Hamilton-Jacobi theory and heat kernels for Grushin operators

Abstract

In this thesis, we study a class of Grushin operators ∆k=∂2/∂x2 + x2k ∂2/∂y2 ; k 2 N+; which are subelliptic operators. We obtain all the geodesics of the subRimannian geometry induced by ∆k. For any two points in the y-axis, there are infinitely many geodesics connecting them; while for any other two points, there are finitely many geodesic connections. The y-axis is the canonical submanifold whose tangent space recovers the missing direction. We generalize Beals-Gaveau-Greiner's complex Hamilton- Jacobi theory to general step subelliptic operators and apply this to the Grushin operators ∆k. We construct the modified action function f and obtain an integral representation of the heat kernel for step 3 Grushin operator (i.e., ∆2). By computing the critical points of f, the small time aymptotics of the heat kernel is obtained using the method of steepest descent, it has a close relation with the geodesics induced by the operator ∆2. For higher step cases, the modified action functions are also given. In the final part, we give some open problems and future work in this direction.