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328 K. Gürlebeck and J. Morais

with values in the reduced quaternions. A serious study of the geometric prop-

erties of monogenic functions requires to consider at ﬁrst the local behavior of such

mappings. This is the main goal of the paper.

The extension of theoretical and practical conformal mapping methods in com-

plex analysis to the higher-dimensional real Euclidean space, particularly in the set-

ting of quaternionic analysis, has originated many questions. Several attempts have

been made to study monogenic functions in R

n+1

by a corresponding differentia-

bility concept or by the existence of a well-deﬁned hypercomplex derivative (see,

e.g., [6, 7, 12, 16, 20]). However, it is not commonly realized that monogenic func-

tions can be characterized via a generalized conformality concept. It is shown by

examples in [4] and later on in [3] that these functions can preserve some of the ge-

ometrical properties like length, distance, or special angles while mapping special

domains onto the ball. Both papers [3, 4] are mainly concerned with generaliza-

tions of the Bergman reproducing kernel approach to numerical mapping problems

analogously to the complex Bergman kernel method of constructing the conformal

mapping from a domain onto the disk.

At this point it is important to remark that, in contrast to the planar case, in spaces

n+1

of dimension n ≥ 2 the set of conformal mappings is restricted to the set of

Möbius tranformations only, at ﬁrst shown by Liouville in 1850 [15] for the three-

dimensional case. We state the result:

Theorem 1 (Liouville’s theorem) Let Ω ⊂ H. A C

-function f :Ω →H is a con-

formal mapping if and only if f is a Möbius transformation.

It is well known that the theory of monogenic functions does not cover the set of

Möbius tranformations in higher dimensions and that the Möbius transformations

are not monogenic. One can only expect that monogenic functions represent certain

quasiconformal mappings. On the other hand, the class of all quasiconformal map-

pings is much bigger than the class of monogenic functions. The question arises if

monogenic functions correspond to a special subclass of quasiconformal mappings.

In [17], Malonek introduced the concept of monogenic-conformal mappings real-

ized by Clifford-valued functions deﬁned in R

n+1

. The main tool in his paper is the

study of relations between special surface and volume differential forms. Together

with the geometric interpretation of the hypercomplex derivative [7], dilatations and

distortions of these mappings can be estimated. This includes the description of

the interplay between the Jacobian determinant and the hypercomplex derivative of

monogenic functions (see [5]).

In this contribution we will characterize monogenic functions by more visible

geometric mapping properties. It is clear that the local mapping properties of a

monogenic function or of a real analytic function are mainly determined by the

behavior of the linear part of their Taylor expansions. In this line of reasoning, we

start by presenting the geometric behavior of the linear part of a monogenic function

with values in the reduced quaternions. As a consequence, we show that monogenic

functions can be deﬁned as mappings which map inﬁnitesimal balls onto explicitly

characterized ellipsoids and vice versa.

Geometric Characterization of M-Conformal Mappings 329

2 Preliminaries

Let {e

, e

} be an orthonormal basis of the Euclidean vector space R

.We

introduce the (quaternionic-)product subject to the following multiplication rules:

= e

; e

,i=1, 2, 3;

= e

=−1; e

=−e

This noncommutative product generates the algebra of real quaternions denoted

by H. The real vector space R

will be identiﬁed with H by means of

a := (a

) ∈R

←→ a := a

∈H,

where a

∈R (i = 0, 1, 2, 3).

We remark that the vector e

is the multiplicative unit element of H.Fromnow

on we will identify e

with 1, and we simply write

a :=a

We shall denote by Sc(a) := a

the scalar part of a and by Vec(a) := a

+ a

its vector part. Analogously to the complex case, the (quaternionic-)

conjugate element of a is

a :=Sc(a) −Vec(a) =a

−a

The norm of a is given by |a|=

√

aa =



and coincides with

its corresponding Euclidean norm as a vector in R

Let us now consider the subset

A :=span

{1, e

, e

}

of the algebra H. Its elements are called reduced quaternions. The real vector space

is to be identiﬁed with A via

x :=(x

) ∈R

←→ x := x

∈A.

We emphasize that A is a real vectorial subspace, but not a subalgebra, of H.

Now, let us consider an open subset Ω of R

with a piecewise smooth boundary.

