Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-12T09:24:35.840Z Has data issue: false hasContentIssue false
  • English
  • Français

Incompleteness of the pressure metric on the Teichmüller space of a bordered surface

Published online by Cambridge University Press:  28 September 2017

BINBIN XU
BINBIN XU*
Affiliation:
Korea Institute for Advanced Study (KIAS), 85 Hoegi-ro, Dongdaemun-gu, 02455 Seoul, Republic of Korea email xubin@kias.re.kr
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that the pressure metric on the Teichmüller space of a bordered surface is incomplete and that a completion can be given by the moduli space of metrics on a graph (dual to a special ideal triangulation of the same bordered surface) equipped with pressure metric. In contrast to the closed surface case, we obtain as a corollary that the pressure metric is not bi-Lipschitz to the Weil–Petersson metric.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 6 , June 2019 , pp. 1710 - 1728
Copyright
© Cambridge University Press, 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Basmajian, A.. The orthogonal spectrum of a hyperbolic manifold. Amer. J. Math. 115(5) (1993), 1139–1159.Google Scholar
Beardon, A. F.. The Geometry of Discrete Groups (Graduate Texts in Mathematics, 91) . Springer, New York, 1995, Corrected reprint of the 1983 original.Google Scholar
Bowen, R.. Periodic orbits for hyperbolic flows. Amer. J. Math. 94 (1972), 1–30.Google Scholar
Bridgeman, M.. Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space. Geom. Topol. 14(2) (2010), 799–831.Google Scholar
Bridgeman, M., Canary, R., Labourie, F. and Sambarino, A.. The pressure metric for Anosov representations. Geom. Funct. Anal. 25(4) (2015), 1089–1179.Google Scholar
Farb, B. and Margalit, D.. A Primer on Mapping Class Groups (Princeton Mathematical Series, 49) . Princeton University Press, Princeton, NJ, 2012.Google Scholar
Guo, R.. On parameterizations of Teichmüller spaces of surfaces with boundary. J. Differential Geom. 82(3) (2009), 629–640.Google Scholar
Livšic, A. N.. Cohomology of dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1296–1320.Google Scholar
Luo, F.. On Teichmüller spaces of surfaces with boundary. Duke Math. J. 139(3) (2007), 463–482.Google Scholar
Masur, H.. Extension of the Weil–Petersson metric to the boundary of Teichmüller space. Duke Math. J. 43(3) (1976), 623–635.Google Scholar
McMullen, C. T.. Thermodynamics, dimension and the Weil–Petersson metric. Invent. Math. 173(2) (2008), 365–425.Google Scholar
Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990), 268pp.Google Scholar
Pollicott, M. and Sharp, R.. A Weil–Petersson type metric on spaces of metric graphs. Geom. Dedicata 172(1) (2014), 229–244.Google Scholar
Sormani, C.. How Riemannian manifolds converge. Metric and Differential Geometry (Progress in Mathematics, 297) . Birkhäuser/Springer, Basel, 2012, pp. 91–117.Google Scholar
Ushijima, A.. A canonical cellular decomposition of the Teichmüller space of compact surfaces with boundary. Comm. Math. Phys. 201(2) (1999), 305–326.Google Scholar
Wolpert, S.. Noncompleteness of the Weil–Petersson metric for Teichmüller space. Pacific J. Math. 61(2) (1975), 573–577.Google Scholar
Wolpert, S. A.. Thurston’s Riemannian metric for Teichmüller space. J. Differential Geom. 23(2) (1986), 143–174.Google Scholar