Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

Sphericalization and flattening


Authors: Zoltán M. Balogh and Stephen M. Buckley
Journal: Conform. Geom. Dyn. 9 (2005), 76-101
MSC (2000): Primary 30F45
DOI: https://doi.org/10.1090/S1088-4173-05-00124-4
Published electronically: November 29, 2005
MathSciNet review: 2179368
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The conformal deformations of flattening and sphericalization of length metric spaces are considered. These deformations are dual to each other if the space satisfies a simple quantitative connectivity property. Moreover, the quasihyperbolic metrics corresponding to the flat and the spherical metrics are bilipschitz equivalent if a weaker connectivity condition is satisfied.


References [Enhancements On Off] (What's this?)

  • [AG] K. Astala and F. W. Gehring, Quasiconformal analogues of theorems of Koebe and Hardy-Littlewood, Mich. Math. J. 32 (1985), 99-107. MR 0777305 (86j:30029)
  • [AK] K. Astala and P. Koskela, Quasiconformal mappings and global integrability of the derivative, J. Anal. Math. 57 (1991), 203-220. MR 1191747 (94c:30026)
  • [BB] Z. M. Balogh and S.M. Buckley, Geometric characterizations of Gromov hyperbolicity, Invent. Math 153 (2003), 261-301. MR 1992014 (2004i:30042)
  • [BK] Z. M. Balogh and P. Koskela, Quasiconformality, quasisymmetry and removability in Loewner spaces, Duke Math. J. 101 (2000), 554-577. MR 1740689 (2001d:30029)
  • [BHK] M. Bonk, J. Heinonen, and P. Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque 270 (2001). MR 1829896 (2003b:30024)
  • [BHR] M. Bonk, J. Heinonen, and S. Rohde, Doubling conformal densities, J. reine angew. Math. 541 (2001), 117-141. MR 1876287 (2002k:30036)
  • [BKo] M. Bonk and P. Koskela, Conformal metrics and the size of the boundary, Amer. J. Math. 124 (2002), 1247-1287. MR 1939786 (2003i:30068)
  • [BKR] M. Bonk, P. Koskela, and S. Rohde, Conformal metrics on the unit ball in Euclidean space, Proc. London Math. Soc. 77 (1998), 635-664. MR 1643421 (99f:30033)
  • [HK] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1-61. MR 1654771 (99j:30025)
  • [He1] D. Herron, Quasiconformal deformations and volume growth, preprint.
  • [He2] D. Herron, Conformal deformations of uniform Loewner spaces, Math. Proc. Camb. Phil. Soc. 136 (2004), 325-360. MR 2040578 (2005i:30029)
  • [Se1] S. Semmes, Bilipschitz mappings and strong $ A_{\infty }$ weights, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 18 (1993), 211-248. MR 1234732 (95g:30032)
  • [Se2] S. Semmes, On the nonexistence of bilipschitz parametrization and geometric problems about $ A_{\infty }$ weights, Rev. Math. Iberoamericana 12 (1996), 345-360.
  • [Ty1] J. T. Tyson, Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Math. 23 (1998), 525-548. MR 1642158 (99i:30038)
  • [Ty2] J. T. Tyson, Metric and geometric quasiconformality in Ahlfors regular Loewner spaces, Conform. Geom. Dyn. 5 (2001), 21-73. MR 1872156 (2002m:30026)
  • [V] J. Väisälä, Lectures on $ n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics 229, Springer-Verlag, Berlin, 1971. MR 0454009 (56:12260)

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 30F45

Retrieve articles in all journals with MSC (2000): 30F45


Additional Information

Zoltán M. Balogh
Affiliation: Departament Mathematik, Universität Bern, Sidlerstrasse 5, 3012, Bern, Schweiz
Email: zoltan@math-stat.unibe.ch

Stephen M. Buckley
Affiliation: Department of Mathematics, National University of Ireland, Maynooth, Co. Kildare, Ireland
Email: sbuckley@maths.nuim.ie

DOI: https://doi.org/10.1090/S1088-4173-05-00124-4
Received by editor(s): October 26, 2004
Received by editor(s) in revised form: September 28, 2005
Published electronically: November 29, 2005
Additional Notes: This research was partially supported by the Swiss Nationalfond and Enterprise Ireland. It was partly conducted during a visit by the second author to the University of Bern; the hospitality of the Mathematics Department was much appreciated.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society