Properness of associated minimal surfaces

TL;DR

This paper demonstrates the existence of proper minimal surfaces associated with null holomorphic curves on open Riemann surfaces, extending the class of known proper minimal immersions with controlled boundary behavior.

Contribution

It constructs proper conformal minimal immersions from any open Riemann surface into , with prescribed boundary conditions on the boundary circle.

Findings

Existence of proper minimal immersions for any open Riemann surface.

Construction of null holomorphic curves with prescribed boundary sets.

Properness of the associated minimal surfaces into .

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 $v \in Z_N.$