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. ------
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