Curvature pinching estimate under the Laplacian G_{2} flow

TL;DR

This paper establishes a curvature pinching estimate for the Laplacian G_{2} flow, linking traceless Ricci curvature, scalar curvature, and Weyl tensor norms, and explores implications for flow long-term behavior.

Contribution

It introduces a new pinching estimate relating key curvature quantities under the Laplacian G_{2} flow, aiding in understanding flow singularities and long-term existence.

Findings

Traceless Ricci curvature is bounded in terms of scalar curvature and Weyl tensor norm.

The C^{1} norm of the Weyl tensor must blow up at a certain rate if scalar curvature remains bounded.

The estimate helps analyze the long-time existence of the Laplacian G_{2} flow.

Abstract

In this paper, we derive a pinching estimate on the traceless Ricci curvature in term of scalar curvature and the C^{1} norm of the Weyl tensor under the Laplacian G_{2} flow for closed G_{2} structures. Then we apply this estimate to study the long time existence of the Laplacian G_{2} flow and prove that the C^{1} norm of the Weyl tensor has to blow up at least at a certain rate under bounded scalar curvature.