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

TL;DR

The authors prove that every open Riemann surface $M$ admits a proper embedding into $\mathbb{R}^4$ by harmonic functions, improving the Greene–Wu bound from $\mathbb{R}^5$ in the surface case. They construct a harmonic map $h:M\to\mathbb{R}^2$ so that, for a simple almost proper holomorphic function $z:M\to\mathbb{C}$, the map $(z,h):M\to\mathbb{R}^4\cong \mathbb{C}\times\mathbb{R}^2$ is a proper embedding, with $h$ equal to the real part of a holomorphic map $M\to\mathbb{C}^2$. The proof combines a mixed holomorphic–harmonic approach with Runge–Mergelyan approximation with interpolation in an inductive construction that preserves injectivity on an exhaustion of $M$ and ensures properness. Consequently, every open orientable Riemannian surface embeds properly into $\mathbb{R}^4$ via a harmonic map, connecting to broader questions about proper holomorphic embeddings into lower-dimensional complex spaces and highlighting a potential path toward the Forster–Bell–Narasimhan conjecture.

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.

Figures (5)

• Figure 4.1: Left: The sets $\Gamma_a$ and $U_a$. Right: The set $K_1$.
• Figure 4.2: The biholomorphic map $\psi^b$, $b\in B_0$.
• Figure 4.3: The set $W$.
• Figure 4.4: The sets $Y$ and $\varpi_a$.
• Figure 4.5: Left: The sets $\hat{\varpi}^b_+$ and $\hat{\varpi}^b_-$. Right: The set $K_2$.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Definition 2.1
• Theorem 2.2
• Lemma 3.1
• Corollary 3.2
• proof : Proof of Corollary \ref{['co:fun']} assuming Lemma \ref{['le:fun']}