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

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 $\mathbb{R}^m \ltimes \mathbb{R}^n$ and $\mathbb{R}^m \ltimes \mathrm{H}^{2n+1}$, 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.


Introduction
Biharmonic functions are important in physics. Aside from continuum mechanics and elasticity theory, the biharmonic equation also makes an appearance in twodimensional 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 [14].
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 [3]. The development of the very last years has changed this and can be traced e.g. in the following publications: [7], [8], [9], [10], [11], [12], [13]. There the authors develop methods for constructing explicit p-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 complexvalued proper p-harmonic functions on (M, g), see Section 3.
We then apply our new method to construct proper p-harmonic functions on the solvable Lie group semidirect products R m ⋉ R n and R m ⋉ H 2n+1 , where H 2n+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.

Preliminaries
Let (M, g) be a smooth manifold equipped with a Riemannian metric g. We complexify the tangent bundle T M 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 follows For two complex-valued functions φ, ψ : (M, g) → C we have the following wellknown product rule τ (φ · ψ) = τ (φ) · ψ + 2 κ(φ, ψ) + φ · τ (ψ), (2.1) where the conformality operator κ is given by κ(φ, ψ) = g(∇φ, ∇ψ).
Definition 2.1. Let p be a positive integer. Then a smooth complex-valued function φ : It should be noted that the harmonic functions are exactly the 1-harmonic and the biharmonic functions are the 2-harmonic ones. In some texts the p-harmonic functions are also called polyharmonic of order p. We also note that if a function is p-harmonic then it is also r-harmonic for any r ≥ p. Hence, one is usually interested in studying functions which are proper p-harmonic.

Complex Isoparametric Functions
A method for constructing proper p-harmonic functions on Riemannian manifolds has recently been developed in [13]. The p-harmonic functions that the authors consider are of the form f • φ, where f : U ⊂ C → C is a holomorphic function defined on an open set U containing φ(M ), and φ : (M, g) → C is an eigenfunction i.e. a smooth complex-valued function such that for some constants λ, µ ∈ C. The construction from [13] can be adapted to the more general setting when φ is a complex isoparametric function, as we will now demonstrate. Classically, isoparametric functions on Riemannian manifolds are defined as realvalued functions φ : (M, g) → R such that the tension field τ and the conformality operator κ satisfy τ (φ) = Φ • φ and κ(φ, φ) = Ψ • φ, for some smooth functions Φ, Ψ. These have been extensively studied due to their beautiful geometric properties, see e.g. [18]. As we are mainly interested in complexvalued functions, the following complex-valued analogue of the classical real-valued isoparametric functions will turn out to be useful. Definition 3.1. Let (M, g) be a Riemannian manifold. Then a smooth complexvalued function φ : M → C is said to be complex isoparametric on M if there exist holomorphic functions Φ, Ψ : U → C defined on some open set U ⊂ C containing φ(M ), such that the tension field τ and the conformality operator κ satisfy As in [13], the gist of constructing p-harmonic functions using isoparametric functions is to study compositions of the form f • φ, where φ : (M, g) → C is complex isoparametric and f : U → C is a holomorphic function defined on some open set U ⊂ C containing φ(M ). By the chain rule (2.2), the tension field of such compositions satisfies With this in hand, one can first obtain a harmonic function f 1 • φ by solving which, by the chain rule, reduces to the complex ordinary differential equation Continuing inductively, we assume that a proper (p − 1)-harmonic function of the form f p−1 •φ has already been constructed. We can then obtain a proper p-harmonic function f p • φ by solving the problem which again reduces to a complex ordinary differential equation, namely More explicitly, we get the following result.
where c ∈ C is non-zero. (2) If Ψ does not vanish identically, letÛ be an open simply connected subset of and define the holomorphic functions f p :Û → C for p ≥ 1 by where c = (c 1 , c 2 ) ∈ C 2 is non-zero. Then in both cases, the composition Remark 3.3. If the isoparametric function φ is real-valued, then one can weaken the assumptions in Theorem 3.2 by only requiring that Φ and Ψ are smooth functions of a real variable andÛ can be taken to be an interval. In the complex-valued case, the requirement thatÛ is simply connected is needed to ensure that the holomorphic antiderivatives are well-defined.
Upon applying Theorem 3.2 to the particular case when the isoparametric function φ is an eigenfunction on (M, g) satisfying (3.1), one sees that the composition which has already been obtained in the aforementioned paper [13].

