# Gaussian upper bounds on Heat kernel

Let $M$ be a complete Riemannian manifold with Ricci curvature bounded below and let $k(t, x, y)$ be the heat kernel of the Laplace-Beltrami operator.

It seems to be standard that for any compact set $K \subset M \times M$ contained in a small enough neighborhood of the diagonal and any $t_0, \varepsilon>0$, one has the bound $$k(t, x, y) \leq C(t_0, \varepsilon, K) \, t^{-n/2} \exp\Bigl( -\frac{1}{(4+\varepsilon)t}d(x, y)^2 \Bigr),$$ uniformly for $(x, y) \in K$ and $0<t\leq t_0$. Here $n = \dim(M)$

In the paper [1] below, the author (Thomas H. Parker) claims to proof the same result above for $\varepsilon = 0$ (Theorem B), which would be certainly great to have. He also says that the result has been established before by methods of stochastic analysis and gives a reference to the paper [2], but I couldn't find that result there$\,^*$.

The problem with paper [1] is, plainly, that the proof is quite long and technical and therefore hard to verify; furthermore there are many things that are somewhat wrong or unaccurate already in the theorem (Thm. A and B are both not exactly true as stated), which does not increase trust in the results.

So my question is if this is a known result. Or probably even known to be wrong? It seems that if it was true, it would be in one of the many papers and surves of Grigor'yan, I would thing... Are there other references then the two given by me?

$\,^*$ The problem with paper [2] is that the author (Molchanov) only uses the sign $\sim$ to notify asymptotics, which can mean different things, and never says what he really means by that sign

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\Edit: I found this result for $\varepsilon = 0$ for compact $M$ in Hsu (Stochastic analysis on manifolds), but with the "wrong" exponent of $t$ in front, namely with $-n + 1/2$ instead of $-n/2$.

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