Curvature tensor under the Ricci-Harmonic flow

TL;DR

This paper proves that under the Ricci-Harmonic flow, a uniform bound on Ricci curvature guarantees a uniform bound on the entire curvature tensor, ensuring controlled geometric evolution.

Contribution

The paper establishes a new curvature estimate linking Ricci bounds to full curvature tensor bounds under the Ricci-Harmonic flow.

Findings

Ricci curvature bounds imply curvature tensor bounds

Uniform bounds are preserved over the flow

Enhances understanding of geometric stability under Ricci-Harmonic flow

Abstract

We prove that if the Ricci curvature is uniformly bounded under the Ricci-Harmonic flow for all times $t$ \in[0, T), then the curvature tensor has to be uniformly bounded as well.