Complex-valued (p,q)-harmonic morphisms from Riemannian manifolds

We introduce the natural notion of (p,q)-harmonic morphisms between Riemannian manifolds. This unifies several theories that have been studied during the last decades. We then study the special case when the maps involved are complex-valued. For these we find a characterisation and provide new non-trivial examples in important cases.


Introduction
The history of harmonic morphisms can be traced back to the pioneering work [7] of Jacobi from 1848. He studies complex-valued functions pulling back harmonic functions in the complex plane C to harmonic functions in the 3-dimensional Euclidean space R 3 . The notion is then generalised to the Riemannian setting in the late 1970s, independently by Fuglede and Ishihara, see [2] and [6]. This has lead to a lively development that can be followed both in [1] and at the regularly up-dated on-line bibliography [5].
Loubeau and Ou study biharmonic morphisms between Riemannian manifolds, see [8] and [9]. These are maps pulling back biharmonic functions to biharmonic functions. In his work [10], Maeta introduces the notion of triharmonic morphisms. These are mappings pulling back triharmonic functions to triharmonic functions.
Recently, Ghandour and Ou introduce the notion of generalised harmonic morphisms between Riemannian manifolds, see [3] and [4]. These are maps pulling back harmonic functions to biharmonic functions. They also find a characterisation for these non-linear objects.
In this work we unify the above notions by defining the (p, q)-harmonic morphisms. These are maps between Riemannian manifolds pulling back q-harmonic functions to p-harmonic functions. Just as the harmonic morphisms and their above mentioned variants, they are solutions to an overdetermined system of non-linear partial differential equations. This means that they have no general existence theory. For this reason it is interesting to develop methods for constructing solutions in particular cases.
In this paper we focus our attention on complex-valued (p, q)-harmonic morphisms from Riemannian manifolds. The aim is to extend the known characterisation to this case and to manufacture new non-trivial examples to this non-linear problem. The explicit examples presented here involve rather demanding computations. They were all tested, by the computer algebra systems Maple and Mathematica, independently.

Preliminaries
Let (M, g) be an m-dimensional Riemannian manifold and T C M be the complexification of the tangent bundle T M of M . We extend the metric g to a complex-bilinear form on T C M . Then the gradient ∇z of a complexvalued function z : (M, g) → C is a section of T C M . In this situation, the well-known complex linear Laplace-Beltrami operator (alt. tension field) τ on (M, g) acts locally on z as follows For two complex-valued functions z, w : (M, g) → C we have the following well-known relation τ (z · w) = τ (z) · w + 2 · κ(z, w) + z · τ (w), (2.1) where the complex bilinear conformality operator κ is given by κ(z, w) = g(∇z, ∇w). Locally this satisfies κ(z, w) = m i,j=1 As a direct consequence of the complex linearity, bi-linearity of the operators τ and κ, respectively, we have the following. Lemma 2.1. Let (M, g) be a Riemannian manifold and z, w : (M, g) → C be two complex-valued functions. Then the tension field τ and the conformality operator κ satisfy τ (z) = τ (z) and κ(z, w) = κ(z,w). (2.2) We are now ready to define the complex-valued proper p-harmonic functions, the main objects of our study. Definition 2.2. For a positive integer p, the iterated Laplace-Beltrami operator τ p is given by τ 0 (z) = z and τ p (z) = τ (τ (p−1) (z)).
We now introduce the natural notion of a (p, q)-harmonic morphism. For (p, q) = (1, 1) this is the classical case of harmonic morphisms introduced by Fuglede and Ishihara, in [2] and [6], independently.
between Riemannian manifolds is said to be a (p, q)-harmonic morphism if, for any q-harmonic function As an immediate consequence of Definition 2.3 we have the following natural composition law.
Another useful consequence of Definition 2.3 is the following.

