The Erwin Schrodinger International Boltzmanngasse

ESI Institute for Mathematical Physics A Wien Austria

A Weierstrass Representation

Theorem for Lorentz Surfaces

Jerzy Julian Konderak

Vienna Preprint ESI Octob er

Supp orted by Federal Ministry of Science and Transp ort Austria

Available via httpwwwesiacat

A WEIERSTRASS REPRESENTATION



THEOREM FOR LORENTZ SURFACES

Jerzy Julian KONDERAK

Octob er

Abstract

We consider functions with values in the algebra of Lorentz num

b ers L and which are dierentiable with resp ect to the algebraic struc

ture of this algebra L as an analogue of the holomorphic functions

Then we apply this functions to prove a Weierstrass representation

3

theorem for Lorentz surfaces immersed in the space R In the pro of

1

we essentially follow the mo del of the complex numb ers We apply

our representation theorem to construct explicit minimal immersions

Intro duction

The study of minimal sub in pseudoRiemannian manifolds is quite

dierent from the Riemannian case In the former case we have to deal with

a hyp erb olic PDEs while in the latter case we have to do with the elliptic

PDEs Hence the dierence b etween pseudoRiemannian and Riemannian is

as b etween the hyp erb olic and elliptic PDSs

The classical Weierstrass representation theorem uses the relationship

b etween holomorphic functions and solutions of certain elliptic PDEs cf

and the references there In the present pap er we consider the algebra

L of Lorentz numb ers which have many similar prop erties to the complex

numb ers The main drawback of L is that it contains zero divisors We

consider Ldierentiable functions and manifolds mo deled on L Analysis



Mathematics Sub ject Classication Primary A Secondary C

over algebras is an old and well develop ed part of mathematics cf and

the references there

Lorentz surfaces are two dimensional real oriented pseudoRiemannian

manifolds They are naturally mo deled on the algebra of Lorentz numb ers

We use Ldierentiable functions to obtain solutions of the minimal sub

3

manifolds in R We prove a global version of the Weierstrass representation

1

theorem for Lorentz surfaces This problem was studied by many authors

using PDEs metho ds cf and the references there however those results

are only of the lo cal typ e An intensive use of the Lorentz numb ers in the

theory of minimal immersions and harmonic maps may b e found in the recent

pap ers by S Erdem cf

Preliminaries

Lorentz numb ers

We consider the algebra of Lorentz numb ers L and recall here some prop erties

of L We would like to stress its geometrical similarity to the eld

of the complex numb ers The algebra L may b e dened as L fu v ju v

Rg with the assumption that the immaginary unit has the prop erty that

2

More precisely in the set L we have two internal op erations sum

and pro duct They are dened as follows

u v u v u u v v

1 1 2 2 1 2 1 2

u v u v u u v v u v u v

1 1 2 2 1 2 1 2 1 2 2 1

With these op erations the set L is an asso ciative commutative algebra over

R with unity this algebra is called the algebra of Lorentz numbers The

algebra L admits the zero divisors they are the numb ers of the typ e u u

where u R n fg We denote by K the set consisting of the zero divisors

and the zero of L

There is naturally dened conjugation in L namely u v u v

Moreover we put Reu v u and Imu v v Let z u v b e an

element of L which is not a zero divisor Then there exists the inverse of z

1

and we have that z z z z The algebra L is isomorphic as an algebra

to R R via the map L R R dened by u v u v u v

1

The inverse of this is given by x y x y x y

The algebra L is a Cliord algebra asso ciated with the space of the reals

equipp ed with the standard scalar pro duct unlike complex numb ers which

is the Cliord algebra asso ciated with the reals equipp ed with minus the

standard pro duct The algebra L may b e also seen as the algebra

asso ciated with the cyclic group Z The name Lorentz numb ers is taken

2

from The other names in use are dual numb ers hyp erb olic numb ers

paracomplex numb ers hyp ercomplex numb ers p erplex numb ers etc cf

Elementary functions over the Lorentz numb ers

We shall consider functions whose domain is contained in L and with values

in L We shall use later these functions to construct examples of minimal

immersions The details ab out these functions are discussed in In what

follows we shall write functions of the variable z in the typewritten charac

ters to distinguish them from the resp ective complex valued functions

