Heat kernels and regularity for rough metrics on smooth manifolds

TL;DR

This paper establishes the existence and regularity of heat kernels for rough metrics on smooth manifolds, showing they are globally continuous and locally Hölder continuous, using advanced parabolic estimates.

Contribution

It introduces methods to prove the existence and regularity of heat kernels for rough metrics, extending classical results to less regular geometric settings.

Findings

Globally continuous heat kernels exist for rough metrics.

Heat kernels are Hölder continuous in space and time.

The approach uses local parabolic Harnack estimates in weighted Sobolev spaces.

Abstract

We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are H\"older continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.