Convex Integration Theory

§1. Historical Remarks Convex Integration theory, first introduced by M. Gromov [17], is one of three general methods in immersion-theoretic topology for solving a broad range of problems in geometry and topology. The other methods are: (i) Removal of Singularities, introduced by M. Gromov and Y. Eliashberg [8]; (ii) the covering homotopy method which, following M. Gromov's thesis [16], is also referred to as the method of sheaves. The covering homotopy method is due originally to S. Smale [36] who proved a crucial covering homotopy result in order to solve the classification problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succes­ sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Conse­ quently, each method has a range of applications to problems in topology that are best suited to its particular insight. For example, a distinguishing feature of Convex Integration theory is that it applies to solve closed relations in jet spaces, including certain general classes of underdetermined non-linear systems of par­ tial differential equations. As a case of interest, the Nash-Kuiper Cl-isometrie immersion theorem ean be reformulated and proved using Convex Integration theory (cf. Gromov [18]). No such results on closed relations in jet spaees can be proved by means of the other two methods.

1 Introduction. - §1 Historical Remarks. - §2 Background Material. - §3 h-Principles. - §4 The Approximation Problem. - 2 Convex Hulls. - §1 Contractible Spaces of Surrounding Loops. - §2 C-Structures for Relations in Affine Bundles. - §3 The Integral Representation Theorem. - 3 Analytic Theory. - §1 The One-Dimensional Theorem. - §2 The C? -Approximation Theorem. - 4 Open Ample Relations in Spaces of 1-Jets. - §1 C°-Dense h-Principle. - §2 Examples. - 5 Microfibrations. - §1 Introduction. - §2 C-Structures for Relations over Affine Bundles. - §3 The C? -Approximation Theorem. - 6 The Geometry of Jet spaces. - §1 The Manifold X? . - §2 Principal Decompositions in Jet Spaces. - 7 Convex Hull Extensions. - §1 The Microfibration Property. - §2 The h-Stability Theorem. - 8 Ample Relations. - §1 Short Sections. - §2 h-Principle for Ample Relations. - §3 Examples. - §4 Relative h-Principles. - 9 Systems of Partial Differential Equations. - §1 Underdetermined Systems. - §2 Triangular Systems. - §3 C1-Isometric Immersions. - 10 Relaxation Theorem. - §1 Filippov s Relaxation Theorem. - §2 C? -Relaxation Theorem. - References. - Index of Notation.