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Transactions of the American Mathematical Society

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



An extremal length problem on a bordered Riemann surface

Author: Jeffrey Clayton Wiener
Journal: Trans. Amer. Math. Soc. 203 (1975), 227-245
MSC: Primary 30A48
MathSciNet review: 0361054
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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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Keywords: Extremal length, harmonic function, Kerékjártó-Stoilöw compactification, canonical exhaustion, double of a surface, regular exhaustion
Article copyright: © Copyright 1975 American Mathematical Society