# Carleman approximation by conformal minimal immersions and directed holomorphic curves

## TL;DR

This paper generalizes Carleman's theorem by showing that continuous functions on divergent Jordan arcs in open Riemann surfaces can be approximated by conformal minimal immersions and directed holomorphic curves, with applications to Plateau problems.

## Contribution

It extends Carleman's approximation theorem to conformal minimal immersions and directed holomorphic curves on Riemann surfaces, including null curves, with control over completeness and properness.

## Key findings

- Approximation of continuous functions by conformal minimal immersions on Jordan arcs.
- Extension of the approximation to null curves and directed holomorphic immersions.
- Application to solving Plateau problems for divergent Jordan curves.

## Abstract

Let $\mathcal{R}$ be an open Riemann surface. In this paper we prove that every continuous function $M \to \mathbb{R}^n$, $n\ge 3$, defined on a divergent Jordan arc $M \subset \mathcal{R}$ can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalization of Carleman's theorem. In fact, we prove that this result remains true for null curves and many other classes of directed holomorphic immersions for which the directing variety satisfies a certain flexibility property. Furthermore, the constructed immersions may be chosen to be complete or proper under natural assumptions on the variety and the continuous map. As a consequence we give an approximate solution to a Plateau problem for divergent Jordan curves in the Euclidean spaces.