Stability of spacelike hypersurfaces in de Sitter space

Zhang, Shicheng

doi:10.1090/S0002-9939-2014-12276-3

Stability of spacelike hypersurfaces in de Sitter space
HTML articles powered by AMS MathViewer

by Shicheng Zhang

Proc. Amer. Math. Soc. 143 (2015), 851-857

DOI: https://doi.org/10.1090/S0002-9939-2014-12276-3

Published electronically: October 16, 2014

PDF | Request permission

Abstract:

This paper discusses compact linear Weingarten space-like hypersurfaces in a de Sitter space We prove that a compact linear Weingarten space-like hypersurface in a de Sitter space is stable if and only if it is totally umbilical.

References

H. Alencar, M. do Carmo, and A. G. Colares, Stable hypersurfaces with constant scalar curvature, Math. Z. 213 (1993), no. 1, 117–131. MR 1217674, DOI 10.1007/BF03025712
João Lucas Barbosa and Manfredo do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984), no. 3, 339–353. MR 731682, DOI 10.1007/BF01215045
J. Lucas Barbosa, Manfredo do Carmo, and Jost Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z. 197 (1988), no. 1, 123–138. MR 917854, DOI 10.1007/BF01161634
João Lucas Marques Barbosa and Antônio Gervasio Colares, Stability of hypersurfaces with constant $r$-mean curvature, Ann. Global Anal. Geom. 15 (1997), no. 3, 277–297. MR 1456513, DOI 10.1023/A:1006514303828
João Lucas Marquês Barbosa and Vladimir Oliker, Stable spacelike hypersurfaces with constant mean curvature in Lorentz space, Geometry and global analysis (Sendai, 1993) Tohoku Univ., Sendai, 1993, pp. 161–164. MR 1361178
Hang Chen and Xianfeng Wang, Stability and eigenvalue estimates of linear Weingarten hypersurfaces in a sphere, J. Math. Anal. Appl. 397 (2013), no. 2, 658–670. MR 2979602, DOI 10.1016/j.jmaa.2012.08.003
Qing Ming Cheng, Complete space-like hypersurfaces of a de Sitter space with $r=kH$, Mem. Fac. Sci. Kyushu Univ. Ser. A 44 (1990), no. 2, 67–77. MR 1072301, DOI 10.2206/kyushumfs.44.67
Qing-Ming Cheng and Susumu Ishikawa, Spacelike hypersurfaces with constant scalar curvature, Manuscripta Math. 95 (1998), no. 4, 499–505. MR 1618206, DOI 10.1007/s002290050043
Shiu Yuen Cheng and Shing Tung Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195–204. MR 431043, DOI 10.1007/BF01425237
Haizhong Li, Young Jin Suh, and Guoxin Wei, Linear Weingarten hypersurfaces in a unit sphere, Bull. Korean Math. Soc. 46 (2009), no. 2, 321–329. MR 2502796, DOI 10.4134/BKMS.2009.46.2.321
Liu Ximin and Deng Junlei, Stable space-like hypersurfaces in the de Sitter space, Arch. Math. (Brno) 40 (2004), no. 2, 111–117. MR 2068685
Ximin Liu and Biaogui Yang, Stable space-like hypersurfaces with constant scalar curvature in generalized Robertson-Walker spacetimes, Balkan J. Geom. Appl. 13 (2008), no. 1, 66–76. MR 2395376