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Leafwise Brownian Motions and Some Function Theoretic Properties of Laminations

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Abstract

We discuss the value distribution of Borel measurable functions which are subharmonic or meromorphic along leaves on laminations. They are called leafwise subharmonic functions or meromorphic functions respectively. We consider cases that each leaf is a negatively curved Riemannian manifold or Kähler manifold. We first consider the case when leaves are Riemannian with a harmonic measure in L.Garnett sense. We show some Liouville type theorem holds for leafwise subharmonic functions in this case. In the case of laminations whose leaves are Kähler manifolds with some curvature condition we consider the value distribution of leafwise meromorphic functions. If a lamination has an ergodic harmonic measure, a variant of defect relation in Nevanlinna theory is obtained for almost all leaves. It gives a bound of the number of omitted points by those functions. Consequently we have a Picard type theorem for leafwise meromorphic functions.

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Acknowledgements

The author would like to thank the referee for several helpful comments.

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Authors and Affiliations

  1. Department Mathematics, Keio University, 3-14-1, Hiyoshi, Yokohama, 223-8521, Japan

    Atsushi Atsuji

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  1. Atsushi Atsuji
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Correspondence to Atsushi Atsuji.

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Partly supported by the Grant-in-Aid for Scientific Research (C) 24540192, Japan Society for the Promotion of Science and Keio Gijuku Academic Development Funds.

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Atsuji, A. Leafwise Brownian Motions and Some Function Theoretic Properties of Laminations. Potential Anal 48, 85–113 (2018). https://doi.org/10.1007/s11118-017-9627-9

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