Some results for the Perelman LYH-type inequality

TL;DR

This paper establishes new gradient estimates and inequalities for the conjugate heat equation under Ricci flow on manifolds with boundary, extending Perelman's results without relying on reduced distance properties.

Contribution

It provides a global Li-Yau gradient estimate, offers an alternative proof of Perelman's inequality, and derives various gradient bounds for the conjugate heat equation on manifolds with boundary.

Findings

Proved a global Li-Yau gradient estimate for the conjugate heat equation.

Reproved Perelman's Li-Yau-Hamilton inequality without using reduced distance.

Established gradient estimates for the Dirichlet fundamental solution.

Abstract

Let $(M,g(t))$, $0\le t\le T$, $\partial M\ne\phi$, be a compact $n$-dimensional manifold, $n\ge 2$, with metric $g(t)$ evolving by the Ricci flow such that the second fundamental form of $\partial M$ with respect to the unit outward normal of $\partial M$ is uniformly bounded below on $\partial M\times [0,T]$. We will prove a global Li-Yau gradient estimate for the solution of the generalized conjugate heat equation on $M\times [0,T]$. We will give another proof of Perelman's Li-Yau-Hamilton type inequality for the fundamental solution of the conjugate heat equation on closed manifolds without using the properties of the reduced distance. We will also prove various gradient estimates for the Dirichlet fundamental solution of the conjugate heat equation.