The Mittag-Leffler theorem for proper minimal surfaces and directed meromorphic curves

2024/11/25 by Antonio Alarcón, Alarcon, Antonio, Tjasa Vrhovnik +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2411.16268

openalex publication_date 2024/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish a Mittag-Leffler-type theorem with approximation and interpolation for meromorphic curves M→ ℂⁿ (n≥ 3) directed by Oka cones in ℂⁿ on any open Riemann surface M. We derive a result of the same type for proper conformal minimal immersions M→ ℝⁿ. This includes interpolation in the poles and approximation by embeddings, the latter if n≥ 5 in the case of minimal surfaces. As applications, we show that complete minimal ends of finite total curvature in ℝ⁵ are generically embedded, and characterize those open Riemann surfaces which are the complex structure of a proper minimal surface in ℝ³ of weak finite total curvature.

Related

• Algebraic approximation and the Mittag-Leffler theorem for minimal\n surfaces
• Uniform Approximation by Complete Minimal Surfaces of Finite Total Curvature in ℝ³
• Properness of associated minimal surfaces
• Embedded minimal surfaces of finite topology
• Interpolation and optimal hitting for complete minimal surfaces with finite total curvature