A nonexistence theorem of proper harmonic morphisms between hyperbolic spaces (Q1851076)

Harmonic morphisms between Riemannian manifolds are harmonic maps which preserve germs of harmonic functions. It is proved that there is no harmonic morphism from any hyperbolic space to a lower-dimensional hyperbolic space which is proper and of class \(C^2\) up to the boundary. The proof is quite short. Besides simple properties of smooth proper maps at the boundary of hyperbolic space, it uses only the harmonicity equation in coordinates and the maximum principle for subharmonic functions.