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Abstract: We study singularities of spacelike, constant non-zero mean curvature CMCsurfaces in the Lorentz-Minkowski 3-space $L^3$. We show how to solve thesingular Bj\-orling problem for such surfaces, which is stated as follows:given a real analytic null-curve $f 0x$, and a real analytic null vectorfield $vx$ parallel to the tangent field of $f 0$, find a conformallyparameterized generalized CMC $H$ surface in $L^3$ which contains this curveas a singular set and such that the partial derivatives $f x$ and $f y$ aregiven by $\frac{\dd f 0}{\dd x}$ and $v$ along the curve. Within the class ofgeneralized surfaces considered, the solution is unique and we give a formulafor the generalized Weierstrass data for this surface. This gives a frameworkfor studying the singularities of non-maximal CMC surfaces in $L^3$. We usethis to find the Bj\-orling data - and holomorphic potentials - whichcharacterize cuspidal edge, swallowtail and cross cap singularities.



Author: David Brander

Source: https://arxiv.org/







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