Extremal quasiconformal mappings with prescribed boundary values

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1. Let R be a Riemann surface whose universal covering space is conformally equivalent to the unit disk. We can regard R as the interior of a Riemann surface with boundary R* whose boundary is as large as possible (see ?3). A quasiconformal map f of R onto another Riemann surface S has a unique continuous extensionf* mapping R* onto S*. Two quasiconformal mapsf and g of R onto S are homotopic modulo the boundary iff* =g* on R* -R and there exists a homotopy between f* and g* which is constant on R*R. If R* R is empty then f is just homotopic to g. Let K be the complex cotangent bundle of R. Then /(f), the Beltrami differential of f, is an element of LO(KK1), the Banach space of all essentially bounded sections of the bundle RK-1. (Locally v eL-(KK-1) is given by v dza/dz,.) L-(KK-1) is the dual of L1(K2), the Banach space of integrable sections of the bundle K2. (Locally e L1(K2) is given by p,P dZ.) Let A(R) denote the closed subspace of L1(K2) of all integrable analytic quadratic differentials on R. By ll((f) I A(R)II we denote the norm of the restriction of /(f), regarded as a linear functional on L1(K2), to the closed subspace A(R). The quasiconformal map f: R -S is extremal if (/3(f) ? (/3(g) ( for all maps g homotopic to f modulo the boundary. The main result is