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

Présenté par : Jean-Pierre Demailly

Onat, Leyla1
1 Department of Mathematics, Faculty of Arts and Sciences, Adnan Menderes University, 09010 Aydın, Turkey
Comptes Rendus. Mathématique, Tome 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
Onat, Leyla 1

1 Department of Mathematics, Faculty of Arts and Sciences, Adnan Menderes University, 09010 Aydın, Turkey 
@article{CRMATH_2010__348_17-18_997_0,
     author = {Onat, Leyla},
     title = {Some characterizations of the {Wulff} shape},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {997--1000},
     publisher = {Elsevier},
     volume = {348},
     number = {17-18},
     year = {2010},
     doi = {10.1016/j.crma.2010.07.028},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2010.07.028/}
}
TY  - JOUR
AU  - Onat, Leyla
TI  - Some characterizations of the Wulff shape
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 997
EP  - 1000
VL  - 348
IS  - 17-18
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2010.07.028/
DO  - 10.1016/j.crma.2010.07.028
LA  - en
ID  - CRMATH_2010__348_17-18_997_0
ER  - 
%0 Journal Article
%A Onat, Leyla
%T Some characterizations of the Wulff shape
%J Comptes Rendus. Mathématique
%D 2010
%P 997-1000
%V 348
%N 17-18
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2010.07.028/
%R 10.1016/j.crma.2010.07.028
%G en
%F CRMATH_2010__348_17-18_997_0
Onat, Leyla. Some characterizations of the Wulff shape. Comptes Rendus. Mathématique, Tome 348 (2010) no. 17-18, pp. 997-1000. doi : 10.1016/j.crma.2010.07.028. http://www.numdam.org/articles/10.1016/j.crma.2010.07.028/

[1] Clarenz, U. The Wulff-shape minimizes an anisotropic Willmore functional, Interf. Free Bound., Volume 6 (2004) no. 3, pp. 351-359

[2] Feeman, G.F.; Husiung, C.C. Characterizations of the Riemannian n-spheres, Amer. J. Math., Volume 81 (1959), pp. 691-708

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

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

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

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

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

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

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

Cité par Sources :