Boundary behavior of conformal deformations

Nieminen, Tomi; Tossavainen, Timo

doi:10.1090/S1088-4173-07-00161-0

Boundary behavior of conformal deformations
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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

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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