Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-01-05954-8
Published electronically: April 24, 2001
MathSciNet review: 1845022
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [BC] R. Bishop and R. Crittenden, Geometry of Manifolds, Academic Press, New York and London, 1964. MR 29:6401
  • [B] P. Buser, A note on the isoperimetric constant, Ann. Sci. Ecole Norm. Sup. 15 (1982), 213-230. MR 84e:58076
  • [CGT] J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geometry 17 (1982), 15-53. MR 84b:58109
  • [G] A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), 135-249. MR 99k:58195
  • [HK1] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1-61. MR 99j:30025
  • [HK2] J. Heinonen and P. Koskela, A note on Lipschitz functions, upper gradients, and the Poincaré inequality, New Zealand J. Math. 28 (1999), 37-42. MR 2000d:46041
  • [H1] I. Holopainen, Nonlinear potential theory and quasiregular mappings on Riemannian manifolds, Ann. Acad. Sci. Fenn. Ser. A I Math. Diss. 74 (1990), 1-45. MR 91e:31029
  • [H2] I. Holopainen, Positive solutions of quasilinear elliptic equations on Riemannian manifolds, Proc. London Math. Soc. (3) 65 (1992), 651-672. MR 94d:58161
  • [H3] I. Holopainen, Volume growth, Green's functions, and parabolicity of ends, Duke Math. J. 97 (1999), 319-346. MR 2000i:58066
  • [Li] P. Li, Curvature and function theory on Riemannian manifolds, Surveys in Diff. Geom. (to appear).
  • [LT] P. Li and L.F. Tam, Green's functions, harmonic functions, and volume comparison, J. Differential Geom. 41 (1995), 277-318. MR 96f:53054
  • [Liu] Z. Liu, Ball covering property and nonnegative Ricci curvature outside a compact set, Differential Geometry: Riemannian Geometry, Proc. Symp. Pure Math., vol. 54 (3), Amer. Math. Soc., Providence, RI, 1993, pp. 459-464.
  • [S] C.-J. Sung, A note on the existence of positive Green's function, J. Funct. Anal. 156 (1998), 199-207. MR 99g:53046

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58J60, 53C20, 31C12

Retrieve articles in all journals with MSC (2000): 58J60, 53C20, 31C12


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: https://doi.org/10.1090/S0002-9939-01-05954-8
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

American Mathematical Society