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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A partial surface variation for extremal schlicht functions
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by T. L. McCoy PDF
Trans. Amer. Math. Soc. 234 (1977), 119-138 Request permission

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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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 234 (1977), 119-138
  • MSC: Primary 30A38
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0473163-1
  • MathSciNet review: 0473163