Papers
Topics
Authors
Recent
Search
2000 character limit reached

The geometry of the Thurston metric: a survey

Published 8 Jul 2023 in math.GT, math.CV, and math.DG | (2307.03874v1)

Abstract: This paper is a survey about the Thurston metric on the Teichm\"uller space. The central issue is the constructions of extremal Lipschitz maps between hyperbolic surfaces. We review several constructions, including the original work of Thurston. Coarse geometry and isometry rigidity of the Thurston metric, relation between the Thurston metric and the Thurston compactification are discussed. Some recent generalizations and developments of the Thurston metric are sketched.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (78)
  1. S.I. Al’ber. On n-dimensional problems in the calculus of variations in the large. In Sov. Math. Dokl, volume 5, page 26, 1964.
  2. The horofunction compactification of Teichmüller spaces of surfaces with boundary. Topology Appl., 208:160–191, 2016.
  3. Generalizing stretch lines for surfaces with boundary. Int. Math. Res. Not. IMRN, (23):18919–18991, 2022.
  4. Francisco Arana-Herrera. Effective mapping class group dynamics, i: Counting lattice points in teichmüller space. Duke Mathematical Journal, 1(1):1–93, 2023.
  5. Lattice point asymptotics and volume growth on Teichmüller space. Duke Math. J., 161(6):1055–1111, 2012.
  6. Assaf Bar-Natan. Geodesic Envelopes in Teichmuller Space Equipped with the Thurston Metric. ProQuest LLC, Ann Arbor, MI, 2022. Thesis (Ph.D.)–University of Toronto (Canada).
  7. Shearing coordinates and convexity of length functions on Teichmüller space. Amer. J. Math., 135(6):1449–1476, 2013.
  8. Mladen Bestvina. A Bers-like proof of the existence of train tracks for free group automorphisms. Fund. Math., 214(1):1–12, 2011.
  9. Francis Bonahon. The geometry of Teichmüller space via geodesic currents. Inventiones mathematicae, 92(1):139–162, 1988.
  10. Francis Bonahon. Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic form. Toulouse. Faculté des Sciences. Annales. Mathématiques. Série 6, 5(2):233–297, 1996.
  11. Parameterizing Hitchin components. Duke Mathematical Journal, 163(15):2935–2975, 2014.
  12. Hitchin characters and geodesic laminations. Acta Math, 218:201–295, 2017.
  13. Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow. arXiv:2102.13124v2, Geometry &\&&  Topology, to appear, 2021.
  14. Thurston’s asymmetric metrics for Anosov representations. arXiv:2210.05292, 2022.
  15. Comparison between Teichmüller and Lipschitz metrics. Journal of the London Mathematical Society. Second Series, 76(3):739–756, 2007.
  16. Transverse Measures and best Lipschitz and Least Gradient maps. October 2020.
  17. Analytic properties of stretch maps and geodesic laminations. May 2022.
  18. Length spectra and degeneration of flat metrics. Inventiones mathematicae, 182(2):231–277, 2010.
  19. Coarse and fine geometry of the Thurston metric. Forum of Mathematics. Sigma, 8:Paper No. e28, 58, 2020.
  20. James Eells, Jr. and J. H. Sampson. Harmonic mappings of Riemannian manifolds. Amer. J. Math., 86:109–160, 1964.
  21. Mirzakhani’s curve counting and geodesic currents, volume 345 of Progress in Mathematics. Birkhäuser/Springer, Cham, [2022] ©2022.
  22. Length functions on currents and applications to dynamics and counting. In In the tradition of Thurston—geometry and topology, pages 423–458. Springer, Cham, [2020] ©2020.
  23. Thurston’s work on surfaces, volume 48 of Mathematical Notes. Princeton University Press, Princeton, NJ, 2012. Translated from the 1979 French original by Djun M. Kim and Dan Margalit.
  24. Moduli spaces of local systems and higher Teichmüller theory. Publications Mathématiques de l’IHÉS, 103:1–211, 2006.
