On the stability for Alexandrov's Soap Bubble Theorem

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. par 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. par We present a stability estimate that states that a compact hypersurface $GasubsetRR^N$ can be contained in a spherical annulus whose interior and exterior radii, say $ ho_i$ and $ ho_e$, satisfy the inequality $$ ho_e - ho_i le C r H - H_0 r^{ au_N}_{L^1 (Ga)}, $$ where $ au_N=1/2$ if $N=2, 3$, and $ au_N=1/(N+2)$ if $Nge 4$. Here, $H$ is the mean curvature of $Ga$, $H_0$ is some reference constant and $C$ is a constant that depends on some geometrical and spectral parameters associated with $Ga$. This estimate improves previous results in the literature under various aspects. par We also present similar estimates for some related overdetermined problems.