Conitinuous leafwise harmonic functions on codimension one transversely isometric foliations

TL;DR

This paper proves that on certain foliated manifolds with a transverse Riemannian structure, all continuous leafwise harmonic functions must be constant along the leaves, revealing a rigidity property of such functions.

Contribution

It establishes a new rigidity result for continuous leafwise harmonic functions on codimension one foliations with transverse Riemannian structures.

Findings

All continuous leafwise harmonic functions are constant on leaves.

The result applies to foliations admitting a transverse dimension one Riemannian foliation.

Provides insight into the structure of harmonic functions in foliated manifolds.

Abstract

Let $\FF$ be a codimension one foliation on a closed manifold $M$ which admits a transverse dimension one Riemannian foliation. Then any continuous leafwise harmonic functions are shown to be constant on leaves.