Introduction to global analysis minimal surfaces in Riemannian manifolds

385 Pages·2017·1.948 MB·English

by  Moore, John Douglas

GRADUATE STUDIES 187
IN MATHEMATICS
Introduction to
Global Analysis
Minimal Surfaces in
Riemannian Manifolds
John Douglas Moore
American Mathematical Society
Introduction to
Global Analysis
Minimal Surfaces in
Riemannian Manifolds

GRADUATE STUDIES 187
IN MATHEMATICS
Introduction to
Global Analysis
Minimal Surfaces in
Riemannian Manifolds
John Douglas Moore
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Dan Abramovich
Daniel S. Freed (Chair)
Gigliola Staffilani
Jeff A. Viaclovsky
2010 Mathematics Subject Classification. Primary 58E12,58E05; Secondary 46T10,
53C20, 55P62.
For additional informationand updates on this book, visit
www.ams.org/bookpages/gsm-187
Library of Congress Cataloging-in-Publication Data
Names: Moore,JohnD.(JohnDouglas),1943-author.
Title: Introductiontoglobalanalysis: minimalsurfacesinRiemannianmanifolds/JohnDouglas
Moore.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2017]|Series: Gradu-
atestudiesinmathematics;volume187|Includesbibliographicalreferencesandindex.
Identifiers: LCCN2017028964|ISBN9781470429508(alk. paper)
Subjects: LCSH: Global analysis (Mathematics) | Minimal surfaces. | Riemannian manifolds. |
Manifolds(Mathematics)|Morsetheory. |AMS:Globalanalysis,analysisonmanifolds–Vari-
ational problems in infinite-dimensional spaces – Applications to minimal surfaces (problems
intwoindependentvariables). msc|Globalanalysis,analysisonmanifolds–Variationalprob-
lemsininfinite-dimensionalspaces–Abstractcriticalpointtheory(Morsetheory,Ljusternik-
Schnirelmann (Lyusternik-Shnirel(cid:2)man) theory, etc.). msc | Functional analysis – Nonlinear
functional analysis – Manifolds of mappings. msc | Differential geometry – Global differen-
tialgeometry–GlobalRiemanniangeometry,including pinching. msc|Algebraictopology–
Homotopytheory–Rationalhomotopytheory. msc
Classification: LCCQA614.M662017|DDC516.3/62–dc23LCrecordavailableathttps://lccn.
loc.gov/2017028964
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Contents
Preface vii
Chapter 1. Infinite-dimensional Manifolds 1
§1.1. A global setting for nonlinear DEs 1
§1.2. Infinite-dimensional calculus 2
§1.3. Manifolds modeled on Banach spaces 17
§1.4. The basic mapping spaces 25
§1.5. Homotopy type of the space of maps 33
§1.6. The α- and ω-Lemmas 39
§1.7. The tangent and cotangent bundles 40
§1.8. Differential forms 44
§1.9. Riemannian and Finsler metrics 49
§1.10. Vector fields and ODEs 53
§1.11. Condition C 55
§1.12. Birkhoff’s minimax principle 60
§1.13. de Rham cohomology 63
Chapter 2. Morse Theory of Geodesics 71
§2.1. Geodesics 71
§2.2. Condition C for the action 76
§2.3. Fibrations and the Fet-Lusternik Theorem 81
§2.4. Second variation and nondegenerate critical points 85
§2.5. The Sard-Smale Theorem 91
§2.6. Existence of Morse functions 95
v
vi Contents
§2.7. Bumpy metrics for smooth closed geodesics 100
§2.8. Adding handles 108
§2.9. Morse inequalities 114
§2.10. The Morse-Witten complex 118
Chapter 3. Topology of Mapping Spaces 125
§3.1. Sullivan’s theory of minimal models 125
§3.2. Minimal models for spaces of paths 131
§3.3. Gromov dimension 138
§3.4. Infinitely many closed geodesics 145
§3.5. Postnikov towers 148
§3.6. Maps from surfaces 155
Chapter 4. Harmonic and Minimal Surfaces 169
§4.1. The energy of a smooth map 169
§4.2. Minimal two-spheres and tori 178
§4.3. Minimal surfaces of arbitrary topology 188
§4.4. The α-energy 204
§4.5. Morse theory for a perturbed energy 216
§4.6. Bubbles 225
§4.7. Existence of minimal two-spheres 239
§4.8. Existence of higher genus minimal surfaces 249
§4.9. Unstable minimal surfaces 256