Complex-valued (2, q)-Harmonic Morphisms
Throughout this work we assume that z : (M, g) → C is a differentiable complex-valued function on a Riemannian manifold and that f : U → C is differentiable and defined on an open subset U of C containing the image z(M ) of z. Further let φ : (M, g) → C be the composition φ = f • z. For this situation we have the following result that later will be employed several times.
Lemma 3.1. Let z : (M, g) → C be a complex-valued function on a Riemannian manifold and F, G : U → C be differentiable functions defined on an open subset U of C containing the image z(M ) of z. Then the tension field τ and the conformality operator κ satisfy . and κ(F (z,z), G(z,z)) = ∂F ∂z · κ(z, G(z,z)) + ∂F ∂z · κ(z, G(z,z)).
Proof. For a point p ∈ M , let {X 1 , . . . , X m } be a local orthonormal frame around p such that ∇ X k X k = 0 at p. Then the conformality operator κ satisfies The statement for the tension field τ follows immediately from the following elementary calculations performed at the point p, where ∇ X k X k = 0.
As a direct consequence of Lemma 3.1, we now see that the tension field τ (φ) of the composition φ = f • z is given by  Since the function f is assumed to be harmonic we have It now follows from Lemma 2.1 and equation (3.1) that τ (φ) = 0 is equivalent to κ(z, z) = 0 and τ (z) = 0. Proof. The condition 1 < q implies from (3.1) that both κ(z, z) = 0 and κ(z,z) = 0 or equivalently that the function z is constant.
The next result is our fundamental tool for analysing the case of (2, q).
For later use, we now reformulate Lemma 3.4 and thereby show that the 2-tension field τ 2 (φ) of φ can be presented in terms of the different partial derivatives of f with coefficients determined by the functions z,z and their various tension fields.
Proof. The statement follows directly by inserting the following identities, and their conjugates, into the formula given in Lemma 3.4. For this see Lemma 2.1.
In their paper [4], the authors introduce the notion of generalised harmonic morphisms between Riemannian manifolds. These are exactly the (2, 1)-harmonic morphisms in the sense of our Definition 2.3. They give a characterisation of these objects between Riemannian manifolds. In general this is rather complicated, see Theorem 2.2 of [4]. In our context, of complex-valued functions, it is the following.
Since the function f is assumed to be harmonic we also have This means that the formulae for the 2-tension field τ 2 (φ), presented in Lemmas 3.1 and 3.5, simplify considerably. The statement is then a direct consequence of the latter.
Remark 3.7. In the case when the Riemannian manifold (M, g) is a surface, i.e. of dimension 2, then the horizontal conformality of φ : M → C and the Cauchy-Riemann equations imply harmonicity. That means that in this case no proper (2, 1)-harmonic morphisms do exist.
In their paper [4] the authors construct the following first known proper (2, 1)-harmonic morphism. This was basically the only known example before this current study. Then Furthermore they introduce several interesting general methods for constructing solutions to our non-linear (2, 1)-problem from Euclidean spaces. The following result is a direct consequence of Corollary 3.1. of [4]. Proof. It is a classical result that any such holomorphic function f is a (1, 1)-harmonic morphism. The statement then is a direct concequence of our Lemma 2.4.
The next result follows directly from Corollary 3.1 of [4]. It can now be proven in exactly the same way as Proposition 3.9. Remark 3.11. The reader should note that the word "proper" does not appear in Proposition 3.10. As we will see later, there is a good reason for this.
From the above calculations of the 2-tension field τ 2 (φ) we now have the following result in the case when (p, q) = (2, 2). This should be compared with Theorem 4.1 of [8] and Theorem 3.3 of [9]. The condition λ 2 τ (z) + dz grad λ 2 = 0 presented there, can in our case be expressed as Since the function f is assumed to be 2-harmonic we also have τ 2 (f ) = ∂ 4 f ∂z 2 ∂z 2 = 0. This means that the formulae for the 2-tension field τ 2 (φ), presented in Lemmas 3.1 and 3.5, simplify considerably. The statement is then an immediate consequence of the latter.
The next statement follows immediately from Proposition 3.2. of [4].
is a (2, 1)-harmonic morphism on the Riemannian product M × N .

New (2, 1)-harmonic morphisms
In this section we present several new proper complex-valued (2, 1)-harmonic morphisms locally defined on Euclidean R n . Example 4.2 shows that such objects can easily be constructed for any dimension n ≥ 4. ' Let φ : U → C be a function defined locally on an open subset U of R 2p \{0}. Then its dual function φ * is the compisition φ * = φ • i p : U → C.
is a proper (2, 1)-harmonic morphism if and only if the complex coefficients satisfy the relation 1 + b 2 4 + · · · + b 2 n = 0. The same applies to the dual function φ * = φ • i p in the case when n = 2p.
In the above Examples 4.2-4.5 we have seen that the constructed complexvalued (2, 1)-harmonic morphisms φ and its dual φ * are both proper. The next three examples show that this is not true in general, see Remark 3.11. Example 4.6. For complex numbers a, b, c, d ∈ C with a 2 + b 2 + c 2 + d 2 = 0, let φ : R 4 \ {0} → C be the proper (2, 1)-harmonic morphism Then its dual function φ * = φ • i 2 is the globally defined (1, 1)-harmonic morphism satisfying This is clearly a (2, 1)-harmonic morphism, but it is not proper.
Then φ is a proper (2, 1)-harmonic morphism and its dual φ * = φ • i 2 is the holomorphic function This is clearly a (2, 1)-harmonic morphism which is not proper.
Then φ is a proper (2, 1)-harmonic morphism and its dual satisfying is holomorphic and hence a (2, 1)-harmonic morphism, but not proper.

