Dupin超曲面的主曲率的重数

作者证明了如下结果 :设M是空间形式的闭定向Dupin超曲面 ,其截面曲率为正 ,M至少有两个不同主曲率 .如果除最小 (或最大 )主曲率外 ,M的其余主曲率均为常数 ,则最小 (或最大 )主曲率的重数大于 2 .     

R~6中的Laguerre等参超曲面的分类

若超曲面的Laguerre形式为零且Laguerre第二基本形式的特征值(称为Laguerre主曲率)为常数,则称超曲面为Laguerre等参超曲面.对R~6中的Laguerre等参超曲面进行了研究,得到了分类定理.     

关于Weingarten超曲面的一个存在性定理

本文研究伪黎曼空间型中一类特殊的Weingarten超曲面的存在性问题.通过引入伪黎曼空间型中旋转超曲面的概念,并给出其主曲率计算公式.得到伪黎曼空间型中旋转型Weingarten超曲面的存在性定理.推广并统一了相关结果.     

共形空间中有两个共形主曲率的共形等参超曲面

研究了共形空间中正则超曲面的共形几何,并在共形等价意义下对有两个共形主曲率的共形等参超曲面作了分类.     

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.     

高斯曲率估计与常数平均曲率超曲面的稳定性

本文利用活动标架法及Ｌａｐｌａｃｉａｎ特征值方法研究了常数平均曲率超曲面的稳定性．给出了三维拟常曲率流形中具有常数平均曲率超曲面的共形度量的高斯曲率之上界估计．证明了三维拟常曲率流形中具有常数平均曲率超曲面上一个单连通有界区域为稳定的充分条件．     

常曲率空间的半平行超曲面和高阶平行超曲面的分类

本文给出了非欧氏常曲率空间Ｎ＾ｎ＋１（Ｃ）的半平行超曲面的分类，并利用此分类定理证明了非欧氏常曲率空间的高阶平行超曲面与平行超曲面的等价性，从而也给出了非欧氏常曲率空间的高阶平行超曲面的分类；     

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。     

常曲率黎曼流形的共形平坦的极小超曲面(英文)

本文研究常曲率黎曼流形 S~(n+1)(c)中的共形平坦的极小超曲面 M~h,证明了下面结果.定理 设 M~h 是 n+1维常曲率黎曼流形 S~(n+1)(c)的共形平坦超曲面(n≥4),则 M~n是常数量曲率的极小超曲面的充要条件是:(1)M~n 的数量曲率 R=(n-1)c 时,M~n 是全测地超曲面,从而也有常曲率 c;(2)M~n 的数量曲率 R≠n(n-1)c 时,c>0和 M~n 局部可约为常曲率黎曼流形S~(n-1)(n/(n-1) c)与直线 R′的乘积.系,设 M~n 是具有非正常曲率 c 的黎曼流形 S~(n+1)(c)的共形平坦超曲面(n≥4),如果M~n 是常数量曲率的极小超曲面,则 M~n 是全测地超曲面。     

双曲空间形式中具有常平均曲率的超曲面(英文)

本文研究了双曲空间形式H~(n+1)(—1)中具有常平均曲率及两个离散主曲率(其中一个主曲率是1-重)的完备连通可定向的n-维超曲面M~n.利用活动标架,得到如果M~n的基本形式的模长满足刚性条件(1.3),那么M~n同构双曲柱面.     

A sharp height estimate for compact spacelike hypersurfaces with constant r-mean curvature in the Lorentz-Minkowski space and application

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.     

Complete Linear Weingarten Spacelike Hypersurfaces Immersed in a Locally Symmetric Lorentz Space

In this paper, we establish a characterization theorem concerning the complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space, whose sectional curvature is supposed to obey certain appropriated conditions. Under a suitable restriction on the length of the second fundamental form, we prove that a such spacelike hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures one of which is simple.     

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.     

Lorentz-Minkowski空间中的乘积空间R~k×H~(n-k)

本文的主要结果是:Lorentz-Minkowski空间中介于两个同心伪圆柱面之间的完备、类空、常平均曲率超曲面必为伪圆柱面,即乘积空间R~k×H~(n-k)．对于常高阶平均曲率的情况,如果截曲率有下界,那么介于两个同心伪球面之间的完备类空超曲面必为伪球面．     

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.     

Extensions of the mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space

In this paper we define one-parameter families of Legendrian double fibrations in the products of pseudo-spheres in Lorentz-Minkowski space which are the extensions of four Legendrian double fibrations in the previous research (Izumiya, 2009 [9]). We show that these are contact diffeomorphic to each other. Moreover, we construct one-parameter families of new extrinsic differential geometries on spacelike hypersurfaces in these pseudo-spheres as applications of such extensions of the Legendrian double fibrations.     

A Weierstrass representation for linear Weingarten spacelike surfaces of maximal type in the Lorentz-Minkowski space

In this work we extend the Weierstrass representation for maximal spacelike surfaces in the 3-dimensional Lorentz-Minkowski space to spacelike surfaces whose mean curvature is proportional to its Gaussian curvature (linear Weingarten surfaces of maximal type). We use this representation in order to study the Gaussian curvature and the Gauss map of such surfaces when the immersion is complete, proving that the surface is a plane or the supremum of its Gaussian curvature is a negative constant and its Gauss map is a diffeomorphism onto the hyperbolic plane. Finally, we classify the rotation linear Weingarten surfaces of maximal type.     

四维Minkowski空间中类空超曲面的双曲高斯映射的奇点

介绍了四维Minkowski空间中类空超曲面的局部理论,定义了类空超曲面上的双曲高斯映射,双曲高度函数及距离平方函数,给出了一些定理的详细证明.介绍了一种证明高度函数是Morse族的新方法并应用Arnold等建立的Lagrange奇点理论对类空超曲面的双曲高斯映射的奇点进行了分类.     

Curvature properties of compact spacelike hypersurfaces in de Sitter space

In this paper we establish a sufficient condition for a compact spacelike hypersurface in de Sitter space to be spherical in terms of a lower bound for the square of its mean curvature. Our result will be a consequence of the maximum principle for the Laplacian operator. We also derive some other applications and consequences of our main result. In particular, we establish another sufficient condition for a compact spacelike hypersurface in de Sitter space to be spherical in terms of a pinching condition for its scalar curvature, as well as in terms of the Ricci curvature and in terms of the higher order mean curvatures     

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.