Large time behavior of the heat kernel

TL;DR

This paper investigates the long-term behavior of the heat kernel on complete Riemannian manifolds with nonnegative Ricci curvature, extending previous results and providing a counterexample to limit behavior consistency.

Contribution

It introduces a new approach using metric blow-downs and Gromov-Hausdorff limits to analyze heat kernel limits, and constructs the first example of inconsistent heat kernel limits.

Findings

Established the large time limit of heat kernels in a broader context.

Generalized P. Li's earlier results on heat kernel behavior.

Provided the first example of a manifold with inconsistent heat kernel limits.

Abstract

In this paper, we study the large time behavior of the heat kernel on complete Riemannian manifolds with nonnegative Ricci curvature, which was studied by P. Li with additional maximum volume growth assumption. Following Y. Ding's original strategy, by blowing down the metric, using Cheeger and Colding's theory about limit spaces of Gromov-Hausdorff convergence, combining with the Gaussian upper bound of heat kernel on limit spaces, we succeed in reducing the limit behavior of the heat kernel on manifold to the values of heat kernels on tangent cones at infinity of manifold with renormalized measure. As one application, we get the consistent large time limit of heat kernel in more general context, which generalizes the former result of P. Li. Furthermore, by choosing different sequences to blow down the suitable metric, we show the first example manifold whose heat kernel has inconsistent limit behavior, which answers an open question posed by P. Li negatively.