Some comments on spacelike minimal surfaces with null polygonal boundaries in $AdS_m$

TL;DR

This paper explores the geometry of spacelike minimal surfaces in anti-de Sitter space with null polygonal boundaries, identifying key holomorphic functions and conjecturing how their zeros encode boundary data.

Contribution

It identifies holomorphic functions for the Pohlmeyer system in AdS4 and proposes a conjecture linking zeros of these functions to boundary conformal invariants.

Findings

Two holomorphic functions for AdS4 minimal surfaces are identified.

A conjecture relates zeros of holomorphic functions to boundary null polygon data.

The Pohlmeyer reduced system involves coupled differential equations for curvature and torsion.

Abstract

We discuss some geometrical issues related to spacelike minimal surfaces in $AdS_m$ with null polygonal boundaries at conformal infinity. In particular for $AdS_4$, two holomorphic input functions for the Pohlmeyer reduced system are identified. This system contains two coupled differential equations for two functions $\alpha (z,\bar z)$ and $\beta (z,\bar z)$, related to curvature and torsion of the surface. Furthermore, we conjecture that, for a polynomial choice of the two holomorphic functions, the relative positions of their zeros encode the conformal invariant data of the boundary null $2n$-gon.