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Lines of minima with no end in Thurston's boundary of Teichmüller space

Author: Yuki Iguchi
Journal: Conform. Geom. Dyn. 16 (2012), 22-43
MSC (2010): Primary 30F45, 30F60, 32G15, 57M15; Secondary 57M50, 32G15, 30F60, 30F45
Published electronically: March 7, 2012
MathSciNet review: 2890254
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \nu ^+$ and $ \nu ^-$ be two measured laminations which fill up a hyperbolic surface. Kerckhoff [Duke Math. J. 65 (1992), 187-213] defines a line of minima as a family of surfaces where convex combinations of the hyperbolic length functions of $ \nu ^+$ and $ \nu ^-$ are minimum. This is a proper curve in the Teichmüller space. We show that there exists a line of minima which does not converge in the Thurston compactification of the Teichmüller space of a compact Riemann surface. We also show that the limit set of the line of minima is contained in a simplex on the Thurston boundary.

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  • [Bu] P. Buser, Geometry and spectra of compact Riemann surfaces. Progress in Mathematics 106, Birkhäuser, Boston (1992). MR 1183224 (93g:58149)
  • [CRS1] Y. Choi, K. Rafi and C. Series, Lines of minima are uniformly quasigeodesics, Pacific J. Math. 237:1 (2008), 21-44. MR 2415205 (2009b:32017)
  • [CRS2] -, Lines of minima and Teichmüller geodesics. Geom. Funct. Anal. 18 (2008), 698-754. MR 2438996 (2010k:30053)
  • [DS] R. Diaz and C. Series, Limit points of lines of minima in Thurston's boundary of Teichmüller space. Alg. Geom. Top. 3 (2003), 207-234. MR 1997320 (2004f:32014)
  • [FLP] A. Fathi, F. Laudenbach, and V. Poénaru, Travaux de Thurston sur les surfaces. Astérisque, Vol. 66-67, Soc. Math. de France (1979). MR 568308 (82m:57003)
  • [GM] F. Gardiner and H. Masur, Extremal length geometry of Teichmüller space. Complex Variables Theory Appl. 16 (1991), no. 2-3, 209-237. MR 1099913 (92f:32034)
  • [Ke] S. Kerckhoff, Lines of minima in Teichmüller space. Duke Math. J. 65 (1992), no. 2, 187-213. MR 1150583 (93b:32027)
  • [Kh] A. Khinchin, Continued fractions. Russian edition, Dover Publications, Mineola, NY (1997). MR 1451873 (98c:11008)
  • [L] A. Lenzhen, Teichmüller geodesics that do not have a limit in $ \mathcal {PMF}$. Geom. and Top. 12 (2008), 177-197. MR 2377248 (2008k:30056)
  • [Le] G. Levitt, Foliations and laminations on hyperbolic surfaces. Topology 22 (1983), 119-135. MR 683752 (84h:57015)
  • [M] B. Maskit, Comparison of extremal and hyperbolic lengths. Ann. Acad. Sci. Fenn. 10 (1985), 381-386. MR 802500 (87c:30062)
  • [Ma] H. Masur, Two boundaries of Teichmüller space. Duke Math. J. 49 (1982), 183-190. MR 650376 (83k:32035)
  • [Mi] Y. Minsky, Extremal length estimates and product regions in Teichmüller space. Duke Math. J. 83 (1996), 249-286. MR 1390649 (97b:32019)
  • [St] K. Strebel, Quadratic Differentials, Springer Verlag, Berlin and New York (1984). MR 743423 (86a:30072)
  • [Wo] S. Wolpert, The length spectra as moduli for compact Riemann surfaces. Ann. Math. 109 (1979), 323-351. MR 528966 (80j:58067)

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Additional Information

Yuki Iguchi
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan

Keywords: Teichmüller space, Thurston’s boundary, Teichmüller geodesic, line of minima.
Received by editor(s): August 2, 2011
Published electronically: March 7, 2012
Additional Notes: The author is partially supported by “Global COE: Computationism as a Foundation for the Sciences”.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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