Boundary behavior of conformal deformations
HTML articles powered by AMS MathViewer
- by Tomi Nieminen and Timo Tossavainen
- Conform. Geom. Dyn. 11 (2007), 56-64
- DOI: https://doi.org/10.1090/S1088-4173-07-00161-0
- Published electronically: May 30, 2007
- PDF | Request permission
Abstract:
We study conformal deformations of the Euclidean metric in the unit ball $\mathbb {B}^{n}$. We assume that the density associated with the deformation satisfies a Harnack inequality and an arbitrary volume growth condition on the isodiametric profile. We establish a Hausdorff (gauge) dimension estimate for the set $E\subset \partial \mathbb {B}^{n}$ where such a deformation mapping can “blow up”. We also prove a generalization of Gerasch’s theorem in our setting.References
- Mario Bonk and Pekka Koskela, Conformal metrics and size of the boundary, Amer. J. Math. 124 (2002), no. 6, 1247–1287. MR 1939786, DOI 10.1353/ajm.2002.0034
- Mario Bonk, Pekka Koskela, and Steffen Rohde, Conformal metrics on the unit ball in Euclidean space, Proc. London Math. Soc. (3) 77 (1998), no. 3, 635–664. MR 1643421, DOI 10.1112/S0024611598000586
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Thomas E. Gerasch, On the accessibility of the boundary of a simply connected domain, Michigan Math. J. 33 (1986), no. 2, 201–207. MR 837578, DOI 10.1307/mmj/1029003349
- Bruce Hanson, Pekka Koskela, and Marc Troyanov, Boundary behavior of quasi-regular maps and the isodiametric profile, Conform. Geom. Dyn. 5 (2001), 81–99. MR 1872158, DOI 10.1090/S1088-4173-01-00076-5
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
- Olli Martio and Raimo Näkki, Boundary accessibility of a domain quasiconformally equivalent to a ball, Bull. London Math. Soc. 36 (2004), no. 1, 115–118. MR 2011985, DOI 10.1112/S0024609303002650
- C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
- William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3
Bibliographic Information
- Tomi Nieminen
- Affiliation: Department of Mathematics and Statistics, Jyväskylä University, P.O. Box 35, FIN-40014 Jyväskylä, Finland
- Email: tominiem@maths.jyu.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): October 20, 2006
- Published electronically: May 30, 2007
- © Copyright 2007 American Mathematical Society
- Journal: Conform. Geom. Dyn. 11 (2007), 56-64
- MSC (2000): Primary 30C65
- DOI: https://doi.org/10.1090/S1088-4173-07-00161-0
- MathSciNet review: 2314242