CUBO A Mathematical Journal Vol.11, No¯ 01, (123–143). March 2009 Regular quaternionic functions and conformal mappings Alessandro Perotti 1 Department of Mathematics, University of Trento, Via Sommarive, 14, I–38050 Povo Trento, Italy. email: perotti@science.unitn.it ABSTRACT In this paper we study the action of conformal mappings of the quaternionic space on a class of regular functions of one quaternionic variable. We consider functions in the kernel of the Cauchy-Riemann operator D = 2 ( ∂ ∂z̄1 + j ∂ ∂z̄2 ) = ∂ ∂x0 + i ∂ ∂x1 + j ∂ ∂x2 − k ∂ ∂x3 , a variant of the Cauchy–Fueter operator. This choice is motivated by the strict relation existing between this type of regularity and holomorphicity w.r.t. the whole class of complex structures on H. For every imaginary unit p ∈ S2, let Jp be the corresponding complex structure on H. Let Holp(Ω, H) be the space of holomorphic maps from (Ω,Jp) to (H,Lp), where Lp is defined by left multiplication by p. Every element of Holp(Ω, H) is regular, but there exist regular functions that are not holomorphic for any p. These properties can be recognized by computing the energy quadric of a function. We show that the energy quadric is invariant w.r.t. three–dimensional rotations of H. As an application, we obtain that every rotation of the space H can be generated by biregular rotations, invertible regular functions with regular inverse. Moreover, we prove that the energy quadric of a regular function can always be diagonalized by means of a three–dimensional rotation. 1Work partially supported by MIUR (PRIN Project “Proprietà geometriche delle varietà reali e complesse") and GNSAGA of INdAM 124 Alessandro Perotti CUBO 11, 1 (2009) RESUMEN En este artículo estudiamos la acción de aplicaciones conforme del espacio de cuater- niones sobre la clase de funciones regulares de una variable cuaternionica. Nosotros consideramos funciones en el kernel del operador de Cauchy–Riemann D = 2 ( ∂ ∂z̄1 + j ∂ ∂z̄2 ) = ∂ ∂x0 + i ∂ ∂x1 + j ∂ ∂x2 − k ∂ ∂x3 , una variante del operador de Cauchy–Fueter. Esta elección es motivada por la relación estricta existente entre este tipo de regularidad y holomorficidad w.r.t. de la clase en- tera de estructuras complejas sobre H. Para todo imaginario unitario p ∈ S2, sea Jp la correspodiente estructura compleja sobre H. Sea Holp(Ω, H) el espacio de aplicaciones holomórficas de (Ω,Jp) a (H,Lp), donde Lp es definido por multiplicación a la izquierda por p. Todo elemento de Holp(Ω, H) es regular, pero existen funciones regulares que no son holomórficas para cualquer p. Estas propiedades pueden ser reconocidas mediante el cálculo de la energía cuadrica de una función. Nosotros mostramos que la energía cuadrica es invariante w.r.t. por rotaciones tres–dimensionales de H. Como aplicación, obtenemos que toda rotación del espacio H puede ser generada por rotaciones bi reg- ulares, funciones regulares invertibles con inversa regular. Además mostramos que la energía cuadrica de una función regular siempre puede ser diagonalizada por una rotación tres–dimensional. Key words and phrases: quaternionic regular functions, hyperholomorphic functions, conformal mappings, Möbius transformations. Math. Subj. Class.: Primary 30G35; Secondary 30A30 1 Introduction. The aim of this paper is to analyze the action of the conformal group of the one–point compacti- fication H∗ of H on a class of regular functions of one quaternionic variable. Let Ω be a smooth bounded domain in C2. Let H be the space of real quaternions q = x0 +ix1 +jx2 +kx3, where i,j,k denote the basic quaternions. We identify H with C 2 by means of the mapping that associates the quaternion q = z1 +z2j with the pair (z1,z2) = (x0 +ix1,x2 +ix3). We consider the class R(Ω) of left–regular (also called hyperholomorphic) functions f : Ω → H in the kernel of the Cauchy–Riemann operator D = 2 ( ∂ ∂z̄1 + j ∂ ∂z̄2 ) = ∂ ∂x0 + i ∂ ∂x1 + j ∂ ∂x2 − k ∂ ∂x3 . This differential operator is a variant of the original Cauchy–Riemann–Fueter operator (cf. for CUBO 11, 1 (2009) Regular quaternionic functions and conformal mappings 125 example [19] and [5, 5]) ∂ ∂x0 + i ∂ ∂x1 + j ∂ ∂x2 + k ∂ ∂x3 . Hyperholomorphic functions have been studied by many authors (see for instance [1, 7, 11, 12, 14, 17, 18]). Many of their properties can be easily deduced from known properties satisfied by Fueter–regular functions, since a function f is regular on Ω if and only if f ◦ γ is Fueter–regular on γ(Ω) = γ−1(Ω), where γ is the reflection of C2 defined by γ(z1,z2) = (z1, z̄2). However, regular functions in the space R(Ω) have some characteristics that are