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 φ : ( M n , g ) → ( N m , h )be a horizontally conformal submersion. We prove that if the horizontal distribution is integrable, then any minimal fiber of φ is volume‐stable. This result is an improved version of the main theorem in [15]. As a corollary, we obtain if φ is a submersive harmonic morphism whose fibers are totally geodesic, and the horizontal distribution is integrable, then any fiber of φ is volume‐stable and so such a map φ is energy‐stable if M is compact. We also show that if φ : ( M n , g ) → ( N m , 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 φ is a harmonic morphism.