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The Ricci-DeTurck flow on complete manifolds

Jing-Bin Cai, Bing Wang

Abstract

Based on the framework of Koch-Lamm and tensor heat kernel estimates, we obtain a uniform proof of the short-time existence, uniqueness, and continuous dependence for Ricci flows starting from a complete Riemannian metric with bounded curvature. A new ingredient is an effective continuous dependence estimate without the assumption of injectivity radius lower bound.

The Ricci-DeTurck flow on complete manifolds

Abstract

Based on the framework of Koch-Lamm and tensor heat kernel estimates, we obtain a uniform proof of the short-time existence, uniqueness, and continuous dependence for Ricci flows starting from a complete Riemannian metric with bounded curvature. A new ingredient is an effective continuous dependence estimate without the assumption of injectivity radius lower bound.
Paper Structure (6 sections, 14 theorems, 246 equations)

This paper contains 6 sections, 14 theorems, 246 equations.

Table of Contents

  1. Introduction
  2. The Ricci-DeTurck operator
  3. Heat kernel estimates
  4. Short-time existence and uniqueness
  5. Continuous dependence
  6. Proof of the main theorem

Key Result

Theorem 1.1

Let $(M^n, g_0)$ be a complete Riemannian manifold with bounded curvature $|Rm|_{g_0} \leq \Lambda_0$. Then there exist constants $T=T(n,\Lambda_0)$, $\delta=\delta(n,\Lambda_0)$ and $C=C(n,\Lambda_0)$ with the following property.

Theorems & Definitions (28)

  • Theorem 1.1: Main theorem
  • Corollary 1.2
  • proof : Outline of the proof of Theorem \ref{['thm:main']}:
  • Proposition 2.1
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 18 more