Vertical limits of graph domains

Hakobyan, Hrant; Šarić, Dragomir

doi:10.1090/proc12780

Vertical limits of graph domains

Authors: Hrant Hakobyan and Dragomir Šarić
Journal: Proc. Amer. Math. Soc. 144 (2016), 1223-1234
MSC (2010): Primary 30F60; Secondary 30C62, 32G15
DOI: https://doi.org/10.1090/proc12780
Published electronically: August 18, 2015
MathSciNet review: 3447674
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

[1] Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0357743
[2] 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
[3] Francis Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), no. 1, 139–162. MR 931208, https://doi.org/10.1007/BF01393996
[4] 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
[5] O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas; Die Grundlehren der mathematischen Wissenschaften, Band 126. MR 0344463
[6] Anna Lenzhen, Teichmüller geodesics that do not have a limit in 𝒫ℳℱ, Geom. Topol. 12 (2008), no. 1, 177–197. MR 2377248, https://doi.org/10.2140/gt.2008.12.177
[7] Howard Masur, Two boundaries of Teichmüller space, Duke Math. J. 49 (1982), no. 1, 183–190. MR 650376
[8] 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, https://doi.org/10.4171/029-1/5
[9] Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706
[10] Dragomir Šarić, Real and complex earthquakes, Trans. Amer. Math. Soc. 358 (2006), no. 1, 233–249. MR 2171231, https://doi.org/10.1090/S0002-9947-05-03651-2
[11] Dragomir Šarić, Geodesic currents and Teichmüller space, Topology 44 (2005), no. 1, 99–130. MR 2104004, https://doi.org/10.1016/j.top.2004.05.001
[12] Dragomir Šarić, Infinitesimal Liouville distributions for Teichmüller space, Proc. London Math. Soc. (3) 88 (2004), no. 2, 436–454. MR 2032514, https://doi.org/10.1112/S0024611503014539
[13] D. Šarić, Thurston's boundary for Teichmüller spaces of infinite surfaces: the geodesic currents and the length spectrum, preprint, available at Arxiv.
[14] 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
[15] 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, https://doi.org/10.1090/S0273-0979-1988-15685-6
[16] 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.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30F60, 30C62, 32G15

Retrieve articles in all journals with MSC (2010): 30F60, 30C62, 32G15

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

DOI: https://doi.org/10.1090/proc12780
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
Article copyright: © Copyright 2015 American Mathematical Society