Curvature pinching estimate under the Laplacian G2 flow

Abstract

In this paper, we derive a pinching estimate for the traceless Ricci curvature in terms 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.