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?



Long-time decay of heat kernel on compact manifolds
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


