Stochastic proof of upper bound for the heat kernel coupled with geometric flow, and Ricci flow

TL;DR

This paper provides a new proof of Gaussian upper bounds for the heat kernel coupled with Ricci flow, using horizontal coupling methods to generalize Harnack inequalities and derive on-diagonal bounds.

Contribution

It introduces a novel proof technique employing horizontal coupling to establish heat kernel bounds under Ricci flow, extending previous methods.

Findings

Established Gaussian upper bounds for heat kernels under Ricci flow.

Derived on-diagonal heat kernel bounds along Ricci flow.

Generalized Harnack inequalities for inhomogeneous heat equations.

Abstract

We give a proof of Gaussian upper bound for the heat kernel coupled with the Ricci ow. Previous proofs by Lei Ni [5] use Harnack inequality and doubling volume property, also the recent proof by Zhang and Cao [6] uses Sobolev type inequality that is conserved along Ricci ow. We will use a horizontal coupling of curve [1] Arnaudon Thalmaier, C., in order to generalize Harnack inequality with power -for inhomogeneous heat equation - introduced by F.Y Wang. In the case of Ricci ow, we will derive on-diagonal bound of the Heat kernel along Ricci ow (and also for the usual Heat kernel on complete Manifold).