A quaternion-valued function or, brieﬂy, an H-valued function is a mapping f :Ω →

H. From now on we will restrict the latter to functions with values in A, i.e., such

that

f(x) =



f(x)





i=1



f(x)



330 K. Gürlebeck and J. Morais

where the coordinate-functions [f]

(i = 0, 1, 2) are real-valued functions in Ω.

Properties such as continuity, differentiability, or integrability are ascribed co-

ordinate-wisely.

For continuously real-differentiable functions f :Ω →A, the operator D deﬁned

Df =

∂f

∂x

∂f

∂x

∂f

∂x

(1)

is called the generalized Cauchy–Riemann operator on R

. It can be understood as

a three-dimensional extension of the classical Cauchy–Riemann operator ∂

.Inthe

same way, we deﬁne the conjugate quaternionic Cauchy–Riemann operator

D by

Df =

∂f

∂x

−e

∂f

∂x

−e

∂f

∂x

, (2)

which is a generalization of the operator ∂

Deﬁnition 1 A continuously real-differentiable A-valued function f is called mono-

genic (resp., antimonogenic) in Ω if Df =0 (resp.,

Df =0) in Ω.

Remark 1 It is not necessary to distinguish between left- and right-monogenic

(resp., antimonogenic) functions in the case of A-valued functions because Df =0

(resp.,

Df =0) implies that fD =0 (resp., fD =0) and vice versa.

Analogously as in the complex one-dimensional case,

D deﬁnes a derivative

of monogenic functions. This was shown for arbitrary dimensions in [7], where

(

D)f was called hypercomplex derivative of f. In the four-dimensional case we

ﬁnd analogous results in [18] and [20].

The generalized Cauchy–Riemann operator (1) and its conjugate (2) factorize the

three-dimensional Laplace operator, that is,

f =DDf =DDf (3)

for f ∈C

, which implies that any monogenic function is also a harmonic function.

This means that quaternionic analysis gives us a reﬁnement of harmonic analysis.

For future use, we also need the connection between quaternionic analysis and

Clifford analysis. We restrict ourselves to the case of C

0,3

, the universal real Clif-

ford algebra of signature (0, 3), constructed over the canonical orthonormal basis

{ε

, ε

} of the Euclidean vector space R

. The basis elements satisfy the

following multiplication rules:

+ε

=−2δ

i,j

,i,j=1, 2, 3;

=ε

,i=1, 2, 3,

where δ

i,j

stands for the Kronecker symbol.

Geometric Characterization of M-Conformal Mappings 331

The skew-ﬁeld of the quaternions can be identiﬁed with the even subalgebra

C

0,3

by setting

∼

−ε

, e

∼

−ε

, e

∼

+ε

As a consequence, starting with the Dirac operator in C

0,3

D =ε

∂

∂y

+ε

∂

∂y

+ε

∂

∂y

we obtain the generalized Cauchy–Riemann operator via the identiﬁcation

C

0,3

y =y

←→ x = y

−y

∈A

and

−ε

∼

Moreover, a vector-valued function

f in C

0,3

is identiﬁed with the A-valued

function f by

f =













←→ f =[f]

−[f]

, (4)

where [

f(y)]

=[f(x)]

(i =0, 1, 2).

These identiﬁcations deﬁne a link between null solutions of the Dirac equation

in C

0,3

and of the Cauchy–Riemann equations in H. It is well known that the

solutions of the Dirac equations are invariant under rotations. These two facts allow

us, under certain conditions, to transpose some mapping properties of our original

A-valued functions under rotations. More precisely, we have:

Lemma 1 Let f be an A-valued monogenic function deﬁned in a domain Ω of R

Then,

f =D



−1



where U is an orthogonal transformation in R

, and

f =[f]

−[f]

3 Monogenic-Conformal Mappings and Their Relation

to Monogenic Functions

Nowadays the concept of conformal mappings is used in complex analysis, includ-

ing applications to elliptic partial differential equations, differential geometry, po-

tential theory, mathematical physics, and calculus of variations. Conformal map-

pings are closely related to the differentiability and analyticity of complex functions

in the way that every conformal mapping is holomorphic.

