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

Antonio Alarcon, Tjasa Vrhovnik

TL;DR

The paper develops a comprehensive Mittag-Leffler theory for directed meromorphic immersions and for proper minimal surfaces on open Riemann surfaces. It combines Mergelyan-type approximation, desingularization via transversality and sprays, and period-control arguments within the Oka framework to achieve approximation, interpolation, and, in favorable dimensions ($n\ge5$), embedding results. These results yield new existence theorems for proper embedded minimal surfaces in $\mathbb{R}^5$ with finite total curvature ends and provide a structural characterization of which open Riemann surfaces support proper minimal surfaces in $\mathbb{R}^3$ of weak finite total curvature. The methods intertwine complex analysis on Riemann surfaces with Oka theory to extend Runge-Weierstrass-type results from holomorphic curves to the broader setting of directed A-immersions and meromorphic A-immersions, with sharp control over poles, flux, and global geometry.

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.

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

TL;DR

The paper develops a comprehensive Mittag-Leffler theory for directed meromorphic immersions and for proper minimal surfaces on open Riemann surfaces. It combines Mergelyan-type approximation, desingularization via transversality and sprays, and period-control arguments within the Oka framework to achieve approximation, interpolation, and, in favorable dimensions (), embedding results. These results yield new existence theorems for proper embedded minimal surfaces in with finite total curvature ends and provide a structural characterization of which open Riemann surfaces support proper minimal surfaces in of weak finite total curvature. The methods intertwine complex analysis on Riemann surfaces with Oka theory to extend Runge-Weierstrass-type results from holomorphic curves to the broader setting of directed A-immersions and meromorphic A-immersions, with sharp control over poles, flux, and global geometry.

Abstract

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

Paper Structure

This paper contains 9 sections, 19 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Mittag-Leffler theorem for proper minimal surfaces
  3. The Mittag-Leffler theorem for directed meromorphic curves
  4. Notation and preliminaries
  5. The Mergelyan theorem for meromorphic $A$-immersions
  6. Desingularizing meromorphic $A$-immersions
  7. The Mittag-Leffler theorem for meromorphic $A$-immersions
  8. The Mittag-Leffler theorem for meromorphic $A$-immersions with fixed components
  9. Global properties and completion of the proof of Theorem \ref{['th:intro-ML-DirMer']}

Key Result

Theorem 1.1

Let $M$ be an open Riemann surface, $\varnothing\neq E\subset M$ be a closed discrete subset, and $V\subset M$ be a locally connected smoothly bounded closed neighbourhood of $E$ all whose connected components are Runge compact sets. Let $n\ge 3$ be an integer and assume that $u:V\setminus E\to\math

Figures (3)

  • Figure 5.1: Sets in the inductive process of the proof.
  • Figure 7.1: Discs $\Delta$ and $\Omega$ in the proof of Lemma \ref{['lemma:almost-properness']}.
  • Figure 7.2: Sets used at the $j$-th step of induction.

Theorems & Definitions (41)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • proof
  • Theorem 1.6
  • Corollary 1.7
  • Definition 2.1
  • Definition 2.2
  • ...and 31 more