A sharp quantitative version of Alexandrov's theorem via the method of moving planes

TL;DR

This paper provides a precise quantitative version of Alexandrov's Soap Bubble Theorem, showing that hypersurfaces with nearly constant mean curvature are close to spheres, with optimal bounds on their deviation.

Contribution

It introduces a sharp, quantitative stability estimate for Alexandrov's theorem using the method of moving planes, extending the classical result to near-spherical surfaces.

Findings

Establishes a bound on the difference of radii for hypersurfaces with small mean curvature oscillation.

Proves that such hypersurfaces are diffeomorphic to spheres.

Shows the hypersurfaces are $C^1$-close to a sphere.

Abstract

We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let $S$ be a $C^2$ closed embedded hypersurface of $\mathbb{R}^{n+1}$, $n\geq1$, and denote by $osc(H)$ the oscillation of its mean curvature. We prove that there exists a positive $\varepsilon$, depending on $n$ and upper bounds on the area and the $C^2$-regularity of $S$, such that if $osc(H) \leq \varepsilon$ then there exist two concentric balls $B_{r_i}$ and $B_{r_e}$ such that $S \subset \overline{B}_{r_e} \setminus B_{r_i}$ and $r_e -r_i \leq C \, osc(H)$, with $C$ depending only on $n$ and upper bounds on the surface area of $S$ and the $C^2$ regularity of $S$. Our approach is based on a quantitative study of the method of moving planes and the quantitative estimate on $r_e-r_i$ we obtain is optimal. As a consequence of this theorem, we also prove that if $osc(H)$ is small then $S$ is diffeomorphic to a sphere and give a quantitative bound which implies that $S$ is $C^1$-close to a sphere.