Synopses & Reviews
This book provides a comprehensive study of convex integration theory in immersion-theoretic topology. Convex integration theory, developed originally by M. Gromov, provides general topological methods for solving the h-principle for a wide variety of problems in differential geometry and topology, with applications also to PDE theory and to optimal control theory. Though topological in nature, the theory is based on a precise analytical approximation result for higher order derivatives of functions, proved by M. Gromov. This book is the first to present an exacting record and exposition of all of the basic concepts and technical results of convex integration theory in higher order jet spaces, including the theory of iterated convex hull extensions and the theory of relative h-principles. A second feature of the book is its detailed presentation of applications of the general theory to topics in symplectic topology, divergence free vector fields on 3-manifolds, isometric immersions, totally real embeddings, underdetermined non-linear systems of PDEs, the relaxation theorem in optimal control theory, as well as applications to the traditional immersion-theoretical topics such as immersions, submersions, k-mersions and free maps. The book should prove useful to graduate students and to researchers in topology, PDE theory and optimal control theory who wish to understand the h-principle and how it can be applied to solve problems in their respective disciplines. ------
Synopsis
1. Historical Remarks Convex Integration theory, ?rst 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 classi?cation problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succ- sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Con- 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 ConvexIntegrationtheoryisthatitappliestosolveclosed relationsinjetspaces, including certain general classes of underdetermined non-linear systems of par- 1 tial di?erential equations. As a case of interest, the Nash-Kuiper C -isometric immersion theorem can be reformulated and proved using Convex Integration theory (cf. Gromov 18]). No such results on closed relations in jet spaces can be proved by means of the other two methods. On the other hand, many classical results in immersion-theoretic topology, such as the classi?cation of immersions, are provable by all three methods.
Synopsis
This is a comprehensive study of convex integration theory in immersion-theoretic topology, providing methods for solving the h-principle for a variety of problems in differential geometry and topology, with application to PDE theory and optimal control theory.
About the Author
David Spring is a Professor of mathematics at the Glendon College in Toronto, Canada.
Table of Contents
1 Introduction 2 Convex Hulls 3 Analytic Theory 4 Open Ample Relations in Spaces of 1-Jets 5 Microfibrations 6 The Geometry of Jet spaces 7 Convex Hull Extensions 8 Ample Relations 9 Systems of Partial Differential Equations 10 Relaxation Theorem References Index Index of Notation