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 $M$ 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 $X(bM)={\bigcup }_{i}X(b{D}_i)$ is a union of pairwise disjoint Jordan curves. This extends a recent result for bordered Riemann surfaces.

Rev. Mat. Iberoam., to appear

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• The Calabi-Yau property of superminimal surfaces in self-dual Einstein four-manifolds
• Holomorphic Legendrian curves in ${\mathbb{CP}}^3$ and superminimal surfaces in ${\mathbb{S}}^4$
• The Calabi-Yau problem for minimal surfaces with Cantor ends
• Complete nonsingular holomorphic foliations on Stein manifolds
• Proper superminimal surfaces of given conformal types in the hyperbolic four-space