Variation of conformal spheres by simultaneous sewing along several arcs
Abstract: Let M be a closed Riemann surface of genus zero, a tree on M with branches , and a point of . A family of neighboring topological surfaces is formed by regarding each as a slit with edges and , and re-identifying p on with on , with vanishing at the endpoints of . We assume the and are such that, under a certain natural choice of uniformizers, the are closed Riemann surfaces of genus zero. Then there exists a unique function mapping conformally onto the complex number sphere, with normalization . Under appropriate smoothness hypotheses on and the , we find the first variation of f as a function of . Further, we obtain smoothness results for f as a function of . The problem is connected with the study of the extremal schlicht functions; that is, the schlicht mappings of the unit disc corresponding to boundary points of the coefficient bodies.
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