On extendibility of a map induced by the Bers isomorphism
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- by Hideki Miyachi and Toshihiro Nogi PDF
- Proc. Amer. Math. Soc. 142 (2014), 4181-4189 Request permission
Abstract:
Let $S$ be a closed Riemann surface of genus $g(\geqq 2)$ and set $\dot {S}=S \setminus \{ \hat {z}_0 \}$. Then we have the composed map $\varphi \circ r$ of a map $r: T(S) \times U \rightarrow F(S)$ and the Bers isomorphism $\varphi : F(S) \rightarrow T(\dot {S})$, where $F(S)$ is the Bers fiber space of $S$, $T(X)$ is the Teichmüller space of $X$ and $U$ is the upper half-plane.
The purpose of this paper is to show that the map $\varphi \circ r:T(S)\times U \rightarrow T(\dot {S})$ has a continuous extension to some subset of the boundary $T(S) \times \partial U$.
References
- Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
- Lipman Bers, Fiber spaces over Teichmüller spaces, Acta Math. 130 (1973), 89–126. MR 430318, DOI 10.1007/BF02392263
- Lipman Bers, Finite-dimensional Teichmüller spaces and generalizations, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 131–172. MR 621883, DOI 10.1090/S0273-0979-1981-14933-8
- Lipman Bers, An inequality for Riemann surfaces, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 87–93. MR 780038
- J. F. Brock, Continuity of Thurston’s length function, Geom. Funct. Anal. 10 (2000), no. 4, 741–797. MR 1791139, DOI 10.1007/PL00001637
- Jeffrey F. Brock, Richard D. Canary, and Yair N. Minsky, The classification of Kleinian surface groups, II: The ending lamination conjecture, Ann. of Math. (2) 176 (2012), no. 1, 1–149. MR 2925381, DOI 10.4007/annals.2012.176.1.1
- Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1183224
- Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR 1215481, DOI 10.1007/978-4-431-68174-8
- E. Klarreich, The boundary at infinity of the curve complex and relative Teichmüller spaces, preprint.
- Christopher J. Leininger, Mahan Mj, and Saul Schleimer, The universal Cannon-Thurston map and the boundary of the curve complex, Comment. Math. Helv. 86 (2011), no. 4, 769–816. MR 2851869, DOI 10.4171/CMH/240
- Bernard Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381–386. MR 802500, DOI 10.5186/aasfm.1985.1042
- Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103–149. MR 1714338, DOI 10.1007/s002220050343
- Katsuhiko Matsuzaki and Masahiko Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. Oxford Science Publications. MR 1638795
- Yair N. Minsky, Teichmüller geodesics and ends of hyperbolic $3$-manifolds, Topology 32 (1993), no. 3, 625–647. MR 1231968, DOI 10.1016/0040-9383(93)90013-L
- Yair N. Minsky, On rigidity, limit sets, and end invariants of hyperbolic $3$-manifolds, J. Amer. Math. Soc. 7 (1994), no. 3, 539–588. MR 1257060, DOI 10.1090/S0894-0347-1994-1257060-3
- Ken’ichi Ohshika, Limits of geometrically tame Kleinian groups, Invent. Math. 99 (1990), no. 1, 185–203. MR 1029395, DOI 10.1007/BF01234417
- Chaohui Zhang, Nonextendability of the Bers isomorphism, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2451–2458. MR 1249895, DOI 10.1090/S0002-9939-1995-1249895-0
Additional Information
- Hideki Miyachi
- Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Machi- kaneyama 1-1, Toyonaka, Osaka, 560-0043, Japan
- MR Author ID: 650573
- Email: miyachi@math.sci.osaka-u.ac.jp
- Toshihiro Nogi
- Affiliation: Osaka City University Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
- Email: toshihironogi8@gmail.com
- Received by editor(s): February 10, 2012
- Received by editor(s) in revised form: June 13, 2012, and January 8, 2013
- Published electronically: August 1, 2014
- Additional Notes: The first author was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (C), 21540177.
The second author was partially supported by the JSPS Institutional Program for Young Research Overseas Visits “Promoting international young researchers in mathematics and mathematical sciences led by OCAMI” - Communicated by: Michael Wolf
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 4181-4189
- MSC (2010): Primary 30F60, 32G15, 20F67
- DOI: https://doi.org/10.1090/S0002-9939-2014-12140-X
- MathSciNet review: 3266988