Embedded complex curves in the affine plane

IF 1 3区 数学 Q1 MATHEMATICS
Antonio Alarcón, Franc Forstnerič
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引用次数: 0

Abstract

This paper brings several contributions to the classical Forster–Bell–Narasimhan conjecture and the Yang problem concerning the existence of proper, almost proper, and complete injective holomorphic immersions of open Riemann surfaces in the affine plane \(\mathbb {C}^2\) satisfying interpolation and hitting conditions. We also show that every compact Riemann surface contains a Cantor set whose complement admits a proper holomorphic embedding in \(\mathbb {C}^2\), and every connected domain in \(\mathbb {C}^2\) admits complete, everywhere dense, injectively immersed complex discs. The focal point of the paper is a lemma saying for every compact bordered Riemann surface, M, closed discrete subset E of \(\mathring{M}=M\setminus bM\), and compact subset \(K\subset \mathring{M}\setminus E\) without holes in \(\mathring{M}\), any \(\mathscr {C}^1\) embedding \(f:M\hookrightarrow \mathbb {C}^2\) which is holomorphic in \(\mathring{M}\) can be approximated uniformly on K by holomorphic embeddings \(F:M\hookrightarrow \mathbb {C}^2\) which map \(E\cup bM\) out of a given ball and satisfy some interpolation conditions.

仿射平面中的嵌入复曲线
本文对经典的 Forster-Bell-Narasimhan 猜想和关于仿射平面 \(\mathbb {C}^2\)中开放黎曼曲面的适当、几乎适当和完全注入全形嵌入的存在性的杨问题做出了一些贡献。我们还证明了每一个紧凑黎曼曲面都包含一个康托集,其补集在\(\mathbb {C}^2\)中允许一个适当的全态嵌入,并且\(\mathbb {C}^2\)中的每一个连通域都允许完整的、无处不密集的、注入浸入的复圆盘。这篇论文的焦点是一个 Lemma,即对于每一个紧凑的有边黎曼曲面 M、\(\mathring{M}=Msetminus bM\) 的封闭离散子集 E,以及\(\mathring{M}\subset \mathring{M}\setminus E\) 中没有洞的紧凑子集,任何\(\mathscr {C}^1\) 的嵌入 \(f. Mhookrightarrow \mathring{M}/setminus E\) 都是完整的:在\(\mathring{M}\)中是全态的,可以在K上通过全态嵌入\(F:M\hookrightarrow \mathbb {C}^2\) 被均匀地近似,全态嵌入\(F:M\hookrightarrow \mathbb {C}^2\)将\(E\cup bM\) 映射出一个给定的球,并且满足一些插值条件。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.10
自引率
10.00%
发文量
99
审稿时长
>12 weeks
期刊介绍: This journal, the oldest scientific periodical in Italy, was originally edited by Barnaba Tortolini and Francesco Brioschi and has appeared since 1850. Nowadays it is managed by a nonprofit organization, the Fondazione Annali di Matematica Pura ed Applicata, c.o. Dipartimento di Matematica "U. Dini", viale Morgagni 67A, 50134 Firenze, Italy, e-mail annali@math.unifi.it). A board of Italian university professors governs the Fondazione and appoints the editors of the journal, whose responsibility it is to supervise the refereeing process. The names of governors and editors appear on the front page of each issue. Their addresses appear in the title pages of each issue.
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