Contents
Preface xi
Chapter 1. Introduction 1
1.1. An overview 1
1.2. Plan of the book 6
Part 1. J-Convexity 9
Chapter 2. J-Convex Functions and Hypersurfaces 11
2.1. Linear algebra 11
2.2. J-convex functions 13
2.3. The Levi form of a hypersurface 15
2.4. Completeness 18
2.5. J-convexity and geometric convexity 19
2.6. Normalized Levi form and mean normal curvature 20
2.7. Examples of J-convex functions and hypersurfaces 22
2.8. Symplectic properties of J-convex functions 25
2.9. Computations in C
n 27
Chapter 3. Smoothing 31
3.1. J-convexity and plurisubharmonicity 31
3.2. Smoothing of J-convex functions 34
3.3. Critical points of J-convex functions 37
3.4. From families of hypersurfaces to J-convex functions 40
3.5. J-convex functions near totally real submanifolds 42
3.6. Functions with J-convex level sets 48
3.7. Normalized modulus of J-convexity 50
Chapter 4. Shapes for i-Convex Hypersurfaces 57
4.1. Main models 57
4.2. Shapes for i-convex hypersurfaces 59
4.3. Properties of i-convex shapes 64
4.4. Shapes in the subcritical case 67
4.5. Construction of special shapes 68
4.6. Families of special shapes 75
4.7. Convexity estimates 83
Chapter 5. Some Complex Analysis 89
5.1. Holomorphic convexity 89
5.2. Relation to J-convexity 90
5.3. Definitions of Stein manifolds 93
5.4. Hartogs phenomena 94
5.5. Grauert’s Oka principle 96
5.6. Coherent analytic sheaves on Stein manifolds 99
5.7. Real analytic manifolds 101
5.8. Real analytic approximations 104
5.9. Approximately holomorphic extension of maps
from totally real submanifolds 107
5.10. CR structures 108
Part 2. Existence of Stein Structures 113
Chapter 6. Symplectic and Contact Preliminaries 115
6.1. Symplectic vector spaces 115
6.2. Symplectic vector bundles 117
6.3. Symplectic manifolds 118
6.4. Moser’s trick and symplectic normal forms 119
6.5. Contact manifolds and their Legendrian submanifolds 122
6.6. Contact normal forms 125
6.7. Real analytic approximations of isotropic submanifolds 127
6.8. Relations between symplectic and contact manifolds 128
Chapter 7. The h-principles 131
7.1. Immersions and embeddings 131
7.2. The h-principle for isotropic immersions 135
7.3. The h-principle for subcritical isotropic embeddings 136
7.4. Stabilization of Legendrian submanifolds 137
7.5. The existence theorem for Legendrian embeddings 139
7.6. Legendrian knots in overtwisted contact manifolds 141
7.7. Murphy’s h-principle for loose Legendrian embeddings 142
7.8. Directed immersions and embeddings 146
7.9. Discs attached to J-convex boundaries 150
Chapter 8. The Existence Theorem 155
8.1. Some notions from Morse theory 155
8.2. Surrounding stable discs 156
8.3. Existence of complex structures 161
8.4. Existence of Stein structures in complex dimension = 2 163
8.5. J-convex surrounding functions 167
8.6. J-convex retracts 171
8.7. Approximating continuous maps by holomorphic ones 174
8.8. Variations on a theme of E. Kallin 181
Part 3. Morse–Smale Theory for J-Convex Functions 185
Chapter 9. Recollections from Morse Theory 187
9.1. Critical points of functions 187
9.2. Zeroes of vector fields 189
9.3. Gradient-like vector fields 192
9.4. Smooth surroundings 198
9.5. Changing Lyapunov functions near critical points 200
9.6. Smale cobordisms 202
9.7. Morse and Smale homotopies 206
9.8. The h-cobordism theorem 210
9.9. The two-index theorem 212
9.10. Pseudo-isotopies 213
Chapter 10. Modifications of J-Convex Morse Functions 215
10.1. Moving attaching spheres by isotropic isotopies 215
10.2. Relaxing the J-orthogonality condition 222
10.3. Moving critical levels 223
10.4. Creation and cancellation of critical points 224
10.5. Carving one J-convex function with another one 225
10.6. Surrounding a stable half-disc 225
10.7. Proof of the cancellation theorem 231
10.8. Proof of the creation theorem 232
Part 4. From Stein to Weinstein and Back 235
Chapter 11. Weinstein Structures 237
11.1. Liouville cobordisms and manifolds 237
11.2. Liouville homotopies 239
11.3. Zeroes of Liouville fields 241
11.4. Weinstein cobordisms and manifolds 243
11.5. From Stein to Weinstein 244
11.6. Weinstein and Stein homotopies 245
11.7. Weinstein structures with unique critical points 249
11.8. Subcritical and flexible Weinstein structures 250
Chapter 12. Modifications of Weinstein Structures 253
12.1. Weinstein structures with given functions 253
12.2. Holonomy of Weinstein cobordisms 256
12.3. Liouville fields near isotropic submanifolds 258
12.4. Weinstein structures near critical points 263
12.5. Weinstein structures near stable discs 265
12.6. Morse–Smale theory for Weinstein structures 267
12.7. Elementary Weinstein homotopies 268
Chapter 13. Existence Revisited 271
13.1. Existence of Weinstein structures 271
13.2. From Weinstein to Stein: existence 273
13.3. Proof of the Stein existence theorems 275
Chapter 14. Deformations of Flexible Weinstein Structures 279
14.1. Homotopies of flexible Weinstein cobordisms 279
14.2. Proof of the first Weinstein deformation theorem 280
14.3. Proof of the second Weinstein deformation theorem 286
14.4. Subcritical Weinstein manifolds are split 288
14.5. Symplectic pseudo-isotopies 292
Chapter 15. Deformations of Stein Structures 295
15.1. From Weinstein to Stein: homotopies 295
15.2. Proof of the first Stein deformation theorem 298
15.3. Homotopies of flexible Stein structures 302
Part 5. Stein Manifolds and Symplectic Topology 305
Chapter 16. Stein Manifolds of Complex Dimension Two 307
16.1. Filling by holomorphic discs 307
16.2. Stein fillings 310
16.3. Stein structures on 4-manifolds 320
Chapter 17. Exotic Stein Structures 323
17.1. Symplectic homology 323
17.2. Exotic Stein structures 325
Appendix A. Some Algebraic Topology 329
A.1. Serre fibrations 329
A.2. Some homotopy groups 331
Appendix B. Obstructions to Formal Legendrian Isotopies 335
Appendix C. Biographical Notes on the Main Characters 343
C.1. Complex analysis 343
C.2. Differential and symplectic topology 348
Bibliography 353
Index 361
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