An Appendix to the Weierstrass Representation of a Maximal Spacelike Surface in $\mathbb{L}^3$

Abstract

In this note we uncover some analytical properties of the Weierstrass pair $\{g,\eta\}$ representing an oriented and connected maximal surface in Minkowski space $\mathbb{L}^3$, proving that $g$ is holomorphic with values in the unit disk $\mathbb{D}$ and $\eta$ is a holomorphic 1-form that never vanishes.