H-principle

In mathematics, 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...?

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Rough idea

Assume we want to find a function [itex]f[itex] on Rm which satisfies a partial differential equation of degree [itex]k[itex], in co-ordinates [itex](u_1,u_2,...,u_m)[itex]. One can rewrite it as

[itex]\Psi(u_1,u_2,...,u_m, J^k_f)=0\!\,[itex]

where [itex]J^k_f[itex] stands for all partial derivatives of [itex]f[itex] up to order [itex]k[itex]. Let us exchange every variable in [itex]J^k_f[itex] for new independent variables [itex]y_1,y_2,...,y_N[itex]. Then our original equation can be thought as a system of

[itex]\Psi^{}_{}(u_1,u_2,...u_m,y_1,y_2,...y_N)=0\!\,[itex]

and some number of equations of the following type

[itex]y_j={\partial y_i\over \partial u_k}.\!\,[itex]

A solution for

[itex]\Psi^{}_{}(u_1,u_2,...u_m,y_1,y_2,...y_N)=0\!\,[itex]

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 [itex]x[itex] and [itex]y[itex] for the location (best choice is the location of mid point of back wheels), and an angle [itex]\alpha[itex] which describes the orientation of the car. The motion of the car satisfies the equation

[itex]\dot x \sin\alpha=\dot y\cos \alpha.\,[itex]

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 [itex]C^0[itex]-dense h-principle.

Ways to prove the h-principle

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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 [itex]f_t[itex] such that [itex]f_0=f[itex], [itex]f_1=-f[itex] and for any [itex]t[itex] we have that [itex]\operatorname{grad}(f_t)[itex] 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 [itex]C^1[itex] isometric embedding of [itex]S^2[itex].

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