Example Let a a L then we have the following p olynomial func

0 n

1 n

tion pz a a z a z dened on L

0 1 n

Example Let z L then we consider the series

n

X

z

n

n=1

It may b e proved that this series converges for all z L Hence we denote

the sum of this series by expz Moreover we have that

h i

expu v expu v expu v expu v expu v

where exp is the classical exp onential function cf

Example We dene the trigonometric functions if z L then we put

exp z exp z

sinz

exp z exp z

cosz

expz expz

sinhz

expz expz

coshz

We proved that sin sinh and cos cosh cf Moreover for each

z u v we have the following explicit expressions

sinhu v sinhu v sinhu v sinhu v

sinhu v

coshu v cosh u v cosh u v coshu v

coshu v

We observe that the resp ective pairs of functions sinh sinh and cosh cosh

coincide when restricted to the reals

Dierentiable functions over the Lorentz numb ers

L has a structure of a two dimensional real vector space and then it has a

natural top ology Let L b e an op en subset z and f L b e a

0

map

Denition It is said that f i s Ldierentiable in z i there exists the

0

following limit

f z f z

0

lim

z z

0

z z

0

z z L nK

0

if it exists we call it t he Lderivative of f in z and denote by f z The

0 0

function f is said Ldierentiable in if f is Ldierentiable at each p oint

of

It is straightforward to prove the following

Observation Let f g be functions dened on with values in L Suppose

that f and g are Ldierentiable in z and a b L Then we have that

0

af z bg z f z g z and f z g z are Ldierentiable in z in the last

0

case we suppose that g z K Moreover we have that

0

af bg z af z bg z

0 0 0

f g z f z g z f z g z

0 0 0 0 0

f z g z f z g z f

0 0 0 0

z

0

2

g g z

0

Remark The dierentiation over algebras was considered by many au

thors The rst pap ers go es back to SLies student Scheers cf a

go o d bibliography ab out the sub ject may b e found in The particu

lar case of the algebra L was studied in in the fties In the recent

times there is renewed interest in the in the Lorentz numb ers analysis cf

Example The p olynomials in z exp sin and cos are Ldierentiable

n n1

maps and we have the following formulas z nz expz expz

sin z cosz cos z sin z

We intro duce the formal derivatives

z u v z u v

In general if f is Ldierentiable in z then it do es not imply that f is

0

continuous in z However if f is Ldierentiable in then f is dieren

0

tiable in the ordinary sense in There is no regularity prop erty for the

Ldierentiable maps and there exist Ldierentiable maps of any class cf

The following is a wellknown fact from the theory of para

complex maps

Theorem Let f L be such that f u v au v bu v and

f

1

a b are of the class C Then f is Ldierentiable in i

z

f

The equation is equivalent to the following version of the Cauchy

z

Riemann equations

a b

u v

a b

v u

An Ldierentiable function F z is called primitive of f i F z f z

for each z in the domain of f It may b e proved that if the domain of f is

simply connected and f is Ldierentiable then there exists a primitive of f

Then we dene Ldierentiable and Lantidierentiable forms as follows

z du dv dz du dv d

Lorentz surfaces

If we have a suciently rich function theory on an algebra then we may de

ne a mo deled lo cally on such algebra and its Cartesian p owers

In particular there is well develop ed theory of paracomplex manifolds cf

and for a bibliography ab out the sub ject We are interested

in the pseudoRiemannian manifolds which corresp ond to the Riemann sur

faces Let M b e a C two dimensional real oriented manifold then M is

called Lorentz surface if there is given an equivalence class of conformal with

p ositive scaling pseudoRiemannian metrics g The metric g as well as all

the other metrics in the equivalence class g are of the signature Equiv

alently the Lorentz surface may b e dened as an oriented manifold equipp ed

with an atlas of charts which consists of orientation preserving maps such

the transition functions are Ldierentiable cf

If M g is a Lorentz surface then there may b e constructed an atlas of

distinguished charts It consists of the charts U such that with resp ect

to the lo cal co ordinates u v determined by the metric g j is equal to

U

2 2 2

du dv for some real valued p ositive function on U It may b e proved

that such atlas consists of Ldierentiable maps For more details ab out the

Lorentz surfaces lo ok to the recent b o ok by T Weinstein cf

Let M b e a Lorentz surface

