Embedded complex curves in the affine plane
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929568802144256 |
|---|---|
| author | Alarcon, Antonio Forstneric, Franc |
| author_facet | Alarcon, Antonio Forstneric, Franc |
| contents | This paper brings several contributions to the classical Forster-Bell-Narasimhan conjecture and the Yang problem concerning the existence of proper and almost proper (hence complete) injective holomorphic immersions of open Riemann surfaces in the affine plane $\mathbb C^2$ satisfying interpolation and hitting conditions. We also show that in every compact Riemann surface there is a Cantor set whose complement admits a proper holomorphic embedding in $\mathbb C^2$. The focal point is a lemma saying the following. Given a compact bordered Riemann surface, $M$, a closed discrete subset $E$ of its interior $\mathring M=M\setminus bM$, a compact subset $K\subset \mathring M\setminus E$ without holes in $\mathring M$, and a $\mathscr C^1$ embedding $f:M\hookrightarrow \mathbb C^2$ which is holomorphic in $\mathring M$, we can approximate $f$ uniformly on $K$ by a holomorphic embedding $F:M\hookrightarrow \mathbb C^2$ which maps $E\cup bM$ out of a given ball and satisfies some interpolation conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_10304 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Embedded complex curves in the affine plane Alarcon, Antonio Forstneric, Franc Complex Variables 32E10, 32E30, 32H02 (Primary), 14H55 (secondary) This paper brings several contributions to the classical Forster-Bell-Narasimhan conjecture and the Yang problem concerning the existence of proper and almost proper (hence complete) injective holomorphic immersions of open Riemann surfaces in the affine plane $\mathbb C^2$ satisfying interpolation and hitting conditions. We also show that in every compact Riemann surface there is a Cantor set whose complement admits a proper holomorphic embedding in $\mathbb C^2$. The focal point is a lemma saying the following. Given a compact bordered Riemann surface, $M$, a closed discrete subset $E$ of its interior $\mathring M=M\setminus bM$, a compact subset $K\subset \mathring M\setminus E$ without holes in $\mathring M$, and a $\mathscr C^1$ embedding $f:M\hookrightarrow \mathbb C^2$ which is holomorphic in $\mathring M$, we can approximate $f$ uniformly on $K$ by a holomorphic embedding $F:M\hookrightarrow \mathbb C^2$ which maps $E\cup bM$ out of a given ball and satisfies some interpolation conditions. |
| title | Embedded complex curves in the affine plane |
| topic | Complex Variables 32E10, 32E30, 32H02 (Primary), 14H55 (secondary) |
| url | https://arxiv.org/abs/2301.10304 |