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Rotational Surfaces in \({\mathbb{L}^3}\) and Solutions of the Nonlinear Sigma Model

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Abstract

The Gauss map of non-degenerate surfaces in the three-dimensional Minkowski space are viewed as dynamical fields of the two-dimensional O(2, 1) Nonlinear Sigma Model. In this setting, the moduli space of solutions with rotational symmetry is completely determined. Essentially, the solutions are warped products of orbits of the 1-dimensional groups of isometries and elastic curves in either a de Sitter plane, a hyperbolic plane or an anti de Sitter plane. The main tools are the equivalence of the two-dimensional O(2, 1) Nonlinear Sigma Model and the Willmore problem, and the description of the surfaces with rotational symmetry. A complete classification of such surfaces is obtained in this paper. Indeed, a huge new family of Lorentzian rotational surfaces with a space-like axis is presented. The description of this new class of surfaces is based on a technique of surgery and a gluing process, which is illustrated by an algorithm.

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Authors and Affiliations

  1. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain

    Manuel Barros & Miguel Ortega

  2. Departamento de Matematicas, Universidad de Córdoba, 14071, Córdoba, Spain

    Magdalena Caballero

Authors
  1. Manuel Barros
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  2. Magdalena Caballero
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  3. Miguel Ortega
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Corresponding author

Correspondence to Miguel Ortega.

Additional information

Communicated by N. A. Nekrasov

This work was partially supported by MEC Grant MTM2007-60731 with FEDER funds and the Junta de Andalucía Grant PO6-FQM-01951.

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Barros, M., Caballero, M. & Ortega, M. Rotational Surfaces in \({\mathbb{L}^3}\) and Solutions of the Nonlinear Sigma Model. Commun. Math. Phys. 290, 437–477 (2009). https://doi.org/10.1007/s00220-009-0850-0

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