Parabolicity, the divergence theorem for δ-subharmonic functions and applications to the liouville theorems for harmonic maps

Atsushi Atsuji

doi:10.2748/tmj/1128703002

Parabolicity, the divergence theorem for δ-subharmonic functions and applications to the liouville theorems for harmonic maps

Atsushi Atsuji

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We show that the parabolicity of a manifold is equivalent to the validity of the ‘divergence theorem’ for some class of δ-subharmonic functions. From this property we can show a certain Liouville property of harmonic maps on parabolic manifolds. Elementary stochastic calculus is used as a main tool.

Original language	English
Pages (from-to)	353-373
Number of pages	21
Journal	Tohoku Mathematical Journal
Volume	57
Issue number	3
DOIs	https://doi.org/10.2748/tmj/1128703002
Publication status	Published - 2005

Keywords

Dirichlet form
Harmonic map
Liouville theorem
Martingale
δ-subharmonic functions

ASJC Scopus subject areas

Mathematics(all)

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10.2748/tmj/1128703002

Cite this

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title = "Parabolicity, the divergence theorem for δ-subharmonic functions and applications to the liouville theorems for harmonic maps",

abstract = "We show that the parabolicity of a manifold is equivalent to the validity of the {\textquoteleft}divergence theorem{\textquoteright} for some class of δ-subharmonic functions. From this property we can show a certain Liouville property of harmonic maps on parabolic manifolds. Elementary stochastic calculus is used as a main tool.",

keywords = "Dirichlet form, Harmonic map, Liouville theorem, Martingale, δ-subharmonic functions",

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AB - We show that the parabolicity of a manifold is equivalent to the validity of the ‘divergence theorem’ for some class of δ-subharmonic functions. From this property we can show a certain Liouville property of harmonic maps on parabolic manifolds. Elementary stochastic calculus is used as a main tool.

KW - Dirichlet form

KW - Harmonic map

KW - Liouville theorem

KW - Martingale

KW - δ-subharmonic functions

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Parabolicity, the divergence theorem for δ-subharmonic functions and applications to the liouville theorems for harmonic maps

Abstract

Keywords

ASJC Scopus subject areas

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