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Almost complex structures on universal coverings of foliations


Author: A. A. Shcherbakov
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 76 (2015), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2015, 137-179
MSC (2010): Primary 32Q30; Secondary 53C12
DOI: https://doi.org/10.1090/mosc/250
Published electronically: November 17, 2015
MathSciNet review: 3467263
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Abstract: We consider foliations of compact complex manifolds by analytic curves. It is well known that if the line bundle tangent to the foliation is negative, then, in general position, all leaves are hyperbolic. The manifold of universal coverings over the leaves passing through some transversal has a natural complex structure. We show that in a typical case this structure can be defined as a smooth almost complex structure on the product of the base by the unit disk. We prove that this structure is quasiconformal on the leaves and that the corresponding $ (1,0)$-forms and their derivatives with respect to the coordinates on the base and in the leaves admit uniform estimates. The derivatives grow no faster than some negative power of the distance to the boundary of the disk.


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Additional Information

A. A. Shcherbakov
Affiliation: A. N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, Moscow, Russia
Email: shcher@elchem.ac.ru

DOI: https://doi.org/10.1090/mosc/250
Keywords: Foliation, Poincar\'e metric, almost complex structure.
Published electronically: November 17, 2015
Additional Notes: This research was supported by RFBR grant no. 10-01-00739 and by RFBR–CRNS grant no. 10-01-93115-CRNS-NTsNIL-a
Article copyright: © Copyright 2015 American Mathematical Society