Variation of conformal spheres by simultaneous sewing along several arcs

Author:
T. L. McCoy

Journal:
Trans. Amer. Math. Soc. **231** (1977), 65-82

MSC:
Primary 30A30

DOI:
https://doi.org/10.1090/S0002-9947-1977-0444922-6

MathSciNet review:
0444922

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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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DOI:
https://doi.org/10.1090/S0002-9947-1977-0444922-6

Article copyright:
© Copyright 1977
American Mathematical Society