The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz-Minkowski space L3

The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz-Minkowski space L3 Isabel Fern´andez ∗Francisco J. L´opez ∗Rabah Souam ∗ Abstract We show that, up to some natural normalizations, the moduli space of singly periodic complete embedded maximal surfaces in the Lorentz-Minkowski space L3 = (R3, dx21 + dx22 − dx23), with fundamental piece having a finite number (n + 1) of singularities, is a real analytic manifold of dimension 3n+4. The underlying topology agrees with the topology of uniform convergence of graphs on compact subsets of {x3 = 0}. 1 Introduction A maximal surface in the Lorentz-Minkowski space L3= (R3, dx2 1+dx2 2−dx2 3) is a spacelike surface with zero mean curvature. It locally maximizes the area functional associated to variations by spacelike surfaces. In a pioneering work, Calabi [2] proved that the affine spacelike planes are the only complete maximal surfaces in L3(Calabi, in fact, showed the analogous result for maximal hypersurfaces in L4and this was later extended to maximal hypersurfaces in Ln,for all n, by Cheng and Yau [3]). Nonetheless, recent works show there is a rich global theory of complete maximal surfaces with singularities in L3([8], [7], [10], [13], [14]). For instance, Umehara and Yamada [14] have obtained results on the global behavior of immersed maximal surfaces having analytic curves of singularities. Of particular interest are the complete embedded maximal surfaces having a closed discrete set of singularities. First it should be remarked that maximal surfaces in L3share some properties with minimal surfaces in the Euclidean space R3.Indeed, Kobayashi [12] gave a Weierstrass type representation in terms of meromorphic data similar to the one of minimal surfaces in R3.Also, both types of surfaces are locally represented as graphs of solutions of ∗Research partially supported by MCYT-FEDER grant number MTM2004-00160. 2000 Mathematics Subject Classification. Primary 53C50; Secondary 58D10, 53C42. Key words and phrases: maximal surfaces, periodic surfaces, conelike singularities 1 Figure 1: The Lorentzian catenoid elliptic operators. An important difference, however, is that the maximal surface equation for graphs in L3may have solutions with isolated singularities and this never happens for the minimal surface equation in R3.Otherwise said, an embedded maximal surface can have an isolated singularity, contrarily to an embedded minimal surface in R3.This is illustrated by the Lorentzian half-catenoid, [12] (see Figure 1). At a singular point around which a maximal surface is embedded in L3,the Gauss curvature blows up, the limit tangent planes become lightlike and the surface is asymptotic to a half lightcone at the singularity (cf. [10], [12] and [8]). For these reasons, such points are called conelike singularities. Embedded complete maximal surfaces with a closed discrete set of singularities are global graphs over any spacelike plane. Conformally, the regular set of such a surface is a Riemann surface minus as many pairwise disjoint closed conformal disks (without accumulation points) as singular points in the surface. A fundamental observation is that the Weierstrass data for the surface extends to the double Riemann surface, [7]. This allows one to work on boundaryless Riemann surfaces. In [8] we developed the theory of embedded complete maximal surfaces in L3of finite type, that is those having a finite number of singularities. We showed, in particular, that the moduli space of such surfaces with n+ 1 ≥2 singularities and vertical limit normal at infinity is a 3n+ 4−dimensional manifold. The next simplest subclass of maximal surfaces in L3with a closed discrete set of singularities consists of the periodic ones with finite type in the quotient. This refers to surfaces that are invariant under a discrete group Gof isometries of L3acting freely and