Let $M$ be a compact Riemannian manifold without boundary, and let $\Delta$ be the Laplace-Beltrami operator on $M$. It is known that for small $t$, let's say, $0< t < t_0(M, g)$, the heat kernel $p(t, x, y)$ satisfies $$ p(t, x, y) \leq C_1 t^{-n/2} e^{\frac{-C_2d(x, y)^2}{t}}. $$

My question is the following. It seems intuitively that if one places certain curvature restrictions on $M$, the constants $C_1, C_2$ could be controlled. Is it known what kind of bounds we can have for $C_1, C_2$ (I am particularly interested in $C_1$) in terms of Ricci or sectional bounds? I looked up some literature, but there is very little I could find on compact manifolds. Any help would be appreciated. Thanks.