Mean Value Inequalities and Conditions to Extend Ricci Flow

TL;DR

This paper explores conditions for extending Ricci flow solutions on closed manifolds, focusing on mean value inequalities and their relation to time slice analysis, with applications to manifolds with nonnegative isotropic curvature.

Contribution

It introduces a systematic approach to mean value inequalities in Ricci flow and links this method to time slice analysis, providing new inequalities for specific curvature conditions.

Findings

Established conditions for Ricci flow extension using mean value inequalities

Connected mean value inequalities with time slice analysis techniques

Derived inequalities for Ricci flow on manifolds with nonnegative isotropic curvature

Abstract

This paper concerns conditions related to the first finite singularity time of a Ricci flow solution on a closed manifold. In particular, we provide a systematic approach to the mean value inequality method, suggested by N. Le and F. He. We also display a close connection between this method and time slice analysis of B. Wang. As an application, we prove several inequalities for a Ricci flow solution on a manifold with nonnegative isotropic curvature.