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Existence and applications of Ricci flows via pseudolocality

Fei He

Abstract

We prove the short-time existence of Ricci flows on complete manifolds with scalar curvature bounded below uniformly, Ricci curvature bounded below by a negative quadratic function, and with almost Euclidean isoperimetric inequality holds locally. In particular, this result applies to manifolds with both Ricci curvature and injectivity radius bounded from below. We also study the short-time behaviour of these solutions which may have unbounded curvature at the initial time, and provide some applications. A key tool is Perelman's pseudolocality theorem.

Existence and applications of Ricci flows via pseudolocality

Abstract

We prove the short-time existence of Ricci flows on complete manifolds with scalar curvature bounded below uniformly, Ricci curvature bounded below by a negative quadratic function, and with almost Euclidean isoperimetric inequality holds locally. In particular, this result applies to manifolds with both Ricci curvature and injectivity radius bounded from below. We also study the short-time behaviour of these solutions which may have unbounded curvature at the initial time, and provide some applications. A key tool is Perelman's pseudolocality theorem.

Paper Structure

This paper contains 4 sections, 20 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Proof of existence
  3. Short-time analysis of the solutions
  4. Applications

Key Result

Theorem 1.1

For every $n$ and $A > 0$, there exist $\delta_0 >0$ and $\epsilon_0 >0$ depending only on $A$ and $n$ with the following property: Suppose $(M^n, g(t)), t\in [0, (\epsilon r)^2]$ is a complete solution of the Ricci flow with bounded curvature, where $0< \epsilon \leq \epsilon_0$ and $r>0$. Let $x_0 and if the $\delta_0-$almost Euclidean isoperimetric inequality holds in $B_{g(0)}(x_0,r)$ with res

Theorems & Definitions (36)

  • Theorem 1.1: Perelman's pseudolocality
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2: Schoen-Yau MR1333601
  • ...and 26 more