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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 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
2.
Knut Smoczyk 《manuscripta mathematica》1998,95(1):225-236
Under the assumption of two a-priori bounds for the mean curvature, we are able to generalize a recent result due to Huisken
and Sinestrari [8], valid for mean convex surfaces, to a much larger class. In particular we will demonstrate that these a-priori
bounds are satisfied for a class of surfaces including meanconvex as well as starshaped surfaces and a variety of manifolds
that are close to them. This gives a classification of the possible singularities for these surfaces in the casen=2. In addition we prove that under certain initial conditions some of them become mean convex before the first singularity
occurs. 相似文献
3.
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
4.
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
5.
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
6.
We use equivariant geometry methods to show the existence of complete
hypersurfaces in euclidean spaces with constant negative
2n-th curvature.Received: 3 July 2002 相似文献
7.
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
8.
Knut Smoczyk 《Calculus of Variations and Partial Differential Equations》1996,4(2):155-170
This paper concerns the deformation by mean curvature of hypersurfaces M in Riemannian spaces Ñ that are invariant under a subgroup of the isometry-group on Ñ. We show that the hypersurfaces contract to this subgroup, if the cross-section satisfies a strong convexity assumption.This forms part of the authors doctoral thesis and was carried out while the author was supported by a scholarship of the Graduiertenkolleg für Geometrie und Mathematische Physik. 相似文献
9.
Knut Smoczyk 《manuscripta mathematica》1998,95(2):225-236
Under the assumption of two a-priori bounds for the mean curvature, we are able to generalize a recent result due to Huisken
and Sinestrari [8], valid for mean convex surfaces, to a much larger class. In particular we will demonstrate that these a-priori
bounds are satisfied for a class of surfaces including meanconvex as well as starshaped surfaces and a variety of manifolds
that are close to them. This gives a classification of the possible singularities for these surfaces in the case n= 2. In addition we prove that under certain initial conditions some of them become mean convex before the first singularity
occurs.
Received: 6 June 1997 / Revised version: 24 October 1997 相似文献
10.
Sven Winklmann 《manuscripta mathematica》2005,118(4):485-499
We consider immersed hypersurfaces :Mn→ℝn+1 with prescribed anisotropic mean curvature . Such hypersurfaces can be characterized as critical points of parametric functionals of the type with an elliptic Lagrangian F depending on normal directions and a smooth vectorfield Q satisfying . We establish curvature estimates for stable hypersurfaces of dimension n≤5, provided F is C3-close to the area integrand. 相似文献
11.
F.E.C. Camargo R.M.B. Chaves L.A.M. Sousa Jr. 《Differential Geometry and its Applications》2008,26(6):592-599
In this paper we give a partially affirmative answer to the following question posed by Haizhong Li: is a complete spacelike hypersurface in De Sitter space , n?3, with constant normalized scalar curvature R satisfying totally umbilical? 相似文献
12.
Ramesh Sharma 《Journal of Geometry》2003,78(1-2):156-167
Contact hypersurfaces of a Kaehler manifold have been
characterized and classified, assuming the second fundamental form to be
Codazzi (in particular, parallel). We have also discussed the special
cases when the ambient space is a (i) Calabi-Yau manifold and (ii) a
complex space-form. 相似文献
13.
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
14.
15.
Pak Tung Ho 《Differential Geometry and its Applications》2008,26(3):273-276
Suppose that M is a compact orientable hypersurface embedded in a compact n-dimensional orientable Riemannian manifold N. Suppose that the Ricci curvature of N is bounded below by a positive constant k. We show that 2λ1>k−(n−1)maxM|H| where λ1 is the first eigenvalue of the Laplacian of M and H is the mean curvature of M. 相似文献
16.
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
17.
18.
We prove that, in Euclidean space, any nonnegatively curved, compact, smoothly immersed hypersurface lies outside the convex hull of its boundary, provided the boundary satisfies certain required conditions. This gives a convex hull property, dual to the classical one for surfaces with nonpositive curvature. A version of this result in the nonsmooth category is obtained as well. We show that our boundary conditions determine the topology of the surface up to at most two choices. The proof is based on uniform estimates for radii of convexity of these surfaces under a clipping procedure, a noncollapsing convergence theorem, and a gluing procedure. 相似文献
19.
Ezio Araujo Costa 《Archiv der Mathematik》2005,85(2):183-189
Let Mn be a complete Riemannian manifold immersed isometrically in the unity Euclidean sphere
In [9], B. Smyth proved that if Mn, n ≧ 3, has sectional curvature K and Ricci curvature Ric, with inf K > −∞, then sup Ric ≧ (n − 2) unless the universal covering
of Mn is homeomorphic to Rn or homeomorphic to an odd-dimensional sphere. In this paper, we improve the result of Smyth. Moreover, we obtain the classification of complete hypersurfaces of
with nonnegative sectional curvature.Received: 11 November 2003 相似文献
20.
For hypersurfaces with regular Weingarten operator in nonflat space forms we study the relations between the intrinsic geometry of the third fundamental form metric and the extrinsic geometry of the hypersurface. We prove a theorema-egregium-type result for this metric and, in particular, give a local classification of hypersurfaces in case of an Einstein structure of this metric.Partially supported by the project 19701003 of NSFC.The geometry groops at TU Berlin and KU Leuven cooperate within the GA DGET program. 相似文献