Biholomorphic isomorphisms between Teichmuller curves

We determine the biholomorphic isomorphisms between Teichmuller curves for torsion free Fuchsian groups. We show that, except in some special cases, a biholomorphic isomorphism between two Teichmuller curves is an allowable mapping; in particular, a biholomorphic automorphism of a Teichmuller curve is induced by an element of the modular group. As a by-product of the proof, we also obtain some topological results on self-mappings of surfaces.