### 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.

Original language | English |
---|---|

Pages (from-to) | 1-29 |

Number of pages | 29 |

Journal | Potential Analysis |

DOIs | |

Publication status | Accepted/In press - 2017 May 4 |

### Fingerprint

### Keywords

- Lamination
- Leafwise Brownian motion
- Nevanlinna theory
- Value distribution theory

### ASJC Scopus subject areas

- Analysis

### Cite this

**Leafwise Brownian Motions and Some Function Theoretic Properties of Laminations.** / Atsuji, Atsushi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Leafwise Brownian Motions and Some Function Theoretic Properties of Laminations

AU - Atsuji, Atsushi

PY - 2017/5/4

Y1 - 2017/5/4

N2 - 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.

AB - 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.

KW - Lamination

KW - Leafwise Brownian motion

KW - Nevanlinna theory

KW - Value distribution theory

UR - http://www.scopus.com/inward/record.url?scp=85018738995&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85018738995&partnerID=8YFLogxK

U2 - 10.1007/s11118-017-9627-9

DO - 10.1007/s11118-017-9627-9

M3 - Article

AN - SCOPUS:85018738995

SP - 1

EP - 29

JO - Potential Analysis

JF - Potential Analysis

SN - 0926-2601

ER -