Mean curvature flow with generic low-entropy initial data

TL;DR

This paper demonstrates that low-entropy closed hypersurfaces in certain dimensions can be perturbed to ensure their mean curvature flow develops only spherical and cylindrical singularities, using a novel density drop approach.

Contribution

It introduces a new density drop technique to analyze mean curvature flow of low-entropy hypersurfaces, extending previous entropy bounds and providing a new proof of regularity results.

Findings

Low-entropy hypersurfaces can be perturbed to have only spherical and cylindrical singularities.

The method applies to all closed surfaces in R^3 with entropy ≤ 2.

A new proof of regularity for area-minimizing hypersurfaces in eight dimensions.

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 $\leq 2$ and to all closed hypersurfaces in $\mathbb{R}^4$ with entropy $\leq \lambda(\mathbb{S}^1 \times \mathbb{R}^2)$. 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).