332 K. Gürlebeck and J. Morais

The extension of conformal mapping methods in C to the higher-dimensional

real Euclidean space, particularly in the setting of quaternionic analysis, has orig-

inated many questions in the last decades. It is not commonly realized that mono-

genic functions can be characterized via a generalized conformality concept. In [17]

the concept of monogenic-conformal mappings described by paravector-valued real

differentiable functions in Ω ⊂R

n+1

with values in the Clifford algebra C

0,n

has

been introduced for the ﬁrst time. These mappings are called in [17] M-conformal

mappings. The relation of this concept with the geometric interpretation of the hy-

percomplex derivative

D allowed it to complete the theory of monogenic func-

tions by providing a still missing geometric characterization of those functions. It

was shown that monogenic-conformal mappings are generalizations of conformal

mappings in the plane. However, they are different from conformal mappings (in

the sense of Gauss) in higher dimensions.

In the following, we will use the notation of [17]. We consider in Ω ⊂ R

n+1

positively oriented differentiable n-dimensional hypersurface S with coherent ori-

ented boundary ∂S.Letz

∗

∈ S be a ﬁxed point, and (S

) a regular sequence of

subdomains which is shrinking to z

∗

as m tends to inﬁnity and whereby z

∗

belongs

to all S

From [17] we take the following theorem:

Theorem 2 Let f be a paravector-valued real differentiable function in Ω ⊂R

n+1

This function realizes locally in the neighborhood of a ﬁxed point z = z

∗

aleftM-

conformal mapping if and only if f is left monogenic, and its left hypercomplex

derivative is different from zero.

The proof of the previous theorem is based in the following integral representa-

tion of the hypercomplex derivative of a monogenic function f:





f =(−1)

lim

m→∞





dμ ∧dz



−1



∂S

(dμf)

in case the limit exists for all possible regular sequences. To be more precise, this

integral is described by the limit of the “quotient” of a 2-form (surface area) and a

3-form (volume) (Sudbery [20]). For more details on this representation, we refer

the reader to [7] and [17].

However, the geometric properties of such a result are not directly visible. In the

next section we will show that the description of monogenic functions can be also

formulated by more accessible geometric mapping properties.

4 Inﬂuence of the Linear Part of a Monogenic Function

Due to the equivalence between hypercomplex differentiability and monogenicity,

the question arises if from this point of view A-valued monogenic functions in R

can also be deﬁned by their geometric mapping properties.

Geometric Characterization of M-Conformal Mappings 333

It is well known from complex analysis that conformal mappings (and also quasi-

conformal mappings, cf. Ahlfors [1]) in the plane are realized in the neighborhood

of a point by all complex functions for which the complex derivative exists and

is not equal to zero. This can be easily shown by looking to the differential of a

nonconstant real differentiable function f : Ω → C deﬁned in a domain Ω ⊂ R

(see, e.g., [17]). This usual treatment includes the description of the connection

between the Jacobian determinant and the derivative of such functions.

On the other hand, it is also known that the Riemann mapping theorem allows one

to deform an (almost) arbitrary domain by a conformal mapping onto a disk (or vice

versa). In particular, it also allows one to prescribe the direction in which the real

axis should be mapped. This behavior is related to conformal (local) properties of

the complex derivative of a complex-valued function. For that, one usually requires

the knowledge of the argument of the derivative at the origin.

To be more precise, assume a general function f : Ω ⊂ R

→ B ⊂ C. We sup-

pose also that the domain B denotes a ball centered at the origin. The ﬁrst derivative

in the origin will give the ﬁrst (linear) approximation of the mapping function f

in terms of a linear mapping represented in the form w = f(0) + zf



(0)(z∈ C),

valid in a neighborhood of z =0. In other words, for a ﬁrst approximation, we re-

quire nothing else than the identity function. Also, it implies that a small ball is

(locally) mapped onto a small ball. However, in the 3D-case the situation is com-

pletely different, since the class of conformal functions is now reduced to the class

of Möbius transformations. A particular important example is the case of f(z) = z

for z = x

∈A. This can be viewed as the counterpart of the monomial

z in C, but unfortunately it is not a monogenic function.

In this line of reasoning we discuss brieﬂy some general aspects concerning a

general real analytic A-valued function f deﬁned in R

. Clearly, one has the decom-

position

f(x) :=f(0) +linf(x) +R(x), (5)

where linf is the linear part (of the Taylor expansion) of f, and R denotes the re-

maining part (degree k ≥2).

