Hausdorff Stability of the Round Two-Sphere Under Small Perturbations of the Entropy

TL;DR

This paper proves that closed surfaces in three-dimensional space with entropy close to that of a perfect sphere are geometrically close to a round sphere, establishing stability under small entropy perturbations.

Contribution

It demonstrates Hausdorff stability of the round two-sphere under small entropy deviations, connecting geometric closeness with entropy measures.

Findings

Surfaces with entropy near the sphere's entropy are Hausdorff close to a sphere.

Quantitative bounds relate entropy difference to geometric proximity.

Supports stability analysis in geometric flows and shape optimization.

Abstract

We show that if a closed surface in $\mathbb{R}^3$ has entropy near to that of the unit two-sphere, then the surface is close to a round two-sphere in the Hausdorff distance.