Harmonic morphisms and subharmonic functions

Let M be a complete Riemannian manifold and N a complete noncompact Riemannian manifold. Let ϕ : M → N be a surjective harmonic morphism. We prove that if N admits a subharmonic function with finite Dirichlet integral which is not harmonic, and ϕ has finite energy, then ϕ is a constant map. Similarly, if f is a subharmonic function on N which is not harmonic and such that | d f | is bounded, and if ∫ M | d ϕ | < ∞ , then ϕ is a constant map. We also show that if N m ( m ≥ 3 ) has at least two ends of infinite volume satisfying the Sobolev inequality or positivity of the first eigenvalue of the Laplacian, then there are no nonconstant surjective harmonic morphisms with finite energy. For p -harmonic morphisms, similar results hold.