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The Calabi–Yau problem for Riemann surfaces with finite genus and countably many ends
Revista Matemática Iberoamericana ( IF 1.2 ) Pub Date : 2020-10-26 , DOI: 10.4171/rmi/1231
Antonio Alarcón 1 , Franc Forstnerič 2
Affiliation  

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 $X\colon M\to\mathbb{R}^3$, extending to a continuous map $X\colon \overline M\to\mathbb{R}^3$, such that $X(bM)=\bigcup_i X(bD_i)$ 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.

中文翻译:

具有有限属和可数多端的黎曼曲面的 Calabi-Yau 问题

在本文中,我们证明如果 $R$ 是一个紧凑的黎曼曲面,并且 $M=R\setminus\bigcup_i D_i$ 是 $R$ 中的一个域,其补集是可数许多成对不相交的光滑有界闭合圆盘的并集,$ D_i$,则有一个完整的共形最小浸入$X\colon M\to\mathbb{R}^3$,扩展到连续映射$X\colon \overline M\to\mathbb{R}^3$,这样 $X(bM)=\bigcup_i X(bD_i)$ 是成对不相交的 Jordan 曲线的并集。特别地,$M$ 是 $\mathbb{R}^3$ 中完全有界极小曲面的复杂结构。这扩展了有限边界黎曼曲面的最新结果。
更新日期:2020-10-26
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