Complete CMC spacelike hypersurfaces immersed in a Lorentzian product space
详细信息 查看全文
- 作者：Cícero P. Aquino ; Henrique F. de Lima ; Eraldo A. Lima Jr.
- 关键词：Primary 53C42 ; Secondary 53B30 ; 53C50 ; Lorentzian product spaces ; Complete spacelike hypersurfaces ; Mean curvature ; Normal hyperbolic angle ; Entire vertical graphs
- 刊名：Archiv der Mathematik
- 出版年：2015
- 出版时间：June 2015
- 年：2015
- 卷：104
- 期：6
- 页码：577-587
- 全文大小：490 KB
- 参考文献：1.Aiyama R.: On the Gauss map of complete space-like hypersurfaces of constant mean curvature in Minkowski space. Tsukuba J. Math. 16, 353-61 (1992)MATH MathSciNet
2.Albujer A.L.: New examples of entire maximal graphs in \({\mathbb{H}^{2} \times \mathbb{R}_{1}}\) , Diff. Geom. Appl. 26, 456-62 (2008)View Article MATH MathSciNet
3.Albujer A.L., Alías L.J.: A local estimate for maximal surfaces in Lorentzian product spaces. Mat. Contemp. 34, 1-0 (2008)MATH MathSciNet
4.Albujer A.L., Alías L.J.: Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces. J. Geom. Phys. 59, 620-31 (2009)View Article MATH MathSciNet
5.Albujer A.L., Alías L.J.: Spacelike hypersurfaces with constant mean curvature in the steady state space. Proc. Amer. Math. Soc. 137, 711-21 (2009)View Article MATH MathSciNet
6.A.L. Albujer, Alías L.J.: Parabolicity of maximal surfaces in Lorentzian product spaces. Math. Z. 267, 453-64 (2011)View Article MathSciNet
7.Alías L.J., Romero A., Sánchez M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in Generalized Robertson-Walker spacetimes. Gen. Relativ. Gravit. 27, 71-4 (1995)View Article MATH
8.Bochner S.: Vector fields and Ricci curvature. Bull. American Math. Soc. 52, 776-97 (1946)View Article MATH MathSciNet
9.M. Berger, P. Gauduchon, and E. Mazet, Le spectre d’une variété Riemannienne, Lectures Notes in Mathematics, Vol. 194, Springer, Berlin, 1971.
10.E. Calabi, Examples of Bernstein problems for some nonlinear equations, Proc. Sympos. Pure Math. 15 (1970), 223-30.
11.A. Caminha, A rigidity theorem for complete CMC hypersurfaces in Lorentz manifolds, Diff. Geom. Appl. 24 (2006), 652-59.
12.Cheng S.Y., Yau S.T.: Maximal spacelike hypersurfaces in the Lorentz-Minkowski space. Ann. of Math. 104, 407-19 (1976)View Article MATH MathSciNet
13.J.M. Espinar and H. Rosenberg, Complete constant mean curvature surfaces and Bernstein type theorems in \({M^{2} \times \mathbb{R}}\) , J. Diff. Geom. 82 (2009), 611-28.
14.Li G., Salavessa I.: Graphic Bernstein results in curved pseudo-Riemannian manifolds. J. Geom. Phys. 59, 1306-313 (2009)View Article MATH MathSciNet
15.de Lima H.F.: On the Gauss map of complete spacelike hypersurfaces with bounded mean curvature in the Minkowski space. Bull. Belgian Math. Soc. Simon Stevin 18, 537-41 (2011)MATH
16.H.F. de Lima and E.A. Lima Jr., Generalized maximum principles and the unicity of complete spacelike hypersurfaces immersed in a Lorentzian product space, Beitr. Algebra Geom. 55 (2014), 59-5.
17.Montiel S.: Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes. Math. Ann. 314, 529-53 (1999)View Article MATH MathSciNet
18.H. Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 (1967), 205-14.
19.B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, 1983.
20.Palmer B.: The Gauss map of a spacelike constant mean curvature hypersurface in Minkowski space. Comment. Math. Helv. 65, 52-7 (1990)View Article MATH MathSciNet
21.Treibergs A.E.: Entire spacelike hypersurfaces of constant mean curvature in Minkowski space. Invent. Math. 66, 39-6 (1982)View Article MATH MathSciNet
22.Xin Y.L.: On the Gauss image of a spacelike hypersurface with constant mean curvature in Minkowski space. Comment. Math. Helv. 66, 590-98 (1991)View Article MATH MathSciNet
23.Yau S.T.(1975) Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28:201-28 - 作者单位：Cícero P. Aquino (1)
Henrique F. de Lima (2)
Eraldo A. Lima Jr. (3)
1. Departamento de Matemática, Universidade Federal do Piauí, Teresina, Piauí, 64049-550, Brazil
2. Departamento de Matemática, Universidade Federal de Campina Grande, Campina Grande, Paraíba, 58429-970, Brazil
3. Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, Ceará, 60455-760, Brazil - 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Mathematics - 出版者：Birkh盲user Basel
- ISSN：1420-8938
文摘
We use Bochner’s formula jointly with the generalized maximum principle of Omori-Yau and an extension of Liouville’s theorem due to Yau in order to show that a complete spacelike hypersurface \({\Sigma^{n}}\) immersed with constant mean curvature in a Lorentzian product space \({\overline{M}^{n+1}=-{\mathbb{R}}{\times}M^{n}}\), whose fiber M n has nonnegative sectional curvature, must be a slice, provided that \({\Sigma^{n}}\) is bounded away from the future (or past) infinity of \({\overline{M}^{n+1}}\) and that its normal hyperbolic angle is bounded. We also study the rigidity of entire vertical graphs with constant mean curvature in such an ambient space.