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

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 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 $X(bM)={\bigcup }_{i}X(b{D}_i)$ 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.