Complete minimal surfaces and harmonic functions

TL;DR

This paper proves that any open Riemann surface with a non-constant harmonic function can be embedded as a complete minimal surface in three-dimensional space, with the harmonic function as its third coordinate.

Contribution

It establishes the existence of complete minimal immersions with prescribed harmonic functions as coordinate functions for any open Riemann surface.

Findings

Constructs complete minimal surfaces with arbitrary conformal structures.

Shows minimal surfaces can have Gauss maps missing two points.

Provides a method to realize harmonic functions as coordinate functions.

Abstract

We prove that for any open Riemann surface $M$ and any non constant harmonic function $h:M \to \mathbb{R},$ there exists a complete conformal minimal immersion $X:M \to \mathbb{R}^3$ whose third coordinate function coincides with $h.$ As a consequence, complete minimal surfaces with arbitrary conformal structure and whose Gauss map misses two points are constructed.