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?


This was proved by S. Y. Cheng, P. Li and S. T. Yau in the paper "On the Upper Estimate of the Heat Kernel of a Complete Riemannian Manifold" when the sectional curvature is bounded between two constants. See here.

  • $\begingroup$ 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 $T$? $\endgroup$ – user91989 Jul 23 '16 at 16:20
  • $\begingroup$ In fact, I read somewhere that for nonpositively curved spaces, one can get a uniform constant $C$ that works for all $T$. $\endgroup$ – user91989 Jul 23 '16 at 16:30
  • 1
    $\begingroup$ 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. $\endgroup$ – MaoWao Aug 5 '16 at 11:52

I think what you need is Theorem 5.5.6 of the book Davies, E. B. Heat kernels and spectral theory. Cambridge Tracts in Mathematics, 92.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.