Low Entropy and the Mean Curvature Flow with Surgery

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 $H-\frac{⟨x,\nu ⟩}{2}>0$ 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.

Revised after referee's report. A number of small errors fixed and minor parts of the argument were altered