The homotopy principle (h-principle) is a very general way to solve partial differential equations PDE (and more generally partial differential relations PDR). The h-principle is good for underdetermined PDE or PDR such as immersion problem, isometric immersions problem and so on.

The theory was started by works of Eliashberg , Gromov and Phillips and was based on earlier results of Hirsch , Kuiper , Nash, Smale...?

Rough idea

Assume we want to find a function f on R^m which satisfies a partial differential equation of degree k, in co-ordinates (u₁,u₂,...,u_m). One can rewrite it as

where stands for all partial derivatives of f up to order k. Let us exchange every variable in for new independent variables y₁,y₂,...,y_N. Then our original equation can be thought as a system of

and some number of equations of the following type

A solution for

is called a non-holomorphic solution, and a solution for the system (which is a solution of our original PDE) is called a holomorphic solution. In order to check if a solution exists, first check if there is a non-holomorphic solution (usually it is quite easy and if not then our original equation did not have any solutions). A PDE satisfies the h-principle if any non-holomorphic solution can be deformed into a holomorphic one in the class of non-holomorphic solutions.

Therefore, once you prove that an equation satisfies h-principle it is really easy to check whether it has solutions. It is surprising that most underdetermined partial differential equations satisfy the h-principle.

The simplest example

A position of a car on the plane is determined by three parameters: two coordinates x and y for the location (best choice is the location of mid point of back wheels), and an angle α which describes the orientation of the car. The motion of the car satisfies the equation

A non-holomorphic solution in this case roughly speaking corresponds to a motion of a car by sliding on the plane. In this case the non-holomorphic solutions are not only homotopic to 'holonomic' ones but also can be arbitrarily well approximated by the holomorphic ones (by going back and forth, like parallel parking in a limited space). This last property is stronger than the general h-principle: it is the so called C⁰-dense h-principle.

Ways to prove the h-principle

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Some paradoxes

Here we list few paradoxical results which can be proved by applying the h-principle:

1. Let us consider functions f on R² without origin f(x)=|x|. Then there is a continuous one parameter family of functions f_t such that f₀ = f, f₁ = - f and for any t we have that grad(f_t) is not zero at any point.

2. Any open manifold admits a (non-complete) Riemannian metric of positive (or negative) curvature.

3. Smale's paradox can be done using C¹ isometric embedding of S².

Related theorems

• Smale's paradox
• Nash embedding theorem