Pinching estimates for solutions of the linearized Ricci flow system in higher dimensions

TL;DR

This paper establishes pinching estimates for solutions to the linearized Ricci flow in higher dimensions, extending known 3D results to dimensions four and above under specific curvature conditions.

Contribution

It generalizes pinching estimates for the linearized Ricci flow to higher dimensions with positive scalar curvature and vanishing Weyl tensor, broadening the scope of previous 3D results.

Findings

Pinching estimates hold in higher dimensions with Weyl tensor vanishing.

Rough pinching estimates are provided without the Weyl tensor condition.

Results extend 3D Ricci flow estimates to higher-dimensional manifolds.

Abstract

We prove pinching estimates for solutions of the linearized Ricci flow system on a closed manifold of dimension $n\geq 4$ with positive scalar curvature and vanishing Weyl tensor. If the vanishing Weyl tensor condition is removed, we only give a rough pinching estimate controlled by some blow-up function in a short time. These results extend the $3$-dimensional case due to Anderson and Chow (2005) \cite{[AC]}.