# Long-time decay of heat kernel on compact manifolds

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?

• 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. Nov 3, 2016 at 12:56
• 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).
– Neal
Nov 3, 2016 at 13:21
• 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. Nov 3, 2016 at 13:25
• 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. Nov 3, 2016 at 13:55
• 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 Nov 3, 2016 at 20:30