Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

Distortion in the spherical metric under quasiconformal mappings


Author: Peter A. Hästö
Journal: Conform. Geom. Dyn. 7 (2003), 1-10
MSC (2000): Primary 30C80
DOI: https://doi.org/10.1090/S1088-4173-03-00088-2
Published electronically: January 23, 2003
MathSciNet review: 1992034
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper contains bounds for the distortion in the spherical metric, that is to say, bounds for the constant of Hölder continuity of mappings $f \colon ({\mathbb R}^n,q) \to ({\mathbb R}^n, q)$ where $q$ denotes the spherical metric. The mappings considered are $K$-quasiconformal ($K\ge 1$) and satisfy some normalizations or restrictions. All bounds are explicit and asymptotically sharp as $K \to 1$.


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

  • 1. G. D. ANDERSON, M. K. VAMANAMURTHY AND M. VUORINEN: Conformal Invariants, Inequalities, and Quasiconformal Maps, J. Wiley, New York, 1997.MR 98h:30033
  • 2. P. BONFERT-TAYLOR: Jørgensen inequality for discrete convergence groups, Ann. Acad. Sci. Fenn. Math. 25 (2000), no. 1, 131-150.MR 2001a:30056
  • 3. R. FEHLMANN AND M. VUORINEN: Mori's Theorem for n-Dimensional Quasiconformal Mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 13 (1988), no. 1, 111-124.MR 90a:30060
  • 4. M. LEHTINEN: Remarks on the maximal dilatation of the Beurling-Ahlfors extension, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 133-139.MR 85j:30039
  • 5. O. LEHTO AND K. I. VIRTANEN: Quasiconformal Mappings of the Plane, 2$^{nd}$ ed., Grundlehren der Mathematischen Wissenschaften, Band 126, Springer Verlag, Berlin-Heidelberg-New York, 1973.MR 49:9202
  • 6. P. TUKIA AND J. V¨AISÄLÄ: Quasiconformal extensions from dimension $n$ to $n+1$, Ann. of Math. 115 (1982), 331-342.MR 84i:30030
  • 7. M. VUORINEN: Conformal Geometry and Quasiregular Mappings, Lecture Notes in Mathematics 1319, Springer-Verlag, Berlin-Heidelberg-New York, 1988.MR 89k:30021

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 30C80

Retrieve articles in all journals with MSC (2000): 30C80


Additional Information

Peter A. Hästö
Affiliation: Department of Mathematics, P.O. Box 4, 00014 University of Helsinki, Finland
Email: peter.hasto@helsinki.fi

DOI: https://doi.org/10.1090/S1088-4173-03-00088-2
Keywords: Spherical chordal metric, distortion, quasiconformal mappings
Received by editor(s): February 11, 2002
Published electronically: January 23, 2003
Additional Notes: Supported in part by The Academy of Finland, Research Contract 12132. I would also like to thank Matti Vuorinen for pointing out this problem to me as well as for advice and suggestions during the process of writing this paper.
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society