Ruled surfaces of non-degenerate third fundamental forms in Minkowski 3-spaces

In this paper, we study ruled surfaces in a Minkowski 3-space satisfying some equation in terms of a position vector field and Laplacian operator with respect to non-degenerate third fundamental form. Furthermore, we give a new example of null scroll in a Minkowski 3-space.     

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.     

Singularities of maximal surfaces

We show that the singularities of spacelike maximal surfaces in Lorentz–Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de Sitter 3-space. To prove these, we shall give a simple criterion for a given singular point on a surface to be a cuspidal cross cap. Dedicated to Yusuke Sakane on the occasion of his 60th birthday.     

The Sub-Parabolic Lines in the Minkowski 3-Space

Results in Mathematics - In this paper, we investigate the focal surfaces obtained by the normal rectilinear congruence in the Minkowski 3-space. The sub-parabolic points and the ridge points are...     

Twisted Surfaces with Null Rotation Axis in Minkowski 3-Space

We examine curvature properties of twisted surfaces with null rotation axis in Minkowski 3-space. That is, we study surfaces that arise when a planar curve is subject to two synchronized rotations, possibly at different speeds, one in its supporting plane and one of this supporting plane about an axis in the plane. Moreover, at least one of the two rotation axes is a null axis. As is clear from its construction, a twisted surface generalizes the concept of a surface of revolution. We classify flat, constant Gaussian curvature, minimal and constant mean curvature twisted surfaces with a null rotation axis. Aside from pseudospheres, pseudohyperbolic spaces and cones, we encounter B-scrolls in these classifications. The appearance of B-scrolls in these classifications is of course the result of the rotation about a null axis. As for the cones in the classification of flat twisted surfaces, introducing proper coordinates, we prove that they are determined by so-called Clelia curves. With a Clelia curve we mean a curve that has linear dependent spherical coordinates.     

Discrete maximal surfaces with singularities in Minkowski space

We describe discrete maximal surfaces with singularities in 3-dimensional Minkowski space and give a Weierstrass type representation for them. In the smooth case, maximal surfaces (spacelike surfaces with mean curvature identically 0) in Minkowski 3-space generally have certain singularities. We give a criterion that naturally describes the “singular set” for discrete maximal surfaces, including a classification of the various types of singularities that are possible in the discrete case.     

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.     

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.     

Nonlinear partial differential equations associated with binormal motions of constant torsion curves in Minkowski 3-space

We give three nonlinear partial differential equations which are associated with binormal motions of constant torsion curves in Minkowski 3-space. We also give B?cklund transformations for these equations, as well as for surfaces swept out by related moving curves. As applications, from some trivial binormal motions we construct some new binormal motions.     

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.     

Timelike Minimal Surfaces via Loop Groups

This work consists of two parts. In Part I, we shall give a systematic study of Lorentz conformal structure from structural viewpoints. We study manifolds with split-complex structure. We apply general results on split-complex structure for the study of Lorentz surfaces.In Part II, we study the conformal realization of Lorentz surfaces in the Minkowski 3-space via conformal minimal immersions. We apply loop group theoretic Weierstrass-type representation of timelike constant mean curvature for timelike minimal surfaces. Classical integral representation formula for timelike minimal surfaces will be recovered from loop group theoretic viewpoint.     

关于三维Minkowski空间中直线汇的一些注记

本文研究了三维Minkowski空间中直线汇的一些性质,特别是关于类时线汇的性质．讨论了线汇基本元素的存在性,并证明了关于三维Minkowski空间中类时线汇的配分参数的一个结果,推广了苏步青1927年的—个成果．     

INTEGRABLE SYSTEM AND SPACELIKE SURFACES WITH PRESCRIBED MEAN CURVATURE IN MINKOWSKI 3-SPACE

1IntroductionItiswell-knownthattheclassicalWeierstrass-EnneperrepresentationformuladescribeschinalsurfacesinEuclidean3-spaceR3intermsoftheirGaussmapsandauxiliaryholomorphicfunctions[1].Moregenerally,aremarkablerepresentationforlllulahasbeendiscoveredbyKenmotsu[2]forarbitrarysurfaCesinR3withnonvallishingmeancurvature,whichdescribesthesesurfacesintermsoftheirGaussmapsandmeancurvaturefunctions.Recently,Konopelchenko[31rediscoveredthisrepresentationformulaindifferentbutequivalentformillconnect…     

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     

INFINITESIMAL DEFORMATIONS OF TIME-LIKE SURFACES IN MINKOWSKI 3-SPACE

In this paper, infinitesimal deformations of time-like surfaces are investigated in Minkowski 3-space R^2,1. It is shown that some given deformations of the time-like surface can be described by 2 1 dimensional integrable systems. Moreover spectral parameters are introduced, and it is proved that deformation families are soliton surfaces‘ families.     

Two non existence results for the self-similar equation in Euclidean 3-space

We prove that the only self-similar surfaces of Euclidean 3-space which are foliated by circles are the self-similar surfaces of revolution discovered by S. Angenent and that the only ruled, self-similar surfaces are the cylinders over planar self-similar curves.     

Classification of Totally Geodesic Surfaces in the Manifold of Directions in Physical Space

The Grassmannian of bivectors over the pseudo-Euclidean Minkowski 4-space is considered and its two-dimensional totally geodesic submanifolds are classified. The family of such surfaces is described in the language of the affine geometry of three-space. Bibliography: 5 titles.     

Timelike Constant Mean Curvature Surfaces with Singularities

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behavior of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a singular geometric Cauchy problem. The solution shows that the generic singularities of the generalized surfaces are cuspidal edges, swallowtails, and cuspidal cross caps.     

On the Surface the Fermi–Walker Derivative in Minkowski 3-Space

In this paper Fermi–Walker derivative and Fermi–Walker parallelism and non-rotating frame concepts are given along the curve lying on the spacelike surface and the timelike surface in Minkowski 3-space. First, we consider a curve lying on the spacelike surface and investigate the Fermi–Walker derivative along the curve. The concepts which Fermi–Walker derivative and its theorems are analyzed along the curve lying on the spacelike surface in Minkowski 3-space. And then we consider a curve lying on the timelike surface and investigate the Fermi–Walker derivative along the curve.     

Space curves of constant breadth in Minkowski 3-space

In this paper, we study spacelike and timelike curves of constant breadth in Minkowski 3-space. We show that in Minkowski 3-space spacelike and timelike curves of constant breadth are normal, helices, and spherical curves in some special cases. Furthermore, we give that the total torsion of a closed spacelike curve of constant breadth is zero while the total torsion of a simple closed timelike curve is equal to ${2\pi n, (n \in Z)}$ .