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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48 .

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An extremal length problem on a bordered Riemann surface
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by Jeffrey Clayton Wiener PDF
Trans. Amer. Math. Soc. 203 (1975), 227-245 Request permission

Abstract:

Partition the contours of a compact bordered Riemann surface $R’$ into four disjoint closed sets ${\alpha _0},{\alpha _1},{\alpha _2}$ and $\gamma$ with ${\alpha _0}$ and ${\alpha _1}$ nonempty. Let $F$ denote the family of all locally rectifiable $1$-chains in $R’ - \gamma$ which join ${\alpha _0}$ to ${\alpha _1}$. The extremal length problem on $R’$ considers the existence of a real-valued harmonic function $u$ on $R’$ which is 0 on ${\alpha _0},1$ on ${\alpha _1}$, a constant on each component ${\nu _k}$ of ${\alpha _2}$ with ${\smallint _{{\nu _k}}}^ \ast du = 0$ and $^ \ast du = 0$ along $\gamma$ such that the extremal length of $F$ is equal to the reciprocal of the Dirichlet integral of $u$, that is, $\lambda (F) = {D_{R’}}{(u)^{ - 1}}$. Let $\bar R$ denote a bordered Riemann surface with a finite number of boundary components and $S$ a compactification of $\bar R$ with the property that $\partial \bar R \subset S$. We consider the extremal length problem on $\bar R$ (as a subset of $S$) when ${\alpha _0},{\alpha _1}$, and ${\alpha _2}$ are relatively closed subarcs of $\partial \bar R$ and when ${\alpha _0},{\alpha _1}$ and ${\alpha _2}$ are closed subsets of $\partial S = (S - \bar R) \cup \partial \bar R$.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 203 (1975), 227-245
  • MSC: Primary 30A48
  • DOI: https://doi.org/10.1090/S0002-9947-1975-0361054-4
  • MathSciNet review: 0361054