An extremal length problem on a bordered Riemann surface

Abstract

Partition the contours of a compact bordered Riemann surface R ′ R’ into four disjoint closed sets α 0 , α 1 , α 2 {\alpha _0},{\alpha _1},{\alpha _2} and γ \gamma with α 0 {\alpha _0} and α 1 {\alpha _1} nonempty. Let F F denote the family of all locally rectifiable 1 1 -chains in R ′ − γ R’ - \gamma which join α 0 {\alpha _0} to α 1 {\alpha _1} . The extremal length problem on R ′ R’ considers the existence of a real-valued harmonic function u u on R ′ R’ which is 0 on α 0 , 1 {\alpha _0},1 on α 1 {\alpha _1} , a constant on each component ν k {\nu _k} of α 2 {\alpha _2} with ∫ ν k ∗ d u = 0 {\smallint _{{\nu _k}}}^ \ast du = 0 and ∗ d u = 0 ^ \ast du = 0 along γ \gamma such that the extremal length of F F is equal to the reciprocal of the Dirichlet integral of u u , that is, λ ( F ) = D R ′ ( u ) − 1 \lambda (F) = {D_{R’}}{(u)^{ - 1}} . Let R ¯ \bar R denote a bordered Riemann surface with a finite number of boundary components and S S a compactification of R ¯ \bar R with the property that ∂ R ¯ ⊂ S \partial \bar R \subset S . We consider the extremal length problem on R ¯ \bar R (as a subset of S S ) when α 0 , α 1 {\alpha _0},{\alpha _1} , and α 2 {\alpha _2} are relatively closed subarcs of ∂ R ¯ \partial \bar R and when α 0 , α 1 {\alpha _0},{\alpha _1} and α 2 {\alpha _2} are closed subsets of ∂ S = ( S − R ¯ ) ∪ ∂ R ¯ \partial S = (S - \bar R) \cup \partial \bar R .