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
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.
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Acknowledgements
The authors wish to thank Prof. S. Sakaguchi (Tohoku University) for bringing to their attention reference [Re2] and for many fruitful discussions.
Remarks 3.8 (iii) and 4.2 (iii) were suggested by the anonymous referee. The authors warmly thank him/her for the nice improvements to this paper.
The paper was partially supported by a grant iFUND-Azione 2 of the Università di Firenze, under a scientific and cultural agreementwith Tohoku University, and by the GNAMPA (first author) and GNSAGA (second author) of the Istituto Nazionale di Alta Matematica (INdAM).
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To Prof. Shigeru Sakaguchi on the occasion of his 60-th birthday
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Magnanini, R., Poggesi, G. On the stability for Alexandrov’s Soap Bubble theorem. JAMA 139, 179–205 (2019). https://doi.org/10.1007/s11854-019-0058-y
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DOI: https://doi.org/10.1007/s11854-019-0058-y