Helicoidal maximal surfaces in Lorentz-Minkowski space

Monatsh. Math. 140, 315–334 (2003)

DOI 10.1007/s00605-003-0111-9

Helicoidal Maximal Surfaces inLorentz-Minkowski Space

By

Pablo Mira1 and Jose A. Pastor2

1 Universidad Polit�eecnica de Cartagena, Cartagena, Spain2 Universidad de Murcia, Espinardo, Spain

Received June 19, 2002; in revised form February 4, 2003Published online November 28, 2003 # Springer-Verlag 2003

Abstract. We investigate helicoidal surfaces in Lorentz-Minkowski space L3. A purely geometricmethod is used for classifying the helicoidal maximal surfaces. This description consists on exposinghelicoidal maximal surfaces as solutions to certain adequate Bj€oorling problems in L3, and shows that amaximal surface is helicoidal if and only if it lies in the associate family of a catenoid in L3. We shallalso give characteristic properties of the catenoids and helicoids in L3.

2000 Mathematics Subject Classification: 53A10, 53C50Key words: Minkowski space, helicoidal surface, Bj€oorling problem, maximal surface, associate

family

1. Introduction

One of the most interesting features of the local surface theory in Lorentz-Minkowski space L3 is that, even though it behaves quite like the classical surfacetheory in Euclidean space, there are sometimes striking differences. Perhaps themost representative one is the existence of a certain type of Lorentzian surfaces,called flat B-scrolls, discovered by Graves [11]. These surfaces are isometric immer-sions from L2 to L3 with non-diagonalizable shape operator. They are flat and mini-mal but not totally geodesic, and have appeared completing many classificationresults for surfaces in L3 [2, 8, 10, 11, 26]. More recently, Dillen and K€uuhnel [9]investigated helicoidal motions in L3 and exhibited a special type of them that doesnot have an Euclidean counterpart. They christened them as cubic screw motions.

Motivated by [9], the present paper studies helicoidal surfaces in L3, that is,semi-Riemannian surfaces in L3 that are invariant under a helicoidal motion group.We will relate flat B-scrolls with cubic screw motions. Actually we shall constructa special type of B-scroll, which we call parabolic null cylinder, that is invariantunder a group of cubic screw motions. Even though from the Euclidean viewpointthis is nothing but a right cylinder over a parabola, it has interesting propertieswhen seen as a helicoidal surface in L3. For instance, whereas a right circularcylinder in L3 is helicoidal for a specific axis, and for every pitch, these parabolicnull cylinders are helicoidal just for one pitch, but for infinitely many axes. Inparticular, they are not rotation surfaces in L3.

Recalling the role that rotation surfaces in L3 have played in the global theoryof surfaces in L3 (see for instance [12, 24]) it seems surprising that no systematicapproach to the geometry of helicoidal surfaces in L3 has been made up to now. Infact, in [9] the authors restrict their study to helicoidal ruled surfaces, and in [22]one only finds also particular cases of helicoidal surfaces.

The other main motivation of this paper comes from the work [3], where a newapproach to the study of maximal surfaces can be found, inspired by the classicalBj€oorling problem for minimal surfaces in R3. Here by a maximal surface we meana spacelike surface in L3 with vanishing mean curvature. The results in [3] and [18]show how to use holomorphic data to characterize certain families of maximalsurfaces defined in terms of curves. From this, we conceived the possibility ofgiving a classification of helicoidal maximal surfaces in L3 by using purely geo-metric arguments, i.e., without solving analytically the differential equationH ¼ 0. In Euclidean space, two classical topics such as Bj€oorling problem andhelicoidal minimal surfaces have never been put together.

The classification result that we have achieved via this method is the following.

Theorem 1.1. A maximal surface in L3 is helicoidal with axis ‘ if and only if itlies in the associate family of a catenoid in L3 with axis ‘.

The associate family of a maximal surface in L3 is defined in Section 4. By acatenoid in L3 we mean a rotation maximal surface in L3 (see [15, 19, 3, 18, 26]).There exist three different catenoids in L3, according to the causal character oftheir axes. We shall review them in Examples 6.1, 6.2 and 6.3. For timelikeversions of these catenoids, see [19, 22, 26].

We have organized this paper as follows. In Section 2 we study orbits of pointsunder helicoidal motion groups, and use them to give global parametrizations forhelicoidal surfaces in L3. In Section 3 we construct the parabolic null cylinders andrelate their properties with that of right circular cylinders, showing some relevantdifferences between their behaviours (Proposition 3.1). Inspired by [5] we use para-bolic null cylinders to give a characterization of helices in L3 with lightlike axis.

From here until the end of this work we deal with helicoidal maximal surfaces.The theory of maximal surfaces in L3 has to start necessarily from Calabi-Bern-stein theorem, which states that the only complete maximal surfaces in L3 arespacelike planes (see [6, 7]). Thus, even though there is available a Weierstrassformula for maximal surfaces in L3, it does not have the importance of its Eucli-dean counterpart. In [3] a holomorphic representation formula for maximal sur-faces is found that is quite adequate for the situation in L3, since it splits the localgeometry of a maximal surface into 1-dimensional objects.

Section 4 will introduce the approach in [3] for the study of maximal surfaces.This method is the key tool of our study, and will be used to prove in Section 5 thefollowing characterization of helicoidal maximal surfaces in L3.

Theorem 1.2. Let G‘;h be a helicoidal motion group in L3 with axis ‘ and pitch h. If‘ is non-lightlike (resp. lightlike and h 6¼ 0), a maximal surface is a piece of a heli-coidal surface invariant under G‘;h if and only if it intersects a right circular cylinder(resp. a parabolic null cylinder) with axis ‘ at a constant angle along a helix G‘;hðpÞ.

316 P. Mira and J. A. Pastor

The notion of intersection at a constant angle is given in Definition 4.2. Inthe proof of this Theorem we will find explicit parametrizations for all helicoidalmaximal surfaces in L3. To do so we do not need to solve a single differentialequation, in contrast with the usual computations of the Euclidean situation.

In the final part of the work we shall prove the classification of helicoidalmaximal surfaces mentioned above (see Theorem 1.1). In addition, followingthe spirit of Theorem 1.2, we will characterize the catenoids and the helicoidsin L3 in Section 6.

To end up, we shall also show in the Appendix, based on [21], that the associ-ate family of a spacelike helicoidal constant mean curvature surface in L3 is madeup by helicoidal surfaces. This fact was stated in [22] but the explanation giventhere is incorrect.

Concerning timelike surfaces in L3 with vanishing mean curvature, there doesnot exist a Bj€oorling-type formula. Indeed, timelike minimal surfaces cannot berepresented in terms of holomorphic data. Thus, the problem of giving a purelygeometric classification (i.e. without the need to solve analytically the differentialequation H ¼ 0) of timelike minimal helicoidal surfaces needs a differentapproach. Any general method for studying timelike minimal surfaces seems tobe of interest, since this theory is not much developed.

Finally, let us indicate that a Bj€oorling-type formula is very likely to exist forspacelike surfaces in L3 that satisfy the equation H ¼ cK, where c2R, since in [1]the authors proved that these surfaces can be represented by holomorphic data. In[4] it is found an extension of the Bj€oorling representation to Rn.

