Variation of conformal spheres by simultaneous sewing along several arcs

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.