Liouville type theorem for transversally harmonic maps

Let (M, F) be a complete foliated Riemannian manifold and all leaves be compact. Let (M', F') be a foliated Riemannian manifold of non-positive transversal sectional curvature. Assume that the transversal Ricci curvature Ric(Q) of M satisfies Ric(Q) >= -lambda(0) at all point x is an element of M and Ric(Q) > -lambda(0) at some point x(0), where lambda(0) is the infimum of the spectrum of the basic Laplacian acting on L-2-basic functions on M. Then every transversally harmonic map phi: M -> M' of finite transversal energy is transversally constant.

Automated summary

This paper investigates the relationship between complete foliated Riemannian manifolds and foliated Riemannian manifolds with non-positive transversal sectional curvature. The main result states that under certain conditions on the transversal Ricci curvature, every transversally harmonic map of finite transversal energy is transversally constant.