$p$-Harmonic and Complex Isoparametric Functions on the Lie Groups ${\mathbb{R}}^m⋉{\mathbb{R}}^n$ and ${\mathbb{R}}^m⋉{\mathrm{H}}^{2n+1}$

Abstract

In this paper we introduce the new notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper $p$-harmonic functions. We then apply this to construct the first known explicit proper $p$-harmonic functions on the Lie group semidirect products ${\mathbb{R}}^m⋉{\mathbb{R}}^n$ and ${\mathbb{R}}^m⋉{\mathrm{H}}^{2n+1}$, where ${\mathrm{H}}^{2n+1}$ denotes the classical $(2n+1)$-dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.