Periodic maximal graphs in the Lorentz-Minkowki space L3

Proceedings of XIII Fall Workshop on Geometry and Physics Murcia, September 20–22, 2004 Publ. de la RSME, Vol. 9 (2005), 149–154 Periodic maximal graphs in the Lorentz-Minkowki space L3 Isabel Fern´ andez1and Francisco J. L´ opez2 1[email protected] 2[email protected] Departamento de Geometr´ıa y Topolog´ıa Universidad de Granada Abstract. We study maximal graphs in the Lorentz-Minkowski space L3 invariant under a discrete group of isometries and having a finite number of singularities in its fundamental piece. We also give a method to construct them, based on the Weierstrass representation for maximal surfaces. Keywords: Maximal surfaces, conelike singularities, periodic surfaces. 2000 Mathematics Subject Classification: 53C50, 53C42, 53A10 1. Introduction Maximal surfaces in a Lorentzian manifold are spacelike surfaces with zero mean curvature. In the Lorentz-Minkowski space L3these surfaces arise as local maxima for the area functional associated to variations of the surface by spacelike surfaces. Also, maximal graphs in L3are the solutions for a quasi-linear elliptic differential equation, and therefore a maximum principle for them is satisfied. As in the case of minimal surfaces in the Euclidean space, maximal surfaces have a conformal representation (Weierstrass representation) in terms of meromorphic data on a Riemann surface. A classical result by Calabi [1] asserts that the unique complete maximal surfaces in L3are the spacelike planes. However, if we allow the existence of singularities, there is a vast theory of complete maximal surfaces, see for example [10], [2], [4], [5], [6]. In this paper we focus our attention on isolated embedded singularities of maximal surfaces, also called conelike singularities (see [7]). If in adittion the surface is complete or proper, it turns out that it is a graph over any spacelike plane of L3. 150 Periodic maximal graphs in L3 We say that a surface is periodic if it is invariant under a group of isometries Gof L3acting properly and freely on L3.This paper develops the main results obtained by the authors in [3] for periodic maximal surfaces in the embedded case. In concrete, we show that the group Gcontains a finite index subgroup G0which is a group of translations of rank 0 (that is, G0={Id}), 1 (singly periodic surfaces), or 2 (doubly periodic surfaces). We also use the Weierstrass representation of maximal surfaces to give a recipe recovering these surfaces. Figure 1: Examples of maximal graphs with isolated singularities 2. Preliminaries 2.1. Spacelike immersion with isolated singularities Through this paper L3will denote the 3-dimensional Lorentz-Minkowski space, that is L3=(R3,dx 2+dy2−dz2),and Ma differentiable surface. An immersion X:M−→L3is said to be spacelike if for any p∈M, the tangent plane TpMwith the induced metric is spacelike, that is to say, the induced metric on Mis Riemannian. This metric induces a conformal structure on M,andsoitbecomesinaRiemannsurface. Let F⊂Mbe a discrete closed subset of a differentiable surface Mand ds2a Riemannian metric in M−F. Take a point q∈F, an open disk D(q) in Msuch that D(q)∩F={q}and an isothermal parameter zfor ds2on D(q)−{q}.Then write ds2=h|dz|2,where h(w)>0 for any w∈z(D(q)−{q}). By definition, the Riemannian metric ds2is singular at qif for any disk D(q) and any parameter zas above, the limit limp→qh(z(p)) vanishes (as a matter of fact, it suffices to check this condition just for one disc and conformal parameter). The metric ds2is said to be singular at Fif it is singular at any point of F. In this case, (M,ds 2) is said to be a Riemannian surface with isolated singularities and Fis the singular set of (M,ds 2). Definition 1. Let X:M→L3be a continuous map. Suppose there is a discrete closed F⊂Msubset such that X|M−Fis a spacelike immersion and (M,ds 2) is a Riemannian surface with isolated