The Semidirect Products
Low-dimensional Lie groups, particularly of dimension three and four, are of great importance in physics. Probably most notably such Lie groups are used as models of spacetime in the theory of general relativity. By Lie's third theorem, there is a bijection between real finite-dimensional Lie algebras and connected simply connected Lie groups of the same dimension. Conveniently, real Lie algebras of dimension less than or equal to six have been classified, see e.g. the recent book [17]. As a consequence, one also obtains classifications of low-dimensional connected simply connected Lie groups.
The study conducted in this paper was initially meant to be a study of p-harmonic functions on the four-dimensional connected simply connected Lie groups. For our purposes, only the Lie groups whose Lie algebras are indecomposable, i.e. not direct products of lower dimensional Lie algebras, are of interest. The reason for this is that the Lie groups whose Lie algebras are decomposable are themselves direct products of lower dimensional Lie groups, and the theory of p-harmonic functions on product manifolds is well-known, see e.g. [8].
According to the classification given for example in [15] and [4], all four-dimensional indecomposable real Lie algebras are semidirect products of one of the following types: where r n denotes the n-dimensional abelian algebra and h 2n+1 denotes the (2n + 1)dimensional Heisenberg algebra. We thus see that the corresponding four-dimensional connected simply connected Lie groups are semidirect products of the form where H 2n+1 is the (2n + 1)-dimensional Heisenberg group. This motivates our interest in studying the more general semidirect products R m ⋉ R n and R m ⋉ H 2n+1 .
In particular, we note that such semidirect products are automatically solvable and hence diffeomorphic to the vector space of the corresponding dimension.

The Semidirect Products
for some family A = (A k ) m k=1 of commuting matrices in gl n (R). Then the semidirect product R m ⋉ µ R n is by definition the smooth manifold R m × R n equipped with the Lie group operation Note that in this case, the family A completely determines the semidirect product R m ⋉ µ R n , so that it is natural to use the more suggestive notation R m ⋉ A R n .
The Lie group R m ⋉ A R n has a faithful linear representation as the matrix group of the representation is needed in general since µ may not be injective, but (parts of) this block may often be omitted. This linear representation induces a natural basis for the corresponding Lie algebra r m ⋉ A r n , namely that consisting of the elements where 1 ≤ k ≤ m, 1 ≤ i ≤ n, and e i are the canonical unit vectors in R n . We now define an inner product on the Lie algebra r m ⋉ A r n by requiring that this basis becomes orthonormal and we extend this inner product to a left-invariant Riemannian metric on the Lie group R m ⋉ A R n . A simple calculation shows that, in the coordinates (t, x) ∈ R m × R n , this metric is given by .
It is easy to see that if φ, ψ : (R m ⋉ A R n , g) → C are two complex-valued functions then the Laplace-Beltrami operator satisfies while the conformality operator is given by Throughout this section we let J 2n denote the block diagonal 2n × 2n matrix defined by Note that this is the standard complex structure on R 2n ∼ = C n . We recall that the Heisenberg group H 2n+1 can be seen as the space R × R 2n equipped with the Lie group operation (ξ, x) ⊞ (η, y) = (ξ + η + 1 2 J 2n x, y , x + y). Its Lie algebra h 2n+1 is nilpotent and has one-dimensional center. More explicitly for a basis {Ξ, X 1 , . . . , X 2n } of h 2n+1 the non-zero Lie brackets are given by In what follows, we consider semidirect products R m ⋉μ H 2n+1 with respect to a specific class of homomorphismsμ : R m → Aut(H 2n+1 ). The reader interested in more details about this can find them in [16]. Letμ : where a : R m → R and µ : R m → GL 2n (R) are given by where A (i,j) ∈ R 2×2 are such that Tr A (i,i) = a for 1 ≤ i ≤ n, and adj(A (i,j) ) denotes the adjugate i.e. the transpose of the cofactor matrix of A (i,j) . In this situation, the Lie group semidirect product R m ⋉μ H 2n+1 is the manifold R m × R × R 2n equipped with the Lie group operation where (t, ξ, x), (s, η, y) ∈ R m × R × R 2n . As in the previous section, such semidirect products are completely determined by the family A, so that it is natural to use the more suggestive notation R m ⋉ A H 2n+1 . A faithful linear representation of the Lie group R m ⋉ A H 2n+1 is given by the matrix group where as before (D k ) ij = δ ik δ jk . Just as in the previous section, the m-dimensional block Exp m k=1 D k t k of the representation is needed in general, but (parts of) it can often be removed. We equip the Lie group R m ⋉ A H 2n+1 with the left-invariant Riemannian metric induced by the inner product on its Lie algebra r m ⋉ A h 2n+1 defined by requiring that the basis An elementary but long calculation shows that this metric can be expressed in the coordinates (t, ξ, x) ∈ R m × R × R 2n as With this in hand, it is easy to see that if φ, ψ : (R m ⋉ A H 2n+1 , g) → C are two complex-valued functions, then the Laplace-Beltrami operator satisfies and the conformality operator satisfies

