Isometric Riemannian manifolds at infinity. (Variétés riemanniennes isométriques à l’infini.) (French) Zbl 0845.58054

Coulhon, Thierry; Saloff-Coste, Laurent

Rev. Mat. Iberoam. 11, No. 3, 687-726 (1995).

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.

Reviewer: V.Oproiu (Iaşi)

MSC:

58J60	Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C23	Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Keywords:

isometry group; isometric graphs; Riemannian manifolds