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1.
In this paper, we classify complete spacelike hypersurfaces in the anti-de Sitter space (n?3) with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we also obtain several rigidity theorems for such hypersurfaces. 相似文献
2.
We classify spacelike hypersurfaces of the de Sitter space with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we prove several rigidity theorems for such hypersurfaces. 相似文献
3.
Hypersurfaces with constant scalar curvature in space forms 总被引:17,自引:0,他引:17
Li Haizhong 《Mathematische Annalen》1996,305(1):665-672
4.
Complete space-like hypersurfaces with constant scalar curvature 总被引:6,自引:0,他引:6
Ximin Liu 《manuscripta mathematica》2001,105(3):367-377
Let M
n
be a complete space-like hypersurface with constant normalized scalar curvature R in the de Sitter space S
n
+
1
1 and denote . We prove that if the norm square of the second fundamental form of M
n
satisfies , then either and M
n
is a totally umbilical hypersurface; or , and, up to rigid motion, M
n
is a hyperbolic cylinder .
Received: 8 February 2001 / Revised version: 27 April 2001 相似文献
5.
6.
Huai-Dong Cao Ying Shen Shunhui Zhu 《Calculus of Variations and Partial Differential Equations》1998,7(2):141-157
We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski
space into the hyperbolic space. As an application, we prove a Bernstein theorem which says that if the image of the Gauss
map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends
the previous theorems obtained by B. Palmer [Pa] and Y.L. Xin [Xin1] where they assume that the image of the Gauss map is
bounded. We also prove a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional
spaces.
Received July 4, 1997 / Accepted October 9, 1997 相似文献
7.
In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to
compact spacelike hypersurfaces which are immersed in de Sitter spaceS
1
n+1
(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant
scalar curvaturen(n−1)r is isometric to a sphere ifr<c.
Research partially Supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and
Culture. 相似文献
8.
Spacelike hypersurfaces with constant scalar curvature 总被引:1,自引:0,他引:1
In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to
compact spacelike hypersurfaces which are immersed in de Sitter space S
n
+1
1(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant
scalar curvature n(n-1)r is isometric to a sphere if r << c.
Received: 18 December 1996 / Revised version: 26 November 1997 相似文献
9.
10.
Henrique F. de Lima 《Differential Geometry and its Applications》2008,26(4):445-455
In this paper we obtain a sharp height estimate concerning compact spacelike hypersurfaces Σn immersed in the (n+1)-dimensional Lorentz-Minkowski space Ln+1 with some nonzero constant r-mean curvature, and whose boundary is contained into a spacelike hyperplane of Ln+1. Furthermore, we apply our estimate to describe the nature of the end of a complete spacelike hypersurface of Ln+1. 相似文献
11.
In this work we obtain a Simons' type inequality for a suitable tensor and apply it in order to obtain some results characterizing umbilical submanifolds and a product of submanifolds in a semi-Riemannian space form. 相似文献
12.
In this paper, a complete space-like hypersurface with constant normal scalar curvature in a locally symmetric Lorentz space is discussed. The rigidity theorem is proved by using the operator □ introduced by S. Y. Cheng and S. T. Yau, and the result is a partially affirmative answer to the question posed by Haizhong Li in 1997. 相似文献
13.
In this paper, we investigate the complete spacelike hypersurfaces with constant mean curvature and two distinct principal
curvatures in an anti-de Sitter space. We give a characterization of hyperbolic cylinder and prove the conjecture in a paper
by L. F. Cao and G. X. Wei [J. Math. Anal. Appl., 2007, 329(1): 408–414]. 相似文献
14.
Let M be a compact oriented minimal
hypersurface of the unit n-dimensional sphere
Sn.
It is known that if the norm squared of the second fundamental form,
, satisfies that
for all
, then M is isometric to a Clifford
minimal hypersurface ([2], [5]). In this paper we will generalize this result
for minimal hypersurfaces with two principal curvatures and dimension greater
than 2. For these hypersurfaces we will show that if the average of the function
is n - 1, then M
must be a Clifford hypersurface.
Received: 24 December 2002 相似文献
15.
Fei-Tsen Liang 《Annali dell'Universita di Ferrara》2002,48(1):189-217
We consider the problem of determining the existence of absolute apriori gradient bounds of nonparametric hypersurfaces of
constant mean curvature in ann-dimensional sphereB
R, 1>R>R
0
(n)
, (R
0
(n)
being a constant depending only onn), without imposing boundary conditions or bounds of any sort.
Sunto Consideriamo il problema di determinare stime a priori di gradienti di ipersuperfici non parametriche di curvatura media costante in una sferan-dimensionaleB R, 1>R>R 0 (n), (R 0 (n) essendo una costante che dipende solo dan), senza imporre condizioni al contorno o limiti di altro tipo.相似文献
16.
Kazuo Akutagawa 《Mathematische Zeitschrift》1987,196(1):13-19
17.
给出了De Sitter空间S1^n 1(1)(n≥3)的类空超英面是半对称的充要条件,决定了S1^n 1(1)(n≥3)的半对称类空超曲面的局部结构,证明了S1^n 1(1)(n≥3)具有常平均曲率的连通完备的半对称类空超曲面或是全脐的,或是具有两上不同主曲率的等参超曲面。 相似文献
18.
Xu Cheng 《Archiv der Mathematik》2006,86(4):365-374
We discuss the non-existence of complete noncompact constant mean curvature hypersurfaces with finite index in a 4- or 5-dimensional
manifold. As a consequence, we obtain that a complete noncompact constant mean curvature hypersurface in
with finite index must be minimal.
Received: 30 May 2005 相似文献
19.
Rafael López Sebastián Montiel 《Calculus of Variations and Partial Differential Equations》1999,8(2):177-190
We give an existence result for constant mean curvature graphs in hyperbolic space . Let be a compact domain of a horosphere in whose boundary is mean convex, that is, its mean curvature (as a submanifold of the horosphere) is positive with respect to the inner orientation. If H is a number such that , then there exists a graph over with constant mean curvature H and boundary . Umbilical examples, when is a sphere, show that our hypothesis on H is the best possible.
Received July 18, 1997 / Accepted April 24, 1998 相似文献
20.
Qintao Deng 《Archiv der Mathematik》2008,90(4):360-373
In this paper, we consider complete hypersurfaces in R
n+1 with constant mean curvature H and prove that M
n
is a hyperplane if the L
2 norm curvature of M
n
satisfies some growth condition and M
n
is stable. It is an improvement of a theorem proved by H. Alencar and M. do Carmo in 1994. In addition, we obtain that M
n
is a hyperplane (or a round sphere) under the condition that M
n
is strongly stable (or weakly stable) and has some finite L
p
norm curvature.
Received: 14 July 2007 相似文献