Gaussian upper heat kernel bounds on closed Riemannian manifolds 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 $c = 1/4$. 
 A: This is related to Li-Yau gradient estimates. I think the usual set up is for a complete manifold with $\textrm{Ric}(M)>-k$. You probably need some Harnark type inequalities for parabolic equations. If you can do this, I think you can also get comparison formulas on the spectral gap associated to the manifold (there is recent work by Xiangjin Xu on this). But all of above things may be very technical, and I am not sure how does it answer your original question. 
One elementary thing that may or may not be helpful is that you may expand the trace of the heat kernel locally using the curvature tensors. If the inequalities you suggest does exist, then it would extend over to the heat trace. So this gives some zeroth approximation whether your suggestion is true (if I recall correctly, the first term involves the scalar curvature, etc).  
A: I think that the closest statement to an answer is contained in the famous paper:
Li, Peter; Yau, Shing Tung, On the parabolic kernel of the Schrödinger operator, Acta Math. 156, 154-201 (1986). ZBL0611.58045.
In particular, in Corollary 3.1, they prove the following:

The estimate is global in time, and the constants are explicit in the proof. Furthermore, $V$ represents the volume of balls of radius $\sqrt{t}$. If you want to make them even more explicit in terms of $t$, you need to add an upper bound on the sectional curvature in order to bound from below the volume of balls with standard comparison theorems, at least for sufficiently small times.
