Vol. 28, No. 2, 1969

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Function-theoretic degeneracy criteria for Riemannian manifolds

Moses Glasner and Richard Emanuel Katz

Vol. 28 (1969), No. 2, 351–356
Abstract

The modulus of a relatively compact set with border consisting of at least two components is a measure of its magnitude with regard to harmonic functions. A divergent modular sum associated with difference sets obtained from an exhaustion of a Riemannian manifold is characteristic of parabolicity. The existence of a divergent minimum modular sum implies that the manifold carries no nonconstant harmonic functions with finite Dirichlet integral.

Mathematical Subject Classification
Primary: 53.72
Secondary: 30.00
Milestones
Received: 13 February 1968
Published: 1 February 1969
Authors
Moses Glasner
Richard Emanuel Katz