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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 0444922
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Abstract: Let M be a closed Riemann surface of genus zero, $ \Gamma $ a tree on M with branches $ {\Gamma _j}$, and $ {p_0}$ a point of $ M - \Gamma $. A family of neighboring topological surfaces $ M(\varepsilon )$ is formed by regarding each $ {\Gamma _j}$ as a slit with edges $ \Gamma _j^ - $ and $ \Gamma _j^ + $, and re-identifying p on $ {\Gamma ^{{ - _j}}}$ with $ p + \varepsilon {\chi _j}(p,\varepsilon )$ on $ \Gamma _j^ + $, with $ {\chi _j}$ vanishing at the endpoints of $ {\Gamma _j}$. We assume the $ {\Gamma _j}$ and $ {\chi _j}$ are such that, under a certain natural choice of uniformizers, the $ M(\varepsilon )$ are closed Riemann surfaces of genus zero. Then there exists a unique function $ f(p,\varepsilon ;{p_0})$ mapping $ M(\varepsilon )$ conformally onto the complex number sphere, with normalization $ f({p_0},\varepsilon ;{p_0}) = 0,f'({p_0},\varepsilon ;{p_0}) = 1$. Under appropriate smoothness hypotheses on $ \Gamma $ and the $ {\chi _j}$, we find the first variation of f as a function of $ \varepsilon $. Further, we obtain smoothness results for f as a function of $ \varepsilon $. 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: http://dx.doi.org/10.1090/S0002-9947-1977-0444922-6
PII: S 0002-9947(1977)0444922-6
Article copyright: © Copyright 1977 American Mathematical Society