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

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 Bjorling 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 df(0)/dx 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 Bjorling data - and holomorphic potentials - which characterize cuspidal edge, swallowtail and cuspidal cross cap singularities.