Minimal spacelike surfaces and the graphic equations in R^4_1
aa r X i v : . [ m a t h . DG ] F e b MINIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONSIN R M. P. DUSSAN, A. P. FRANCO FILHO, R. S. SANTOS
Abstract.
In this paper we study an extension of the Bernstein Theorem for minimalspacelike surfaces of the four dimensional Minkowski vector space form and we obtainthe class of those surfaces which are also graphics and have non-zero Gauss curvature.That is the class of entire solutions of a system of two elliptic non-linear equations thatis an extension of the equation of minimal graphic of R . Therefore, we prove that theso-called Bernstein property does not hold in general for the case of graphic spacelikesurfaces in R . In addition, we also obtain explicitly the conjugated minimal spacelikesurface, and identify the necessary conditions to extend continuously a local solution ofthe generalized Cauchy-Riemann equations. Keywords : Minimal spacelike surface, Bernstein Theorem, Weierstrass representation
MSC : 53C42; 53C50 1.
Introduction
One relevant classic result in the context of the global geometry of spacelike surfacesit is the Bernstein Theorem, which assures that if a minimal surface in the Euclidean 3-dimensional space E is an entire graphic of a function f : Ω ⊂ R → R , then it is a plane.Or equivalently, for S being a regular surface of E and for a fixed direction span { ∂ } and a system of coordinates ( O, x, y, z ), such that in those coordinates ∂ = (0 , , If S is a minimal surface and the orthogonalprojection in the coordinate plane ( O, x, y ) is 1-1 and onto, then the surface is a plane. In the context lorentzian, it is well known the Cabali-Bernstein Theorem which es-tablishes that in the 3-dimensional Minkowski space R the only entire minimal graphic { ( f ( x, y ) , x, y ) | ( x, y ) ∈ Ω ⊂ R } are the spacelike planes. One can see the E. Calabi workin [2] as a transposition of the Bernstein Theorem for R , where the fixed direction is atimelike unit vector.After the Bernstein and Calabi-Bernstein results, several authors have shown interestin these global results, and hence in the literature are found several works proving theBernstein property from different viewpoints, providing diverse extensions or new proofsof those theorems.Although in codimension one the Bernstein property is hold, it is worth pointing outthat the property may be not hold in codimension bigger that one. That is the casein codimension two, where the Kommerell work ([6]) considers minimal surface in the Euclidean 4-dimensional space R , and proves that graphic of entire holomorphic functiongives minimal surfaces such that its projection in the plane ( O, x, y ) is 1-1, onto and itsGauss curvature is not zero.Motived by the results above and on the influence of the works of J. C. C. Nitsche ([7])and of T. Rad´o ([10]), we show through of this paper that the Bernstein property doesnot hold for spacelike surfaces in the 4-dimensional Minkowski space R . More than it, inthis paper we also provide answers to the question whether it is possible to establish someextension of the Bernstein Theorem for those kind of surfaces. Since the inner productused in this case is undefined, we need to consider two cases: fixing a timelike plane ora spacelike plane. So, explicitly, we work on answering the following question, which is ageneralization type of the Bernstein and Kommerell theorems: Are there complex functions, not necessarily holomorphic, defined in the whole plane,whose graphic spacelike surface associated to orthogonal projection on a timelike plane oron a spacelike plane in R , is onto and with non-zero Gauss curvature? Through of this paper we answer the previous question. In fact, we obtain two classesof minimal entire graphic spacelike surfaces in R of type ( A ( x, y ) , x, y, B ( x, y )) and( x, A ( x, y ) , B ( x, y ) , y ), for A ( x, y ) , B ( x, y ) being smooth functions to real-valued, for whichthere exist points with non-zero Gauss curvature. We call the graphics above, as the firstand second type, respectively. Our technique involves the use of a Weierstrass representa-tion involving three holomorphic functions a complex-valued a ( w ) , b ( w ) and µ ( w ). Thatrepresentation allows us to show that for getting the graphic minimal spacelike surfacesthe holomorphic functions a and b have to be proportional complexes if the graphic is offirst type, or inversely proportional complexes with the imaginary part different from zeroif the graphic is the second type. Moreover, if the functions a and b are assumed to be de-fined in whole the complex plane C , we find classes of graphic surfaces of first and secondtype which are entire and minimal with Gauss curvature different from zero. Therefore ourtheorems 4.5 and 4.8 provide explicit examples which prove that the Bernstein propertydoes not hold in general for spacelike surfaces in R .Carrying out our study of the spacelike surfaces in R , we also obtain explicitly theconjugate minimal spacelike surface using the Weierstrass representation. In addition,we identify under what conditions we can guarantee that a non-isothermic neighborhoodcan be extended to the entire complex plane. That is done using the generalized Cauchy-Riemman equations on neighborhood in non-isothermic coordinates. So, our work can beseen as an extension of the program developed by T. Rad´o in [10].In this paper we also give several examples of graphic minimal spacelike surfaces in R with Gauss curvature non-zero, and we find conditions to construct graphic minimalspacelike surfaces which have a new type of singularities, it called lightlike singularities,as defined by Kobayashi in [5]. Those singularities are points where the tangent plane ofthe surface is also tangent to the lightcone of R . INIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONS IN R Finally, in the last section of this paper, we construct a θ -family of minimal spacelikesurfaces in R which transports a minimal first type graphic surface in E to a associatedminimal first type graphic surface in L . That allows us to conclude, as expected, thatthe Bernstein Theorem holds if and only if the Calabi Theorem holds.For obtaining our results, we use the integral representation of the spacelike surfaces in R , and of the adaptation of [4] to the Minkowski space R . The details of this adaptationcan be found in the article of authors M.P. Dussan, A. P. Franco Filho and P. Sim˜oes ([3]).Moreover, we pay attention to the Kobayashi work in [5], where he used the technique ofWeierstrass representation to find several examples of minimal spacelike surfaces R andto find new type of singularities for these surfaces. Those singularities are points where asmanifold these surfaces are defined but where the metric vanishes. That means in thosepoints the tangent plane of S is also tangent to the lightcone of R . The Helicoid is abeautiful example that we can find in [5].2. Basic Facts and Notations
The Minkowski space R will be the 4-dimensional real space R equipped with thebilinear form called of Lorentzian product, which is given by h ( a, b, c, d ) , ( t, x, y, z ) i = − at + bx + cy + dz. A spacelike plane V ⊂ R is a 2-dimensional vector subspace where h v, v i > v = 0 of the plane V . A timelike plane T ⊂ R is a 2-dimensional vector subspace wherethere exists a timelike vector t ∈ T , that means that h t, t i <
0, and an other spacelikevector n ∈ T such that h n, n i > h t, n i = 0.We say that timelike plane T is the orthogonal complement of the spacelike plane V ,denoted by the symbols V = T ⊥ and T = V ⊥ , if h x, y i = 0 for all x ∈ T and y ∈ V. The following proposition is very useful throughout this work, it establishes a specialorthonormal basis for each timelike plane. We denote by ∂ the vector (1 , , , Proposition . For each spacelike plane V E equipped with a orthonormal basis { e , e } , the (unique) timelike plane T = V ⊥ has an orthonormal basis { τ, ν } satisfyingthe following conditions:1. h τ, τ i = − and h τ, ∂ i < . That means τ is timelike, future directed unit vectorof T .2. h ν, ν i = 1 with h ν, ∂ i = 0 . That means that ν is a vector into the -dimensionalsubspace { } × R ⊂ R , which we will identify with the Euclidean -dimensional vectorspace E .3. h τ, ν i = 0 and for all other orthonormal basis { t, n } of T we have that τ ≤ | t | .4. The orthonormal basis { τ, e , e , ν } , in this order, is positive relative to the Minkowskireferential { ∂ , ∂ , ∂ , ∂ } given by the canonical basis of R . M. P. DUSSAN, A. P. FRANCO FILHO, R. S. SANTOS
Proof.
