Heat kernel estimates on Harnack manifolds and beyond

Abstract

On a Riemannian manifold, $M$, the heat kernel is a smooth function on $(0,+\infty )\times M\times M$, $(t,x,y)\mapsto p(t,x,y)$, and the shape of this function depends on the properties of $M$. This article pays particular attention to the long-time, large-scale behavior of the heat kernel and its relation to the global geometry of $M$. When does the heat kernel look like a bell curve? If it does not, what does it look like and why? To answer such questions, one needs tools to obtain sharp two-sided estimates for the heat kernel in terms of the time variable $t>0$ and basic geometric quantities depending on $x,y\in M$. Under what assumptions on $M$, can one hope to obtain such bounds?