p-Harmonic
Functions on R m ⋉ R n and R m ⋉ H 2n+1 As we have seen in the preceding sections, there are a lot of similarities between the Laplace-Beltrami operators and the conformality operators on the Lie groups R m ⋉ R n and R m ⋉ H 2n+1 so that here we will study these in parallel. To simplify the statements of our results we fix the following notation.
Notation 5.1. Let G be a Lie group, either R n or H 2n+1 . If G = R n the indices i, j are assumed to satisfy 1 ≤ i, j ≤ n and in the case G = H 2n+1 the condition 1 ≤ i, j ≤ 2n. In both cases we have 1 ≤ k ≤ m, unless otherwise specified. Throughout this section A = (A k ) k will denote a commuting family of inveritable real matrices which are of dimensions n × n if G = R n and of dimensions 2n × 2n if G = H 2n+1 . In the latter case, we also assume that each member of the family A is of the form (4.3). Finally, we will use the following notation and the conformality operator is given by regardless of the choice of the Lie group G.
Our first result is now the following one of general variable separation.
Proposition 5.2. Let φ, ψ : R m ⋉ A G → C be two complex-valued functions such that φ depends only on t ∈ R m while τ α (ψ) is independent of t for all α ≥ 0. Then the tension field of their product φ · ψ satisfies the identity In particular, if φ and ψ are proper p-harmonic and proper r-harmonic on R m ⋉ A G, respectively, then their product φ · ψ is proper (p + r − 1)-harmonic on R m ⋉ A G.

Remark 5.3.
This result is reminiscent of the variable separation statement on product manifolds, which can be found in Lemma 6.1 of [8]. For our result here we need the additional assumption that τ α (ψ) is independent of t for all α ≥ 0. This assumption is superfluous for direct products of manifolds, but it is essential in our cases.
Proof. Since φ is a function of t, we see by the formulae (4.1) and (4.4) for the Laplace-Beltrami operator, that τ β (φ) remains to be a function of t for all β ≥ 0. Since by assumption, the tension field τ α (ψ) is independent of t for all α ≥ 0, it follows from the relations (4.2) and (4.5) for the conformality operator, that κ(τ β (φ), τ α (ψ)) = 0, for all α, β ≥ 0. The first part of our statement now follows easily by induction combined with the product rule (2.1) for the Laplace-Beltrami operator. For the final claim of the result notice that the identity we have just proven implies that so that the product φ · ψ is indeed proper (p + r − 1)-harmonic.
Constructing proper p-harmonic functions ψ : R m ⋉ A G → C such that τ α (ψ) is independent of t for α ≥ 0 seems difficult in general, if not impossible. However, we note that any harmonic function on R m ⋉ A G which is independent of t trivially satisfies this condition. On the other hand, we can use our main Theorem 3.2 to construct proper p-harmonic functions depending only on t. We summarize these thoughts in the following statement.

