r -Harmonic and Complex Isoparametric Functions on the Lie Groups $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ R
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r‑Harmonic and Complex Isoparametric Functions on the Lie Groups ℝm ⋉ ℝn and ℝm ⋉ H2n+1 Sigmundur Gudmundsson1 · Marko Sobak1 Received: 6 April 2020 / Accepted: 2 September 2020 / Published online: 21 September 2020 © The Author(s) 2020
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 ⋉ 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. Keywords Biharmonic functions · Solvable Lie groups Mathematics Subject Classification 31B30 · 53C43 · 58E20
1 Introduction Biharmonic functions are important in physics. Aside from continuum mechanics and elasticity theory, the biharmonic equation also makes an appearance in two-dimensional hydrodynamics problems involving Stokes flows of incompressible Newtonian fluids. A comprehensive review of this fascinating history of biharmonic functions can be found in the article [11]. On this subject, the literature is vast. With only very few exceptions, the domains are either surfaces or open subsets of flat Euclidean space, see, for example, [2]. The development of the very last years has changed this and can be traced, e.g., in the following publications: [4–10]. There the authors develop methods for constructing explicit r-harmonic functions on the classical Lie groups and even some symmetric spaces. In this paper, we introduce the notion of complex isoparametric functions on a Riemannian manifold (M, g), see Definition 3.1. It turns out that together with the so-called eigenfunctions they provide us with a method for manufacturing complex-valued proper r-harmonic functions on (M, g), see Sect. 3. * Sigmundur Gudmundsson [email protected] Marko Sobak [email protected] 1
Mathematics, Faculty of Science, University of Lund, Box 118, 221 Lund, Sweden
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Annals of Global Analysis and Geometry (2020) 58:477–496
We then apply our new method to construct proper r-harmonic functions on the solvable Lie group semidirect products ℝm ⋉ ℝn and ℝm ⋉ H2n+1 , where H2n+1 denotes the classical (2n + 1)-dimensional Heisenberg group. The study of these particular Lie groups is motivated by the fact that all four-dimensional irreducible Lie groups are, up to isomorphism, semidirect products of one of these two types.
2 Preliminaries Let (M, g) be a smooth manifold equipped with a Riemannian metric g. We complexify the tangent bundle TM of M to T ℂ M and extend the metric g to a complex bilinear form on T ℂ M . Then, the gradient ∇𝜙 of a complex-valued function 𝜙 ∶ (M, g) → ℂ is a section of T ℂ M . In this situation, the well-known linear Laplace--Beltrami operator (alt. tension field) 𝜏 on (M, g) acts locally on 𝜙 as
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