三维Lorentz空间形式的共形群

研究了三维Lorentz空间形式R1^3,S1^2,H1^3的共形群,通过计算得到R1^3,S1^3,H1^3的共形群的具体表达形式,为进一步研究三维Lorentz空间式上的共形几何奠定基础．     

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in S4, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them the interesting duality theorem holds. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.     

Global geometry and topology of spacelike stationary surfaces in the 4-dimensional Lorentz space

For spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space, one can naturally introduce two Gauss maps and a Weierstrass-type representation. In this paper we investigate the global geometry of such surfaces systematically. The total Gaussian curvature is related with the surface topology as well as the indices of the so-called good singular ends by a Gauss–Bonnet type formula. On the other hand, as shown by a family of counterexamples to Osserman?s theorem, finite total curvature no longer implies that Gauss maps extend to the ends. Interesting examples include the deformations of the classical catenoid, the helicoid, the Enneper surface, and Jorge–Meeks? k-noids. Each family of these generalizations includes embedded examples in the 4-dimensional Lorentz space, showing a sharp contrast with the 3-dimensional case.     

三维洛仑兹空间形式中的类光曲线

讨论三维洛仑兹空间形式中的类光曲线粒子模型,研究依赖于粒子轨道的Cartan曲率的作用,找出了沿着极值曲线的Killing场,通过Killing场构造合适的坐标系,用积分求出极值曲线的参数表达式.     

On constant slope spacelike surfaces in 3-dimensional Minkowski space

A spacelike surface in the Minkowski 3-space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface. In this work, we study such surfaces and classify all of them.     

Conformal isoparametric hypersurfaces with two distinct conformal principal curvatures in conformal space

The conformal geometry of regular hypersurfaces in the conformal space is studied.We classify all the conformal isoparametric hypersurfaces with two distinct conformal principal curvatures in the conformal space up to conformal equivalence.     

Laguerre homogeneous surfaces in R~3

In this paper,we study oriented surfaces of R3 in the context of Laguerre geometry.We construct Laguerre invariants on the non-Dupin developable surfaces,which determine the surfaces up to a Laguerre transformation.Finally,we classify the Laguerre homogeneous surfaces in R3 under the Laguerre transformation groups.     

三维空间中的共形曲线论

本文对单位球面S~3中的曲线在Mbius变换下的性质进行了研究.给S~3中的曲线定义了一套由共形弧长参数、共形曲率和共形挠率组成的Mbius不变参数系统,并证明这个系统决定S~3中所有光滑曲线在Mbius变换下的分类.     

Invariant surfaces in the homogeneous space Sol with constant curvature

A surface in homogeneous space is said to be an invariant surface if it is invariant under some of the two 1‐parameter groups of isometries of the ambient space whose fix point sets are totally geodesic surfaces. In this work we study invariant surfaces that satisfy a certain condition on their curvatures. We classify invariant surfaces with constant mean curvature and constant Gaussian curvature. Also, we characterize invariant surfaces that satisfy a linear Weingarten relation.     

Equivariant Willmore surfaces in conformal homogeneous three spaces

The complete classification of homogeneous three spaces is well known for some time. Of special interest are those with rigidity four which appear as Riemannian submersions with geodesic fibres over surfaces with constant curvature. Consequently their geometries are completely encoded in two values, the constant curvature, c$c$, of the base space and the so called bundle curvature, r$r$. In this paper, we obtain the complete classification of equivariant Willmore surfaces in homogeneous three spaces with rigidity four. All these surfaces appear by lifting elastic curves of the base space. Once more, the qualitative behaviour of these surfaces is encoded in the above mentioned parameters (c,r)$(c,r)$. The case where the fibres are compact is obtained as a special case of a more general result that works, via the principle of symmetric criticality, for bundle-like conformal structures in circle bundles. However, if the fibres are not compact, a different approach is necessary. We compute the differential equation satisfied by the equivariant Willmore surfaces in conformal homogeneous spaces with rigidity of order four and then we reduce directly the symmetry to obtain the Euler Lagrange equation of 4r²$4r^2$-elasticae in surfaces with constant curvature, c$c$. We also work out the solving natural equations and the closed curve problem for elasticae in surfaces with constant curvature. It allows us to give explicit parametrizations of Willmore surfaces and Willmore tori in those conformal homogeneous 3-spaces.     