§4.10. An application to curvature and topology 267
Chapter 5. Generic Metrics 277
§5.1. Bumpy metrics for minimal surfaces 277
§5.2. Local behavior of minimal surfaces 281
§5.3. The two-variable energy revisited 292
§5.4. Minimal surfaces without branch points 308
§5.5. Minimal surfaces with simple branch points 318
§5.6. Higher order branch points 334
§5.7. Proof of the Transversal Crossing Theorem 347
§5.8. Branched covers 349
Bibliography 357
Index 365
Preface
This book is devoted to giving the foundations for a partial Morse theory of
minimal surfaces in Riemannian manifolds. It is based upon lecture notes
for graduate courses on “Topics in Differential Geometry”, given at the
University of California, Santa Barbara, during the fall quarter of 2009 and
again in the spring of 2014, but it also includes several topics not treated in
these courses.
It might be helpful to start with a description of the goal of our presen-
tation. Morse theory is concerned with the relationship between the critical
pointsofasmoothfunctiononamanifoldandthetopologyofthatmanifold.
It might have developed in three main stages.
Thefirststageinourfictionalhistorywouldhavebeenfinite-dimensional
Morse theory, which relates critical points of proper nonnegative functions
on finite-dimensional manifolds to the homology of these manifolds via the
Morseinequalities. The foundationswerelaidinMarstonMorse’sfirstland-
mark article on what is now known as Morse theory [Mor25], but Morse
quickly turned his attention to problems from the calculus of variations,
which ultimately became part of the infinite-dimensional theory. Never-
theless, finite-dimensional Morse theory became one of the primary tools
for studying the topology of finite-dimensional manifolds and had many
successes, including the celebrated h-cobordism theorem of Smale [Mil65],
crucial for the classification of manifolds in high dimensions. Modern expo-
sitions of finite-dimensional Morse theory often construct a chain complex
from the free abelian group generated by the critical points of a “generic”
function, the boundary being defined by orbits of the gradient flow which
connect the critical points. The homology of this chain complex, called the
vii
viii Preface
Morse-Witten complex, is isomorphic to the usual integer homology of the
manifold.
What might have been the second stage, the Morse theory of geodesics,
formed the core of what Morse [Mor34] called “the calculus of variations in
the large”. Motivated to some extent by earlier work on celestial mechanics
by Poincar´e and Birkhoff, Morse studied the calculus of variations for the
lengthfunctionoractionfunctiononthespaceofpathsjoiningtwopointsin
a Riemannian manifold (or the space of closed paths in a Riemannian man-
ifold), the critical points of these functions being geodesics. His idea was to
approximate the infinite-dimensional space of paths by a finite-dimensional
manifold of very high dimension and then apply finite-dimensional Morse
theory to this approximation. This approach is explained in Milnor’s clas-
sical book on Morse theory [Mil63] and produced many striking results in
the theory of geodesics in Riemannian geometry, such as the theorem of
Serre [Ser51] that any two points on a compact Riemannian manifold can
be joined by infinitelymany geodesics. It also provided the first proof of the
Bott periodicity theorem from homotopy theory. A Morse theory of closed
geodesics, representing periodic motion in certain mechanical systems, was
also constructed. One might regard the Morse theory of geodesics as an
application of topology to the study of ordinary differential equations, in
particular, to those equations which like the equation for geodesics arise
from classical mechanics.
