Partial Differential Relations (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 9)

The classical theory of partial differential equations is rooted in physics, where equations (are assumed to) describe the laws of nature. Law abiding functions, which satisfy such an equation, are very rare in the space of all admissible functions (regardless of a particular topology in a function space). Moreover, some additional (like initial or boundary) conditions often insure the uniqueness of solutions. The existence of these is usually established with some apriori estimates which locate a possible solution in a given function space. We deal in this book with a completely different class of partial differential equations (and more general relations) which arise in differential geometry rather than in physics. Our equations are, for the most part, undetermined (or, at least, behave like those) and their solutions are rather dense in spaces of functions. We solve and classify solutions of these equations by means of direct (and not so direct) geometric constructions. Our exposition is elementary and the proofs of the basic results are selfcontained. However, there is a number of examples and exercises (of variable difficulty), where the treatment of a particular equation requires a certain knowledge of pertinent facts in the surrounding field. The techniques we employ, though quite general, do not cover all geometrically interesting equations. The border of the unexplored territory is marked by a number of open questions throughout the book. 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 Pt. 1. A Survey Of Basic Problems And Results. Solvability And The Homotopy Principle -- Homotopy And Approximation -- Singularities And Non-singular Maps -- Localization And Extension Of Solutions -- Pt. 2. Methods To Prove The H-principle. Removal Of Singularities -- Continuous Sheaves -- Inversion Of Differential Operators -- Convex Integration -- Pt. 3. Isometric C{592}-immersions. Isometric Immersions Of Riemannian Manifolds -- Isometric Immersions In Low Codimension -- Isometric C{592}-immersions Of Pseudo-riemannian Manifolds -- Symplectic Isometric Immersions. Mikhael Gromov. Includes Index. Bibliography: P. [350]-357.