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

## Univerza v Ljubljani

We show that if R is a compact Riemann surface and M is a domain in R whose complement is a union of countably many pairwise disjoint smoothly bounded closed discs, then M is the complex structure of a complete bounded minimal surface in \(\mathbb{R}^3\). More precisely, we prove that there is a complete conformal minimal immersion \(X:M→\mathbb{ℝ}^3\) extending to a continuous map from the closure of \(M\) such that \(X(\partial M)\) is a union of pairwise disjoint Jordan curves. This extends a result for finite bordered Riemann surfaces proved in 2015. (Joint work with Antonio Alarcon.)