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 a broadly accessible exposition of the principle and its applications. The essence ofthe $h$-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 $C $-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.