Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T07:00:00.483Z Has data issue: false hasContentIssue false

Definitions for a Class of Plane Quasiconformal Mappings

Published online by Cambridge University Press:  22 January 2016

F. W. Gehring*
Affiliation:
University of Michigan, Ann Arbor, Michigan, Harvard University, Cambridge Massachusetts
Rights & Permissions [Opens in a new window]

Extract

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.

This report is a survey of some of the many different ways of characterizing a class of plane quasiconformal mappings. This class was considered by Ahlfors [4] in his treatment of the Teichmiiller problem, and it has been studied rather extensively in the last ten years.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1967

References

[1] Agard, S. B., Topics in the theory of quasiconformal mappings, University of Michigan dissertation, 1965.Google Scholar
[2] Agard, S. B. and Gehring, F. W., Angles and quasiconformal mappings, Proc. London Math. Soc. (3) 14 A (1965), pp. 121.Google Scholar
[3] Ahlfors, L. V., Conformal mapping, Lecture notes transcribed by Osserman, R. at Oklahoma, A. and College, M., 1951.Google Scholar
[4] Ahlfors, L. V., On quasiconformal mappings, J. d’Analyse Math. 3 (1954), pp. 158.Google Scholar
[5] Ahlfors, L. V., Quasiconformal reflections, Acta Math. 109 (1963), pp. 291301.CrossRefGoogle Scholar
[6] Ahlfors, L. V., Extension of quasiconformal mappings from two to three dimensions, Proc. Nat. Acad. Sci. USA 51 (1964), pp. 768771.CrossRefGoogle ScholarPubMed
[7] Ahlfors, L. V. and Bers, L., Riemann’s mapping theorem for variable metrics, Ann. Math. 72 (1960), pp. 385404.Google Scholar
[8] Bers, L., On a theorem of Mori and the definition of quasiconformality, Trans. Amer. Math. Soc. 84 (1957), pp. 7884.Google Scholar
[9] Bers, L., The equivalence of two definitions of quasiconformal mappings, Comm. Math. Helv. 37 (1962), pp. 348154.Google Scholar
[10] Beurling, A. and Ahlfors, L. V., The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), pp. 125142.CrossRefGoogle Scholar
[11] Gehring, F. W., The definitions and exceptional sets for quasiconformal mappings f Ann. Acad. Sci. Fenn. 281 (1960), pp. 128.Google Scholar
[12] Gehring, F. W., Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), pp. 353393.CrossRefGoogle Scholar
[13] Gehring, F. W., Extension of quasiconformal mappings in three space, J. d’Analyse Math. 14 (1965), pp. 171182.Google Scholar
[14] Gehring, F. W. and Lehto, O., On the total differentiability of functions of a complex variable, Ann. Acad. Sci. Fenn. 272 (1959), pp. 1-9,Google Scholar
[15] Gehring, F. W. and Väisälä, J., On the geometric definition for quasiconformal mappings, Comm. Math. Helv. 36 (1961), pp. 1932.Google Scholar
[16] Hersch, J., Longueurs extrémales et theorie des fonctions, Comm. Math. Helv. 29 (1955), pp. 301337.Google Scholar
[17] Kelingos, J. A., Characterizations of quasiconformal mappings in terms of harmonic and hyperbolic measure, Ann. Acad. Sci. Fenn. 368 (1965), pp. 116.Google Scholar
[18] Lehto, O. and Virtanen, K. I., Quasikonforme Abbildungen, Springer-Verlag, Berlin-Heidelberg-New York 1965.CrossRefGoogle Scholar
[19] Loewner, C., On the conformal capacity in space, J. Math. Mech. 8 (1959), pp. 411414.Google Scholar
[20] Mori, A., On quasi con formality and pseudo-analyticity, Trans. Amer. Math. Soc. 84 (1957), pp. 5677.Google Scholar
[21] Pesin, I. N., Metric properties of Q-quasiconformal mappings, Mat. Sbornik 40 (82) (1956), pp. 281294 (Russian).Google Scholar
[22] Pfluger, A., Quasikonforme Abbildungen und logarithmische Kapazität, Ann. Inst. Fourier Grenoble 2 (1951). pp. 6980.Google Scholar
[23] Pfluger, A., Über die Äquivalenz der geometrischen und der analytischen Definition quasikonformer Abbildungen, Comm. Math. Helv. 33 (1959), pp. 2333.Google Scholar
[24] Reich, E., On a characterization of quasiconformal mappings, Comm. Math. Helv. 37 (1962), pp. 4448.Google Scholar
[25] Renggli, H., Quasiconformal mappings and extremal lengths, Amer. J. Math. 86 (1964), pp. 6369.Google Scholar
[26] Taari, O., Charakteristerung der Quasikonformität mit Hilfe der Winkelverzerrung, Ann. Acad. Sci. Fenn. 390 (1966), pp. 143.Google Scholar
[27] Väisälä, J., On quasiconformal mappings in space, Ann. Acad. Sci. Fenn. 298 (1961), pp. 136.Google Scholar