2. Helicoidal Surfaces in L3

The Lorentz-Minkowski space L3 is defined as the vector space R3 endowedwith the Lorentzian metric

h ; i ¼ dx21 þ dx2

2 � dx23;

where ðx1; x2; x3Þ are canonical coordinates in R3. Recall that, given u, v2L3, thecross-product u� v2L3 is defined so that the relation hu� v; xi ¼ detðu; v; xÞholds for all x2L3. A helicoidal motion group is a non trivial 1-parameter groupof rigid motions of L3, and any element of such a group is called a helicoidalmotion of L3. Here by trivial cases we mean pure translation groups. Every heli-coidal motion group is completely determined by an axis ‘ and a pitch h2R. IfG‘;h denotes the helicoidal motion group with axis ‘ and pitch h, by making anadequate choice of coordinates in L3 we can always suppose that ‘ is the x1-axis,the x3-axis or the null line spanned by ð1; 0; 1Þ, depending on whether ‘ is space-like, timelike or lightlike. If p ¼ ða; b; cÞ2L3, the image of p under any helicoidalmotion of the 1-parameter group G‘;h is (see [9, 22])

�sðpÞ ¼1 0 0

0 cosh s sinh s

0 sinh s cosh s

0@

1A

a

b

c

0@

1Aþ h

s

0

0

0@

1A if ‘ is spacelike; ð2:1Þ

Helicoidal Maximal Surfaces in Lorentz-Minkowski Space 317

�sðpÞ ¼cos s � sin s 0

sin s cos s 0

0 0 1

0@

1A

a

b

c

0@

1Aþ h

0

0

s

0@

1A if ‘ is timelike; ð2:2Þ

�sðpÞ ¼1� s2

2s s2

2

�s 1 s

� s2

2s 1þ s2

2

0@

1A

a

b

c

0@

1Aþ h

s3

3� s

s2

s3

3þ s

0@

1A if ‘ is lightlike; ð2:3Þ

where in any case s2R. If we choose h ¼ 0 we obtain a rotation about the axis ‘.We remark that the helicoidal motions with lightlike axis are referred to in [9] ascubic screw motions.

The orbit of a point p2L3 under G‘;h is denoted by G‘;hðpÞ. This orbit is a curvethat can be parametrized as�ðsÞ ¼ �sðpÞ, where �s is as in (2.1), (2.2) or (2.3). Thus�ðsÞ always has constant speed and constant causal character. For later use we notethat if �s is as in (2.3), the vector field �00 is lightlike if and only if h ¼ 0.

A surface in L3 is a smooth immersion X : M2 ! L3 of a connected 2-dimen-sional orientable manifold so that M2 inherits via X a non-degenerate metric. Thismetric will also be denoted by h ; i, and it must be Riemannian or Lorentzian. Weshall call surfaces of this kind spacelike and timelike, respectively. Since we areassuming surfaces in L3 to be orientable, we can always choose a unit normalvector field N globally defined on M2. This normal field N can be regarded as amap N : M2 ! H2

þ if the surface is spacelike, and as a map N : M2 ! S21 if the

surface is timelike. Here H2þ ¼ fx2L3 : hx; xi ¼ �1; x3 5 0g is the hyperbolic

space and S21 ¼ fx2L3 : hx; xi ¼ 1g is the de Sitter space. This map N will be

called in any of the two cases the Gauss map of the surface.A surface in L3 is said to be a helicoidal surface if its image is invariant under

the action of a certain helicoidal motion group. It comes clear from the definitionthat if a helicoidal surface invariant under G‘;h meets the axis ‘ at some point, andif ‘ is not lightlike, then the whole axis is contained in the surface. Hence, thesehelicoidal surfaces are globally made up by pieces of helicoidal surfaces not meet-ing the axis, and glued together along this axis. So, from now on we will assumethat helicoidal surfaces with non-lightlike axis do not meet the axis.

As a first step in order to establish global parametrizations for helicoidalsurfaces in L3 we note the following fact about orbits in L3.

Lemma 2.1. Let G‘;hðpÞ be an orbit in L3.

(1) Assume that ‘ is not lightlike, p 2= ‘, and define � � L3 as the only planesuch that ‘ � � and p2�. If � is spacelike (resp. timelike), then G‘;hðpÞ meets aplane �0 � L3 with ‘ � �0 if and only if �0 is spacelike (resp. timelike).

(2) Assume that ‘ is spacelike, h 6¼ 0, and let � be a plane in L3 perpendicularto ‘. Then G‘;hðpÞ meets � exactly at one point.

(3) Assume that ‘ is lightlike, h 6¼ 0, and define � � L3 as the only degenerateplane such that ‘ � �. Then G‘;hðpÞ meets � exactly at one point.

Proof. For the first claim we consider the case in where ‘, � are both spacelike,since all the others can be treated similarly. Then we suppose that ‘ is the x1-axis


and � is the x1, x2-plane. So p ¼ ða; b; 0Þ and the orbit G‘;hðpÞ is parametrized as�ðsÞ ¼ ðaþ hs; b cosh s; b sinh sÞ. If �0 is any spacelike plane in L3 containing ‘,there exists a certain v2�0 \ f‘g? with hv; vi ¼ b2. Therefore there is aunique s02R so that v ¼ ð0; b cosh s0; b sinh s0Þ. In this manner we find that�ðs0Þ2�0.

The second claim is immediate. Finally, for the third one, we assume that ‘ islightlike, say for instance the line spanned by ð1; 0; 1Þ, and that h 6¼ 0. The onlydegenerate plane in L3 that contains ‘ is the x1 ¼ x3-plane, and q2L3 lies in � ifand only if q ¼ ð�; �; �Þ, �; �2R. If p ¼ ða; b; cÞ the orbit G‘;hðpÞ is given by�ðsÞ ¼ �sðpÞ, with �s as in (2.3), and it is easy to check that �ðs0Þ lies in � if andonly if s0 ¼ ða� cÞ=2h. This concludes the proof. &

In Euclidean space every helicoidal surface invariant under a screw motiongroup G with axis ‘ is the orbit of a plane curve � under G, so that the plane inwhich � lies contains the axis ‘. The key fact of this construction is the transitivitycondition that every orbit GðpÞ meets all planes containing the axis.

In L3 the above Lemma shows that this transitivity condition is no longer trueunless the axis is timelike. Thus, not every helicoidal surface in L3 can be con-structed following the above Euclidean procedure. Nevertheless, we can still getthe following

Lemma 2.2. Let � � L3 be a helicoidal surface invariant under G‘;h.

(1) Assume that ‘ is spacelike and h 6¼ 0, choose � � L3 a plane perpendicularto ‘ and let � ¼ � \�. Then � is a connected regular curve and � ¼ G‘;hð�Þ.

(2) Assume that ‘ is timelike, choose � � L3 a plane containing ‘ and let� ¼ � \�. Then � is made up by connected pieces of regular curves, and if �0

is any connected component of � we have � ¼ G‘;hð�0Þ.(3) If ‘ is lightlike, h 6¼ 0 and � � L3 is the only degenerate plane that con-

tains ‘, then � ¼ � \� is a connected regular curve and � ¼ G‘;hð�Þ.Proof. In any of the three assertions we can consider a point p2� \�. Then

G‘;hðpÞ is a curve in � that is not contained in �. Therefore the plane � meets �transversally along �, and � is made up by connected pieces of regularcurves. Thus an open piece of � is given by G‘;hð�Þ. Moreover, since from Lemma2.1 any orbit G‘;hðqÞ meets �, we find that � ¼ G‘;hð�Þ. If the axis ‘ is nottimelike, each orbit G‘;hðqÞ meets � exactly once. Thus � is connected, sinceso is �. So, it remains only to check the connectedness part for the case in which‘ is timelike.