properly and such that the quotient surfaces are embedded and have a finite number of singularities in the flat and complete 3-dimensional Lorentzian manifold L3/G. In [7], the first and second author proved fundamental facts about the global geometry of these surfaces. In particular, they classified the discrete groups G⊂Iso(L3) for which L3/G contains complete maximal surfaces of finite type. If L3/G is orientable and orthochronous (i.e the elements of Gpreserve the orientation and the future time direction) and contain a complete embedded maximal surface Sof finite type then Gis a group of spacelike translations of rank one or two and Sis an annulus of finite conformal type or a torus, respectively. In this paper, we study the moduli space of (embedded) singly periodic maximal surfaces in L3having finite type in the quotient space. That is, we consider the group < T > generated by a spacelike translation Tin L3and complete embedded maximal surfaces with a finite number of singularities in L3/ < T > . Several examples of this kind were 2 constructed in [7] (see figure 2). First note that up to an ambient isometry of L3and rescaling, we can suppose that T= (1,0,0).In this case any complete embedded maximal surface with a finite number of singularities in L3/h(1,0,0)iis a graph over the cylinder {x3= 0} ⊂ L3/h(1,0,0)i.We also normalize so that one of the ends of the surface is asymtpotic to {x3= 0, x2≥0}.Our main result then says that: The space Mnof marked entire maximal graphs over the cylinder {x3= 0} in L3/ < (1,0,0) >having n+ 1 ≥2conelike singularities (the mark is an ordering of the set of singularities) and an end asymptotic to {x3= 0, x2≥0}, is a real analytic manifold of dimension 3n+ 4.A global coordinate system is given by the ordered sequence of points in the mark and the normal to the second end. This space is a (n+ 1)!-sheeted covering of the space Gnof (non marked) entire maximal graphs over the cylinder {x3= 0}in L3/h(1,0,0)i,having n+1 ≥2conelike singularities and an end asymptotic to {x3= 0, x2≥0}. The underlying topology of Gnis equivalent to the uniform convergence of graphs on compact subsets of the cylinder {x3= 0}. We have organized our paper as follows: Section 2 contains some preliminaries about the local behavior of maximal surfaces around isolated singularities and the global behavior of complete maximal surfaces of finite type in the quotient space L3/ < T > . Section 3 is devoted to the proof of the main theorem. Our approach relies on algebraic geometry tools: we define some natural bundles on the moduli space Tnof once punctured marked circular domains with n+ 1 boundary components (a mark is an ordering of the boundary circles), and introduce a spinorial bundle Snassociated to the moduli space of Weierstrass data of surfaces in the space of graphs with n+ 1 singularities. The convergence in Mn means convergence of marked conformal structures in Tnand of Weierstrass data. 2 Preliminaries We denote by C,Dthe extended complex plane C∪{∞} and the unit disc {z∈C:|z|< 1},respectively. Throughout this paper, L3will denote the three dimensional Lorentz-Minkowski space (R3,h,i),where h,i=dx2 1+dx2 2−dx3 3.By definition, a coordinate system (y1, y2, y3) in L3 is said to be a (2,1)-coordinate system if the Lorentzian metric is given by dy2 1+dy2 2−dy3 3. We say that a vector u∈R3− {0}is spacelike, timelike or lightlike if kuk2:= hu,uiis positive, negative or zero, respectively. When uis spacelike, kukis chosen non negative. The vector 0is spacelike by definition. A plane in L3is spacelike, timelike or lightlike if the induced metric is Riemannian, non degenerate and indefinite or degenerate, respectively. We call H2={(x1, x2, x3)∈R3:x2 1+x2 2−x2 3=−1}the hyperbolic sphere in L3 of constant intrinsic curvature −1.Note that H2has two connected components H2 +:= H2∩{x3≥1}and H2 −:= H2∩{x3≤ −1}.The stereographic projection σfor H2is defined 3 as follows: