Complete cohomogeneity one solitons for Bryant’s closed G2-Laplacian flow. Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics. September 14, 2020
We will report briefly on some of our recent results, numerical investigations (and resulting conjectures) and work still-in-progress on complete noncompact cohomogeneity one solitons in Bryant’s closed G2-Laplacian flow (G2-solitons for short). Our results include the construction of new complete noncompact (asymptotically conical) shrinking and expanding (gradient) G2-solitons on the total space of the bundle of anti-self dual 2-forms on S^4 and CP^2; we conjecture the existence of a 1-parameter family of asymptotically conical steady (gradient) G2-solitons on the total space of the bundle of anti-self dual 2-forms of CP^2 and describe some analytic and numerical evidence supporting this conjecture. In other more widely studied geometric flows (e.g. Ricci flow, codimension 1 mean curvature flow and Lagrangian mean curvature flow) solitons are key to understanding singularity formation and for attempting to continue the flow after the first singular time. We discuss very briefly how our results reveal both some similarities and differences with the behaviour of solitons currently known in Ricci flow and Kahler-Ricci flow.