On the stability for Alexandrov’s soap bubble theorem.(English)Zbl 1472.53013

Summary: 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 principle. In [Indiana Univ. Math. J. 26, 459–472 (1977; Zbl 0391.53019)], R. C. 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 $$\Gamma \subset \mathbb{R}^N$$ can be contained in a spherical annulus whose interior and exterior radii, say $$\rho_i$$ and $$\rho_e$$, satisfy the inequality $\rho_e - \rho_i \leq C \| H - H_0\|_{L^1(\Gamma)}^{\tau_N},$ where $$\tau_N = 1/2$$ if $$N = 2, 3$$, and $$\tau_N = 1/(N + 2)$$ if $$N \geq 4$$. 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.

MSC:

 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 49Q10 Optimization of shapes other than minimal surfaces

Zbl 0391.53019
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