To do so, let us choose two connected components �0, �00 of �, and supposethat G‘;hð�0Þ \ �00 6¼ ;. Then there is some s02R such that �s0

ð�0Þ \ �00 6¼ ;, andso �s0

ð�0Þ � �. This s0 satisfies �s0ð�0Þ � �00, since otherwise �00 would admit a

proper connected extension in �. By interchanging �0 and �00 we observe that�s0ð�0Þ ¼ �00. As a consequence, any connected component �0 of � satisfies

�0 � G‘;hð�0Þ, or alternatively G‘;hð�0Þ \ �0 ¼ ;. However, since � is connectedand G‘;hð�Þ ¼ � this last situation is not possible. Therefore we get that� � G‘;hð�0Þ, and finally that G‘;hð�0Þ ¼ �. This finishes the proof of Lemma2.2. &


Let us remark that if ‘ is lightlike, the only plane orthogonal to ‘ that meets ‘ isprecisely the only degenerate plane in which ‘ lies. By means of Lemma 2.2 we cangive explicit global parametrizations for all helicoidal surfaces in L3. The followingtable collects all possible parametrizations of helicoidal surfaces � in L3, accordingto the causal character of the axis ‘. In that table, and after making an adequatechoice of coordinates in L3, we are assuming that ‘ is the x1-axis, the x3-axis or theline spanned by ð1; 0; 1Þ, and that the plane � of Lemma 2.2 is the x2, x3-plane, thex1, x3-plane or the x1 ¼ x3-plane, respectively. To gain clarity, we have intentionallyomitted the cases of rotation surfaces with spacelike or lightlike axis, since in thosesituations a global parametrization as in the table is not generally available. How-ever, rotation surfaces in L3 are well known [12, 13, 15, 19, 22, 26].

3. Helices, Cylinders, Parabolic Null Cylinders

Inspired in [16], we define a circle in L3 as a non-linear orbit G‘;hðpÞ, beingp 2= ‘ and h ¼ 0. Thus circles are always spacelike or timelike planar curves withnon-zero constant curvature (see [16]).

By a right circular cylinder we shall mean a right cylinder in L3 with non nullgeneratrix ‘ and whose directrix is a circle lying in a plane perpendicular to ‘. Theright circular cylinders defined over circles in timelike (resp. spacelike) planes arecommonly called hyperbolic (resp. Lorentzian) cylinders. A right circular cylinderis expressed in suitable coordinates of L3 as x2

1 � x23 ¼ �R2, x2

1 þ x22 ¼ R2 or

x21 � x2

3 ¼ R2, being R> 0. The first one is a spacelike CMC surface, while theothers are timelike CMC surfaces.

If a right circular cylinder � has non null generatrix ‘, it is easy to convinceoneself that � is invariant under all helicoidal motion groups with axis ‘, inde-pendently of the pitch h. In this way, the right circular cylinders can be consideredas the most simple helicoidal surfaces with non lightlike axis in L3.

If we consider the problem of finding a helicoidal surface with lightlike axishaving the above invariance property, we get the non existence of a right circularcylinder counterpart. Nevertheless, we can still make the following construction:let ‘ be a lightlike axis in L3, and let � be a plane orthogonal to ‘ such that‘ \� 6¼ ;. Now, choose ‘0 a straight line parallel to ‘, orthogonal to � and thatmeets �. For any h 6¼ 0 we can consider the helicoidal surface P � L3 given by theorbit of ‘0 under G‘;h (note that a right circular cylinder can be constructed in thisway). This surface may be parametrized in suitable coordinates of L3 as

Xðs; tÞ ¼ �sð�ðtÞÞ; being �s as in ð2:3Þ and �ðtÞ ¼ ðt; b; tÞ; b2R: ð3:1ÞAnother parametrization for P is given by

Yðu; tÞ ¼ 1

2

��2 u3

3� uþ 2t; �u2 þ 2b; �2 u3

3þ uþ 2t

�; ð3:2Þ

being � ¼ 1=2h. This fact can be easily checked by noting that in the abovecoordinates P is the quadric

�ðx3 � x1Þ2 ¼ 2ðx2 � bÞ: ð3:3Þ


Recalling the definition of flat B-scroll over a null curve (see [11], [5]), we easilysee from (3.2) that P is a flat B-scroll with B ¼ ð1; 0; 1Þ over the null curve�ðuÞ ¼ Yðu; 0Þ. The metric of the immersion (3.2) is d�2 ¼ �2dudt, and its(non-diagonalizable) shape operator is

A ¼ 0 ��0 0

� �:

Motivated by all of this, we shall call P a parabolic null cylinder with pseudo-radius � and axis ‘. In spite of all the above analogies between parabolic nullcylinders and right circular cylinders, we shall show next some properties ofparabolic null cylinders as helicoidal surfaces in L3 that do not have an Euclideancounterpart.

Proposition 3.1. Let P be a parabolic null cylinder with pseudo-radius � andaxis ‘, and let � denote the only degenerate plane in which ‘ lies. Then

(1) P is a parabolic null cylinder with pseudo-radius � and axis ‘0, being ‘0 anyaxis parallel to ‘ (i.e. pointing in the B-direction) and lying in �.

(2) P is congruent in L3 to any other parabolic null cylinder.(3) P is invariant under G‘;h only for h ¼ 1=2�.(4) P is made up by all orbits of pitch h ¼ 1=2� and axis parallel to ‘ lying in

� that pass through a point p2P.

Proof. We shall assume that P is given by (3.3) and that ‘ is the axis spanned byð1; 0; 1Þ. Then every axis ‘0 � � parallel to ‘ is of the form fx1 ¼ x3; x2 ¼ �g. Ifwe make a translation in L3 of vector ð0; �; 0Þ we obtain that P is given in the newcoordinates as

�ðx3 � x1Þ2 ¼ 2ðx2 � ðbþ �ÞÞ:

This shows that P is a parabolic null cylinder with axis ‘0 and pseudo-radius �, andproves the first assertion.

To prove the second one, we first note that P is congruent to the parabolic nullcylinder S0 of equation �ðx3 � x1Þ2 ¼ 2x2 and that, by composing if necessarywith a reflection about the x1, x3-plane, we shall assume that � is positive. Therotation of L3 about the x2-axis

cosh� 0 sinh�0 1 0

sinh� 0 cosh�

0@

1A

with angle � ¼ � log ð ffiffiffi�p Þ takes P0 to the quadric ðx3 � x1Þ2 ¼ 2x2. Thus P iscongruent to this last quadric. Since this can be done for every B-scroll in L3, wehave established the second claim.

The third property is immediate. Finally, for the last one we simply have tonote that if we parametrize P as in (3.1), the curve �ðsÞ ¼ Xðs;msÞ is an orbit withpitch h and axis fx1 ¼ x3; x2 ¼ �mg passing through ð0; b; 0Þ. &


Remark 3.2. From the first of the above properties we infer that the axis of aparabolic null cylinder P is not unique, but the direction to which it points is. Thisdirection is precisely the B-direction of the B-scroll, that is, the direction of therulings of P.

Now, let � be a spacelike curve parametrized with constant speed. We will saythat � is a Frenet curve if it admits a Frenet trihedron TðsÞ ¼ ftðsÞ ¼�0ðsÞ=k�0ðsÞk; nðsÞ; bðsÞg at every point. This means that nðsÞ is collinear with�00ðsÞ, bðsÞ ¼ tðsÞ� nðsÞ, where � is the cross product in L3, and TðsÞ is anorthonormal basis of L3 for all s. It is clear that a spacelike curve of constant speedis a Frenet curve if and only if its acceleration vector is always spacelike ortimelike. For a detailed description of frame fields along timelike and null curvesin L3, [5] is a good reference.

