On spacelike and timelike minimal surfaces in $AdS_n$

TL;DR

This paper analyzes spacelike and timelike minimal surfaces in anti-de Sitter space using a Pohlmeyer reduction, deriving differential equations, and establishing the non-existence of certain flat spacelike surfaces beyond known solutions.

Contribution

It provides a unified reduction framework for both surface types, characterizes their differences, and proves the non-existence of flat spacelike minimal surfaces in higher-dimensional AdS spaces.

Findings

No flat spacelike minimal surfaces in AdS_n for n≥4 beyond four cusp solutions.

A parameterization of flat timelike minimal surfaces in AdS_5 using two chiral fields.

Derived differential equations for both timelike and spacelike minimal surfaces.

Abstract

We discuss timelike and spacelike minimal surfaces in $AdS_n$ using a Pohlmeyer type reduction. The differential equations for the reduced system are derived in a parallel treatment of both type of surfaces, with emphasis on their characteristic differences. In the timelike case we find a formulation corresponding to a complete gauge fixing of the torsion. In the spacelike case we derive three sets of equations, related to different parameterizations enforced by the Lorentzian signature of the metric in normal space. On the basis of these equations, we prove that there are no flat spacelike minimal surfaces in $AdS_n, n\geq 4$ beyond the four cusp surfaces used in the Alday-Maldacena conjecture. Furthermore, we give a parameterization of flat timelike minimal surfaces in $AdS_5$ in terms of two chiral fields.