Overview

In differential geometry and topology one often deals with systems of partial differential equations as well as partial differential inequalities that have infinitely many solutions whatever boundary conditions are imposed. It was discovered in the 1950s that the solvability of differential relations (i.e., equations and inequalities) of this kind can often be reduced to a problem of a purely homotopy-theoretic nature. One says in this case that the corresponding differential relation satisfies the $h$-principle. Two famous examples of the $h$-principle, the Nash-Kuiper $C^1$-isometric embedding theory in Riemannian geometry and the Smale-Hirsch immersion theory in differential topology, were later transformed by Gromov into powerful general methods for establishing the $h$-principle. The authors cover two main methods for proving the $h$-principle: holonomic approximation and convex integration. The reader will find that, with a few notable exceptions, most instances of the $h$-principle can be treated by the methods considered here. A special emphasis is made on applications to symplectic and contact geometry. The present book is the first broadly accessible exposition of the theory and its applications, making it an excellent text for a graduate course on geometric methods for solving partial differential equations and inequalities. Geometers, topologists, and analysts will also find much value in this very readable exposition of an important and remarkable topic. This second edition of the book is significantly revised and expanded to almost twice of the original size. The most significant addition to the original book is the new part devoted to the method of wrinkling and its applications. Several other chapters (e.g., on multivalued holonomic approximation and foliations) are either added or completely rewritten.

Full Product Details

Author:   K. Cieliebak ,  Y. Eliashberg ,  N. Mishachev
Publisher:   American Mathematical Society
Imprint:   American Mathematical Society
Edition:   2nd Revised edition
Volume:   239
Weight:   0.421kg
ISBN:  

9781470461058

ISBN 10:   1470461056
Pages:   363
Publication Date:   31 May 2024
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

Intrigue Holonomic approximation: Jets and holonomy Thom transversality theorem Holonomic approximation Applications Multivalued holonomic approximation Differential relations and Gromov's $h$-principle: Differential relations Homotopy principle Open Diff $V$-invariant differential relations Applications to closed manifolds Foliations Singularities and wrinkling: Singularities of smooth maps Wrinkles Wrinkles submersions Folded solutions to differential relations The $h$-principle for sharp wrinkled embeddings Igusa functions The homotopy principle in symplectic geometry: Symplectic and contact basics 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 to microflexibility Microflexible $\mathfrak{A}$-invariant differential relations Further applications to symplectic geometry Convex integration: One-dimensional convex integration Homotopy principle for ample differential relations Directed immersions and embeddings First order linear differential operators Nash-Kuiper theorem Bibliography Index.

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Author Information

K. Cieliebak, University of Augsburg, Germany. Y. Eliashberg, Stanford University, CA. N. Mishachev, Lipetsk Technical University, Russia.

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