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 Λ n−2 . In particular, 2-convexity is not assumed. Next we show the surgery flow with just the initial convexity assumption H − x,ν 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 Λ n−2 are isotopic to the round n-sphere.

Multiplicity one for min-max theory in compact manifolds with boundary and its applications

“…Based on this fact, there has been much research on low entropy mean curvature flow; cf. [5,6,43]. In contrast, our theorem implies that large entropy mean curvature flow may have very complicated tangent flows, which addresses the complication of mean curvature flow singularities.…”