Skip to main content
Log in

A foliation of the ball by complete holomorphic discs

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Alarcón, A.: Complete complex hypersurfaces in the ball come in foliations. ArXiv e-prints, (2018). arXiv:1802.02004

  2. Alarcón, A., Forstnerič, F.: Every bordered Riemann surface is a complete proper curve in a ball. Math. Ann. 357(3), 1049–1070 (2013)

    Article  MathSciNet  Google Scholar 

  3. Alarcón, A., Forstnerič, F.: Null curves and directed immersions of open Riemann surfaces. Invent. Math. 196(3), 733–771 (2014)

    Article  MathSciNet  Google Scholar 

  4. Alarcón, A., Forstnerič, F.: The Calabi-Yau problem for Riemann surfaces with finite genus and countably many ends. Rev. Mat. Iberoam., to appear. arXiv e-prints. arXiv:1904.08015

  5. Alarcón, A., Forstnerič, F.: New complex analytic methods in the theory of minimal surfaces: a survey. J. Aust. Math. Soc. 106(3), 287–341 (2019)

    Article  MathSciNet  Google Scholar 

  6. Alarcón, A., Globevnik, J.: Complete embedded complex curves in the ball of \({\mathbb{C}}^2\) can have any topology. Anal. PDE 10(8), 1987–1999 (2017)

    Article  MathSciNet  Google Scholar 

  7. Alarcón, A., Globevnik, J., López, F.J.: A construction of complete complex hypersurfaces in the ball with control on the topology. J. Reine Angew. Math. 751, 289–308 (2019)

    Article  MathSciNet  Google Scholar 

  8. Alarcón, A., López, F.J.: Null curves in \(\mathbb{C}^3\) and Calabi-Yau conjectures. Math. Ann. 355(2), 429–455 (2013)

    Article  MathSciNet  Google Scholar 

  9. Alarcón, A., López, F.J.: Complete bounded embedded complex curves in \(\mathbb{C}^2\). J. Eur. Math. Soc. (JEMS) 18(8), 1675–1705 (2016)

    Article  MathSciNet  Google Scholar 

  10. Charpentier, S., Kosiński, Ł.: Construction of labyrinths in pseudoconvex domains. arXiv e-prints, (Jul 2019). arXiv:1907.02803

  11. Drinovec Drnovšek, B.: Complete proper holomorphic embeddings of strictly pseudoconvex domains into balls. J. Math. Anal. Appl. 431(2), 705–713 (2015)

    Article  MathSciNet  Google Scholar 

  12. Forstnerič, F.: Noncritical holomorphic functions on Stein manifolds. Acta Math. 191(2), 143–189 (2003)

    Article  MathSciNet  Google Scholar 

  13. Forstnerič, F., Rosay, J.-P.: Approximation of biholomorphic mappings by automorphisms of \({\mathbb{C}}^n\). Invent. Math. 112(2), 323–349 (1993). (Erratum: Invent. Math., 118(3):573–574, 1994)

    Article  MathSciNet  Google Scholar 

  14. Forstnerič, F.: Stein manifolds and holomorphic mappings (The homotopy principle in complex analysis), volume 56 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 2nd edn. Springer, Cham (2017)

  15. Globevnik, J.: A complete complex hypersurface in the ball of \({\mathbb{C}}^N\). Ann. of Math. (2) 182(3), 1067–1091 (2015)

    Article  MathSciNet  Google Scholar 

  16. Globevnik, J.: Holomorphic functions unbounded on curves of finite length. Math. Ann. 364(3–4), 1343–1359 (2016)

    Article  MathSciNet  Google Scholar 

  17. Jones, P.W.: A complete bounded complex submanifold of \({ C}^{3}\). Proc. Am. Math. Soc. 76(2), 305–306 (1979)

    MATH  Google Scholar 

  18. Stout, E.L.: Polynomial Convexity, Volume 261 of Progress in Mathematics. Birkhäuser Boston Inc, Boston (2007)

  19. Yang, P.: Curvatures of complex submanifolds of \({ C}^{n}\). J. Differ. Geom. 12(4), 499–511 (1977). (1978)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

A. Alarcón is supported by the State Research Agency (SRA) and European Regional Development Fund (ERDF) via the Grant no. MTM2017-89677-P, MICINN, Spain. F. Forstnerič is supported by the research program P1-0291 and the research Grant J1-9104 from ARRS, Republic of Slovenia.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Antonio Alarcón.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alarcón, A., Forstnerič, F. A foliation of the ball by complete holomorphic discs. Math. Z. 296, 169–174 (2020). https://doi.org/10.1007/s00209-019-02430-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-019-02430-6

Keywords

Mathematics Subject Classification

Navigation