Spinorial representation of surfaces in Lorentzian homogeneous spaces of dimension 3

TL;DR

This paper develops a spinorial representation for surfaces in 3D Lorentzian homogeneous spaces, unifying various surface correspondences and extending classical results to Lorentzian geometry.

Contribution

It introduces a novel spinorial framework for surfaces in Lorentzian homogeneous spaces, including a representation theorem and new surface correspondences.

Findings

Representation theorem for surfaces in (, au) spaces

Recovery of Calabi correspondence between minimal and maximal surfaces

Establishment of Lawson type correspondence in pseudo-hyperbolic space

Abstract

We find a spinorial representation of a Riemannian or Lorentzian surface in a Lorentzian homogeneous space of dimension $3.$ We in particular obtain a representation theorem for surfaces in the $\mathbb{L}(\kappa,\tau)$ spaces. We then recover the Calabi correspondence between minimal surfaces in $\mathbb{R}^3$ and maximal surfaces in $\mathbb{R}_1^3$, and obtain a new Lawson type correspondence between CMC surfaces in $\mathbb{R}_1^3$ and in the 3-dimensional pseudo-hyperbolic space $\mathbb{H}_1^{3}.$