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Periodic maximal surfaces in the Lorentz–Minkowski space $${\mathbb{L}^3}$$

Periodic maximal surfaces in the Lorentz–Minkowski space $${\mathbb{L}^3}$$ A maximal surface $${\mathcal{S}}$$ with isolated singularities in a complete flat Lorentzian 3-manifold $$\mathcal{N}$$ is said to be entire if it lifts to a (periodic) entire multigraph $${\sim\mathcal{S}}$$ in $${\mathbb{L}^3}$$ . In addition, $${\mathcal{S}}$$ is called of finite type if it has finite topology, finitely many singular points and $${\sim\mathcal{S}}$$ is a finitely sheeted multigraph. Complete (or proper) maximal immersions with isolated singularities in $${\mathcal{N}}$$ are entire, and entire embedded maximal surfaces in $${\mathcal{N}}$$ with a finite number of singularities are of finite type. We classify complete flat Lorentzian 3-manifolds carrying entire maximal surfaces of finite type, and deal with the topology, Weierstrass representation and asymptotic behavior of this kind of surfaces. Finally, we construct new examples of periodic entire embedded maximal surfaces in $${\mathbb{L}^3}$$ with fundamental piece having finitely many singularities. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Periodic maximal surfaces in the Lorentz–Minkowski space $${\mathbb{L}^3}$$

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References (27)

Publisher
Springer Journals
Copyright
Copyright © 2006 by Springer-Verlag
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-006-0087-y
Publisher site
See Article on Publisher Site

Abstract

A maximal surface $${\mathcal{S}}$$ with isolated singularities in a complete flat Lorentzian 3-manifold $$\mathcal{N}$$ is said to be entire if it lifts to a (periodic) entire multigraph $${\sim\mathcal{S}}$$ in $${\mathbb{L}^3}$$ . In addition, $${\mathcal{S}}$$ is called of finite type if it has finite topology, finitely many singular points and $${\sim\mathcal{S}}$$ is a finitely sheeted multigraph. Complete (or proper) maximal immersions with isolated singularities in $${\mathcal{N}}$$ are entire, and entire embedded maximal surfaces in $${\mathcal{N}}$$ with a finite number of singularities are of finite type. We classify complete flat Lorentzian 3-manifolds carrying entire maximal surfaces of finite type, and deal with the topology, Weierstrass representation and asymptotic behavior of this kind of surfaces. Finally, we construct new examples of periodic entire embedded maximal surfaces in $${\mathbb{L}^3}$$ with fundamental piece having finitely many singularities.

Journal

Mathematische ZeitschriftSpringer Journals

Published: Dec 22, 2006

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