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 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 [2], 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.

**[1]**R. Courant,*Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces*, Interscience Publishers, Inc., New York, N.Y., 1950. Appendix by M. Schiffer. MR**0036317****[2]**T. L. McCoy,*Variation of conformal spheres by simultaneous sewing along several arcs*, Trans. Amer. Math. Soc.**231**(1977), no. 1, 65–82. MR**0444922**, https://doi.org/10.1090/S0002-9947-1977-0444922-6**[3]**A. C. Schaeffer and D. C. Spencer,*Coefficient Regions for Schlicht Functions*, American Mathematical Society Colloquium Publications, Vol. 35, American Mathematical Society, New York, N. Y., 1950. With a Chapter on the Region of the Derivative of a Schlicht Function by Arthur Grad. MR**0037908****[4]**Menahem Schiffer and Donald C. Spencer,*Functionals of finite Riemann surfaces*, Princeton University Press, Princeton, N. J., 1954. MR**0065652**

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

Article copyright:
© Copyright 1977
American Mathematical Society