Every meromorphic function is the Gauss map of a conformal minimal surface

TL;DR

This paper demonstrates that any meromorphic function on an open Riemann surface can serve as the Gauss map for a conformal minimal surface in Euclidean space, expanding the understanding of minimal surface representations.

Contribution

It establishes that every meromorphic function can be realized as the Gauss map of a conformal minimal immersion, generalizing to higher dimensions and analyzing the space of such immersions.

Findings

Every meromorphic function is the Gauss map of some conformal minimal surface.

Any conformal minimal immersion can be deformed into a flat one.

The connected components of the space of conformal minimal immersions are characterized.

Abstract

Let $M$ be an open Riemann surface. We prove that every meromorphic function on $M$ is the complex Gauss map of a conformal minimal immersion $M\to\mathbb{R}^3$ which may furthermore be chosen as the real part of a holomorphic null curve $M\to\mathbb{C}^3$. Analogous results are proved for conformal minimal immersions $M\to\mathbb{R}^n$ for any $n>3$. We also show that every conformal minimal immersion $M\to\mathbb{R}^n$ is isotopic through conformal minimal immersions $M\to\mathbb{R}^n$ to a flat one, and we identify the path connected components of the space of all conformal minimal immersions $M\to\mathbb{R}^n$ for any $n\ge 3$.