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Boundary behavior of conformal deformations. (English) Zbl 1133.30321

Summary: 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.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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[1] Mario Bonk and Pekka Koskela, Conformal metrics and size of the boundary, Amer. J. Math. 124 (2002), no. 6, 1247 – 1287. · Zbl 1018.30016
[2] 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. · Zbl 0916.30017 · doi:10.1112/S0024611598000586
[3] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801
[4] Thomas E. Gerasch, On the accessibility of the boundary of a simply connected domain, Michigan Math. J. 33 (1986), no. 2, 201 – 207. · Zbl 0605.30020 · doi:10.1307/mmj/1029003349
[5] Bruce Hanson, Pekka Koskela, and Marc Troyanov, Boundary behavior of quasi-regular maps and the isodiametric profile, Conform. Geom. Dyn. 5 (2001), 81 – 99 (electronic). · Zbl 1051.30021 · doi:10.1090/S1088-4173-01-00076-5
[6] 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. · Zbl 0819.28004
[7] 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. · Zbl 1040.30010 · doi:10.1112/S0024609303002650
[8] C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. · Zbl 0204.37601
[9] William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. · Zbl 0692.46022
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