**Graduate Studies in Mathematics**

Volume: 187;
2017;
368 pp;
Hardcover

MSC: Primary 58;
Secondary 46; 53; 55

Print ISBN: 978-1-4704-2950-8

Product Code: GSM/187

List Price: $83.00

AMS Member Price: $66.40

MAA Member Price: $74.70

**Electronic ISBN: 978-1-4704-4317-7
Product Code: GSM/187.E**

List Price: $83.00

AMS Member Price: $66.40

MAA Member Price: $74.70

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#### Supplemental Materials

# Introduction to Global Analysis: Minimal Surfaces in Riemannian Manifolds

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*John Douglas Moore*

During the last century, global analysis was
one of the main sources of interaction between geometry and
topology. One might argue that the core of this subject is Morse
theory, according to which the critical points of a generic smooth
proper function on a manifold \(M\) determine the homology of the
manifold.

Morse envisioned applying this idea to the calculus of variations,
including the theory of periodic motion in classical mechanics, by
approximating the space of loops on \(M\) by a
finite-dimensional manifold of high dimension. Palais and Smale
reformulated Morse's calculus of variations in terms of
infinite-dimensional manifolds, and these infinite-dimensional
manifolds were found useful for studying a wide variety of nonlinear
PDEs.

This book applies infinite-dimensional manifold theory to the Morse
theory of closed geodesics in a Riemannian manifold. It then
describes the problems encountered when extending this theory to maps
from surfaces instead of curves. It treats critical point theory for
closed parametrized minimal surfaces in a compact Riemannian manifold,
establishing Morse inequalities for perturbed versions of the energy
function on the mapping space. It studies the bubbling which occurs
when the perturbation is turned off, together with applications to the
existence of closed minimal surfaces. The Morse-Sard theorem is used
to develop transversality theory for both closed geodesics and closed
minimal surfaces.

This book is based on lecture notes for graduate courses on “Topics
in Differential Geometry”, taught by the author over several
years. The reader is assumed to have taken basic graduate courses in
differential geometry and algebraic topology.

#### Readership

Graduate students and researchers interested in differential geometry.

#### Reviews & Endorsements

This book provides a thoughtful introduction to classical geometric applications of global analysis in the context of geodesics and minimal surfaces...it would be a good choice of textbook for a graduate topics course that provides a more classical overview of the area.

-- Renato G. Bettiol, Mathematical Reviews

#### Table of Contents

# Table of Contents

## Introduction to Global Analysis: Minimal Surfaces in Riemannian Manifolds

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Chapter 1. Infinite-dimensional Manifolds 116
- 1.1. A global setting for nonlinear DEs 116
- 1.2. Infinite-dimensional calculus 217
- 1.3. Manifolds modeled on Banach spaces 1732
- 1.4. The basic mapping spaces 2540
- 1.5. Homotopy type of the space of maps 3348
- 1.6. The 𝛼- and 𝜔-Lemmas 3954
- 1.7. The tangent and cotangent bundles 4055
- 1.8. Differential forms 4459
- 1.9. Riemannian and Finsler metrics 4964
- 1.10. Vector fields and ODEs 5368
- 1.11. Condition C 5570
- 1.12. Birkhoff’s minimax principle 6075
- 1.13. de Rham cohomology 6378

- Chapter 2. Morse Theory of Geodesics 7186
- 2.1. Geodesics 7186
- 2.2. Condition C for the action 7691
- 2.3. Fibrations and the Fet-Lusternik Theorem 8196
- 2.4. Second variation and nondegenerate critical points 85100
- 2.5. The Sard-Smale Theorem 91106
- 2.6. Existence of Morse functions 95110
- 2.7. Bumpy metrics for smooth closed geodesics 100115
- 2.8. Adding handles 108123
- 2.9. Morse inequalities 114129
- 2.10. The Morse-Witten complex 118133

- Chapter 3. Topology of Mapping Spaces 125140
- Chapter 4. Harmonic and Minimal Surfaces 169184
- 4.1. The energy of a smooth map 169184
- 4.2. Minimal two-spheres and tori 178193
- 4.3. Minimal surfaces of arbitrary topology 188203
- 4.4. The 𝛼-energy 204219
- 4.5. Morse theory for a perturbed energy 216231
- 4.6. Bubbles 225240
- 4.7. Existence of minimal two-spheres 239254
- 4.8. Existence of higher genus minimal surfaces 249264
- 4.9. Unstable minimal surfaces 256271
- 4.10. An application to curvature and topology 267282

- Chapter 5. Generic Metrics 277292
- 5.1. Bumpy metrics for minimal surfaces 277292
- 5.2. Local behavior of minimal surfaces 281296
- 5.3. The two-variable energy revisited 292307
- 5.4. Minimal surfaces without branch points 308323
- 5.5. Minimal surfaces with simple branch points 318333
- 5.6. Higher order branch points 334349
- 5.7. Proof of the Transversal Crossing Theorem 347362
- 5.8. Branched covers 349364

- Bibliography 357372
- Index 365380
- Back Cover Back Cover1385