Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness

Abstract

We develop foundational theory for the Laplacian flow for closed G2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound on Λ(x,t)=(|∇T(x,t)|g(t)2+|Rm(x,t)|g(t)2)12will imply bounds on all covariant derivatives of Rm and T. (2). We show that Λ (x, t) will blow up at a finite-time singularity, so the flow will exist as long as Λ (x, t) remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.