The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space |
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Authors: | Isabel Fernández Francisco J López Rabah Souam |
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Institution: | (1) Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain;(2) Institut de Mathématiques de Jussieu-CNRS, Université Paris 7, 75251 Paris Cedex 05, France |
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Abstract: | We prove that a complete embedded maximal surface in = (3, dx12 + dx22-dx32) with a finite number of singularities is an entire maximal graph with conelike singularities over any spacelike plane, and so, it is asymptotic to a spacelike plane or a half catenoid. We show that the moduli space of entire maximal graphs over {x3=0} in with n+12 singular points and vertical limit normal vector at infinity is a 3n+4-dimensional differentiable manifold. The convergence in means the one of conformal structures and Weierstrass data, and it is equivalent to the uniform convergence of graphs on compact subsets of {x3=0}. Moreover, the position of the singular points in 3 and the logarithmic growth at infinity can be used as global analytical coordinates with the same underlying topology. We also introduce the moduli space of marked graphs with n+1 singular points (a mark in a graph is an ordering of its singularities), which is a (n+1)-sheeted covering of . We prove that identifying marked graphs differing by translations, rotations about a vertical axis, homotheties or symmetries about a horizontal plane, the corresponding quotient space is an analytic manifold of dimension 3n–1. This manifold can be identified with a spinorial bundle associated to the moduli space of Weierstrass data of graphs in .Mathematics Subject Classification (2000): 53C50, 58D10, 53C42First and second authors research partially supported by MEC-FEDER grant number MTM2004-00160Second and third authors research partially supported by Consejería de Educación y Ciencia de la Junta de Andalucía and the European Union. |
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