Comparison of period coordinates andTeichmüller distances

We show that when two unit-area quadratic differentials are -close with respect to good systems of period coordinates and lie over a compact subset K of the moduli space of Riemann surfaces M g;n , then their underlying Riemann surfaces are C ˛-close in the Teichmüller metric.Here, ˛depends only on the genus g and the number of marked points, while C depends on K. 30F60 1. Introduction and statement of main result 2451 2. Sketch of proof 2455 3. Teichmüller spaces and quadratic differentials 2456 4. Quadratic differentials on the sphere 2464 5. Perturbing quadratic differentials on the sphere 2481 6. Building the quasiconformal map 2492 Appendix A. Local finiteness of period coordinate systems 2498 Appendix B. ı-clusters and the Euclidean metric 2501