Rough isometries, and combinatorial approximations of geometries of non- compact Riemannian manifolds
We study a map \(\phi\) : \(X\to Y\), not necessarily continuous, between non-compact Riemannian manifolds X and Y, which satisfies the following two conditions: (i) for some \(\epsilon >0\), the \(\epsilon\)-neighborhood of the image of \(\phi\) in Y coincides with Y; (ii) for all \(x_ 1,x_ 2\in X\), the inequalities \[ a^{-1} d(x_ 1,x_ 2)-b\leq d(\phi (x_ 1),\phi (x_ 2))\leq a d(x_ 1,x_ 2)+b \] hold, where \(a\geq 1\) and \(b\geq 0\) are suitable constants, and d denotes the distance functions on X and Y. We call such a map a rough isometry. The main theorems of the paper suggest that some attributes of non- compact Riemannian manifolds are inherited through rough isometries. More precisely, we show that the volume growth rates and the validity of isoperimetric inequalities of non-compact complete Riemannian manifolds are preserved by rough isometries, under the additional conditions on the manifolds that the Ricci curvatures are bounded from below and the injectivity radii are positive. We also prove the Liouville theorem for positive harmonic functions on a complete Riemannian manifold which is roughly isometric to the Euclidean space and satisfies the above conditions on the curvature and the injectivity radius. In the proofs of these theorems, we frequently employ the method of combinatorial approximation of Riemannian geometry: We approximate ''continuous'' geometry of a non-compact Riemannian manifold by combinatorial geometry of a suitable discrete subset of the manifold.
scientific article; zbMATH DE number 3883148