r-Harmonic and complex isoparametric functions on the Lie groups Rm⋉Rn and Rm⋉H2n+1

Sigmundur Gudmundsson; Marko Sobak

doi:10.1007/s10455-020-09736-3

r-Harmonic and complex isoparametric functions on the Lie groups R^m⋉Rⁿ and R^m⋉H²ⁿ⁺¹

Sigmundur Gudmundsson, Marko Sobak

Lund University

Research output: Contribution to journal › Article › peer-review

Abstract

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

Original language	English
Pages (from-to)	477-496
Number of pages	20
Journal	Annals of Global Analysis and Geometry
Volume	58
Issue number	4
Early online date	2020 Sept 21
DOIs	https://doi.org/10.1007/s10455-020-09736-3
Publication status	Published - 2020 Nov 1

Subject classification (UKÄ)

Geometry

Free keywords

Biharmonic functions
Solvable Lie groups

Access to Document

10.1007/s10455-020-09736-3Licence: CC BY

Cite this

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abstract = "In this paper we introduce the notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper r-harmonic functions. We then apply this to construct the first known explicit proper r-harmonic functions on the Lie group semidirect products Rm⋉ Rn and Rm⋉ H 2n+1, where H 2n+1 denotes the classical (2 n+ 1) -dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.",

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N2 - In this paper we introduce the notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper r-harmonic functions. We then apply this to construct the first known explicit proper r-harmonic functions on the Lie group semidirect products Rm⋉ Rn and Rm⋉ H 2n+1, where H 2n+1 denotes the classical (2 n+ 1) -dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.

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

KW - Biharmonic functions

KW - Solvable Lie groups

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DO - 10.1007/s10455-020-09736-3

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