singularities in F, where ds2 is the metric induced by X. Isabel Fern´ andez and Francisco J. L´ opez 151 Then, Xis said to be a spacelike immersion with (isolated) singularities at F, and X(M) a spacelike surface with (isolated) singularities at X(F). The following lemma describes the behavior of a spacelike immersion around an isolated singularity. Lemma 1 ([3]).Let X:M→L3be a spacelike immersion with isolated singularities and Πa spacelike plane. Label π:L3→Πas the Lorentzian orthogonal projection. Then, h:= π◦Xis a branched local homeomorphism and its branch points correspond to the locally non embedded singularities of X. As a consequence, if Xis an embedding locally around the singular points and is proper, then X(M)is a graph over any spacelike plane (in particular X is an embedding). The same conclusion holds if we replace proper by complete. 2.2. Maximal surfaces A maximal immersion X:M→L3is a spacelike immersion with vanishing mean curvature. The notion of maximal immersion with (isolated) singularities is defined analogously. If X:M→L3is a (everywhere regular) maximal immersion, it is known that there exist a meromorphic map gwith |g| = 1 and a holomorphic 1-form φ3defined on the Riemann surface Msatisfying that the vectorial 1-form Φ=(φ1,φ 2,φ 3):=(i 2(1 g−g)φ3,−1 2(1 g+g)φ3,φ 3) is holomorphic, non vanishing and without real periods in M.Moreover, up to a translation Xis given by X(p)=Rep p0Φ,where p0is an arbitrary point. Either the pair (g,φ3) or the vectorial 1-form Φ is called the Weierstrass representation of the maximal immersion X. As mentioned before, we will focus our attention in embedded surfaces, and therefore, we will consider only embedded singularities. For a more general treatment of non-embedded maximal surfaces with isolated singularities see [3]. As a consequence of Lemma 1, any proper (or complete) maximal surface with isolated embedded singularities is a graph over any spacelike plane. The behaviour of a maximal immersion around embedded isolated singularities is well known (see for example [4], [7]). As a matter of fact, if D is a disc around such a singularity p, then D\{p}is conformally equivalent to an annulus A,and the Gauss map of the immersion becomes lightlike at the boundary component of Acorresponding to the singularity. Moreover, around X(p)thesurfaceX(M) is asymptotic to a component of the light cone at X(p).For this reason, isolated embedded singularities of maximal surfaces are also called conelike singularities. 152 Periodic maximal graphs in L3 Definition 2. We say that a maximal graph with isolated singularities X: M→L3is G-periodic if X(M) is invariant under a discrete subgroup Gof isometries acting freely and properly of L3.We say that Xis singly (resp. doubly) periodic if Gis a group of translations of rank one (resp. two). If in addition the quotient of the singular set of Xunder the relation induced by Gis finite we say that Xis of finite type. Let X:M→L3be a maximal graph of finite type and label F={pα: α∈Λ}⊂Mas its singular set. Taking into account the local behavior around the singularities described above and the results about the Koebe uniformization given in [8], we can deduce that the Riemann surface M\F is biholomorphic to a circular domain C\∪ α∈ΛDα,where Dαare pairwise disjoint closed discs in Cwhose boundaries γα:= ∂(Dα),α∈Λ,correspond to the singularities. In this setting, we label M0:= C\∪α∈ΛInt(Dα)(1) and we refer to it as the conformal support of X. The conformal reparameterization X0:C\∪α∈ΛDα→L3extends to M0by putting X0(γα)=X(pα). 