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1.
We prove a Bernstein type theorem for constant mean curvature hypersurfaces in ℝ
n+1 under certain growth conditions for n ⩽ 3. Our result extends the case when M is a minimal hypersurface in the same condition.
相似文献
2.
Miguel Ortega Juan de Dios Pérez Young Jin Suh 《Czechoslovak Mathematical Journal》2006,56(2):377-388
In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space
form M
m
(c), c ≠ 0 as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional
curvature given in [11]. 相似文献
3.
Jui-Tang Ray Chen 《Annals of Global Analysis and Geometry》2009,36(2):161-190
This article concerns the structure of complete noncompact stable hypersurfaces M
n
with constant mean curvature H > 0 in a complete noncompact oriented Riemannian manifold N
n+1. In particular, we show that a complete noncompact stable constant mean curvature hypersurface M
n
, n = 5, 6, in the Euclidean space must have only one end. Any such hypersurface in the hyperbolic space with
, respectively, has only one end. 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.
We classify hypersurfaces of the hyperbolic space ?n+1(c) with constant scalar curvature and with two distinct principal curvatures. Moreover, we prove that if Mn is a complete hypersurfaces 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 ≥ c. Additionally, we prove two rigidity theorems for such hypersurfaces. 相似文献
6.
7.
We classify the hypersurfaces of revolution in euclidean space whose second fundamental form defines an abstract pseudo-Riemannian
metric of constant sectional curvature. In particular we find such piecewise analytic hypersurfaces of classC
2 where the second fundamental form defines a complete space of constant positive, zero, or negative curvature. Among them
there are closed convex hypersurfaces distinct from spheres, in contrast to a theorem of R. Schneider (Proc. AMS 35, 230–233,
(1972)) saying that such a hypersurface of classC
4 has to be a round sphere. In particular, the sphere is notII-rigid in the class of all convexC
2-hypersurfaces. 相似文献
8.
We classify the hypersurfaces of revolution in euclidean space whose second fundamental form defines an abstract pseudo-Riemannian
metric of constant sectional curvature. In particular we find such piecewise analytic hypersurfaces of class C
2
where the second fundamental form defines a complete space of constant positive, zero, or negative curvature. Among them
there are closed convex hypersurfaces distinct from spheres, in contrast to a theorem of R. Schneider (Proc. AMS 35, 230–233,
(1972)) saying that such a hypersurface of class C
4
has to be a round sphere. In particular, the sphere is not II-rigid in the class of all convex C
2
-hypersurfaces.
Received 11 October 1994; in final form 26 April 1995 相似文献
9.
In this paper, we completely classify complete hypersurfaces inR
4 with constant mean curvature and constant scalar curvature.The project was supported by NNSFC, FECC, and CPF. 相似文献
10.
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
11.
In this paper we discuss rotational hypersurfaces in and more specifically rotational hypersurfaces with periodic mean curvature function. We show that, for a given real analytic function H(s) on , every rotational hypersurface M in with mean curvature H(s) can be extended infinitely in the sense that all coordinate functions of the generating curve of M are defined on all of as well. For rotational hypersurfaces with periodic mean curvature we present a criterion characterizing the periodicity of such hypersurfaces in terms of their mean curvature function. We also discuss a method to produce families of periodic rotational hypersurfaces where each member of the family has the same mean curvature function. In fact, given any closed planar curve with curvature κ, we prove that there is a family of periodic rotational hypersurfaces such that the mean curvature of each element of the family is explicitly determined by κ. Delaunay's famous result for surfaces of revolution with constant mean curvature is included here as the case where n=3 and κ is constant. 相似文献
12.
Gil Solanes 《Transactions of the American Mathematical Society》2006,358(3):1105-1115
We give an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces. As a tool, we obtain variation formulas in integral geometry with interest in its own.
13.
This paper gives a classification of complete hypersurfaces with nonzero constant mean curvature and constant quasi-Gauss-Kronecker curvature in the hyperbolic space H4(-1),whose scalar curvature is bounded from below. 相似文献
14.
Bao Fu Wang 《数学学报(英文版)》2009,25(8):1353-1362
Given a bounded convex domain Ω with C∞ boundary and a function ψ∈C∞(δΩ), Li-Simon-Chen can construct an Euclidean complete and W-complete convex hypersurface M with constant affine Gauss-Kronecker curvature, and they guess the M is also affine complete. In this paper, we give a confirmation answer. 相似文献
15.
Yi Bing SHEN Xiao Hua ZHU 《数学学报(英文版)》2005,21(3):631-642
By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1+1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n+1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1+1 are considered. 相似文献
16.
Adrian Butscher 《Annals of Global Analysis and Geometry》2009,36(3):221-274
Four constructions of constant mean curvature (CMC) hypersurfaces in
\mathbb Sn+1{\mathbb {S}^{n+1}} are given, which should be considered analogues of ‘classical’ constructions that are possible for CMC hypersurfaces in Euclidean
space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an
equator, are shown to exist for all the values of the mean curvature. Second, a hypersurface is constructed which consists
of two chains of spheres winding around a pair of orthogonal equators, showing that Delaunay-like hypersurfaces can be fused
together in a symmetric manner. Third, a Delaunay-like handle can be attached to a generalized Clifford torus of the same
mean curvature. Finally, two generalized Clifford tori of equal but opposite mean curvature of any magnitude can be attached
to each other by symmetrically positioned Delaunay-like ‘arms’. This last result extends Butscher and Pacard’s doubling construction
for generalized Clifford tori of small mean curvature. 相似文献
17.
The purpose of this paper is to study compact or complete spacelike hypersurfaces with constant normalized scalar curvature in a locally symmetric Lorentz space satisfying some curvature conditions. We give an optimal estimate of the squared norm of the second fundamental form of such hypersurfaces. Furthermore, the totally umbilical hypersurfaces are characterized. 相似文献
18.
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
19.
Zhongmin Shen 《manuscripta mathematica》2002,109(3):349-366
We construct infinitely many two-dimensional Finsler metrics on 𝕊2 and 𝔻2 with non-zero constant flag curvature. They are all not locally projectively flat.
Received: 10 October 2001 / Revised version: 15 July 2002 相似文献
20.
本文研究了单位球中的数量曲率满足r=aH+b的完备超曲面的问题.利用极值原理的方法,获得了超曲面的一个刚性结果,推广了这一类具有常中曲率或者常数量曲率超曲面的结果. 相似文献