A local curvature estimate for the Ricci-harmonic flow on complete Riemannian manifolds

TL;DR

This paper establishes local $L^p$ estimates for the Riemannian curvature under Ricci-harmonic flow on complete noncompact manifolds, leading to conditions for flow extension beyond finite time.

Contribution

It provides new local curvature estimates for Ricci-harmonic flow and demonstrates flow extendibility under bounded Ricci curvature and finite time interval.

Findings

Bounded $L^p$ norm of Riemannian curvature under flow

Local boundedness of curvature via De Giorgi-Nash-Moser iteration

Flow can be extended past finite time if Ricci curvature is bounded

Abstract

In this paper we consider the local $L^p$ estimate of Riemannian curvature for the Ricci-harmonic flow or List's flow introduced by List \cite{List2005} on complete noncompact manifolds. As an application, under the assumption that the flow exists on a finite time interval $[0,T)$ and the Ricci curvature is uniformly bounded, we prove that the $L^p$ norm of Riemannian curvature is bounded, and then, applying the De Giorgi-Nash-Moser iteration method, obtain the local boundedness of Riemannian curvature and consequently the flow can be continuously extended past $T$.