A LIOUVILLE TYPE THEOREM FOR HARMONIC MORPHISMS

Abstract

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of nonpositive scalar curvature. Let μ0 be the least eigenvalue of the Laplacian acting on L2-functions on M . We show that if RicM ≥ −μ0 at all x ∈ M and either RicM > −μ0 at some point x0 or Vol(M) is infinite, then every harmonic morphism φ : M → N of finite energy is constant.