Fundamental solution of the heat equation on an arbitrary Riemannian manifold

Abstract

We consider upper estimates for the Green function of the heat equation on an arbitrary smooth connected Riemannian manifold M. We define the Green function G(t, x; y) as the limit of the Green functions G~ for the precompact domains ~CM as ~ ~ M. If the manifold M has boundary, then we will always assume the Neumann homogeneous condition to be fulfilled on the boundary and also consider only those ~ for which a~ is transversed to 3M. Let us denote the geodesic distance between two points x, y~M by Ix yJ and the geodesic ball of radius r with center at x by Br x. If N is a submanifold, then we will denote its volume, corresponding to the dimension, by JN I . THEOREM I. Let the following isoperimetric inequality be fulfilled in a precompact geodesic ball Bpx: for each domain Q ~ Box that has smooth boundary 3Q, transversal to aM,