Partial differential relations

Cover......Page 1
Title page......Page 3
Copyright page......Page 4
Foreword......Page 5
Contents......Page 7
1.1.1 Jets, Relations, Holonomy......Page 11
1.1.2 The Cauchy-Riemann Relation, Oka\'s Principle and the Theorem of Grauert......Page 14
1.1.3 Differentiable Immersions and the $h$-Principle of Smale-Hirsch......Page 16
1.1.4 Osculating Spaces and Free Maps......Page 18
1.1.5 Isometric Immersions of Riemannian Manifolds and the Theorems of Nash and Kuiper......Page 20
1.2.1 Classification of Solutions by Homotopy and the Parametric $h$-Principle......Page 23
1.2.2 Density of the $h$-Principle in the Fine Topologies......Page 28
1.2.3 Functionally Closed Relations......Page 32
1.3.1 Singularities as Differential Relations......Page 36
1.3.2 Genericity, Transversality and Thorn\'s Equisingularity Theorem......Page 40
1.4.1 Local Solutions of Differential Relations......Page 45
1.4.2 The $h$-Principle for Extensions; Flexibility and Micro-flexibility......Page 49
1.4.3 Ordinary Differential Equations and \"Zero-Dimensional\" Relations......Page 54
1.4.4 The $h$-Principle for the Cauchy Extension Problem......Page 56
2.1.1 Immersions and $k$-Mersions $V \\to \\mathbb{R}^q$ for $q > k$......Page 58
2.1.2 Immersions and Submersions $V \\to W$......Page 62
2.1.3 Folded Maps $V^n \\to W^q$ for $q \\leq n$......Page 64
2.1.4 Singularities and the Curvature of Smooth Maps......Page 71
2.1.5 Holomorphic Immersions of Stein Manifolds......Page 75
2.2 Continuous Sheaves......Page 84
2.2.1 Flexibility and the $h$-Principle for Continuous Sheaves......Page 85
2.2.2 Flexibility and Micro-flexibility of Equivariant Sheaves......Page 88
2.2.3 The Proof of the Main Flexibility Theorem......Page 90
2.2.4 Equivariant Microextensions......Page 94
2.2.5 Local Compressibility and the Proof of the Microextension Theorem......Page 97
2.2.6 An Application: Inducing Euclidean Connections......Page 103
2.2.7 Non-flexible Sheaves......Page 108
2.3.1 Linearization and the Linear Inversion......Page 124
2.3.2 Basic Properties of Infinitesimally Invertible Operators......Page 127
2.3.3 The Nash (Newton-Moser) Process......Page 131
2.3.4 Deep Smoothing Operators......Page 133
2.3.5 The Existence and Convergence of Nash\'s Process......Page 141
2.3.6 The Modified Nash Process and Special Inversions of the Operator $\\mathcal{D}$......Page 149
2.3.7 Infinite Dimensional Representations of the Group $\\Diff(V)$......Page 155
2.3.8 Algebraic Solution of Differential Equations......Page 158
2.4.1 Integrals and Convex Hulls......Page 178
2.4.2 Principal Extensions of Differential Relations......Page 184
2.4.3 Ample Differential Relations......Page 190
2.4.4 Fiber Connected Relations and Directed Immersions......Page 193
2.4.5 Directed Embeddings and the Relative $h$-Principle......Page 199
2.4.6 Convex Integration of Partial Differential Equations......Page 204
2.4.7 Underdetermined Evolution Equations......Page 205
2.4.8 Triangular Systems of P.D.E......Page 208
2.4.9 Isometric $C^1$-Immersions......Page 211
2.4.10 Isometric Maps with Singularities......Page 217
2.4.11 Equidimensional Isometric Maps......Page 224
2.4.12 The Regularity Problem and Related Questions in the Convex Integration......Page 229
3.1.1 Nash\'s Twist and Approximate Immersions; Isometric Embeddings into $\\mathbb{R}^q$......Page 231
3.1.2 Isometric Immersions $V^n \\to W^q$ for $q \\geq (n + 2)(n + 5)/2$......Page 234
3.1.3 Convex Cones in the Space of Metrics......Page 241
3.1.4 Inducing Forms of Degree $d > 2$......Page 242
3.1.5 Immersions with a Prescribed Curvature......Page 245
3.1.6 Extensions oflsometric Immersions......Page 250
3.1.7 Isometric Immersions $V^n \\to W^q$ for $q \\geq (n + 2)(n + 3)/2$......Page 257
3.1.8 Isometric Cylinders $V^n \\times \\mathbb{R} \\to W^q$ for $q \\geq (n + 2)(n + 3)/2$......Page 260
3.1.9 Non-free Isometric Maps......Page 264
3.2 Isometric Immersions in Low Codimension......Page 269
3.2.1 Parabolic Immersions......Page 270
3.2.2 Hyperbolic Immersions......Page 279
3.2.3 Geometric Obstructions to Isometric $C^2$-Immersions $V^2 \\to \\mathbb{R}^3$......Page 289
3.2.4 Isometric $C^\\infty$-Immersions $V^2 \\to \\mathbb{R}^q$ for $3 \\leq q leq 6......Page 299
3.3 Isometric $C^\\infty$-Immersions of Pseudo-Riemannian Manifolds......Page 316
3.3.1 Local Pseudo-Riemannian Immersions......Page 317
3.3.2 Global Immersions......Page 322
3.3.3 Immersions with a Prescribed Curvature and the $C^1$-Approximation......Page 326
3.3.4 Isotropic Maps and Non-unique Isometric Immersions......Page 331
3.3.5 Isometric $C^\\infty$-Immersions $V^n \\to W^q$ for $q \\geq [n(n + 3)/2]+2$......Page 334
3.4 Symplectic Isometric Immersions......Page 337
3.4.1 Immersions of Exterior Forms......Page 338
3.4.2 Symplectic Immersions and Embeddings......Page 343
3.4.3 Contact Manifolds and Their Immersions......Page 348
3.4.4 Basic Problems in the Symplectic Geometry......Page 350
References......Page 360
Author Index......Page 369
Subject Index......Page 371