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The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space $\l^3$
Differential Geometry · Submitted 19 Nov 2003
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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 $3n-1.$
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | result 0.99 | 0.93 | 0.79 | asserted 1.00 |
| s1 | 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. | contribution 0.99 | 0.78 | 0.05 | asserted 0.60 |
| s2 | 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. | result 0.99 | 0.95 | 0.18 | asserted 1.00 |
| s3 | 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 $3n-1.$ | result 0.62 | 0.95 | 0.13 | asserted 1.00 |
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Original 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 $3n-1.$
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