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.     

Minimal surfaces and particles in 3-manifolds

We consider 3-dimensional anti-de Sitter manifolds with conical singularities along time-like lines, which is what in the physics literature is known as manifolds with particles. We show that the space of such cone-manifolds is parametrized by the cotangent bundle of Teichmüller space, and that moreover such cone-manifolds have a canonical foliation by space-like surfaces. We extend these results to de Sitter and Minkowski cone-manifolds, as well as to some related “quasifuchsian” hyperbolic manifolds with conical singularities along infinite lines, in this later case under the condition that they contain a minimal surface with principal curvatures less than 1. In the hyperbolic case the space of such cone-manifolds turns out to be parametrized by an open subset in the cotangent bundle of Teichmüller space. For all settings, the symplectic form on the moduli space of 3-manifolds that comes from parameterization by the cotangent bundle of Teichmüller space is the same as the 3-dimensional gravity one. The proofs use minimal (or maximal, or CMC) surfaces, along with some results of Mess on AdS manifolds, which are recovered here in a different way, using differential-geometric methods and a result of Labourie on some mappings between hyperbolic surfaces, that allows an extension to cone-manifolds.      

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.     

Complex analysis,maximal immersions and metric singularities

This paper introduces a complex analysis for the wave equation and for a singular second-order partial differential equation. As a main application of this complex analysis we construct type changing zero mean curvature immersions into Minkowski space. We also prove the existence of isothermal coordinates on a Lorentzian surface using this complex analysis and characterize flat maximal surfaces by their Gauss image. Finally we study the metric singularities of maximal immersions and semi Riemannian manifolds in general.     

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.     

三维Minkowski空间中非类光曲线的双曲达布像和从切高斯曲面

本文主要给出了三维Minkowski空间中非类光曲线的双曲达布像和从切高斯曲面的奇点分类,并且建立了奇点和曲线几何不变量之间的联系,其中曲线几何不变量与曲线同螺线切触的阶数密切相关.     

Some classification of surfaces of revolution in Minkowski 3-space

We study the surfaces of revolution with the non-degenerate second fundamental form in Minkowski 3-space. In particular, we investigate the surfaces of revolution satisfying an equation in terms of the position vector field and the 2nd-Laplacian in Minkowski 3-space. As a result, we give some new examples of the surfaces of revolution with light-like axis in Minkowski 3-space.     

Tubular surfaces of center curves on spacelike surfaces in Lorentz‐Minkowski 3‐space

In this paper, we introduce three kinds of tubular surfaces associated to original center curves γ lying in spacelike surfaces in Lorentz‐Minkowski 3‐space. It is demonstrated that these tubular surfaces can occur some singularities and the types of these singularities can be characterised by several invariants, respectively. Some interesting relations between the contacts of original curve γ with osculating model surfaces, the contacts of γ with slices, and the singularities of three kinds of surfaces are further revealed. Several examples are presented to explain the theoretical results.     

Conservation laws of second order for the Born-Infeld equation and other related equations

We describe a class of quasilinear partial differential equations of second order with two independent variables in the general case of mixed type for which we construct conservation laws of second order which are quadratic with respect to the second derivatives. As examples, we present similar conservation laws for the Born-Infeld equation, for the equations of minimal and maximal surfaces in Minkowski space, and for the classical equation of minimal surfaces.     

Heat flow for p-harmonic maps with small initial data

We study developing singularities for surfaces of rotation with free boundaries and evolving under volume-preserving mean curvature flow. We show that singularities form a finite, discrete set along the axis of rotation. We prove a monotonicity formula and conclude that type I singularities are asymtotically cylindrical.     

Projective Minkowski set

The Minkowski set or the central symmetry set (CSS) of a smooth curve Γ on the affine plane is the envelope of chords connecting pairs of points such that the tangents to Γ at them are parallel. Singularities of CSS are of interest, in particular, for applications (for example, in computer graphics). A generalization of the Minkowski set is considered in the paper, namely, the projective Minkowski set with respect to a line on the plane; in the case of general position, we describe its singularities and the bifurcation set of lines corresponding to lines defining the projective Minkowski set having singularities being more degenerate than those of the Minkowski set for a generic line.     

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.     

Hyperbolic ends with particles and grafting on singular surfaces

We prove that any 3-dimensional hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particles, it follows that any 3-dimensional convex GHM de Sitter spacetime with particles also admits a unique foliation by constant Gauss curvature surfaces. We prove that the grafting map from the product of Teichmüller space with the space of measured laminations to the space of complex projective structures is a homeomorphism for surfaces with cone singularities of angles less than π, as well as an analogue when grafting is replaced by “smooth grafting”.     

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.     

The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz–Minkowski space $$\mathbb{L}^3$$

We show that, up to some natural normalizations, the moduli space of singly periodic complete embedded maximal surfaces in the Lorentz–Minkowski space , with fundamental piece having a finite number (n + 1) of singularities, is a real analytic manifold of dimension 3n + 4. The underlying topology agrees with the topology of uniform convergence of graphs on compact subsets of {x ₃ = 0}.      

Taut,pseudotaut and equisingularly rigid singularities

The aim of this paper is to find all plane curve singularities that are taut resp. pseudotaut. It turns out that this problem coincides with the determination of equisingularly rigid singularities. The latter one is achieved in the irreducible case by explicit construction of nontrivial deformations usiing analytical invariants of the Puiseux expansion introduced by Kasner and Zariski, in the reducible case with a cohomological criterion for the triviality of Wahl's functor ES of equisingular deformations of a resolution. Equisingular rigidity is the same as K-zero- or unimodality with discrete parameter. An application is the determination of all equisingularly rigid double points of surfaces, which are just the stabilizations of equisingularly rigid plane curve singularities.     

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.     

Isomorphisms of certain correlated probability spaces and of discrete memoryless correlated sources

Summary In an earlier paper [5], we defined a sufficient set of invariants for the isomorphy of discrete memoryless correlated sources with maximal correlation <1. Using the structure of isomorphisms of certain correlated probability spaces, we give here a sufficient set of invariants also for the case of maximal correlation equal to 1. We show, in particular, that two discrete memoryless stationary correlated sources with maximal correlation 1 may be isomorphic in a non-trivial way.