We need to define the vector τ , therefore all the statements of the propositionfollow immediately. In fact, we take τ as being(1) τ = 1 p e ) + ( e ) ( ∂ + e e + e e ) , where e i = −h ∂ , e i i for i = 1 ,
2. It is trivial to see that h τ, τ i = − h τ, e i i = 0 for i = 1 ,
2, and that τ is directed future. Since by the assumption V E , we have thetimelike plane generated by { ∂ , τ } . Then we take ν to be the unique unit vector of theline span { ∂ , τ } ∩ E such that { τ, e , e , ν } is a positive basis.Now, assuming that we have other orthonormal basis { t, n } for T we can take theLorentz transformation given by t = cosh ϕ τ + sinh ϕ ν and n = sinh ϕ τ + cosh ϕ ν, assumed that t >
0. Since −h t, ∂ i = − cosh ϕ h τ, ∂ i it follows then that t > τ . (cid:3) A Semi-rigid frame.
Let R = E ⊕ T be given by the directed sum of a spacelikeplane E and its orthogonal complement T , which is a timelike plane. Definition . A semi-rigid referential of the Minkowski space R associated to adirected sum R = E ⊕ T , is a positive basis { l , e , e , l } of R satisfying the followingconditions:1. E = span { e , e } and T = span { l , l } .2. { e , e } is an orthonormal basis for E .3. { l , l } is a null basis for T such that l = 1 = l . Proposition . If we have two semi-rigid referential { l , e , e , l } and { ˜ l , ˜ e , ˜ e , ˜ l } ,associated to the directed sum R = E ⊕ T with T = E ⊥ , then l = ˜ l and l = ˜ l . Therefore the complex numbers given by a ( l ) = l + il − l and b ( l ) = l + il l are univocally determined by the directed sum R = E ⊕ T . Proof.
In the Lorentz plane T with induced orientation by ∂ , there exists only twoindependent lightlike directions L and L . Therefore adding the condition h L , ∂ i = − h L , ∂ i , we obtain the unique basis { l , l } for T given by 3. of the Definition 2.2. (cid:3) INIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONS IN R Corollary . The matrix associated to the set of the semi-rigid referentials of thedirected sum R = E ⊕ T , is given by M ( ϑ ) = ϑ sin ϑ − sin ϑ cos ϑ
00 0 0 1 for ϑ ∈ R . Moreover, M ( ϑ ) is a 1-parameter sub-group of the Minkowski group of isometry of R , and all geometric facts that we will see in this work, are invariant by this sub-group.Indeed, we will see that the complex functions a ( l ) and b ( l ) determine the geometricproperties of minimal spacelike surfaces of R . Proposition . The frame associated to the vector subspace E can be taken in termsof a ( p ) and b ( p ) , namely, e ( p ) = W ( p ) + W ( p )2 | − a ( p ) b ( p ) | and e ( p ) = W ( p ) − W ( p )2 i || − a ( p ) b ( p ) || , where W ( p ) = ( a ( p ) + b ( p ) , a ( p ) b ( p ) , i (1 − a ( p ) b ( p )) , a ( p ) − b ( p )) with h W ( p ) , W ( p ) i = 2 | − a ( p ) b ( p ) | Spacelike Surfaces in R . Definition . A spacelike surface S ⊂ R is a smooth -dimensional sub-manifold ofthe topological real vector space R that at each point p ∈ S its tangent plane T p S relativeto the lorentz product of R is a spacelike plane.A spacelike parametric surface of R is a two parameters map ( U, X ) from a connectedopen subset U ⊂ R into R , such that the topological subspace X ( U ) is a spacelike surface.Henceforward we will assume that ( X ( U ) , X − ) is a chart of a complete atlas for aspacelike surface S of R . Let (( x, y ) , U ) be a connected and simply connected open subset of the Euclidean plane R . If X ( x, y ) = ( X ( x, y ) , X ( x, y ) , X ( x, y ) , X ( x, y )) is a spacelike parametric surfaceof R then, we have a metric tensor induced by the lorentzian semi-metric of R given by g = X i,j h D i X, D j X i dx i ⊗ dx j , and the second quadratic form of S = X ( U ) is a quadratic symmetric 2-form B = X i,j Ψ ij dx i ⊗ dx j , M. P. DUSSAN, A. P. FRANCO FILHO, R. S. SANTOS that is given by covariant partial derivative by the formula D ij X − X k Γ kij D k X = Ψ ij . From the definition of the Christoffel symbols Γ kij it follows that h Ψ ij , D k X i ≡
0. Settinga pair of pointwise orthonormal vectors for the normal bundle
N S given by τ ( x, y ) and ν ( x, y ), where τ ( x, y ) is a future directed timelike unit vector and ν ( x, y ) is a spacelikeunit vector, we can assume that h ν ( x, y ) , (1 , , , i ≡
0. Therefore we haveΨ ij = h ij τ + n ij ν where by definition h ij = −h D ij X, τ i and n ij = h D ij X, ν i . Since dim( N p S ) = 2 we need to define the normal connection for S , which is given bya covariant vector γ = P γ k dx k where γ k = h D k τ , ν i = h D k ν, τ i . Next we will display this set of structural equations for S = X ( U ), equation (2) beingthe Gauss equation, (3) and (4) corresponding to Weingarten equations for S . Namely, D ij X − X k Γ kij D k X = h ij τ + n ij ν (2) D k τ = X m h km D m X + γ k ν (3) D k ν = − X m n km D m X + γ k τ. (4) Definition . We say that the surface S = X ( U ) is a minimal surface if and onlyif H S = 12 X Ψ ij g ij = 0 . The vector field H S is called the mean curvature vector of S .It follows from equations (2) that an equivalent definition for minimal surfaces is thecondition H S = X g ij ( D ij X − X Γ kij D k X ) = (∆ g X , ∆ g X , ∆ g X , ∆ g X ) = 0 , where ∆ g is the Laplace-Beltrami operator over S = ( U, g ) . Next we observe that one can associate a Riemann surface to S . In fact, from the wellknow theorem which assures that any spacelike surface admits an isothermic coordinateatlas, that means, there is a parametrization f ( w ) = ( f ( w ) , f ( w ) , f ( w ) , f ( w )) , w = u + iv ∈ U ′ ⊂ C , INIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONS IN R such that f ( U ′ ) ⊂ S = X ( U ) and the induced metric tensor is g = λ dwdw , or moreexplicitly h f u , f u i = λ = h f v , f v i and h f u , f v i = 0 . Since f w = ( f u − if v ), we extend the bilinear form of R to a complex bilinear formover C ≡ R + i R , namely, h X + iY , A + iB i = h X, A i − h
Y, B i + i ( h X, B i + h Y, A i . Hence it implies that(5) h f w , f w i = 0 and h f w , f w i = h f w , f w i = λ / . Now, if we have two isothermic charts ( U ′ , f ) and ( V, h ) for S then, when makes sense,the overlapping map is a holomorphic function, and so we can see M = ( S, A ) as aRiemann surface equipped with the conformal atlas A , and such that the induced metrictensor ds = λ ( w ) | dw | is a compatible metric for the Riemann surface M .Finally, we note that does not exist compact spacelike surfaces in R , so from now M will be either the disk D = { z ∈ C : zz < } that is a hyperbolic Riemann surface , or the complex plane C which is a parabolic Riemann surface, since we are assuming that M is a connected and simply connected Riemann surface. Moreover, if h ( z ( w )) = f ( w )then from chain rule it follows f w ( w ) = h z ( z ( w )) dzdw ( w ) and h f w , f w i = h h z , h z i (cid:12)(cid:12)(cid:12)(cid:12) dzdw (cid:12)(cid:12)(cid:12)(cid:12) . A solution for the equations (5).
Expanding in its coordinates we have that theequation (5) becomes − ( f w ) + ( f w ) + ( f w ) + ( f w ) = 0 . Denoting the complex derivate of the components f iw by Z i and assuming that Z − iZ =0, we have Z − Z Z − iZ Z + Z Z − iZ = Z + iZ Z − iZ . Defining by a = Z + Z Z − iZ , b = Z − Z Z − iZ and µ = Z − iZ , we obtain that the derivate f w can be represented by f w = µW ( a, b ) where W ( a, b ) = ( a + b, ab, i (1 − ab ) , a − b ) , from ( a, b ) ∈ F ( M, C ) × F ( M, C ) in C . M. P. DUSSAN, A. P. FRANCO FILHO, R. S. SANTOS
Moreover, we have that λ = 2 h f w , f w i = 4 µµ (1 − ab )(1 − ab ) . Therefore, µ = 0 and1 − ab = 0 are the conditions to obtain a surface without singularities in its metric.Now, since we can write W ( a, b ) = ( a, , i, a ) + b (1 , a, − ia, −
1) we obtain the caseswhere happens ( Z + iZ )( Z − iZ ) = 0 through of the expressions f w = η ( a, , i, a ) and f w = ξ (1 , a, − ia, − Z = 0 = Z we obtain f w = η (0 , , i,
0) which canbe identified with the plane { } × R × { } .The following lemma is an extension to R of a theorem obtained by Monge: Lemma . For a λ -isothermic spacelike parametric surface ( U, f ) the following state-ment are equivalent:(i) The surface f ( U ) is minimal, H f ( w ) ≡ .(ii) The maps µ, a, b are holomorphic functions from U into C . Proof.
