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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

**[BC]**Richard L. Bishop and Richard J. Crittenden,*Geometry of manifolds*, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR**0169148****[B]**Peter Buser,*A note on the isoperimetric constant*, Ann. Sci. École Norm. Sup. (4)**15**(1982), no. 2, 213–230. MR**683635****[CGT]**Jeff Cheeger, Mikhail Gromov, and Michael Taylor,*Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds*, J. Differential Geom.**17**(1982), no. 1, 15–53. MR**658471****[G]**Alexander Grigor′yan,*Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds*, Bull. Amer. Math. Soc. (N.S.)**36**(1999), no. 2, 135–249. MR**1659871**, 10.1090/S0273-0979-99-00776-4**[HK1]**Juha Heinonen and Pekka Koskela,*Quasiconformal maps in metric spaces with controlled geometry*, Acta Math.**181**(1998), no. 1, 1–61. MR**1654771**, 10.1007/BF02392747**[HK2]**Juha Heinonen and Pekka Koskela,*A note on Lipschitz functions, upper gradients, and the Poincaré inequality*, New Zealand J. Math.**28**(1999), no. 1, 37–42. MR**1691958****[H1]**Ilkka Holopainen,*Nonlinear potential theory and quasiregular mappings on Riemannian manifolds*, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes**74**(1990), 45. MR**1052971****[H2]**Ilkka Holopainen,*Positive solutions of quasilinear elliptic equations on Riemannian manifolds*, Proc. London Math. Soc. (3)**65**(1992), no. 3, 651–672. MR**1182105**, 10.1112/plms/s3-65.3.651**[H3]**Ilkka Holopainen,*Volume growth, Green’s functions, and parabolicity of ends*, Duke Math. J.**97**(1999), no. 2, 319–346. MR**1682233**, 10.1215/S0012-7094-99-09714-4**[Li]**P. Li,*Curvature and function theory on Riemannian manifolds*, Surveys in Diff. Geom. (to appear).**[LT]**Peter Li and Luen-Fai Tam,*Green’s functions, harmonic functions, and volume comparison*, J. Differential Geom.**41**(1995), no. 2, 277–318. MR**1331970****[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]**Chiung-Jue Anna Sung,*A note on the existence of positive Green’s function*, J. Funct. Anal.**156**(1998), no. 1, 199–207. MR**1632905**, 10.1006/jfan.1997.3235

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:
http://dx.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