σ:C− {|z|= 1} −→ H2;z→2Im(z) |z|2−1,−2Re(z) |z|2−1,|z|2+ 1 |z|2−1, where σ(∞) = (0,0,1). By definition, an isometry in L3is said to be orthochronous if its associated linear isometry preserves H2 +(and so H2 −). In other words, it preserves the future direction. In the sequel, Nwill denote a complete flat 3-dimensional Lorentzian manifold (i.e., a 3-dimensional differential manifold endowed with a flat metric of index one). It is well known that the universal isometric covering of Nis L3(see for example [16],[15]).Thus N can be regarded as the quotient of L3under the action of a discrete group Gof isometries acting freely and properly on L3. In what follows, Mwill denote a differentiable surface. An immersion X:M −→ N is spacelike if the tangent plane at any point is spacelike, that is to say, the induced metric on Mis Riemannian. In this case, S=X(M) is said to be a spacelike surface in N.If N=L3/G, where Gis a (possibly trivial) group of translations acting freely and properly on L3,the locally well defined Gauss map N0of Xassigns to each point of Ma point of H2.A connectedness argument gives that N0is globally well defined and N0(M) lies, up to a Lorentzian isometry, in H2 −.This means that Mis orientable. A maximal immersion X:M −→ N is a spacelike immersion with null mean curvature. In this case, S=X(M) is said to be a maximal surface in N.Using isothermal parameters, Mcan be endowed with a conformal structure. In the orientable case, Mbecomes a Riemann surface. Theorem 2.1 (Weierstrass representation of maximal surfaces in L3[12]) Let X: M → L3be a conformal maximal immersion of a Riemann surface. Then gdef =σ−1◦N0 is a meromorphic function, and there exists a holomorphic 1-form φ3defined on Msuch that (i) the 1-forms given by φ1=i 2(1 g−g)φ3and φ2=−1 2(1 g+g)φ3are holomorphic on M (ii) the induced Riemannian metric on Mis given by ds2=|φ1|2+|φ2|2− |φ3|2=1 41 |g|− |g|)2|φ3|2 (iii) for any closed curve γin Mwe have Re Rγ(φ1, φ2, φ3) = 0 (iv) up to a translation, the immersion is given by X=Re RP0(φ1, φ2, φ3),where P0∈M is an arbitrary point. Conversely, given gand φ3a meromorphic function and a holomorphic 1-form on M, respectively, such that (i),(ii)and (iii)are satisfied, then (iv)defines a conformal maximal immersion of Min L3. 4 Remark 2.1 (Weierstrass data of maximal surfaces in translational spaces) The Weierstrass data (φ1, φ2, φ3)of a maximal surface in L3are invariant by translations. Therefore, maximal surfaces in a quotient L3/G, where Gis a group of translations acting properly and freely, also have a Weierstrass representation as above except that the condition (iii)is replaced by the following one: (iii′)for any closed curve γin Mthe translation of associated vector Re Rγ(φ1, φ2, φ3) is an element of the group G. Definition 2.1 Let X:M → N be a topological embedding of a smooth surface in a Lorentzian 3-manifold and F⊂ M a closed discrete subset. We say that Xis a maximal embedding with singular set Fif X|M−Fis a maximal embedding and the induced metric on M − Fconverges to zero at any point of F. In this case we say that S=X(M)is a maximal surface in Nwith singularities at X(F). Lemma 2.1 (structure of embedded singularities in L3[8], [4]) Let X:D → L3 be a maximal embedding defined on an open disk D,with an isolated singularity at q∈ D. Then D − {q}is conformally equivalent to {z∈C:r < |z|<1},for some r > 0, and if X0:{r < |z|<1} → L3is a conformal reparameterization of X, then X0extends analytically to Ar:= {1≤ |z|<1}by setting X0({|z|= 1}) = P0:= X(q). The Weierstrass data (g, φ3)of X0satisfy: gis injective and |g|= 1 on {|z|= 1}, and φ3(z)6= 0,|z|= 1.In particular, X0reflects analytically about {|z|= 1}to the mirror surface A∗ r:= {z∈C: 1 ≤ |z|<1/r},verifying g◦J= 1/gand J∗(φ3) = −φ3,where J(z) = 1/z is the mirror involution. Moreover for any spacelike plane Πplane containing P0the Lorentzian orthogonal projection π:X(D)→Πis a local homeomorphism and X(D)is