Maximum principle and convergence of fundamental solutions for the Ricci flow

TL;DR

This paper establishes a maximum principle for linear parabolic equations on evolving non-compact manifolds and proves the convergence and uniqueness of fundamental solutions for the conjugate heat equation under Ricci flow, clarifying key aspects used in Ricci flow analysis.

Contribution

It provides new maximum principles, convergence results, and detailed proofs of fundamental solutions for the conjugate heat equation in Ricci flow, including results previously used without proof by Perelman.

Findings

Convergence of Neumann Green functions to fundamental solutions.

Uniqueness of fundamental solutions under exponential decay.

Detailed proof of convergence of fundamental solutions in Ricci flow sequences.

Abstract

In this paper we will prove a maximum principle for the solutions of linear parabolic equation on complete non-compact manifolds with a time varying metric. We will prove the convergence of the Neumann Green function of the conjugate heat equation for the Ricci flow in $B_k\times (0,T)$ to the minimal fundamental solution of the conjugate heat equation as $k\to\infty$. We will prove the uniqueness of the fundamental solution under some exponential decay assumption on the fundamental solution. We will also give a detail proof of the convergence of the fundamental solutions of the conjugate heat equation for a sequence of pointed Ricci flow $(M_k\times (-\alpha,0],x_k,g_k)$ to the fundamental solution of the limit manifold as $k\to\infty$ which was used without proof by Perelman in his proof of the pseudolocality theorem for Ricci flow.