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

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.

To appear in J. Eur. Math. Soc. (JEMS)