Long time existence and bounded scalar curvature in the Ricci-harmonic flow

TL;DR

This paper investigates the long-term existence of the Ricci-harmonic flow by analyzing scalar curvature and Weyl tensor, extending previous Ricci flow results, and also examines integral bounds of Riemann curvature in four dimensions.

Contribution

It extends Cao's results on Ricci flow to Ricci-harmonic flow and generalizes Simon's work on Riemann curvature bounds in four dimensions.

Findings

Established conditions for long-time existence based on scalar curvature and Weyl tensor.

Extended integral bounds of Riemann curvature to Ricci-harmonic flow in four dimensions.

Generalized previous results from Ricci flow to Ricci-harmonic flow.

Abstract

In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao's result \cite{Cao2011} in the Ricci flow. In dimension four, we also study the integral bound of the "Riemann curvature" for the Ricci-harmonic flow generalizing a recently result of Simon \cite{Simon2015}.