We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space L3. We show how to solve the singular Björling problem for such surfaces, which is stated as follows: given a real analytic null-curve f0(x), and a real analytic null vector field v(x) parallel to the tangent field of f0, find a conformally parameterized (generalized) CMC H surface in L3 which contains this curve as a singular set and such that the partial derivatives fx and fy are given by df0/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 L3. We use this to find the Björling data – and holomorphic potentials – which characterize cuspidal edge, swallowtail and cuspidal cross cap singularities.
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