Every nonflat conformal minimal surface is homotopic to a proper one

TL;DR

This paper proves that any nonflat conformal minimal surface in Euclidean space can be continuously deformed into a proper one, with additional injectivity and embedding properties possible in higher dimensions, linking minimal surfaces to holomorphic null embeddings.

Contribution

It establishes homotopy equivalences between nonflat conformal minimal immersions and proper embeddings, extending to holomorphic null embeddings and general holomorphic immersions directed by Oka cones.

Findings

Every nonflat conformal minimal immersion is homotopic to a proper one.

In dimensions n≥5, such immersions can be made injective, yielding proper embeddings.

Connections are made between minimal surfaces and holomorphic null embeddings.

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 $\mathbb{C}^n$.