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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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2 The simplest example 3 Ways to prove h-principle 4 Some paradoxes 5 Related theorems |
Assume we want to find a function on Rm which satisfy a partial differential equation of degree , in co-ordinates one can rewrite it as
Rough idea
where stands for all partial derivatives of up to order . Let us exchange every variable in for new independent variables
.
Then our original equation can be thought as a system ofand some number of equations of the following type
A solution for
is called a non-holomorphic solution and for the system (which is a solution of our original PDE) holomorphic solution.
In order to check whether a solution exist we first check whether 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.
A position of a car on the plane is determined by three parameters: two coordinates and for the location (best choice is the location of mid point of back wheels), and an angle which describes the orientation of the car. The motion of the car satisfies the equation
The simplest example
A non-holomorphic solution in this case 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 arbitrary well approximated by the holomorphic ones (by going back and forth as for parallel parking in a limited space).
This last property is stronger than the general h-principle: it is called the -dense h-principle.
The following is a list of paradoxical results which can be proved by applying the h-principle:
1. Let us consider functions f on a R2 without origin f(x)=|x|. Then there is continuous one parameter family of functions such that ,
and for any we have that grad 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 isometric embedding of .Ways to prove h-principle
......Some paradoxes