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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
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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.

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Additional Information

Tomi Nieminen
Affiliation: Department of Technology, Jyväskylä University of Applied Sciences, P.O. Box 207, FIN-40101 Jyväskylä, Finland

Timo Tossavainen
Affiliation: Department of Teacher Education, University of Joensuu, P.O. Box 86, FIN-57101 Savonlinna, Finland

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.