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Ilyas Khan: Lectures

June 7, 2020 (Jointly with Alec Payne)
TITLE: Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G2-Laplacian Flow

ABSTRACT: The Laplacian flow is a natural geometric flow which deforms closed G2-structures on 7-manifolds. This flow could be a major avenue to insight into manifolds with G2-holonomy. However as with many geometric flows, singularities are expected to form in finite time. Self-similar soliton solutions to the flow are expected to play a significant role in the analysis of these singularities. In this talk, we consider self-shrinking solitons (these are necessarily noncompact) with prescribed asymptotics on their ends. In particular, we consider the important class of asymptotically conical (AC) shrinkers, the first examples of which were recently constructed by Haskins-Nordstrom. We describe our proof of the uniqueness of AC gradient shrinking solitons for the Laplacian flow of closed G2-structures: If two gradient shrinking solitons to Laplacian flow are asymptotic to the same closed G2-cone, then their G2-structures are equivalent, and in particular, the two solitons are isometric. This is joint work of Mark Haskins, Ilyas Khan and Alec Payne.

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