# The Calabi-Yau problem for Riemann surfaces with finite genus and countably many ends

## TL;DR

This paper proves that certain Riemann surfaces with countably many ends can be realized as complete bounded minimal surfaces in three-dimensional space, extending previous results to more complex topologies.

## Contribution

It extends the class of Riemann surfaces known to admit complete bounded minimal immersions to include those with countably many ends.

## Key findings

- Constructs complete conformal minimal immersions for the specified Riemann surfaces.
- Ensures the boundary images form disjoint Jordan curves.
- Generalizes previous results for bordered Riemann surfaces.

## Abstract

In this paper, we show that if $R$ is a compact Riemann surface and $M=R\setminus\,\bigcup_i D_i$ is a domain in $R$ whose complement is a union of countably many pairwise disjoint smoothly bounded closed discs $D_i$, then $M$ is the complex structure of a complete bounded minimal surface in $\mathbb R^3$. We prove that there is a complete conformal minimal immersion $X:M\to\mathbb R^3$ extending to a continuous map $X:\overline M\to\mathbb R^3$ such that $X(bM)=\bigcup_i X(bD_i)$ is a union of pairwise disjoint Jordan curves. This extends a recent result for bordered Riemann surfaces.