# Revista Matemática Iberoamericana

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**Published online first: 2020-10-26**

**DOI: 10.4171/rmi/1231**

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

Antonio Alarcón^{[1]}and Franc Forstnerič

^{[2]}(1) Universidad de Granada, Spain

(2) University of Ljubljana, Slovenia

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 there is a complete conformal minimal immersion $X\colon M\to\mathbb{R}^3$, extending to a continuous map $X\colon \overline M\to\mathbb{R}^3$, such that $X(bM)=\bigcup_i X(bD_i)$ is a union of pairwise disjoint Jordan curves. In particular, $M$ is the complex structure of a complete bounded minimal surface in $\mathbb{R}^3$. This extends a recent result for finite bordered Riemann surfaces.

*Keywords: *Riemann surface, minimal surface, Calabi–Yau problem

Alarcón Antonio, Forstnerič Franc: The Calabi–Yau problem for Riemann surfaces with finite genus and countably many ends. *Rev. Mat. Iberoam.* Electronically published on October 26, 2020. doi: 10.4171/rmi/1231 (to appear in print)