Mean Curvature and Volume of Extrinsic Spheres in Hypersurfaces of Real Space Forms

Given a hypersurface P^n-1 in a real space form of constantcurvature b, , we have obtained a lower bound for the norm of the mean curvature normal vector field of extrinsicspheres in P^n-1 in terms of the mean curvature of the geodesic spheres in , with the same radius, and the meancurvature of P^n-1, characterizing too the equality.     

On Ruled Real Hypersurfaces in Complex Space Forms

Ruled real hypersurfaces of complex space forms are investigated by using the fact that such hypersurfaces can be constructed by moving a 1-codimensional complex totally geodesic submanifold of the ambient space along a curve. Among other results, a classification of minimal ruled real hypersurfaces and an example of a homogeneous ruled real hypersurface are given.     

A Rigidity Theorem for Compact Hypersurfaces with an Upper Bound for the Ricci Curvature

We focus our attention on compact hypersurfaces with Ricci curvature bounded from above and we give a sufficient condition for them to be spherical. This generalizes and completes previous results.     

Classifications of Isoparametric Hypersurfaces in Randers Space Forms

In this paper, we give the complete classifications of isoparametric hypersurfaces in Randers space forms. By studying the principal curvatures of anisotropic submanifolds in a Randers space (N, F) with the navigation data (h, W), we find that a Randers space form (N, F, dμ_BH) and the corresponding Riemannian space (N, h) have the same isoparametric hypersurfaces, but in general, their isoparametric functions are different. We give a necessary and sufficient condition for an isoparametric function of (N, h) to be isoparametric on (N, F, dμ_BH), from which we get some examples of isoparametric functions.     

三维空间形式中常平均曲率曲面的稳定性

本文利用共形度量高斯曲率的估计研究了三维空间形式N^3(C）中具常平均曲率曲面的区域稳定性。     

A Characterization of Quadric Constant Mean Curvature Hypersurfaces of Spheres

Let be an immersion of a complete n-dimensional oriented manifold. For any v∈ℝ ⁿ⁺², let us denote by ℓ _v :M→ℝ the function given by ℓ _v (x)=〈φ(x),v〉 and by f _v :M→ℝ, the function given by f _v (x)=〈ν(x),v〉, where is a Gauss map. We will prove that if M has constant mean curvature, and, for some v≠0 and some real number λ, we have that ℓ _v =λ f _v , then, φ(M) is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface M ⁿ in which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to 2n+4. A. Brasil Jr. was partially supported by CNPq, Brazil, 306626/2007-1.     

Curvature Estimates for Strongly Stable Domains on Surfaces with Constant Mean Curvature in Space Forms

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Space-like Isoparametric Hypersurfaces in Lorentzian Space Forms

The purpose of this paper is to make clear the so-called Nomizu problem, whether it is possible to find examples of space-like isoparametric hypersurfaces in H ₁ ⁿ⁺¹ with more than two distinct principal curvatures. It is proved that a space-like isoparametric hypersurface in H ₁ ⁿ⁺¹ or S ₁ ⁿ⁺¹ can have at most two distinct principal curvatures. The authors present the classification and explicit analytic expressions of such type of isoparametric hypersurfaces. This paper was translated from J. Nanchang Univ. Nat. Sci. Ed., 2004, 28(2): 113–117     

Pseudosymmetric Lightlike Hypersurfaces in Indefinite Sasakian Space Forms

We study pseudosymmetric lightlike hypersurfaces of an indefinite Sasakian space form, tangent to the structure vector field. We obtain sufficient conditions for a lightlike hypersurface to be pseudosymmetric, pseudoparallel and Ricci-pseudosymmetric in an indefinite Sasakian space form. We also find certain conditions for a pseudosymmetric lightlike hypersurface of an indefinite Sasakian space form to be totally geodesic and check the effect of Weyl projective pseudosymmetry conditions on the geometry of a lightlike hypersurface of an indefinite Sasakian space form. Moreover we give some physical interpretations of pseudo-symmetry conditions.     

Hypersurfaces with Constant Mean Curvature in Space Forms

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实空间形式中超曲面的刚性定理

本文给出实空间形式中超曲面的一个刚性定理,同时也得到求面中子流形的刚性结果．     

Ribaucour Transformations for Hypersurfaces in Space Forms

The theory of Ribaucour transformations for hypersurfaces in space forms is established. For any such hypersurface M, that admits orthonormal principal vector fields, it was shown the existence of a totally umbilic hypersurface locally associated to M by a Ribaucour transformation. A method of obtaining linear Weingarten surfaces in a three-dimensional space form is provided. By applying the theory, a new one-parameter family of complete constant mean curvature (cmc) surfaces in the unit sphere, locally associated to the flat torus, is obtained. The family contains a class of complete cmc cylinders in the sphere. In particular, one gets a family of complete minimal surfaces and minimal cylinders, locally associated to the Clifford torus.Mathematics Subject Classifications (2000): 53C20.     

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

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).

     

空间形式中具常数量曲率的子流形的内蕴刚性

Under the assumption that the normalized mean curvature vector is parallel in the normal bundle, by using the generalized ChengYau＇s self-adjoint differential operator, here we obtain some rigidity results for compact submanifolds with constant scalar curvature and higher codimension in the space forms.     

从不定度量空间形式到不定度量空间形式的等距浸入

设Msm(c)是等距浸入在2m-1维不定度量空间形式Ns2m-1((?))(c<(?))中的m维不定度量空间形式．若Msm(c)是极小的,我们证明Msm必定是有同一个指标s的2m-1维伪球面中的平坦子流形．我们还用孤立子理论给出了Ns2m-1中平坦的指标为s的子流形与系统之间的对应．     

Ricci Curvature of Certain Submanifolds in Kenmotsu Space Forms

In this paper, we obtain some sharp inequalities between the Ricci cur- vature and the squared mean curvature for bi-slant and semi-slant submanifolds in Kenmotsu space forms. Estimates of the scalar curvature and the k-Ricci curvature, in terms of the squared mean curvature, are also proved respectively.     

Minimizing Constant Mean Curvature Hypersurfaces in Hyperbolic Space

We study the constant mean curvature (CMC) hypersurfaces in whose asymptotic boundaries are closed codimension-1 submanifolds in . We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area-minimizing, convex hull property, exchange roundoff trick to CMC hypersurface context. We also give a generic uniqueness result for CMC hypersurfaces in hyperbolic space.     

余辛流形的半不变子流形的Ricci曲率

本文研究了余辛流形的半不变子流形,得到了这类子流形的Ricci曲率与平均曲率平方之间的—个不等式,并讨论了等式成立的充分必要条件．