It follows from the Laplace-Beltrami operator that ∆ M f i ( w ) = λ ( f i ( w )) ww =0 for i = 0 , , , . (cid:3) An integral representation.
Let (
U, X ) be a spacelike parametric surface of R where X ( x, y ) = ( X ( x, y ) , X ( x, y ) , X ( x, y ) , X ( x, y ))and U ⊂ R is a simply connected domain. Then, the vector 1-form given by dX = ∂X∂x dx + ∂X∂y dy is exact and therefore closed. So, the integral equation associated to ( U, X ) is(6) X ( x, y ) = X ( x , y ) + Z ( x,y )( x ,y ) ∂X∂x dx + ∂X∂y dy. Moreover, each solution of equation (6) is a spacelike parametric surface (
U, X ) if itholds E = h X x , X x i > , G = h X y , X y i > , F = h X x , X y i and EG − F > . From Definition 2.7 and Lemma 2.8, we obtain:
Corollary . Let U ⊂ R a simply connected domain. If ( U, X ) is a minimal space-like parametric surface which is solution of the integral equation (6), then each coordinatefunction of X ( x, y ) is a harmonic real-valued function on U . Proof.
Indeed, the Laplace-Beltrami operator ∆ M is a tensorial operator defined bycontraction of the Gauss equation (2), as follows in Definition 2.7. (cid:3) INIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONS IN R We note that with isothermic local coordinates the integral representation (6) is usuallycalled of Weierstrass integral equation, namely, f ( w ) = p + 2 ℜ Z ww µ ( ξ ) W ( a ( ξ ) , b ( ξ )) dξ, where f w ( w ) is the solution of equation (5) in Subsection 2.2.2.5. The structural equations with isothermic parameters.
Let (
U, f ) be a para-metric sub-surface of X ( M ) given with isothermic parameters w = u + iv such that h f w , f w i = 0 and h f w , f w i = λ /
2. In this case we have the following version of structuralequations (2), (3) and (4) for minimal surfaces.
Lemma . Let ( U, f ) be a λ -isothermic coordinates system for a minimal surface ( M, X ) of R . We have the following structural equations in w = u + iv ∈ U : τ w = σf w + Γ ν and ν w = χf w + Γ τ (7) f ww = 2 λ w λ f w + σλ τ − χλ ν and f ww = 0(8) Γ( w ) = h τ w , ν i = −h ν w , τ i = γ ( w ) − iγ ( w )2(9) Proof.
We start showing equation (8). For that we take f ww = Af w + Bf w + Cτ + Dν ,and assume that equations (7) and (9) are the definition of the functions associated tothe normal connection for ( U, f ).From h f w , f w i = 0 it follows h f ww , f w i = 0, therefore B = 0. From h f w , f w i = λ / h f ww , f w i + h f w , f ww i = λ w λ , and, since f ww = 0 we obtain A = 2 λ w λ . Now, from h f w , τ i = 0 we have that h f ww , τ i + h f w , τ w i = 0, therefore we obtain C = σ λ .Analogously one has D = − χ λ . So we have showed equation (8).The definition of the functions σ and χ is obtained by equations (7), that from h f w , τ i =0 and from the minimal condition for ( M, f ) it follows that h τ w , f w i + h τ, f ww i = 0. Thusthe tangential component of τ w is σf w . Then, we take the equations (7) as a definitionof the functions associated to the normal connection of ( M, f ). Equation (9) defines thefunction Γ. (cid:3) Two types of Graphics for Minimal Surfaces of R First, let us recall that R has topological structure and differential structure of theEuclidean space R .If R ( u, v ) = ( ϕ ( u, v ) , ψ ( u, v )) is a function from U ⊂ R in R , we can see as a graphicof R the set of point of R such thatgraphic(R) = { (( u, v ) , ( ϕ ( u, v ) , ψ ( u, v ))) ∈ R : ( u, v ) ∈ U ⊂ R } . Since we can choose four equivalent positions for the timelike axis in R , we only need topick two of those positions to get all the possibilities of graphic surfaces. In fact:Fixing the signature of R by ( − , +1 , +1 , +1) we take by definition:(1) The first type of graphic surfaces as given by X ( x, y ) = ( A ( x, y ) , x, y, B ( x, y )) where ( x, y ) ∈ U ⊂ R . (2) The second type of graphic surfaces as given by X ( x, y ) = ( x, A ( x, y ) , B ( x, y ) , y ) where ( x, y ) ∈ U ⊂ R . We will always assume that the functions A and B are C ∞ ( U ), U is a connected andsimply connected open subset of R and that X ( U ) is a spacelike surface of R . Proposition . A minimal graphic surface (first or second type) of R satisfies thefollowing system of equations (10) (cid:26) g D A − g D A + g D A = 0 g D B − g D B + g D B = 0 where g = P ij g ij du i du j is the positive defined metric tensor associated to the surface S = X ( U ) .The system of equations (10) only says that A and B are harmonic functions of theRiemann surface ( U, X ) . Proof.
Taking the matrix representation of metric tensor and its inverse tensor[ g ij ] = (cid:20) E FF G (cid:21) , [ g ij ] = 1 EG − F (cid:20) G − F − F E (cid:21) , one has, from Definition 2.7, that the mean curvature vector is given by2 H X = 1 EG − F ( G Ψ − F Ψ + E Ψ ) . Now, for each type of surface we take a pointwise basis { N , N } for its normal bundle,as follows.If X ( x, y ) = ( A ( x, y ) , x, y, B ( x, y )) we take the orthogonal vectors N = (1 , A x , A y ,
0) and N = (0 , B x , B y , − , and so in this case, D ij X = ( D ij A, , , D ij B ).If X ( x, y ) = ( x, A ( x, y ) , B ( x, y ) , y ) we take the orthogonal vectors N = ( A x , , , − A y ) and N = ( B x , , , − B y ) . Then in this case D ij X = (0 , D ij A, D ij B, (cid:3) INIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONS IN R Our first example corresponds to minimal spacelike surfaces, which are graphic surfacesof the first type defined in the whole plane R . Example . For each harmonic function θ : R −→ R the maps X ( x, y ) = ( θ ( x, y ) , x, y, θ ( x, y )) or X ( x, y ) = ( θ ( x, y ) , x, y, − θ ( x, y )) are both minimal spacelike parametric surfaces, locally isometric to the Euclidean plane R , and therefore flat surfaces.In fact, it assumes the first expression of X ( x, y ) . Since X x = ( θ x , , , θ x ) and X y =( θ y , , , θ y ) it follows h X x , X x i = 1 = h X y , X y i with h X x , X y i = 0 . Now, by assumption ∆ θ = θ xx + θ yy = 0 , it follows that H X ( x, y ) = (0 , , , .We also observe that, according the notation of Subsection 2.3, this class of surfacescorresponds to when Z + iZ = 0 , with Z − Z = 0 and Z = 0 , where Z i are thecomponents in the representation X w ( w ) = ( θ w , , i , θ w ) . Moreover we can write these parametric surfaces as follows: For A = B = θ ( x, y ) we have that X ( x, y ) = (0 , x, y,
0) + θ ( x, y )( ∂ + ∂ ) , therefore X ( R ) is a subset of adegenerated hyperplane, and this shows that its normal curvature vanishes identically. The Example 1 shows that we need a formula of the second quadratic form in terms offunctions µ , a and b . That formula was already obtained in Theorem 3.3 from [3], so werewrite next. Lemma . Let f w = µW ( a, b ) , where a and b are holomorphic functions from M into C . The second quadratic form in complex notation is given by (11) ( f ww ) ⊥ = µa w − ab L ( b ) + µb w − ba L ( a ) , where L ( b ) and L ( a ) are future directed lightlike vectors given by L ( b ) = (1 + bb, b + b, − i ( b − b ) , − bb ) and L ( a ) = (1 + aa, a + a, − i ( a − a ) , − aa ) . It follows from Lemma 3.2 the next corollary.
