广义de Sitter空间中法向Lorentzian超曲面的nullcone Gauss映射的奇点

利用奇点理论研究了广义de Sitter空间中具有Lorentzian法空间的一类超曲面.介绍了这类超曲面的局部微分几何,定义了nullcone Gauss映射及nullcone高度函数族,进而研究了nullcone高度函数族的性质及nullcone高斯映射的几何意义,最后研究了这类超曲面的通有性质.     

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.     

n维Anti-de Sitter空间中类空超曲面的类时Gauss像的奇点

本文利用奇点理论研究了n维Anti-deSitter空间中的类空超曲面,介绍了类空超曲面的局部微分几何,定义了类时Anti-deSitterGauss像及Anti-deSitter高度函数,并进一步的利用Anti-deSitter高度函数族和流形间的切触理论研究了类时Anti-deSitterGauss像的几何意义及类空超曲面的通有性.最后研究了类空超曲面的AdS-Monge型.     

3维双曲空间中曲面的双曲Gauss映照和法Gauss映照

本文导出了3维双曲空间中曲面的双曲Gauss映照和法Gauss映照的关系,发现了一般的曲面由双曲Gauss映照和平均曲率函数唯一确定,并证明了双曲Gauss映照所满足的二阶线性椭圆方程,给出了两种形式的关于双曲Gauss映照的三阶非线性偏微分方程(组)的一个解.     

Compact Homogeneous Hypersurfaces in Hyperbolic Space

In this paper we prove that an n-dimensional compact homogeneous Riemannian manifold isometrically immersed in the hyperbolic space Hⁿ⁺¹ is isometric to a sphere.AMS Subject Classification (1990): 53A07, 53C42, 57S15     

Cohomogeneity One Hypersurfaces of the Hyperbolic Space

In this paper we extend to the hyperbolic space results firstly obtainedin Podestà and Spiro (Ann. Global Anal. Geom. 13(2) (1995), 169–184) for compact cohomogeneity, Riemannian manifoldsimmersed as hypersurfaces of the Euclidean space and in Seixas(Ph.D. Thesis, 1996), wherethe complete case was studied. Both works give sufficient conditionsfor the hypersurface to be of revolution. We study cohomogeneity, complete hypersurfaces of the hyperbolic space and prove a similar resultof Seixas (Ph.D. Thesis, 1996), obtaining also a class of nonrotational examples.     

The value distribution of the hyperbolic Gauss map

In this paper, we investigate the hyperbolic Gauss map of a complete CMC-1 surface in , and prove that it cannot omit more than four points unless the surface is a horosphere.

     

Immersed surfaces of prescribed Gauss curvature into Minkowski space

Given a positive real valued function on the disc, we will immerse the disc into three dimensional Minkowski space in such a way that Gauss curvature at the image point of is . Our approach lies on the construction of Gauss map of surfaces.

     

Radial Graphs with Constant Mean Curvature in the Hyperbolic Space

The existence is proved of radial graphs with constant mean curvature in the hyperbolic space H ⁿ⁺¹ defined over domains in geodesic spheres of H ⁿ⁺¹ whose boundary has positive mean curvature with respect to the inward orientation.     

Singularities of linear systems and the Waring problem

The Waring problem for homogeneous forms asks for additive decomposition of a form into powers of linear forms. A classical problem is to determine when such a decomposition is unique. In this paper we answer this question when the degree of is greater than the number of variables. To do this we translate the algebraic statement into a geometric one concerning the singularities of linear systems of with assigned singularities.

     

Products of Hyperbolic Metric Spaces

Let (X _i d _i ), i=1,2, be proper geodesic hyperbolic metric spaces. We give a general construction for a 'hyperbolic product' X ₁× _h X ₂ which is itself a proper geodesic hyperbolic metric space and examine its boundary at infinity.     

5维伪欧氏空间中3维类时子流形的奇点分类

主要通过定义在指标数为2的5维伪欧氏空间中的3维类时子流形M上的类时高斯映射和类时高度函数,并研究M与管状超曲面CM的奇点分类.     

Willmore子流形的共形高斯映射

本文研究球空间中子流形的共形高斯映射,用Moebius不变量刻划了该映射 为调和映射的条件.作为特例,指出球空间的2维子流形的共形高斯映射是调和映射 当且仅当该子流形是Willmore子流形.     

Maps Between Complex Hyperbolic Surfaces

We study examples of surjective holomorphic maps between complex hyperbolic surfaces that are not covering maps. The induced homomorphisms on the fundamental groups have infinite kernel, thus are examples of the nonstandard homomorphisms found by Mostow. We also study the singularities of these maps.     

A Parametrization of Isometric Immersions between Hyperbolic Spaces

We parametrize the space of isometric immersions of the hyperbolic n -space into the hyperbolic ( n +1)-space by a family of properly chosen (at most) countable n -tuples of real-valued functions. This is an answer to the problem posed by Nomizu in 1973. We also construct a one-parameter family of deformable isometric immersions from the quotient spaces of the hyperbolic 2-plane modulo and some totally discontinuous subgroups of isometries.     

广义de Sitter空间中的类时超曲面的切触性质

从切触几何及Legendrian奇点理论的角度研究了广义de sitter空间中的类时超曲面的切触性质及gdS-高斯像的奇点的分类和几何意义.     

Hyperbolic Hilbert space

The hyperbolic complex space is one class of non-Euclidean spaces with continuous singular points. It corresponds with Minkowski space, and it has the characteristic that the space-time direction is different in nature. Regard the hyperbolic complex space as original spaces. We can abstract a class of the hyperbolic inner product space and the hyperbolic Hilbert space.     

Supercritical Elliptic Equation in Hyperbolic Space

In this paper, we study the following semi-linear elliptic equation $$-Δ_H^nu=|u|^{p-2}u,\qquad\qquad (0.1)$$ in the whole Hyperbolic space $\mathbb{H}^n$,where n ≥ 3, p › 2n/(n-2). We obtain some regularity results for the radial singular solutions of problem (0.1). We show that the singular solution $u^∗$ with $lim_{t → 0}(sinht)^{\frac{2}{p-2}}⋅u(t)=±(\frac{2}{p-2}(n-2-\frac{2}{p-2})^{\frac{1}{p-2}}$ belongs to the closure (in the natural topology given by $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N))$ of the set of smooth classical solutions to the Eq. (0.1). In contrast, we also prove that any oscillating radial solutions of (0.1) on $\mathbb{H}^N$\{0} fails to be in the space $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N)$.     

A Remark on Isolated Singularities at the Free Boundary of Harmonic Maps

Using a simple lemma on harmonic functions it is shown how well known results can be combined to exclude isolated singularities at the free boundary of two-dimensional weakly harmonic maps.