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.
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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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Roth, J., Upadhyay, A. On compact anisotropic Weingarten hypersurfaces in Euclidean space. Arch. Math. 113, 213–224 (2019). https://doi.org/10.1007/s00013-019-01315-8
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DOI: https://doi.org/10.1007/s00013-019-01315-8
Keywords
- Wulff shape
- Weingarten hypersurfaces
- Anisotropic mean curvature
Mathematics Subject Classification
- 53C42
- 53A07
- 49Q10