much of the method we used

then was peculiar to the case

k = λ .

We have developed

now new estimates, more powerful and more general, with which we have extended the regularity theory to the dimensions in theorem A.

This is the first general

existence theorem of this type for regular minimal surfaces when

k

dimensional

k > 2.

Generally speaking, there are two large parts to the logical development.

In chapters 3 and 4 we derive

one part, a very general existence theory for minimal surfaces, applicable on arbitrary compact riemannian manifolds in all dimensions and codimensions.

Chapters

5, 6, and 7 compose the second part, a regularity theory, in which derive the special estimates necessary to establish the existence theorem A.

The general existence

theory has its roots in [AF1] and [AF2].

Almgren

demonstrated [AF1] that on a compact manifold

M , the

homotopy groups of the integral cycle groups on isomorphic to the homology groups of

M .

M

are

This led him

to construct a variational calculus in the large analogou to that of Marston Morse, from which he concluded [AF2] that

M

supports a nonzero stationary integral varifold

in all dimensions not exceeding dim(M).

Here we

construct a similar variational calculus and prove what turns out to be a critical extension; namely, that

M

supports nonzero stationary integral varifolds with an additional variational property, which we have called almost minimizing (3.1).

Intuitively one considers an

almost minimizing varifold to be one which may be approxi mated arbitrarily closely by integral currents which are themselves very nearly locally area minimizing.

Almost

minimizing varifolds are principal objects of our investi gation.

The origins of the concept are quite natural

(cf. 1.1 and 1.2).

The main existence theorem is the

following. THEOREM B.

GENERAL EXISTENCE THEOREM FOR MINIMAL

SURFACES ON RIEMANNIAN MANIFOLDS (4.11). k < η , every compact

η

For each

dimensional riemannian

manifold of class 4 supports a nonzero

k dimensional

stationary integral varifold which, at each point in

the manifold. is almost minimizing in all

small annular neighborhoods of that point. The study of almost minimizing varifolds began in the first place because Almgren's theorem on the existence of stationary integral varifolds is inadequate to settle the question of existence of regular minimal surfaces on manifolds.

This is because varifolds which are

only stationary and integral have in general essential singularities, possibly of positive measure.

If, in

addition, the varifold is almost minimizing, then it possesses strong local stability properties which yield estimates on the singular sets.

In particular, these

estimates imply the singular set is empty for hypersurfaces of

η

dimensional manifolds, 3 < η < 6, which is theorem A. Thus, our regularity theory depends on careful

analysis of stable surfaces (minimal surfaces whose second variation of area is nonnegative), a class of minimal surfaces which has been vigorously investigated in recent years (see [SJ], [LHB2], or [SSY], for example). For our purposes the salient property of stable surfaces

5 is t h a t t h e i r g e o m e t r i c c o n f i g u r a t i o n s a r e

considerably

m o r e r e s t r i c t e d than t h o s e of g e n e r a l m i n i m a l

surfaces.

T h e p r i n c i p a l d e s c r i p t i v e r e s u l t is t h i s . THEOREM C. DECOMPOSITION THEOREM FOR HYPERSURFACES (6.3). (or m o r e g e n e r a l l y ,

If M

and

is

s u b m a n i f o l d o f c l a s s 5 of p l a n a r ) , then a s t a b l e of

M

STABLE

k

a (k+1)-dimensional and

M

is

dimensional

sufficiently

submanifold

lying s u f f i c i e n t l y n e a r a c o n e is the

d i s j o i n t u n i o n of m i n i m a l g r a p h s of

functions

over a single k-plane. T h e p r o o f o f t h e d e c o m p o s i t i o n t h e o r e m (as w e l l a s

theorems

D and E below) depends on a strong pointwise curvature

estimate

f o r s t a b l e s u r f a c e s d u e to S c h o e n , S i m o n , and Yau [ S S Y ] .

A

d e r i v a t i o n o f t h i s e s t i m a t e b a s e d o n [SSY] is in c h a p t e r 5 . W e also prove an interesting compactness property of stable manifolds. THEOREM D. COMPACTNESS THEOREM FOR REGULAR HYPERSURFACES

(7.5).

If

and

STABLE M

is

a compact (k+1)-dimensiona1 riemannian manifold of class 5 ,

then the s p a c e o f u n i f o r m l y m a s s

bounded, stable, k dimensional integral varifolds

6 on

M

weak

w i t h r e g u l a r s u p p o r t is c o m p a c t in the topology.

If in

and if

C

is a

k

dimensional cone

w h i c h is s t a b l e a n d r e g u l a r (except at the v e r t e x ) ,

t h e n it is w e l l k n o w n that If

C

must be a hyperplane.

, then w e g e n e r a l i z e this as f o l l o w s . THEOREM E . DISK THEOREM FOR STABLE CONES If

, and if

C

i n t e g r a l v a r i f o l d in (C

is

a

k

(7.6).

dimensional

w h i c h is a c o n e

n e e d n o t h a v e r e g u l a r s u p p o r t ) , then t h e r e is

a sequence of stable

k

dimensional

integral

v a r i f o l d s w i t h r e g u l a r s u p p o r t c o n v e r g i n g to if and o n l y if

C

C

is a h y p e r p l a n e . p o s s i b l y w i t h

multiplicity. F i n a l l y w e o b t a i n an i n t e r e s t i n g e x i s t e n c e and

regularity

t h e o r e m a n a l o g o u s to the c l a s s i c t h e o r e m s of M o r s e - T o m p k i n s and Shiffman

[SM].

T H E O R E M F . E X I S T E N C E O F M I N I M A L M A N I F O L D S OF G E N E R A L CRITICAL TYPE

(7.14).

