On the hyperbolic length and quasiconformal mappings

Abstract

Let ϕ : R → S be a K-quasiconformal mapping of a hyperbolic Riemann surface R to another S. It is important to see how the hyperbolic structure is changed by ϕ. S. Wolpert (1979, The length spectrum as moduli for compact Riemann surfaces. Ann. of Math. 109, 323–351) shows that the length of a closed geodesic is quasi-invariant. Recently, A. Basmajian (2000, Quasiconformal mappings and geodesics in the hyperbolic plane, in The Tradition of Ahlfors and Bers, Contemp. Math. 256, 1–4) gives a variational formula of distances between geodesics in the upper half-plane. In this article, we improve and generalize Basmajian's result. We also generalize Wolpert's formula for loxodromic transformations.