Holomorphic mappings into Teichmüller spaces

Tanigawa, Harumi

doi:10.1090/S0002-9939-1993-1113649-4

Holomorphic mappings into Teichmüller spaces
HTML articles powered by AMS MathViewer

by Harumi Tanigawa PDF

Proc. Amer. Math. Soc. 117 (1993), 71-78 Request permission

Abstract:

Let $T(F)$ be the Teichmüller space of a finitely generated Fuchsian group $F$ of the first kind. We shall show that there exists no nontrivial family of holomorphic proper mappings of the unit disc into $T(F)$ with a complex analytic parameter. Using this result, we shall investigate a family of quasi-conformal mappings with a complex analytic parameter whose Beltrami differentials vanish on a fixed set of positive area.

References

Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404. MR 115006, DOI 10.2307/1970141
Lipman Bers, On boundaries of Teichmüller spaces and on Kleinian groups. I, Ann. of Math. (2) 91 (1970), 570–600. MR 297992, DOI 10.2307/1970638
Lipman Bers, Fiber spaces over Teichmüller spaces, Acta Math. 130 (1973), 89–126. MR 430318, DOI 10.1007/BF02392263
Lipman Bers, On iterates of hyperbolic transformations of Teichmüller space, Amer. J. Math. 105 (1983), no. 1, 1–11. MR 692104, DOI 10.2307/2374379
Lipman Bers and H. L. Royden, Holomorphic families of injections, Acta Math. 157 (1986), no. 3-4, 259–286. MR 857675, DOI 10.1007/BF02392595
Yôichi Imayoshi and Hiroshige Shiga, A finiteness theorem for holomorphic families of Riemann surfaces, Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 11, Springer, New York, 1988, pp. 207–219. MR 955842, DOI 10.1007/978-1-4613-9611-6_{1}5
Irwin Kra, On spaces of Kleinian groups, Comment. Math. Helv. 47 (1972), 53–69. MR 306485, DOI 10.1007/BF02566788
Irwin Kra, Canonical mappings between Teichmüller spaces, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 143–179. MR 598682, DOI 10.1090/S0273-0979-1981-14870-9
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
Bernard Maskit, On boundaries of Teichmüller spaces and on Kleinian groups. II, Ann. of Math. (2) 91 (1970), 607–639. MR 297993, DOI 10.2307/1970640
Junjiro Noguchi, Moduli spaces of holomorphic mappings into hyperbolically imbedded complex spaces and locally symmetric spaces, Invent. Math. 93 (1988), no. 1, 15–34. MR 943922, DOI 10.1007/BF01393686
V. V. Savin, The moduli of Riemann surfaces, Dokl. Akad. Nauk SSSR 196 (1971), 783–785 (Russian). MR 0311902
Hiroshige Shiga, On analytic and geometric properties of Teichmüller spaces, J. Math. Kyoto Univ. 24 (1984), no. 3, 441–452. MR 766636, DOI 10.1215/kjm/1250521274
Hiroshige Shiga, Characterization of quasidisks and Teichmüller spaces, Tohoku Math. J. (2) 37 (1985), no. 4, 541–552. MR 814080, DOI 10.2748/tmj/1178228594