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

Published online by Cambridge University Press:  15 March 2011

DAVID BRANDER
DAVID BRANDER*
Affiliation:
Department of Mathematics, Matematiktorvet, Building 303 S, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark. e-mail: [email protected]
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 3 , May 2011 , pp. 527 - 556
Copyright
Copyright © Cambridge Philosophical Society 2011

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