Low Entropy and the Mean Curvature Flow with Surgery

TL;DR

This paper extends the mean curvature flow with surgery to certain low-entropy, mean convex hypersurfaces without requiring 2-convexity, and shows these flows lead to topological classification of self-shrinkers.

Contribution

It introduces a new extension of the mean curvature flow with surgery for low-entropy hypersurfaces without 2-convexity assumptions.

Findings

Flow with surgery applies to hypersurfaces with entropy less than Λ_{n-2}

Smooth self-shrinkers with low entropy are topologically spheres

Flow preserves certain convexity conditions during evolution

Abstract

In this article, we extend the mean curvature flow with surgery to mean convex hypersurfaces with entropy less than $\Lambda_{n-2}$. In particular, 2-convexity is not assumed. Next we show the surgery flow with just the initial convexity assumption $H - \frac{\langle x, \nu \rangle}{2} > 0$ is possible and as an application we use the surgery flow to show that smooth $n$-dimensional closed self shrinkers with entropy less than $\Lambda_{n-2}$ are isotopic to the round $n$-sphere.