Rough isometries and the parabolicity of Riemannian manifolds
A rough isometry is a map \(\phi\) : \(X\to Y\), not necessarily continuous, between metric spaces X and Y, for which the following two conditions are satisfied: (i) for some \(\epsilon >0\), the \(\epsilon\)-neighborhood of the image of \(\phi\) in Y coincides with Y; (ii) there are constants \(a\geq 1\) and \(b\geq 0\) such that \(a^{-1}d(x_ 1,x_ 2)-b\leq d(\phi x_ 1,\phi x_ 2)\leq a d(x_ 1,x_ 2)+b,\) \(x_ 1,x_ 2\in X.\) X is said to be roughly isometric to Y if there is a rough isometry \(\phi\) : \(X\to Y\). The notion of rough isometry was introduced in the author's previous paper in J. Math. Soc. Japan 37, 391-413 (1985; Zbl 0564.53015)], and it was seen that some geometric attributes of non- compact complete Riemannian manifolds, such as volume growth rate and the validity of isoperimetric inequalities, are inherited through rough isometries. In the present paper, the author shows that this is also the case with the parabolicity. By definition, a Riemannian manifold is said to be parabolic if there are no positive superharmonic functions other than constant ones, and the main result of the paper is the following theorem: Let X and Y be complete Riemannian manifolds with the Ricci curvatures bounded below and with the positive injectivity radii, and assume that X is roughly isometric to Y. Then X is parabolic if so is Y.
scientific article; zbMATH DE number 3923605