Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T16:13:16.954Z Has data issue: false hasContentIssue false
  • English
  • Français

POLYGONAL QUASICONFORMAL MAPPINGS AND CHORD-ARC CURVES

Part of: Geometric function theory Riemann surfaces

Published online by Cambridge University Press:  02 November 2016

SHENGJIN HUO ,
SHENGJIAN WU  and
HUI GUO
SHENGJIN HUO*
Affiliation:
Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China email [email protected]
SHENGJIAN WU
Affiliation:
LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, China email [email protected]
HUI GUO
Affiliation:
College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China email [email protected]
*
✉ [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we show that a polygonal quasiconformal mapping always corresponds to a chord-arc curve. Furthermore, we find that the set of curves corresponding to polygonal quasiconformal mappings is path connected in the set of all bounded chord-arc curves.

Keywords

chord-arc curvepolygonal quasiconformal mappingconformal welding

MSC classification

Primary: 30C62: Quasiconformal mappings in the plane
Secondary: 30C75: Extremal problems for conformal and quasiconformal mappings, other methods 30F60: Teichmüller theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 1 , February 2017 , pp. 66 - 72
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Astala, K. and González, M. J., ‘Chord-arc curves and the Beurling transform’, Invent. Math. 205 (2015), 57–81.CrossRefGoogle Scholar
Astala, K. and Zinsmeister, M., ‘Teichmüller spaces and BMOA’, Math. Ann. 289 (1991), 613–625.CrossRefGoogle Scholar
Falconer, K. J. and Marsh, D. T., ‘Classification of quasicircles by Hausdorff dimension’, Nonlinearity 2 (1989), 489–493.CrossRefGoogle Scholar
Gardiner, F. P., Teichmüller Theory and Quadratic Differentials (Wiley, New York, 1987).Google Scholar
Garnett, J., Bounded Analytic Functions (Academic Press, London–New York, 1981).Google Scholar
Garnett, J. B. and Marshall, D. E., Harmonic Measure (Cambridge University Press, Cambridge, 2005).CrossRefGoogle Scholar
Huo, S. J., Tang, S. A. and Wu, S. J., ‘Hausdorff dimension of quasi-circles of polygonal mappings and its application’, Sci. China Math. 56(5) (2013), 1033–1040.CrossRefGoogle Scholar
Katznelson, Y., Nag, S. and Sullivan, D. P., ‘On conformal welding homeomorphisms associated to Jordan curves’, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), 293–306.CrossRefGoogle Scholar
Mateu, J., Orobitg, J. and Verdera, J., ‘Extra cancellation of even Calderón–Zygmund operators and quasiconformal mappings’, J. Math. Pures Appl. (9) 91 (2009), 402–431.CrossRefGoogle Scholar
Reich, E. and Strebel, K., ‘Extremal quasiconformal mappings with prescribed boundary values’, in: Contributions to Analysis, A Collection of Papers Dedicated to Lipman Bers (Academic Press, New York, 1974), 375–391.Google Scholar
Schechter, M., Principles of Functional Analysis (Academic Press–Harcourt Brace Jovanovich, New York–London, 1973).Google Scholar
Strebel, K., ‘Extremal quasiconformal mappings’, Results Math. 10 (1986), 168–210.CrossRefGoogle Scholar