Conformal geometry of surfaces in Lorentzian space forms

We study the conformal geometry of an oriented space-like surface in three-dimensional Lorentzian space forms. After introducing the conformal compactification of the Lorentzian space forms, we define the conformal Gauss map which is a conformally invariant two parameter family of oriented spheres. We use the area of the conformal Gauss map to define the Willmore functional and derive a Bernstein type theorem for parabolic Willmore surfaces. Finally, we study the stability of maximal surfaces for the Willmore functional.Dedicated to Professor T.J. WillmoreSupported by an FPPI Postdoctoral Grant from DGICYT Ministerio de Educación y Ciencia, Spain 1994 and by a DGICYT Grant No. PB94-0750-C02-02     

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.     

A kind of helicoidal surfaces in 3-dimensional Minkowski space

In this paper we constructed a helicoidal surface with a light-like axis with prescribed mean curvature or Gauss curvature given by smooth function in 3-dimensional Minkowski space and solved an open problem left by Beneki, Kaimakamis, and Papantoniou in [J. Math. Anal. Appl. 275 (2002) 586-614].     

三维欧氏空间中的一类Weingarten螺旋面

通过求解相关的非线性常微分方程,构造了三维欧氏空间中主曲率之差为常数的螺旋面,并证明这类曲面的广泛存在性.     

A complete classification of parallel surfaces in three-dimensional homogeneous spaces

We complete the classification of surfaces with parallel second fundamental form in all three-dimensional homogeneous spaces. The second named author is a postdoctoral researcher of the Research Foundation—Flanders (FWO).     

Maximal surfaces in a certain 3-dimensional homogeneous spacetime

A generalized integral representation formula for spacelike maximal surfaces in a certain 3-dimensional homogeneous spacetime is obtained. This spacetime has a solvable Lie group structure with left invariant metric. The normal Gauß map of maximal surfaces in the homogeneous spacetime is discussed and the harmonicity of the normal Gauß map is studied.     

Critical curves for the total normal curvature in surfaces of 3-dimensional space forms

A variational problem closely related to the bending energy of curves contained in surfaces of real 3-dimensional space forms is considered. We seek curves in a surface which are critical for the total normal curvature energy (and its generalizations). We start by deriving the first variation formula and the corresponding Euler–Lagrange equations of these energies and apply them to study critical special curves (geodesics, asymptotic lines, lines of curvature) on surfaces. Then, we show that a rotation surface in a real space form for which every parallel is a critical curve must be a special type of a linear Weingarten surface. Finally, we give some classification and existence results for this family of rotation surfaces.     

Minkowski空间中保高斯映射的共形形变

设Ｍｎ为Ｒｉｅｍａｎｎ流形，给定类空浸入：Ｍｎ→Ｒｎ，ｐ，如果存在另一个类空浸入：Ｍｎ→Ｒｎ，ｐ，使与在共形对应之下且对应点的地空间平行，则称类空子流形是可保高斯映射共形形变的．本文给出可保高斯映射共形形变的充要条件．对ｎ＝２，ｐ＝１的情形，如果上述形变是同向的，我们分类了曲面；如果是反向的，我们用主曲率满足的方程来描述．     

Biharmonic curves in 3-dimensional Sasakian space forms

We show that every proper biharmonic curve in a 3-dimensional Sasakian space form of constant holomorphic sectional curvature H is a helix (both of whose geodesic curvature and geodesic torsion are constants). In particular, if H ≠  1, then it is a slant helix, that is, a helix which makes constant angle α with the Reeb vector field with the property . Moreover, we construct parametric equations of proper biharmonic herices in Bianchi–Cartan–Vranceanu model spaces of a Sasakian space form.