On the Mean Curvature Evolution of Two-Convex Hypersurfaces

Abstract

We study the mean curvature evolution of smooth, closed, two-convex hypersurfaces in $\mathbb{R}^{n+1}$ for $n \ge 3$. Within this framework we effect a reconciliation between the flow with surgeries—recently constructed by Huisken and Sinestrari in G. Huisken & C. Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces,—and the wellknown weak solution of the level-set flow: we prove that the two solutions agree in an appropriate limit of the surgery parameters and in a precise quantitative sense. Our proof relies on geometric estimates for certain $L^p$-norms of the mean curvature which are of independent interest even in the setting of classicalmean curvature flow. We additionally show how our construction can be used to pass these estimates to limits and produce regularity results for the weak solution.