Alec Payne (NCSU): Closed G2-Structures with Negative Ricci Curvature

In this talk, we survey G2-geometry and discuss recent rigidity results for closed G2-structures. Closed G2-structures are cross product structures on 7-manifolds that are analogous to symplectic structures. Every known construction of a compact manifold with G2-holonomy begins with finding a suitable closed G2-structure and perturbing it to be Ricci-flat. The goal of this talk is to present recent results on the rigidity of closed G2-structures subject to Ricci curvature restrictions. Firstly, we show that no compact manifold admits a closed G2-structure with negative Ricci curvature. This is surprising since negative Ricci curvature is a very weak curvature condition. Secondly, we show that there is no closed G2-structure on a noncompact manifold whose metric is complete and has sufficiently negatively pinched Ricci curvature. This settles the so-called G2-Goldberg conjecture in the noncompact setting, generalizing work of Cleyton-Ivanov and Bryant in the compact setting.