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

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

2003-11-19

arxiv.org/abs/math/0311330v2

Mathematics

Differential Geometry

32 pages, 4 figures, corrected typos, former Theorem 3.3 (now Theorem 2.2) modified

Scientific paper

We show that a complete embedded maximal surface in the 3-dimensional Lorentz-Minkowski space $L^3$ with a finite number of singularities is, up to a Lorentzian isometry, an entire graph over any spacelike plane asymptotic to a vertical half catenoid or a horizontal plane and with conelike singular points. We study the space $G_n$ of entire maximal graphs over $\{x_3=0\}$ in $L^3$ with $n+1 \geq 2$ conelike singularities and vertical limit normal vector at infinity. We show that $G_n$ is a real analytic manifold of dimension $3n+4,$ and the coordinates are given by the position of the singular points in $R^3$ and the logarithmic growth at the end. We also introduce the moduli space $M_n$ of {\em 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 $G_n.$ 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 $M_n$ is an analytic manifold of dimension $3n-1.$

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