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Convexity of complements of limit sets for holomorphic foliations on surfaces

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Abstract

Let \({\mathcal {F}}\) be a holomorphic foliation on a compact Kähler surface with hyperbolic singularities and no foliation cycle. We prove that if the limit set of \({\mathcal {F}}\) has zero Lebesgue measure, then its complement is a modification of a Stein domain. This applies for the case of suspensions of Kleinian representations, answering a question asked by Brunella. The proof consists in building, in several steps, a metric of positive curvature for the normal bundle of \({\mathcal {F}}\) near the limit set. Then we construct a proper strictly plurisubharmonic exhaustion function for the complement of the limit set by extending Brunella’s method to our singular context. The arguments hold more generally when the limit set is thin, a property relying on Brownian motion.

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Acknowledgements

We thank Alano Ancona for pointing the Ref. [6] on the thin property and Misha Lyubich for discussions about that property for Julia sets. We also thank the Research in Paris program of Institut Henri Poincaré for the very nice working conditions offered to us during fall 2021.

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  1. AGM, UMR CNRS 8088, CY Cergy Paris Université, Paris, France

    Bertrand Deroin

  2. IRMAR, UMR CNRS 6625, Université de Rennes, Rennes, France

    Christophe Dupont & Victor Kleptsyn

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  1. Bertrand Deroin
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  2. Christophe Dupont
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Correspondence to Christophe Dupont.

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Deroin, B., Dupont, C. & Kleptsyn, V. Convexity of complements of limit sets for holomorphic foliations on surfaces. Math. Ann. 388, 2727–2753 (2024). https://doi.org/10.1007/s00208-023-02590-1

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