Denition If we have a form on a Lorentz surface M with values

in L then it is said of type i lo cally with resp ect to a distingished

chart it is of the form dz for some Lvalued function We say that is

Ldierentiable i the co ecient function is Ldierentiable on M This

denition do es not dep end on the choice of a distinguished chart

1

Let a b R b e a piecewise C curve and a form on M with

values in L Then there is dened the integral of along in the following

R R

b

dt It is clear that such an integral may not exists b ecause way

a

may not b e integrable in a reasonable sense It is enough to assume that

the form is continuous to assure the existence of such an integral

Observation If the function f z has a primitive F z in a domain

R

f z dz F b F a and in particular this integral does not then

depend on the path connecting points a and b

For details ab out the integrals of Lvalued form lo ok to

A representation theorem

3 3

Let R b e the vector space R equipp ed with the scalar pro duct of

1

signature In the other words if x x x x y y y y

1 2 3 1 2 3

3 3

R then x y x y x y x y On the space R we have the natural

1 1 2 2 3 3

1

orientation given by the canonical form dx dx dx We observe that

1 2 3

dx e dx e dx e

1 2 3

1 2 3

3

where e e e is the canonical basis of R and is the musical isomorphism

1 2 3

3 3

b etween R and R dened by Hence the canonical orientation of

1 1

3

R is given by the ordered triple e e e There is given the Ho dge star

1 2 3

1

3 3 3

op erator R R R determined by the identity

1 1 1

e e e

1 2 3

3 3

It is straightforward to verify that if x x x x R for each R

1 2 3

1 1

and y y y y then

1 2 3

e e e

1 2 3

x y x x x

1 2 3

y y y

1 2 3

3

dened by X Y X Y Hence we have the natural vector pro duct in R

1

cf

Let M g b e a two dimensional oriented pseudoRiemannian manifold

3

b e an immersion Then X is said to b e an isometric and let X M R

1

immersion if g X the map X is said to b e a conformal immersion

if g is conformal to X with a p ositive scaling function

Supp ose that X is an isometric immersion For such an immersion there

is dened the unit normal vector eld N which is spacelike the second

fundamental form the Gauss curvature and the mean curvature H The

surface M is said to b e minimal if H cf

We observe that M g is a Lorentz surface Consider a lo cal distin

guished chart U and the co ordinates u v determined by this chart

2 2 2

Then in these co ordinates we have that g j du dv It means that

U

X X X X X X

2

and

u v u u v v

for some p ositive valued function on U For each real valued function f on

M there is dened the Lapalcian of f by the formula f d df Then

it is straightforward to prove that

2

f

f

2

z z

Moreover we have that X H N This gives immediately the following

observation

X

is Ldierentiable Observation An immersion X is minimal i

z

Proof In fact X is minimal i H ie

2

f X

X

2 2

z z z z

X

which is equivalant to the prop erty that is Ldierentiable

z

X

X

1

in particular Let X X X X and we put

1 1 2 3

z z

X X

3 2

and

3 2

z z

3

then Observation If X is a minimal immersion of M in R

1

2 2 2

3 2 1

Proof In fact we have that

2 2

X X X X

1 1 2 2

2 2 2

3 2 1

u v u v

2

X X

3 3

u v

X X X X X X

1 2 3 1 2 3

u u u u u u

X X X X X X

1 2 3 2 3 1

v v v v v v

X X X X X X

1 2 3 1 2 3

u v u v u v

X X X X

u u v v

X X

u v

which proves On the other hand we have that

X X

z z

X X X X

u v u v

X X X X

u u v v

X X

2

u u

is always p ositive and is proved Hence it follows that

Observation Suppose that X is minimal Then the formula

X

dz j

U

z

denes a global Ldierentiable form on M

Proof It follows from the prop erties of Ldierentiable maps that the def

inition of is wellp osed that is it do es not dep end on the choice of the

distinguished co ordinates on M It is formally the same pro of as in the

complex case

Theorem Let M g be a connected Lorentz surface together with distin

guished local charts Let be Ldierentiable forms on M

1 2 3

such that if has the fol lowing expression dz j in a local

j j

distinguished chart then

2 2 2

i

3 2 1

ii

1 2 3

1 2 3

Moreover we assume that

R

z

iii the integrals Re j does not depend on the choice of a

j

z

0

piecewise dierentiable curve connecting a xed point z of M and any

0

point z of M

Then the mapping dened by the fol lowing formula

Z Z Z

z z z

X z Re Re Re

1 2 3

z z z

0 0 0

is a minimal conformal immersion

Proof We essentially rep eat the comments given in Of course the condi

