A partial surface variation for extremal schlicht functions
Author: T. L. McCoy
Journal: Trans. Amer. Math. Soc. 234 (1977), 119-138
MSC: Primary 30A38
Erratum: Trans. Amer. Math. Soc. 240 (1978), 393.
MathSciNet review: 0473163
Abstract: Let a topological sphere be formed from by dissecting the circumference into finitely many pairs of disjoint arcs, identifying and in opposite directions and making further identifications among the endpoints. If there exists a meromorphic function , real and non-negative on 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 to effect the metric identification of the pairs . In a natural way, Q-polygons become Riemann surfaces, thus can be mapped conformally onto the number sphere. When Q is of the form , then the corresponding mapping functions, suitably normalized, become the extremal schlicht functions for the coefficient body [3, p. 120].
Suppose that for a given dissection of there is a family of consistent meromorphic functions. For Q sufficiently smooth as a function of , we study the variation of the corresponding normalized mapping functions , using results of , and show smoothness of f as a function of . Specializing Q to the form above, we deduce the existence of smooth submanifolds of and obtain a variational formula for the extremal schlicht functions corresponding to motion along these submanifolds.
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