- 09/10/2023: Closed G2-Structures with Negatively Pinched Ricci Curvature
- 06/07/2022: Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G2-Laplacian Flow
September 10, 2023
TITLE: Closed G2-Structures with Negatively Pinched Ricci Curvature
ABSTRACT: The Goldberg conjecture is a well-known question about whether compact almost Kähler, Einstein manifolds must be Kähler. The analogue of the Goldberg conjecture for G_2-geometry asks whether an Einstein closed G_2-structure must be torsion-free, and this was proven for compact manifolds by Cleyton-Ivanov and Bryant. In this talk, we discuss how this result extends to noncompact manifolds. In general, we find that there is no closed G_2-structure on a noncompact manifold whose metric is complete and has sufficiently negatively pinched Ricci curvature.
June 7, 2022 (Jointly with Ilyas Khan)
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.