Singularities of spacelike constant mean curvature surfaces in Lorentz-Minkowski space

TL;DR

This paper develops a method to construct and analyze singularities of spacelike constant mean curvature surfaces in Lorentz-Minkowski space, providing explicit formulas and characterizations for various types of surface singularities.

Contribution

It introduces a solution to the singular Björling problem for non-zero CMC surfaces in Lorentz-Minkowski space, including formulas for Weierstrass data and singularity classification.

Findings

Unique solutions for singular CMC surfaces given Björling data

Explicit formulas for Weierstrass data of singular surfaces

Characterization of cuspidal edge, swallowtail, and cross cap singularities

Abstract

We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space $L^3$. We show how to solve the singular Bj\"orling problem for such surfaces, which is stated as follows: given a real analytic null-curve $f_0(x)$, and a real analytic null vector field $v(x)$ parallel to the tangent field of $f_0$, find a conformally parameterized (generalized) CMC $H$ surface in $L^3$ which contains this curve as a singular set and such that the partial derivatives $f_x$ and $f_y$ are given by $\frac{\dd f_0}{\dd x}$ and $v$ along the curve. Within the class of generalized surfaces considered, the solution is unique and we give a formula for the generalized Weierstrass data for this surface. This gives a framework for studying the singularities of non-maximal CMC surfaces in $L^3$. We use this to find the Bj\"orling data -- and holomorphic potentials -- which characterize cuspidal edge, swallowtail and cross cap singularities.