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

TL;DR

This paper classifies rotational Lorentzian surfaces in Minkowski space by linking their Gauss maps to solutions of the nonlinear sigma model, revealing a new family of such surfaces through a surgical construction method.

Contribution

It provides a complete classification of rotational solutions in the nonlinear sigma model context, introducing a novel family of Lorentzian surfaces via a surgical and gluing technique.

Findings

Complete classification of rotational solutions in the sigma model.

Introduction of a new family of Lorentzian rotational surfaces.

Use of surgical and gluing methods for surface construction.

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.