Stability and integrability of horizontally conformal maps and harmonic morphisms

In this article, we study stability of minimal fibers and integrability of horizontal distribution for horizontally conformal maps and harmonic morphisms. Let phi:(Mn,g)(Nm,h) be a horizontally conformal submersion. We prove that if the horizontal distribution is integrable, then any minimal fiber of phi is volume-stable. This result is an improved version of the main theorem in [15]. As a corollary, we obtain if phi is a submersive harmonic morphism whose fibers are totally geodesic, and the horizontal distribution is integrable, then any fiber of phi is volume-stable and so such a map phi is energy-stable if M is compact. We also show that if phi:(Mn,g)(Nm,h) is a horizontally conformal map from a compact Riemannian manifold M into an orientable Riemannian manifold N which is horizontally homothetic, and if the pull-back of the volume form of N is harmonic, then the horizontal distribution is integrable and phi is a harmonic morphism.