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Holomorphic families of geodesic discs in infinite dimensional Teichmüller spaces

Published online by Cambridge University Press:  22 January 2016

Harumi Tanigawa
Harumi Tanigawa*
Affiliation:
Department of Mathematics, Nagoya University, Nagoya 464-01, Japan
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The theory of quasiconformal mappings plays an important role in Teichmüller theory. The Teichmüller spaces of Riemann surfaces are defined as quotient spaces of the spaces of Beltrami differentials, and the Teichmüller distances are defined to measure quasiconformal deformations between the Riemann surfaces representing points in the Teichmüller spaces. The Teichmüller spaces are complex Banach manifolds equipped with natural complex structures such that the canonical projections are holomorphic. It is known (see Gardinar [4]) that the Teichmüller distance, defined independently of the complex structures, coincides with the Kobayashi distance.

In spite of the naturality of the definition of a Teichmüller space as a quotient of Beltrami differentials, for given two Beltrami differentials it is hard to determine whether they are equivalent or not. For this reason, it is not trivial to describe geodesic lines with respect to the Teichmüller-Kobayashi metric.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 127 , September 1992 , pp. 117 - 128
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1992

References

[ 1 ] Abikoff, W., On boundaries of Teichmüller spaces and on Kleinian groups III, Acta. Math., 134 (1975), 217–247 CrossRefGoogle Scholar
[ 2 ] Bers, L., On boundaries of Teichmüller spaces and on Kleinian groups I, Ann. of Math., 91 (1970), 570–600.Google Scholar
[ 3 ] Bers, L. and Royden, H. J., Holomorphic families of injections, Acta. Math., 157 (1986), 159–186.Google Scholar
[ 4 ] Gardiner, F., Teichmüller theory and quadratic differentials, John Wiley and Sons, 1987.Google Scholar
[ 5 ] Gehring, F., Quasiconformal mappings which hold the real axis pointwise fixed, Mathematical essays dedicated to A. J. Macintyre, Ohio Univ Press, 1970, pp. 145–158.Google Scholar
[ 6 ] Hensgen, W., Some remarks on boundary values of vector-valued harmonic and analytic functions, Arch. Math., 57 (1991), 88–96.Google Scholar
[ 7 ] Kobayashi, S., Hyperbolic Manifolds and Holomorphic Mappings, Dekker, 1970.Google Scholar
[ 8 ] Kra, I., On Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math., 146 (1981), 231–270 CrossRefGoogle Scholar
[ 9 ] Lehto, O., Univalent functions and Teichmüller spaces, Springer-Verlag, 1985.Google Scholar
[10] Maitani, F., On the rigidity of an end under conformal mappings preserving the infinite homology bases, (preprint).Google Scholar
[11] Maskit, B., On boundaries of Teichmüller spaces and on Kleinian groups II, Ann. of Math., 91 (1970), 607–639.CrossRefGoogle Scholar
[12] Matsuzaki, K., A characterization of extended Schottky type groups with a remark to Ahlfors’ conjecture, J. Math. Kyoto Univ., 31 (1991), 259–264.Google Scholar
[13] Rudin, W., Function theory in the unit ball of Cn , Springer Verlag, 1980.Google Scholar
[14] Shiga, H., On analytic and geometrties of Teichmüller spaces, J. Math. Kyoto Univ., 24 (1984), 441–452.Google Scholar
[15] Shiga, H., Characterization of quasidisks and Teichmüller spaces, Tôhoku Math. J., 37 (1985), 541–552.Google Scholar
[16] Taniguchi, M., On the rigidity of an infinite Riemann surface, Complex Variables, 14 (1990), 161–167.Google Scholar
[17] Tanigawa, H., Holomorphic mappings into Teichmüller spaces, (to appear in Proc. A. M. S.).Google Scholar
[18] Tanigawa, H., Rigidity and boundary behavior of holomorphic mappings, (to appear in J. Math. Soc. Japan).Google Scholar
[19] Thurston, W., The geometry and topology of 3-manifold, Lecture Notes, Princeton Univ., 1978–1979.Google Scholar
[20] Zhong, L., Non-uniqueness of geodesies in infinite dimensional Teichmüller spaces, Complex Variables, 16 (1991), 261–272.Google Scholar