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Geometry and Topology Seminar: Alec Payne, Duke University, The Structure of Shrinking Solutions to G2-Laplacian Flow

November 30, 2022 | 11:00 am - 12:00 pm EST

In this talk, we will survey G2-structures, which are cross product structures on 7-manifolds, and we will discuss recent developments on a natural geometric flow of G2-structures called Laplacian flow. The Laplacian flow was introduced by Robert Bryant as a tool to explore the geometry of G2-structures on 7-manifolds and to construct examples of G2-holonomy manifolds. Manifolds with G2-holonomy are particularly special since these are the only known source of nonflat Ricci-flat metrics on compact, simply-connected manifolds of odd dimension. As with many geometric flows, the most important feature of the flow is its singularities, and self-similarly shrinking solutions are expected to play a significant role in the analysis of singularities of Laplacian flow. We will focus on the important class of asymptotically conical (AC) self-similarly shrinking solutions for Laplacian flow, the first examples of which were constructed recently by Haskins-Nordstrom. We will 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 with Mark Haskins and Ilyas Khan.

SAS Hall 4201

Hybrid with Zoom:
Meeting ID: 915 3371 7619
Passcode: 976581


November 30, 2022
11:00 am - 12:00 pm EST
Event Category:


SAS 4201