Comptes Rendus
no. 17-18
Differential Geometry
Some characterizations of the Wulff shape
[Sur certaines caractérisations des formes de Wulff]

Présenté par : Jean-Pierre Demailly

Leyla Onat1
1 Department of Mathematics, Faculty of Arts and Sciences, Adnan Menderes University, 09010 Aydın, Turkey
Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 997-1000.

Étant donné une fonction positive F sur Sn qui vérifie une condition de convexité convenable, nous considérons la r-ième courbure moyenne anisotrope pour les hypersurfaces de Rn+1 qui est une généralisation de la r-ième courbure moyenne usuelle Hr. En utilisant une formule intégrale de type Minkowski pour les hypersurfaces compactes due à H.J. He et H. Li, nous introduisons de nouvelles caractérisations des formes de Wulff.

For a positive function F on Sn which satisfies a suitable convexity condition, we consider the r-th anisotropic mean curvature for hypersurfaces in Rn+1 which is a generalization of the usual r-th mean curvature Hr. By using an integral formula of Minkowski type for compact hypersurface due to H.J. He and H. Li, we introduce some new characterizations of the Wulff shape.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.07.028
Leyla Onat 1

1 Department of Mathematics, Faculty of Arts and Sciences, Adnan Menderes University, 09010 Aydın, Turkey 
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Leyla Onat. Some characterizations of the Wulff shape. Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 997-1000. doi : 10.1016/j.crma.2010.07.028. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.028/

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[3] G.H. Hardy; J.E. Littlewood; G. Polya Inequalities, Cambridge University Press, Cambridge, 1934

[4] Y.J. He; H. Li Integral formula of Minkowski type and new characterization of the Wulff shape, Acta Math. Sinica, Volume 24 (2008), pp. 697-704

[5] C.C. Hsiung Some integral formulas for closed hypersurfaces, Math. Scand., Volume 2 (1954), pp. 86-294

[6] M. Koiso; B. Palmer Geometry and stability of surfaces with constant anisotropic mean curvature, Indiana Univ. Math. J., Volume 54 (2005), pp. 1817-1852

[7] B. Palmer Stability of the Wulff shape, Proc. Amer. Math. Soc., Volume 126 (1998), pp. 3661-3667

[8] R. Reilly The relative differential geometry of nonparametric hypersurfaces, Duke Math. J., Volume 43 (1976), pp. 705-721

[9] J. Taylor Crystalline variational problems, Bull. Amer. Math. Soc., Volume 84 (1978), pp. 568-588

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