Introduction to Global Analysis: Minimal Surfaces in Riemannian Manifolds (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 187)

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. Cover......Page 1 Title page......Page 4 Contents......Page 6 Preface......Page 8 1.1. A global setting for nonlinear DEs......Page 16 1.2. Infinite-dimensional calculus......Page 17 1.3. Manifolds modeled on Banach spaces......Page 32 1.4. The basic mapping spaces......Page 40 1.5. Homotopy type of the space of maps......Page 48 1.6. The ��- and ��-Lemmas......Page 54 1.7. The tangent and cotangent bundles......Page 55 1.8. Differential forms......Page 59 1.9. Riemannian and Finsler metrics......Page 64 1.10. Vector fields and ODEs......Page 68 1.11. Condition C......Page 70 1.12. Birkhoff’s minimax principle......Page 75 1.13. de Rham cohomology......Page 78 2.1. Geodesics......Page 86 2.2. Condition C for the action......Page 91 2.3. Fibrations and the Fet-Lusternik Theorem......Page 96 2.4. Second variation and nondegenerate critical points......Page 100 2.5. The Sard-Smale Theorem......Page 106 2.6. Existence of Morse functions......Page 110 2.7. Bumpy metrics for smooth closed geodesics......Page 115 2.8. Adding handles......Page 123 2.9. Morse inequalities......Page 129 2.10. The Morse-Witten complex......Page 133 3.1. Sullivan’s theory of minimal models......Page 140 3.2. Minimal models for spaces of paths......Page 146 3.3. Gromov dimension......Page 153 3.4. Infinitely many closed geodesics......Page 160 3.5. Postnikov towers......Page 163 3.6. Maps from surfaces......Page 170 4.1. The energy of a smooth map......Page 184 4.2. Minimal two-spheres and tori......Page 193 4.3. Minimal surfaces of arbitrary topology......Page 203 4.4. The ��-energy......Page 219 4.5. Morse theory for a perturbed energy......Page 231 4.6. Bubbles......Page 240 4.7. Existence of minimal two-spheres......Page 254 4.8. Existence of higher genus minimal surfaces......Page 264 4.9. Unstable minimal surfaces......Page 271 4.10. An application to curvature and topology......Page 282 5.1. Bumpy metrics for minimal surfaces......Page 292 5.2. Local behavior of minimal surfaces......Page 296 5.3. The two-variable energy revisited......Page 307 5.4. Minimal surfaces without branch points......Page 323 5.5. Minimal surfaces with simple branch points......Page 333 5.6. Higher order branch points......Page 349 5.7. Proof of the Transversal Crossing Theorem......Page 362 5.8. Branched covers......Page 364 Bibliography......Page 372 Index......Page 380 Back Cover......Page 385 Applies infinite-dimensional manifold theory to the Morse theory of closed geodesics in a Riemannian manifold. This volume then describes the problems encountered when extending this theory to maps from surfaces instead of curves. It also treats critical point theory for closed parametrized minimal surfaces in a compact Riemannian manifold.