# Convex Integration Theory: Solutions to the h-principle in geometry and topology

Springer Science & Business Media, 2 dec. 2010 - 213 pagina's
§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.

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### Inhoudsopgave

 CHAPTER 1 INTRODUCTION 1 CHAPTER 2 CONVEX HULLS 19 CHAPTER 3 ANALYTIC THEORY 33 CHAPTER 4 OPEN AMPLE RELATIONS IN 1JET SPACES 49 CHAPTER 5 MICROFIBRATIONS 70 CHAPTER 6 THE GEOMETRY OF JET SPACES 87 CHAPTER 7 CONVEX HULL EXTENSIONS 101
 CHAPTER 8 AMPLE RELATIONS 121 CHAPTER 9 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 165 CHAPTER 10 RELAXATION THEORY 200 References 207 INDEX 210 INDEX OF NOTATION 212 Copyright

### Over de auteur (2010)

David Spring is a Professor of mathematics at the Glendon College in Toronto, Canada.