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?

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