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

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

2009-12-30

arxiv.org/abs/0912.5081v2

Math. Proc. Camb. Phil. Soc. (2011), 150, 527--556

Mathematics

Differential Geometry

28 pages, 2 figures. Version 2: Figure 2 added

Scientific paper

10.1017/S030500411100007

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.

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