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Topology and dynamics of laminations in surfaces of general type


Authors: Bertrand Deroin and Christophe Dupont
Journal: J. Amer. Math. Soc. 29 (2016), 495-535
MSC (2010): Primary 32V30, 37C85; Secondary 37F75, 37C45
DOI: https://doi.org/10.1090/jams832
Published electronically: June 2, 2015
MathSciNet review: 3454381
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Abstract: We study Riemann surface laminations in complex algebraic surfaces of general type. We focus on the topology and dynamics of minimal sets of holomorphic foliations and on Levi-flat hypersurfaces. We begin by providing various examples. Then our first result shows that Anosov Levi-flat hypersurfaces do not embed in surfaces of general type. This allows one to classify the possible Thurston's geometries carried by Levi-flat hypersurfaces in surfaces of general type. Our second result establishes that minimal sets in surfaces of general type have a large Hausdorff dimension as soon as there exists a simply connected leaf. For both results, our methods rely on ergodic theory: we use harmonic measures and Lyapunov exponents.


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Additional Information

Bertrand Deroin
Affiliation: Ecole Normale Supérieure, DMA, UMR CNRS 8553, 45 rue d’Ulm, 75230 Paris Cedex 05, France
Email: bertrand.deroin@ens.fr

Christophe Dupont
Affiliation: Université de Rennes 1, IRMAR, UMR CNRS 6625, 35042 Rennes Cedex, France
Email: christophe.dupont@univ-rennes1.fr

DOI: https://doi.org/10.1090/jams832
Keywords: Riemann surfaces laminations, minimal set, Levi-flat hypersurface, harmonic measure, Lyapunov exponent, surface of general type, Thurston's geometry
Received by editor(s): May 11, 2012
Received by editor(s) in revised form: July 31, 2013, November 28, 2014, and March 9, 2015
Published electronically: June 2, 2015
Additional Notes: The research leading to these results has received funding from the European Research Council under the European Community’s seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No. FP7-246918, and from the ANR projects Dynacomplexe ANR-07-JCJC-0006 and LAMBDA, ANR-13-BS01-0002.
The first author was supported by the ANR projects 08-JCJC-0130-01 and 09-BLAN-0116.
The second author was supported by the ANR project 07-JCJC-0006-01.
Dedicated: To the memory of Marco Brunella
Article copyright: © Copyright 2015 American Mathematical Society