Let $M$ be a compact Riemannian manifold, and $K(t;z,w)$ the heat kernel associated to the usual Laplace-Beltrami operator on functions. There are results of the form
$$K(t;z,z) \leq \frac{C_M}{f_z(t)}.$$
Where the constant $C_M$ is implicitly dependent on the manifold $M$ and $f_z(t)$ is an explicit function in the time, such as $$f_z(t) = \text{vol}(B_{\sqrt{t}}(z)).$$
Question Are bounds that explicitly depend on the geometry of the manifold (in the form of injectivity radius, Ricci curvature, volume and so on) known? And bounds of the form above that also have the "right" large time behavior?
Edit: Setting $\alpha=1$ and $\epsilon = \frac{1}{2}$ in Corollary 3.1 of "On the parabolic kernel of the Schrödinger operator" (Li, Yau), gives the notable bound: $$K(t;z,z) \leq \frac{c_{\text{dim}(M)}}{\text{vol}(B_{\sqrt{t}}(z))}e^{d_{\text{dim}(M)} K t}.$$ Where $M$ is a complete manifold without boundary with Ricci curvature bounded from below by $-K$, and $c_{\text{dim}(M)}, d_{\text{dim}(M)}$ are universal constants only dependent on the dimension of $M$. While this is a bound of the required form it is very unsharp in the situation: $K\neq 0$, $\text{vol}(M) < \infty$, and $t$ large.
Moreover the article "Sharp explicit lower bounds of heat kernels" (Wang), treats in paragraph $5$ the case of compact manifolds (and many other cases in the preceding paragraph), but it seems to be only treating lower bounds.
