# A foliation of the ball by complete holomorphic discs

@article{Alarcn2019AFO, title={A foliation of the ball by complete holomorphic discs}, author={Antonio Alarc{\'o}n and Franc Forstneri{\vc}}, journal={Mathematische Zeitschrift}, year={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.

#### 7 Citations

Wild holomorphic foliations of the ball

- Mathematics
- 2021

We prove that the open unit ball Bn of C n (n ≥ 2) admits a nonsingular holomorphic foliation F by closed complex hypersurfaces such that both the union of the complete leaves of F and the union of… Expand

The Calabi–Yau problem for Riemann surfaces with finite genus and countably many ends

- Mathematics
- 2019

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… Expand

Minimal surfaces in Euclidean spaces by way of complex analysis

- Mathematics
- 2021

This is an extended version of my plenary lecture at the 8th European Congress of Mathematics in Portorož on 23 June 2021. The main part of the paper is a survey of recent applications of… Expand

Construction of labyrinths in pseudoconvex domains

- Mathematics
- 2019

We build in a given pseudoconvex (Runge) domain D of $${\mathbb {C}}^N$$ C N an $$\mathcal O(D)$$ O ( D ) -convex set $$\Gamma $$ Γ , every connected component of which is a holomorphically… Expand

HOLOMORPHIC LEGENDRIAN CURVES IN CP AND SUPERMINIMAL SURFACES IN S4

- 2019

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective 3-space CP, both from open and compact Riemann surfaces, and we prove that the space… Expand

Foliations on the open $3$-ball by complete surfaces

- Mathematics
- 2019

When is a manifold a leaf of a complete closed foliation on the open unit ball? We give some answers to this question.

Holomorphic Legendrian curves in $\mathbb{CP}^3$ and superminimal surfaces in $\mathbb S^4$

- Mathematics
- 2019

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective $3$-space $\mathbb{CP}^3$, both from open and compact Riemann surfaces, and we prove… Expand

#### References

SHOWING 1-10 OF 34 REFERENCES

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

- Mathematics
- 2013

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… 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 any… Expand

Complete proper holomorphic embeddings of strictly pseudoconvex domains into balls

- Mathematics
- 2015

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 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

Complete complex hypersurfaces in the ball come in foliations.

- Mathematics
- 2018

In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ is a level set of a noncritical holomorphic function on… Expand

A complete complex hypersurface in the ball of C^N

- Mathematics
- 2014

In 1977 P.Yang asked whether there exist complete immersed complex submanifolds g : M^k --> C^N with bounded image. A positive answer is known for holomorphic curves (k=1) and partial answers are… Expand

Stein Manifolds and Holomorphic Mappings: The Homotopy Principle in Complex Analysis

- Mathematics
- 2011

Preliminaries. - Stein Manifolds. - Stein Neighborhoods and Holomorphic Approximation. - Automorphisms of Complex Euclidean Spaces. - Oka Manifolds. - Elliptic Complex Geometry and Oka Principle. -… Expand

A construction of complete complex hypersurfaces in the ball with control on the topology

- Mathematics
- 2015

Given a closed complex hypersurface $Z\subset \mathbb{C}^{N+1}$ $(N\in\mathbb{N})$ and a compact subset $K\subset Z$, we prove the existence of a pseudoconvex Runge domain $D$ in $Z$ such that… Expand

Holomorphic functions unbounded on curves of finite length

- Mathematics
- 2014

Given a pseudoconvex domain , we prove that there is a holomorphic function f on D such that the lengths of paths $$p:\ [0,1] \rightarrow D$$p:[0,1]→D along which $$\mathfrak {R}f$$Rf is bounded… Expand

The Calabi–Yau problem for Riemann surfaces with finite genus and countably many ends

- Mathematics
- 2019

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… Expand