tion iii assures that the mapping X is well dened Condition i and ii

in a lo cal distinguished co ordinates means that

X X X X X X

u u v v u v

X X X X

u u v v

Equations and are equivalent to the prop erty that X is a conformal

immersion Since

X

dz

1 2 3

z

and is Ldierentiable then it follows that X is a minimal immersion with

resp ect to the induced metric

Condition iii of the ab ove theorem is usually said that has no real

periods

The dierentials satisfying conditions i and ii of Theorem

1 2 3

may b e characterized similarly as in the complex case Supp ose that

2

K for all x M and T M with Then we put

3 x x

1

and g

2 3

2 3

We observe that is an Ldierentiable an form and g is Ldierentiable

function on M Now we may recover as functions of and g In

1 2 3

fact we have that

g

1

2

g

2

2

g

3

Then we have that

1 1 2 2 3 3

2

g g

Then it follows that satisfy i and ii of Theorem i

1 2 3

and Img

Then as a conclusion we have the following theorem

Theorem Let M g be a connected Lorentz surface an L dieren

and tiable form on M and g an L dierentiable map such that

1 1

2 2

Img Suppose also that the forms g g g have no

2 2

3

real periods We x a point z M then the map X M R dened by

0

1

Z Z Z

z z z

2 2

g Re g X z Re g Re

z z z

0 0 0

is a conformal minimal immersion

The dierentials and satisfying i of Theorem may b e de

1 2 3

scrib ed in a dierent way from that one in Supp ose that

2 3 x

K for all x M and T M with Then we put

x

1

and g

2 3

2 3

1 1

2 2

Then we recover the forms g g g

1 2 3

2 2

and Img These forms satisfy ii of Theorem i again

Another way of describing forms and is the following supp ose

1 2 3

that K for all x M and T M with Then we

2 3 x x

put

1

and g

2 3

2 3

1

2 2

Then we recover the forms g g g

1 2 3

2 2

and check that

2

g g

1 2 3 1 2 3

and g g Hence condition ii of Theorem is satised i

This gives the following version of our representation theorem

Theorem Let M g be a connected Lorentz surface an L dieren

tiable form on M and g an Ldierentiable map such that and

1

2 2

g g Suppose also that the forms g g and g

2 2

3

have no real periods We x a point z M then the map X M R

0

1

dened by

Z Z Z

z z z

2 2

g Re g X z Re g Re

z z z

0 0 0

is a conformal minimal immersion

Remark If or then and the immersion

2 3 2 3 1

dened by the formula has its values in a plane parallel to the plane

x

1

Remark Let us consider a map given by and supp ose that

g x g x then x is a singular p oint of X On the other hand if

and a p oint is not singular then the pullback metric X is conformal

to g

We apply Theorem to construct examples of minimal immersions

Example An analogue of the helicoid We put M fz L Rez

g g z expz and expz dz Then b oth g and are L

dierentiable Moreover

expz expz expRez

and then on the other hand

g z g z expRez

for z L We apply and get

dz sinz dz cosz dz

1 2 3

Then we observe that the ab ove forms are Ldierentiable in a simply

connected domain Hence have no real nor immaginary p erio ds

1 2 3

We have that

Z Z Z

z z z

z cosz sinz

1 2 3

0 0 0

Then we apply formula and get

X z Re z cosz sinz

Imz Recosz Imsinz

Then applying Example we get the following explicit formula for the min

imal immersion

sinhu v sinhu v coshu v cosh u v

X u v v

Example An analogue of the catenoid Supp ose that M L

expz dz and g z expz Then for each z L we have that

expRez g g expRez

Then we observe that the following forms

g dz

1

2

g sinz dz

2

2

g cosz dz

3

are Ldierentiable in whole L and then without p erio ds Integrating we get

that

Z Z Z

z z z

z cosz sinz

1 2 3

0 0 0

Then applying formula we get the following conformal minimal immer

sion

X z Re z cosz sinz

Rez Recosz Imsinz

Applying Excercise we get that

sinhu v sinhu v cosh u v coshu v

X u v u

Example An analogue of Ennep ers surface We consider M fz L

z z g and put g z z dz Then for each z M we have

g z g z

Hence using substitution we obtain

g z dz

1

2 2

g z dz

2

2 2

g z dz

3

The ab ove forms are Ldierentiable in whole L hence without the real nor

immaginary p erio ds Hence we have that

Z Z Z

z z z

3 3 2

z z z z z

1 1 3

0 0 0

Then from formula we get that the induced conformal minimal immer

sion is

2 3 3

z z z z z

X z Re Re Im

The the explicit formula for this surface is the following

3 2 3 2 2 2

u u uv v v u v u v

X u v

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3

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