New stability results for spheres and Wulff shapes

  • 1 Laboratoire d’Analyse et de Mathématiques Appliquées, 77454, Marne-la-Vallée, France

Abstract

We prove that a closed convex hypersurface of the Euclidean space with almost constant anisotropic first and second mean curvatures in the Lp-sense is W2,p-close (up to rescaling and translations) to the Wulff shape. We also obtain characterizations of geodesic hyperspheres of space forms improving those of and .

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  • [1] C. De Lellis, S. Müller: Optimal rigidity estimates for nearly umbilical surfaces. J. Di. Geom. 69 (1) (2005) 75–110.

  • [2] A. De Rosa, S. Gioffrè: Quantitative stability for anisotropic nearly umbilical hypersurfaces. arXiv:1705.09994 (2017)

  • [3] S. Gioffrè: A W 2,p-estimate for nearly umbilical hypersurfaces. arXiv:1612.08570 (2016)

  • [4] Y. He, H. Li: Integral formula of Minkowski type and new characterization of the Wulff shape. Acta Math. Sinica 24 (4) (2008) 697–704.

  • [5] D. Hoffman, D. Spruck: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm. Pure Appl. Math. 27 (1974) 715–727.

  • [6] J. H. Michael, L. M. Simon: Sobolev and mean-value inequalities on generalized submanifolds of Rn. Comm. Pure Appl. Math. 26 (1973) 361–379.

  • [7] L. Onat: Some characterizations of the Wulff shape. C. R. Math. Acad. Sci. Paris 348 (17-18) (2010) 997–1000.

  • [8] D. Perez: On nearly umbilical hypersurfaces. Ph.D. thesis, Universität Zürich (2011)

  • [9] J. Roth: Extrinsic radius pinching for hypersurfaces of space forms. Diff. Geom. Appl. 25 (5) (2007) 485–499.

  • [10] J. Roth: Rigidity results for geodesic spheres in space forms. In: Differential Geometry, Proceedings of the VIIIth International Colloquium in Differential Geometry, Santiago de Compostela. World Scientific (2009) 156–163.

  • [11] J. Roth: Une nouvelle caractérisation des sphères géodésiques dans les espaces modèles. Compte-Rendus – Mathématique 347 (19-20) (2009) 1197–1200.

  • [12] J. Roth: A remark on almost umbilical hypersurfaces. Arch. Math. (Brno) 49 (1) (2013) 1–7.

  • [13] J. Roth, J. Scheuer: Explicit rigidity of almost-umbilical hypersurfaces. (2015). arXiv preprint arXiv:1504.05749

  • [14] P. Topping: Relating diameter and mean curvature for submanifolds of Euclidean space. Comment. Math. Helv. 83 (3) (2008) 539–546.

  • [15] J.Y. Wu, Y. Zheng: Relating diameter and mean curvature for Riemannian submanifolds. Proc. Amer. Math. Soc. 139 (11) (2011) 4097–4104.

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