The Royden boundary of a Riemannian manifold
Moses Glasner, Richard Katz · Illinois Journal of Mathematics · 1970
An ever present theme in function-theory is the study of a given family of harmonic functions in terms of their boundary values.This theme persists for investigations on open Riemann surfaces which are not embedded in larger ones.In the absence of a natural boundary an ideal boundary can be tailored to suit the study of a particular type of harmonic function.Royden [9] showed that an open Riemann surface can always be compactified in "such a fashion that the HBD-functions (harmonic, bounded, Dirichlet-finite) have continuous extensions and are determined by their behavior on the harmonic boundary, a small subset of the "new" points.Nakai [7] discovered that this same harmonic boundary serves as the basis for a representation theory for the HD-functions.It is natural to ask how much of this theory can be extended to Riemannian manifolds.Not only do the known proofs rely on strictly Riemann surface techniques but the Royden algebra which determines the compactification is of an essentially different nature in higher dimensions.This difference lies in the following result of Nakai [6]' quasi-conformally equivalent Riemann surfaces have isomorphic Royden algebras but only quasi-isometrically equiva- lent manifolds have this property (also cf.[8]).In this paper we show that the theory can be carried over in its entirety.In Sections 1 and 2 we introduce the fundamentals.We establish the maximum principle for HD-functions in terms of their values on the harmonic boundary in Sections 4 and 5.The Royden-Nakai decomposition theorem for Riemann- ian manifolds is given in Sections 6 and 7 and Section 8 contains some easy consequences.At this point the theory on Riemann surfaces easily generalizes and we conclude in Section 9 by simply stating two results.For a complete account we refer the reader to the forthcoming monograph of Sario and Nakai [10] on Riemann surfaces.The authors have been informed that Loeb and Walsh [3] have obtained similar results for Banach sublattices of HB-functions in the axiomatic setting.1. Let R be a noncompact orientable Riemannian manifold.We focus our attention on the Tonelli functions on R, the vector-lattice of continuous real- valued functions on R with locally square integrable first partial derivatives.The Dirichlet integral of Tonelli functions over relatively compact regions 2