Corollary . The second quadratic form of a minimal spacelike surface ( U, f ) islightlike type if and only if a w = 0 or b w = 0 . Therefore in this case, the Gauss curvature K ( f ) = 0 and the surface is contained in a degenerated hyperplane.Reciprocally, if the Gauss curvature K ( f ) = 0 then the second quadratic form is lightliketype or it is zero, ( f ww ) ⊥ = 0 . Now, we apply equations (10) for graphic minimal surfaces in E and L . We give theexplicit equation for each case.For the first type:(1) When A ( x, y ) ≡ E given by an unique function B ( x, y ): f ( x, y ) = (0 , x, y, B ( x, y )) ∈ E , with the induced metric tensor over f ( U ) as a spacelike surface of R . Then system (10)becomes to the equation(12) (1 + B y ) B xx − B x B y B xy + (1 + B x ) B yy = 0 , which is called the equation of minimal graphic for smooth surface of the Euclidean space R ≡ E . In this case Bernstein showed that if U = R then the solution of equation (12)is a plane.(2) When B ( x, y ) ≡ E given by an unique function A ( x, y ): f ( x, y ) = ( A ( x, y ) , x, y, ∈ L , with the induced metric tensor over f ( U ) as a spacelike surface of R . System (10)becomes to the equation(13) (1 − A y ) A xx + 2 A x A y A xy + (1 − A x ) A yy = 0 with A x < A x + A y < , which is called the equation of minimal graphic for smooth surface of the Lorentzian space L . For this case, Calabi showed that if U = R then the solution of equation (13) is aplane.Now we turn our attention for graphic minimal spacelike surfaces of the second type,given by the representation f ( x, y ) = ( x, A ( x, y ) , B ( x, y ) , y ). In this case,(3) When B ( x, y ) ≡ A ( x, y ): f ( x, y ) = ( x, A ( x, y ) , , y )) ∈ L , with the induced metric tensor over f ( U ) as a spacelike surface of R . Then system (10)becomes to the equation(14) (1 + A y ) A xx − A x A y A xy + ( − A x ) A yy = 0 with A x > A y + 1 , and, we will say that this equation is the equation for graphic of second type of minimalsmooth surface of L .4. About the Extension of Local Solutions of the Graphic Equations
In this section we study whether it is possible to extend to whole the complex plane C the local solutions for the graphic equations given in system (10).We start identifying a formula for the Gauss curvature of the surface. In fact, for f w = µW ( a, b ) where ( U, f ) is a minimal spacelike surface of R , with holomorphic functions a ( w ), b ( w ), µ ( w ), we know that the expression for the Gauss curvature is given by K ( f ) = − ∆ ln λ λ = − λ ∆ ln λ. INIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONS IN R Now, since λ = 4 µµ (1 − ab )(1 − ab ) and ∆ = 4 ∂ ww , we obtain K ( f ) = − (ln(1 − ab )(1 − ab )) ww µµ (1 − ab )(1 − ab ) . Since (ln(1 − ab )(1 − ab )) ww = − a w (cid:18) b − ab (cid:19) w − b w (cid:18) a − ba (cid:19) w , it follows that(15) K ( f ) = ℜ ( a w b w (1 − ab ) ) µµ (1 − ab ) (1 − ab ) . First case . We will focus our attention to find surfaces given by X ( x, y ) = ( A ( x, y ) , x, y, B ( x, y )) for all ( x, y ) ∈ R , satisfying the equations (10), which means that X ( R ) = S is a minimal surface of R .So a question arises: Is there a non-flat solution to this problem ?For answering that question we proceed as follows. First, we construct a pointwise basisfor the normal bundle. In fact, it takes the vector fields N and N , along S = X ( R ),used in the proof of Proposition 3.1, namely, N = (1 , A x , A y ,
0) and N = (0 , − B x , − B y , . Then we have the following proposition.
Proposition . The spacelike Gauss map ν ( x, y ) for the minimal surface S ⊂ R isgiven by ν ( x, y ) = 1 p B x ) + ( B y ) (0 , − B x , − B y , . Proof.
We only need to see if the orientation of { N , N } and the orientation of { ∂ , ∂ } are compatible each other. The compatibly orientations follow from the projectedvectors ( N , , , N ) = ∂ and ( N , , , N ) = ∂ . (cid:3) Corollary . The Gauss map ν : S −→ S ⊂ E is such that ν = 1 p B x ) + ( B y ) > . In other words, ν ( S ) is the (open) north hemisphere of the Riemann sphere S . Now we assume that we have a local representation (
U, f ) such that f ( U ) ⊂ S and f w = µ ( a + b, ab, i (1 − ab ) , a − b ) , where a, b, µ are holomorphic functions from U into C , and U is a connected and simplyconnected open subset of C . Then the normal bundle has a pointwise basis of lightlike vectors { L ( a ) , L ( b ) } like in Lemma 3.2, which allows, in easier form, to compute thefourth component of the spacelike Gauss map ν ( a, b ), as follows. Lemma . For an isothermic local representation ( U, f ) such that f ( U ) ⊂ S we have (16) ν ( a, b ) = 1 | − ab | p | a | p | b | (1 − | ab | ) . Moreover, the maximal extension of holomorphic functions a, b , is conditioned by theinequalities: (17) | − ab | 6 = 0 and | ab | = 1 . Proof.
Taking the normalization of the vector N given by N ( a, b ) = 11 + bb L ( b ) −
11 + aa L ( a )one gets ν ( a, b ) since N ( a, b ) = 0. Therefore, we obtain the component ν ( a, b ) given in(16) and the inequalities (17). (cid:3) Now we observe that the first inequality in (17) is the functional area √ EG − F = | µ | | − ab | . Then for our purposes, we will find a necessary and sufficient conditionto obtain a maximal extension of the function √ EG − F . Hence if we assume theintegrating factor being constant µ = 1, we need just to consider the maximal extensionof | − ab | .For achieving that goal we give the next corollary, which follows from Liouville Theo-rem and Theorem 4.3, since for a ( w ) b ( w ) being an entire bounded function, it must beconstant. Corollary . If a ( w ) and b ( w ) can be extended for whole the plane C , then thereexists a constant c ∈ C such that a ( w ) b ( w ) = c . Hence it follows as direct consequence of Corollary 4.4, that if a ( w ) = b ( w ) or a ( w ) = − b ( w ) for all w ∈ C , then a ( w ) = √ c . That means that ( C , f ) is a spacelike plane of R .Moreover from the Corollary 4.4, we can also construct an example of a minimal surface( C , f ), which is a graphic with Gauss curvature K ( f ) = 0. This means a set of points p of the surface such that the condition K ( p ) = 0 is not satisfied on the entire plane C .Even more, now we are abled to prove our next result which provides a general class ofexamples of entire graphic minimal surfaces of first type such that the Gauss curvature K ( f ) = 0. Theorem . Let a = a ( w ) be a holomorphic function defined in whole the plane C such that a ( w ) = 0 for each w ∈ C . Let c = α + iβ ∈ C \ { , , − } such that α + β = 1 ,and it takes the holomorphic function b ( w ) = ca ( w ) from C in C . Then the surface given INIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONS IN R by (18) f ( w ) = X + 2 ℜ Z w (cid:18) a ( ξ ) + ca ( ξ ) , c, i (1 − c ) , a ( ξ ) − ca ( ξ ) (cid:19) dξ, is a minimal surface of R , which is a graphic surface of type X ( x, y ) = ( A ( x, y ) , x, y, B ( x, y )) through of the transformation of coordinates given by x w = (1 + c ) and y w = i (1 − c ) .Moreover, assuming that a ( w ) is not a constant function then, there exists a point p ∈ S such that K ( p ) = 0 . Hence the surface can not be contained in hyperplanes of R . Proof.
Taking x ( u, v ) = 2[(1 + α ) u − βv ] and y ( u, v ) = 2[ βu + ( α − v ] we get theequation of the coordinates change, namely,(19) (cid:20) uv (cid:21) = 12[ α + β − (cid:20) α − β − β α (cid:21) (cid:20) xy (cid:21) . Therefore, since a ( w ) and b ( w ) are holomorphic functions and α + β = 1, we obtainthat equation (18) represents a graphic minimal surfaces of first type.Since the metric is given by λ = 4 | − ac/a | follows that ∆ ln λ = 0 in points where a w ( w ) = 0. Then, since K ( f ) = − λ ∆ ln λ , it follows that in those points happen K ( f ) = 0.Next, by integration we can obtain the components functions A ( w ) = f ( w ) and B ( w ) = f ( w ), and through of the coordinate transformation given by the equation(19) we obtain the explicit representation as graphic surface.To finish, we see the real spacial property of surface S . In fact, it supposes that thereis a vector v = ( v , v , v , v ) ∈ R such that h v, f w i = 0. Then from the equality − v ( a + b ) + v (1 + ab ) + iv (1 − ab ) + v ( a − b ) = 0, we obtain( v − v ) a − ( v + v ) b + ( v + iv ) + ab ( v − iv ) = 0 . It defines T = v − v , S = v + v , Z = v + iv , then we obtain ( T a + Z ) + b ( aZ − S ) = 0 , which implies that b = T a + ZS − aZ = ca if and only if T = 0 = S and c = − ZZ .
