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 et al. (J Funct Anal 271(9):2604–2630, 2016) who gave local curvature estimates for the Ricci flow on complete manifolds and then provided a new elementary proof of Sesum’s result (Sesum in Am J Math 127(6):1315–1324, 2005), 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 and Wei’s (Geom Funct Anal 27(1):165–233, 2017) 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_{g(t)} is equal to the Laplacian of R_{g(t)}, plus an extra term which can be written as the difference of two nonnegative quantities.