We can define the curvature � and torsion � of a spacelike Frenet curvein the obvious way. If both �, � are constant, the Frenet equations can be eas-ily integrated. Here we are interested in the following modified version of thisfact.

Proposition 3.3. Let �ðsÞ : I ! L3 be a spacelike Frenet curve in L3 parame-trized with constant speed, and let �, � denote its curvature and its torsion,respectively. Then the following statements are equivalent:

(i) The curve �ðsÞ is an orbit G‘;hðpÞ, with p 2= ‘ and ‘ non lightlike.(ii) � and � are both constant, with j�j ¼ j� j only if �00ðsÞ is timelike.

(iii) The curve �ðsÞ is a geodesic in a right circular cylinder with non lightlikeaxis.

Moreover, the axis of the cylinder in (iii) coincides with the axis of the heli-coidal motion group G‘;h in (i).

The proof of this result is analogous to that of its Euclidean counterpart, and sowe omit it. This proposition as well as the next one have been motivated by someresults in [5].

Theorem 3.4. Let �ðsÞ : I ! L3 be a spacelike Frenet curve in L3 parame-trized with constant speed, and let �, � denote its curvature and its torsion,respectively. Then the following statements are equivalent:

(i) The curve �ðsÞ is an orbit G‘;hðpÞ with ‘ lightlike and h 6¼ 0.(ii) � and � are both constant with j�j ¼ j� j> 0 and �00ðsÞ spacelike.

(iii) The curve �ðsÞ is a geodesic in a parabolic null cylinder.Moreover, the axis of the helicoidal motion group G‘;h in (i) points in the B-

direction of the parabolic null cylinder in (iii).

Proof. A straightforward computation proves that (i) implies (ii) whereas theuniqueness and existence theorem for Frenet curves in L3 gives the converse. Thus,let us prove that (i) implies (iii). Let �ðsÞ be an orbit G‘;hðpÞ with ‘ lightlike. Thehypothesis h 6¼ 0 together with Lemma 2.1 allow us to express �ðsÞ as

�ðsÞ ¼�

aþ bsþ h

�s3

3� s

�; bþ hs2; aþ bsþ h

�s3

3þ s

��


with q ¼ ða; b; aÞ the unique point where the orbit meets the plane x1 ¼ x3. Thiscurve is contained in the parabolic null cylinder P with � ¼ 1=2h given byðx3 � x1Þ2 ¼ 4hðx2 � bÞ. A direct computation shows that �00 is orthogonal to Pimplying that � is a geodesic in this surface.

Finally, let us prove that (iii) implies (i). Let P be a parabolic null cylinderparametrized by (3.1). The geodesics of P are then images under X of straightlines in the parameter plane R2 and, as we showed in Proposition 3.1, suchcurves are orbits with null axis pointing in the B-direction of P. This completesthe proof. &

Remark 3.5. If G‘;hðpÞ is a spacelike orbit as in the above theorem, it isstraightforward to check that both � ,� do not depend on the pitch h. This factshows once more how the geometric properties of helicoidal motions with lightlikeaxis do not have Euclidean analogues. As a simple consequence of this fact and ofthe fundamental theorem for Frenet curves in L3 we get that two spacelike orbitsG‘;hðpÞ and G‘;h0 ðpÞ with different pitches h, h0 6¼ 0 that pass through the samepoint p are congruent in L3.

We shall call any curve satisfying the conditions of the previous theorems ahelix in L3. The axis ‘ is called the axis of the helix.

In spite of the above theorems, let us remark that, opposite to the Euclideancase, not every spacelike orbit is a helix, since there are two kinds of orbits that arenot Frenet curves. The first one consists of circles in degenerate planes. Thisfamily has been treated in [16], where it is given a definition of constant curvaturefor spacelike curves in degenerate planes in a way that makes circles become theonly such curves of constant curvature. The second family consists of orbitsG‘;hðpÞ, being ‘ spacelike and p 2= ‘ a point of a degenerate plane � in which ‘is contained. Any such spacelike orbit lies in �, and it does not have constantcurvature with the above definition.

To end up this Section we point out the following fact about helicoidal surfacesin L3, which has inspired Theorem 1.2.

Lemma 3.6. Let � � L3 be a helicoidal surface in L3, invariant under G‘;h,and choose a helix G‘;hðpÞ in �. If ‘ is non-lightlike (resp. lightlike and h 6¼ 0), then� meets the right circular cylinder (resp. any parabolic null cylinder) with axis ‘in which G‘;hðpÞ lies at a constant angle along G‘;hðpÞ.

Table 1. Global parametrizations in suitable coordinate systems for helicoidal surfaces � in L3,invariant under G‘;h

‘ spacelike, h 6¼ 0 Xðs; tÞ ¼ �sð�ðtÞÞ, where�ðtÞ ¼ ð0; aðtÞ; bðtÞÞ;�s as in ð2:1Þ

�

‘ timelike Xðs; tÞ ¼ �sð�ðtÞÞ, where�ðtÞ ¼ ðaðtÞ; 0; bðtÞÞ;�s as in ð2:2Þ

�

‘ lightlike, h 6¼ 0 Xðs; tÞ ¼ �sð�ðtÞÞ, where�ðtÞ ¼ ðaðtÞ; bðtÞ; aðtÞÞ;�s as in ð2:3Þ

�


Proof. Straightforward computations for each of the cases in Table 1, takinginto account that G‘;hðpÞ will be a geodesic for the right circular cylinder (resp. theparabolic null cylinder). &

4. Maximal Surfaces and Bjorling Problem

A spacelike surface : M2 ! L3 is said to be a maximal surface if the meancurvature of the immersion vanishes identically. Since M2 is Riemannian, wecan choose isothermal coordinates s, t around any point of M2. Therefore everyspacelike surface in L3 may be seen, at least locally, as a conformal map : � � C�! L3. Here by writing � � C we want to emphasize that the complexparameter z ¼ sþ it provides isothermal parameters s, t for �. It turns out that ifwe are given a spacelike surface of the type : � � C�! L3, then this surface ismaximal if and only if the coordinate functions 1, 2, 3 are harmonic functionsin the usual Euclidean sense.

It is well known that every simply connected maximal surface in L3 can beexpressed as the real part of a holomorphic curve f : � � C! C3 such thathf 0; f 0i ¼ f 01

2 þ f 022 � f 03

2 ¼ 0 and jf 01j2 þ jf 02j

2 � jf 03j2 > 0: If : � � C! L3 is

the maximal surface represented by the holomorphic curve f , then f ¼ eif repre-sents again a maximal surface for all 2R (see [25]). The family of maximalsurfaces f ¼ Re fg is called the associate family of the maximal surface . Twomaximal surfaces are called associate if each of them lies in the associate familyof the other. The surface � ¼ �=2 is called the adjoint surface of .

Next we turn our attention to Bj€oorling problem in L3. Following [3], theBj€oorling problem for maximal surfaces in L3 consists on the following: let � :I�! L3 be a regular spacelike analytic curve in L3, and let V : I�! L3 be a unittimelike analytic vector field along � such that h�0;Vi� 0. Construct a maximalsurface in L3 containing � whose Gauss map along � is given by V.

This Bj€oorling problem turns out to have a unique solution. The specific result,which was proved in [3], states what follows.

