Suppose M is a smooth manifold, and we have two Riemannian metrics on M, say g and h, with g bigger than h (i.e. for every tangent vector at every point, the norm according to g is bigger than the norm according to h). Suppose $H_g$ and $H_h$ are the heat kernels on (M, g) and (M, h) respectively. Does ${\frac{H_g(x, y, t)} {H_g(x, x, t)} \leq \frac {H_h(x, y, t)}{H_h(x, x, t)}}$ necessarily hold for all x, y, and t?

This intuitively makes sense: if you have something diffusing on M, starting at the point x, it should diffuse "faster" with the metric h because distances are smaller, and so after time t it seems you should have a greater proportion of the diffusing substance at y compared to at x. On $\mathbb{R^n}$ this is trivially true, also I've tested it out numerically for a variety of metrics on the circle and it seems to hold. But a proof eludes me.

While I am curious about the general case, I'm especially interested when the manifolds in question are quotients of $\mathbb{R^n}$ by a lattice, with the corresponding flat metrics, because an affirmative answer in this special case can be used to solve this problem.

Here is a graph of the heat kernel for a range of translation-invariant metrics on the circle, scaled to 1 at the origin:

As you can see, heat kernels for smaller metrics are everywhere above those for larger ones, so this hypothesis numerically seems to hold here. Also, I haven't displayed them here but I've also tried a few non-translation-invariant metrics without having found a counterexample.