ON THE UPPER ESTIMATE OF THE HEAT KERNEL OF A COMPLETE RIEMANNIAN MANIFOLD

Let M be a complete non-compact Riemannian manifold whose sectional curvature is bounded between two constants -k and K. Then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in Euclidean space. The purpose of this paper is to give a justification of such a statement. In [5], J. Cheeger and the third author found a lower estimate of the heat kernel by "comparing" it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold. This lower estimate is sharp if we insist the dependence should be on the lower bound of the Ricci curvature alone. It remains to give an upper estimate of the heat kernel. One does not, however, expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold. In fact, the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior. This is reflected by the fact that the Laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case. What we will prove here is that in any case, the heat kernel has to decay in a manner similar to the Euclidean heat kernel. Thus we will prove that for any constant C > 4, there exists C1 depending on C, T, the bound of the curvature of M and x so that for all t E [0, T]