Remarks on Complete Spacelike Hyper-surfaces with Constant Mean Curvature in the de Sitter Space

Let M be an n(≥3)-dimensional completely non-compact spacelike hypersurface in the de Sitter space S1 (n+1) (1) with constant mean curvature and non negative sectional curvature. It is proved that M is isometric to a hyperbolic cylinder or an Euclidean space if H ≥1. When 2(n-1)~(1.2)/n < H < 1, there exists a complete rotation hypersurfaces which is not a hyperbolic cylinder.     

Conformal CMC-Surfaces in Lorentzian Space Forms

Let Q3 be the common conformal compactification space of the Lorentzian space forms R13, S13 and H13. We study the conformal geometry of space-like surfaces in Q3. It is shown that any conformal CMC-surface in Q3 must be conformally equivalent to a constant mean curvature surface in R13, S13 or H13. We also show that if x : M→Q3 is a space-like Willmore surface whose conformal metric g has constant curvature K, then either K = - 1 and x is conformally equivalent to a minimal surface in R13, or K = 0 and x is conformally equivalent to the surface H1(1/(2~(1/2)))×H1(1/(2~(1/2))) in H13.     

Laguerre minimal surfaces in ℝ3

Laguerre geometry of surfaces in R^3 is given in the book of Blaschke, and has been studied by Musso and Nicolodi, Palmer, Li and Wang and other authors. In this paper we study Laguerre minimal surface in 3-dimensional Euclidean space R^3. We show that any Laguerre minimal surface in R^3 can be constructed by using at most two holomorphic functions. We show also that any Laguerre minimal surface in R^3 with constant Laguerre curvature is Laguerre equivalent to a surface with vanishing mean curvature in the 3-dimensional degenerate space R0^3.     

Time-like Willmore surfaces in Lorentzian 3-space

Let R13 be the Lorentzian 3-space with inner product (, ). Let Q3 be the conformal compactification of R13, obtained by attaching a light-cone C∞ to R13 in infinity. Then Q3 has a standard conformal Lorentzian structure with the conformal transformation group O(3,2)/{±1}. In this paper, we study local conformal invariants of time-like surfaces in Q3 and dual theorem for Willmore surfaces in Q3. Let M (?) R13 be a time-like surface. Let n be the unit normal and H the mean curvature of the surface M. For any p ∈ M we define S12(p) = {X ∈ R13 (X - c(P),X - c(p)) = 1/H(p)2} with c(p) = P 1/H(p)n(P) ∈ R13. Then S12 (p) is a one-sheet-hyperboloid in R3, which has the same tangent plane and mean curvature as M at the point p. We show that the family {S12(p),p ∈ M} of hyperboloid in R13 defines in general two different enveloping surfaces, one is M itself, another is denoted by M (may be degenerate), and called the associated surface of M. We show that (i) if M is a time-like Willmore surface in Q3 with non-degenerate associated surface M, then M is also a time-like Willmore surface in Q3 satisfying M = M; (ii) if M is a single point, then M is conformally equivalent to a minimal surface in R13.     

Closed Strong Spacelike Curves,Fenchel Theorem and Plateau Problem in the 3-Dimensional Minkowski Space

The authors generalize the Fenchel theorem for strong spacelike closed curves of index 1 in the 3-dimensional Minkowski space, showing that the total curvature must be less than or equal to 2π. Here the strong spacelike condition means that the tangent vector and the curvature vector span a spacelike 2-plane at each point of the curve γ under consideration. The assumption of index 1 is equivalent to saying that γ winds around some timelike axis with winding number 1. This reversed Fenchel-type inequality is proved by constructing a ruled spacelike surface with the given curve as boundary and applying the Gauss-Bonnet formula. As a by-product, this shows the existence of a maximal surface with γ as the boundary.     

Classification of Lagrangian Surfaces of Curvature e in Non-flat Lorentzian Complex Space Form M12（4ε）

It is well known that a totally geodesic Lagrangian surface in a Lorentzian complex space form M12（4ε） of constant holomorphic sectional curvature 4s is of constant curvature 6. A natural question is ＂Besides totally geodesic ones how many Lagrangian surfaces of constant curvature εin M12（46） are there？＂ In an earlier paper an answer to this question was obtained for the case e = 0 by Chen and Fastenakels. In this paper we provide the answer to this question for the case ε≠0. Our main result states that there exist thirty-five families of Lagrangian surfaces of curvature ε in M12（4ε） with ε ≠ 0. Conversely, every Lagrangian surface of curvature ε≠0 in M12（4ε） is locally congruent to one of the Lagrangian surfaces given by the thirty-five families.     

