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Archiv der Mathematik

, Volume 113, Issue 2, pp 213–224 | Cite as

On compact anisotropic Weingarten hypersurfaces in Euclidean space

  • Julien Roth
  • Abhitosh UpadhyayEmail author
Article
  • 54 Downloads

Abstract

We show that, up to homotheties and translations, the Wulff shape \(\mathcal {W}_F\) is the only compact embedded hypersurface of the Euclidean space satisfying \(H_r^F=aH^F+b\) with \(a\geqslant 0\), \(b>0\), where \(H^F\) and \(H_r^F\) are, respectively, the anisotropic mean curvature and anisotropic r-th mean curvature associated with the function \(F:\mathbb {S}^n\longrightarrow \mathbb {R}_+^*\). Further, we show that if the \(L^2\)-norm of \(H_r^F-aH^F-b\) is sufficiently close to 0, then the hypersurface is close to the Wulff shape for the \(W^{2,2}\)-norm.

Keywords

Wulff shape Weingarten hypersurfaces Anisotropic mean curvature 

Mathematics Subject Classification

53C42 53A07 49Q10 

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Notes

Acknowledgements

The second author is supported by National Post-doctoral fellowship of Science and Engineering Research Board (File no. PDF/2017/001165), India. He would also like to express his thanks to Professor Patrice Philippon (DR CNRS, resp. LIA IFPM) for providing necessary support to stay in France and to the Laboratoire d’Analyse et de Mathématiques Appliquées, Marne-la-Vallée, for local hospitality during the preparation of this paper.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Laboratoire d’Analyse et de Mathématiques AppliquéesUPEM-UPEC CNRSMarne-la-ValléeFrance
  2. 2.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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