Let $M$ be a closed Riemannian manifold, and let $h(t, x, y)$ denote the heat kernel on $M$. We know that there exists *short time* upper Gaussian heat kernel bounds of the following kind:
$$
h(t, x, y) \leq Ct^{-n/2}e^{-\frac{c\text{ dist}(x, y)^2}{t}}.
$$

My question is, is it known how the constant $c$ on the exponent behaves with respect to curvature conditions on $M$? For example, do we have, let's say, estimates of the following kind: if $g_1, g_2$ are two metrics on $M$, and $\text{Ric}$ denotes Ricci curvature, then $\text{Ric}_1 \leq \text{Ric}_2$, then $h_1(t, x, y) \leq h_2(t, x, y)$? I am guessing that if the volume grows faster, the heat kernel will decay faster (as then the "heat" will have more "area" to cover). It seems intuitive that such things should be true, but I would really appreciate a reference if they are actually true.

Edit: Now I have realized that the previous version of my question is a little vague, after parsing through a bit of Fabrice Baudoin's blog, and the comments and answers below. **Now, I would like to ask the following specific question: in addition to short time, if we are also looking at near-diagonal ($x$ and $y$ are close), then what happens to the constant $C$?** Are there estimates in terms of curvature? From this MO post, it is clear that one can take $c = 1/4$.

sectionalcurvatureupperbound, but for times smaller than the injectivity radius (and this is the best you can hope for). $\endgroup$ – Raziel Mar 18 at 11:23