Heat kernel estimates on connected sums of parabolic manifolds

TL;DR

This paper derives precise two-sided estimates for the heat kernel on connected sums of parabolic manifolds, showing that the on-diagonal behavior depends on the end with maximal volume growth, with explicit examples provided.

Contribution

It provides the first matching two-sided heat kernel estimates for connected sums of parabolic manifolds satisfying Li-Yau estimates, highlighting the influence of volume growth on heat kernel behavior.

Findings

On-diagonal heat kernel bounds depend on maximal volume growth end.

Explicit bounds are given for connected sums like R^2#R^2 and R^1#R^2.

The results extend heat kernel estimates to parabolic manifold connected sums.

Abstract

We obtain matching two sided estimates of the heat kernel on a connected sum of parabolic manifolds, each of them satisfying the Li-Yau estimate. The key result is the on-diagonal upper bound of the heat kernel at a central point. Contrary to the nonparabolic case (which was settled in [15]), the on-diagonal behavior of the heat kernel in our case is determined by the end with the maximal volume growth function. As examples, we give explicit heat kernel bounds on the connected sums $R^2#R^2$ and $R^1#R^2$ where $R^1 = R_+\timesS^1$.