Let $M$ be a compact Riemannian manifold and let $V \in C^\infty(M)$. Consider the operator $\Delta + V$ and let $p_t(x, y)$ be the corresponding heat kernel. If $\Delta + V$ is a positive operator such that the first eigenvalue is larger than $\varepsilon >0$, we know that the heat kernel satisfies the estimate $$ p_t(x, y) \leq C e^{-t\varepsilon}$$ for some constant $C>0$ and all $t>1$.

Question: Can I make the constant $C$ depend only on the volume and the dimension of $M$? That is, does there exist a constant $C = C(n, v)$ such that for all $n$-dimensional Riemannian manifolds $M$ with fixed volume $v$ and all functions $V \in C^\infty(M)$ such that $\Delta+V$ is bounded below by $\varepsilon>0$, the corresponding heat kernel satisfies the above bound?

Note that without restriction on the volume, this cannot work, since the heat kernel for $V=0$ converges to the function $1/\sqrt{v}$, where $v$ is the volume of $M$.

\Edit: if the answer to the question is no, what are "good" (necessary or sufficient) geometric assumptions that assure such a bound? In other words, what other explicit geometric data should $C$ depend on?

  • $\begingroup$ Intuitively, I would expect not; if the manifold is very "narrow" near a point then you would expect heat to move away more slowly, and your conditions don't control that. For a possible counterexample, what if you consider a flat torus which is the quotient of $[0,n] \times [0,1/n]$, and let's say $V=1$? I would be surprised if you can uniformly bound $p_1(x,y)$ over all such tori. $\endgroup$ Nov 3, 2016 at 12:56
  • 2
    $\begingroup$ In the case $V=0$ the answer is certainly "no," the long-time decay of the heat kernel should go as $e^{-\lambda_1 t}$ where $\lambda_1$ is the first nonzero eigenvalue. An estimate with the Cheeger constant should make @NateEldredge's intuition precise. I am not so familiar with cases when there's a potential, but I imagine there are Cheeger-type estimates one could employ (the underlying intuition is to correctly choose a test function that's orthogonal to constants and then use min-max and the coarea formula). $\endgroup$
    – Neal
    Nov 3, 2016 at 13:21
  • 2
    $\begingroup$ But if $M$ is compact, then there should not be any decay at all, right? In fact, it should converge to the constant $1/\sqrt(v)$, where $v$ is the volume. $\endgroup$ Nov 3, 2016 at 13:25
  • 1
    $\begingroup$ However, I don't want to bound $\lambda_1$ (suppose that we already have such a bound), instead I would like to conclude a bound on the heat kernel from such a bound on the first eigenvalue. $\endgroup$ Nov 3, 2016 at 13:55
  • 1
    $\begingroup$ Have you looked at the paper of Carlen,Kusuoka and Stroock in Annales Institut Henri Poincare Probabilites et Statistiques 23 no S2 1987 245-287 $\endgroup$ Nov 3, 2016 at 20:30

1 Answer 1


The paper of Carlen , Kusuoka and Stroock relates Nash inequality to diagonal bounds for the Heat Kernel for both short times and all time.They also explain a method of E B Davies to convert diagonal bounds of the heat kernel to off diagonal bounds.The paper is in Annales Institut Henri Poincare Probabilites et Statistiques 23 no S2 1987 pages 245-287


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

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