Harmonic functions on R-covered foliations and group actions on the circle

TL;DR

This paper investigates harmonic functions on R-covered foliations and group actions on the circle, establishing conditions under which these functions are constant along leaves, thus extending the Liouville property to new geometric contexts.

Contribution

It proves that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property, with related results for R-covered foliations and group actions.

Findings

Codimension-one foliated bundles over negatively curved manifolds have the Liouville property.

Results extend to R-covered foliations and discrete group actions.

Harmonic functions are constant along leaves under these conditions.

Abstract

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.