Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness

TL;DR

This paper establishes fundamental estimates, uniqueness, and compactness results for the Laplacian flow of closed G_2 structures, providing key tools for future geometric analysis and understanding singularities and solitons.

Contribution

It introduces Shi-type derivative estimates, proves uniqueness results, and develops a compactness theorem for the Laplacian flow of closed G_2 structures, advancing the theoretical framework.

Findings

Shi-type derivative estimates for curvature and torsion tensors

Finite-time blow-up of the flow at singularities

Existence of compact soliton solutions

Abstract

We develop foundational theory for the Laplacian flow for closed G_2 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 $\Lambda(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$ will imply bounds on all covariant derivatives of Rm and T. (2). We show that $\Lambda(x,t)$ will blow up at a finite-time singularity, so the flow will exist as long as $\Lambda(x,t)$ 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). (5). Finally, we study compact soliton solutions of the Laplacian flow.