Under the above hypotheses, we claim that the linear part of f is also an A-

valued function naturally related to the hypercomplex derivative of f at the origin.

On the other hand, linf is a general linear function or, more explicitly, a general A-

valued linear function of three real variables and nine real parameters. Therefore, its

coordinates [linf]

(i =0, 1, 2) are given by



linf(x)



= a



linf(x)



= b

, (6)



linf(x)



= c

where, as described, the coefﬁcients a

(i =0, 1, 2) are real valued.

Remark 2 From now on we will restrict ourselves only to mappings from the interior

of a ball into the interior of a special domain of R

334 K. Gürlebeck and J. Morais

Remark 3 If the function is vanishing at the origin, then each monogenic approx-

imation polynomial begins with the linear expression linf. It is clear then that the

invariance of the origin guarantees that the interior of the ball will be mapped to the

interior of the image domain.

We will show in this section that monogenic functions are closely related to spe-

cial ellipsoids in the neighborhood of a given point. As is well known, ellipsoids play

an important role in many disciplines as simple effective object models. Among their

applications which enjoy a great variety, there are the following two closely related

classes: ellipsoidal models are used both directly to capture shape and indirectly as

bounding approximations. Nowadays, in medical areas, ellipsoids are also used to

model the shape, volume, or area of anatomical parts, such as the heart, spine, and

blood vessels.

Ellipsoids can be represented by its center and the lengths of the semiaxes to-

gether with the orientation. An ellipsoid E

∗

with semiaxes parallel to the coordinate

directions is then generated by the quadratic form

∗



) :



, (7)

where α, β, and γ are the lengths of the semiaxes.

On the other hand, if the axes are rotated to a general position, then the equation

for an ellipsoid E can be written as

+bx

+cx

+dx

+ex

+fx

=1, (8)

where a,b,c,d,e, and f are new constants. Those six constants indicate the shape

and the size of the ellipsoid E and the directions of its axes. After a rotation it is

clear that the directions of the axes remain orthogonal.

Remark 4 The absence of linear terms in (8) means that we keep the origin at the

center of the ellipsoid. We use this simpliﬁcation because the displacement of its

center will not add anything essential.

Later on we will also use the representation of an ellipsoid by a symmetric matrix.

Denoting by E the referred matrix, an equivalent deﬁnition of a general ellipsoid E

is given by

E :=



X :X

EX =1,X=(x

)



In this sense, the geometric properties of E are reﬂected in algebraic proper-

ties of the corresponding symmetric matrix E.Ifλ

and ν

are the eigenvalues and

eigenvectors of E, then the principal axes of the ellipsoid are parallel to ν

, and the

corresponding lengths of the semiaxes are given by |λ

−1

. For convenience, in what

follows, we will restrict ourselves to ellipsoids centered at the origin. There is no

loss of generality in this assumption because we are interested only in the shape of

our ellipsoids.

Geometric Characterization of M-Conformal Mappings 335

After the above preparations, we are now ready to characterize the mapping prop-

erties of the linear part of an arbitrary A-valued monogenic function. In the fol-

lowing we consider the image of a ball under the mapping linf, being a positively

oriented differentiable surface with coherent oriented boundary.

Theorem 3 Let linf be an A-valued linear function deﬁned in a domain Ω of R

with nonvanishing Jacobian determinant. Then, the function linf is monogenic if and

only if it maps a ball onto an ellipsoid with the property that the reciprocal of the

length of one semiaxis is equal to the sum of the reciprocals of the lengths of the

other two semiaxes.

Proof Taking into account identiﬁcation (4), we start to write the linear function

linf in the form

linf(x) =



linf(x)



−



linf(x)



−



linf(x)



, (9)

where the coordinates [linf]

(i = 0, 1, 2) are given as in (6). We begin to prove the

necessary condition. By the monogenicity of linf, it holds

⎧

⎪

⎨

⎪

⎩

=−(b

(10)

For X = (x

)

,letA be the matrix which represents linf, i.e., such that

(1, −e

, −e

)AX=linf(x).

The matrix A is symmetric and is given by

A =



−(b



Now, as is well known from Linear Algebra, there exists a real orthogonal matrix

C such that

AC =D =diag(λ

,λ

where λ

,λ

, and λ

are the eigenvalues of A (real-valued and different from zero).

