We show that any rotationally symmetric Riemannian manifold has the L1-Liouville property for harmonic functions, i.e., any integrable harmonic function on it must be identically constant. We also give a characterization of a manifold which carries a non-constant L1 nonnegative subharmonic function.
"Uniqueness of L1 harmonic functions on rotationally symmetric Riemannian manifolds." Kodai Math. J. 37 (1) 1 - 15, March 2014. https://doi.org/10.2996/kmj/1396008245