Theorem 4.1 (Solution to the Bj€oorling problem). Let � : I�! L3 be a regularanalytic spacelike curve in L3, and let V : I�! L3 be a timelike analytic unitvector field along � such that h�0;Vi� 0. There exists a unique maximal surfacewhose image contains �ðIÞ and such that its Gauss map along � is V . Thismaximal surface is explicitly given by

ðs; tÞ ¼ Re�ðzÞ � Im

ðz

s0

VðwÞ��0ðwÞdw: ð4:1Þ

Here z ¼ sþ it, �ðzÞ, VðzÞ are holomorphic extensions of �ðsÞ, VðsÞ over a certainsimply connected open set � � C containing I, and s02 I is fixed but arbitrary.

The uniqueness part means the following: two maximal surfaces that satisfy theconditions of the Theorem must overlap over a non-empty open set that contains�ðIÞ. Since maximal surfaces are real analytic, this amounts to say that bothmaximal surfaces determine the same inextendible maximal surface. Besides, letus note that the integral in formula (4.1) neither depends on s0 nor on the chosenpath between s0 and z.


Let us recall (see [20]) that if v, w2L3 are timelike vectors, there exists aunique ’5 0 such that jhv;wij ¼ cosh’jvjjwj. In the same way, if v is spacelikeand w is timelike, there exists a unique ’5 0 such that jhv;wij ¼ sinh’jvjjwj.This last property is trivial.

Definition 4.2. Let : M2�! L3 be a spacelike surface that intersects a time-like (resp. spacelike) surface � � L3 along a spacelike curve �ðsÞ. We say that thesurface intersects � with constant angle ’5 0 provided jhNðsÞ;N�ðsÞij� sinh’(resp. jhNðsÞ;N�ðsÞij � cosh’). Here NðsÞ and N�ðsÞ stand, respectively, for theGauss map of the surface at �ðsÞ and the Gauss map of � at �ðsÞ.

In the case where � is timelike and ’ ¼ 0 we say that the surface intersects �orthogonally.

The next result is a consequence of Theorem 4.1, and can be seen as thesolution to a Bj€oorling-type problem that fits the conditions of the kind of resultswe want to treat.

Theorem 4.3. Let � : I�! L3 be a regular spacelike analytic curve of a semi-Riemannian analytic surface � � L3, and choose ’> 0. There exist locally exactlytwo maximal surfaces in L3 that intersect � along � with constant angle ’, andboth of them can be explicitly constructed. If � is a non-degenerate plane, thosetwo maximal surfaces are congruent in L3.

Proof. If � is spacelike with Gauss map N�, we can suppose reversing theorientation of � if necessary that hNðsÞ;N�ðsÞi ¼ � cosh’. This conditiontogether with hN;Ni ¼ �1 show that

NðsÞ ¼ cosh’N�ðsÞ þ "sinh’

j�0ðsÞj �0ðsÞ�N�ðsÞ; ð4:2Þ

where " ¼ �1. Applying Theorem 4.1 we find exactly two maximal surfaces thatsatisfy the required conditions, depending on the sign of ". In the case where � is aplane, N�ðsÞ is constant and we may assume, composing if necessary with asymmetry with respect to �, that " ¼ 1. Hence the two maximal surfaces arecongruent in L3.

If � is timelike the proof keeps the same pattern. Here the analogue to equation(4.2) is

NðsÞ ¼ sinh’N�ðsÞ þ "cosh’

j�0ðsÞj �0ðsÞ�N�ðsÞ; " ¼ �1; ð4:3Þ

where we have assumed that hN;N�i ¼ sinh’. &

We note that an easy modification of the above proof yields the following fact:let � : I�! L3 be a regular spacelike analytic curve in L3 contained in a timelikeanalytic surface � � L3. There exists a unique maximal surface in L3 that inter-sects � orthogonally along �. This maximal surface is given explicitly by

ðs; tÞ ¼ Re�ðzÞ � Im

ðz

j�0jðwÞN�ðwÞdw: ð4:4Þ


Here N�ðsÞ stands for the Gauss map of � at �ðsÞ, and j�0jðzÞ denotes the holo-morphic extension of the analytic function j�0ðsÞj : I ! Rþ.

This last result as well as Theorem 4.1 were proved in [18], but we haveexposed them here again for the convenience of the reader.

5. Proof of Theorem 1.2

Thanks to Lemma 3.6 we have to check only the converse. For this let us startwith a spacelike helix �ðsÞ in L3 that will be assumed to be a geodesic of thespacelike right circular cylinder C� x2

2 � x23 ¼ �b2. Thus �ðsÞ ¼ ðaþ hs;

b sinh s; b cosh sÞ. Consider a maximal surface in L3 that intersects C along �ðsÞwith constant angle ’5 0, that is, such that hNC;Ni� � cosh’. Here NC, N arerespectively the Gauss map of C and of the maximal surface. Note thatNCðsÞ ¼ ð0; sinh s; cosh sÞ. Then, taking into account (4.2), we get

NðsÞ��0ðsÞ ¼ ð�b cosh’; h cosh s cosh’þ A sinh s; h sinh s cosh’þ A cosh sÞ;where A ¼ �ðh2 þ b2Þ1=2

sinh’, � ¼ �1. With this, by making use of formula(4.1) it is found that the maximal surface that we are considering is an open pieceof the maximal surface given by

1ðs; tÞ ¼ aþ hsþ bt cosh’;

2ðs; tÞ ¼ b sinh s cos t � ðh cosh’ cosh sþ A sinh sÞ sin t;

3ðs; tÞ ¼ b cosh s cos t � ðh cosh’ sinh sþ A cosh sÞ sin t:

ð5:1Þ

Finally, if � denotes a helicoidal motion of L3 with spacelike axis ‘ as the x1-axisand pitch h, one easily verifies from the above expression and (2.1) that�ððs; tÞÞ ¼ ðsþ ; tÞ. This means that the maximal surface is invariant underG‘;h and we are done.

For the other two possible situations of spacelike helices in L3 with non-light-like axis we proceed similarly. In both cases the helices are geodesics in timelikeright circular cylinders, and so we have to use equation (4.2) instead of (4.2). Weget

1ðs; tÞ ¼ aþ hs� bt sinh’;

2ðs; tÞ ¼ b cosh s cos t � ðh sinh’ sinh sþ A cosh sÞ sin t;

3ðs; tÞ ¼ b sinh s cos t � ðh sinh’ cosh sþ A sinh sÞ sin t; ð5:2Þfor the helix �ðsÞ ¼ ðaþ hs; b cosh s; b sinh sÞ, where A ¼ � cosh’ðh2 � b2Þ1=2

,� ¼ �1, and

1ðs; tÞ ¼ a cos s cosh t � ðh sinh’ sin sþ A cos sÞ sinh t;

2ðs; tÞ ¼ a sin s cosh t þ ðh sinh’ cos s� A sin sÞ sinh t;

3ðs; tÞ ¼ bþ hsþ at sinh’;

ð5:3Þ

for �ðsÞ ¼ ða cos s; a sin s; bþ hsÞ, being A ¼ � cosh’ða2 � h2Þ1=2, � ¼�1.

