Isometric Riemannian manifolds at infinity
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.
scientific article; zbMATH DE number 838782
scientific article