Introduction to the H-principle
Yakov Eliashberg, N. Mishachev - Collection Graduate Studies in Mathematics
Résumé
One of the most powerful modern methods of solving partial differential equations is Gromov's h-principle. It has also been, traditionally, one of the most difficult to explain. This book is the first broadly accessible exposition of the principle and its applications. The essence of the h-principle is the reduction of problems involving partial differential relations to problems of a purely homotopy-theoretic nature. Two famous examples of the h-principle are the Nash-Kuiper $C1$-isometric embedding theory in Riemannian geometry and the Smale-Hirsch immersion theory in differential topology. Gromov transformed these examples into a powerful general method for proving the h-principle. Both of these examples and their explanations in terms of the h-principle are covered in detail in the book. The authors cover two main embodiments of the principle: holonomic approximation and convex integration. The first is a version of the method of continuous sheaves. The reader will find that, with a few notable exceptions, most instances of the h-principle can be treated by the methods considered here. There are, naturally, many connections to symplectic and contact geometry. The book would be an excellent text for a graduate course on modern methods for solving partial differential equations. Geometers and analysts will also find much value in this very readable exposition of an important and remarkable technique.
Sommaire
- Intrigue
- Thom transversality theorem
- Holonomic approximation
- Applications
- Homotopy principle
- Open Diff V-invariant differential relations
- Applications to closed manifolds
- Symplectic and contact structures on open manifolds
- Symplectic and contact structures on closed manifolds
- Embeddings into symplectic and contact manifolds
- Microflexibility and holonomic mathcal{R}-approximation
- First applications of microflexibility
- Microflexible mathfrak{U}-invariant differential relations
- Further applications to symplectic geometry
- Homotopy principle for ample differential relations
- Directed immersions and embeddings
- First order linear differential operators
- Nash-Kuiper theorem
Caractéristiques techniques
PAPIER | |
Éditeur(s) | American Mathematical Society (AMS) |
Auteur(s) | Yakov Eliashberg, N. Mishachev |
Collection | Graduate Studies in Mathematics |
Parution | 08/08/2002 |
Nb. de pages | 198 |
Format | 18 x 26 |
Couverture | Relié |
Poids | 620g |
Intérieur | Noir et Blanc |
EAN13 | 9780821832271 |
ISBN13 | 978-0-821-83227-1 |
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