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A Weierstrass Representation Theorem for Lorentz Surfaces 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 submanifolds 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 isomorphism 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 group 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 manifold 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
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