Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 

 

Conformal metrics on the unit ball: The Gehring-Hayman property and the volume growth


Authors: Tomi Nieminen and Timo Tossavainen
Journal: Conform. Geom. Dyn. 13 (2009), 225-231
MSC (2010): Primary 30C65
Published electronically: October 28, 2009
MathSciNet review: 2558992
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We continue the study of conformal metrics on the unit ball in Euclidean space. We assume that the density $ \rho $ associated with the metric satisfies a Harnack inequality and then consider how much we can relax the volume growth condition from that in [Proc. London Math. Soc. Vol. 77 (3) (1998), 635-664] so that the Gehring-Hayman property still holds along the radii, i.e., if a boundary point can be accessed via a path with $ \rho $-length $ M<\infty $, then the $ \rho $-length of the corresponding radius is bounded by $ CM$. It turns out that if the path is inside a Stolz cone, then this result holds irrespective of the volume growth condition. Moreover, even if the path is not inside a Stolz cone, we are able to relax the volume growth condition for large $ r$, and still conclude that the corresponding radius is $ \rho $-rectifiable. This observation leads to a new estimate on the size of the boundary set corresponding to the $ \rho $-unrectifiable radii.


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


Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 30C65

Retrieve articles in all journals with MSC (2010): 30C65


Additional Information

Tomi Nieminen
Affiliation: Department of Technology, Jyväskylä University of Applied Sciences, P.O. Box 207, FIN-40101 Jyväskylä, Finland
Email: tomi.nieminen@jamk.fi

Timo Tossavainen
Affiliation: Department of Teacher Education, University of Joensuu, P.O. Box 86, FIN-57101 Savonlinna, Finland
Email: timo.tossavainen@joensuu.fi

DOI: https://doi.org/10.1090/S1088-4173-09-00202-1
Keywords: Boundary, conformal metrics, Gehring-Hayman property, quasiconformal mapping
Received by editor(s): June 28, 2009
Published electronically: October 28, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.