New stability results for spheres and Wulff shapes

Julien Roth

Communications in Mathematics (2018)

  • Volume: 26, Issue: 2, page 153-167
  • ISSN: 1804-1388

Abstract

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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 [Ro1] and [Ro].

How to cite

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Roth, Julien. "New stability results for spheres and Wulff shapes." Communications in Mathematics 26.2 (2018): 153-167. <http://eudml.org/doc/294422>.

@article{Roth2018,
abstract = {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 [Ro1] and [Ro].},
author = {Roth, Julien},
journal = {Communications in Mathematics},
keywords = {Hypersurfaces; Anisotropic mean curvatures; Wulff Shape; Almost umibilcal},
language = {eng},
number = {2},
pages = {153-167},
publisher = {University of Ostrava},
title = {New stability results for spheres and Wulff shapes},
url = {http://eudml.org/doc/294422},
volume = {26},
year = {2018},
}

TY - JOUR
AU - Roth, Julien
TI - New stability results for spheres and Wulff shapes
JO - Communications in Mathematics
PY - 2018
PB - University of Ostrava
VL - 26
IS - 2
SP - 153
EP - 167
AB - 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 [Ro1] and [Ro].
LA - eng
KW - Hypersurfaces; Anisotropic mean curvatures; Wulff Shape; Almost umibilcal
UR - http://eudml.org/doc/294422
ER -

References

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  2. Rosa, A. De, Gioffrè, S., Quantitative stability for anisotropic nearly umbilical hypersurfaces, 2017, arXiv:1705.09994. (2017) MR3896120
  3. Gioffrè, S., A W 2 , p -estimate for nearly umbilical hypersurfaces, 2016, arXiv:1612.08570. (2016) 
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  8. Perez, D., On nearly umbilical hypersurfaces, 2011, Ph.D. thesis, Universität Zürich. (2011) 
  9. Roth, J., 10.1016/j.difgeo.2007.06.017, Diff. Geom. Appl., 25, 5, 2007, 485-499, (2007) MR2351425DOI10.1016/j.difgeo.2007.06.017
  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, (2009) MR2523501
  11. Roth, J., 10.1016/j.crma.2009.09.012, Compte-Rendus - Mathématique, 347, 19-20, 2009, 1197-1200, (2009) MR2567002DOI10.1016/j.crma.2009.09.012
  12. Roth, J., 10.5817/AM2013-1-1, Arch. Math. (Brno), 49, 1, 2013, 1-7, (2013) MR3073010DOI10.5817/AM2013-1-1
  13. Roth, J., Scheuer, J., Explicit rigidity of almost-umbilical hypersurfaces, 2015, arXiv preprint arXiv:1504.05749. (2015) MR3919552
  14. Topping, P., 10.4171/CMH/135, Comment. Math. Helv., 83, 3, 2008, 539-546, (2008) MR2410779DOI10.4171/CMH/135
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