3. Main Results Theorem 2 ([3]).Let X:M→L3be a G-periodic maximal graph of finite type. Then the subgroup G0of Gconsisting of the positive and orthochronous (that is, preserving H2 +)isometriesofG, which is a finite index subgroup of G, is either the identity or a group of spacelike translations of rank 1 or 2. Our aim now is to describe the global behavior of the G-periodic maximal graphs of finite type when Gis one of the three groups given in the above theorem in terms of its Weierstrass data. Thus, let X:M→L3be as in the statement of the theorem, and consider its conformal support M0and the conformal reparameterization X0:M0→ L3(see Equation (1)). Since the isometries in Gpreserves the singular set we can regard Gas group of transformations in M0.So, we can consider the induced immersion ˆ X0:ˆ M0=M0/G →L3/G. It follows that ˆ X0is a complete maximal immersion with a finite number of singularities. Moreover, since Mis simply connected, ˆ X0(ˆ M0) is an embedded surface in L3/G. If in addition Gis a translational group, the Weierstrass data of X0can be also pushed out to ˆ M0. Observe that the Riemann surface with boundary ˆ M0=M0/G is biholomorphic to Σ \∪k j=0Int(Dj),where Djare pairwise disjoint closed discs and Isabel Fern´ andez and Francisco J. L´ opez 153 Σ=Cif G={Id},Σ=C∗in the singly periodic case, and Σ is a torus in the doubly periodic case. In order to use the tools we need for our purposes is useful to work with boundaryless surfaces, for this reason we introduce the notion of the double surface of the conformal support. This surface is nothing but the quotient of ˆ M0∪ˆ M∗ 0by identifying their boundary components, ∂(ˆ M0)≡∂(ˆ M∗ 0),where ˆ M∗ 0is the mirror surface associated to ˆ M0(see [9] for more details). It follows that the Weierstrass data Φ can be extended holomorphically to the double surface Sand satisfy J∗(Φ) = −Φ,where J:S→Sis the mirror involution, that maps each point of ˆ M0into its mirror image and vice versa (observe that the fixed point set of Jcoincides with ∂(ˆ M0)). Theorem 3 ([3]).Let Sbe a compact Riemann surface of genus k≥0and J:S→Sbe an antiholomorphic involution having k+1 pairwise disjoint Jordan curves of fixed points γ0,...,γ k.Suppose also that S\∪k j=0γjhas two connected components, namely Ωand J(Ω),any one of them homeomorphic (and so biholomorphic) to a circular domain1in the extended complex plane Cor in a torus T. Consider a meromorphic vectorial 1-form Φ=(φ1,φ 2,φ 3)defined on S, non vanishing, with J∗(Φ) = −Φand having poles at F∞∪J(F∞),where F∞⊂Ωconsists of one (and in this case the poles are double) or two points (and in this case the poles are simple) if Ω⊂Cand F∞=∅if Ω⊂T. Finally, label Gas the group of translations of vectors {Re γΦ:γ∈ H1(Ω0,Z)},where Ω0is the quotient surface obtained from Ω\F∞by identifying each component γjof ∂(Ω) to a point qj/∈Ω,j=0,...,k (qj=qhfor j=h). Then, Ghas rank 0 (if F∞has 1 point), 1 (if F∞contains 2 points) or 2 (F∞=∅), and the map ˆ X0: Ω \F∞→L3/G, ˆ X0=Re(Φ), is well defined and provides a complete maximal surface with k+1 singular points (namely ˆ X0(γj),j=0,...,k) whose lifting to L3is a G-periodic maximal graph with k+1singular points in its fundamental piece. Conversely, any such surface can be obtained in this way. From the behaviour of the Weierstrass data described above we can deduce the asymptotic behaviour of the lifted periodic surface in L3.It turns out that if G={Id}the surface is asymptotic at infinity to either half catenoid or a spacelike plane, and in the singly periodic case the surface is asymptotic to 1that is, an open domain bounded by analytical circles 154 Periodic maximal graphs in L3 two spacelike half planes. In the doubly periodic case the resulting surface is contained in a slab (see [4], [3] for a detailed proof). Examples of surfaces constructed using the above representation (for example, the surfaces in Figure 1) can be found in [3]. Acknowledgments This work has been partially supported by Spanish MEC-FEDER grant number MTM2004-00160. References [1] E. Calabi. Examples of the Bernstein problem for some nonlinear equations. Proc. Symp. Pure Math., vol. 15 (1970), 223-230. [2] F.J.M. Estudillo and A. Romero. 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