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Quantitative stability for hypersurfaces with almost constant curvature in space forms

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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 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.

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Acknowledgements

The authors wish to thank Harold Rosenberg for suggesting the problem studied in this paper. The paper was completed, while the second author was visiting the Department of Mathematics of the ETH in Zürich, and he wishes to thank the institute for hospitality and support.

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Authors and Affiliations

  1. Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133, Milan, Italy

    Giulio Ciraolo

  2. Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134, Florence, Italy

    Alberto Roncoroni

  3. Dipartimento di Matematica “Giuseppe Peano”, Università degli Studi di Torino, Via Carlo Alberto 10, 10123, Turin, Italy

    Luigi Vezzoni

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  1. Giulio Ciraolo
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  2. Alberto Roncoroni
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  3. Luigi Vezzoni
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Correspondence to Alberto Roncoroni.

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The first two authors have been supported by GNAMPA of INdAM. The third author has been supported by GNSAGA of INdAM.

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Ciraolo, G., Roncoroni, A. & Vezzoni, L. Quantitative stability for hypersurfaces with almost constant curvature in space forms. Annali di Matematica 200, 2043–2083 (2021). https://doi.org/10.1007/s10231-021-01069-7

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