Thus, from ZZ = − c and v − v = 0 = v + v , it follows that v R . Contradiction. (cid:3) So from Theorem 4.5 we can construct a classe of minimal graphic surfaces of first type,whose Gauss curvature is not null in some points of the surface. That means the classicBernstein theorem does not hold in this case. Next we give some particular examples ofthat fact.
Example . For a simple example, we take a = e w and c = 2 . Then according to The-orem 4.5 we can take b = ca = 2 e − w and X = 2( − , , , , to have the parametrization f ( w ) = 2(( e u − e − u ) cos v, u, v, ( e u + 2 e − u ) cos v ) . Therefore taking the coordinates transformation given by x = 6 u and y = 2 v , we get thegraphic parametrization given by X ( x, y ) = (2( e x − e − x ) cos( y , x, y, e x + 2 e − x ) cos( y , for which there are points such that the Gaussian curvature is not zero. In fact, it is justto take α and β such that α = cos(2 y ) and β = sin(2 y ) , that means, c = e iy . Example . In this example we use Theorem 4.5 to construct minimal graphic surfacesof first type. We start assuming a ( w ) = e w and b ( w ) = e iθ a ( w ) for θ ∈ (0 , π ) . Since | c | = | e iθ | = 2 , the condition α + β = 1 is hold. Then W ( a, b ) is given by W ( a, b ) = ( e w + 2 e iθ e − w , e iθ , i (1 − e iθ ) , e w − e iθ e − w ) . Now we take the factor of integration µ = 1 , to obtain the integral representation (18)given by f ( w ) = 2 ℜ Z w ( e ξ + 2 e iθ e − ξ , e iθ , i (1 − e iθ ) , e ξ − e iθ e − ξ ) dξ, more explicitly (20) f ( u, v ) = 2( e u cos v − e − u (cos v cos θ + sin v sin θ ) , (1 + 2 cos θ ) u − v sin θ, ( − θ ) v + 2 u sin θ, e u cos v + 2 e − u (cos v cos θ + sin v sin θ )) . Hence making the coordinates transformation x w = 1 + 2 e iθ and y w = i (1 − e iθ ) , we get x = 2[(1 + 2 cos θ ) u − v sin θ ] and y = 2[( − θ ) v + 2 u sin θ ] . Thus the minimal graphic surface is given by X ( x, y ) = ( A ( x, y ) , x, y, B ( x, y )) , where thefunctions A ( x, y ) , B ( x, y ) are given by the first and fourth component of formula (20) with u = 16 ((2 cos θ − x + 2 y sin θ ) and v = 16 ( − x sin θ + (1 + 2 cos θ ) y ) . We observe that since a w = e w never vanishes, all the points of the graphic surface aresuch that K ( p ) = 0 . Second case . We will focus our attention to find surfaces given by X ( x, y ) = ( x, A ( x, y ) , B ( x, y ) , y ) for all ( x, y ) ∈ R , satisfying the equations (10). That means that X ( R ) = S is a graphic minimal surfaceof R of second type.So a question arises: Is there a non-flat solution to this problem ?For answering that question we proceed as before, constructing first a pointwise basisfor the normal bundle.
INIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONS IN R Let us take the attitude matrix of dX :[ dX ] t = (cid:20) A x B x A y B y (cid:21) . The unit spacelike Gauss map ν = ν ( x, y ) is given by ν ( x, y ) = 1 p J + ( B x ) + ( A x ) (0 , B x , − A x , J )for J = ∂ ( A,B ) ∂ ( x,y ) = A x B y − A y B x . Since we can not control the functions ν i for i = 1 , ,
3, we will work with the Weierstrassform f w = µ ( a + b, ab, i (1 − ab ) , a − b )and the transformation of coordinates(21) x w = µ ( a + b ) and y w = µ ( a − b ) , where x w y w − x w y w = 2 | µ | ( ab − ab ) . Lemma . It considers the transformation of coordinates given by equations (21).Then Jacobian function x w y w − x w y w = 2 | µ | ( ab − ab ) does not vanish in a domain U ⊂ M if and only if, for each w ∈ U , (22) a ( w ) = 0 = b ( w ) and ℑ ( a ( w ) b ( w ) ) = 0 . A maximal extension of holomorphic functions a, b is conditioned by the inequalities(22) and by | − ab | 6 = 0 . Proof.
First we observe that a ( w ) = 0 = b ( w ) is a necessary condition. Moreover, foreach w ∈ U , − i ℑ ( a ( w ) b ( w ) ) = a ( w ) b ( w ) − a ( w ) b ( w ) = a ( w ) b ( w ) − a ( w ) b ( w ) b ( w ) b ( w ) . Hence, since the Jacobian function does not vanish, it follows that ℑ ( a ( w ) b ( w ) ) = 0. Theconversely follows immediately. (cid:3) From Lemma 4.6 and from Little Picard Theorem, it follows the next corollary.
Corollary . It assumes that the holomorphic functions a ( w ) and b ( w ) can beextended for whole the plane C . Then there exists a constant c ∈ C \ { , , − } such that b ( w ) = ca ( w ) .Moreover, as consequence, if f w is such that f w = µ ( a (1 + c ) , ca , i (1 − ca ) , a (1 − c )) then x ( w ) = 2 ℜ (cid:18) (1 + c ) Z w µ ( ξ ) a ( ξ ) dξ (cid:19) and y ( w ) = 2 ℜ (cid:18) (1 − c ) Z w µ ( ξ ) a ( ξ ) dξ (cid:19) . Taking P ( w ) + iQ ( w ) = R w µ ( ξ ) a ( ξ ) dξ and c = α + iβ we obtain x ( u, v ) = 2[(1 + α ) P ( w ) − βQ ( w )] and y ( w ) = 2[(1 − α ) P ( w ) + βQ ( w )] . Proof.
Since for a ( w ) = 0 = b ( w ), with a ( w ) b ( w ) entire and such that ℑ ( a ( w ) b ( w ) ) = 0 (Lemma4.6), the map a ( w ) b ( w ) does not cover whole the complex plane, then from Little PicardTheorem, it follows that a ( w ) b ( w ) is constant. Under the hypotheses that constant can not be0, 1 neither -1. (cid:3) Remark . We observe that Corollary 4.7 has a weakness because while in Theorem 5.7the equation (19) gives us the inversion function which is linear, and which we can use toconstruct the graphic over whole the complex plane C , Corollary 4.7 can not guarantee thatwe have a graphic over all complex plane, since it could exist ramifications. For instance,taking a ( w ) = e w and µ = 1 , we obtain P ( u, v ) = e u cos v and Q ( u, v ) = e u sin v . So, x ( u, v ) = 2[(1 + α ) e u cos v − βe u sin v ] and y ( u, v ) = 2[(1 − α ) e u cos v + βe u sin v ] , whichare periodic functions in the variable v . In the next theorem we answer the question whether there exist a non-flat solutionwhich is entire graphic surface of second type. In fact, we argue that if a = a ( w ) is agiven holomorphic function defined in whole C and such that a ( w ) = 0, then we can takethe holomorphic function µ ( w ) = a ( w ) and take also f w = µW ( a ( w ) , ca ( w )), with constant c ∈ C \ { , , − } . Then next we will show that in this case, it can exist points in thesurface such that the Gauss curvature is not zero. Theorem . Let a = a ( w ) be a holomorphic function defined in whole the plane C such that a ( w ) = 0 for each w ∈ C . For c = α + iβ ∈ C \ R we take b ( w ) = ca ( w ) and µ ( w ) = a ( w ) . Then the surfaces given by (23) f ( w ) = X + 2 ℜ Z w (cid:18) c, a ( ξ ) + ca ( ξ ) , i (cid:18) a ( ξ ) − ca ( ξ ) (cid:19) , − c (cid:19) dξ, are minimal surfaces of R , which represent graphic of type X ( x, y ) = ( x, A ( x, y ) , B ( x, y ) , y ) ,where the transformation of coordinates is given by x w = (1 + c ) and y w = (1 − c ) .Moreover, in this case, the Gauss curvature K ( f )( w ) = 0 if and only if a w ( w ) = 0 .Therefore, assuming that a = a ( w ) is not a constant function, there exists p ∈ S suchthat K ( p ) = 0 . Again, there is not a hyperplane containing the surface S . Proof.
