Proper harmonic embeddings of open Riemann surfaces into ℝ⁴

2022/06/07 by Antonio Alarcón, Alarcon, Antonio, Francisco J. Lopez +1
Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2206.03566

openalex publication_date 2022/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that every open Riemann surface admits a proper embedding into ℝ⁴ 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.

Related

• Proper harmonic maps from hyperbolic Riemann surfaces into the Euclidean plane
• Embedding Riemann surfaces with isolated punctures into the complex\n plane
• Surfaces in ℝ⁷ obtained from harmonic maps in S⁶
• New examples of r-harmonic immersions into the sphere
• Computation of harmonic functions on higher genus surfaces