Pseudolocality theorems of Ricci flows on incomplete manifolds

TL;DR

This paper establishes pseudolocality theorems for Ricci flows on incomplete manifolds, demonstrating short-time existence under certain conditions and characterizing Euclidean space via isoperimetric inequalities.

Contribution

It proves pseudolocality theorems for Ricci flows on incomplete manifolds and applies these results to show existence and rigidity results.

Findings

Short-time existence of Ricci flows on incomplete manifolds with bounded scalar curvature.

Characterization of Euclidean space via isoperimetric inequalities and nonnegative scalar curvature.

Abstract

In this paper we study the pseudolocality theorems of Ricci flows on incomplete manifolds. We prove that if a ball with its closure contained in an incomplete manifold has the small scalar curvature lower bound and almost Euclidean isoperimetric constant, or almost Euclidean local $\boldsymbol{\nu}$ constant, then we can construct a solution of Ricci flow in the ball which have the pseudolocality property. We also give two applications. First, we prove the short-time existence of Ricci flows on complete manifolds with scalar curvature bounded below uniformly and almost Euclidean isoperimetric inequality holds locally. Second, we show that any complete manifold with nonnegative scalar curvature and Euclidean isoperimetric inequality must be isometric to the Euclidean space.