P-Harmonic morphisms, minimal foliations, and rigidity of metrics

Abstract

We classify p-harmonic morphisms of twisted product type from complete simply connected manifolds and polynomial p-harmonic morphisms and holomorphic p-harmonic morphisms between Euclidean spaces. We also characterize those p-harmonic functions f:(M,g)→ R whose level hypersurfaces produce minimal foliations of ( M, g) generalizing Baird–Eells’ results on harmonic morphisms. Among applications, we show that Nil space ( R 3,g Nil ) and Sol space ( R 3,g Sol ) admit many 1-harmonic submersions and hence many foliations by minimal surfaces. We also prove that if a complete conformally flat non-flat metric g U = F −2∑ i=1 m d x i 2 on a connected open subset U of R m admits one Riemannian or m−1 minimal coordinate plane foliations, then ( U, g U ) must be hyperbolic space ( H m , x m −2∑ i=1 m d x i 2) up to a homothety.