asymptotic near P0to a half light cone with vertex at P0. The point P0is said to be a conelike singularity of X(D). Remark 2.2 The universal covering of a complete flat Lorentzian 3-dimensional manifold is isometric to L3(cf. [15]). Therefore the above Lemma extends to the more general context of complete flat Lorentzian 3-manifolds. As a consequence of the previous lemma, if X:M → N is a complete embedded maximal surface with a closed discrete set F⊂ M of singularities, where Nis complete and flat, then M − Fis conformally equivalent to Σ − ∪p∈FDp,where Σ is a Riemann surface and the {Dp}p∈Fare closed pairwise disjoint conformal disks with no accumulation in Σ. We also have that the conformal reparameterization X: Σ − ∪p∈FDp→ N extends analytically to M0:= Σ − ∪p∈FInt(Dp),by putting X(∂Dp) = X(p) for each p∈F. In the sequel we will refer to M0as the conformal support of the embedding X. We also say that M0is the conformal support of the maximal surface X(M)⊂ N . In particular, if N=L3or N=L3/G, where Gis a translational group, the Weierstrass data (φ1, φ2, φ3) extend analytically to M0. 5 We denote by M∗ 0the mirror surface of M0and by Nthe double surface, that is, N=M0∪ M∗ 0with the identification ∂(M0)≡∂(M∗ 0).Moreover, we call J:N→N the antiholomorphic mirror involution, and observe that the fixed point set of Jcoincides with ∂(M0).By Lemma 2.1, the Weierstrass data Φ := (φ1, φ2, φ3) can be extended by Schwartz reflection to Nsatisfying J∗(Φ) = −Φ. For the sake of simplicity, complete maximal embedded surfaces with a finite set of singularities in Nwill be called CMF surfaces. Definition 2.2 (flux of a closed curve) Let S⊂ N be an oriented CMF surface, and let X:M0→ N be a conformal reparameterization of S. For any closed curve γ(s)in M0parameterized by the arclength, we label νas its unit conormal vector so that {ν, γ′}is positive with respect to the orientation in M0.The flux vector of the curve γis defined as F(γ) := Zγ ν(s)ds Since Xis harmonic it follows from Stokes theorem that F(γ) depends only on the homology class of γin M0.If N=L3or L3/G, where Gis translational, and we denote by Φ = (φ1, φ2, φ3) the Weierstrass data of X, it is easy to check that F(γ) = ImZγ Φ We define the flux at a conelike singularity q∈Sas the flux along any curve homotopic to the boundary component γ0of M0corresponding to q. It can be checked hat the flux at conelike singularity is always a timelike vector (see [11]). Definition 2.3 (CMSF surfaces) A complete embedded maximal surface ˜ S⊂L3with a closed discrete set of singularities is said to be singly periodic of finite type if: •˜ Sis invariant under the free and proper action of an infinite cyclic group Gof isometries of L3 •˜ S/G is a CMF surface in L3/G. In the sequel ˜ Swill be called for short a CMSF surface. Remark 2.3 There is a natural connection between CMSF surfaces in L3and CMF surfaces in quotients L3/G, where Gis cyclic. As stated in the above definition a CMSF surface determines a CMF surface in the corresponding space L3/G. Conversely, Theorem 2.2 below will show that the universal covering of a CMF surface in L3/G is a CMSF surface in L3. 6 2.1 Representation of CMSF surfaces In the sequel we will denote by huithe cyclic group generated by the translation of vector u∈L3. Theorem 2.2 ([7]) Let Sbe a CMF surface with n+ 1 singular points in L3/G, where G is a cyclic group of isometries of L3acting properly and freely. Then •The CMSF surface ˜ Sobtained by lifting Sto L3is an entire graph over any spacelike plane. •The group Gis generated by a spacelike translation T. •The conformal support M0of Sis C∗− ∪n i=0Int(Di)where the Di, i = 0,...,n are pairwise disjoint closed (circular) disks. The associated double surface Nis N− {0,∞, J(0), J(∞)},where Nis a compact Riemann surface of genus n, the points 0,∞ ∈ Ncorrespond to the ends of M0and Jdenotes the mirror involution. •If X:M0→L3/G is a conformal parameterization of S, X