Local curvature estimates for the Laplacian flow

Abstract

In this paper we give local curvature estimates for the Laplacian flow on closed G_2-structures under the condition that the Ricci curvature is bounded along the flow. The main ingredient consists of the idea of Kotschwar-Munteanu-Wang who gave local curvature estimates for the Ricci flow on complete manifolds and then provided a new elementary proof of Sesum's result, and the particular structure of the Laplacian flow on closed G_2-structures. As an immediate consequence, this estimates give a new proof of Lotay-Wei's result which is an analogue of Sesum's theorem. The second result is about an interesting evolution equation for the scalar curvature of the Laplacian flow of closed G_2-structures. Roughly speaking, we can prove that the time derivative of the scalar curvature R_t is equal to the Laplacian of R_t, plus an extra term which can be written as the difference of two nonnegative quantities.