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A note on Ricci flow from small curvature concentration and a Morrey-type condition

Albert Chau, Adam Martens

Abstract

In \cite{ChauMartens} the authors proved the long-time existence of Ricci flow starting from complete bounded curvature Riemannian manifolds with scale-invariant integral curvature bounded by a dimensional constant times the inverse of the Sobolev constant. We generalize this result by replacing the bounded curvature assumption with the assumption that $g$ is only equivalent to a complete bounded curvature metric $h$ while satisfying a Morrey-type condition on the gradient of $g$ relative to $h$: a local integral condition on the covariant derivative $\nabla_h g$. The Morrey-type condition was first considered in \cite{LeeLiu} in the context of Ricci flow on non-compact manifolds, and in particular allows the possibility for $g$ to have unbounded curvature on $M$. As in \cite{ChauMartens}, our long-time solution enjoys curvature decay estimates implying in particular that $M$ is diffeomorphic to $\mathbb{R}^n$.

A note on Ricci flow from small curvature concentration and a Morrey-type condition

Abstract

In \cite{ChauMartens} the authors proved the long-time existence of Ricci flow starting from complete bounded curvature Riemannian manifolds with scale-invariant integral curvature bounded by a dimensional constant times the inverse of the Sobolev constant. We generalize this result by replacing the bounded curvature assumption with the assumption that is only equivalent to a complete bounded curvature metric while satisfying a Morrey-type condition on the gradient of relative to : a local integral condition on the covariant derivative . The Morrey-type condition was first considered in \cite{LeeLiu} in the context of Ricci flow on non-compact manifolds, and in particular allows the possibility for to have unbounded curvature on . As in \cite{ChauMartens}, our long-time solution enjoys curvature decay estimates implying in particular that is diffeomorphic to .

Paper Structure

This paper contains 3 sections, 3 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Upper-Continuity of $L^p$ curvature
  3. Proof of Theorem \ref{['thmmain']}

Key Result

Theorem 1

For any $n,A>0$, there exists $C_0(n,A),\sigma(n,A)>0$ such that the following holds. Let $(M^n,g)$ be a complete n-dimensional Riemannian manifold satisfying all the following estimates: Then there exists a complete long-time Ricci flow $g(t)$, $t\in [0,\infty)$ with initial condition $g(0)=g$ which satisfies all the following scaling-invariant estimates on the solution:

Theorems & Definitions (7)

  • Theorem : ChenEricChauMartensChanChenLeeChanHuangLeeMartensScalarLeeLee
  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2.1
  • proof
  • proof : Proof of Theorem \ref{['thmmain']}