applies each boundary circle ∂(Di)to a singular point of S. •The Weierstrass data Φ := (φ1, φ2, φ3)of Xcan be extended by Schwarz reflection to N, satisfying: J∗(Φ) = −Φ. Moreover Φhas simple poles at 0,∞, J(0) and J(∞),and the topological ends of S are of Scherk type, that is to say, asymptotic to spacelike flat half cylinders in L3/G. •Tcan be chosen as the translation of vector u=Re RγΦ,where γ⊂ M0is a closed loop around 0. Conversely given M0:= C∗− ∪n i=0Int(Di)where the Di, i = 0,...,n are pairwise disjoint closed (circular) disks, define Nand Jas before and take a Weierstrass data Φon Nsatisfying J∗(Φ) = −Φand having simple poles at the ends. Then X:M0→L3/huiX=Re ZP0 Φ where u=Re RγΦ,defines a complete embedded maximal surface with n+ 1 singularities and its universal covering is a CMSF surface in L3invariant under the group hui. 2.2 Uniqueness of CMSF surfaces Theorem 2.3 (Uniqueness) Let Sidenote a CMF surface in L3/huiwith singular points qi 1,...,qi n∈L3/hui, i = 1,2where u∈ {x3= 0}, u 6= 0.Suppose S1and S2are contained in {x3≥0} ⊂ L3/hui,have the same limit normal directions at the ends and that q1 j=q2 j, j= 1,...,n. Then S1=S2. 7 Figure 2: Examples of CMSF surfaces Proof : The proof is based on the maximum principle. A regular maximal surface in L3can be represented locally as a graph x3=u(x1, x2) of a smooth function u, with u2 x1+u2 x2<1, satisfying the equation: (1 −u2 x1)ux2x2+ 2ux1ux2ux1x2+ (1 −u2 x2)ux1x1= 0. The maximum principle for elliptic quasilinear equations then gives the following geometric maximum principles for maximal surfaces: Let ˜ S1and ˜ S2be two maximal embedded surfaces (possibly with boundary) in L3which intersect tangentially at a point p. Suppose that locally, around p,˜ S1is above ˜ S2,that is to say, u1≥u2where uidenotes the function defining the graph ˜ Si, i = 1,2.Then ˜ S1=˜ S2 locally around pif one of the following hypotheses holds: •pis an interior point of ˜ S1and ˜ S2, •pis a boundary point of ˜ S1and ˜ S2and ∂˜ S1and ∂˜ S2are tangent at p. In either case, by analyticity of solutions of elliptic equations, we also infer that the two graphs ˜ S1and ˜ S2coincide whenever they are simultaneously defined. It is important to emphasize that this local statement only works for maximal graphs without singularities. Consider now S1and S2as in the statement of the theorem. Since uis a horizontal vector, vertical translations are well defined isometries of L3/hui.For any t∈R,put Si(t) = Si+ (0,0, t), i = 1,2. ¿From our assumptions and the asymptotic behavior of the ends given in Theorem 2.2, we deduce that, for t > 0 big enough, S1(t)> S2,that is to say S1(t)∩S2=∅and S1(t) is above S2.Let t0= inf{t > 0 : S1(t)> S2}.We are going to prove that t0= 0. Suppose t0>0.If S1(t0)∩S26=∅then S1(t0) and S2have a contact point different from the singularities. But then the interior maximum principle implies that S1(t0) = S2, which is absurd. Assume now that S1(t0)∩S2=∅(contact at infinity). Because the ends of the surfaces are asymptotic to spacelike flat half cylinders, then for ǫ > 0 small enough, S1(t0−ǫ)∩S2 is a non empty compact real 1-dimensional analytic manifold containing a Jordan curve Γ spanning two parallel annular ends without singular points E1⊂S1(t0−ǫ) and E2⊂S2, with E1∩E2= Γ.Let F1=RΓν1and F2=RΓν2the flux along Γ in S1(t0−ǫ) and S2resp. (see definition 2.2). It is not hard to see that Fiis orthogonal to uand to the limit normal 8 vector at the end Ei.Moreover, hFi, Fii=hu, ui,and so we infer that F1=F2.However by the boundary maximum principle, the third coordinate of ν1is strictly bigger than that of ν2along Γ,which is a contradiction. This proves that t0= 0,and reversing the argument, that S1=S2. ✷ Corollary 2.1 The group of ambient isometries preserving a CMF surface Sin L3/hui coincides with: •the group of orthochronous (i.e., preserving the future direction) ambient isometries leaving invariant the set of its singularities and