Functional Analysis and Its Applications

, Volume 53, Issue 3, pp 237–239 | Cite as

Uniformization of Foliations with Hyperbolic Leaves and the Beltrami Equation with Parameters

  • A. A. ShcherbakovEmail author


We consider foliations of compact complex manifolds by analytic curves. We suppose that the line bundle tangent to the foliation is negative. We show that, in the generic case, there exists a finitely smooth homomorphism holomorphic on the fibers and mapping fiberwise the manifold of universal coverings over the leaves passing through a given transversal B onto some domain with continuous boundary in B × ℂ depending on the leaves. The problem can be reduced to the study of the Beltrami equation with parameters on the unit disk in the case when the derivatives of the corresponding Beltrami coefficient grow no faster than some negative power of the distance to the boundary of the disk.

Key words

foliation Beltrami equation almost complex structure 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work was supported by RFBR grant 16-01-00748.


  1. [1]
    M. Brunella, in: Holomorphic Dynamical Systems, Lecture Notes in Math., Springer-Verlag, 2010, 105–165.Google Scholar
  2. [2]
    A. Verjovsky, in: Contemp. Math., vol. 58, part III, Amer. Math. Soc., Providence, RI, 1987, 233–253.Google Scholar
  3. [3]
    A. A. Glutsyuk, Trudy MIAN, 213 (1997), 90–111; English transl.: Proc. Steklov Inst. Math., 213 (1996), 83–103.Google Scholar
  4. [4]
    A. A. Glutsyuk, C. R. Math. Acad. Sci. Paris, 334:6 (2002), 489–494.MathSciNetCrossRefGoogle Scholar
  5. [5]
    T.-C. Dinh, V.-A. Nguyen, and N. Sibony, in: Frontiers in complex dynamics (In celebration of John Milnor’s 80th birthday), Princeton Univ. Press, Princeton, NJ, 2014, 569–592.CrossRefGoogle Scholar
  6. [6]
    Yu. S. Ilyashenko, Mat. Sb., 88(130):4(8) (1972), 558–577; English transl.: Math. USSR-Sb., 17:4 (1972), 551–569.MathSciNetGoogle Scholar
  7. [7]
    Yu. S. Ilyashenko, Trudy Mat. Inst. Steklov., 254 (2006), 196–214; English transl.: Proc. Steklov Inst. Math., 254 (2006), 184–200.Google Scholar
  8. [8]
    Yu. S. Ilyashenko, Topol. Methods Nonlinear Anal., 11:2 (1998), 361–373.MathSciNetCrossRefGoogle Scholar
  9. [9]
    G. Calsamiglia, B. Deroin, S. Frankel, and A. Guillot, J. Eur. Math. Soc., 15:3 (2013), 1067–1099.MathSciNetCrossRefGoogle Scholar
  10. [10]
    A. Lins Neto, Bol. Soc. Bras. Mat., Nova Ser., 31:3 (2000), 351–366.CrossRefGoogle Scholar
  11. [11]
    A. A. Shcherbakov, Trudy Moskov. Mat. Obshch., 76 (2015), 153–205; English transl.: Trans. Moscow Math. Soc., 76:2 (2015), 137–179.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2019

Authors and Affiliations

  1. 1.Institute of Physical Chemistry and Electro ChemistryRussian Academy of SciencesMoscowRussia

Personalised recommendations