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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Volume growth and parabolicity


Authors: Ilkka Holopainen and Pekka Koskela
Journal: Proc. Amer. Math. Soc. 129 (2001), 3425-3435
MSC (2000): Primary 58J60, 53C20, 31C12
Published electronically: April 24, 2001
MathSciNet review: 1845022
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Abstract:

We characterize $p$-parabolicity of a noncompact complete Riemannian manifold $M$ in terms of the volume growth of $M$ under very weak assumptions on $M$. Some of the results also apply to the setting of metric measure spaces.


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Additional Information

Ilkka Holopainen
Affiliation: Department of Mathematics, University of Helsinki, P.O. Box 4 (Yliopistonkatu 5), FIN-00014 Helsinki, Finland
Email: ilkka.holopainen@helsinki.fi

Pekka Koskela
Affiliation: Department of Mathematics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland
Email: pkoskela@math.jyu.fi

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05954-8
PII: S 0002-9939(01)05954-8
Keywords: Volume growth, harmonic function, Green's function, parabolicity
Received by editor(s): December 1, 1999
Received by editor(s) in revised form: April 3, 2000
Published electronically: April 24, 2001
Additional Notes: The first author’s work was supported by the Academy of Finland, projects 6355 and 44333
The second author’s work was supported by the Academy of Finland, project 39788
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2001 American Mathematical Society