Quantitative stability for hypersurfaces with almost constant curvature in space forms

Giulio Ciraolo; Alberto Roncoroni; Luigi Vezzoni

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arXiv · 1812.00775

Quantitative stability for hypersurfaces with almost constant curvature in space forms

Giulio Ciraolo·Alberto Roncoroni·Luigi Vezzoni

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Abstract

The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures $f(k_1,\ldots,k_{n-1})$ satisfying suitable conditions. In this paper we give sharp quantitative estimates of proximity to a single sphere for Alexandrov Soap Bubble Theorem in space forms when the curvature operator $f$ is close to a constant. Under an assumption that prevents bubbling, the proximity to a single sphere is quantified in terms of the oscillation of the curvature function $f$. Our approach provides a unified picture of quantitative studies of the method of moving planes in space forms.

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Giulio Ciraolo, Alberto Roncoroni, Luigi Vezzoni. 2018-11-30. Quantitative stability for hypersurfaces with almost constant curvature in space forms. https://arxiv.org/abs/1812.00775

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