The space of Lorentzian flat tori in anti-de Sitter 3-space

TL;DR

This paper characterizes the space of Lorentzian flat tori immersed in anti-de Sitter 3-space, solving longstanding open problems and describing these tori via pairs of special closed curves in the hyperbolic plane.

Contribution

It provides a comprehensive description of isometric immersions of Lorentzian planes and flat tori into anti-de Sitter space, addressing open questions from 1981.

Findings

Complete classification of Lorentzian flat tori in anti-de Sitter space.

Description of immersions via pairs of closed wave front curves.

Resolution of open problems posed by Dajczer and Nomizu.

Abstract

We describe the space of isometric immersions from the Lorentz plane $L^2$ into the 3-dimensional anti-de Sitter space, and solve several open problems of this context raised by M. Dajczer and K. Nomizu in 1981. We also obtain from the above result a description of the space of Lorentzian flat tori isometrically immersed in the anti de Sitter space in terms of pairs of closed curves with wave front singularities in the hyperbolic plane satisfying some compatibility conditions.