Function-theoretic degeneracy criteria for Riemannian manifolds

Moses Glasner and Richard Emanuel Katz

Vol. 28 (1969), No. 2, 351–356

DOI: 10.2140/pjm.1969.28.351

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