Properness of associated minimal surfaces

Abstract:

For any open Riemann surface $\mathcal {N}$ and finite subset $\mathfrak {Z}\subset \mathbb {S}^1=\{z\in \mathbb {C} |\;|z|=1\},$ there exist an infinite closed set $\mathfrak {Z}_{\mathcal {N}} \subset \mathbb {S}^1$ containing $\mathfrak {Z}$ and a null holomorphic curve $F=(F_j)_{j=1,2,3}:\mathcal {N}\to \mathbb {C}^3$ such that the map \[ \mathfrak {Y}:\mathfrak {Z}_{\mathcal {N}}\times \mathcal {N} \to \mathbb {R}^2,\quad \mathfrak {Y}(\mathfrak {v},P)=\textrm {Re}\big (\mathfrak {v} (F_1,F_2)(P)\big ), \] is proper.

In particular, $\textrm {Re}\big (\mathfrak {v} F\big ):\mathcal {N} \to \mathbb {R}^3$ is a proper conformal minimal immersion properly projecting into $\mathbb {R}^2\equiv \mathbb {R}^2\times \{0\} \subset \mathbb {R}^3,$ for all $\mathfrak {v} \in \mathfrak {Z}_{\mathcal {N}}.$