Laguerre Isopararmetric Hypersurfaces in R^4

Let x : M → Rn be an umbilical free hypersurface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C , and a Laguerre second fundamental form B which are invariants of x under Laguerre transformation group. A hypersurface x is called Laguerre isoparametric if its Laguerre form vanishes and the eigenvalues of B are constant. In this paper, we classify all Laguerre isoparametric hypersurfaces in R4 .     

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.     

On complete submanifolds with parallel mean curvature in negative pinched manifolds

A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n p)-dimensional manifold Nn p with negative sectional curvature is proved. For given positive integers n(≥ 2), p and for a constant H satisfying H ＞ 1 there exists a negative number τ(n,p, H) ∈ (-1, 0) with the property that if the sectional curvature of N is pinched in [-1, τ(n,p, H)], and if the squared length of the second fundamental form is in a certain interval, then Nn p is isometric to the hyperbolic space Hn p(-1). As a consequence, this submanifold M is congruent to Sn(1/ H2-1) or theVeronese surface in S4(1/√H2-1).     

On Möbius form and Möbius isoparametric hypersurfaces

An umbilic-free hypersurface in the unit sphere is called MSbius isoparametric if it satisfies two conditions, namely, it has vanishing MSbius form and has constant MSbius principal curvatures. In this paper, under the condition of having constant MSbius principal curvatures, we show that the hypersurface is of vanishing MSbius form if and only if its MSbius form is parallel with respect to the Levi-Civita connection of its MSbius metric. Moreover, typical examples are constructed to show that the condition of having constant MSbius principal curvatures and that of having vanishing MSbius form are independent of each other.     

Weingarten rotation surfaces in 3-dimensional de Sitter space

In the 3-dimensional de Sitter Space , a surface is said to be a spherical (resp. hyperbolic or parabolic) rotation surface, if it is a orbit of a regular curve under the action of the orthogonal transformations of the 4-dimensional Minkowski space which leave a timelike (resp. spacelike or degenerate) plane pointwise fixed. In this paper, we give all spacelike and timelike Weingarten rotation surfaces in .     

de Sitter空间中具有常均曲率的类空超曲面

设M是deSitter空间中具有常均曲率的类空超曲面,本文给出M是全脐的或是等参的一些条件．     

Constant Angle Property and Canonical Principal Directions for Surfaces in {\mathbb{M}^{2}(c) \times {\mathbb{R}_{1}}}

In this paper, we study surfaces in Lorentzian product spaces ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ . We classify constant angle spacelike and timelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ . Moreover, complete classifications of spacelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ and timelike surfaces in ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ with a canonical principal direction are obtained. Finally, a new characterization of the catenoid of the 3rd kind is established, as the only minimal timelike surface with a canonical principal direction in Minkowski 3–space.     

三维 Anti-de Sitter空间中常平均曲率旋转曲面

本文主要构造三维 Anti-de Sitter空间中单参数族的常平均曲率旋转曲面.     

Classification of Timelike Constant Slope Surfaces in 3-Dimensional Minkowski Space

A 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 paper, we give a complete classification of timelike constant slope surfaces in the three dimensional Minkowski space.     

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.     

广义de Sitter空间中的类时超曲面

利用奇点理论研究广义de Sitter空间中的类时超曲面.介绍类时超曲面的局部微分几何,定义了广义de Sitter高斯像及广义de Sitter高度函数,研究广义deSitter高度函数族的性质及广义de Sitter高斯像的几何意义,介绍了一种证明高度函数为Morse族的新方法.最后研究了类时超曲面的通有性质.     

Timelike surfaces of constant mean curvature ±1 in anti-de Sitter 3-space ℍ3 1(−1)

It is shown that timelike surfaces of constant mean curvature ± in anti-de Sitter 3-space ?³ ₁(?1) can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in ?SL₂? via Bryant type representation formulae. These Bryant type representation formulae are used to investigate an explicit one-to-one correspondence, the so-called Lawson–Guichard correspondence, between timelike surfaces of constant mean curvature ± 1 and timelike minimal surfaces in Minkowski 3-space E ³ ₁. The hyperbolic Gauß map of timelike surfaces in ?³ ₁(?1), which is a close analogue of the classical Gauß map is considered. It is discussed that the hyperbolic Gauß map plays an important role in the study of timelike surfaces of constant mean curvature ± 1 in ?³ ₁(?1). In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gauß map and timelike surface of constant mean curvature ± 1 in ?³ ₁(?1) is studied.     

De Sitter空间中紧致类空超曲面的积分公式及其Goddard猜想的应用

本文对ｄｅＳｉｔｅｒ空间Ｓｎ＋１１（１）中紧致类空超曲面建立Ｍｉｎｋｏｗｓｋｉ型公式，并给出其到ｒ－次（ｒ＝１，２，…，ｎ－１）平均曲率为常数超曲面的应用．当ｒ＝１时，我们得到Ｍｏｎｔｉｅｌ的结果．