A Sharp Lower Bound for the Entropy of Closed Hypersurfaces up to Dimension Six

TL;DR

This paper proves that for dimensions 2 to 6, the round sphere uniquely minimizes entropy among all closed smooth hypersurfaces, confirming a conjecture and extending results to singular self-shrinkers using weak mean curvature flow.

Contribution

It confirms the conjecture that spheres minimize entropy among closed hypersurfaces up to dimension six and extends the result to include singular self-shrinkers, using a novel weak mean curvature flow approach.

Findings

Round sphere uniquely minimizes entropy for 2 ≤ n ≤ 6.

Extension of entropy minimization to compact singular self-shrinkers.

Provides a new proof of the main result in prior work.

Abstract

In [5], Colding-Ilmanen-Minicozzi-White showed that within the class of closed smooth self-shrinkers in $\mathbb{R}^{n+1}$, the entropy is uniquely minimized at the round sphere. They conjectured that, for $2\leq n\leq 6$, the round sphere minimizes the entropy among all closed smooth hypersurfaces. Using an appropriate weak mean curvature flow, we prove their conjecture. For these dimensions, our approach also gives a new proof of the main result of [5] and extends its conclusions to compact singular self-shrinkers.