Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Holomorphic motions and quasicircles


Author: Gaven J. Martin
Journal: Proc. Amer. Math. Soc. 141 (2013), 3911-3918
MSC (2000): Primary 30C62, 37F30, 30C75
DOI: https://doi.org/10.1090/S0002-9939-2013-11684-9
Published electronically: July 23, 2013
MathSciNet review: 3091780
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a new application of the theory of holomorphic motions to the study of the distortion of holomorphic maps of disks and annuli establishing sharp distortion theorems.


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

  • 1. K. Astala, T. Iwaniec and G.J. Martin, Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton Mathematical Series, 48. Princeton University Press, Princeton, NJ, 2009. MR 2472875 (2010j:30040)
  • 2. K. Astala, T. Iwaniec, G.J. Martin and J. Onninen, Extremal mappings of finite distortion, Proc. London Math. Soc., 91 (2005), 655-702. MR 2180459 (2006h:30016)
  • 3. A.F. Beardon and D. Minda, The hyperbolic metric and geometric function theory, Quasi-conformal mappings and their applications, 9-56, Narosa, New Delhi, 2007. MR 2492498 (2011c:30108)
  • 4. F.P. Gardiner, Y. Jiang and Z. Wang, Holomorphic motions and related topics, Geometry of Riemann surfaces, 156-193, London Math. Soc. Lecture Note Ser., 368, Cambridge Univ. Press, Cambridge, 2010. MR 2665009 (2011j:37087)
  • 5. Y. Jiang, Holomorphic motions, Fatou linearization, and quasiconformal rigidity for parabolic germs, Michigan Math. J., 58 (2009), 517-534. MR 2595552 (2011h:37075)
  • 6. S.L. Krushkal, Quasiconformal extensions and reflections, Handbook of complex analysis: geometric function theory, 2, 507-553, Elsevier, Amsterdam, 2005. MR 2121866 (2006a:30018)
  • 7. R. Kühnau, Möglichst konforme Spiegelung an einer Jordankurve [Extremal conformal reflection in a Jordan curve], Jahresber. Deutsch. Math.-Verein., 90 (1988), 90-109. MR 939755 (89d:30023)
  • 8. R. Mané, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup., 16 (1983), 193-217. MR 732343 (85j:58089)
  • 9. G. J. Martin, The distortion theorem for quasiconformal mappings, Schottky's theorem and holomorphic motions, Proc. Amer. Math. Soc., 125 (1997), 1095-1103. MR 1363178 (97g:30017)
  • 10. G. J. Martin, Topology and distortion, preprint.
  • 11. D. Partyka, The maximal dilatation of Douady and Earle extension of a quasisymmetric automorphism of the unit circle, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 44 (1990), 45-57 (1991). MR 1157878 (93h:30035)
  • 12. E. Reich and K. Strebel, Extremal plane quasiconformal mappings with given boundary values, Bull. Amer. Math. Soc., 79 (1973), 488-490. MR 0338362 (49:3127)
  • 13. Z. Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc., 111 (1991), 347-355. MR 1037218 (91f:58078)
  • 14. Strebel, K., Extremal Quasiconformal Mappings, Results Math., 10 (1986), 168-210. MR 869809 (88a:30048)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 30C62, 37F30, 30C75

Retrieve articles in all journals with MSC (2000): 30C62, 37F30, 30C75


Additional Information

Gaven J. Martin
Affiliation: Institute for Advanced Study, Massey University, Auckland, New Zealand
Email: g.j.martin@massey.ac.nz

DOI: https://doi.org/10.1090/S0002-9939-2013-11684-9
Received by editor(s): September 27, 2011
Received by editor(s) in revised form: January 23, 2012
Published electronically: July 23, 2013
Additional Notes: This research was supported by the Marsden Fund. The author thanks Aalto University and the University of Helsinki, where this research was initiated
Communicated by: Mario Bonk
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society