Palais and Smale were able to provide an elegant reformulation of the
Morse theory of geodesics in the language of infinite-dimensional Hilbert
manifolds [PS64]. They showed that the action function on the infinite-
dimensionalmanifoldof paths satisfies“ConditionC”, a conditionreplacing
“proper” in the finite-dimensional theory, and they showed that Condition
C is sufficient for the development of Morse theory in infinite dimensions.
This became a standard approach to the Morse theory of geodesics during
the last several decades of the twentieth century, and we will describe it in
some detail later.
OnemightregardthethirdstageofMorsetheoryasencompassingmany
strands, but our viewpoint is to focus on techniques for applying Morse
theory to nonlinear elliptic partial differential equations coming from the
calculus of variations in which the domain is a two-dimensional compact
surface. Morse himself hoped to apply the ideas of his theory to what
is arguably the central case—the partial differential equations for minimal
surfaces—andhefocusedonthecaseinwhichtheambientspaceisEuclidean
space. The first steps in this direction were taken by Morse and Tompkins,
as well as Shiffman, who established the theorem that if a simple closed
curve in Euclidean space R3 bounds two stable minimal disks, it bounds a
Preface ix
third,whichisnotstable. Thisprovidedaversionoftheso-called“mountain
pass lemma” for minimal disks in Euclidean space. The results of Morse,
Tompkins, and Shiffman suggested that Morse inequalities should hold for
minimal surfaces in Euclidean space with boundary constrained to lie on
a given Jordan curve and, indeed, such inequalities were later established
under appropriate hypotheses (as explained, for example, in [JS90]).
But the most natural extension of the Morse theory of closed geodesics
in Riemannian manifolds to partial differential equations would be a Morse
theory of closed two-dimensional minimal surfaces in a general curved am-
bient Riemannian manifold M, instead of the ambient Euclidean space used
in the classical theory of minimal surfaces with boundary. The generaliza-
tion to ambient Riemannian manifolds with arbitrary curvature introduces
complexity and requires new techniques. Unfortunately, if Σ is a connected
compact surface, it becomes awkward to extend the finite-dimensional ap-
proximationprocedure—soeffectiveinthetheoryofgeodesics—tothespace
Map(Σ,M)ofmappingsfromΣtoM. Onemighthopeforabetterapproach
based upon the theory of infinite-dimensional manifolds, as developed by
Palais and Smale, but a serious problem is encountered: the standard en-
ergy function
E : Map(Σ,M) −→ R,
used in the theory of harmonic maps and parametrized minimal surfaces,
fails to satisfy the compactness condition, the Condition C which Palais
and Smale had used as a foundation for their theory, when Map(Σ,M) is
completed with respect to a norm strong enough to lie within the space of
continuous functions.
To get around this difficulty, Sacks and Uhlenbeck introduced the α-
energy [SU81], [SU82] in which α > 1 is a parameter, a perturbation of
the usual energy which is defined on the completion of Map(Σ,M) with
respect to a Banach space norm which is both weak enough to satisfy Con-
ditionCandstrongenoughfortheα-energytobeaC2 functiononaBanach
space having the same homotopy type as the space of continuous maps from
Σ to M. We can express the fact that such a completion exists by saying
thattheα-energylieswithin“Sobolevrange”. Theα-energyapproachesthe
usualenergyastheparameterαintheperturbationgoestoone, andwecan
thereforesaythattheusualenergyonmapsfromcompactsurfacesis“onthe
borderofSobolevrange”, approachableby Morsetheoryviaapproximation.
Using this approximation via the α-energy, Sacks and Uhlenbeck were able
to establish many striking results in the theory of minimal surfaces in Rie-
mannian manifolds, including the fact that any compact simply connected
Riemannianmanifoldcontainsatleastonenonconstantminimaltwo-sphere,
which parallels the classical theorem of Fet and Lyusternik stating that any