The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space

Abstract

We prove that a complete embedded maximal surface in L3 with a finite number of sin- gularities 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 Gn of entire maximal graphs over {x3 = 0} in L3 with n + 1 + 2 singular points and vertical limit normal vector at infinity is a 3n + 4-dimensional differentiable manifold. The convergence in Gn 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 R3 and the logarithmic growth at infinity can be used as global analytical coordinates with the same underlying topology. We also introduce the moduli space Mn 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 Gn. 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 ˆM n is an analytic manifold of dimension 3n−1. This manifold can be identified with a spinorial bundle Sn associated to the moduli space of Weierstrass data of graphs in Gn.