Illinois Journal of Mathematics

Conformal metrics and boundary accessibility

Tomi Nieminen

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Abstract

We study conformal metrics on the unit ball of Euclidean space. We prove an extension of a theorem originally due to Gerasch on the broadly accessibility of the boundary points of a domain quasiconformally equivalent to a ball. We also show that our result is close to optimal. Our abstract approach leads to new results also for the boundary behavior of (quasi)conformal mappings.

Article information

Source
Illinois J. Math., Volume 53, Number 1 (2009), 25-38.

Dates
First available in Project Euclid: 22 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1264170837

Digital Object Identifier
doi:10.1215/ijm/1264170837

Mathematical Reviews number (MathSciNet)
MR2584933

Zentralblatt MATH identifier
1191.30010

Subjects
Primary: 30C65: Quasiconformal mappings in $R^n$ , other generalizations

Citation

Nieminen, Tomi. Conformal metrics and boundary accessibility. Illinois J. Math. 53 (2009), no. 1, 25--38. doi:10.1215/ijm/1264170837. https://projecteuclid.org/euclid.ijm/1264170837


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References

  • D. R. Adams and L. I. Hedberg, Function spaces and potential theory, Springer, Berlin, 1996.
  • M. Bonk, P. Koskela and S. Rohde, Conformal metrics on the unit ball in euclidean space, Proc. London Math. Soc. (3) 77 (1998), 635–664.
  • H. Federer, Geometric measure theory, Springer, New York, 1969.
  • T. E. Gerasch, On the accessibility of the boundary of a simply connected domain, Michigan Math. J. 33 (1986), 201–207.
  • S. Hencl and P. Koskela, Quasihyperbolic boundary conditions and capacity: Uniform continuity of quasiconformal mappings, J. Anal. Math. 96 (2005), 19–35.
  • P. Koskela and T. Nieminen, Homeomorphisms of finite distortion: Discrete length of radial images, Math. Proc. Cambridge Philos. Soc. 144 (2008), 197–205.
  • P. Koskela and T. Nieminen, Uniform continuity of quasiconformal mappings and conformal deformations, Conform. Geom. Dyn. 12 (2008), 10–17.
  • P. Koskela and S. Rohde, Hausdorff dimension and mean porosity, Math. Ann. 309 (1997), 593–609.
  • P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Univ. Press, 1995.
  • O. Martio and R. Näkki, Boundary accessibility of a domain quasiconformally equivalent to a ball, Bull. London. Math. Soc. 36 (2004), 114–118.
  • T. Nieminen, Generalized mean porosity and dimension, Ann. Acad. Sci. Fenn. Math. 31 (2006), 143–172.
  • T. Nieminen and T. Tossavainen, Boundary behavior of conformal deformations, Conform. Geom. Dyn. 11 (2007), 56–64.
  • C. A. Rogers, Hausdorff measures, Cambridge Univ. Press, 1970.
  • E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970.