Topology and dynamics of laminations in surfaces of general type. (English) Zbl 1334.32014
Summary: 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.
MSC:
| 32V30 | Embeddings of CR manifolds |
| 37F75 | Dynamical aspects of holomorphic foliations and vector fields |
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