Furthermore, Newton’s identities (Vieta’s formulae) (see [21]) relate the eigenvalues

of A in the following way:

+λ

= tr(A) =0,

= det(A).

Hereby tr(A) and det(A) denote the trace and determinant of the matrix A,re-

spectively.

336 K. Gürlebeck and J. Morais

In the following, let us assume without loss of generality that det(A) > 0. For

det(A) < 0, the proof is similar. Hence, we have to consider two different cases:

1. If λ

> 0, then from the previous conditions we have that λ

and λ

are both

negative numbers.

2. If λ

< 0, one can easily see that λ

and λ

must have different signs.

Then, from the relations

=(−λ

) +(−λ

)>0,

=(−λ

) +(−λ

)>0,

=(−λ

) +(−λ

)>0,

respectively, we get the conditions for the lengths of the semiaxes α, β, and γ .This

means that one can express the reciprocal of the length of one semiaxis as the sum

of the reciprocals of the lengths of the other two semiaxes. To end with the proof

of the necessary condition, notice that the maximum of the eigenvalues gives the

minimum value of the lengths of the semiaxes.

For the sufﬁcient condition, having in mind Lemma 1, the main idea is to rotate

an arbitrary ellipsoid (with the described property of the semiaxes) such that the

directions of its semiaxes y

(0)

, y

(1)

, y

(2)

coincide with the directions of the stan-

dard coordinate system (y

). Such an ellipsoid is given by (7). The referred

property of the semiaxes is invariant under rotations. It is a direct consequence that

Dlinf = 0 and that linf must be monogenic. For more details, we refer the reader

to [8]. 

Remark 5 The composition of a linear function with a translation allows one to

extend the result to an ellipsoid centered at an arbitrary point ˜x. This composition

preserves the monogenicity.

The result of Theorem 3 allows us to describe monogenic functions as a special

class of quasiconformal mappings. If we visualize quasiconformal mappings in R

by points, given by the lengths of the semiaxes of the associated ellipsoids, then

the monogenic functions (with nonvanishing Jacobian determinant) can be seen as

a two-dimensional manifold in R

. This manifold is deﬁned by the relation between

the semiaxes as in Theorem 3 and visualized in Fig. 1.

Next, one has to show that Theorem 3 can be generalized to arbitrary real-analytic

functions which have the described local mapping properties. A monogenic function

with nonvanishing linear part will map in the small balls to the special class of

ellipsoids. Nonvanishing linear part means that all directional ﬁrst derivatives of

the function are different from zero. Equivalently this can be characterized by the

Jacobian determinant and the hypercomplex derivative:

Geometric Characterization of M-Conformal Mappings 337

Fig. 1 Geometric

characterization of

monogenic functions

Theorem 4 (See [5]) Let f be an H-valued monogenic function deﬁned in a domain

Ω of R

. Then we have

detJ

(0) =0 ⇐⇒ ∃ p :|p|=1 ∧Df







z=0

=0.

This theorem can be considered as a higher-dimensional analogue of the con-

dition in complex analysis, saying that f



) = 0 for a holomorphic function f .

For more details and further relations to the hypercomplex derivative and quasi-

conformal mappings, see the paper [5].

Theorem 5 Let f be an A-valued real-analytic function deﬁned in a domain Ω

of R

with nonvanishing Jacobian determinant. Then, the function f is monogenic

if and only if it maps locally a ball onto an ellipsoid with the property that the

reciprocal of the length of one semiaxis is equal to the sum of the reciprocals of the

lengths of the other two semiaxes.

Proof The necessary condition can be found in [8]. For the sufﬁcient condition, we

suppose that the real-analytic function f maps the unit ball to an ellipsoid with the

referred property. It is clear that the function f can be expanded in a real Taylor

series in a neighborhood of an arbitrary point a ∈Ω:

f(x) = f(a) +

+∞



n=1



|α|=n

−a

)

−a

)

−a

)

∂

+α

∂x

f(a)

= f(a) +linf(x −a) +R(x −a).

For simplicity, we take a =(0, 0, 0). We then have

f(x) =f(0) +linf(x) +R(x).

Now, we deal with the property of monogenicity of the mapping f. We discuss

this property only locally for a neighborhood of the point x =0. Without loss of gen-

erality (removing higher-order terms from the Taylor expansion described above),