Isometric Riemannian manifolds at infinity
type: Item
instance of: scholarly article
title: Isometric Riemannian manifolds at infinity
zbMATH Open document ID: 0845.58054
DOI: 10.4171/RMI/190
author: Thierry Coulhon
author: Laurent Saloff-Coste
published in: Revista Matemática Iberoamericana
publication date: 1996-09-17T00:00:00Z
full work available at URL: https://eudml.org/doc/39498
review text: The authors study some properties of noncompact Riemannian manifolds, depending only on the geometry at infinity. The considered manifold \(M\) satisfies a certain inequality \((P)_{\text{loc}}\) of Poincaré type and, as a metric space, the inequality \((DV)_{\text{loc}}\): \(V(x,2r)\leq C_r V(x,r)\), \(x\in M\), \(V(x,r)\) -- the volume of the disk of ray \(r\), centered in \(x\). The authors associate to a manifold satisfying the conditions \((P)_{\text{loc}}\) and \((DV)_{\text{loc}}\) a weighted, countable, locally finite, connected graph and show that some inequalities of Poincaré and Sobolev type are transmitted to these associated graphs. They obtain the stability at infinity under the isometries of several properties of the manifolds. It is obtained that a manifold with minorated Ricci curvature which is isometric at infinity to a manifold of positive or zero Ricci curvature or to a Lie group of polynomial growth of volume, do not have positive trivial harmonic functions.
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 838782
zbMATH Keywords: isometry group
zbMATH Keywords: isometric graphs
zbMATH Keywords: Riemannian manifolds
reviewed by: Vasile Oproiu
MaRDI profile type: Publication
Recommended article: On isotropy irreducible Riemannian manifolds
Recommended article: On the geometry of Riemannian manifolds with a Lie structure at infinity
Recommended article: Isometric immersions of Riemannian manifolds
Recommended article: Q3753058
Recommended article: Q3414170
Recommended article: Q4011217
Recommended article: Q4324535
Recommended article: Isoperimetric Inequalities in Riemannian Manifolds
Recommended article: Q3817243
description: scientific article; zbMATH DE number 838782