Let (M,F) be a compact codimension-one foliated manifold whose leaves are equipped with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of F. If every such function is constant on leaves we say that (M,F) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. Related results for R-covered foliations, as well as for discrete group actions and discrete harmonic functions, are also established.
Fenley, Sergio; Feres, Renato; and Parwani, Kamlesh, "Harmonic functions on R-covered foliations and group actions on the circle" (2009). Faculty Research and Creative Activity. 22.