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

TL;DR

This paper classifies and analyzes the space of complete embedded maximal surfaces with isolated singularities in 3D Lorentz-Minkowski space, revealing their structure as real analytic manifolds with explicit parameters.

Contribution

It introduces the space of entire maximal graphs with conelike singularities, showing it forms a real analytic manifold and describing its moduli space structure.

Findings

The space of such maximal graphs is a real analytic manifold of dimension 3n+4.

The moduli space of marked graphs is a (n+1)-sheeted covering of the space of graphs.

Identification of symmetries yields a manifold of dimension 3n-1.

Abstract

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 $3n-1.$