more intimately related to the theory of holomorphic functions of two complex variables. In particular, the space R(Ω) contains the spaces of holomorphic maps with respect to any constant complex structure. This is no longer true if we adopt the original definition of Fueter regularity (see Section 2 for more details). Let J1,J2 be the complex structures on the tangent bundle TH ≃ H defined by left multipli- cation by i and j. Let J∗ 1 ,J∗ 2 be the dual structures on the cotangent bundle T∗H ≃ H and set J∗ 3 = J∗ 1 J∗ 2 . For every complex structure Jp = p1J1 + p2J2 + p3J3 (p a imaginary unit in the unit sphere S2), let ∂p = 1 2 ( d + pJ∗ p ◦ d ) be the Cauchy–Riemann operator with respect to the structure Jp. Let us define Holp(Ω, H) = Ker ∂p, the space of holomorphic maps from (Ω,Jp) to (H,Lp), where Lp is the complex structure defined by left multiplication by p. Then every element of Holp(Ω, H) is regular. These subspaces do not fill the whole space of regular functions (cf. [13]). This result is a consequence of a criterion of Jp–holomorphicity, based on the concept of energy quadric of a regular function (cf. Section 3.2 for exact definitions). In Section 4 we come to conformal transformations. >From a theorem of Liouville, the general conformal mapping of H∗ is the composition of a sequence of translations, dilations, rotations and inversions. It can be written as a quaternionic Möbius transformation, i.e. a fractional linear map of the form LA(q) = (aq + b)(cq + d) −1, with A ∈ GL(2, H). For properties of these maps, see for example [2], [5]§6.2, [11] and [19] and the references cited in those papers. Given a function f ∈ C1(Ω) and a conformal transformation LA, let f A be the function fA(q) = (cγ(q) + d)−1 |cγ(q) + d|2 f(L′ γ(A) (q)), where L′ γ(A) (q) = γ ◦ LA ◦ γ(q). In Theorem 3, we prove that f is regular on Ω if and only if f A is regular on Ω′ = (L′ γ(A) ) −1 (Ω). Moreover, (fA)B = fAB for every A,B ∈ GL(2, H). The first property can be deduced from Theorem 6 of Sudbery [19] using the reflection γ. We are interested also in the action of conformal mappings on the energy quadric and on the holomorphicity properties of the maps. For a general conformal transformation LA, the energy 126 Alessandro Perotti CUBO 11, 1 (2009) and, a fortiori, the energy quadric of a regular function is not conserved. However, we show that three–dimensional rotations of H (those which fix the real numbers) conserve the energy quadric (for translations this it is a trivial fact). Let a ∈ H, a 6= 0. Let rota(q) = aqa −1 be the three–dimensional rotation of H defined by a. In Theorem 4, we prove that the function fa = rotγ(a) ◦ f ◦ rota is regular on Ωa = rot−1 a (Ω) if and only if f is regular on Ω. Moreover, the energy density of fa is E(fa) = E(f) ◦ rota and the matrix function M(f) (for f regular M(f) is the energy quadric, cf. Section 3) transforms in the following way M(fa) = Qa(M(f) ◦ rota)Q T a , where Qa ∈ SO(3) is the orthogonal matrix associated to the rotation rotγ(a) of the space R 3 = 〈i,j,k〉. This formula has many consequences. It allows to obtain (Corollary 3) that fa is Jp– holomorphic if and only if f is Jp′ –holomorphic, with p ′ = rot −1 γ(a) (p). Moreover, we get (Corollary 4) that the energy quadric of a regular function can always be diagonalized by means of a three– dimensional rotation. Finally, we obtain a biregularity result about rotations (Proposition 2 and Corollary 5). We prove that every three-dimensional rotation is the composition of (at most) two three-dimensional biregular rotations, and that every four-dimensional rotation is the composition of two biregular rotations. 2 Notations and definitions 2.1 Fueter regular functions We identify the space C2 with the set H of quaternions by means of the mapping that associates the pair (z1,z2) = (x0 +ix1,x2 +ix3) with the quaternion q = z1 +z2j = x0 +ix1 +jx2 +kx3 ∈ H. A quaternionic function f = f1 + f2j ∈ C 1 (Ω) is (left) regular (or hyperholomorphic) on Ω if Df = 2 ( ∂ ∂z̄1 + j ∂ ∂z̄2 ) = ∂f ∂x0 + i ∂f ∂x1 + j ∂f ∂x2 − k ∂f ∂x3 = 0 on Ω. We will denote by R(Ω) the space of regular functions on Ω. With respect to this definition of regularity, the space R(Ω) contains the identity mapping and every holomorphic mapping (f1,f2) on Ω (with respect to the standard complex structure) defines a regular function f = f1 + f2j. We recall some properties of regular functions, for which we refer to the papers of Sudbery[19], Shapiro and Vasilevski[17] and Nōno[12]: 1. The complex components are both holomorphic or both non–holomorphic. CUBO 11, 1 (2009) Regular quaternionic functions and conformal mappings 127 2. Every regular function is harmonic. 