ON LENGTH DISTORTIONS WITH RESPECT TO QUADRATIC DIFFERENTIAL METRICS

SUN, ZONGLIANG

Glasgow mathematical journal, Tome 56 (2014) no. 3, pp. 681-689

Résumé

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.

Détail

DOI : 10.1017/S001708951400010X

Mots-clés : Primary 30F45, Secondary 51M25