On the stability for Alexandrov's Soap Bubble theorem

TL;DR

This paper investigates how close a hypersurface with nearly constant mean curvature is to a perfect sphere, providing quantitative stability estimates that improve upon previous results and apply to related overdetermined problems.

Contribution

It introduces new stability estimates for Alexandrov's Soap Bubble theorem, quantifying the hypersurface's proximity to a sphere based on mean curvature deviations, with improved bounds over prior work.

Findings

Hypersurfaces with near-constant mean curvature are contained in a small spherical annulus.

The stability estimate depends on the dimension, with specific exponents for N=2,3 and N≥4.

The results extend to certain overdetermined problems, enhancing understanding of geometric rigidity.

Abstract

Alexandrov's Soap Bubble theorem dates back to $1958$ and states that a compact embedded hypersurface in $\mathbb{R}^N$ with constant mean curvature must be a sphere. For its proof, A.D. Alexandrov invented his reflection priciple. In $1982$, R. Reilly gave an alternative proof, based on integral identities and inequalities, connected with the torsional rigidity of a bar. In this article we study the stability of the spherical symmetry: the question is how much a hypersurface is near to a sphere, when its mean curvature is near to a constant in some norm. We present a stability estimate that states that a compact hypersurface $\Gamma\subset\mathbb{R}^N$ can be contained in a spherical annulus whose interior and exterior radii, say $\rho_i$ and $\rho_e$, satisfy the inequality $$ \rho_e - \rho_i \le C \Vert H - H_0 \Vert^{\tau_N}_{L^1 (\Gamma)}, $$ where $\tau_N=1/2$ if $N=2, 3$, and $\tau_N=1/(N+2)$ if $N\ge 4$. Here, $H$ is the mean curvature of $\Gamma$, $H_0$ is some reference constant and $C$ is a constant that depends on some geometrical and spectral parameters associated with $\Gamma$. This estimate improves previous results in the literature under various aspects. We also present similar estimates for some related overdetermined problems.