3. If Ω is pseudoconvex, every complex harmonic function is the complex component of a regular function on Ω. 4. The space R(Ω) of regular functions on Ω is a right H–module with integral representation formulas. 5. f is regular ⇔ ∂f1 ∂z̄1 = ∂f2 ∂z2 , ∂f1 ∂z̄2 = − ∂f2 ∂z1 . We note that a function f = f1 + f2j is regular on Ω if and only if its Jacobian matrix has the form J(f) = ( ∂(f1,f2, f̄1, f̄2) ∂(z1,z2, z̄1, z̄2) ) =   a1 −b̄2 −c̄2 −c1 a2 b̄1 c̄1 −c2 −c2 −c̄1 ā1 −b2 c1 −c̄2 ā2 b1   at every z ∈ Ω, where a = ( ∂f1 ∂z1 , ∂f2 ∂z1 ) , b = ( ∂f̄2 ∂z̄2 ,− ∂f̄1 ∂z̄2 ) , c = ( ∂f̄2 ∂z1 ,− ∂f̄1 ∂z1 ) = − ( ∂f1 ∂z̄2 , ∂f2 ∂z̄2 ) . We shall call a matrix of this form a regular matrix. Note that a regular matrix can have rank 0, 2, 3 or 4 but not rank 1. A definition equivalent to regularity has been given by Joyce[6] in the setting of hypercomplex manifolds. Joyce introduced the module of q–holomorphic functions on a hypercomplex manifold. A hypercomplex structure on the manifold H is given by the complex structures J1,J2 on TH ≃ H defined by left multiplication by i and j. Let J∗ 1 ,J∗ 2 be the dual structures on T∗H ≃ H. In complex coordinates    J∗ 1 dz1 = idz1, J ∗ 1 dz2 = idz2 J∗ 2 dz1 = −dz̄2, J ∗ 2 dz2 = dz̄1 J∗ 3 dz1 = idz̄2, J ∗ 3 dz2 = −idz̄1 where we make the choice J∗ 3 = J∗ 1 J∗ 2 , which is equivalent to J3 = −J1J2. In real coordinates, the action of the structures is the following    J1dx0 = −dx1, J1dx2 = −dx3, J2dx0 = −dx2, J2dx1 = dx3, J3dx0 = dx3, J3dx1 = dx2. A function f is regular if and only if f is q–holomorphic, i.e. df + iJ∗ 1 (df) + jJ∗ 2 (df) + kJ∗ 3 (df) = 0. In complex components f = f1 + f2j, we can rewrite the equations of regularity as ∂f1 = J ∗ 2 (∂f2). 128 Alessandro Perotti CUBO 11, 1 (2009) The original definition of regularity given by Fueter (cf. [19] or [5]) differs from that adopted here by a real coordinate reflection. Let γ be the transformation of C2 defined by γ(z1,z2) = (z1, z̄2). Then a C 1 function f is regular on the domain Ω if and only if f ◦ γ is Fueter–regular on γ(Ω) = γ−1(Ω), i.e. it satisfies the differential equation ( ∂ ∂x0 + i ∂ ∂x1 + j ∂ ∂x2 + k ∂ ∂x3 ) (f ◦ γ) = 0 on γ−1(Ω). 2.2 Biregular functions A quaternionic function f ∈ C1(Ω) is called biregular if f is invertible and f, f−1 are regular. If this property holds locally, f is called locally biregular. These functions were studied in [8], [9] and [15]. The class BR(Ω) of biregular functions is closed with respect to right multiplication by a non- zero quaternion, but it is not closed with respect to composition or sum: even if f + g is invertible and f,g ∈ BR(Ω), the sum can be not biregular. 2.2.0.1 Examples 1. Every biholomorphic map (f1,f2) on Ω defines a biregular function f = f1 + f2j. 2. The identity function is biregular on H. More generally, the affine functions f(q) = qa + b, a ∈ H∗, b ∈ H, are biregular on H. 3. f = z̄1 + z̄2j ∈ R(H), f −1 = f ∈ BR(H). 4. The function f = z1 + z2 + z̄1 + (z1 + z2 + z̄2)j is regular, but the inverse map f−1 = 1 3 (z1 + z2 + z̄1 − 2z̄2 + (z1 + z2 − 2z̄1 + z̄2)j) is not regular. Note that in this case the Jacobian determinant is negative. This cannot happen for a biregular function (cf. [15]). 2.3 Holomorphic functions w.r.t. a complex structure Jp Let Jp = p1J1 +p2J2 +p3J3 be the orthogonal complex structure on H defined by a unit imaginary quaternion p = p1i + p2j + p3k in the sphere S 2 = {p ∈ H | p2 = −1}. In particular, J1 is the standard complex structure of C2 ≃ H. Let Cp = 〈1,p〉 be the complex plane spanned by 1 and p and let Lp be the complex structure defined on T∗Cp ≃ Cp by left multiplication by p. If f = f 0 + if1 : Ω → C is a Jp–holomorphic CUBO 11, 1 (2009) Regular quaternionic functions and conformal mappings 129 function, i.e. df0 = J∗ p (df1) or, equivalently, df + iJ∗ p (df) = 0, then f defines a regular function f̃ = f0 + pf1 on Ω. We can identify f̃ with a holomorphic function f̃ : (Ω,Jp) → (Cp,Lp). We have Lp = Jγ(p), where γ(p) = p1i + p2j − p3k. More generally, we can consider the space of holomorphic maps from (Ω,Jp) to (H,Lp) Holp(Ω, H) = {f : Ω → H of class C 1 | ∂pf = 0 on Ω} = Ker ∂p where ∂p is the Cauchy–Riemann operator with respect to the structure Jp ∂p = 1 2 ( d + pJ∗ p ◦ d ) . These functions will be called Jp–holomorphic maps on Ω. For any positive orthonormal basis {1,p,q,pq} of H (p,q ∈ S2), let f = f1 + f2q be the decomposition of f with respect to the orthogonal sum H = Cp ⊕ (Cp)q. Let f1 = f 0 + pf1, f2 = f 2 + pf3, with f0,f1,f2,f3 the real components of f w.r.t. the basis {1,p,q,pq}. Then the equations of regularity can be rewritten in complex form as ∂pf1 = J ∗ q (∂pf2), where f2 = f 2 − pf3 and ∂p = 1 2 ( d − pJ∗ p ◦ d ) . Therefore every f ∈ Holp(Ω, H) is a regular function on Ω. Remark 1. 1. The identity map belongs to the space Holi(Ω, H)∩Holj(Ω, H) but not to Holk(Ω, H). 2. For every p ∈ S2, Hol−p(Ω, H) = Holp(Ω, H). 3. Every Cp–valued regular function is a Jp–holomorphic function. 4. If f ∈ Holp(Ω, H) ∩ Holq(Ω, H), with p 6= ±q, then f ∈ Holr(Ω, H) for every r = αp+βq ‖αp+βq‖ (α,β ∈ R) in the circle of S2 generated by p and q. If the almost complex structure Jp is not constant, i.e. not compatible with the hyperkähler structure of H, we get a similar result. Note that in this case the structure is not necessarily integrable. Let f ∈ C1(Ω) and assume that p = p(z) ∈ S2 varies continuously with z in Ω. We will say that p is f-equivariant if f(z) = f(z′) implies p(z) = p(z′) (z,z′ ∈ Ω). This property allows to define p∗ : f(Ω) → S2 such that p∗ ◦ f = p on Ω. In [15], the following result was proved. Proposition 1. If f ∈ C1(Ω) satisfies the equation ∂p(z)f = 1 2 [ df(z) + p(z)J∗ p(z) ◦ df(z) ] = 0 (1) at every z ∈ Ω, then f is a regular function on Ω. If, moreover, the structure p is f-equivariant and p∗ admits a continuous extension to an open set U ⊇ f(Ω), then f is a (pseudo)holomorphic map from (Ω,Jp) to (U,Lp∗ ). 130 Alessandro Perotti CUBO 11, 1 (2009) Example 1. f(z) = z̄1 + z 2 2 + z̄2j is regular on H. On Ω = H \ {z2 = 0} f is holomorphic w.r.t. the almost complex structure Jp, where p(z) = 1√ |z2|2+|z2|4 ( |z2| 2i + (Im z2)j − (Re z2)k ) . Note that p(z) can not be continued to H as a continuous map. Also the inverse map f−1(z) = z̄1 − z 2 2 + z̄2j is regular on H. Then f is biregular on H. But f is also (pseudo)biholomorphic on Ω: f(Ω) = Ω and f−1 : (Ω,Jp′ ) → (H,Lp′◦f) is holomorphic, where p′(z) = 1√ |z2|2+|z2|4 ( |z2| 2i − (Im z2)j + (Re z2)k ) . Note that Lp∗ = Lp◦f−1 = Jp′ at f(z) and Lp′◦f = Jp at z ∈ Ω. 3 A criterion for holomorphicity 3.1 Energy and regularity In [13] it was proved that on every domain Ω there exist regular functions that are not Jp- holomorphic for any p. A similar result was obtained by Chen and Li[3] for the larger class of q-maps between hyperkähler manifolds. The criterion for holomorphicity is based on an energy-minimizing property of holomorphic maps. The energy density (w.r.t. the euclidean metric) of a function f : Ω → H, of class C1(Ω), is given by E(f) = 1 2 ‖df‖2 = 1 2 tr(J(f)J(f) T ). After integration on Ω, we get the energy of f ∈ C1(Ω): EΩ(f) = 1 2 ∫ Ω E(f)dV. Using ideas from [10] and [3], it was proved in [13] that a regular function f ∈ C1(Ω) minimizes energy in the homotopy class constituted by maps u with u|∂Ω = f|∂Ω which are homotopic to f relative to ∂Ω: Now we introduce the Lichnerowicz invariants. Let A(f) = (aαβ) be the 3 × 3 matrix with entries the real functions aαβ = −〈Jα,f ∗Liβ 〉, where (i1, i2, i3) = (i,j,k). For f ∈ C 1 (Ω), we set AΩ(f) = ∫ Ω A(f) dV and MΩ(f) = 1 2 ((tr AΩ(f))I3 − AΩ(f)) , where I3 denotes the identity matrix. We recall the criterion for regularity and holomorphicity proved in [13]. CUBO 11, 1 (2009) Regular quaternionic functions and conformal mappings 131 Theorem 1. 1. MΩ(f) is a relative homotopy invariant of f. 2. f is regular on Ω if and only if EΩ(f) = tr MΩ(f). 3. If f ∈ R(Ω), then MΩ(f) is symmetric and positive semidefinite. 4. If f ∈ R(Ω), then f belongs to some space Holp(Ω, H) (for a constant structure Jp) if and only if det MΩ(f) = 0. 5. f ∈ Holp(Ω, H) if and only if Xp = (p1,p2,p3) is a unit vector in the kernel of MΩ(f). >From the criterion it can be seen that almost all regular functions are not holomorphic with respect to any constant complex structure Jp. Example 2. f = z̄1 + z2 + z̄2j is Jp-holomorphic, with p = 1√ 5 (i − 2k), since on the unit ball B (with normalized unit volume) EB(f) = 3 and MB(f) =   2 0 1 0 1 2 0 1 0 1 2   . Example 3. f = z1 + z2 + z̄1 + (z1 + z2 + z̄2)j is regular, but not holomorphic: EB(f) = 6 and MB(f) =   2 0 0 0 2 0 0 0 2   . Example 4. f = z̄1 + z̄2j is regular and has matrix MB(f) =   2 0 0 0 0 0 0 0 0   of rank one. This means that f ∈ Holj(H, H) ∩ Holk(H, H). Example 5. The identity mapping belongs to the space Holi(H, H) ∩ Holj(H, H) = ⋂ p∈ Holp(H, H). Example 6 (Nonlinear case). f = |z1| 2 − |z2| 2 + z̄1z̄2j has energy EB(f) = 2 on the unit ball. The matrix MB(f) is MB(f) =   4 3 0 0 0 1 3 0 0 0 1 3   . Therefore f is regular but not holomorphic w.r.t. any constant complex structure Jp. 132 Alessandro Perotti CUBO 11, 1 (2009) 3.2 The energy quadric In [15], a pointwise version of the criterion for holomorphicity was established. Theorem 2. Let Ω be connected and f ∈ C1(Ω). Consider the matrix of real functions on Ω M(f) = 1 2 ((tr A(f))I3 − A(f)) . 1. f is regular on Ω if and only if E(f) = tr M(f) at every point z ∈ Ω. 2. If f ∈ R(Ω), then M(f) is symmetric and positive semidefinite. 3. If f ∈ R(Ω), then det M(f) = 0 on Ω if and only if there exists an open, dense subset Ω ′ ⊆ Ω on which f satisfies equation (1) for some function p(z) : Ω′ → S2. Moreover, if det M(f) = 0 and p(z) is f-equivariant, p∗ ◦ f = p and p∗ extends continuously to an open set U ⊇ f(Ω), then f is a (pseudo)holomorphic map from (Ω′,Jp) to (U,Lp∗ ). Let a = ( ∂f1 ∂z1 , ∂f2 ∂z1 ) , b = ( ∂f̄2 ∂z̄2 ,− ∂f̄1 ∂z̄2 ) , c = ( ∂f̄2 ∂z1 ,− ∂f̄1 ∂z1 ) , d = − ( ∂f1 ∂z̄2 , ∂f2 ∂z̄2 ) . Then the energy density is given by E(f) = |a|2+|b|2+|c|2+|d|2. A lengthy but straightforward computation gives the following expression for the matrix M(f): M(f) =   |c|2 + |d|2 Im(〈d,a〉 − 〈c,b〉) Re(〈d,a〉 + 〈c,b〉) Im(〈c,a〉 − 〈d,b〉) 1 2 |a − b|2 + 1 2 |c − d|2 − Im(〈a,b〉 + 〈c,d〉) Re(〈c,a〉 + 〈d,b〉) − Im(〈a,b〉 − 〈c,d〉) 1 2 |a + b|2 + 1 2 |c − d|2   . Then E(f) = tr M(f) if and only if c = d, i.e. f is regular. In this case the matrix M(f) becomes M(f) =   2|c|2 Im〈c,a − b〉 Re〈c,a + b〉 Im〈c,a − b〉 1 2 |a − b|2 − Im〈a,b〉 Re〈c,a + b〉 − Im〈a,b〉 1 2 |a + b|2   . Definition 1. For a regular function f on Ω, the family of positive semi-definite quadrics with matrices {M(f)(z) |z ∈ Ω} will be called the energy quadric of f. Remark 2. If f is invertible, then every p(z) is f-equivariant. If p is a constant complex structure, then p is f-equivariant for every f. Remark 3. If f is (real) affine, M(f) is a constant matrix. If f is not affine, det M(f) = 0 on Ω does not imply that det MΩ(f) = 0, but Theorems 1 and 2 imply that the converse is true. CUBO 11, 1 (2009) Regular quaternionic functions and conformal mappings 133 Example 7. The function f(z) = z̄1 + z 2 2 + z̄2j is regular (also biregular, cf. Example 1) on H. We have E(f) = 2 + 4|z2| 2, M(f) = 2   1 − Im z2 Re z2 − Im z2 |z2| 2 0 Re z2 0 |z2| 2   . Then the energy quadric of f is singular on H. On the domain Ω′ = H \{z2 = 0}, where M(f) has maximum rank 2, the kernel of M(f) is spanned by the vector X = (|z2| 2, Im z2,− Re z2). Then f is Jp-holomorphic on Ω ′, with p(z) = 1√ |z2|2+|z2|4 ( |z2| 2i + (Im z2)j − (Re z2)k ) . On the unit ball B, EB(f) = 10 3 and the matrix MB(f) = ∫ B M(f)dV =   2 0 0 0 2 3 0 0 0 2 3   is non-singular. Therefore, f is not Jq-holomorphic for any constant complex structure Jq. Example 8. The function f = |z1| 2 − |z2| 2 + z̄1z̄2j introduced in Example 6 has energy density 3|z|2 and energy quadric with matrix M(f) =   2|z|2 0 0 0 1 2 |z|2 0 0 0 1 2 |z|2   . Therefore f is regular but not holomorphic w.r.t. any almost complex structure Jp. Note that det M(f) = 1 2 |z|6 vanishes only at the origin. In [15], it was shown that if f ∈ BR(Ω) is a biregular function, then there exists an open, dense subset Ω′ ⊆ Ω, and an almost complex structure p(z) on Ω′, such that f : (Ω′,Jp) → (f(Ω ′ ),Lp∗ ) is a holomorphic map, with holomorphic inverse f−1 : (f(Ω′),Jp′ ) → (Ω ′,Lp′◦f). Here p = p1i + p2j + p3k : Ω ′ → S2, p∗ = p ◦ f−1 and p′ = p1i + p2j − p3k. In particular, any such map f preserves orientation. 4 Regular functions and conformal mappings In this section we are going to analyze the action of the conformal group of H on regular functions. Some of the results we describe can be deduced from [19] Theorem 6 using the reflection γ(z1,z2) = 134 Alessandro Perotti CUBO 11, 1 (2009) (z1, z̄2) introduced in §2.1, but here we want to investigate also the action on the energy quadric and the holomorphicity properties of the maps. We recall some definitions and properties of conformal and orientation preserving mappings of the one–point compactification Ĥ of H, for which we refer to [2], [5]§6.2, [11] and [19] and to the references cited in those papers. The Dieudonné determinant of a quaternionic matrix A = [ a b c d ] is the real non–negative number detH(A) = √ |a|2|d|2 + |b|2|c|2 − 2Re(cābd̄). It satisfies Binet property detH(AB) = detH(A)detH(B) and a 2 × 2 matrix A is (left and right) invertible if and only if detHA 6= 0. Then we can consider the general linear group GL(2, H) = { A = [ a b c d ] quaternionic matrix of order 2 | detHA 6= 0 } . A theorem of Liouville tells that the general conformal transformation of H∗ is a quaternionic Möbius transformation, i.e. a fractional linear map of the form LA(q) = (aq + b)(cq + d) −1, for A ∈ GL(2, H). The matrix A is determined by LA up to a real scalar multiple. For every pair of matrices A,B ∈ GL(2, H), LA ◦ LB = LAB. We have also the alternative representation of conformal mappings L′ A (q) = (qc + d)−1(qa + b), detHĀ 6= 0. Theorem 3. Given a function f ∈ C1(Ω) and a conformal transformation LA(q) = (aq + b)(cq + d)−1, let fA be the function fA(q) = (cγ(q) + d)−1 |cγ(q) + d|2 f(L′ γ(A) (q)), where γ(A) = [ γ(a) γ(b) γ(c) γ(d) ] . Then f is regular on Ω if and only if fA is regular on Ω′ = (L′ γ(A) ) −1 (Ω). Moreover, (fA)B = fAB for every A,B ∈ GL(2, H). Proof. We deduce the first statement from the result of Sudbery (cf. [19] Theorem 6), since f ∈ R(Ω) iff F = f ◦ γ is Fueter–regular on γ(Ω). This last condition is equivalent to the Fueter– regularity of the transformed function FA(p) = (cp + d)−1 |cp + d|2 F(LA(p)) CUBO 11, 1 (2009) Regular quaternionic functions and conformal mappings 135 on (LA) −1 (γ(Ω)). Note that this function differs from the one given by Sudbery by a real constant factor. We then obtain that f is regular iff FA ◦ γ is regular. We have FA ◦ γ(q) = (cγ(q) + d)−1 |cγ(q) + d|2 f ◦ γ ◦ LA ◦ γ(q) = f A (q), since γ◦LA◦γ(q) = L ′ γ(A) (q). Now we come to the last statement of the theorem. Let B = [ a′ b′ c′ d′ ] and C = AB = [ a′′ b′′ c′′ d′′ ] . Then (fA)B(q) = (c′γ(q) + d′)−1 |c′γ(q) + d′|2 fA(L′ γ(B) (q)) (c′γ(q) + d′)−1 |c′γ(q) + d′|2 (cγ(L′ γ(B) (q)) + d)−1 |cγ(L′ γ(B) (q)) + d|2 f((L′ γ(A) ◦ L′ γ(B) )(q)) Let q′ = γ(q). The last statement of the theorem follows from the equalities L′ γ(A) ◦ L′ γ(B) = (γ ◦ LA ◦ γ) ◦ (γ ◦ LB ◦ γ) = γ ◦ LAB ◦ γ = L ′ γ(AB) and (c′q′ + d′) (cγ(L′ γ(B) (q)) + d) = (q′c′ + d′) ((q′c′ + d′)−1(q′a′ + b′)c̄ + d̄) (q′a′ + b′)c̄ + (q′c′ + d′)d̄ = q′(a′c̄ + c′d̄) + b′c̄ + d′d̄ c′′q′ + d′′ Remark 4. If t is a non–zero real number, ftA = t−3fA. Then fA depends only for a real scalar multiple on the matrix chosen to represent the conformal transformation LA. We can also restrict the choice of the matrix to the subgroup SL(2, H) = {A ∈ GL(2, H) | detH(A) = 1}. In this case, the same conformal transformation gives rise to two functions, fA and f−A = −fA. Every conformal transformation is the composition of a sequence of translations, dilations, rotations and inversions. In order to illustrate the preceding theorem, we now apply it to these basic cases. Example 9. The inversion q 7→ q−1 corresponds to the matrix A = [ 0 1 1 0 ] (up to a real scalar multiple) and transforms a regular f ∈ R(Ω) into finv(q) = γ(q)−1 |q|2 f(q−1), regular on Ω′ = {q ∈ H | q−1 ∈ Ω}. 136 Alessandro Perotti CUBO 11, 1 (2009) Example 10. In particular, the inverted function of the constant function f = 1 2π2 is the Cauchy– Fueter kernel for the module of regular functions G(q) = G(z1 + z2j) = 1 2π2 z̄1 − z̄2j |z|4 . Example 11. A translation q 7→ q + b corresponds to the matrix A = [ 1 b 0 1 ] . The transformed function is fA(q) = f(L′ γ(A) (q)) = f(q + γ(b)). Example 12. A dilation q 7→ aq, a 6= 0 real, has matrix A = [ a 0 0 1 ] . A function f transforms into fA(q) = f(qa). Example 13. Given two unit quaternions a,d ∈ H, the diagonal matrix A = [ a 0 0 d ] induces the four–dimensional rotation q 7→ aqd−1. Given a regular function f on Ω, the function fA(q) = d−1f(γ(d)−1qγ(a)) is regular on Ω′ = γ(d)Ωγ(a)−1. Example 14. The quaternionic Cayley transformation ψ(q) = (q + 1)(1 − q)−1 maps diffeomor- phically the unit ball B to the right half–space H+ = {q ∈ H | Re(q) > 0} (see [2] for geometric properties of ψ). It transforms regular functions f on H+ into fψ(q) = 23/2 (1 − γ(q))−1 |1 − γ(q)|2 f(ψ(q)), regular on B. The inverse mapping ψ−1(q) = (q − 1)(1 + q)−1 transforms f ∈ R(B) into fψ −1 (q) = 23/2 (1 + γ(q))−1 |1 + γ(q)|2 f(ψ−1(q)) ∈ R(H+). The factor 23/2 in the formulas has been chosen to get (fψ)ψ −1 = f. If we take the identity map, which is regular on H, as f, from Theorem 3 we get the following: Corollary 1. For every conformal transformation LA(q) = (aq + b)(cq + d) −1, the function (cγ(q) + d)−1 |cγ(q) + d|2 L′ γ(A) (q), is regular on {q ∈ H | cγ(q) + d 6= 0}. CUBO 11, 1 (2009) Regular quaternionic functions and conformal mappings 137 4.1 The quadric energy of rotated regular functions A unit quaternion d defines the three–dimensional rotation q 7→ rotd(q) := dqd −1, which gives rise to the function (cf. Example 13) fA(q) = d−1f(γ(d)−1qγ(d)), where A is the scalar matrix A = [ d 0 0 d ] . Taking d = γ(a)−1 and multiplying by γ(a)−1 on the right, we obtain the function fa = rotγ(a) ◦ f ◦ rota. From Theorem 3 we immediately get: Corollary 2. Let f ∈ C1(Ω) and let a ∈ H, a 6= 0. Let rota(q) = aqa −1 be the three–dimensional rotation of H defined by a. Then the function fa = rotγ(a) ◦ f ◦ rota is regular on Ωa = rot−1 a (Ω) = a−1Ωa if and only if f is regular on Ω. Remark 5. The rotated function fa has the following properties: 1. (fa)b = fab and (f + g)a = fa + ga. 2. (fa)a −1 = f. 3. f−a = fa. 4. If b ∈ H, then (fb)a = fa rotγ(a)(b). Now we analyze the action of rotations on the energy quadric. We obtain in this way a new proof of the preceding result and we get new holomorphicity properties of rotated regular functions. Theorem 4. Let f ∈ C1(Ω) and let a ∈ H, a 6= 0. Let fa = rotγ(a) ◦ f ◦ rota. Then the energy density of fa is E(fa) = E(f) ◦rota and the matrix function M(f) defined in Section 3 transforms in the following way M(fa) = Qa(M(f) ◦ rota)Q T a , where Qa is the orthogonal matrix in SO(3) associated to the rotation rotγ(a) of the space 〈i,j,k〉. Before coming to the theorem, we prove a simple result about holomorphicity of rotations. Lemma 1. For every p ∈ S2, the three-dimensional rotation rota(q) = aqa −1 is a holomorphic map from (H,Jγ(p)) to (H,Lrota(p)). Proof. Let B = {p,p′,pp′} be a positive orthonormal base of R3 = 〈i,j,k〉. Let Xp = (p1,p2,p3), Xp′ = (p ′ 1 ,p′ 2 ,p′ 3 ), Xr = (r1,r2,r3), with r = pp ′ = r1i + r2j + r3k. Given the transition matrix A 138 Alessandro Perotti CUBO 11, 1 (2009) with columns Xp,Xp′,Xr, the coordinates x ′ α (α = 1, 2, 3) of q = x0 + x1i + x2j + x3k w.r.t. B are given by the product (x′ 1 x′ 2 x′ 3 ) T = AT (x1 x2 x3) T . Then x′ 1 = ∑ α pαxα, x ′ 2 = ∑ α p′ α xα, x ′ 3 = ∑ α rαxα. >From this we get that the functions g1 = x0 +x ′ 1 rota(p) and g2 = x ′ 2 +x′ 3 rota(p) are holomorphic from (H,Jγ(p)) to (H,Lrota(p)), since Jγ(p)(dx0) = (p1J1 + p2J2 − p3J3)(dx0) = − ∑ α pαdxα = −dx ′ 1 and Jγ(p)(dx ′ 2 ) = ∑ α p′ α (p1J1 + p2J2 − p3J3)(dxα) = ∑ α pαp ′ α dx0 − (p2p ′ 3 − p3p ′ 2 )dx1 − (p3p ′ 1 − p1p ′ 3 )dx2 − (p1p ′ 2 − p2p ′ 1 )dx3 = −r1dx1 − r2dx2 − r3dx3 = −dx ′ 3 . The lemma now follows from the equality rota(q) = a(x0 + x ′ 1 p + x′ 2 p′ + x′ 3 r)a−1 = (x0 + x ′ 1 rota(p)) + (x ′ 2 + x′ 3 rota(p))rota(p ′ ) = g1 + g2 rota(p ′ ) If in the preceding lemma p is replaced by γ(p), we get that the map rota(q) is holomorphic also from (H,Jp) to (H,Lrota(γ(p))) = (H,Jp′ ), where p ′ = γ(rota(γ(p))) = γ(a) −1pγ(a) = rot−1 γ(a) (p). Replacing a with γ(a) we also get that rotγ(a) is holomorphic from (H,Lp′ ) = (H,Jrota(γ(p))) to (H,Lrotγ(a)(p′)) = (H,Lp). Then we can draw a commutative diagram with holomorphic vertical maps (H,Jp′ ) f // (H,Lp′ ) rotγ(a) �� (H,Jp) rota OO f a // (H,Lp) (2) Proof of Theorem 4. Let J be the real Jacobian matrix of f ◦rota. Then the real Jacobian matrix of fa is the product QaJ. It follows that E(f a ) = 1 2 tr(QaJJ TQT a ) = 1 2 tr(JJT ) = E(f ◦ rota). A similar computation gives E(f ◦ rota) = E(f) ◦ rota. For the second statement of the theorem, it is sufficient to prove the equality A(fa) = Qa(A(f) ◦ rota)Q T a , (3) CUBO 11, 1 (2009) Regular quaternionic functions and conformal mappings 139 for the matrix functions A(f) and A(fa) defined in Section 3, since then the matrices A(fa) and A(f) ◦ rota have the same trace and therefore Qa(M(f) ◦ rota)Q T a = 1 2 (tr A(f) ◦ rota) I3 − 1 2 A(fa) = 1 2 (tr A(fa)I3 − A(f a )) = M(fa). It remains to prove (3). Let p = p1i + p2j + p3k ∈ S 2 and p′ = rot−1 γ(a) (p). Let us define the p–holomorphic energy of f Ip(f) = 1 2 ‖df + Lp ◦ df ◦ Jp‖ 2 = 1 2 ‖df + pdf ◦ Jp‖ 2 = 2‖∂pf‖ 2. Then we obtain, as in [3], E(f) + 〈Jp,f ∗Lp〉 = 1 4 Ip(f). If X = (p1,p2,p3), then XA(fa)XT = ∑ α,β pαpβaαβ = −〈 ∑ α pαJα, (f a ) ∗ ∑ β pβLiβ 〉 = −〈Jp, (f a ) ∗Lp〉 = E(f a ) − 1 4 Ip(f a ). Now let X′ = (p′ 1 ,p′ 2 ,p′ 3 ) = XQa. A similar computation gives XQaA(f ◦ rota)Q T a XT = X′A(f ◦ rota)X ′T = E(f) ◦ rota − 1 4 Ip′ (f) ◦ rota. >From the first statement of the theorem and the arbitrariness of X, equation (3) is equivalent to the equality, for any p ∈ S2, of the holomorphic energies Ip′ (f) ◦ rota = Ip(f a ). (4) >From Lemma 1 (cf. diagram (2)) and rotational invariance of the norm we get 2Ip(f a ) = ‖dfa + Lp ◦ df a ◦ Jp‖ 2 = ‖rotγ(a) ◦ df ◦ drota + Lp ◦ rotγ(a) ◦ df ◦ drota ◦ Jp‖ 2 = ‖rotγ(a) ◦ df ◦ drota + rotγ(a) ◦ Lp′ ◦ df ◦ Jp′ ◦ drota‖ 2 = ‖df + Lp′ ◦ df ◦ Jp′‖ 2 ◦ rota = 2Ip′ (f) ◦ rota. Then the equality (4) is true and the theorem is proved. Corollary 3. Let f ∈ C1(Ω) and let a ∈ H, a 6= 0. Let fa = rotγ(a) ◦ f ◦ rota. Let Qa ∈ SO(3) be associated to the rotation rotγ(a) of the space 〈i,j,k〉. Then 1. f is regular on Ω if and only if fa is regular on Ωa = rot−1 a (Ω) = a−1Ωa. 140 Alessandro Perotti CUBO 11, 1 (2009) 2. fa is Jp–holomorphic if and only if f is Jp′ –holomorphic, with p ′ = rot −1 γ(a) (p). 3. If f ∈ C1(Ω), then (cf. Theorem 1) MΩa (f a ) = QaMΩ(f)Q T a . Proof. 1) From Theorem 4 we get that tr M(fa) = tr M(f) ◦ rota and E(f a ) = E(f) ◦ rota. The first statement follows from Theorem 2, which tells that f is regular iff E(f) = tr M(f). 2) It is an immediate consequence of equality (4), since a function is Jp–holomorphic iff its p–holomorphic energy vanishes. 3) It follows easily by integration of M(fa) on Ωa. Corollary 4. For every f ∈ R(Ω), there exists a ∈ H, a 6= 0, such that the matrices M(fa) and MΩa (f a ) are diagonal, with non–negative entries. Proof. It follows immediately from Theorems 4 and 2, since when f is regular M(f) is symmetric and positive semidefinite. Remark 6. For a general conformal transformation LA, the energy and, a fortiori, the energy quadric of a regular function is not conserved. For example, the constant function 1 has zero energy, while E(2π2G) 6= 0 and 1inv = 2π2G (cf. Example 10). The same happens for Jp–holomorphicity. For example, the identity function is in the spaces Holi(H) and Holj(H), while idinv(q) = γ(q)−1q−1 |q|2 ∈ R(H \ {0}) is not holomorphic w.r.t. any structure Jp. This can be seen by computing the energy quadric M(idinv). Since det M(idinv) = 32/|q|30 is always non–zero, it follows from Theorem 2 that idinv is not Jp–holomorphic, for any p (even non–constant). The rank of id inv is three, because its image is contained in the space 〈1, i,j〉, and the function can not have rank less than three, otherwise its quadric energy would have zero determinant (cf. [15] Theorem 7). A simpler example is given again by the function 1inv, since the energy quadric of the kernel G is M(G) = 2/|q|8I3. 4.2 Biregular rotations If in Theorem 4 and its corollaries we take as f the identity map we get the following: Proposition 2. For every a ∈ H, a 6= 0, the three–dimensional rotation rotγ(a)a is a bireg- ular function on H, with energy quadric M(rotγ(a)a) of rank 1. This means that rotγ(a)a is holomorphic w.r.t. a circle of structures p ∈ S2. More precisely, rotγ(a)a ∈ Holp(H) for every p ∈ 〈rotγ(a)(i),rotγ(a)(j)〉 ∩ S 2. CUBO 11, 1 (2009) Regular quaternionic functions and conformal mappings 141 Proof. We have rotγ(a)a = id a (cf. Theorem 4). Then M(rotγ(a)a) = QaM(id)Q T a = Qa   0 0 0 0 0 0 0 0 2  Q T a has rank 1. Its kernel gives the structures with respect to which the rotation is holomorphic. From Corollary 3(2), these structures are generated by rotγ(a)(i) and rotγ(a)(j), since id ∈ Holi(H) ∩ Holj(H). Biregularity follows from (γ(a)a)−1 = a−1γ(a−1), which implies the equality (ida)−1 = idγ(a −1 ) ∈ R(H). Remark 7. Not every rotation is a regular function, since the quaternion γ(a)a is a reduced quaternion, with fourth component zero. These quaternion numbers correspond to rotations of R 3 = 〈i,j,k〉 with axis orthogonal to the k axis. However, every quaternion is the product of two reduced quaternions and the map a 7→ γ(a)a is surjective from H to the space Hr of reduced quaternions. The surjectivity of a 7→ γ(a)a can be seen explicitly, or can be deduced from a property of the regular function idinv (cf. Remark 6). Its restriction to the unit sphere S3 is the map q 7→ γ(q̄)q̄ ∈ S3 ∩ Hr. It is surjective since id inv has rank three. Corollary 5. 1. The left–multiplication map la′ (q) = a ′q is biregular for every reduced quater- nion a′ = γ(a)a 6= 0. 2. Every three-dimensional rotation is the composition of two three-dimensional biregular rota- tions. 3. Every four-dimensional rotation is the composition of two biregular rotations. Proof. 1) la′ (q) = γ(a)aq = rotγ(a)a(q)(a −1γ(a)−1) has the same regularity and holomorphicity properties of rotγ(a)a, since R(Ω) is a right H–module for every Ω and ∂p(fb) = (∂pf)b for every f and every b ∈ H. 2) It follows from what has been said in the above remark: if c = a′b′, with a′ = γ(a)a, b′ = γ(b)b ∈ Hr, then rotc = rota′ ◦ rotb′ = rotγ(a)a ◦ rotγ(b)b. 3) A four–dimensional rotation rotc,d(q) = cqd −1, with |cd−1| = 1, can be decomposed as rotc,d(q) = cqc −1 (cd−1) = rotc(q) (cd −1 ) = (rota′ ◦ rotb′ )(q) (cd −1 ), where c = a′b′ as before. Let f(q) = rota′ (q) (cd −1 ) ∈ BR(H). Then rotc,d = f ◦ rotb′ . The pair of biregular functions in the corollary can be chosen in the same space Holp(H). This comes from Proposition 2, because the two great circles of complex structures in S2 coincide 142 Alessandro Perotti CUBO 11, 1 (2009) or intersect in two antipodal points defining a space Holp(H). 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