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Uniqueness of Ricci flow with scaling invariant estimates

Man-Chun Lee

Abstract

In this work, we prove uniqueness for complete non-compact Ricci flow with scaling invariant curvature bound. This generalizes the earlier work of Chen-Zhu, Kotschwar and covers most of the example of Ricci flows with unbounded curvature. In dimension three, we use it to show that complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique, extending the strong uniqueness Theorem of Chen. This is based on solving Ricci-harmonic map heat flow in unbounded curvature background.

Uniqueness of Ricci flow with scaling invariant estimates

Abstract

In this work, we prove uniqueness for complete non-compact Ricci flow with scaling invariant curvature bound. This generalizes the earlier work of Chen-Zhu, Kotschwar and covers most of the example of Ricci flows with unbounded curvature. In dimension three, we use it to show that complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique, extending the strong uniqueness Theorem of Chen. This is based on solving Ricci-harmonic map heat flow in unbounded curvature background.

Paper Structure

This paper contains 9 sections, 15 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Some Preliminaries on Ricci flows
  3. Distance distortion under $c/t$
  4. Ricci-harmonic map heat flow
  5. Existence in bounded curvature
  6. Ricci-DeTurck flow and a-priori estimates
  7. Localized Maximum Principle
  8. Existence of Ricci-harmonic map heat flow
  9. Uniqueness of Ricci flows

Key Result

Theorem 1.1

Suppose $(M,g_0)$ is a complete non-compact manifold. If $g(t)$ and $\tilde{g}(t)$ are complete solution to Ricci flow on $M\times [0,T]$ such that $g(0)=\tilde{g}(0)=g_0$ and on $M\times (0,T]$ for some ${\alpha}>0$, then $g(t)\equiv \tilde{g}(t)$ on $M\times [0,T]$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Corollary 1.1
  • Lemma 2.1: Corollary 3.3 in SimonTopping2023
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.1
  • proof
  • Theorem 2.1
  • ...and 26 more