Boundary behavior of functions on complete manifolds of negative curvature
type: Item
instance of: scholarly article
title: Boundary behavior of functions on complete manifolds of negative curvature
zbMATH Open document ID: 0686.31004
author: Hitoshi Arai
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1989-01-01T00:00:00Z
review text: We study harmonic and h-harmonic functions on complete negatively curved manifolds, and their behavior near its sphere at infinity. Let M be a complete, simply connected Riemannian manifolds, which sectional curvatures are bounded by two negative constants. For such manifold, \textit{M. T. Anderson} and \textit{R. Schoen} proved that the Martin compactification of M is homeomorphic to one of Eberlein and O'Neill, and that positive harmonic functions on M satisfy the so-called Fatou phenomena [Ann. Math., II. Ser. 121, 429-461 (1985; Zbl 0587.53045)]. In our paper we introduce a class of approach regions and prove local Fatou phenomena of harmonic and h-harmonic functions on M. Furthermore, we investigate their fine, semifine, nontangential and admissible limits at the sphere at infinity. Our results extend also Theorem 4 in \textit{A. Ancona} [Ann. Math., II. Ser. 125, 495-536 (1987; Zbl 0652.31008)].
Mathematics Subject Classification ID: 31B25
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 31D05
zbMATH DE Number: 4124070
zbMATH Keywords: negatively curved manifolds
zbMATH Keywords: Fatou phenomena
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227828
OpenAlex ID: W2022764371
cites work: Negatively curved manifolds, elliptic operators, and the Martin boundary
cites work: Positive harmonic functions on complete manifolds of negative curvature
cites work: Harmonic analysis on negatively curved manifolds. I
cites work: Limites angulaires et limites fines
cites work: Q4751075
cites work: Q3344779
cites work: Differential equations on riemannian manifolds and their geometric applications
cites work: Q3342857
cites work: Visibility manifolds
cites work: Fatou-Naïm-Doob limit theorems in the axiomatic system of Brelot
cites work: Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel
cites work: Positive Harmonic Functions on Lipschitz Domains
cites work: Boundary behavior of harmonic functions in non-tangentially accessible domains
cites work: Q3751848
cites work: Riemannian geometry.
cites work: Fine Convergence and Admissible Convergence for Symmetric Spaces of Rank One
cites work: Q5607484
cites work: Fine and Nontangential Convergence on an NTA Domain
DOI: 10.2748/TMJ/1178227828
description: scientific article; zbMATH DE number 4124070