Crickets, zippers and the Bers universal Teichmüller space

Astala, K.; Gehring, F. W.

doi:10.1090/S0002-9939-1990-1028037-6

Crickets, zippers and the Bers universal Teichmüller space
HTML articles powered by AMS MathViewer

by K. Astala and F. W. Gehring PDF

Proc. Amer. Math. Soc. 110 (1990), 675-687 Request permission

Abstract:

This paper contains a simple characterization of the conformal mappings $f$ of the unit disk $U$ whose Schwarzians lie in the closure of the Bers universal Teichmüller space. A second characterization is given and three examples are studied for the case where $f(U)$ is the complement of a quasiarc.

References

Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
Kari Astala, Self-similar zippers, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, pp. 61–73. MR 955808, DOI 10.1007/978-1-4613-9602-4_{4}
K. Astala and F. W. Gehring, Injectivity, the BMO norm and the universal Teichmüller space, J. Analyse Math. 46 (1986), 16–57. MR 861687, DOI 10.1007/BF02796572
Clifford J. Earle and James Eells Jr., On the differential geometry of Teichmüller spaces, J. Analyse Math. 19 (1967), 35–52. MR 220923, DOI 10.1007/BF02788708
Barbara Brown Flinn, Jordan domains and the universal Teichmüller space, Trans. Amer. Math. Soc. 282 (1984), no. 2, 603–610. MR 732109, DOI 10.1090/S0002-9947-1984-0732109-2
F. W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353–393. MR 139735, DOI 10.1090/S0002-9947-1962-0139735-8
F. W. Gehring, Univalent functions and the Schwarzian derivative, Comment. Math. Helv. 52 (1977), no. 4, 561–572. MR 457701, DOI 10.1007/BF02567390
F. W. Gehring, Spirals and the universal Teichmüller space, Acta Math. 141 (1978), no. 1-2, 99–113. MR 499134, DOI 10.1007/BF02545744
Walter Hengartner and Glenn Schober, On Schlicht mappings to domains convex in one direction, Comment. Math. Helv. 45 (1970), 303–314. MR 277703, DOI 10.1007/BF02567334
Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0
William P. Thurston, Zippers and univalent functions, The Bieberbach conjecture (West Lafayette, Ind., 1985) Math. Surveys Monogr., vol. 21, Amer. Math. Soc., Providence, RI, 1986, pp. 185–197. MR 875241, DOI 10.1090/surv/021/15
P. Tukia and J. Väisälä, Extension of embeddings close to isometries or similarities, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 153–175. MR 752401, DOI 10.5186/aasfm.1984.0914

Lectures on

-dimensional quasiconformal mappings

Jussi Väisälä, Quasisymmetric embeddings in Euclidean spaces, Trans. Amer. Math. Soc. 264 (1981), no. 1, 191–204. MR 597876, DOI 10.1090/S0002-9947-1981-0597876-7