Convex Integration Theory

1. Front Matter

   Pages i-viii

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2. Introduction

   • David Spring

   Pages 1-18

3. Convex Hulls

   • David Spring

   Pages 19-32

4. Analytic Theory

   • David Spring

   Pages 33-48

5. Open Ample Relations in 1-Jet Spaces

   • David Spring

   Pages 49-69

6. Microfibrations

   • David Spring

   Pages 71-86

7. The Geometry of Jet Spaces

   • David Spring

   Pages 87-99

8. Convex Hull Extensions

   • David Spring

   Pages 101-120

9. Ample Relations

   • David Spring

   Pages 121-164

10. Systems of Partial Differential Equations

    • David Spring

    Pages 165-199

11. Relaxation Theory

    • David Spring

    Pages 201-206

12. Back Matter

    Pages 207-213

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§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.

• Differential topology
• Manifold
• Topology
• differential geometry
• equation
• function
• geometry
• theorem

"Spring's book makes no attempt to include all topics from convex integration theory or to uncover all of the gems in Gromov's fundamental account, but it will nonetheless (or precisely for that reason) take its place as a standard reference for the theory next to Gromov's towering monograph and should prove indispensable for anyone wishing to learn about the theory in a more systematic way."

--- Mathematical Reviews

• Department of Mathematics, Glendon College, Toronto, Canada

  David Spring

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