An extremal length problem on a bordered Riemann surface

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 $\partial S = (S - \bar R) \cup \partial \bar R$.