### 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 |
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Pages (from-to) | 353-373 |

Number of pages | 21 |

Journal | Tohoku Mathematical Journal |

Volume | 57 |

Issue number | 3 |

Publication status | Published - 2005 Sep |

### Fingerprint

### Keywords

- δ-subharmonic functions
- Dirichlet form
- Harmonic map
- Liouville theorem
- Martingale

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Parabolicity, the divergence theorem for δ-subharmonic functions and applications to the Liouville theorems for harmonic maps.** / Atsuji, Atsushi.

Research output: Contribution to journal › Article

*Tohoku Mathematical Journal*, vol. 57, no. 3, pp. 353-373.

}

TY - JOUR

T1 - Parabolicity, the divergence theorem for δ-subharmonic functions and applications to the Liouville theorems for harmonic maps

AU - Atsuji, Atsushi

PY - 2005/9

Y1 - 2005/9

N2 - 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.

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 - δ-subharmonic functions

KW - Dirichlet form

KW - Harmonic map

KW - Liouville theorem

KW - Martingale

UR - http://www.scopus.com/inward/record.url?scp=27444447621&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27444447621&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:27444447621

VL - 57

SP - 353

EP - 373

JO - Tohoku Mathematical Journal

JF - Tohoku Mathematical Journal

SN - 0040-8735

IS - 3

ER -