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

TL;DR

This paper proves a Mittag-Leffler-type theorem for meromorphic curves and proper minimal surfaces in complex and real Euclidean spaces, enabling approximation, interpolation, and embedding results with applications to minimal surface ends.

Contribution

It introduces a new Mittag-Leffler-type theorem for meromorphic curves and minimal surfaces, extending approximation and interpolation techniques in complex and real settings.

Findings

Proper minimal ends of finite total curvature in R^5 are generically embedded.

Characterization of Riemann surfaces that support proper minimal surfaces of weak finite total curvature.

Establishment of approximation and interpolation results for meromorphic curves and minimal immersions.

Abstract

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