Finally, let us consider the case of a spacelike helix with lightlike axis ‘ andpitch h 6¼ 0. The helix �ðsÞ is an orbit G‘;hðpÞ of a point p lying in � ¼ f‘g?. We


shall assume that ‘ is the line spanned by ð1; 0; 1Þ. If we denote p ¼ ða; b; aÞ, itfollows from (2.3) that

�ðsÞ ¼�

aþ bsþ h

�s3

3� s

�; bþ hs2; aþ bsþ h

�s3

3þ s

��: ð5:4Þ

Since �ðsÞ is spacelike we have �4hb> 0. Let P be a parabolic null cylinder thatcontains �ðsÞ as a geodesic. Then the Gauss map NP of P along �ðsÞ isNP ¼ �00=j�00j ¼ ðs; 1; sÞ. Consider a maximal surface in L3 that meets P at aconstant angle ’5 0 along �ðsÞ. Thus we have hN;NPi� sinh’, where hereNðsÞ is the Gauss map of the maximal surface along �ðsÞ. At this point it followsfrom equation (4.3) after some calculations that

NðsÞ��0ðsÞ ¼ ðAs2 þ Bsþ C; 2Asþ B;As2 þ Bsþ bCCÞ;being A ¼ �h sinh’, B ¼ � cosh’ð�4hbÞ�1=2

, C ¼ ðbþ hÞ sinh’, bCC ¼ðb� hÞ sinh’, and � ¼ �1. Note that C � bCC ¼ �2A. Denoting �ðsÞ ¼Ð s

NðrÞ��0ðrÞdr, the holomorphic extension �ðzÞ of �ðsÞ is

�ðzÞ ¼�

A

3z3 þ B

2z2 þ Cz;Az2 þ Bz;

A

3z3 þ B

2z2 þ bCCz

�: ð5:5Þ

Glancing at equation (4.1) we find that the maximal surface we are considering isan open piece of the one given by

ðs; tÞ ¼ Re�ðzÞ � Im �ðzÞ; ð5:6Þwhere �ðzÞ is a holomorphic extension of (5.4) and �ðzÞ is as in (5.5). Let �

denote a helicoidal motion of L3 with axis ‘ and pitch h given by (2.3), beingh 6¼ 0 as above. Then some long but straightforward computations show thatðsþ ; tÞ ¼ �ððs; tÞÞ, where ðs; tÞ is given by (5.6). This tells thatthe maximal surface is invariant under G‘;h. Thus the proof of Theorem 1.2 iscompleted.

6. Examples, Applications and Proof of Theorem 1.1

To begin with, let us recall the maximal rotation surfaces and the maximalruled surfaces in L3, called catenoids and helicoids, respectively. Here a ruledsurface is a spacelike surface in L3 that can be parametrized as Xðs; rÞ ¼�ðsÞ þ rVðsÞ, where �ðsÞ is a spacelike regular curve and VðsÞ is a nowherevanishing spacelike vector field orthogonal to �. We shall follow [3, 18].

Example 6.1. Any maximal rotation surface with spacelike axis is congruent toa hyperbolic catenoid, which is parametrized as

ðsþ itÞ ¼ ARe ðz;�i sin z; cos zÞ: ð6:1Þ

Here A> 0, and ðsþ itÞ is defined on � ¼ ð��=2; �=2Þ�R � C. The adjointsurface of the hyperbolic catenoid is called the helicoid of the second kind. It is theonly ruled surface for which V 0ðsÞ is spacelike and non-zero (see [3]).


Example 6.2. Any maximal rotation surface with timelike axis is congruent toan elliptic catenoid, parametrized as

ðsþ itÞ ¼ A Reð sinh z;�i cosh z; zÞ; ð6:2Þwhere A> 0 and ðsþ itÞ is defined whenever s> 0. Its adjoint surface is calledthe helicoid of the first kind. It is the only ruled surface for which V 0ðsÞ is timelike.The image of the helicoid of the first kind is an open piece of the classical helicoidin R3.

Example 6.3. Any maximal rotation surface with null axis is congruent to aparabolic catenoid. A conformal parametrization of this surface with respect to thecanonical null frame F of L3, given by

F ¼ fðffiffiffi2p

=2; 0;�ffiffiffi2p

=2Þ; ðffiffiffi2p

=2; 0;ffiffiffi2p

=2Þ; ð0; 1; 0Þg; ð6:3Þis

ðsþ itÞ ¼ A

�s;� t2

2ðsþ BÞ þ 1

6s3 þ B

2s2 þ B2

2s; tðsþ BÞ

�: ð6:4Þ

Here A> 0, B2R and ðsþ itÞ is defined whenever s> �B. This surface isclassically known as Geiser’s surface (see [23]), when it is seen in complexifiedEuclidean space (we thank the referee for pointing out this fact). The adjointsurface to the parabolic catenoid is a ruled surface, called Cayley’s ruled surface.It is the only ruled maximal surface such that V 0ðsÞ is lightlike (see [3]).

It comes clear that all helicoids in L3 are helicoidal surfaces. Moreover, thethree different kinds of helicoids correspond to the three different types of heli-coidal motions in L3, depending on the causal character of the axis ‘. Thus it isnatural to refer to the helicoid of the second kind (resp. the helicoid of the firstkind) as the helicoid with spacelike axis (resp. the helicoid with timelike axis).Nevertheless, we shall keep naming Cayley’s ruled surface in that way, since itappears in other geometric contexts asides from L3 (see [9]).

Proof of Theorem 1.1: In order to prove the sufficient condition, it is shown inthe Appendix that every associate surface of a helicoidal maximal surface (parti-cularly of a catenoid) with axis ‘ is helicoidal with axis ‘.

Conversely, suppose that we are given a helicoidal maximal surface with nonlightlike axis ‘. From Theorem 1.2 we can parametrize this surface as (5.1), (5.2)or (5.3) depending on the causal character of ‘ and of the cylinder � in which oneof its orbits lies. In any of this cases, if is an associate surface of the maximalsurface : � � C! L3, then every s-curve ðs; t0Þ is a helix with axis ‘. Thusthe surface will actually be a rotation surface if and only if the translational partof in the direction of the axis ‘ does not depend on s.

If ðs; tÞ is given by (5.1), the first coordinate of ðs; tÞ does not depend on s ifand only if ¼ cot �1ðb cosh’=hÞ. Thus for this particular choice of the surface is a rotation surface. Therefore ðs; tÞ lies in the associate family of a catenoid.

If ðs; tÞ is given by (5.2) (resp. (5.3)), the same arguments show that ðs; tÞ isa catenoid for the choice ¼ cot �1ðb sinh’=hÞ (resp. ¼ cot �1ð�a sinh’=hÞ).


In this way we find that every helicoidal maximal surface with non lightlike axis ‘lies in the associate family of a catenoid with axis ‘.

Finally, suppose that we are given a helicoidal maximal surface ðs; tÞ invar-iant under G‘;h, being ‘ lightlike and h 6¼ 0. If ðs; tÞ is any associate surface ofðs; tÞ, from the Appendix we know that every s-curve �ðsÞ ¼ ðs; t0Þ is againan orbit with axis ‘. Moreover, �ðsÞ will be a circle if and only if �00ðsÞ islightlike. Let us denote �ðsÞ ¼ ðs; t0Þ, being ðs; tÞ as in (5.6). Then

�00ðsÞ ¼ 2ðhs� At0; h; hs� At0Þ:If �ðs; tÞ is the adjoint surface of ðs; tÞ and ��ðsÞ ¼ �ðs; t0Þ, we get from theCauchy-Riemann equations

ð��Þ00ðsÞ ¼ � @2

@s@tðs; t0Þ ¼ ð2ht0 þ 2Asþ B; 2A; 2ht0 þ 2Asþ BÞ: ð6:5Þ

Thus, if ðs; tÞ is an associate surface of ðs; tÞ, we get that ðs; tÞ is a rotationsurface if and only if �ðsÞ ¼ cos �ðsÞ � sin ��ðsÞ is a circle, that is, if andonly if ð�Þ00ðsÞ is lightlike. Since

hð�Þ00ðsÞ; ð�Þ00ðsÞi ¼ 4ðh cos � A sin Þ2;we find that is a rotation surface for ¼ cot �1ð� sinh’Þ, taking into accountthat A ¼ �h sinh’. Thus every helicoidal maximal surface with lightlike axis ‘lies in the associate family of a parabolic catenoid with axis ‘, and the proof iscompleted. &

From this result we find explicit nice parametrizations for all helicoidal max-imal surfaces in L3. In fact, every helicoidal maximal surface Xðs; tÞ with axis ‘ isgiven by Xðs; tÞ ¼ cos Xcatðs; tÞ � sin Xhelðs; tÞ, where Xcat is a catenoid withaxis ‘ and Xhel ¼ ðXcatÞ� is its adjoint helicoid.