By integration we obtain x = 2 ℜ (((1 + α ) + iβ )( u + iv )) = 2[(1 + α ) u − βv ]and y = 2[(1 − α ) u + βv ]. That means,(24) (cid:20) uv (cid:21) = 14 β (cid:20) β βα − α + 1 (cid:21) (cid:20) xy (cid:21) . Since a, b and µ are holomorphic functions, formula (23) in the ( x, y )-coordinates, rep-resents a graphic minimal surface of second type. Moreover, since the Gauss curvature INIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONS IN R is given by K ( f ) = − λ ∆ ln λ where λ = 4 | aa − c | , it follows that ∆ ln λ = 0 in pointswhere a w ( w ) = 0. Hence in those points K ( f ) = 0.Next, by integration we can obtain the components functions A ( w ) = f ( w ) and B ( w ) = f ( w ), and through of the transformation of coordinate we get the explicitrepresentation as graphic surface.Finally we note that it is needed to assume c R , since we can not have x w y w − x w y w =0. It is also impossible to obtain a timelike vector v ∈ R such that h v, f w i = 0, so wehave the real spacial property of the surface in R . (cid:3) So from Theorem 4.8 one can construct a classe of minimal graphic surfaces of secondtype, whose Gauss curvature is not null in any point of the surface. That means theproperty of Bernstein does not hold in this case. The following explicit example illustratesthis fact.
Example . We use Theorem 4.8 to construct second type of minimal graphic surfaces.Let a = e w and b = e iθ a for θ ∈ (0 , π ) . Then the expression of W ( a, b ) is W ( a, b ) = ((1 + e iθ ) e w , e iθ e w , i (1 − e iθ e w ) , (1 − e iθ ) e w ) . Taking the factor µ ( w ) = e − w , the integral representation (23) is given by f ( w ) = 2 ℜ Z w (cid:0) e iθ , e − ξ + e iθ e ξ , i ( e − ξ − e iθ e ξ ) , − e iθ (cid:1) dξ, or more explicitly (25) f ( u, v ) = 2((1 + cos θ ) u − v sin θ, − e − u cos v + e u (cos v cos θ − sin v sin θ ) , − e − u sin v + e u (sin v cos θ + cos v sin θ ) , (1 − cos θ ) u + v sin θ )) . Now making the transformation of coordinates x w = 1 + e iθ and y w = 1 − e iθ , we get x = 2[(1 + cos θ ) u − v sin θ ] and y = 2[(1 − cos θ ) u + v sin θ ] , and hence the graphic minimal surface is given by X ( x, y ) = ( x, A ( x, y ) , B ( x, y ) , y ) wherethe functions A ( x, y ) and B ( x, y ) are given by the second and third component of formula(25) with u = x + y v = 14 sin θ [( − θ ) x + (1 + cos θ ) y ] . Finally we observe that since a w = e w never vanishes, for all the points of the surface onegets that K ( p ) = 0 . The construction of the conjugated surface ( M, Y )We dedicate this section for looking the explicit expression of the conjugated surfaceto a minimal spacelike surface (
M, X ) of R , using the Weierstrass notation. We startdefining a special operator on tangent bundle T S to a surface, as follows.
Definition . Let ( M, X ) be a spacelike surface with line element ds ( X ) = Edx +2 F dxdy + Gdy , and T S be its tangent bundle, where, pointwise, { X x ( p ) , X y ( p ) } is a baseof T p S . Let J : T S −→ T S be the function given by (26) J ( V ) = 1 √ EG − F ( h X x , V i X y − h X y , V i X x ) . Proposition . Let J : T S −→ T S be the function given by the equation (26). Then ∀ V ∈ T S, the following equations are satisfied: h V, J ( V ) i = 0 , h J ( V ) , J ( V ) i = h V, V i and J ( J ( V )) = − V. Proof.
The first equation follows from √ EG − F h V, J ( V ) i = h X x , V ih X y , V i −h X y , V ih X x , V i = 0. For getting second equation we take the values of J in the basis,namely,(27) J ( X x ) = 1 √ EG − F ( EX y − F X x ) and J ( X y ) = 1 √ EG − F ( F X y − GX x ) . Then h J ( X x ) , J ( X x ) i = E, h J ( X y ) , J ( X y ) i = G and h J ( X x ) , J ( X y ) i = F. Now we note that from the pointwise bi-linearity of <, > , it follows the pointwise linearityof J . Therefore if V = aX x + bX y the second equation of the proposition holds.The third equation follows directly from the linearity and from the facts J ( J ( X x )) = − X x and J ( J ( X y )) = − X y . (cid:3) We observe that if S = ( M, X ) be a spacelike surface of R , the vector 1-form associatedto S is given by β = X x dx + X y dy . Therefore, by definition J ( β ) is the 1-form given by(28) J ( β ) = J ( X x ) dx + J ( X y ) dy. Next we related the operator J with the special normal frame { τ, ν } in R .Let l = X ( v , v , v ) be the exterior product in R of a set of vectors { v , v , v } . Bydefinition, since Ω( R ) = ( − dx ) ∧ dx ∧ dx ∧ dx is the volume form, then l is definedby h l, w i = Ω( R )( v , v , v , w ) , ∀ w ∈ R . Then the J operator is equivalent to J ( V ) = X ( τ, ν, V ). Theorem . Let S = ( M, X ) be a spacelike surface of R and let β = X x dx + X y dy be the vector -form associated to S . Then (29) J ( β ) = − F dx − Gdy √ EG − F X x + Edx + F dy √ EG − F X y . The -form J ( β ) is closed if and only if ( M, X ) is a minimal spacelike surface. INIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONS IN R Proof.
The equation (29) follows from equations (27) and (28).For the second statement, we use the representation of the operator J as an exteriorproduct, to obtain J ( β ) = X ( τ, ν, X x dx + X y dy ) = τ × ν × β. Now, since dβ = 0, we get the exterior derivative dJ ( β ) = (( dτ ) × ν × β )+( τ × ( dν ) × β ).Next we will calculate explicitly dJ ( β ). For that, we use dτ = τ x dx + τ y dy , dν = ν x dx + ν y dy , the Weingarten formulas (3), (4) and the anti-commutative properties of theexterior product in R and of the exterior product of 1-forms, to obtain d ( J β ) = ( h + h )( X x × ν × X y ) dx ∧ dy + ( n + n )( τ × X x × X y ) dx ∧ dy. Since X x × ν × X y = −√ EG − F τ and τ × X x × X y = √ EG − F ν one gets(30) dJ ( β ) = − H X √ EG − F dx ∧ dy. Hence it follows from equation (30) that, dJ ( β ) = 0 if and only if ( M, X ) is minimal. (cid:3)
Theorem 5.3 allows us to establish the next corollary which shows the explicit expressionof the minimal conjugate spacelike surface (
M, Y ) in R . It comes from the fact that since J ( β ) is a closed 1-form in a connected simply-connected open subset of C then it is exact. Corollary . Let M be a connected and simply connected open subset of the plane C , and let ( M, X ) be a solution of the minimal graphic equations (10). The integralrepresentation (6) can be extended to Z = X + iY ∈ C by (31) Z ( x, y ) = Z ( x , y ) + Z zz β + iJ ( β ) , where (32) Y ( x, y ) = Y ( x , y ) + Z zz − F dx − Gdy √ EG − F X x + Edx + F dy √ EG − F X y . Moreover, Y gives us the parametrization of the conjugated minimal spacelike surface ( M, Y ) of R . Proof.
