Convex Integration Theory

Solutions to the h-principle in geometry and topology
1. Auflage 2010
ISBN: 978-3-0348-0059-4
Verlag: Birkhäuser

Buch, Englisch, 213 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 710 g

Reihe: Modern Birkhäuser Classics

§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.

Zielgruppe

Research

Autoren/Hrsg.

Spring, David

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Topologie Analytische Topologie

Inhaltsverzeichnis

Convex Hulls.- Analytic Theory.- Open Ample Relations in 1-Jet Spaces.- Microfibrations.- The Geometry of Jet Spaces.- Convex Hull Extensions.- Ample Relations.- Systems of Partial Differential Equations.- Relaxation Theory.

Über Autor(innen)

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