给定主曲率函数的一类特殊曲面的位置向量场

本文给出了 Ｒ３,Ｒ２．１内给定主曲率函数的一类特殊曲面在定义域范围内的位置向量场,从而完满地解决了［１］中所讨论的问题．     

Lorentz-Minkowski空间中旋转W超曲面

1841年,D elaunay获得如下定理:如果在一平面上沿定直线滚动一条二次圆锥直线,然后将其焦点的轨迹绕定直线旋转,则所得到的曲面具有常数平均曲率,反之,所有旋转常数平均曲率曲面(除球面外)都有如此构造.本文将以上的D elaunay定理推广到Lorentz-M inkow sk i空间Rn1 1中类空的Sm型旋转W超曲面.     

给定主曲率函数的曲面存在性定理

本文给出了Ｒ３，Ｒ２，１内给定主曲率函数的一类特殊曲面的局部、整体存在性定理的一个充要条件．     

宇宙时空中具有给定平均曲率的类空超曲面

本文研究了宇宙时空中类空超曲面的推广平均曲率流,通过估计发展超曲面的高度函数、梯度函数和曲率函数等几何量,得到了一类极限类空超曲面,其平均曲率等于给定函数.     

Lorentz空间中常平均曲率类空超曲面

本文证明了ｎ＋１维Ｌｏｒｅｎｔｚ空Ｌｎ＋１中以Ｓｎ－１（ｒ）为边界的紧致常平均曲率类空超曲面只有 Ｂｎ（ｒ）和超伪球面盖．对于 Ｒｎ＋１中的紧致常平均曲率超曲面，当高斯映照像落在一个半球面内时，得到相应的唯一性结果．     

R~3内给定Gauss曲率函数的旋转曲面的存在性定理

本文在R~3内给出了具有给定平均值零的Gauss曲率函数的旋转曲面的存在性定理。     

具有常主曲率对称函数的超曲面

Let M be a compact hypersurface is an(n 1)-dimensional complete constant curvature space N(c),If Ricci curvature of Mis not less than max {0,(n-1)c} and there is a constant main curvature function in M,then M can be classified completly,This is the Liebmann theorem in the widest sense so far.The methods used in this paper can be used to generalize a class of theorems with non-negative (of positive)sectional curvature conditions.     

复射影空间中曲率齐性实超曲面

本文讨论复射影空间中曲率齐性实超曲面，在适当条件下证明了它与复射影空间中等参超曲面等价，因此得到它的局部结构。     

De Sitter空间中半对称的类空超曲面

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。     

De Sitter空间中具有常均曲率的类空超曲面（Ⅱ）

设Ｍ＾ｎ是Ｄｅ Ｓｉｔｔｅｒ空间Ｓ１＾ｎ＋１中具有常数平均曲率且第二基本形式长度的平方为常数的完备类空超曲面，若Ｓ≤２（ｎ－１）＾１／２，则Ｍ＾ｎ是等参超曲面。     

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).

     

Complete spacelike hypersurfaces with constant mean curvature in anti-de Sitter space

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].     

On the volume and the Gauss map image of spacelike hypersurfaces in de Sitter space

In this paper we prove that a complete spacelike hypersurface in de Sitter space such that its image under the Gauss map is contained in a hyperbolic geodesic ball of radius is necessarily compact and its -dimensional volume satisfies , where denotes the volume of a unitary round -sphere. We also characterize the case where these inequalities become equalities. As an application of our result, we also conclude that Goddard's conjecture is true under the assumption that the hyperbolic image of the hypersurface is bounded.

     

On spacelike hypersurfaces with two distinct principal curvatures in Lorentzian space forms

We investigate the spacelike hypersurfaces in Lorentzian space forms (n?4) with two distinct non-simple principal curvatures without the assumption that the (high order) mean curvature is constant. We prove that any spacelike hypersurface in Lorentzian space forms with two distinct non-simple principal curvatures is locally conformal to the Riemannian product of two constant curved manifolds. We also obtain some characterizations for hyperbolic cylinders in Lorentzian space forms in terms of the trace free part of the second fundamental form.     

On the complete linear Weingarten spacelike hypersurfaces with two distinct principal curvatures in Lorentzian space forms

We deal with complete linear Weingarten spacelike hypersurfaces immersed in a Lorentzian space form, having two distinct principal curvatures. In this setting, we show that such a spacelike hypersurface must be isometric to a certain isoparametric hypersurface of the ambient space, under suitable restrictions on the values of the mean curvature and of the norm of the traceless part of its second fundamental form. Our approach is based on the use of a Simons type formula related to an appropriated Cheng–Yau modified operator jointly with some generalized maximum principles.     

Rotational hypersurfaces of periodic mean curvature

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.     

Complete hypersurfaces in a 4-dimensional hyperbolic space

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.