Introduction to the H-principle
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.
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Thom Transversality Theorem
Differential Relations and Gromovs iPrinciple
Open Diff VInvariant Differential Relations
The Homotopy Principle in Symplectic Geometry
Embeddings into Symplectic and Contact Manifolds
Microflexibility and Holonomic KApproximation
First Applications of Microflexibility
Further Applications to Symplectic Geometry
OneDimensional Convex Integration
Homotopy Principle for Ample Differential Relations
Directed Immersions and Embeddings
First Order Linear Differential Operators
Symplectic and Contact Structures on Open Manifolds
Symplectic and Contact Structures on Closed Manifolds
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