Proposition 5.4.
(1) Define the complex-valued function ψ : where a, b, v i , B ij ∈ C are not all zero and the coefficients B ij form a symmetric matrix such that Then ψ is proper harmonic on R m ⋉ A G.
where c = (c 1 , c 2 ) ∈ C 2 is non-zero. Then for each k = 1, . . . , m the function φ k is proper p k -harmonic on R m ⋉ A G. Furthermore, their product Proof. To prove part (i) we note that since B is symmetric we have confirming the result for both choices of the Lie group G.
For part (ii) we note that τ (t k ) = −ω k and κ(t k , t k ) = 1, so that for each k = 1, . . . , m the coordinate function t k is isoparametric. The first claim then follows from Theorem 3.2 after an easy calculation. The second claim follows by noting that which can be proven analogously to Proposition 5.2.
The matrix B in part (i) of our Proposition 5.4 can of course be taken to be zero.
The following examples show though that in important cases this is not necessary.
Example 5.5. Consider the four-dimensional Lie group This is the semidirect product R ⋉ A R 3 with respect to the family A consisting of the single matrix   Then following Proposition 5.2, combined with Proposition 5.4, the function ) is proper p-harmonic on G 4.4 for any non-zero elements c ∈ C 2 and a ∈ C 4 . △ Example 5.6. For α ∈ [−1, 1] consider the following four-dimensional Lie group This is the semidirect product R ⋉ A H 3 with respect to the family A consisting of the single matrix 1 0 0 α .
Then Proposition 5.2, in conjunction with Proposition 5.4, implies that the function defined by is proper p-harmonic on G α 4.8 for any non-zero elements c ∈ C 2 and a ∈ C 5 . △ We now proceed by constructing a non-trivial class of complex isoparametric functions on R m ⋉ A G. These functions will depend only on the variables t and x i.e. they will be independent of the variable ξ if G = H 2n+1 . Here we will use the well-known fact that the elements of a commuting family of matrices always possess a common eigenvector.
Proposition 5.7. Let v be a common eigenvector of the commuting family A T = (A T k ) k and λ = (λ 1 , . . . , λ m ) be the vector consisting of the corresponding eigenvalues. Then the complex-valued function φ : as well as where the final equality follows from the fact that v is an eigenvector of µ(t) T = Exp k A T k t k with eigenvalue e λ,t by assumption.
Having constructed complex isoparametric functions, we can now apply Theorem 3.2 to generate examples of p-harmonic functions. As one might suspect, the antiderivatives from Theorem 3.2 can not be computed explicitly in general. In what follows, we consider several specific cases where this is possible.
Example 5.8. The four-dimensional Lie group All the eigenvalues of A T are 0 and the eigenspace is the one-dimensional space spanned by the unit vector e 3 . Thus, the function φ from Proposition 5.7 is simply the coordinate function (t, x) → x 3 satisfying τ (φ) = 0 and κ(φ, φ) = 1.
A simple application of Theorem 3.2 then shows that the function is proper p-harmonic on G 4.1 . By construction this function also satisfies the condition from Proposition 5.2 in that its Laplacian of any order is independent of t. Hence, if p, r, q are positive integers such that r+q −1 = p, Proposition 5.2 combined with part (ii) Proposition 5.4 implies that the function defined by is proper p-harmonic on G 4.1 for any non-zero a, c ∈ C 2 . △ Example 5.9. The celebrated Thurston geometry Sol 3 is the solvable Lie group Examples of proper p-harmonic functions on this Lie group have already been constructed in [8] and [12]. We can view this Lie group as the semidirect