Mean curvature flow with generic low-entropy initial data

Abstract

We prove that sufficiently low-entropy closed hypersurfaces can be perturbed so that their mean curvature flow encounters only spherical and cylindrical singularities. Our theorem applies to all closed surfaces in ${\mathbb{R}}^3$ with entropy $\le 2$ and to all closed hypersurfaces in ${\mathbb{R}}^4$ with entropy $\le \lambda ({\mathbb{S}}^1\times {\mathbb{R}}^2)$. When combined with recent work of Daniels-Holgate, this strengthens Bernstein-Wang's low-entropy Schoenflies-type theorem by relaxing the entropy bound to $\lambda ({\mathbb{S}}^1\times {\mathbb{R}}^2)$. Our techniques, based on a novel density drop argument, also lead to a new proof of generic regularity result for area-minimizing hypersurfaces in eight dimensions (due to Hardt-Simon and Smale).

arXiv comment18 pages. Final version, to appear in Duke Mathematical Journal