Embedded minimal surfaces in $\mathbb{R}^n$

TL;DR

This paper proves approximation and embedding results for conformal minimal immersions of open Riemann surfaces into Euclidean spaces, establishing new methods and tools like a Mergelyan approximation theorem for such immersions.

Contribution

It introduces a Mergelyan approximation theorem for conformal minimal immersions into n and demonstrates that every open Riemann surface can be properly embedded into 5.

Findings

Any conformal minimal immersion into n (nb5 5) can be approximated by embeddings.

Every open Riemann surface admits a proper conformal minimal embedding into 5.

A new Mergelyan approximation theorem for conformal minimal immersions is established.

Abstract

In this paper, we prove that every confomal minimal immersion of an open Riemann surface into $\mathbb{R}^n$ for $n\ge 5$ can be approximated uniformly on compacts by conformal minimal embeddings. Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into $\mathbb{R}^5$. One of our main tools is a Mergelyan approximation theorem for conformal minimal immersions to $\mathbb{R}^n$ for any $n\ge 3$ which is also proved in the paper.