Heat kernel estimates on manifolds with ends with mixed boundary condition

TL;DR

This paper establishes two-sided heat kernel estimates on manifolds with ends and mixed boundary conditions, extending prior work to include Dirichlet conditions using harmonic functions and $h$-transform techniques.

Contribution

It extends existing heat kernel estimates to manifolds with mixed boundary conditions, incorporating Dirichlet boundaries through a novel harmonic function construction.

Findings

Two-sided heat kernel estimates are obtained for manifolds with ends and mixed boundary conditions.

The results generalize previous estimates by including Dirichlet boundary conditions.

A new method using harmonic functions and $h$-transform is developed for the proof.

Abstract

We obtain two-sided heat kernel estimates for Riemannian manifolds with ends with mixed boundary condition, provided that the heat kernels for the ends are well understood. These results extend previous results of Grigor'yan and Saloff-Coste by allowing for Dirichlet boundary condition. The proof requires the construction of a global harmonic function which is then used in the $h$-transform technique.