• Sun, Zongliang (Department of Mathematics, Shenzhen University)
  • Received : 2014.08.27
  • Accepted : 2015.01.14
  • Published : 2016.03.31


In this paper, we study the relations between the Thurston metric and the hyperbolic metric on a closed surface of genus $g{\geq}2$. We show a rigidity result which says if there is an inequality between the marked length spectra of these two metrics, then they are isotopic. We obtain some inequalities on length comparisons between these metrics. Besides, we show certain distance distortions under conformal graftings, with respect to the $Teichm{\ddot{u}}ller$ metric, the length spectrum metric and Thurston's asymmetric metrics.


complex projective structure;hyperbolic metric;marked length spectrum;$Teichm{\ddot{u}}ller$ space;Thurston metric


  1. L. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand Reinhold, Princeton, N.J., 1966.
  2. C. Croke, A. Fathi, and J. Feldman, The marked length-spectrum of a surface of nonpositive curvature, Topology 31 (1992), no. 4, 847-855.
  3. M. Duchin, C. Leininger, and K. Rafi, Length spectra and degeneration of flat metrics, Invent. Math. 182 (2010), no. 2, 231-277.
  4. D. Dumas and M.Wolf, Projective structures, grafting and measured laminations, Geom. Topol. 12 (2008), no. 1, 351-386.
  5. A. Fathi, F. Laudenbach, and V. Poenaru, Travaux de Thurston sur les surfaces, Asterisque, Vol. 66-67, Soc. Math. France, Paris, 1979.
  6. Y. Kamishima and Ser P. Tan, Deformation spaces on geometric structures, Aspects of low-dimensional manifolds, 263299, Adv. Stud. Pure Math., 20, Kinokuniya, Tokyo, 1992.
  7. S. Kerckhoff, The asymptotic geometry of Teichmuller space, Topology 19 (1980), no. 1, 23-41.
  8. S. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235-265.
  9. I. Kim, Complex projective structures and the marked length rigidity, RIMS Kokyuroku (Kyoto University) 1104 (1999), 153-159.
  10. J. Lehner, On the $A_q({\Gamma}){\subset}B_q({\Gamma})$ conjecture for infinitely generated groups, in Discontinuous groups and Riemann surfaces (Proc. Conf. Univ. Maryland, College Park, Md., 1973), pp. 283-288. Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N. J., 1974.
  11. Z. Li, Teichmuller metric and length spectrum of Riemann surface, Sci. Sinica Ser. A 29 (1986), no. 3, 265-274.
  12. Z. Li, Length spectrums of Riemann surfaces and the Teichmuller metric, Bull. London Math. Soc. 35 (2003), no. 2, 247-254.
  13. L. Liu, Z. Sun, and H. Wei, Topological equivalence of metrics in Teichmuller space, Ann. Acad. Sci. Fenn. Math. 33 (2008), no. 1, 159-170.
  14. B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381-386.
  15. C. McMullen, Complex earthquakes and Teichmuller theory, J. Amer. Math. Soc. 11 (1998), no. 2, 283-320.
  16. S. Nag, The complex analytic theory of Teichmuller spaces, John Wiley & Sons, New York, 1988.
  17. J.-P. Otal, Le spectre marque des longueurs des surfaces a courbure negative, Ann. of Math. (2) 131 (1990), 151-162.
  18. T. Sorvali, The boundary mapping induced by an isomorphism of covering groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 526 (1972), 1-31.
  19. T. Sorvali, On Teichmuller space of tori, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 1, 7-11.
  20. K. Strebel, Quadratic Differentials, Springer-Verlag, New York, 1984.
  21. H. Tanigawa, Grafting, harmonic maps and projective structures on surfaces, J. Differential Geom. 47 (1997), no. 3, 399-419.
  22. W. Thurston, Minimal stretch maps between hyperbolic surfaces, preprint, 1998.
  23. S. Wolpert, The length spectra as moduli for compact Riemann surfaces, Ann. of Math. 109 (1979), no. 2, 323-351.