Level-Set and Surgery Approaches to Brendle-Huisken G-Flow

Abstract:

In the 2-Convex setting Huisken-Sinestrari developed a surgery technique for mean curvature flow and were able to obtain classification results. However surgery depends on a non-canonical choice of surgery parameters, in contrast to the unique weak solution to the level-set setting for mean curvature flow derived by Spruck-Evans and Chen-Giga-Goto. Independently Lauer and Head were able to show that as we take our surgery paramaters to be arbitrarily large that these two solutions agree.

The surgery results for mean curvature flow do not extend to the Riemannian setting. In order to tackle this Brendle-Huisken inspired by Andrews work on harmonic mean curvature flow introduced G-flow. We derive viscosity solutions for the G-flow and apply Lauer's argument for a reconciliation between the G-flow with surgeries and the viscosity solution.