Abstract
Given a positive function F on S n which satisfies a convexity condition, we introduce the r-th anisotropic mean curvature M r for hypersurfaces in ℝn+1 which is a generalization of the usual r-th mean curvature H r . We get integral formulas of Minkowski type for compact hypersurfaces in R n+1. We give some new characterizations of the Wulff shape by the use of our integral formulas of Minkowski type, in case F = 1 which reduces to some well-known results.
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References
Clarenz, U.: The Wulff-shape minimizes an anisotropicWillmore functional, Interfaces and Free Boundaries, 6, 35–359 (2004)
Koiso, M., Palmer, B.: Geometry and stability of surfaces with constant anisotropic mean curvature. Indiana Univ. Math. J., 54, 1817–1852 (2005)
Palmer, B.: Stability of the Wulff shape. Proc. Amer. Math. Soc., 126, 3661–3667 (1998)
Süss, W.: Zur relativen Differentialgeometrie. V., Tôhoku Math. J., 30, 202–209 (1929)
Taylor, J.: Crystalline variational problems. Bull. Amer. Math. Soc., 84, 568–588 (1978)
Reilly, R.: The relative differential geometry of nonparametric hypersurfaces. Duke Math. J., 43, 705–721 (1976)
Hsiung, C. C.: Some integral formulas for closed hypersurfaces. Math. Scand., 2, 286–294 (1954)
Simon, U.: Minkowskische integralformeln und ihre Anwendungen in der Differentialgeometrie in Grossen. Math. Ann., 173, 307–321 (1967)
Yano, K.: Integral formulas in Riemannian geometry, Marcel, Dekker, N. Y, 1970
Choe, J.: Sufficient conditions for constant mean curvature surfaces to be round. Math. Ann., 323, 143–156 (2002)
Li, H., Chen, W. H.: Integral formulas for compact spacelike hypersurfaces in de Sitter space and their applications to Goddard’s conjecture. Acta Mathematica Sinica, New Series, 14, 285–288 (1998)
Li, H.: Hypersurfaces with constant scalar curvature in space forms. Math. Ann., 305, 665–672 (1996)
Li, H.: Global rigidity theorems of hypersurfaces. Ark. Math., 35, 327–351 (1997)
Barbosa, J. L. M., Colares, A. G.: Stability of hypersurfaces with constant r-mean curvature. Ann. Global. Anal. Geom., 15, 277–297 (1997)
Winklmann, S.: A note on the stability of the Wulff shape. Arch. Math., 87 272–279 (2006)
Montiel, S., Ros, A.: Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures, in Lawson, B. and Tenenblat, K. (eds), Differential Geometry, Pitman Monographs, Vol. 52, Longman, Essex, 1991, pp. 279–296
Hardy, G. H., Littlewood, J. E., Polya, G.: Inequalities, Cambridge Univ. Press, London, 1934
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The first author is supported partially by Tianyuan Fund for Mathematics of NSFC (Grant No. 10526030)
The second author is supported partially by Grant No. 10531090 of the NSFC and by Doctoral Program Foundation of the Ministry of Education of China (2006)
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He, Y.J., Li, H.Z. Integral formula of Minkowski type and new characterization of the Wulff shape. Acta. Math. Sin.-English Ser. 24, 697–704 (2008). https://doi.org/10.1007/s10114-007-7116-6
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DOI: https://doi.org/10.1007/s10114-007-7116-6
Keywords
- Wulff shape
- F-Weingarten operator
- anisotropic principal curvature
- r-th anisotropic mean curvature
- integral formula of Minkowski type