Knotted holomorphic discs in

Baader, Sebastian; Kutzschebauch, Frank; Wold, Erlend Fornaess

doi:10.1515/crelle.2010.079

Cited by 4 publications

(9 citation statements)

References 8 publications

(18 reference statements)

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“…In 2010, Baader, Kutzschebauch and Wold [28] used Fatou-Bieberbach domains to construct the first known example of a knotted properly embedded holomorphic disc in C 2 . Their result was motivated by the problem, raised by Kirby, whether proper holomorphic embeddings of C or the unit disc into C 2 can be topologically knotted.…”

Section: A Domain ωmentioning

confidence: 99%

“…Their result was motivated by the problem, raised by Kirby, whether proper holomorphic embeddings of C or the unit disc into C 2 can be topologically knotted. While the first problem remains open, the second one was solved in the affirmative in [28]. The proof uses wellbehaved Fatou-Bieberbach domains in C 2 constructed by Globevnik in [96], containing small perturbations of the bidisc, and the existence of knotted holomorphic discs in the bidisc.…”

Section: A Domain ωmentioning

confidence: 99%

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The first thirty years of Andersen-Lempert theory

Forstnerič¹,

Kutzschebauch²

2021

Preprint

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In this paper we expose the impact of the fundamental discovery, made by Erik Andersén and László Lempert in 1992, that the group generated by shears is dense in the group of holomorphic automorphisms of a complex Euclidean space of dimensions n > 1. In three decades since its publication, their groundbreaking work led to the discovery of several new phenomena and to major new results in complex analysis and geometry involving Stein manifolds and affine algebraic manifolds with many automorphisms. The aim of this survey is to present the focal points of these developments, with a view towards the future. Dedicated to LászlóLempert in honour of his 70th birthday CONTENTS 1. Introduction 2. Stein manifolds with density properties 2.1. Density property 2.2. Volume density property 2.3. Relative density properties 2.4. Fibred density properties 2.5. Symplectic density property 3. Automorphisms with given jets 4. Fatou-Bieberbach domains 5. Twisted complex lines in C n and nonlinearizable automorphisms 6. Embedding open Riemann surfaces in C 2 7. Complex manifolds exhausted by Euclidean spaces 8. Stein manifolds with the density property and Oka manifolds 9. Complete complex submanifolds 10. An application in 3-dimensional topology 11. The recognition problem for complex Euclidean spaces References

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Section: A Domain ωmentioning

confidence: 99%

Section: A Domain ωmentioning

confidence: 99%

The first thirty years of Andersen-Lempert theory

Forstnerič¹,

Kutzschebauch²

2021

Preprint

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“…In another direction, Baader et al [17] constructed an example of a properly embedded disc in C 2 whose image is topologically knotted, thereby answering a questions of Kirby. It is unknown whether there exists a knotted proper holomorphic embedding C ֒→ C 2 , or an unknotted proper holomorphic embedding D ֒→ C 2 of the disc.…”

Section: Automorphisms Of Euclidean Spaces and Wild Embeddingsmentioning

confidence: 99%

Holomorphic Embeddings and Immersions of Stein Manifolds: A Survey

Forstnerič

2018

Springer Proceedings in Mathematics &Amp; Statistics

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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. We include a new result saying that every continuous map X → Y between Stein manifolds is homotopic to a proper holomorphic embedding provided that dim Y > 2 dim X and we allow a homotopic deformation of the Stein structure on X.

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“…We note that this last step is nontrivial because the open 4-ball is not biholomorphically equivalent to C 2 . Instead, we achieve this re-embedding by situating our initial complex curves inside of Fatou-Bieberbach domains (cf [BKW10]); see §5 for details. This re-embedding may change the surfaces' isotopy types, but the more robust obstruction used to establish Theorem C persists, distinguishing these curves' images in C 2 .…”

Section: Introductionmentioning

confidence: 99%

“…Rudolph showed that such a surface can be realized as a compact piece of an algebraic curve. Crucially, this surface depends not only on the braid but on the quasipositive factorization itself, allowing for subtle control of the resulting surface (c.f., [BKW10,BVHM18,Hay19,Oba20]). We show that the disks from Figure 1 arise from "exotic" braid factorizations.…”

Section: Introductionmentioning

confidence: 99%