p-Harmonic and Complex Isoparametric Functions on the Lie Groups ℝ^m \ltimes ℝⁿ and ℝ^m \ltimes H²ⁿ⁺¹

2020/03/23 by Gudmundsson, Sigmundur, Sobak, Marko
#31B30 #53C43 #58E20 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2003.10194

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 ℝ^m \ltimes ℝⁿ and ℝ^m \ltimes H²ⁿ⁺¹, where H²ⁿ⁺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.

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