# Heat kernel upper bounds on a complete Riemannian manifold

Let $M$ be a complete Riemannian manifold, and $p(t, x, y)$ denotes its heat kernel. I am trying to find sufficient conditions for when the following holds: $$p(t, x, y) \leq Ct^{-n/2}, \forall x, y, t > 0.$$

In particular, I am interested in the following question: does lower Ricci bounds imply the above heat kernel bounds?

• If I understand this correctly, the bounds given there work for $x, y \in M$ and for $t \in (0, T]$, and the constant $C$ depends at least on $T$. Is there a hope of obtaining such bounds for a uniform $C$ independent of $T$? – user91989 Jul 23 '16 at 16:20
• In fact, I read somewhere that for nonpositively curved spaces, one can get a uniform constant $C$ that works for all $T$. – user91989 Jul 23 '16 at 16:30
• Indeed, for nonnegative Ricci-curvature the heat kernel behaves like $Ct^{-n/2}\exp(-c d(x,y)^2/t)$. See Li, Yau. On the parabolic kernel of the Schrödinger operator. – MaoWao Aug 5 '16 at 11:52