In the final part of this work we will make use of Theorem 1.2 and of thegeneral solution to Bj€oorling problem to characterize the helicoids and the cate-noids of L3. The first result in this direction is the following.

Theorem 6.4. Every maximal surface that meets orthogonally a right circularcylinder (resp. a parabolic null cylinder) with axis ‘ along a non-planar helixG‘;hðpÞ is a piece of a helicoid (resp. of Cayley’s ruled surface) with axis ‘.

Proof. Let : � � C! L3 be a maximal surface in the conditions of theresult. The orthogonality hypothesis excludes the case in which the helix lies ina spacelike right circular cylinder. If the helix G‘;hðpÞ has spacelike (resp. timelike,light like) axis, the maximal surface is parametrized in suitable coordinates, keep-ing the notations of Section 5, by (5.2) (resp. (5.3), (5.6)) for the choice ’ ¼ 0.Thus, if ‘ is spacelike, the surface is given by ðs; tÞ ¼ �ðsÞ þ f ðtÞVðsÞ, where

�ðsÞ ¼ ðaþ hs; 0; 0Þ; VðsÞ ¼ ð0; cosh s; sinh sÞ; f ðtÞ ¼ b cos t � A sin t:

In this way ðs; tÞ is a maximal ruled surface, and thus a helicoid with spacelikeaxis ‘. The same argument shows that if ‘ is timelike, ðs; tÞ is a helicoid withtimelike axis ‘.


Finally, if ‘ is lightlike the condition ’ ¼ 0 implies A ¼ C ¼ bCC ¼ 0in (5.5). The maximal surface is then parametrized as ðs; tÞ ¼ �ðsÞ þ f ðtÞVðsÞfor

�ðsÞ ¼ �ðsÞ; VðsÞ ¼ ðs; 1; sÞ; f ðtÞ ¼ Bt � ht2;

with the notations of Section 5. It follows that ðs; tÞ is a ruled surface with nullaxis, thus a Cayley’s ruled surface with axis ‘. &

As a consequence we get that every maximal surface containing a helix as anasymptotic curve is a ruled surface. Moreover, its rulings are orthogonal to the axisof the helix.

Next we will extend Definition 4.2 to degenerate planes. If a spacelike surfaceintersects a degenerate plane � along a regular curve �ðsÞ, we say that it intersects� with constant angle if hNðsÞ; vi is constant for a null vector v2�. It is immedi-ate that this property is independent of our choice of v.

A curve �ðsÞ on a spacelike surface is a line of curvature if �0ðsÞ is aprincipal direction at �ðsÞ for all s. Lines of curvature are characterized by theequation �0 ¼ �N 0 for a certain �2C1ðIÞ. The following result, whose proofis straightforward, characterizes the planar lines of curvature of a spacelikesurface.

Lemma 6.5. Let � : I�! L3 be a regular curve of a spacelike surface S in L3.Assume that �ðIÞ is contained in a plane � of L3. Then � is a line of curvature of Sif and only if S intersects � along � with constant angle.

Next we give a characterization of catenoids in L3. Actually the case of light-like axis will complete in some sense Theorem 1.2, in where we assumed thathelicoidal surfaces with lightlike axis were not rotation surfaces.

Theorem 6.6. Every maximal surface in L3 that contains a circle of a timelike(resp. spacelike, degenerate) plane � as a line of curvature is a piece of a hyper-bolic catenoid (resp. elliptic catenoid, parabolic catenoid) whose axis is orthogo-nal to �.

Proof. Let �ðsÞ be a spacelike circle in L3 contained in a non-degenerate plane�. Then � is perpendicular to the right circular cylinder � in which �ðsÞ lies. If amaximal surface contains �ðsÞ as a planar line of curvature, from Lemma 6.5 thismaximal surface meets � along �ðsÞ at a constant angle. Applying Theorem 1.2 itfollows that the maximal surface is a piece of a catenoid whose axis is orthogonalto �. Thus it is a hyperbolic catenoid if � is timelike, and an elliptic catenoid if �is spacelike.

Let now �ðsÞ be a circle of a degenerate plane � in L3, that will be assumed tobe the x1 ¼ x3-plane. We can parametrize �ðsÞ as (see [16])

�ðsÞ ¼ �ðs2=2; s; s2=2Þ; � 6¼ 0:

Let � be a maximal surface containing �ðsÞ as a planar line of curvature, anddenote by NðsÞ the Gauss map of � along �ðsÞ. Then there is some ’2Rnf0g


such that hNðsÞ; ð1; 0; 1Þi ¼ ’. Putting this relation together with h�0;Ni� 0 andhN;Ni� �1 we obtain

NðsÞ ¼ �1

2’ð1þ ’2ðs2 � 1Þ; 2’2s; 1þ ’2ðs2 þ 1ÞÞ:

Therefore

�ðsÞ ¼ðs

NðrÞ��0ðrÞdr ¼ ��2’

�’2

�s3

3� s

�� s; ’2s2; ’2

�s3

3þ s

�� s

�:

From the general solution to Bj€oorling problem, the maximal surface � is given byðs; tÞ ¼ Re�ðzÞ � Im �ðzÞ. In this way � is parametrized as

1ðs; tÞ ¼�

2’ð’ðs2 � t2Þ � tð1þ ’2Þ þ ’2ðs2t � t3=3ÞÞ;

2ðs; tÞ ¼ �sð1þ ’tÞ;

3ðs; tÞ ¼�

2’ð’ðs2 � t2Þ � tð1� ’2Þ þ ’2ðs2t � t3=3ÞÞ:

Figure 1. Associate family of a hyperbolic catenoid in L3. The final surface in this deformation is ahelicoid with timelike axis, whose image is an open set of the classical helicoid in R3. All the other

figures represent helicoidal maximal surfaces with timelike axis


Finally, if � denotes a rotation of the null axis as in (2.3) for h ¼ 0, it is easilyverified that ðsþ ; tÞ ¼ �ððs; tÞÞ. Thus ðs; tÞ is a rotation surface, that is, it isa parabolic catenoid. &

To end up we treat the question of finding all maximal surfaces that contain ahelix as a geodesic, thus generalizing a result in [3]. It comes clear that if a helixG‘;hðpÞ is a geodesic of a maximal surface, then the circular cylinder in whichG‘;hðpÞ lies is spacelike, and the maximal surface is given by (5.1) for ’ ¼ 0. Sowe have

Corollary 6.7. Let : � � C! L3 be a maximal surface containing a helixG‘;hðpÞ as a geodesic. Then G‘;hðpÞ is given in suitable coordinates by �ðsÞ ¼ðhs; b sinh s; b cosh sÞ and ¼ bXcat þ hXhel. Here Xcat ¼ Reð�iz; sinh z;cosh zÞ is a hyperbolic catenoid with axis ‘ and Xhel ¼ ðXcatÞ� is its adjointhelicoid.

