Vertical limits of graph domains
HTML articles powered by AMS MathViewer
- by Hrant Hakobyan and Dragomir Šarić PDF
- Proc. Amer. Math. Soc. 144 (2016), 1223-1234 Request permission
Abstract:
We consider the limiting behavior of Teichmüller geodesics in the universal Teichmüller space $T(\mathbb {H})$. Our main result states that the limits of the Teichmüller geodesics in the Thurston’s boundary of $T(\mathbb {H})$ may depend on both vertical and horizontal foliation of the corresponding holomorphic quadratic differential.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
- Lars V. Ahlfors, Lectures on quasiconformal mappings, 2nd ed., University Lecture Series, vol. 38, American Mathematical Society, Providence, RI, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. MR 2241787, DOI 10.1090/ulect/038
- Francis Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), no. 1, 139–162. MR 931208, DOI 10.1007/BF01393996
- Albert Fathi, François Laudenbach, and Valentin Poénaru, Thurston’s work on surfaces, Mathematical Notes, vol. 48, Princeton University Press, Princeton, NJ, 2012. Translated from the 1979 French original by Djun M. Kim and Dan Margalit. MR 3053012
- O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463, DOI 10.1007/978-3-642-65513-5
- Anna Lenzhen, Teichmüller geodesics that do not have a limit in ${\scr {PMF}}$, Geom. Topol. 12 (2008), no. 1, 177–197. MR 2377248, DOI 10.2140/gt.2008.12.177
- Howard Masur, Two boundaries of Teichmüller space, Duke Math. J. 49 (1982), no. 1, 183–190. MR 650376
- Jean-Pierre Otal, About the embedding of Teichmüller space in the space of geodesic Hölder distributions, Handbook of Teichmüller theory. Vol. I, IRMA Lect. Math. Theor. Phys., vol. 11, Eur. Math. Soc., Zürich, 2007, pp. 223–248. MR 2349671, DOI 10.4171/029-1/5
- Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706, DOI 10.1007/978-3-662-02770-7
- Dragomir Šarić, Real and complex earthquakes, Trans. Amer. Math. Soc. 358 (2006), no. 1, 233–249. MR 2171231, DOI 10.1090/S0002-9947-05-03651-2
- Dragomir Šarić, Geodesic currents and Teichmüller space, Topology 44 (2005), no. 1, 99–130. MR 2104004, DOI 10.1016/j.top.2004.05.001
- Dragomir Šarić, Infinitesimal Liouville distributions for Teichmüller space, Proc. London Math. Soc. (3) 88 (2004), no. 2, 436–454. MR 2032514, DOI 10.1112/S0024611503014539
- D. Šarić, Thurston’s boundary for Teichmüller spaces of infinite surfaces: the geodesic currents and the length spectrum, preprint, available at Arxiv.
- Kurt Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR 743423, DOI 10.1007/978-3-662-02414-0
- William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6
- J. $\rm {V\ddot {a}is\ddot {a}l\ddot {a}}$, Lectures on n-dimensional quasiconformal mappings. Lecture Notes in Mathematics, 229. Springer-Verlag, Berlin-New York, 1971.
Additional Information
- Hrant Hakobyan
- Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
- Email: hakobyan@math.ksu.edu
- Dragomir Šarić
- Affiliation: Department of Mathematics, Queens College of CUNY, 65-30 Kissena Boulevard, Flushing, New York 11367 – and – Mathematics PhD Program, The CUNY Graduate Center, 365 Fifth Avenue, New York, New York 10016-4309
- Email: Dragomir.Saric@qc.cuny.edu
- Received by editor(s): September 9, 2014
- Received by editor(s) in revised form: September 10, 2014, and March 10, 2015
- Published electronically: August 18, 2015
- Additional Notes: The first author was partially supported by Kansas NSF EPSCoR Grant NSF68311
The second author was partially supported by National Science Foundation grant DMS 1102440 and by the Simons Foundation Collaboration Grant for Mathematicians 2011 - Communicated by: Jeremy Tyson
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1223-1234
- MSC (2010): Primary 30F60; Secondary 30C62, 32G15
- DOI: https://doi.org/10.1090/proc12780
- MathSciNet review: 3447674