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analytic function Beltrami differential bordered Riemann surface boundary point boundary values bounded closed curve closure complex dilatation conclude conformal mapping conformal structure conformally equivalent Conversely countable cover transformation group covering S,a definition denote differential of G equivalence classes exists extended plane finite number finite plane fixed point free follows that f function f fundamental domain Green's function group G harmonic function harmonic measure Hence homotopic hyperbolic identity mapping implies integral invariant isomorphic Lemma Let f Let G limit function mapping f mapping theorem mappings of G maximal dilatation maximum principle metric modulo monodromy theorem non-constant non-euclidean normal open set parabolic parametric disc Perron family plane domain point peS Proof properly discontinuous quasiconformal mappings relatively compact domain Riemann surface simply connected singularity smooth covering surface strong maximum principle structure H subgroup of F subharmonic function topology triple S,f,a unit disc universal covering surface unlimited covering surface