Skip to main content
SpringerLink
Log in

Heat Diffusions on Holomorphic Foliations With Non-Degenerate Singularities

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

Consider a Brody hyperbolic foliation with non-degenerate singularities on a compact complex manifold. We show that the leafwise heat diffusions and the abstract heat diffusions coincide. In particular, this will imply that the abstract heat diffusions are unique.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

Not applicable.

References

  1. Lins Neto, A., Soares, M.G.: Algebraic solutions of one-dimensional foliations. J. Differential Geom. 43(3), 652–673 (1996)

    Article  MathSciNet  Google Scholar 

  2. Jouanolou, J.-P.: Équations de Pfaff algébriques. (French) [Algebraic Pfaffian equations] Lecture Notes in Mathematics, 708. Springer, Berlin, (1979). v+255 pp

  3. Lins Neto, A.: Uniformization and the Poincaré metric on the leaves of a foliation by curves. Bol. Soc. Brasil. Mat. (N.S.) 31(3), 351–366 (2000)

  4. Glutsyuk, A.: Hyperbolicity of the leaves of a generic one-dimensional holomorphic foliation on a nonsingular projective algebraic variety. (Russian) Tr. Mat. Inst. Steklova213 (1997), Differ. Uravn. s Veshchestv. i Kompleks. Vrem., 90–111; translation in Proc. Steklov Inst. Math.213(2), 83–103 (1996)

  5. Dinh, T.-C., Nguyên, V.-A., Sibony, N.: Entropy for hyperbolic Riemann surface laminations II. In: A. Bonifant, M. Lyubich, S. Sutherland, editors. Frontiers in Complex Dynamics: a volume in honor of John Milnor’s 80th birthday, p. 593–622, (2014), Princeton University Press

  6. Dinh, T.-C., Sibony, N.: Some open problems on holomorphic foliation theory. Acta Math. Vietnam. 45(1), 103–112 (2020)

    Article  MathSciNet  Google Scholar 

  7. Fornæss, J.E., Sibony, N.: Riemann surface laminations with singularities. J. Geom. Anal. 18(2), 400–442 (2008)

    Article  MathSciNet  Google Scholar 

  8. Nguyên, V.-A.: Ergodic theory for Riemann surface laminations: a survey. Geometric complex analysis, 291–327, Springer Proc. Math. Stat., 246, Springer, Singapore, 2018

  9. Nguyên, V.-A.: Ergodic theorems for laminations and foliations: recent results and perspectives. Acta Math. Vietnam. 46(1), 9–101 (2021)

    Article  MathSciNet  Google Scholar 

  10. Rebelo, J.: On closed currents invariant by holomorphic foliations. I. Mosc. Math. J. 13(1), 123–185, 190 (2013)

  11. Garnett, L.: Foliations, the ergodic theorem and Brownian motion. J. Funct. Anal. 51(3), 285–311 (1983)

    Article  MathSciNet  Google Scholar 

  12. Dinh, T.-C., Nguyên, V.-A., Sibony, N.: Unique ergodicity for foliations on compact Kähler surfaces. Duke Math. J. 171(13), 2627–2698 (2022)

    MathSciNet  Google Scholar 

  13. Nguyên, V.-A.: Oseledec multiplicative ergodic theorem for laminations. Mem. Amer. Math. Soc.246(1164), ix+174 (2017). ISBN: 978-1-4704-2253-0; 978-1-4704-3637-7

  14. Chen, Z.: Directed harmonic currents near non-hyperbolic linearized singularities. Ergodic Theo. Dynam. Syst. 43(7), 2228–2257 (2023)

    Article  Google Scholar 

  15. Nguyên, V.-A.: Singular holomorphic foliations by curves. III: Zero Lelong numbers. Math. Ann. (2023). https://doi.org/10.1007/s00208-023-02618-6

  16. Dinh, T.-C., Nguyên, V.-A., Sibony, N.: Heat equation and ergodic theorems for Riemann surface laminations. Math. Ann. 354(1), 331–376 (2012)

    Article  MathSciNet  Google Scholar 

  17. Ilyashenko, Y., Yakovenko, S.: Lectures on analytic differential equations. Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008. xiv+625 pp

  18. Lins Neto, A., Canille Martins, J.C.: Hermitian metrics inducing the Poincaré metric, in the leaves of a singular holomorphic foliation by curves. Trans. Amer. Math. Soc. 356(7), 2963–2988 (2004)

    Article  MathSciNet  Google Scholar 

  19. Bacher, F.: Poincaré metric of holomorphic foliations with non-degenerate singularities. Internat. J. Math. 34(10), 22 (2023). (2350059)

  20. Chavel, I.: Eigenvalues in Riemannian geometry. Including a chapter by Burton Randol. With an appendix by Jozef Dodziuk. Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL, 1984. xiv+362 pp. ISBN: 0-12-170640-0

  21. Alkateeb, M., Rebelo, J.: Examples of harmonic foliated currents and singular Levi-flats on the projective plane. arXiv:2304.03744, (2023)

  22. Skoda, H.: Prolongement des courants, positifs, fermés de masse finie. (French) [Extension of closed, positive currents of finite mass]. Invent. Math. 66(3), 361–376 (1982)

  23. Brezis, H.: Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris, 1983. xiv+234 pp. ISBN: 2-225-77198-7

Download references

Acknowledgements

The author is supported by the Labex CEMPI (ANR-11-LABX-0007-01) and by the project QuaSiDy (ANR-21-CE40-0016).

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques Paul Painlevé, Université de Lille, CNRS U.M.R. 8524, 59655, Villeneuve d’Ascq Cedex, France

    François Bacher

Authors
  1. François Bacher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to François Bacher.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bacher, F. Heat Diffusions on Holomorphic Foliations With Non-Degenerate Singularities. J Geom Anal 34, 31 (2024). https://doi.org/10.1007/s12220-023-01485-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-023-01485-6

Keywords

Mathematics Subject Classification

Search

Navigation