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Topology and Complex Structures of Leaves of Foliations by Riemann Surfaces

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Abstract

We study conformal structure and topology of leaves of singular foliations by Riemann surfaces.

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Notes

  1. We recall that a Riemann surface \(Y\subset X\) is weakly directed by \({\mathcal {L}}\) if \(Y\setminus E\) is locally contained in a leaf.

  2. We recall that this means that \(f(\mathbb C)\setminus E\) is locally contained in leaves.

  3. By this we mean that \(\rho \) is continuous (resp. upper or lower semi-continuous) on X at all points \(p\in L\), if L is any leaf without holonomy. In what follows, this will always be our meaning of the word “continuous”.

  4. The proof is the same as that of [10] Theorem 15; if there is no such \(c>0\) one obtains an image of \(\mathbb C\) weakly directed by \({\mathcal {L}}\) using Brody’s reparametrization Lemma.

  5. In the situation at hand, we can also see this directly in two ways. First, the current T is of order zero, and since for any function g defined on E, we can solve the equation \(\partial \overline{\partial }\varphi |_E=g\), the condition that \(\partial \overline{\partial }T=0\) implies that the mass is not concentrated on E. Second, the current cannot have compact support in a domain that carries a strictly plurisubharmonic function \(\phi \), since we would get \(\langle T,i\partial \overline{\partial }(\chi \phi )\rangle >0\) for a suitable cutoff function \(\chi \)

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Acknowledgements

The authors would like to thank the referee for a very thorough reading, and many useful suggestions for improvements. They would also like to M. Kapovich for providing them with a reference to the Klein–Maskit Combination Theorem in connection to the construction in Example 6.6., as well as for other useful comments.

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Authors and Affiliations

  1. Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405, Orsay, France

    Nessim Sibony

  2. Department of Mathematics, University of Oslo, Blindern, PO-BOX 1053, Oslo, Norway

    Nessim Sibony & Erlend Fornæss Wold

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  1. Nessim Sibony
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Correspondence to Erlend Fornæss Wold.

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In memory of Gennadi M. Henkin.

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Erlend Fornæss Wold is supported by NRC Grant Number 240569. Part of this work was done during the international research programme “Several Complex Variables and Complex Dynamics” at the Centre for Advanced Study at the Academy of Science and Letters in Oslo during the academic year 2016/2017.

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Sibony, N., Wold, E.F. Topology and Complex Structures of Leaves of Foliations by Riemann Surfaces. J Geom Anal 30, 2593–2614 (2020). https://doi.org/10.1007/s12220-017-9975-0

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