A partial surface variation for extremal schlicht functions

Abstract:

Let a topological sphere be formed from $|z| \leqslant 1$ by dissecting the circumference into finitely many pairs $({I_j},{I’_j})$ of disjoint arcs, identifying ${I_j}$ and ${I’_j}$ in opposite directions and making further identifications among the endpoints. If there exists a meromorphic function $Q(z)$, real and non-negative on $|z| = 1$ and satisfying certain consistency conditions with respect to the dissection (given in detail in our Introduction), then one forms a Q-polygon by using the element of length i $ds = \sqrt {Q(z)} \;dz/z$ to effect the metric identification of the pairs ${I_j},{I’_j}$. In a natural way, Q-polygons become Riemann surfaces, thus can be mapped conformally onto the number sphere. When Q is of the form $Q(z) = \Sigma _{j = - N}^N{B_j}{z^j}$, then the corresponding mapping functions, suitably normalized, become the extremal schlicht functions for the coefficient body ${V_{N + 1}}$ [3, p. 120]. Suppose that for a given dissection of $|z| = 1$ there is a family $Q(z,t)$ of consistent meromorphic functions. For Q sufficiently smooth as a function of $\varepsilon$, we study the variation of the corresponding normalized mapping functions $f(p,\varepsilon )$, using results of [2], and show smoothness of f as a function of $\varepsilon$. Specializing Q to the form above, we deduce the existence of smooth submanifolds of $\partial {V_{N + 1}}$ and obtain a variational formula for the extremal schlicht functions corresponding to motion along these submanifolds.