# Interpolation by conformal minimal surfaces and directed holomorphic curves

## TL;DR

This paper establishes methods for interpolating conformal minimal surfaces and directed holomorphic curves on open Riemann surfaces, allowing precise control over values, Gauss maps, flux, and topological properties.

## Contribution

It introduces new interpolation techniques for conformal minimal surfaces and directed holomorphic curves, including jet-interpolation, properness, and injectivity conditions.

## Key findings

- Interpolation on discrete sets is possible for conformal minimal immersions.
- Interpolating immersions can be made complete, proper, and one-to-one under certain conditions.
- The results extend to directed holomorphic immersions, including null curves.

## Abstract

Let $M$ be an open Riemann surface and $n\ge 3$ be an integer. We prove that on any closed discrete subset of $M$ one can prescribe the values of a conformal minimal immersion $M\to\mathbb{R}^n$. Our result also ensures jet-interpolation of given finite order, and hence, in particular, one may in addition prescribe the values of the generalized Gauss map. Furthermore, the interpolating immersions can be chosen to be complete, proper into $\mathbb{R}^n$ if the prescription of values is proper, and one-to-one if $n\ge 5$ and the prescription of values is one-to-one. We may also prescribe the flux map of the examples. We also show analogous results for a large family of directed holomorphic immersions $M\to\mathbb{C}^n$, including null curves.