Differential Harnack estimates for conjugate heat equation under the Ricci flow

TL;DR

This paper establishes new localized and global differential Harnack inequalities for positive solutions to the conjugate heat equation under Ricci flow, improving previous results by relaxing curvature conditions and extending to noncompact cases.

Contribution

It introduces generalized Harnack estimates for the conjugate heat equation coupled with Ricci flow, including nonlinear cases with potential, under weaker curvature assumptions.

Findings

Derived localized and global Harnack inequalities for Ricci flow

Extended Li-Yau type estimates to broader settings

Applicable to noncompact manifolds

Abstract

We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$ \Delta - R(x,t)$", where $R$ is the scalar curvature of the Ricci flow, which is well generalised to the case of nonlinear heat equation with potential. Our estimates improve on some well known results by weakening the curvature constraints. As a by product, we obtain some Li-Yau type differential Harnack estimate. The localized version of our estimate is very useful in extending the results obtained to noncampact case.