  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.
