Abstract
ABSTRACTLet R be a Riemann surface of finite type , and a Riemann surface obtained by removing distinct points from R. We assume that the Riemann surfaces are equipped with the hyperbolic metric. For a closed geodesic on R, let be the shortest closed geodesic on which is freely homotopic to on R. In this paper, we give an explicit upper bound for the hyperbolic length of on depending only on and the hyperbolic length of on R.
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