The Calabi-Yau problem for Riemann surfaces with finite genus and countably many ends
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2019
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913568119586816 |
|---|---|
| author | Alarcon, Antonio Forstneric, Franc |
| author_facet | Alarcon, Antonio Forstneric, Franc |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1904_08015 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The Calabi-Yau problem for Riemann surfaces with finite genus and countably many ends Alarcon, Antonio Forstneric, Franc Differential Geometry Complex Variables 53A10, 53C42, 32B15, 32H02 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. |
| title | The Calabi-Yau problem for Riemann surfaces with finite genus and countably many ends |
| topic | Differential Geometry Complex Variables 53A10, 53C42, 32B15, 32H02 |
| url | https://arxiv.org/abs/1904.08015 |