Introduction to the h-principle / Y. Eliashberg, N. Mishachev.

Format:

Book

Author/Creator:

Eliashberg, Y., 1946- author.

Mishachev, N. (Nikolai M.), 1952- author.

Series:

Graduate studies in mathematics ; v. 48.

Graduate studies in mathematics, 1065-7339 ; volume 48

Language:

English

Subjects (All):

Geometry, Differential.

Differentiable manifolds.

Differential equations--Numerical solutions.

Differential equations.

Physical Description:

1 online resource (198 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2002]

Language Note:

English

Summary:

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 arecovered 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 solvingpartial differential equations. Geometers and analysts will also find much value in this very readable exposition of an important and remarkable technique.

Contents:

""Symplectic and contact structures on closed manifolds""""Embeddings into symplectic and contact manifolds""; ""Microflexibility and holonomic â??-approximation""; ""First applications of microflexibility""; ""Microflexible -invariant differential relations""; ""Further applications to symplectic geometry""; ""Part IV. 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""; ""Back Cover""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 199-202) and index.

Description based on print version record.

ISBN:

1-4704-1796-0