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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Conformal metrics on the unit ball: The Gehring-Hayman property and the volume growth
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by Tomi Nieminen and Timo Tossavainen PDF
Conform. Geom. Dyn. 13 (2009), 225-231 Request permission

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
  • 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
  • Received by editor(s): June 28, 2009
  • Published electronically: October 28, 2009
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Conform. Geom. Dyn. 13 (2009), 225-231
  • MSC (2010): Primary 30C65
  • DOI: https://doi.org/10.1090/S1088-4173-09-00202-1
  • MathSciNet review: 2558992