Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows

TL;DR

This paper develops dimension-free Harnack inequalities for conjugate heat equations on evolving Riemannian manifolds, providing new tools for analyzing heat kernels and functional inequalities under geometric flows.

Contribution

It introduces a probabilistic derivative formula for conjugate semigroups on evolving manifolds and derives dimension-free Harnack inequalities with applications to heat kernel bounds.

Findings

Established a dimension-free Harnack inequality for conjugate heat equations.

Derived heat kernel upper bounds for moving metrics.

Obtained log-Sobolev inequalities via supercontractivity of the semigroup.

Abstract

Let $M$ be a differentiable manifold endowed with a family of complete Riemannian metrics $g(t)$ evolving under a geometric flow over the time interval $[0,T[$. In this article, we give a probabilistic representation for the derivative of the corresponding conjugate semigroup on $M$ which is generated by a Schr\"{o}dinger type operator. With the help of this derivative formula, we derive fundamental Harnack type inequalities in the setting of evolving Riemannian manifolds. In particular, we establish a dimension-free Harnack inequality and show how it can be used to achieve heat kernel upper bounds in the setting of moving metrics. Moreover, by means of the supercontractivity of the conjugate semigroup, we obtain a family of canonical log-Sobolev inequalities. We discuss and apply these results both in the case of the so-called modified Ricci flow and in the case of general geometric flows.