\(p\)-harmonic morphisms, minimal foliations, and rigidity of metrics (Q556170)

The paper contains a mixture of results about \(p\)-harmonic morphisms. First, there are two classification results, which are variations of their harmonic (i.e., \(p=2\)) counterparts: \(p\)-harmonic morphisms of twisted product type from complete simply connected manifolds are classified [generalizing \textit{M.~Svensson}, J. Lond. Math. Soc. (2) 68, 781--794 (2003; Zbl 1062.53020)], as well as polynomial and holomorphic \(p\)-harmonic morphisms [generalizing \textit{S.~Gudmundsson} and \textit{R.~Sigurdsson}, Potential Anal. 2, 295--298 (1993; Zbl 0783.58015)]. Moreover, \(p\)-harmonic functions (including the case \(p=1\)) whose level surfaces produce minimal foliations are characterized, generalizing results from [\textit{P.~Baird} and \textit{J.~Eells}, Lect. Notes Math. 894, 1--25 (1981; Zbl 0485.58008)]. It is also proven that if a complete conformally non-flat metric \(g=F^{-2}\sum_{i=1}^m dx_i^2\) on an open subset of \(\mathbb{R}^m\) admits one Riemannian or \(m{-}1\) minimal coordinate plane foliations, then it must be the hyperbolic metric up to a homothety.