A Bernstein theorem for complete spacelike hypersurfaces of constant mean curvature in Minkowski space

Author:
Baoqiang Wu

Journal:
Proc. Amer. Math. Soc. **132** (2004), 211-215

MSC (2000):
Primary 53C21, 53C42

DOI:
https://doi.org/10.1090/S0002-9939-03-07045-X

Published electronically:
June 5, 2003

MathSciNet review:
2021264

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove a general Bernstein theorem on the complete spacelike constant mean curvature hypersurfaces in Minkowski space. The result generalizes the previous result of Cao-Shen-Zhu (1998) and Xin (1991). The proof again uses the fact that the Gauss map of a constant mean curvature hypersurface is harmonic, which was proved by K. T. Milnor (1983), and the maximum principle of S. T. Yau (1975).

**1.**E. Calabi, Examples of Bernstein problems for some nonlinear equations,*Proc. Sympos. Pure Appl. Math.***15**(1968), 223-230. MR**41:8806****2.**H.-D. Cao, Y. Shen and S. Zhu, A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space,*Calc. Var. Partial Differential Equations***7**(1998) 141-157. MR**99h:53007****3.**S. Y. Cheng and S. T. Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces,*Ann. of Math.***104**(1976), 407-419. MR**55:4063****4.**S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications,*Comm. Pure Appl. Math.***28**(1975), 333-354. MR**52:6608****5.**H. Choi and A. Treibergs, Gauss maps of spacelike mean curvature hypersurfaces of Minkowski space,*J. Differential Geom.***32**(1990), 775-817. MR**92e:58051****6.**K. T. Milnor, Harmonic maps and classical surface theory in Minkowski -space.*Trans. Amer. Math. Soc.***208**(1983), 161-185. MR**85e:58037****7.**B. Palmer, The Gauss map of a spacelike constant mean curvature hypersurface of Minkowski space,*Comment. Math. Helvetici.***65**(1990), 52-57. MR**91a:53014****8.**Y.-L. Xin, On the Gauss image of a spacelike hypersurface with constant mean curvature in Minkowski space,*Comment. Math. Helvetici.***66**(1991), 590-598. MR**93d:53016****9.**S. T. Yau, Harmonic functions on complete Riemannian manifolds,*Comm. Pure Appl. Math.***28**(1975), 201-228. MR**55:4042**

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Additional Information

**Baoqiang Wu**

Affiliation:
Department of Mathematics, Xuzhou Normal University, Xuzhou 221009, People’s Republic of China

Email:
bqwu@pub.xz.jsinfo.net

DOI:
https://doi.org/10.1090/S0002-9939-03-07045-X

Keywords:
Constant mean curvature,
spacelike hypersurface,
Minkowski space

Received by editor(s):
May 31, 2002

Received by editor(s) in revised form:
August 15, 2002

Published electronically:
June 5, 2003

Additional Notes:
This research was partially supported by a JNSF grant

Communicated by:
Bennett Chow

Article copyright:
© Copyright 2003
American Mathematical Society