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Geometric and Functional Analysis

, Volume 27, Issue 1, pp 165–233 | Cite as

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

  • Jason D. LotayEmail author
  • Yong Wei
Open Access
Article

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
$$\Lambda(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$$
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) 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.

Keywords and phrases

Laplacian flow G2 structure Shi-type estimates Uniqueness Compactness Soliton 

Mathematics Subject Classification

53C44 53C25 53C10 

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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