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Geometry
Volume 2013 (2013), Article ID 718272, 7 pages
http://dx.doi.org/10.1155/2013/718272
Research Article

Hypersurfaces with Null Higher Order Anisotropic Mean Curvature

1School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China
2Research Institute of Mathematics and Applied Mathematics, Shanxi University, Taiyuan 030006, China

Received 18 April 2013; Accepted 11 June 2013

Academic Editor: Reza Saadati

Copyright © 2013 Hua Wang and Yijun He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. E. Brothers and F. Morgan, “The isoperimetric theorem for general integrands,” The Michigan Mathematical Journal, vol. 41, no. 3, pp. 419–431, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. U. Clarenz, “The Wulff shape minimizes an anisotropic Willmore functional,” Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, vol. 6, no. 3, pp. 351–359, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. Koiso and B. Palmer, “Geometry and stability of surfaces with constant anisotropic mean curvature,” Indiana University Mathematics Journal, vol. 54, no. 6, pp. 1817–1852, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Koiso and B. Palmer, “Stability of anisotropic capillary surfaces between two parallel planes,” Calculus of Variations and Partial Differential Equations, vol. 25, no. 3, pp. 275–298, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. Koiso and B. Palmer, “Anisotropic capillary surfaces with wetting energy,” Calculus of Variations and Partial Differential Equations, vol. 29, no. 3, pp. 295–345, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. M. Koiso and B. Palmer, “Uniqueness theorems for stable anisotropic capillary surfaces,” SIAM Journal on Mathematical Analysis, vol. 39, no. 3, pp. 721–741, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  7. F. Morgan, “Planar Wulff shape is unique equilibrium,” Proceedings of the American Mathematical Society, vol. 133, no. 3, pp. 809–813, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. B. Palmer, “Stability of the Wulff shape,” Proceedings of the American Mathematical Society, vol. 126, no. 12, pp. 3661–3667, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. E. Taylor, “Crystalline variational problems,” Bulletin of the American Mathematical Society, vol. 84, no. 4, pp. 568–588, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. C. Reilly, “The relative differential geometry of nonparametric hypersurfaces,” Duke Mathematical Journal, vol. 43, no. 4, pp. 705–721, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. L. Cao and H. Li, “r-minimal submanifolds in space forms,” Annals of Global Analysis and Geometry, vol. 32, no. 4, pp. 311–341, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. H. Li, “Hypersurfaces with constant scalar curvature in space forms,” Mathematische Annalen, vol. 305, no. 4, pp. 665–672, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. Montiel and A. Ros, “Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures,” in Differential geometry, B. Lawson and K. Tenenblat, Eds., vol. 52, pp. 279–296, Longman, Harlow, UK, 1991. View at Zentralblatt MATH · View at MathSciNet
  14. A. Ros, “Compact hypersurfaces with constant higher order mean curvatures,” Revista Matemática Iberoamericana, vol. 3, no. 3-4, pp. 447–453, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. T. Hasanis and D. Koutroufiotis, “A property of complete minimal surfaces,” Transactions of the American Mathematical Society, vol. 281, no. 2, pp. 833–843, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. H. Alencar and K. Frensel, “Hypersurfaces whose tangent geodesics omit a nonempty set,” in Pitman Monographs, vol. 52, pp. 1–13, Surveys in Pure and Applied Mathematics, 1991. View at Zentralblatt MATH · View at MathSciNet
  17. H. Alencar and M. Batista, “Hypersurfaces with null higher order mean curvature,” Bulletin of the Brazilian Mathematical Society, vol. 41, no. 4, pp. 481–493, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. H. Alencar, Hipersuperfícies Mnimas de ℝ2m Invariantes por SO(m), SO(m) [Doctor thesis], IMPA-Brazil, 1988.
  19. Y. He, H. Li, H. Ma, and J. Ge, “Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures,” Indiana University Mathematics Journal, vol. 58, no. 2, pp. 853–868, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer, New York, NY, USA, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  21. Z. Shen, Lectures on Finsler Geometry, World Scientific, Singapore, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  22. J. L. M. Barbosa and A. G. Colares, “Stability of hypersurfaces with constant r-mean curvature,” Annals of Global Analysis and Geometry, vol. 15, no. 3, pp. 277–297, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  23. R. C. Reilly, “Variational properties of functions of the mean curvatures for hypersurfaces in space forms,” Journal of Differential Geometry, vol. 8, pp. 465–477, 1973. View at Zentralblatt MATH · View at MathSciNet
  24. J. Hounie and M. L. Leite, “The maximum principle for hypersurfaces with vanishing curvature functions,” Journal of Differential Geometry, vol. 41, no. 2, pp. 247–258, 1995. View at Zentralblatt MATH · View at MathSciNet
  25. J. Hounie and M. L. Leite, “Two-ended hypersurfaces with zero scalar curvature,” Indiana University Mathematics Journal, vol. 48, no. 3, pp. 867–882, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. A. Caminha, “On spacelike hypersurfaces of constant sectional curvature lorentz manifolds,” Journal of Geometry and Physics, vol. 56, no. 7, pp. 1144–1174, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet