Proper biharmonic maps and (2,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2,1)$$\end{document}-harmonic morphisms from some wild geometries

In this work we construct a variety of new complex-valued proper biharmonic maps and (2, 1)-harmonic morphisms on Riemannian manifolds with non-trivial geometry. These are solutions to a non-linear system of partial differential equations depending on the geometric data of the manifolds involved.


Introduction
The concept of a harmonic morphism φ : (M, g) → (N, h), between Riemannian manifolds, was introduced by Fuglede and Ishihara in the late 1970s independently, see [2] and [6].These are maps pulling back local real-valued harmonic functions on N to harmonic functions on M .These objects have an interesting connection with the geometry of the manifolds involved and have lead to vibrant research activities, as can be traced in the excellent work [1], by Baird and Wood, and the regularly updated online bibliography [5], maintained by the second author.
Recently, the notion was generalised to (p, q)-harmonic morphisms, pulling back real-valued q-harmonic functions on N to p-harmonic functions on M , see [3].The case of (2, 1) had earlier been studied in [4] under the name generalised harmonic morphisms.In [3], the authors characterise complexvalued (p, q)-harmonic morphisms φ : (M, g) → C in terms of a heavily non-linear system of partial differential equations.They also provide methods for producing explicit solutions in the case when the domain (M, g) is the m-dimensional Euclidean space.
The principal aim of this work is to extend the study to complex-valued (2, 1)-harmonic morphisms from Riemannian manifolds (M, g).We model our manifolds M as open subsets of R m , equipped with a Riemannian metric g of a particular form, see Section 3. We then investigate when the natural projection Φ : (M, g) → C onto the first two coordinates is horizontally conformal, harmonic and even biharmonic.This leads to a non-linear system of partial differential equations involving the geometric data on (M, g).We then find several explicit solutions and thereby construct metrics g turning the projection Φ into a proper biharmonic map and even a proper (2, 1)harmonic morphism.By this we construct the first known complex-valued (2, 1)-harmonic morphisms from Riemannian manifolds with non-trivial geometry.Since the problem is invariant under conformal changes of the metric on (N, h) this provides local solutions to any Riemann surface, see Proposition 7.2.

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 ∇φ of a complexvalued function φ : (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 φ as follows For two complex-valued functions φ, ψ : (M, g) → C we have the following well-known relation where the complex bilinear conformality operator κ is given by κ(φ, ψ) = g(∇φ, ∇ψ).Locally this satisfies We are now ready to define the complex-valued proper p-harmonic functions.

Some Rather Wild Geometries
The aim of this section is to describe a particular collection of Riemannian manifolds (M, g) investigated in this work.As far as we know they have not been studied before in the geometric literature.We present formulae for their sectional curvatures to show that the geometry is here far from being trivial.We then give two concrete examples that turn out to be useful later on.
For an open subset M of R m , let λ, λ 1 , . . ., λ m : M → R be C 3 -functions such that λ = λ 1 = λ 2 and equip the manifold M with the Riemannian metric g of the special form For the tangent bundle T M of (M, g) we have the following global orthonormal frame When appropriate, we shall by x = (x, y, x 3 , . . ., x m ) denote the canonical coordinates (x 1 , . . ., x m ) on R m and set where the subscript x j means the partial derivative with respect to the j-th coordinate function.A standard computation shows that for the sectional curvature K In particular, for the horizonal section X ∧ Y we have Let Φ : (R m , g) → C be the horizontally conformal submersion with dilation e λ : R m → R + .For the tangent bundle T M we have the following orthogonal decomposition T M = H ⊕ V into its horizontal and vertical subbundles H and V, respectively, where We now present two Riemannian manifolds (M, g) with non-trivial geometry.Later in this work, we then show that the complex-valued function Φ : (M, g) → C is proper biharmonic in these and other similar cases.
Then the sectional curvature function K of the manifold (M 3 , g) satisfies If we assume that A, B, θ ∈ Ω(M ), rather than A, B, θ ∈ R, then the geometry of (M, g) runs rather wild.The formulae for the sectional curvature K become far too extensive to be included in this work.For explicit proper biharmonic maps in that general case, see Example 5.2.
Then a standard computation shows that the sectional curvatures of (M, g) fulfill If we assume that A, B, Ψ ∈ Ω(M ), rather than A, B, Ψ ∈ R, then the formulae for the sectional curvature K become very complicated, including partial derivatives of these functions.For explicit proper (2, 1)-harmonic morphisms in that general case, see Example 7.1 4. The tension Fields τ (Φ) and τ 2 (Φ) Our first principal aim is to construct Riemannian manifolds (M, g), of the form introduced in Section 3, such that the horizontally conformal submersion Φ : (M, g) → C with is a proper biharmonic map.For this purpose we now want to determine the tension field τ (Φ) and the bitension field τ 2 (Φ) of Φ, respectively.Lemma 4.1.Let (M m , g) be a Riemannian manifold, as defined above, with the orthonormal basis {X 1 , . . ., X m } for the tangent bundle T M .Then its Levi-Civita connection satisfies Proof.The statement follows from the following computation with dilation e λ : M → R + .Then we have the following relation Proof.It follows from Lemma 4.1 and the fact that the differential dΦ satisfies dΦ(X With the next result we provide a formula for the tension field τ (Φ) of the horizontally conformal submersion Φ.
The tension field τ (Φ) of Φ is defined by the well-known formula For the first part, we have For the second part, we now employ Lemma 4.1 and yield The statement is now an immediate consequence of the above calculations.is harmonic and hence a harmonic morphism if and only if (f x , f y ) = 0.
Proof.This is an immediate consequence of Proposition 4.3 and the characterisation of harmonic morphisms, proven by Fuglede and Ishihara in [2] and [6], respectively.
After determining the tension field τ (Φ), we now turn our attention to the bitension field τ 2 (Φ).
With the next result we present a formula for the bitension field τ 2 (Φ) of the horizontally conformal submersion Φ.
Proof.The bitension field τ 2 (Φ) of the C 4 -map Φ is given by First we notice that for k = 1, 2 we have Differentiating once more gives For the second part of the bitension field τ 2 (Φ) of Φ we now yield We now easily obtain the stated result by adding the terms.
In the Section 4 we have derived explicit formulae for the tension fields τ (Φ) and τ 2 (Φ).This leads to a system of non-linear partial differential equations for the pair of functions (λ, f ).We are now interested in constructing Riemannian metrics g, on open subsets M of R m , turning the horizontally conformal submersion Φ : (M, g) → C into proper biharmonic maps i.e. finding explicit solutions (λ, f ) to the system Let us first consider the case when λ = α ∈ Ω(M ) i.e. independent of the two first coordinates x and y.Then the differential operators D 1 and D 2 simplify to As a first example we have the following which clearly gives solutions to the system under consideration.
It is clear that the choice of M and A, B, α, β ∈ Ω(M ) can lead to rather non-trivial geometries (M, g).
If we now assume that the function f : M → R is independent of the coordinate y then we have that The ordinary differential equation can easily be integrated to for some A, B ∈ Ω(M ).Integrating yet again, we finally obtain defined on the appropriate open subset M of R m and with β ∈ Ω(M ).From this we see that under the above mentioned assumptions and modulo the functions A, B, β ∈ Ω(M ), the solution is uniquely determined.This leads to the following.
Example 5.2.For an open subset M of R m and A, B, α, β, θ ∈ Ω(M ) let the functions λ, f : M → R be defined by If A = 0, then the associated horizontally conformal map Φ : M → C is proper biharmonic.
By the seperation of variables, one easily yields the next two families of solutions.
Example 5. 3.For an open subset M of R m and A, B, C, D, α, β ∈ Ω(M ) let the functions λ, f : M → R be defined by If A 2 + B 2 = 0 then the associated horizontally conformal map Φ : (M, g) → C is proper biharmonic.
We also yield the following examples without assuming the condition λ ∈ Ω(M ).A, B, C, D, α, β ∈ Ω(M ) let the functions λ, f : M → R be defined by If r = 0, then the associated horizontally conformal map Φ : (M, g) → C is proper biharmonic.
Example 5.7.For an open subset M of R m and A, B, r, α, β, θ ∈ Ω(M ) let the functions λ, f : M → R be defined by If r = 0, then the associated horizontally conformal map Φ : (M, g) → C is proper biharmonic.
Example 5.8.For an open subset M of R m and A, α, β, θ ∈ Ω(M ) let the functions λ, f : M → R be defined by Then the associated horizontally conformal map Φ : (M, g) → C is proper biharmonic.
Example 5.9.For an open subset M of R m and A, α, β, θ ∈ Ω(M ) let the functions λ, f : M → R be defined by Then the associated horizontally conformal map Φ : (M, g) → C is proper biharmonic if and only if A = 1.
Definition 6.3.Let λ, f : M → R be differentiable functions on an open subset M of R m with coordinates x = (x, y, x 3 , . . ., x m ).Then we define the non-linear partial differential operators D 3 , D 4 by Proof.It follows from Lemma 6.2 that τ (Φ 2 ) = 0 and Then the following computations provide the result.

Explicit (2, 1)-Harmonic Morphisms
In Sections 4 and 6, we have defined the partial differential operators D 1 , D 2 , D 3 and D 4 .We will now use these to construct explicit proper (2, 1)-harmonic morphisms Φ : (M, g) → C. We will then show how these can be employed to produce a large variety of concrete proper biharmonic maps.
Example 7.1.For an open subset M of R m and A, B, α, β ∈ Ω(M ) let the functions λ, f : M → R be defined by The next result is a reformulation of Proposition 3.9 of [3], see also Corollary 3.1 of [4].Together with Example 7.1 it is a useful tool for manufacturing a large variety of proper (2, 1)-harmonic morphisms (M, g) → N 2 , to Riemann surfaces on the non-trivial manifolds constructed there.
Proposition 7.2.Let (M, g) be a Riemannian manifold, N 2 be a Riemann surface and φ : M → C be a proper (2, 1)-harmonic morphism.Further, let F : U → N 2 be a non-constant holomorphic function defined on an open subset of C containing φ(M ).Then the composition F • φ : (M, g) → N 2 is a proper (2, 1)-harmonic morphism, in particular a proper biharmonic map.
Every complex-valued harmonic function, locally defined in the plane C, is the sum of a holomorphic and an anti-holomorphic one.This leads us to the next statement.The stated result now follows from these calculations.

Definition 4 . 5 .
For an open subset M of R m , let λ, f : M → R be differentiable functions on M with coordinates x = (x, y, x 3 , . . ., x m ).Then we define the non-linear partial differential operators D 1 , D 2 by

Example 5 . 1 .
For an open subset M of R m and A, B, α, β ∈ Ω(M ) let the functions λ, f : M → R be defined by

Example 5 . 4 .
0 then the associated horizontally conformal map Φ : (M, g) → C is proper biharmonic.For an open subset M of R m and A, B, C, D, α, β ∈ Ω(M ) let the functions λ, f : M → R be defined by

Example 5 . 5 .
For an open subset M of R m and

Proposition 7. 3 .
Let (M, g) be a Riemannian manifold and φ : M → C be a submersive (2, 1)-harmonic morphism.Further, let F, G : U → C be holomorphic functions defined on an open subset U of C containing φ(M ) and ψ = F + Ḡ.Then the composition ψ • φ : (M, g) → C is a biharmonic map.It is proper if and only if