Rotational surfaces in \({\mathbb{L}^3}\) and solutions of the nonlinear sigma model. (English) Zbl 1182.53057
Summary: 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 one-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.
MSC:
| 53C43 | Differential geometric aspects of harmonic maps |
| 53A05 | Surfaces in Euclidean and related spaces |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 53B30 | Local differential geometry of Lorentz metrics, indefinite metrics |
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