Convex and Two-Convex Hypersurfaces Along Exterior Flows

Abstract

We study both convex and two-convex hypersurfaces embedded in Euclidean space. In 2009 Huisken and Sinestrari were able to classify surfaces undergoing two-convex mean curvature flow using the surgery algorithm Hamilton used for Ricci flow. Recently Head and Lauer were able to independently show that as we take our surgery parameter to infinity, that it will converge to the weak solution of level-set mean curvature flow as studied by Spruck-Evans as well as Chen, Giga and Goto. Ideally one would like to extend the surgery algorithm to the Riemannian setting, however, two-convexity is not preserved for mean curvature flow. Instead we look at Brendle-Huisken G-flow, which does preserve two-convexity. In their paper they were then able to use the Huisken-Sinestrari surgery algorithm for mean curvature flow for the Huisken-Brendle G-flow in both the Euclidean and Riemannian setting. We explicitly develop the details implied by Brendle and Huiskens paper, and address some adjustments which must be made to the arguments from the mean curvature case. In order to do so we make adjustments to the gradient estimate of Brendle and Huisken. We are then required to make some more adjustments to prove the Neck Detection Lemma and the Neck Continuation Theorem. To prove the latter theorem we also need a lower bound for the time needed between two consecutive surgeries which we provide. We then describe an interesting observation for the level-set equation of any extrinsic flow. We discover that the level-set PDE will always contain the infinity Laplacian which has well known unique viscosity solutions which have recently been shown by Evans and Smart to be everywhere differentiable. In the last section of the Appendix we prove a reconciliation result between the flow with surgeries and the viscosity solution, however this depends on explicitly obtaining a viscosity solution for G-flow.