Some comments on spacelike minimal surfaces with null polygonal boundaries in $Ad{S}_m$

We discuss some geometrical issues related to spacelike minimal surfaces in $Ad{S}_m$ with null polygonal boundaries at conformal infinity. In particular for $Ad{S}_4$, 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.