Estimating the trace-free Ricci tensor in Ricci flow

TL;DR

This paper investigates conditions under which the trace-free Ricci tensor can be controlled during Ricci flow, revealing that it depends on other curvature components without initial data assumptions.

Contribution

It establishes a new relationship showing the trace-free Ricci tensor is controlled by other curvature components in Ricci flow on compact manifolds, without initial data constraints.

Findings

Trace-free Ricci tensor controlled by other curvature components

Control holds without initial data hypotheses

Provides insight into curvature behavior during Ricci flow

Abstract

An important and natural question in the analysis of Ricci flow singularity formation in dimensions four and above is as follows: What are the weakest conditions that provide control of the norm of the Riemann curvature tensor? In this short note, we show that on a compact manifold, the trace-free Ricci tensor is controlled in a precise fashion by the other components of the irreducible decomposition of the curvature tensor, without any hypotheses on the initial data.