Ruled surfaces of Weingarten type in Minkowski 3-space, II

In our previous paper of the same title, we did not study the ruled surfaces of Weingarten type M : x(s, t)=α(s)+t β (s) in Minkowski 3-space with vector fields β and β′ along the base curve β such that β is nowhere null but β′ is null everywhere. We here fulfill our project by investigating this remaining case.     

Ruled Weingarten surfaces in Minkowski 3-space

We characterize all ruled surfaces in Minkowski 3-space with a relation between the Gauss and mean curvature (Weingarten surfaces). It turns out that, except if the rulings are in a null direction, these are given by Lorentzian screw motions of straight lines. However, if the rulings are always in a null direction, then every ruled surface is Weingarten. Received: 9 February 1998 / Revised version: 20 December 1998     

Mannheim partner curves in 3-space

In this paper, we study Mannheim partner curves in three dimensional space. We obtain the necessary and sufficient conditions for the Mannheim partner curves in Euclidean space and Minkowski space , respectively. Some examples are also given. Supported by NSFC, No.10371013; Joint Research of NSFC and KOSEF, NEU     

Flat surfaces in the hyperbolic 3-space

In this paper we give a conformal representation of flat surfaces in the hyperbolic 3-space using the complex structure induced by its second fundamental form. We also study some examples and the behaviour at infinity of complete flat ends. Received: 18 September 1997     

Lightlike surfaces of spacelike curves in de Sitter 3-space

The Lorentzian space form with the positive curvature is called de Sitter space which is an important subject in the theory of relativity. In this paper we consider spacelike curves in de Sitter 3-space. We define the notion of lightlike surfaces of spacelike curves in de Sitter 3-space. We investigate the geometric meanings of the singularities of such surfaces. Work partially supported by Grant-in-Aid for formation of COE. ‘Mathematics of Nonlinear Structure via Singularities’     

Holomorphic representation of constant mean curvature surfaces in Minkowski space: Consequences of non-compactness in loop group methods

We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space R2,1. The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group SU₂ with SU1,1. The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an open dense subset, after doubling the size of the real form SU1,1, and prove several results concerning the behavior of the surface as the boundary of this open set is encountered. We then use the generalized Weierstrass representation to create and classify new examples of spacelike CMC surfaces in R2,1. In particular, we classify surfaces of revolution and surfaces with screw motion symmetry, as well as studying another class of surfaces for which the metric is rotationally invariant.     

Introducing Weingarten cyclic surfaces in R3

In this paper, Cyclic surfaces are introduced using the foliation of circles of curvature of a space curve. The conditions on a space curve such that these cyclic surfaces are of type Weingarten surfaces or HK-quadric surfaces are obtained. Finally, some examples are given and plotted.     

A characterization of Weingarten surfaces in hyperbolic 3-space

We study 2-dimensional submanifolds of the space \({\mathbb{L}}({\mathbb{H}}^{3})\) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in ?³ orthogonal to the geodesics of Σ.We prove that the induced metric on a Lagrangian surface in \({\mathbb{L}}({\mathbb{H}}^{3})\) has zero Gauss curvature iff the orthogonal surfaces in ?³ are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in \({\mathbb{L}}({\mathbb{H}}^{3})\) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in ?³.     

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.     

Position vectors of spacelike general helices in Minkowski 3-space

In this paper, the position vectors of a spacelike general helix with respect to the standard frame in Minkowski space are studied in terms of the Frenet equations. First, a vector differential equation of third order is constructed to determine the position vectors of an arbitrary spacelike general helix. In terms of solution, we determine the parametric representation of the general helices from the intrinsic equations. Moreover, we give some examples to illustrate how to find the position vectors of spacelike general helices with a spacelike and timelike principal normal vector.     

Polynomial translation surfaces of Weingarten types in Euclidean 3-space

In this paper, we classify polynomial translation surfaces in Euclidean 3-space satisfying the Jacobi condition with respect to the Gaussian curvature, the mean curvature and the second Gaussian curvature.     

Constant mean curvature surfaces of annular type

We consider large solutions of annular type to the volume constrained Douglas problem. They are conformally immersed H-surfaces. By rescaling we set the volume functional at one while the boundary curves shrink to the origin. We show that the solutions become spherical in a precise manner. Spherical bubbling may fail if the conformality condition is dropped. We also discuss the rotationally symmetric annular solutions to the H-surface equation and consider some illustrative examples. Received: 2 May 2000 / Accepted: 23 January 2001 / Published online: 4 May 2001     

Special Weingarten surfaces foliated by circles

In this paper we study surfaces in Euclidean 3-space foliated by pieces of circles that satisfy a Weingarten condition of type aH + bK = c, where a,b and c are constant, and H and K denote the mean curvature and the Gauss curvature respectively. We prove that such a surface must be a surface of revolution, one of the Riemann minimal examples, or a generalized cone. Authors’ address: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain     

Ruled Weingarten surfaces in the Galilean space

In this paper we study ruled Weingarten surfaces in the Galilean space. Weingarten surfaces are surfaces having a nontrivial funcional relation between their Gaussian and mean curvature. We describe some further examples of Weingarten surfaces.      

Isolated singularities of the prescribed mean curvature equation in Minkowski 3-space

We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski 3-space.     

Uniqueness and non-existence theorems for conformally invariant equations

By using the equivalent integral form for the Q-curvature equation, we generalize the well-known non-existence results on two-dimensional Gaussian curvature equation to all dimensional Q-curvature equation. Somehow, we introduce a new approach to Q-curvature equation which is higher order and even pseudo-differential equation. As a by-product, we do classify the solutions for Q=1 solutions on Sⁿ as well as on Rⁿ with necessary growth rate assumption.     

Space-like loxodromes on rotational surfaces in Minkowski 3-space

We find all space-like loxodromes on rotational surfaces which have space-like meridians or time-like meridians, respectively by using a relevant Lorentzian angle in Minkowski 3-space. To understand loxodromes better, we draw some pictures of them via Mathematica computer program.     

Hypersurfaces of restricted type in Minkowski space

A submanifold M _n ^r of Minkowski space is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of to the tangent space of M _n ^r at every point of M _n ^r . In this paper we completely classify hypersurfaces of restricted type in . More precisely, we prove that a hypersurface of is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S ^k × , S _k ¹ × , H ^k × , S _n ¹ , H ⁿ , with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium.     

On a quasilinear elliptic equation and a Riemannian metric invariant under Möbius transformations

Summary The main result of this paper is the determination of maximal solutions of the equation u =u ^{(n + 2)/(n – 2)} arising from the conformal change of a flat metric in a domain (in euclidean space) to a metric of negative scalar curvature. These solutions are obtained on the upper half ball (and therefore on anything conformally equivalent to it). There are a couple of applications using basic comparison principle arguments. Namely, the associated metric on a domain induced by a maximal solution is shown to be comparable to the quasihyperbolic metric for nice domains. Also an analogue of the isoperimetric inequality is found for the harmonic radius.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth