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

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    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.
    Original languageEnglish
    JournalMathematical Proceedings of the Cambridge Philosophical Society
    Pages (from-to)527-556
    Publication statusPublished - 2011


    • Constant mean curvature surfaces
    • Bjorling problem
    • singularities


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