Lorentz harmonic maps into the hyperbolic plane and spacelike surfaces in anti-de Sitter 3-space

TL;DR

This paper develops a loop group method to relate Lorentz harmonic maps into the hyperbolic plane with spacelike surfaces in anti-de Sitter 3-space, including explicit solutions and a geometric Cauchy problem.

Contribution

It introduces a DPW-type representation for Lorentz harmonic maps and establishes a correspondence with spacelike surfaces, solving the geometric Cauchy problem explicitly.

Findings

Explicit solutions for Lorentz harmonic maps using potentials.

A correspondence between harmonic maps and spacelike surfaces in anti-de Sitter space.

Constructive method for the geometric Cauchy problem for constant curvature surfaces.

Abstract

We study the relationship between Lorentz harmonic maps into the hyperbolic plane and spacelike surfaces in anti-de Sitter 3-space. Using loop group techniques, we develop a DPW-type representation for Lorentz harmonic maps and provide an explicit solution of the associated Cauchy problem in terms of a pair of potentials. We then establish a correspondence between Lorentz harmonic maps and spacelike immersions in anti-de Sitter space, identifying conditions under which a harmonic map arises as the Gauss map of a surface. In the nondegenerate case, this leads to a one-parameter family of spacelike surfaces of constant Gauss curvature, together with explicit reconstruction formulas. We also analyze the degenerate case, where the Gauss map fails to be an immersion, and show that additional data are required to recover the surface. Finally, we formulate and solve the geometric Cauchy problem for spacelike surfaces of constant curvature in anti-de Sitter space, providing a constructive method to recover surfaces from prescribed initial data. This establishes a direct link between the analytic theory of Lorentz harmonic maps and the geometry of surfaces in Lorentzian space forms.