Heat kernel upper bounds on a complete non-compact manifold

Let M be a smooth connected non-compact geodesically complete Riemannian manifold, ? denote the Laplace operator associated with the Riemannian metric, n = 2 be the dimension of M. Consider the heat equation on the manifoldut - ?u = 0,where u = u(x,t), x I M, t > 0. The heat kernel p(x,y,t) is by definition the smallest positive fundamental solution to the heat equation which exists on any manifold (see [Ch], [D]). The purpose of the present work is to obtain uniform upper bounds of p(x,y,t) which would clarify the behaviour of the heat kernel as t ? +8 and r = dist(x,y) ? +8.