Rotational Surfaces in $\mathbb{L}^3$ and Solutions in the Nonlinear Sigma Model
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.
Cite
@article{arxiv.0705.2206,
title = {Rotational Surfaces in $\mathbb{L}^3$ and Solutions in the Nonlinear Sigma Model},
author = {Manuel Barros and Magdalena Caballero and Miguel Ortega},
journal= {arXiv preprint arXiv:0705.2206},
year = {2009}
}