Properness of associated minimal surfaces

Abstract

We prove that for any open Riemann surface $N$ and finite subset $Z\subset {\mathbb{S}}^1=\{z\in \mathbb{C}||z|=1\},$ there exist an infinite closed set $Z_N\subset {\mathbb{S}}^1$ containing $Z$ and a null holomorphic curve $F=(F_j{)}_{j=1,2,3}:N\to {\mathbb{C}}^3$ such that the map $Y:Z_N\times N\to {\mathbb{R}}^2,$ $Y(v,P)=Re(v(F_1,F_2)(P)),$ is proper. In particular, $Re(vF):N\to {\mathbb{R}}^3$ is a proper conformal minimal immersion properly projecting into ${\mathbb{R}}^2={\mathbb{R}}^2\times \{0\}\subset {\mathbb{R}}^3,$ for all $v\in Z_N.$

17 pages, 5 figures

References in corpus (1)

• Uniform Approximation by Complete Minimal Surfaces of Finite Total Curvature in ${\mathbb{R}}^3$