Abstract
AbstractIn 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 $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$
R
m
⋉
R
n
and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$
R
m
⋉
H
2
n
+
1
, where $$\mathrm {H}^{2n+1}$$
H
2
n
+
1
denotes the classical $$(2n+1)$$
(
2
n
+
1
)
-dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.
Publisher
Springer Science and Business Media LLC
Subject
Geometry and Topology,Analysis
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