Leafwise Brownian Motions and Some Function Theoretic Properties of Laminations

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)1-29
Number of pages29
JournalPotential Analysis
DOIs
Publication statusAccepted/In press - 2017 May 4

Fingerprint

Lamination
Brownian motion
Leaves
Meromorphic Function
Subharmonic Function
Harmonic Measure
Value Distribution
Nevanlinna Theory
Borel Functions
Liouville Type Theorem
Ergodic Measure
Subharmonics
Meromorphic
Measurable function
Riemannian Manifold
Defects
Curvature
Theorem

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.

In: Potential Analysis, 04.05.2017, p. 1-29.

Research output: Contribution to journalArticle

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