Chemistry - H-principle

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...?

Contents

1 Rough idea

2 The simplest example

3 Ways to prove the h-principle

4 Some paradoxes

5 Related theorems

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 $(u 1, u 2,..., u m)$ . One can rewrite it as

$\Psi(u_1,u_2,...,u_m, J^k_f)=0\!\,$

where $J^k_f$ stands for all partial derivatives of $f$ up to order $k$ . Let us exchange every variable in $J^k_f$ for new independent variables $y 1, y 2,..., y N$ . Then our original equation can be thought as a system of

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

and some number of equations of the following type

$y_j={\partial y_i\over \partial u_k}.\!\,$

A solution for

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

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

$\dot x \sin\alpha=\dot y\cos \alpha.$

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 $C 0$ -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 $f 0 = f$ , $f 1 = - f$ 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 $C 1$ isometric embedding of $S 2$ .

Related theorems

Categories: Partial differential equations