product The eigenvalues of A T are 1 and −1, and the corresponding eigenvectors are e 1 and e 2 , respectively. In view of Proposition 5.7 we get an isoparametric function for each eigenvector, namely both of which satisfy τ (φ i ) = φ i and κ(φ i , φ i ) = φ 2 i + 1. Applying Theorem 3.2 to these isoparametric functions, we find that the functions are proper p-harmonic on Sol 3 for any non-zero c ∈ C 2 . △ The reader should note that in both Examples 5.8 and 5.9 the eigenvector v is non-isotropic i.e. v, v = 0. This is of course expected since the eigenvector is real in both cases. However, if the eigenvector v turns out to be complex, then it could happen that it is isotropic. In this case we see directly that the function φ from Proposition 5.7 becomes an eigenfunction cf. (3.1). In fact, a quick inspection of the proof of Proposition 5.7 shows that in this case the vector λ can be taken arbitrarily, rather than requiring it to consist of eigenvalues.
Corollary 5.10. Let v be a common eigenvector of the commuting family A T = (A T k ) k and suppose that v is isotropic. Then for any complex vector ν ∈ C m the function φ : is an eigenfunction on R m ⋉ A G satisfying τ (φ) = ν, ν + ω φ and κ(φ, φ) = ν, ν φ 2 .
For this situation the antiderivatives from Theorem 3.2 have already been computed in [13], cf. also (3.2).
Example 5.11. Consider the following interesting four-dimensional Lie group This is the semidirect product R 2 ⋉ A R 2 , where the family A consists of the two commuting matrices Their two common eigenvectors are v ± = (1, ±i), both of which are isotropic. Hence, we see from Corollary 5.10 that for any ν = (ν 1 , ν 2 ) ∈ C 2 , the functions are eigenfunctions on G 4.10 with In the case when v is isotropic, we can take the real and the imaginary part of the function φ from Proposition 5.7 to obtain even more isoparametric functions.
Proposition 5.12. Let v be a common eigenvector of the commuting family A T = (A T k ) k and let λ = (λ 1 , . . . , λ m ) be the vector consisting of the corresponding eigenvalues. Further suppose that the eigenvector v is isotropic and define the function x . Then the real part φ 1 = Re φ and the imaginary part φ 2 = Im φ of φ are isoparametric functions on R m ⋉ A G with Proof. The result follows by direct calculations and Proposition 5.7.
Example 5.13. For α ≥ 0, consider the four-dimensional Lie group G α 4.9 given by  e αt x 1 cos(t)−x 2 sin(t) 2 e αt ξ 0 0 e αt cos(t) e αt sin(t) We can view this group as the semidirect product R ⋉ A H 3 with respect to the family A consisting of the single matrix The two eigenvectors of A T are v ± = (1, ±i) and the corresponding eigenvalues are λ ± = α ± i. Note that the eigenvectors v ± are isotropic. In what follows, it suffices to only consider the eigenvector v = (1, i) and its eigenvalue λ = α+i. The functions φ 1 and φ 2 from Proposition 5.12 are then given by satisfying τ (φ i ) = 3α 2 φ i and κ(φ i , φ i ) = α 2 φ 2 i + 1. We now apply Theorem 3.2 to these isoparametric functions.
In the case when α = 0 an easy calculation of the antiderivatives from Theorem 3.2 shows that for any positive integer p and any non-zero c ∈ C 2 , the functions are proper p-harmonic on G 0 4.9 . By construction, these functions also satisfy the condition from Proposition 5.2 so we may also multiply them by a function depending only on t to obtain even more examples of proper p-harmonic functions cf. Example 5.8.
In the case when α = 0 the functions f p from Theorem 3.2 do not seem to possess a nice closed formula. However, let us at least note that for any non-zero c ∈ C 2 the functions given by