# Every bordered Riemann surface is a complete proper curve in a ball

@article{Alarcn2013EveryBR,
title={Every bordered Riemann surface is a complete proper curve in a ball},
author={A. Alarc{\'o}n and F. Forstneri{\vc}},
journal={Mathematische Annalen},
year={2013},
volume={357},
pages={1049-1070}
}
• Published 2013
• Mathematics
• Mathematische Annalen
We prove that every bordered Riemann surface admits a complete proper holomorphic immersion into a ball of $$\mathbb C ^2$$, and a complete proper holomorphic embedding into a ball of $$\mathbb C ^3$$.

#### Figures from this paper

A foliation of the ball by complete holomorphic discs
• Mathematics
• 2019
We show that the open unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$ $(n>1)$ admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs.
Complete proper holomorphic embeddings of strictly pseudoconvex domains into balls
We construct a complete proper holomorphic embedding from any strictly pseudoconvex domain with $\mathcal{C}^2$-boundary in $\mathbb{C}^n$ into the unit ball of $\mathbb{C}^N$, for $N$ large enough,Expand
Complete embedded complex curves in the ball of $\mathbb{C}^2$ can have any topology
• Mathematics
• 2016
In this paper we prove that the unit ball $\mathbb{B}$ of $\mathbb{C}^2$ admits complete properly embedded complex curves of any given topological type. Moreover, we provide examples containing anyExpand
Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves
• Mathematics
• 2015
In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$ for any $n\ge 3$.Expand
Boundary continuity of complete proper holomorphic maps
We show that there is no complete proper holomorphic map from the open disc U in C to the bidisc UxU which extends continuously to the closed disc.
Complete bounded embedded complex curves in C^2
• Mathematics
• 2013
We prove that any convex domain of C^2 carries properly embedded complete complex curves. In particular, we exhibit the first examples of complete bounded embedded complex curves in C^2
Proper superminimal surfaces of given conformal types in the hyperbolic four-space
Let $H^4$ denote the hyperbolic four-space. Given a bordered Riemann surface, $M$, we prove that every smooth conformal superminimal immersion $\overline M\to H^4$ can be approximated uniformly onExpand
Holomorphic Embeddings and Immersions of Stein Manifolds: A Survey
In this paper we survey results on the existence of holomorphic embeddings and immersions of Stein manifolds into complex manifolds. Most of them pertain to proper maps into Stein manifolds. WeExpand
Noncritical holomorphic functions on Stein spaces
We prove that every reduced Stein space admits a holomorphic function without critical points. Furthermore, any closed discrete subset of such a space is the critical locus of a holomorphic function.Expand
Null curves and directed immersions of open Riemann surfaces
• Mathematics
• 2014
In this paper we study holomorphic immersions of open Riemann surfaces into C^n whose derivative lies in a conical algebraic subvariety A of C^n that is smooth away from the origin. ClassicalExpand

#### References

SHOWING 1-10 OF 32 REFERENCES
Bordered Riemann surfaces in C2
• Mathematics
• 2009
Abstract We prove that the interior of any compact complex curve with smooth boundary in C 2 admits a proper holomorphic embedding into C 2 . In particular, if D is a bordered Riemann surface whoseExpand
Embedding Certain Infinitely Connected Subsets of Bordered Riemann Surfaces Properly into ℂ2
We prove that certain infinitely connected domains D in a bordered Riemann surface, which admits a holomorphic embedding into C2, admit a proper holomorphic embedding into C2. We also prove thatExpand
Embeddings of infinitely connected planar domains into C^2
• Mathematics
• 2011
We prove that every circled domain in the Riemann sphere admits a proper holomorphic embedding to C^2. Our methods also apply to circled domains with punctures, provided that all but finitely many ofExpand
Proper holomorphic embeddings of Riemann surfaces with arbitrary topology into $\mathbb{C}^2$
• Mathematics
• 2011
We prove that given an open Riemann surface $N,$ there exists an open domain $M\subset N$ homeomorphic to $N$ which properly holomorphically embeds in $\mathbb{C}^2.$ Furthermore, $M$ can be chosenExpand
Existence of proper minimal surfaces of arbitrary topological type
• Mathematics
• 2009
Consider a domain D in R^3 which is convex (possibly all R^3) or which is smooth and bounded. Given any open surface M, we prove that there exists a complete, proper minimal immersion f : M --> D.Expand
HYPERBOLIC COMPLETE MINIMAL SURFACES WITH ARBITRARY TOPOLOGY
We show a method to construct orientable minimal surfaces in Et3 with arbitrary topology. This procedure giares complete examples of two different kinds: surfaces whose Gauss map omits four points ofExpand
Holomorphic curves in complex spaces
• Mathematics
• 2006
We study the existence of topologically closed complex curves normalized by bordered Riemann surfaces in complex spaces. Our main result is that such curves abound in any noncompact complex spaceExpand
Complete bounded holomorphic curves immersed in c2 with arbitrary genus
• Mathematics
• 2008
In (MUY), a complete holomorphic immersion of the unit disk D into C 2 whose image is bounded was constructed. In this paper, we shall prove existence of com- plete holomorphic null immersions ofExpand
Stein Manifolds and Holomorphic Mappings: The Homotopy Principle in Complex Analysis
Preliminaries. - Stein Manifolds. - Stein Neighborhoods and Holomorphic Approximation. - Automorphisms of Complex Euclidean Spaces. - Oka Manifolds. - Elliptic Complex Geometry and Oka Principle. -Expand
Null curves and directed immersions of open Riemann surfaces
• Mathematics
• 2014
In this paper we study holomorphic immersions of open Riemann surfaces into C^n whose derivative lies in a conical algebraic subvariety A of C^n that is smooth away from the origin. ClassicalExpand