Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Stability of spacelike hypersurfaces in de Sitter space


Author: Shicheng Zhang
Journal: Proc. Amer. Math. Soc. 143 (2015), 851-857
MSC (2010): Primary 53C42, 53A10; Secondary 53B30, 53C50
DOI: https://doi.org/10.1090/S0002-9939-2014-12276-3
Published electronically: October 16, 2014
MathSciNet review: 3283671
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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, https://doi.org/10.1007/BF03025712
  • [2] João Lucas Barbosa and Manfredo do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984), no. 3, 339-353. MR 0735120, https://doi.org/10.1007/BF01215045
  • [3] 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 0917854, https://doi.org/10.1007/BF01161634
  • [4] 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, https://doi.org/10.1023/A:1006514303828
  • [5] 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
  • [6] 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, https://doi.org/10.1016/j.jmaa.2012.08.003
  • [7] 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, https://doi.org/10.2206/kyushumfs.44.67
  • [8] Qing-Ming Cheng and Susumu Ishikawa, Spacelike hypersurfaces with constant scalar curvature, Manuscripta Math. 95 (1998), no. 4, 499-505. MR 1618206, https://doi.org/10.1007/s002290050043
  • [9] Shiu Yuen Cheng and Shing Tung Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195-204. MR 0431043
  • [10] 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, https://doi.org/10.4134/BKMS.2009.46.2.321
  • [11] Ximin Liu and Deng Junlei, Stable space-like hypersurfaces in the de Sitter space, Arch. Math. (Brno) 40 (2004), no. 2, 111-117. MR 2068685
  • [12] 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C42, 53A10, 53B30, 53C50

Retrieve articles in all journals with MSC (2010): 53C42, 53A10, 53B30, 53C50


Additional Information

Shicheng Zhang
Affiliation: School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, Jiangsu 221116, People’s Republic of China
Email: zhangshicheng@jsnu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2014-12276-3
Keywords: de Sitter space, spacelike hypersurface, stability, linear Weingarten hypersurfaces
Received by editor(s): January 21, 2013
Received by editor(s) in revised form: May 22, 2013, and May 26, 2013
Published electronically: October 16, 2014
Additional Notes: This project was supported by the National Natural Science Foundation of China (No. 61271002) and PAPD of Jiangsu Higher Education Institutions.
Dedicated: This paper is dedicated to our advisors
Communicated by: Lei Ni
Article copyright: © Copyright 2014 American Mathematical Society