Neck Detection for Two-Convex Hypersurfaces Embedded in Euclidean Space undergoing Brendle-Huisken G-Flow

TL;DR

This paper develops a method for detecting neck regions in two-convex hypersurfaces evolving under Brendle-Huisken G-flow, extending surgery techniques from mean curvature flow to this new nonlinear flow setting.

Contribution

It adapts gradient estimates and neck detection techniques to the G-flow, enabling classification of two-convex hypersurfaces under this flow.

Findings

Neck detection method for G-flow established

Extension of surgery algorithm to G-flow

Classification results for two-convex hypersurfaces

Abstract

Recently Brendle-Huisken introduced a fully nonlinear flow $G$. Their aim was to extend the surgery algorithm of Huisken-Sinestrari, into the Riemannian setting. The aim of this paper is to go through the details on how to perform neck detection for a closed, embedded hypersurface $M_0$ in $\mathbb{R}^{n+1}$ undergoing this $G$-flow. In order to do this we make some adjustments to Brendle and Huiskens gradient estimate, after we have done this we can go on to argue as in Huisken-Sinestrari's paper Mean curvature flow with surgeries of two-convex hyperusrfaces, in order to classify two-convex surfaces undergoing $G$-flow.