Appendix

Let X : � � C! L3 be a spacelike surface with constant mean curvature Hand metric ds2 ¼ e�jdzj2, and let us denote by ’ its Hopf differential (see [21] forthe details). The associate surface to X with angle , X : � � C! L3, is the only(up to congruences in L3) spacelike surface with constant mean curvature H,metric ds2 and Hopf differential ’ ¼ ei’. This concept generalizes the one ofassociate maximal surfaces (see [25, pg. 185]). The family fX : 2 ½0; 2�Þg con-tains all isometric immersions of ð�; ds2Þ in L3 with constant mean curvature H.

Now consider an isometry � : �! � of ð�; ds2Þ. It is easy to check thatX � � is associate with angle to X � �. Suppose that � comes from a selfcongruence of X, that is, there is a rigid motion � of L3 such thatX � � ¼ � � X. Then the immersions X � � and � � X are both associate toX � � with angle , and thus they are congruent in L3. In this way there is a rigidmotion � of L3 such that

X � � ¼ � � X;

that is, � also comes from a self congruence of X. This will be used to show thefollowing result.

Proposition 6.8. The associate family of a spacelike helicoidal CMC surface inL3 is made up by helicoidal surfaces.

Proof. If X is a spacelike helicoidal CMC surface in L3, it admits a continuous1-parameter group of isometries f�� : �2Rg, each of which comes from a selfcongruence of X. From the above comments, each X associate to X has a con-tinuous 1-parameter group of self congruences in L3. Since the 1-parametergroups of rigid motions of L3 are helicoidal motion groups, we get that X ishelicoidal. &

In the case where H ¼ 0 all the associate surfaces to a given maximalsurface can be given explicitly. It is easy to verify in this situation that if X is ahelicoidal maximal surface in L3 with spacelike (resp. timelike, lightlike) axis,


every associate maximal surface to X is helicoidal with spacelike (resp. timelike,lightlike) axis. More precisely, if �ðsÞ ¼ Xðs; t0Þ is an orbit of a maximal surfaceX : � � C! L3 invariant under G‘;h, and ��ðsÞ ¼ X�ðs; t0Þ, then from the proofof Proposition 6.8 it follows that ��ðsÞ is an orbit. A standard use of the Cauchy-Riemann equations as in (6.5) shows that the axis of ��ðsÞ is ‘. Thus X� ishelicoidal with axis ‘. Once here, the fact that every associate surface X to Xhas axis ‘ follows directly, taking into account that X ¼ cos X � sin X�.

The question of isometric deformations of timelike CMC surfaces in L3 wastreated in [14]. It is shown there that the timelike right circular cylinders withspacelike axis and timelike axis, respectively, belong to the same 1-parameterfamily of isometric CMC timelike surfaces. Thus the assertion that we maderegarding the axes of associate maximal surfaces does not hold for the timelikecase.

Acknowledgements. The authors are grateful to the referees of this article for their valuable com-ments and suggestions in order to improve it.

References

[1] Aledo JA, Galvez JA, A Weierstrass representation for maximal linear Weingarten spacelikesurfaces in the Lorentz-Minkowksi space. Preprint

[2] Alıas LJ, Ferrandez A, Lucas P, Mero~nno MA (1998) On the Gauss map of B-scrolls. Tsukuba JMath 22: 371–377

[3] Alıas LJ, Chaves RMB, Mira P (2003) Bj€oorling problem for maximal surfaces in Lorentz-Minkowski space. Math Proc Cambridge Philos Soc 134(2): 289–316

[4] Alıas LJ, Mira P (2002) A Schwarz-type formula for minimal surfaces in Euclidean space Rn. CRAcad Sci Paris, Ser I 334: 389–394

[5] Barros M, Ferrandez A, Lucas P, Mero~nno MA (2001) General helices in the three-dimensionalLorentzian space forms. Rocky Mount J Math 31: 373–388

[6] Calabi E (1970) Examples of Bernstein problems for some nonlinear equations. Proc Symp PureMath 15: 223–230

[7] Cheng SY, Yau ST (1976) Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces.Ann Math 104: 407–419

[8] Choi SM, Ki U, Suh YJ (1998) On the Gauss map of null scrolls. Tsukuba J Math 22: 273–279[9] Dillen F, K€uuhnel W (1999) Ruled Weingarten surfaces in Minkowski 3-space. Manuscripta Math

98: 307–320[10] Ferrandez A, Lucas P (1992) On surfaces in the 3-dimensional Lorentz-Minkowski space. Pacific

J Math 152: 93–100[11] Graves LK (1979) Codimension one isometric immersions between Lorentz spaces. Trans Amer

Math Soc 252: 367–392[12] Hano J, Nomizu K (1983) On isometric immersions of the hyperbolic plane into the Lorentz-

Minkowski space and the Monge-Ampere equation of a certain type. Math Ann 262:245–253

[13] Hano J, Nomizu K (1984) Surfaces of revolution with constant mean curvature in Lorentz-Minkowski space. Tohoku Math J 36: 427–437

[14] Inoguchi J (1998) Timelike surfaces of constant mean curvature in Minkowski 3-space. Tokyo JMath 21: 141–152

[15] Kobayashi O (1983) Maximal surfaces in the 3-dimensional Minkowski space L3. Tokyo J Math 6:297–309

[16] L�oopez FJ, L�oopez R, Souam R (2000) Maximal surfaces of Riemann type in Lorentz-Minkowskispace L3. Michigan Math J 47: 469–497

[17] Milnor TK (1990) Associate harmonic immersions in 3-space. Michigan Math J 37: 179–190[18] Mira P (2001) Construction of maximal surfaces in the Lorentz-Minkowski space. In: Mladenov I,

Naber G (eds) Geometry, Integrability and Quantization III, pp 337–350. Sofia: Coral Press[19] McNertney LV (1980) One parameter families of surfaces with constant curvature in Lorentz 3-

space. PhD Thesis, Brown Univ


[20] O’Neill B (1983) Semi-Riemannian Geometry. New York: Academic Press[21] Palmer B (1990) Spacelike constant mean curvature surfaces in pseudo-Riemannian space forms.

Ann Global Anal Geom 8: 217–226[22] Sasahara N (2000) Spacelike helicoidal surfaces with constant mean curvature in Minkowski

3-space. Tokyo J Math 23: 477–502[23] Strubecker K (1959) Differentialgeometrie III, Theorie der Fl€aachenkrummung. Berlin: Walter de

Gruyter[24] Treibergs AE (1982) Entire spacelike hypersurfaces of constant mean curvature in Minkowski

space. Invent Math 66: 39–56[25] Weinstein T (1996) An Introduction to Lorentz Surfaces. Berlin: Walter de Gruyter[26] Van Der Woestjine I (1990) Minimal surfaces of the 3-dimensional Minkowksi space. In: Boyon

M et al (eds) Geometry and Topology of Submanifolds II, pp 344–369. Singapore: WorldScientific

Authors’ addresses: P. Mira, Departamento de Matematica Aplicada y Estadıstica, Universidad Poli-tecnica de Cartagena, E-30203 Cartagena, Murcia, Spain, e-mail: [email protected]; J. A. Pastor,Departamento de Matematicas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain, e-mail:[email protected]

334 P. Mira and J. A. Pastor: Helicoidal Maximal Surfaces in Lorentz-Minkowski Space

Helicoidal maximal surfaces in Lorentz-Minkowski space

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