| Title:
|
New stability results for spheres and Wulff shapes (English) |
| Author:
|
Roth, Julien |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
26 |
| Issue:
|
2 |
| Year:
|
2018 |
| Pages:
|
153-167 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We prove that a closed convex hypersurface of the Euclidean space with almost constant anisotropic first and second mean curvatures in the $L^p$-sense is $W^{2,p}$-close (up to rescaling and translations) to the Wulff shape. We also obtain characterizations of geodesic hyperspheres of space forms improving those of \cite {Ro1} and \cite {Ro}. (English) |
| Keyword:
|
Hypersurfaces |
| Keyword:
|
Anisotropic mean curvatures |
| Keyword:
|
Wulff Shape |
| Keyword:
|
Almost umibilcal |
| MSC:
|
53A10 |
| MSC:
|
53C42 |
| idZBL:
|
Zbl 07058962 |
| idMR:
|
MR3898200 |
| . |
| Date available:
|
2019-05-07T09:26:42Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147653 |
| . |
| Reference:
|
[1] Lellis, C. De, Müller, S.: Optimal rigidity estimates for nearly umbilical surfaces.J. Diff. Geom., 69, 1, 2005, 75-110, MR 2169583, 10.4310/jdg/1121540340 |
| Reference:
|
[2] Rosa, A. De, Gioffrè, S.: Quantitative stability for anisotropic nearly umbilical hypersurfaces.2017, arXiv:1705.09994. MR 3896120 |
| Reference:
|
[3] Gioffrè, S.: A $W^{2,p}$-estimate for nearly umbilical hypersurfaces.2016, arXiv:1612.08570. |
| Reference:
|
[4] He, Y., Li, H.: Integral formula of Minkowski type and new characterization of the Wulff shape.Acta Math. Sinica, 24, 4, 2008, 697-704, MR 2393162, 10.1007/s10114-007-7116-6 |
| Reference:
|
[5] Hoffman, D., Spruck, D.: Sobolev and isoperimetric inequalities for Riemannian submanifolds.Comm. Pure Appl. Math., 27, 1974, 715-727, Zbl 0295.53025, MR 0365424, 10.1002/cpa.3160270601 |
| Reference:
|
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| Reference:
|
[7] Onat, L.: Some characterizations of the Wulff shape.C. R. Math. Acad. Sci. Paris, 348, 17-18, 2010, 997-1000, MR 2721788, 10.1016/j.crma.2010.07.028 |
| Reference:
|
[8] Perez, D.: On nearly umbilical hypersurfaces.2011, Ph.D. thesis, Universität Zürich. |
| Reference:
|
[9] Roth, J.: Extrinsic radius pinching for hypersurfaces of space forms.Diff. Geom. Appl., 25, 5, 2007, 485-499, MR 2351425, 10.1016/j.difgeo.2007.06.017 |
| Reference:
|
[10] Roth, J.: Rigidity results for geodesic spheres in space forms.Differential Geometry, Proceedings of the VIIIth International Colloquium in Differential Geometry, Santiago de Compostela, 2009, 156-163, World Scientific, MR 2523501 |
| Reference:
|
[11] Roth, J.: Une nouvelle caractérisation des sphères géodésiques dans les espaces modèles.Compte-Rendus - Mathématique, 347, 19-20, 2009, 1197-1200, MR 2567002, 10.1016/j.crma.2009.09.012 |
| Reference:
|
[12] Roth, J.: A remark on almost umbilical hypersurfaces.Arch. Math. (Brno), 49, 1, 2013, 1-7, MR 3073010, 10.5817/AM2013-1-1 |
| Reference:
|
[13] Roth, J., Scheuer, J.: Explicit rigidity of almost-umbilical hypersurfaces.2015, arXiv preprint arXiv:1504.05749. MR 3919552 |
| Reference:
|
[14] Topping, P.: Relating diameter and mean curvature for submanifolds of Euclidean space.Comment. Math. Helv., 83, 3, 2008, 539-546, MR 2410779, 10.4171/CMH/135 |
| Reference:
|
[15] Wu, J.Y., Zheng, Y.: Relating diameter and mean curvature for Riemannian submanifolds.Proc. Amer. Math. Soc., 139, 11, 2011, 4097-4104, MR 2823054, 10.1090/S0002-9939-2011-10848-7 |
| . |
