Kairen Cai, Huiqun Xu, "Spacelike hypersurfaces in de Sitter space with constant higher-order mean curvature", International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 019545, 6 pages, 2006. https://doi.org/10.1155/IJMMS/2006/19545
Spacelike hypersurfaces in de Sitter space with constant higher-order mean curvature
The authors apply the generalized Minkowski formula to set up a spherical theorem. It is shown that a compact connected hypersurface with positive constant higher-order mean curvature for some fixed , , immersed in the de Sitter space must be a sphere.
- Q.-M. Cheng and S. Ishikawa, “Spacelike hypersurfaces with constant scalar curvature,” Manuscripta Mathematica, vol. 95, no. 4, pp. 499–505, 1998.
- J. Eells Jr. and J. H. Sampson, “Harmonic mappings of Riemannian manifolds,” American Journal of Mathematics, vol. 86, pp. 109–160, 1964.
- L. Gȧrding, “An inequality for hyperbolic polynomials,” Journal of Mathematics and Mechanics, vol. 8, pp. 957–965, 1959.
- S. Montiel, “An integral inequality for compact spacelike hypersurfaces in de Sitter space and applications to the case of constant mean curvature,” Indiana University Mathematics Journal, vol. 37, no. 4, pp. 909–917, 1988.
- S. Montiel and A. Ros, “Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures,” in Differential Geometry. Proceedings Conference in Honor of Manfredo do Carmo, vol. 52 of Pitman Monogr. Surveys Pure Appl. Math., pp. 279–296, Longman Scientific & Technical, Harlow, 1991.
- B. O'Neill, Semi-Riemannian Geometry. With Applications to Relativity, vol. 103 of Pure and Applied Mathematics, Academic Press, New York, 1983.
- R. C. Reilly, “Applications of the Hessian operator in a Riemannian manifold,” Indiana University Mathematics Journal, vol. 26, no. 3, pp. 459–472, 1977.
- A. Ros, “Compact hypersurfaces with constant higher order mean curvatures,” Revista Matemática Iberoamericana, vol. 3, no. 3-4, pp. 447–453, 1987.
Copyright © 2006 Hindawi Publishing Corporation. 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.