×

zbMATH — the first resource for mathematics

Limit points of lines of minima in Thurston’s boundary of Teichmüller space. (English) Zbl 1066.32020
Let \(S\) be a hyperbolic surface and let \(\mu\) and \(\nu\) be two measured geodesic laminations that fill up \(S\). We denote by \(l_{\mu}\) and \(l_{\nu}\) the geodesic length functions defined by \(\mu\) and \(\nu\) on the Teichmüller space \(\mathcal T\) of \(S\). S. P. Kerckhoff observed in [Duke Math. J. 65, No. 2, 187–213 (1992; Zbl 0771.30043)] that for any \(s\in(0,1)\) the function \((1-s)l_{\mu}+ sl_{\nu}\) has a unique global minimum \(m_s\) on \(\mathcal T\), and he defined the line of minima \({\mathcal L}_{\mu,\nu}\) to be the family of these minima. He stated the following fact: choosing the support of \(\mu\) to be a union of at least two disjoint closed geodesics, one can obtain an example of a line of minima \({\mathcal L}_{\mu,\nu}\) which does not converge to the projective equivalence class of \(\mu\) in Thurston’s compactification \({\mathcal PMF}\) of \(\mathcal T\).
In this paper, the authors give a proof of that statement. They show taking line of minima \({\mathcal L}_{\mu,\nu}\) with \(\mu=\sum_ {i=1}^n a_i\alpha_i\), where \(\alpha_i\) is a simple closed geodesic and \(a_i>0\) for all \(i=1,\ldots, n\), then the point \(m_s\) converges as \(s\to 0\) to the projective class of the lamination \(\sum \alpha_i\), and not to that of \(\sum a_i\alpha_i\). They also prove that if \(\mu\) is uniquely ergodic, then \(\lim_{s\to 0} m_s=[\mu]\in{\mathcal PMF}\). The results parallel results of Howard Masur, obtained in his paper [Duke Math. J. 49, 183–190 (1982; Zbl 0508.30039)], in which he proves a similar behaviour for Teichmüller geodesics.

MSC:
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F60 Teichmüller theory for Riemann surfaces
PDF BibTeX XML Cite
Full Text: DOI EMIS EuDML arXiv
References:
[1] A J Casson, S A Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts 9, Cambridge University Press (1988) · Zbl 0649.57008
[2] F Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. \((2)\) 124 (1986) 71 · Zbl 0671.57008 · doi:10.2307/1971388
[3] R Díaz, C Series, Examples of pleating varieties for twice punctured tori, Trans. Amer. Math. Soc. 356 (2004) 621 · Zbl 1088.30043 · doi:10.1090/S0002-9947-03-03179-9
[4] A Fahti, P Laudenbach, V Poénaru, Travaux de Thurston sur les surfaces, Astérisque 66, Société Mathématique de France (1979) 284
[5] , Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics 83, Birkhäuser (1990) · Zbl 0731.20025
[6] S P Kerckhoff, Earthquakes are analytic, Comment. Math. Helv. 60 (1985) 17 · Zbl 0575.32024 · doi:10.1007/BF02567397 · eudml:139997
[7] S P Kerckhoff, The Nielsen realization problem, Ann. of Math. \((2)\) 117 (1983) 235 · Zbl 0528.57008 · doi:10.2307/2007076
[8] S P Kerckhoff, Lines of minima in Teichmüller space, Duke Math. J. 65 (1992) 187 · Zbl 0771.30043 · doi:10.1215/S0012-7094-92-06507-0
[9] H Masur, Two boundaries of Teichmüller space, Duke Math. J. 49 (1982) 183 · Zbl 0508.30039 · doi:10.1215/S0012-7094-82-04912-2
[10] R C Penner, J L Harer, Combinatorics of train tracks, Annals of Mathematics Studies 125, Princeton University Press (1992) · Zbl 0765.57001
[11] J P Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque (1996) · Zbl 0855.57003
[12] M Rees, An alternative approach to the ergodic theory of measured foliations on surfaces, Ergodic Theory Dynamical Systems 1 (1981) · Zbl 0539.58018
[13] C Series, An extension of Wolpert’s derivative formula, Pacific J. Math. 197 (2001) 223 · Zbl 1065.30044 · doi:10.2140/pjm.2001.197.223
[14] C Series, On Kerckhoff minima and pleating loci for quasi-Fuchsian groups, Geom. Dedicata 88 (2001) 211 · Zbl 1005.30032 · doi:10.1023/A:1013171204254
[15] C Series, Limits of quasi-Fuchsian groups with small bending, Duke Math. J. 128 (2005) 285 · Zbl 1081.30038 · doi:10.1215/S0012-7094-04-12823-4
[16] W P Thurston, The Geometry and Topology of Three-Manifolds, lecture notes, Princeton University (1980)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.