Abstract

We develop the concept of “bridged extremal distance” between disjoint sets X and Z on the boundary of a finitely connected domain G; that is, the extremal length of the family of curves connecting X and Z which are allowed to stop at a component of the “bridge” Y = ∂G ∖ (X ∪ Z) and re-emerge from any other point of that component. We connect bridged extremal distance with the extremal problem of “minimal extremal distance”, and express it in terms of the period matrix associated with the harmonic measures of the boundary components of G. Then, in direct analogy to Ahlfors and Beurling’s extremal length interpretation of logarithmic capacity, we use bridged extremal distance to give an extremal length interpretation of “maximal capacity”.

Mathematical Subject Classification 2000

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