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 at most 2 and to all closed hypersurfaces in ${\mathbb{R}}^4$ with entropy at most $\mathit{\lambda }({\mathbb{S}}^1\times {\mathbb{R}}^2)$. When combined with recent work of Daniels and Holgate, this strengthens Bernstein and Wang’s low-entropy Schoenflies-type theorem by relaxing the entropy bound to $\mathit{\lambda }({\mathbb{S}}^1\times {\mathbb{R}}^2)$. 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).