A partial surface variation for extremal schlicht functions

Author:
T. L. McCoy

Journal:
Trans. Amer. Math. Soc. **234** (1977), 119-138

MSC:
Primary 30A38

DOI:
https://doi.org/10.1090/S0002-9947-1977-0473163-1

Erratum:
Trans. Amer. Math. Soc. **240** (1978), 393.

MathSciNet review:
0473163

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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.

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*Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces*, Interscience Publishers, Inc., New York, N.Y., 1950. Appendix by M. Schiffer. MR**0036317** - T. L. McCoy,
*Variation of conformal spheres by simultaneous sewing along several arcs*, Trans. Amer. Math. Soc.**231**(1977), no. 1, 65–82. MR**444922**, DOI https://doi.org/10.1090/S0002-9947-1977-0444922-6 - A. C. Schaeffer and D. C. Spencer,
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Article copyright:
© Copyright 1977
American Mathematical Society