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arXiv · 1904.08015

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

Antonio Alarcon·Franc Forstneric
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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.

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BibTeXRIS

Antonio Alarcon, Franc Forstneric. 2019-04-16. The Calabi-Yau problem for Riemann surfaces with finite genus and countably many ends. https://doi.org/10.4171/rmi%2F1231

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