Journal of the
Korean Mathematical Society JKMS

We prove that if a given complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincar\'e inequality and the finite covering condition at infinity on each end, then every positive harmonic function on the manifold is asymptotically constant at infinity on each end. This result is a direct generalization of those of Yau and of Li and Tam.

Keywords: rough isometry, Harnack inequality, asymptotically constant, parabolicity, capacity, end, harmonic function, harmonic map, Sobolev's inequality, Poincar\'e inequality, finite covering, volume doubling, Liouville theorem

MSC numbers: 31C05, 31C20, 53C21, 58E20, 58G03, 58G20