On the stability for Alexandrov's Soap Bubble theorem

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 $${\rho }_e-{\rho }_i\le C\Vert H-H_0{\Vert }_{L^1(\Gamma )}^{{\tau }_N},$$ 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.

20 pages, dedicated to prof. Shigeru Sakaguchi on the occasion of his $60^{th}$ birthday. The paper has been accepted by Journal d'Analyse Mathématiques. This amended version incorporates referee's suggestions