Every nonflat conformal minimal surface is homotopic to a proper one
Tjasa Vrhovnik
TL;DR
This work proves that every nonflat conformal minimal immersion $u:M\to\mathbb{R}^n$ from an open Riemann surface into $\mathbb{R}^n$ with $n\ge3$ is homotopic, through nonflat conformal minimal immersions, to a proper one; for $n\ge5$ the endpoint can be chosen injective, yielding a proper embedding. Moreover, $u$ is homotopic to the real part of a proper holomorphic null curve $M\to\mathbb{C}^n$ with prescribed flux, and the authors extend the approach to directed holomorphic immersions by Oka cones in $\mathbb{C}^n$. The proof hinges on a new local isotopy lemma (controlling nearby CMIs on compact sets) and a careful inductive Runge-type construction across an exhaustion of $M$, employing period-dominating sprays and delicate period-cancellation via an implicit-function argument. These results advance the understanding of the homotopy type and properness of minimal surfaces, and they provide a robust framework for obtaining proper directed holomorphic curves in greater generality.
Abstract
Given an open Riemann surface $M$, we prove that every nonflat conformal minimal immersion $M\to\mathbb{R}^n$ ($n\geq 3$) is homotopic through nonflat conformal minimal immersions $M\to\mathbb{R}^n$ to a proper one. If $n\geq 5$, it may be chosen in addition injective, hence a proper conformal minimal embedding. Prescribing its flux, as a consequence, every nonflat conformal minimal immersion $M\to\mathbb{R}^n$ is homotopic to the real part of a proper holomorphic null embedding $M\to\mathbb{C}^n$. We also obtain a result for a more general family of holomorphic immersions from an open Riemann surface into $\mathbb{C}^n$ directed by Oka cones in $\mathbb{C}^n$.