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 $X(bM)=\bigcup_i X(bD_i)$ is a union of pairwise disjoint Jordan curves. This extends a recent result for bordered Riemann surfaces.

Theorems & Definitions (14)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Theorem 1.5
• Example 1.6
• Proposition 1.7
• proof
• Theorem 1.8
• Lemma 2.1
• proof