Since J ( dY ) = J ( J ( dX ) = − dX = − ( X x dx + X y dy ) is a closed vector 1-form,from Theorem 5.3 it follows that H Y ( p ) = 0 for each p ∈ M . (cid:3) Example . Let X ( x, y ) = (0 , x cos y, x sin y, y ) be a parametrizated Helicoid of E .The conjugated minimal spacelike surface, given by equation (32) with Y (0 ,
0) = (0 , , , ,is the Catenoid given in coordinates by Y ( x, y ) = (0 , −√ x sin y, √ x cos y, ln( x + √ x )) . In fact, from X x = (0 , cos y, sin y, and X y = (0 , − x sin y, x cos y, it follows that E = 1 , F = 0 and G = 1 + x . Now from the integral equation (32) we obtain dY = 1 √ x (0 , − x sin y, x cos y, dx − √ x (0 , cos y, sin y, dy. Hence by integrating Y x = √ x (0 , − x sin y, x cos y, and Y y = −√ x (0 , cos y, sin y, ,we get the Catenoid surface ( R , Y ( x, y )) .Moreover, if x ≥ we have the part corresponding to Y ≥ and, if x ≤ we have thepart corresponding to Y ≤ . Both surfaces ( R , X ( x, y )) and its conjugated ( R , Y ( x, y )) are ramified.Finally, if we make x = sinh u and y = v , we obtain ˜ X ( u, v ) = (0 , sinh u cos v, sinh u sin v, v ) and ˜ Y ( u, v ) = (0 , − cosh u sin v, cosh u cos v, u ) , in the isothermic coordinates ( u, v ) . As it is expected it follows ˜ X u = − ˜ Y v and ˜ X v = ˜ Y u . Next example shows an applicability of the J operator. Example . Let X ( x, y ) = ( x cosh y, x sinh y, f ( x ) , be a graphic type of hyperbolicrotation in R in hyperbolic polar coordinates. Since X x = (cosh y, sinh y, f ′ ( x ) , and X y = ( x sinh y, x cosh y, , , we get E ( x, y ) = − f ′ ( x )) > , F ( x, y ) = 0 , G ( x, y ) = x > , W = x p ( f ′ ) − . Hence, a needed condition for obtaining a minimal spacelike surface Y , in terms of theoperator J , is dJ ( β ) = dJ ( dX ) = 0 . Then the Y -coordinate gives us the equation xf ′ √ − f ′ ) = k or more specifically ( k − x )( f ′ ) = k with k ∈ R − { } , | x | < | k | . Now integrating, one obtains f ( x ) = b + ( ± k ) arcsin ( x/k ) . Assuming k > and b = 0 , we get the parametric surface X ( x, y ) = ( x cosh y, x sinh y, k.arcsin ( x/k ) , . If we take x = k sin u and y = v , we get the correspondent minimal parametric surfacewith isothermic parameters given by f ( u, v ) = k (sin u cosh v, sin u sinh v, u, , where E ( u, v ) = k sin u = G ( u, v ) and with lightlike singularities for f u ( u, v ) when u = nπ , n ∈ Z . INIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONS IN R In similar way we get the surface given by g ( u, v ) = k (cos u cosh v, cos u sinh v, v, , which is a minimal ruled surface with the same metric tensor, it is a type of hyperbolichelicoid surface of R . Generalized Cauchy-Riemann equations over ( M, X ) . In this subsection wecontinue studying the local geometry of the spacelike surfaces in R . In particular in thisfirst part, we identify the generalized Cauchy-Riemann type equations over the surface( M, X ) when the parameters are not isothermic, and then we obtain the needed conditionsto extend in continua way any local solution of those equations.For starting, we observe that if we have a sub-surface f ( U ) ⊂ X ( M ) with isothermicparameters w = ( u, v ) ∈ U such that X ( x, y ) = f ( u ( x, y ) , v ( x, y )), then ∂X∂x = ∂u∂x f u + ∂v∂x f v and ∂X∂y = ∂u∂y f u + ∂v∂y f v . Lemma . For each local solution of the equations (33) ∂w∂y = α ( x, y ) ∂w∂x where α ( x, y ) = F ( x, y ) + i p E ( x, y ) G ( x, y ) − F ( x, y ) E ( x, y ) , in a neighborhood U ⊂ M of a point p ∈ M , there exists a parametric isothermic sub-surface ( U, f ) of ( M, X ) such that X ( x, y ) = f ( u ( x, y ) , v ( x, y )) . Moreover, αα = GE . Proof.
Let W = √ EG − F be the area function in coordinates z = x + iy ∈ U .Taking the operator J , since J ( f u ) = f v , J ( f v ) = − f u , it follows J ( X x ) = u x J ( f u ) + v x J ( f v ) = u x f v − v x f u . Hence by equation (27) one gets u x f v − v x f u = EW ( u y f u + v y f v ) − FW ( u x f u + v x f v ) . From this last equation, we obtain the following equations with matrix representation(34) (cid:20) u y v y (cid:21) = (cid:20) F/E − W/EW/E F/E (cid:21) (cid:20) u x v x (cid:21) and (cid:20) u x v x (cid:21) = EG (cid:20) F/E W/E − W/E F/E (cid:21) (cid:20) u y v y (cid:21) . Now we observe that the square matrices of order 2 × u y + iv y = F + iWE ( u x + iv x ) , that is equation (33). (cid:3) We note that if the ( x, y )-coordinates are already isothermic coordinates then α = i andso equations (33) for ( M, X ) becomes to the classic expression of the Cauchy Riemannequations, namely, u y = − v x and u x = v y . So we will call equations (34) or (33) as thegeneralized Cauchy-Riemann equations for ( M, X ).Then as expected we have the following definition-corollary.
Corollary . A smooth function h = ϕ + iψ : S −→ C is generalized holomorphicover the Riemann surface S = X ( M ) if and only if (35) (cid:20) ϕ y ψ y (cid:21) = (cid:20) F/E − W/EW/E F/E (cid:21) (cid:20) ϕ x ψ x (cid:21) or ∂h∂y = α ∂h∂x , where W = √ EG − F . Proof.
Since in an isothermic neighborhood ( U, ˜ h ) the function ˜ h ( u, v ) is holomorphicin the sense of complex variable if and only if h ( x, y ) is holomorphic over S , we have ∂h∂x = ∂ ˜ h∂u ( u x + iv x ) and ∂h∂y = ∂ ˜ h∂u ( u y + iv y ) , because i ˜ h u = ˜ h v holds for C -holomorphic functions. Therefore h y = αh x follows fromthe definition of the function α ( x, y ). The conversely is immediate. (cid:3) We note that for smooth function h = ϕ + iψ : U ⊂ S −→ C is a generalized holomorphicif and only if in isothermic coordinates ( u, v ) the harmonic functions ϕ, ψ are conjugatedharmonic functions satisfying the usual Cauchy-Riemann equations.If we use the operator J , we can also give an equivalent definition, namely: h is ageneralized holomorphic function if and only if dJ ( ϕ ( x, y )) = dψ ( x, y ) and dJ ( ψ ( x, y )) = − dϕ ( x, y ) . They are generalized harmonic functions conjugated each other.Next we are interested in relating the isothermic neighborhood (
U, f ) with the Weier-strass datas a ( w ) and b ( w ) for graphic spacelike surfaces in R .In fact, fixing the semi-rigid referential associated to ( M, X ) given by M = { l ( b ( p )) , e ( p ) , e ( p ) , l ( a ( p )) } where e ( p ) = 1 √ E ∂X∂x and e = J ( e ) = 1 √ E J ( X x ) , we obtain the next result. Proposition . Let S = ( M, X ) be a solution of the minimal graphic equation in R and ( U, f ) be a given locally isothermic sub-surface of S . Let r ( u, v ) be a real-valued INIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONS IN R function and M ( ϑ ) = { l ( b ) , e , e , l ( a ) } ( u,v ) be the semi-rigid referential associated to f w ( w ) = µ ( w ) W ( a ( w ) , b ( w )) = r ( w )(ˆ e ( w ) − i ˆ e ( w )) . Then the following relation is hold: ˆ e ( w ) − i ˆ e ( w ) = (cos ϑ e + sin ϑ e ) − i ( − sin ϑ e + cos ϑ e )) = e iϑ ( e − ie ) . From Proposition 5.7 it follows that if the coordinates (
M, X ), (
U, f ) and ( ˜
U , ˜ f ) arounda point p ∈ f ( U ) ∩ ˜ f ( ˜ U ) are related by the equations X ( x, y ) = f ( u ( x, y ) , v ( x, y )) = f ◦ Φ( x, y ) , X ( x, y ) = ˜ f (˜ u ( x, y ) , ˜ v ( x, y )) = ˜ f ◦ ˜Φ( x, y ) , then the transition functions are given by(36) f ◦ Φ( x, y ) = ˜ f ◦ ˜Φ therefore Ψ = ˜Φ ◦ Φ − = ˜ f − ◦ f. Now, applying the Proposition 5.7, we obtain:1ˆ r f w = e i ˆ φ (ˆ e − i ˆ e ) with 1˜ r ˜ f ˜ w = e i ˜ φ (˜ e − i ˜ e ) , which imply that the angle functions are related each other by the equation:(37) ˆ φ ( u, v ) − ˜ φ ◦ Ψ( u, v ) = ˆ ϑ ( u, v ) − ˜ ϑ ◦ Ψ( u, v ) . Now we have the following facts, which come from equation (37).(1)
If two holomorphic functions agree with each other along a Jordan arc, then theyagree with each other along all connected component of this arc .From (1) we obtain.(2) If ( U, f ) and ( ˜ U , ˜ f ) agree with each other along an Jordan arc in S , they agree witheach other along the open subset f ( U ) ∩ ˜ f ( ˜ U ).(3) The overlapping or transition map between two isothermic coordinates system for aspacelike surface of R are holomorphic function in sense of complex analysis .(4) Each holomorphic function h : U ⊂ C −→ V ⊂ C can be seen as a pointwise C -linear transformation dh z : T z C −→ T h ( z ) C that preserves oriented angles . Lemma . The angle function ˜ ϑ − ϑ determines the transition map of ( U, f ) and ( ˜ U , ˜ f ) for two isothermic parametrizations of the neighborhood f ( U ) ∩ ˜ f ( ˜ U ) ⊂ S , around p ∈ S . From Lemma 5.8 it follows the next result about the extension of local solutions ofequation (33).
