Journal of the
Korean Mathematical Society JKMS

We prove that if a complete manifold is roughly isometric to a manifold on which a suitable volume growth rate, the Sobolev inequality and the Poincar\'e inequality hold, then the Harnack inequality, hence the Liouville theorem, for positive harmonic function is valid. In particular, if a complete $m$-manifold is roughly isometric to an $n$-manifold of non-negative Ricci curvature with $m \ge n$, then the Harnack inequality, hence the Liouville theorem, for positive harmonic function is valid. Our result is a generalization of Kanai's, and it also partly generalizes the results of Saloff-Coste.s

Keywords: rough isometry, Harnack inequality, Liouville theorem, harmonic function

MSC numbers: 53C21, 58G03