If

,

( k - 1 ) - d i m e n s i o n a l i n t e g r a l c y c l e in are

k

dimensional integral currents,

C

is a

[MT]

7 and

and

locally minimize area among all

integral currents with boundary exists a that

V

C , then t h e r e

k dimensional integral varifold

V

such

is s t a t i o n a r y w i t h r e s p e c t to d e f o r m a t i o n s

w i t h c o m p a c t s u p p o r t in

spt

_is a

dimensional, real analytic, minimal

k

submanifold

of N o w w e s u m m a r i z e the h i s t o r y of the q u e s t i o n of the e x i s t e n c e of a compact c a s e of

n k = 1

k

d i m e n s i o n a l m i n i m a l s u r f a c e s on

dimensional manifold and

n

M

.

The classic

a r b i t r a r y w a s s e t t l e d in 1 9 5 1

b y L y u s t e r n i k and F e t [ L F ] , w h o showed t h a t a n a r b i t r a r y compact manifold always supports a closed possibly with self-intersections. S c h n i r e l m a n n [LS] p r o v e d that if

geodesic,

In 1929 L u s t e r n i k and M

is two d i m e n s i o n a l

a n d s i m p l y c o n n e c t e d , then t h e r e e x i s t o n

M

at

least

three closed geodesies without self-intersections.

Recently

t h e r e h a s b e e n c o n s i d e r a b l e i n t e r e s t in s h o w i n g t h a t f o r arbitrary

,

M

supports many closed

geodesies.

There are many interesting results with various geometric and topological hypotheses on

M ; the case

k = 1

has

become a subfield of its own.

For an exhaustive discussion

and bibliography, see [KW]. When

k = 2 , the first general theorem was the

predecessor of theorem A in [PJ2], as described above. Since then there have been two major developments on the existence of minimal immersions of 2-manifolds. Sacks and Uhlenbeck [SU] have studied minimal immersions of spheres, the main result being that if if the universal covering space of

M

then there exists a smooth mapping of

η > 3

and

is not contractible,

2

S

into

M

which

(except perhaps at a finite number of branch points) is a conformal minimal immersion, possibly with selfintersections.

When

η = 3 , there will be no branch points.

In a second development, Schoen and Yau [ SY] have proved that if

S

is a riemann surface,

f : S - M

and the induced map of fundamental groups

f

f

is continuous, :

(S) --

L

L

(M)

is injective, then there is a branched minimal immersion g : S -> M such that

g

#

= f t

and

g

among all maps with the same action on

minimizes induced area w^(S).

These authors

have applied this result to analyze the topology of manifolds with nonnegative scalar curvature.

Other results include the following.

We have already

mentioned the theorem of Almgren [AF2] on the existence of stationary varifolds on arbitrary manifolds. important theorem is that if

k = n-1 < 6

dimensional homology group of

M

and the

k

with coefficients in

the integers does not vanish, then minimal hypersurface.

Another

M

supports a closed

This follows from the methods of

[FH1, chapter 5] applied to a homologically area minimizing representative of a nonzero on

M

(cf. 7.2).

k dimensional homology class

Also, in [LHBl], Lawson explicitly

constructs examples to show that

3 M=S

supports

closed two dimensional minimal submanifolds of arbitrarily high genus. Insofar as possible, our presentation is selfcontained.

We have included, in particular, such techniques

as we need from differential geometry and topology (4.6 excepted).

Regarding geometric measure theory, we have

not been so self-contained; Federer's exhaustive treatment [FH1] makes what would be a lengthy effort redundant at best.

One important topic, varifolds, has appeared since

the publication [FH1]; the best reference for this is the comprehensive monograph [AWl].

We have listed in

2.4 those theorems about varifolds which we need, so that it is not strictly necessary to have [AW1] in hand in order to follow our arguments. We might say a few words about chapter 1.

Although

the complete development of the monograph is lengthy and not always easy, the fundamental ideas are natural and simple.

Chapter 1 is an informal description of our

methods, largely by illustrative examples.

It is also

a good source of examples and counterexamples for specific questions in the theory.

We hope the reader finds it

useful. It is a pleasure to thank Professor F. J. Almgren, Jr., for helpful discussions.

I am grateful to Mrs. Diane

Strazzabosco for typing much of this manuscript. REMARK ADDED IN PROOF.

As described above, the

curvature estimates of [SSY] for stable surfaces were essential in the regularity (theorem A), and also in theorems C, D, and E.

The dimension restriction 2 < k < 5

in these theorems reflected a corresponding restriction in [SSY].

Now R. Schoen and L. Simon have derived more

general curvature estimates in a form applicable to stable k dimensional hypersurfaces for all positive integers k

11 (Regularity of stable minima1 hypersurfaces. p r e p r i n t ) . Among other things, they obtain a suitably extension of the decomposition theorem

formulated

(theorem C) and

the compactness theorem for stable hypersurfaces valid for dimensions

as w e l l .

(theorem

By combining

their

new curvature estimates w i t h the general existence (theorem B) and our continuation arguments

theorem

for almost

m i n i m i z i n g v a r i f o l d s in c h a p t e r 7 , S c h o e n a n d S i m o n the r e g u l a r i t y to d i m e n s i o n s T H E O R E M . If

as and

M

extend

follows.

is. a c o m p a c t

(kH-l)-dimensional r i e m a n n i a n m a n i f o l d of c l a s s then

M

s u p p o r t s a. n o n z e r o . k d i m e n s i o n a l .

stationary, integral varifold

V

is a. c o m p a c t set

such that

S c spt

for which

there

i s a. k d i m e n s i o n a l m i n i m a l s u b m a n i f o l d o f c l a s s