A Generalised Construction Method
The main purpose of this section is to prove Theorem 5.2 which is a wide generalisation of Proposition 3.13.  Proof. It follows from Lemma 5.1 that Φ is horizontally conformal i.e. κ(Φ, Φ) = 0. For the tension field τ (Φ) of Φ we have With this at hand, we can now calculate the 2-tension field τ 2 (Φ) of Φ as follows.

Complex-valued (3, q)-Harmonic Morphisms
In this section we present a formula for the 3-tension field τ 3 (φ), of the composition φ = f • z. It turns out that, just as in the case of (2, q), horizontal conformality, i.e. κ(z, z) = 0, is a necessary condition. Elementary but rather tedious calculations provide the following useful result.
Theorem 6.2. A complex-valued function z : (M, g) → C from a Riemannian manifold is a (3, 1)-harmonic morphism if and only if κ(z, z) = 0, Proof. The method used here is exactly the same as that we have employed in the proof of Theorem 3.6 employing the fact that f is harmonic i.e.
Our next result gives a characterisation in the complex-valued (3, 3)-case. This recovers a particular case of Corollary 6.2 of the interesting work [10] of Maeta. Proof. Here we use exactly the same method as above, utilising Lemma 6.1 and the fact that in this case we have In this section we investigate the p-tension field τ p (φ) of the composition φ = f • z and derive several consequences from the condition τ p (φ) = 0 i.e. of φ being p-harmonic. It turns out that τ p (φ) takes the following form where the coefficients c p jk : U → C are differentiable functions involving various tension fields and conformality operators of the functions z andz.
We have already presented the tension fields τ (φ), τ 2 (φ) and τ 3 (φ) of φ. When calculating the 4-tension field τ 4 (φ) a clear pattern comes to light. These calculations are far too extensive to be presented here. For the ptension field τ p (φ) we have the following result.
The coefficients c p jk are symmetric with respect to complex conjugation i.e. c p jk =c p kj and This leads to the following general result which should be compared with Theorems 3.2, 3.6, 3.12, 6.2, 6.4 and 6.5 above.
Proof. The function z : (M, g) → C is a (p, q)-harmonic morphism if and only if, for any q-harmonic function f : U → C defined on an open subset U of C containing the image z(M ) of z, the p-tension field τ p (φ) of the composition φ = f • z vanishes. Since the function f is assumed to be q-harmonic we know that τ q (f ) = ∂ 2q f ∂z q ∂z q = 0. According to Lemma 7.1 we also have τ p (z) = τ p (z) = 0 and κ(z, z) = κ(z,z) = 0.
If we now plug these indentities into the expression for τ p (φ) this simplifies considerably to where c p pp = 2 p · κ(z,z) p . Hard work then shows that the remaining coefficients satisfy The rest follows by the same method as applied in the proof of Theorem 3.12.
Remark 7.3. In the paper [10], Maeta presents his interesting Conjecture 7.6. In our language his statement is: "A (p, p)-harmonic morphism is characterized as a special horizontally weakly conformal 2p-harmonic map." In our Theorem 7.2 we study the special case of complex-valued (p, p)harmonic morphisms. We obtain a characterisation of these objects and show that they are both horizontally conformal and 2p-harmonic, as Maeta suggests. But additionally, they must satisfy several rather non-trivial conditions. They can therefore rightly be said to be "special horizontally weakly conformal 2p-harmonic maps".
8. The Inversion about the unit sphere S 2p−1 in R 2p In this section we investigate the inversion i p : R 2p \ {0} → R 2p \ {0} about the unit sphere S 2p−1 in R 2p . Then the map i p is horizontally conformal and p-harmonic.
The fact that the map i p is proper p-harmonic is a direct consequence of the following repeated application of Lemma 8.2. Then the tension field τ (φ) of φ satisfies τ (φ) = n(n − p) |x| n+2 · φ.
Proof. First we notice that ∂ ∂x j |x| n = n x j |x| n−2 .

Two Conjectures
We conclude this work with three conjectures that have come to our minds while working on this project. Conjecture 9.1. Let p ∈ Z + be a positive integer and i p = (F 1 , F 2 , . . . , 2p) : be the inversion about the unit sphere S 2p−1 in R 2p . Then z : R 2p \ {0} → C with z = a 1 F 1 + a 2 F 2 · · · + a 2p F 2p is a complex-valued (p, p)-harmonic morphism for any non-zero element a = (a 1 , a 2 , . . . , a 2p ) in C 2p .
Our rather extensive computer calculations show that this Conjecture 9.1 is true in the cases when p = 1, 2, 3, 4, but the statement seems to be difficult to prove in general.
No proper (2, 1)-harmonic morphism is known to exist from the three dimensional Euclidean spaces R 3 , not even locally. For this we have the following.

Acknowledgements
The first author would like to thank the Department of Mathematics at Lund University for its great hospitality during her time there as a postdoc.