Proper harmonic embeddings of open Riemann surfaces into $\mathbb{R}^4$

TL;DR

This paper proves that every open Riemann surface can be properly embedded into four-dimensional space using harmonic functions, improving previous results by reducing the embedding dimension by one.

Contribution

It establishes a new minimal dimension for proper harmonic embeddings of open Riemann surfaces into Euclidean space, extending classical results from Greene and Wu.

Findings

Every open Riemann surface admits a proper harmonic embedding into R^4.

Reduces the known embedding dimension from five to four.

Builds on and improves classical embedding theorems from 1975.

Abstract

We prove that every open Riemann surface admits a proper embedding into $\mathbb{R}^4$ by harmonic functions. This reduces by one the previously known embedding dimension in this framework, dating back to a theorem by Greene and Wu from 1975.