On Euclidean and Teichmüller Distances

Let (X1, q1) and (X2, q2) be Riemann surfaces with quadratic differentials whose associated,flat metrics have unit area and are -close with respect to a metric built from good systems,of period coordinates. Assume X1, X2 lie over a compact subset K of the moduli space of,Riemann surfaces Mg,n. We show that X1 and X2 are Cα-close in the Teichm¨uller metric.,Here, α depends only on the genus g and the number of marked points n, while C depends,on K. To achieve this, we analyze the uniformization maps of neighborhoods of colliding of,singularities of the flat metrics associated to q1 and q2 and build an explicit quasiconformal,map between X1 and X2.