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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References
Alexandrov, A.D.: A characteristic property of spheres. Ann. Mat. Pura Appl. 58, 303–315 (1962)
De Lima, E.: A note on compact Weingarten hypersurfaces embedded in \({\mathbb{R}}^{n+1}\). Arch. Math. (Basel) 111, 669–672 (2018)
De Rosa, A., Gioffrè, S.: Quantitative stability for anisotropic nearly umbilical hypersurfaces. J. Geom. Anal. https://doi.org/10.1007/s12220-018-0079-2 (2018)
De Rosa, A., Gioffrè, S.: Absence of bubbling phenomena for non convex anisotropic nearly umbilical and quasi Einstein hypersurfaces. arXiv:1803.09118
He, Y., Li, H.: Integral formula of Minkowski type and new characterization of the Wulff shape. Acta Math. Sin. 24(4), 697–704 (2008)
He, Y., Li, H., Ma, H., Ge, J.: Compact embedded hypersurfaces with constant higher order anisotropic mean curvature. Indiana Univ. Math. J. 58, 853–868 (2009)
Hsiang, W., Teng, Z., Yu, W.: New examples of constant mean curvature immersions of (2k–1)-spheres into Euclidean 2k-space. Ann. Math. 117(3), 609–625 (1983)
Hsiung, C.C.: Some integral formulas for closed hypersurfaces. Math. Scand. 2, 286–294 (1954)
Kapouleas, N.: Constant mean curvature surfaces constructed by fusing Wente tori. Invent. Math. 119(3), 443–518 (1995)
Korevaar, N.J.: Sphere theorems via Alexandrov constant Weingarten curvature hypersurfaces: appendix to a note of A. Ros. J. Differ. Geom. 27, 221–223 (1988)
Ros, A.: Compact hypersurfaces with constant scalar curvature and a congruence theorem. J. Differ. Geom. 27(2), 215–223 (1988)
Ros, A.: Compact hypersurfaces with constant higher order mean curvatures. Rev. Mat. Iberoamericana 3(3–4), 447–453 (1987)
Roth, J.: Rigidity results for geodesic spheres in space forms. In: Differential Geometry, Proceedings of the VIII International Colloquium. World Scientific, Singapore, pp. 156–163 (2009)
Roth, J.: New stability results for spheres and Wulff shapes. Comm. Math. 26(2), 153–167 (2018)
Wente, H.: Counterexample to a conjecture of H. Hopf. Pac. J. Math. 121(1), 193–243 (1986)
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