  25. Dual Teichmüller and lamination spaces. arXiv preprint math/0510312, 2005.
  26. Moduli spaces of convex projective structures on surfaces. Advances in Mathematics, 208(1):249–273, 2007.
  27. M. Gromov. Hyperbolic manifolds, groups and actions. In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume 97 of Ann. of Math. Stud., pages 183–213. Princeton Univ. Press, Princeton, N.J., 1981.
  28. Maximally stretched laminations on geometrically finite hyperbolic manifolds. Geometry & Topology, 21(2):693–840, 2017.
  29. The marked length spectrum of anosov manifolds. Annals of Mathematics, 190(1):321–344, 2019.
  30. Richard S. Hamilton. Harmonic maps of manifolds with boundary. Lecture Notes in Mathematics, Vol. 471. Springer-Verlag, Berlin-New York, 1975.
  31. Philip Hartman. On homotopic harmonic maps. Canadian J. Math., 19:673–687, 1967.
  32. The earthquake metric on Teichmüller space. in preparation.
  33. The infinitesimal and global thurston geometry of Teichmüller space. November 2021.
  34. Yi Huang and Athanase Papadopoulos. Optimal lipschitz maps on one-holed Tori and the Thurston metric theory of Teichmüeller space. arXiv:1909.03639, 2019.
  35. John H Hubbard. Teichmüller theory and applications to geometry, topology, and dynamics, volume 2. Matrix Editions, 2016.
  36. Jürgen Jost. Harmonic maps between surfaces, volume 1062 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1984.
  37. Bounded combinatorics and the Lipschitz metric on Teichmüller space. Geometriae Dedicata, 159:353–371, 2012.
  38. The shadow of a Thurston geodesic to the curve graph. Journal of Topology, 8(4):1085–1118, 2015.
  39. Zhong Li. Length spectrums of Riemann surfaces and the Teichmüller metric. Bull. London Math. Soc., 35(2):247–254, 2003.
  40. Almost-isometry between the Teichmüller metric and the length-spectrum metric on reduced moduli space for surfaces with boundary. Trans. Amer. Math. Soc., 369(9):6429–6464, 2017.
  41. L. Liu and W. Su. Almost-isometry between Teichmüller metric and length-spectrum metric on moduli space. Bull. Lond. Math. Soc., 43(6):1181–1190, 2011.
  42. Geometry of the complex of curves. I. Hyperbolicity. Invent. Math., 138(1):103–149, 1999.
  43. Yair N Minsky. Harmonic maps, length, and energy in Teichmüller space. Journal of Differential Geometry, 35(1):151–217, 1992.
  44. Yair N. Minsky. Extremal length estimates and product regions in Teichmüller space. Duke Math. J., 83(2):249–286, 1996.
  45. Maryam Mirzakhani. Ergodic theory of the earthquake flow. International Mathematics Research Notices, 2008(9):1073–7928, year=2008, publisher=OUP.
  46. Jean-Pierre Otal. Le spectre marqué des longueurs des surfaces à courbure négative. Annals of Mathematics, 131(1):151–162, 1990.
  47. Huiping Pan. Local Rigidity of the Teichmüller space with the Thurston metric. Science China Mathematics, https://doi.org/10.1007/s11425-021-2020-0, 2023.
  48. Envelopes of the Thurston metric on Teichmüller space. in preparation.
  49. Ray structures on Teichmüller space. arXiv:2206.01371, June 2022.
  50. A. Papadopoulos and W. Su. On the Finsler structure of Teichmüller’s metric and Thurston’s metric. Expo. Math., 33(1):30–47, 2015.
  51. Athanase Papadopoulos. On Thurston’s boundary of Teichmüller space and the extension of earthquakes. Topology Appl., 41(3):147–177, 1991.
  52. Athanase Papadopoulos, editor. Handbook of Teichmüller theory. Vol. III, volume 17 of IRMA Lectures in Mathematics and Theoretical Physics. European Mathematical Society (EMS), Zürich, 2012.
  53. The Weil-Petersson symplectic structure at Thurston’s boundary. Transactions of the American Mathematical Society, 335(2):891–904, 1993.