Proposition . Let w, ˜ w two local solutions of equation (33), around a point p ∈ S ,with w y = αw x and ˜ w y = α ˜ w x . If w x = ˜ w x then w y = ˜ w y .Therefore, all local solution of the equation (33) can be continuously extended whenever E ( x, y ) > and √ EG − F ( x, y ) > . Proof.
The conclusions are immediate. (cid:3)
Next we prove that the solutions w = ( u, v ) of the generalized Cauchy-Riemann equa-tions (34) or (33), are of the form of Nitsche type functions (equation (8), page 23 of[7]). Theorem . The solution for equations (34) are given by Nitsche type functions,that means (38) u = u ( x, y ) = x + R zz Edx + F dyW v = v ( x, y ) = y + R zz F dx + GdyW . Moreover, from equations (38), it is possible to obtain global isothermic coordinates ( U, f ) for the surface S = X ( M ) . Proof.
In fact, since ∂u∂x = W + EW , ∂u∂y = FW , ∂v∂x = FW , ∂v∂y = W + GW the matrix equation (34) is satisfied. In fact, remembering that W + F = EG , we obtain (cid:20) F/W ( W + G ) /W (cid:21) = (cid:20) F/E − W/EW/E F/E (cid:21) (cid:20) ( E + W ) /WF/W (cid:21) . Now, since the solution for equations (34) are in the form (38), we obtain that the localisothermic parameters ( u, v ) can be extended globally for the surface S since the conditionsof Proposition 5.9 are hold. (cid:3) We highlight in this moment that our generalized Cauchy-Riemann equations (34)and its solutions in (38) can be applicated when we want to construct the conjugateminimal spacelike surface (
M, Y ) (32), since the solutions (38) involve terms of the localparametrization of (
M, Y ).Finally we have the following corollary for equations of minimal graphic surfaces in R . Corollary . If S = ( R , X ) is a solution of the minimal graphic equation (10)then, for all p ∈ S , the functions a ( w ) and b ( w ) satisfy either b ( p ) = ca ( p ) with c / ∈ {− , } or a ( p ) b ( p ) = c with ℑ ( c ) = 0 for some constant c ∈ C . The Bernstein Theorem and the Calabi Theorem follows from that c = 1 and c = − and for the second type of surfaces from ℑ ( c ) = 0 .Finally, if as a submanifold of the topological vector space R there exists S = ( R , X ) such that with the induced metric of R , is a spacelike graphic solution in connected andsimply connected open subset M ⊂ C , with the condition that in some point p ∈ S thefollowing statement fails:“ either b ( p ) = ca ( p ) with c / ∈ {− , } or a ( p ) b ( p ) = c with ℑ ( c ) = 0 and for someconstant c ∈ C ”,then the points X ( x, y ) where EG − F = 0 , are points such that the tangent planes of X ( R ) are tangent to the lightcone of R . INIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONS IN R So from the second part of Corollary 5.11, we have found conditions to create graphicminimal spacelike surfaces which have new type of singularities, it called lightlike singu-larities, as defined by Kobayashi in [5]. Those singularities are points where the tangentplane of the surface is also tangent to the lightcone of R .6. A Particular Family of Minimal Surfaces of R In this section we construct examples of minimal spacelike surfaces in R which arevery close related to surfaces in E and L .For the representation f w = µ ( a + b, ab, i (1 − ab ) , a − b ) with µ, a, b holomorphicfunctions from M into C , with M being connected and simply connected open subset ofthe complex plane, we assume the relation b = ae iθ for a parameter θ ∈ R . Definition . A θ -family is a set of minimal surfaces defined on a connected andsimply connected domain M ⊂ C , linking each other by a parameter θ ∈ R , given by thefollowing equation (39) F ( θ ; w ) = P + 2 ℜ Z ww µ ( ξ ) (cid:0) (1 + e iθ ) a ( ξ ) , e iθ a ( ξ ) , i (1 − e iθ a ( ξ )) , (1 − e iθ ) a ( ξ ) (cid:1) dξ. When θ = 0 we say that the surface of L , given by X ( w ) = F (0; w ) , is the initial surfaceof the family, and when θ = π we say that the surface of E , given by Y ( w ) = F ( π ; w ) , isthe associated surface of the initial surface of the family. Lemma . For a θ -family ( M, F ( θ ; w )) of minimal spacelike isothermic parametricsurfaces in R the equations that related the initial surface ( M, X ) and the associatedsurface ( M, Y ) , are given by: (40) ∂Y ∂w = ∂X ∂w , ∂Y ∂w = − i ∂X ∂w and ∂Y ∂w = i ∂X ∂w . Proof.
The equations of lemma follows from X w = µ (2 a, a , i (1 − a ) ,
0) and Y w = µ (0 , − a , i (1 + a ) , a ). (cid:3) Now we construct an example for these equations:
Example . Let ( M, X ) be the minimal spacelike surface of L given, in isothermicparameters, by X ( u, v ) = ( u, sinh u cos v, sinh u sin v, . Since X u = (1 , cosh u cos v, cosh u sin v, and X v = (0 , − sinh u sin v, sinh u cos v, weobtain λ ( X ) = sinh u . We assume that ( u, v ) ∈ M for u > .Therefore, it follows X w = (1 , cosh w, − i sinh w, . To obtain the associated surfacewe find the functions a ( w ) and µ ( w ) . In fact, since µa = 12 , µ (1 + a ) = cosh w , iµ (1 − a ) = − i sinh w , it follows that µ ( w ) = e − w and a ( w ) = e w .For obtaining the associated surface ( M, Y ) , we use Y w = µ (0 , − a , i (1 + a ) , a ) ,and so the surface is such Y w = (0 , − sinh w, i cosh w, . Hence the holomorphic integralcurve is given by ˜ Y ( w ) = 12 (0 , − cosh w, i sinh w, w ) . Thus, the real part of ˜ Y gives us a Catenoid of E parametrized by Y ( u, v ) = (0 , − cosh u cos v, − cosh u sin v, u ) com λ ( Y ) = cosh u. Now, we look for the representation of those two associated surfaces as graphics of firsttype. In fact, for ( M, X ) and the representation P ( x, y ) = ( A ( x, y ) , x, y, : It takes x = sinh u cos v and y = sinh u sin v. Therefore sinh u = p x + y . For ( M, X ) , we obtain that function A in the graphicrepresentation is given by A ( x, y ) = ln( p x + y + p x + y + 1) . For ( M, Y ) and the representation Q ( p, q ) = (0 , p, q, B ( p, q )) : It takes p = − cosh u cos v and q = − cosh u sin v. So, cosh u = p p + q . For ( M, Y ) we obtain the function B as given by B ( p, q ) = ln( p p + q + p p + q − . Example 7, and equations linking the initial surface (
M, X ) and its associated surface(
M, Y ) in the θ -family, suggest the following result. Lemma . For the associated surfaces of the θ -family given by X ( w ) = ( A ( x ( w ) , y ( w )) , x ( w ) , y ( w ) , and Y ( w ) = (0 , p ( w ) , q ( w ) , B ( p ( w ) , q ( w )) , the Jacobian functions of the transformation of coordinates, are related by ∂ ( x, y ) ∂ ( u, v ) = ∂ ( p, q ) ∂ ( u, v ) . Proof.
From equations of associated surfaces (40) it follows that p w = − iy w and q w = ix w . Then p w q w − p w q w = x w y w − x w y w , which implies the relation i [ p u q v − p v q u ] = i [ y v x u − y u x v ] . (cid:3) Finally, from Lemma 6.3 and from our version of the Nitsche equations for transforma-tion of coordinates (38), we obtain the following result.
INIMAL SPACELIKE SURFACES AND THE GRAPHIC EQUATIONS IN R Theorem . The θ -family transports minimal first type graphic solutions P ( x, y ) =( A ( x, y ) , x, y, to minimal associated graphic solutions Q ( p, q ) = (0 , p, q, B ( p, q )) pre-serving the domain dom ( A ) = dom ( B ) = M .If M = C then P ( C ) and Q ( C ) are spacelike planes of R .We can say that “the Bernstein theorem holds if and only if the Calabi theorem holds”. Acknowledgments
The first author’s research was supported by Projeto Tem´aticoFapesp n. 2016/23746-6. S˜ao Paulo. Brazil. This paper is part of the Ph.D. thesis ofR.S. Santos [9], which was presented in Universidade de S˜ao Paulo, Brazil, in February2021.
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