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.

We find approximate solutions (chord–arc curves) for the system of equations of geodesics in Ω∩ℍ for every Denjoy domain Ω, with respect to both the Poincaré and the quasi-hyperbolic metrics. We also prove that these chord–arc curves are uniformly close to the geodesics. As an application of these results, we obtain good estimates for the lengths of simple closed geodesics in any Denjoy domain, and we improve the characterization in a 1999 work by Alvarez et al. on Denjoy domains satisfying the linear isoperimetric inequality.