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(k1, … , kn-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 optimally 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.

Quantitative stability for hypersurfaces with almost constant curvature in space forms / G. Ciraolo, A. Roncoroni, L. Vezzoni. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - 200:5(2021 Oct), pp. 2043-2083. [10.1007/s10231-021-01069-7]

Quantitative stability for hypersurfaces with almost constant curvature in space forms

G. Ciraolo
Primo
;
2021-10

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(k1, … , kn-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 optimally 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.
Alexandrov Soap Bubble Theorem; Mean curvature; Method of the moving planes; Pinching; Quantitative stability; Space forms geometry
Settore MAT/05 - Analisi Matematica
Settore MAT/03 - Geometria
8-mar-2021
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/860550
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