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 ℝN with constant mean curvature must be a sphere. For its proof, A. D. Alexandrov invented his reflection principle. In 1977, 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 near is a hypersurface 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 Γ ⊂ ℝN can be contained in a spherical annulus whose interior and exterior radii, say ρi and ρe, satisfy the inequality $$\rho_e - \rho_i \leq C \parallel{H - H_0}\parallel_{L^1(\Gamma)}^{\tau_N},$$ where τN = 1/2 if N = 2, 3, and τN = 1/(N + 2) if N ≥ 4. Here, H is the mean curvature of Γ, H0 is some reference constant, and C is a constant that depends on some geometrical and spectral parameters associated with Γ. This estimate improves previous results in the literature under various aspects. We also present similar estimates for some related overdetermined problems.