Vol. 23, No. 3, 1967

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An extremal length criterion for the parabolicity of Riemannian spaces

Wellington Ham Ow

Vol. 23 (1967), No. 3, 585–590

It is the purpose of this paper to show that a given Riemannian space satisfying a regularity condition is parabolic if and only if the extremal distance of a fixed ball in the space from the ideal boundary of the space is infinite. We will also show that the harmonic modulus of a space bounded by two sets of boundary components coincides with the extremal distance between the two sets.

Mathematical Subject Classification
Primary: 53.72
Received: 30 January 1967
Published: 1 December 1967
Wellington Ham Ow