Geodesic
ON LENGTH DISTORTIONS WITH RESPECT TO QUADRATIC DIFFERENTIAL METRICS
Glasgow mathematical journal, Tome 56 (2014) no. 3, pp. 681-689

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper, we consider the question about length distortions under quasiconformal mappings with respect to quadratic differential metrics. More precisely, let X and Y be closed Riemann surfaces with genus at least 2, and f: X → Y being a K-quasiconformal mapping. Given two quadratic differential metrics |q1| and |q2| with unit areas on X and Y respectively, whether there exists a constant $\mathcal C$ depending only on K such that $\frac{1}{\mathcal C} l_{q_1} (\gamma) \leq l_{q_2} (f(\gamma)) \leq \mathcal C l_{q_1} (\gamma)$ holds for any simple closed curve γ ⊂ X. Here lqi(α) denotes the infimum of the lengths of curves in the homotopy class of α with respect to the metric |qi|, i = 1, 2. We give positive answers to this question, including the aspects that the desired constant ${\mathcal C}$ explicitly depends on q1, q2 and K, and that the constant $\mathcal C$ is universal for all the quantities involved.
DOI : 10.1017/S001708951400010X
Mots-clés : Primary 30F45, Secondary 51M25 
SUN, ZONGLIANG. ON LENGTH DISTORTIONS WITH RESPECT TO QUADRATIC DIFFERENTIAL METRICS. Glasgow mathematical journal, Tome 56 (2014) no. 3, pp. 681-689. doi: 10.1017/S001708951400010X
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[1] 1.Ahlfors, L., Lectures on quasiconformal mappings (Van Nostrand Reinhold Princeton, NJ, 1966). Google Scholar

[2] 2.Duchin, M., Leininger, C. and Rafi, K., Length spectra and degeneration of flat metrics, Invent. Math. 182 (2) (2010), 231–277. Google Scholar | DOI

[3] 3.Fathi, A., Laudenbach, F. and Poenaru, V., translated by D. Kim and D. Margalit, Thurston's work on surfaces (Mathematical Notes, 48, Princeton University Press, Princeton, 2012). Google Scholar

[4] 4.Gardiner, F., Teichmüller theory and quadratic differentials (John Wiley, New York, NY, 1987). Google Scholar

[5] 5.Hamenstädt, U., Closed Teichmüller geodesics in the thin part of moduli space, arXiv:math/0511349 (2009). Google Scholar

[6] 6.Imayoshi, Y. and Taniguchi, M., An introduction to Teichmüller space (translated and revised from the Japanese by the authors) (Springer-Verlag, Tokyo, Japan, 1992). Google Scholar

[7] 7.Kerckhoff, S., The asymptotic geometry of Teichmüller space, Topology 19 (1) (1980), 23–41. Google Scholar | DOI

[8] 8.Kontsevich, M. and Zorich, A., Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (3) (2003), 631–678. Google Scholar | DOI

[9] 9.Strebel, K., Quadratic differentials (Springer-Verlag, New York, NY, 1984). Google Scholar | DOI

[10] 10.Thurston, W., On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (2) (1988), 417–431. Google Scholar

[11] 11.Wolpert, S., The length spectra as moduli for compact Riemann surfaces, Ann. Math. 109 (2) (1979), 323–351. Google Scholar | DOI

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