  54. On Teichmüller’s metric and Thurston’s asymmetric metric on Teichmüller space. In Handbook of Teichmüller theory. Vol. I, volume 11 of IRMA Lect. Math. Theor. Phys., pages 111–204. Eur. Math. Soc., Zürich, 2007.
  55. Some Lipschitz maps between hyperbolic surfaces with applications to Teichmüller theory. Geometriae Dedicata, 161:63–83, 2012.
  56. Deforming hyperbolic hexagons with applications to the arc and the Thurston metrics on Teichmüller spaces. Monatshefte für Mathematik, 182(4):913–939, 2017.
  57. Combinatorics of train tracks, volume 125 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1992.
  58. Robert C Penner. Universal constructions in Teichmüller theory. Advances in Mathematics, 98(2):143–215, 1993.
  59. Kasra Rafi. A characterization of short curves of a Teichmüller geodesic. Geom. Topol., 9:179–202, 2005.
  60. Geodesic currents and counting problems. Geom. Funct. Anal., 29(3):871–889, 2019.
  61. J. H. Sampson. Some properties and applications of harmonic mappings. Ann. Sci. École Norm. Sup. (4), 11(2):211–228, 1978.
  62. Jenya Sapir. An extension of the Thurston metric to projective filling currents. arXiv preprint arXiv:2210.08130, 2022.
  63. Dragomir Šarić. Circle homeomorphisms and shears. Geometry & Topology, 14(4):2405–2430, 2010.
  64. Circle homeomorphisms with square summable diamond shears. arXiv preprint arXiv:2211.11497, 2022.
  65. On univalent harmonic maps between surfaces. Invent. Math., 44(3):265–278, 1978.
  66. Richard M. Schoen. Analytic aspects of the harmonic map problem. In Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), volume 2 of Math. Sci. Res. Inst. Publ., pages 321–358. Springer, New York, 1984.
  67. The Weil-Petersson and Thurston symplectic forms. Duke Math. J., 108(3):581–597, 2001.
  68. Weixu Su. Problems on the Thurston metric. In Handbook of Teichmüller theory. Vol. V, volume 26 of IRMA Lect. Math. Theor. Phys., pages 55–72. Eur. Math. Soc., Zürich, 2016.
  69. Ivan Telpukhovskiy. Masur’s criterion does not hold in the thurston metric. arXiv:1903.00845v2, 2022.
  70. Guillaume Théret. On the negative convergence of Thurston’s stretch lines towards the boundary of Teichmüller space. Annales AcademiæScientiarum Fennicæ. Mathematica, 32(2):381–408, 2007.
  71. Guillaume Théret. Convexity of length functions and Thurston’s shear coordinates. arXiv:1408.5771, 2014.
  72. William P. Thurston. Earthquakes in two-dimensional hyperbolic geometry. In Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), volume 112 of London Math. Soc. Lecture Note Ser., pages 91–112. Cambridge Univ. Press, Cambridge, 1986.
  73. William P Thurston. The Geometry and Topology of Three-Manifolds: With a Preface by Steven P. Kerckhoff, volume 27. American Mathematical Society, 2022.
  74. William P. Thurston. Minimal stretch maps between hyperbolic surfaces. In Collected works of William P. Thurston with commentary. Vol. I. Foliations, surfaces and differential geometry, pages 533–585. Amer. Math. Soc., Providence, RI, [2022] ©2022. 1986 preprint, 1998 eprint.
  75. Karen Vogtmann. On the geometry of outer space. Bulletin of the American Mathematical Society, 52(1):27–46, 2015.
  76. Cormac Walsh. The horoboundary and isometry group of Thurston’s Lipschitz metric. In Handbook of Teichmüller theory. Vol. IV, volume 19 of IRMA Lect. Math. Theor. Phys., pages 327–353. Eur. Math. Soc., Zürich, 2014.
  77. Michael Wolf. The Teichmüller theory of harmonic maps. Journal of Differential Geometry, 29(2):449–479, 1989.
  78. Michael Wolf. Measured foliations and harmonic maps of surfaces. Journal of Differential Geometry, 49(3):437–467, 1998.
Citations (1)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Authors (2)

  1. Huiping Pan 
  2. Weixu Su 

Collections

Sign up for free to add this paper to one or more collections.

Sign Up