Harnack Estimates for Ricci Flow on a Warped Product

TL;DR

This paper develops Harnack estimates for Ricci flow on warped product manifolds with Ricci-flat fibers, extending Perelman's differential Harnack inequality using monotone formulas.

Contribution

It introduces new Harnack estimates for Ricci flow on warped products with Ricci-flat fibers, generalizing previous results.

Findings

Derived analog of Perelman's differential Harnack inequality

Established estimates for heat conjugate fundamental solutions

Extended monotone formula techniques to warped product settings

Abstract

In this paper, we study the Ricci flow on closed manifolds equipped with warped product metric $(N\times F,g_{N}+f^2 g_{F})$ with $(F,g_{F})$ Ricci flat. Using the framework of monotone formulas, we derive several estimates for the adapted heat conjugate fundamental solution which include an analog of G. Perelman's differential Harnack inequality.