Topology of Closed Hypersurfaces of Small Entropy

TL;DR

This paper proves that all closed hypersurfaces in four-dimensional space with entropy below a certain threshold are topologically equivalent to a three-sphere, using a weak mean curvature flow and surgery techniques.

Contribution

It establishes a topological classification for low-entropy closed hypersurfaces in our dimensions, extending understanding of their geometric structure.

Findings

All such hypersurfaces are diffeomorphic to or our-dimensional space.

The entropy threshold is the same as that of the round cylinder or our-dimensional space.

The method involves a weak mean curvature flow with surgery.

Abstract

We use a weak mean curvature flow together with a surgery procedure to show that all closed hypersurfaces in $\mathbb{R}^4$ with entropy less than or equal to that of $\mathbb{S}^2\times \mathbb{R}$, the round cylinder in $\mathbb{R}^4$, are diffeomorphic to $\mathbb{S}^3$.