preserving the set of normal directions at the ends in case the ends of Sare not parallel •the group of ambient isometries leaving invariant the set of its singularities and the limit normal vector at the ends in case Shas parallel ends. 3 The space of CMSF surfaces In this section we are going to study the moduli space of CMSF surfaces in L3.By Remark 2.3, this space can be identified with the space of CMF surfaces in quotients L3/G, where G=huiand uis a spacelike vector. Moreover, as shown in Theorem 2.2, we can restrict ourselves to the case of CMF graphs over spacelike flat cylinders in L3/hui. First, we have to introduce some normalizations. Let ˜ Sbe a CMSF surface invariant by a spacelike translation T. Up to a isometry in L3and rescaling we will always suppose T(p) = p+ (1,0,0).From Theorem 2.2 we know that S:= ˜ S/h(1,0,0)iis a CMF graph over the cylinder {x3= 0} ⊂ L3/h(1,0,0)iwith flat ends. Up to a hyperbolic rotation in L3/h(1,0,0)iwe can suppose that one of them is asymptotic to the half cylinder {x3= 0, x2≥0}. In the sequel we denote by Gnthe space of CMF graphs in L3/h(1,0,0)iover the cylinder {x3= 0}with one of their ends, which will be denoted E1in the sequel, asymptotic to {x3= 0, x2≥0}and having n+ 1 singularities, n≥1.We will always suppose that all S∈Gnare oriented by the past directed normal. Note that the limit normal to the second end E2of Slies in H2∩ {x3<0, x1= 0}.The latter set is identified, through a suitable stereographic projection, with the real interval ] −1,+1[. Let S∈Gnand label Fas its set of singularities. By definition, a mark in Sis an ordering m= (q0, q1,...,qn)∈L3/h(1,0,0)in+1 of the points in F, and we say that (S, m) is a marked graph. We denote by Mnthe space of marked graphs and define the two following maps: s1:Mn→Gnand s2:Mn→L3/h(1,0,0)in+1×]−1,1[ s1(S, m) = Ss2(S, m) = (m, c) 9 z∈U(v) = Ω(v)− {0,∞}∪Ω(v)∗− {Jv(0), Jv(∞)}∪a0(v),we infer that h(z)∈R∗, for any zsatisfying |z−c0(v)|=r0(v) = c0(v)−1.Then, define φ0 3(x) := 1 h(1) φ, and observe that the equations (φ0 3(x)) = D·Jv(D) ∞ · Jv(∞)and h0 3(1) = 1 (3) characterize φ0 3(x) as meromorphic 1-form on N(v). Given χ=x, q0, ε0∈ Sn×L3/h(1,0,0)i× {−1,1}it is easy to check that there exist unique θχ∈ {|z|= 1}and rχ∈Rsuch that, for φ3(χ) = rχφ0 3(x) and gχ=θχg0 x,the map Xχ:Ω(v)→L3/h(1,0,0)i, Xχ(z) := q0+ Real Zz 1i 2(1 gχ −gχ),−1 2(1 gχ +gχ),1φ3(χ) is well defined and determines a CMF graph Sχ:= XχΩ(v)− {0,∞}∈Gnsatisfying X(a0(v)) = q0and ǫ0is the sign of the third coordinate of the flux along a0(v). Defining the mark mχby qj=Xχ(aj(v)), i = 0,...,n, it is now clear that E−1(χ) = {(Sχ,mχ)},and so, Eis bijective. ✷ 3.2 Structure of the spinorial bundle Sn In the previous subsection we have identified Mnwith the space Sn×L3/h(1,0,0)i×{−1,1}. Our aim now is to show that Sn,and so Mn,has a natural structure of differentiable manifold. Theorem 3.1 (Structure of the spinorial bundle) The space Snis a smooth real (3n+ 1)-dimensional submanifold of Divnand the map v:Sn→ Tn,v(v, D) = vis a finite covering. Proof : The fact Sn6=∅follows from the existence of CMSF surfaces with an arbitrary number of singularities in the quotient (see [7]) and Proposition 3.1. The key step of this proof is that Sndoes not contain any special divisor (see [6]). Consider the differentiable map H:Divn→ Jngiven by H(v, D) = v, 2ϕv(D)−ϕv0·Jv(0) · ∞ · Jv(∞)−T(v), and note that Sn={(v, D)∈Divn:H(v, D) = (v, 0)}.In order to prove that Snis a differentiable submanifold of Divn,it suffices to check that dHqis bijective at any point q of Sn. 16 Let q0:= (v0, D0) be an arbitrary point of Sn.Observe that dHq0is bijective if and only if the map H0:Divn(Ω(v0)) → J (v0) D7→ ϕv0(D)−ϕv0(D0) is a local diffeomorphism at D0. We are going to write the expression of H0in local coordinates around D0∈Divn(Ω(v0)) and H0(D0) = 0 ∈ J (v0).To do this, write D0=zm1 1·...·zms s∈Divn(Ω(v0)),and denote by (Uj, wj:= z−zj) the conformal parameter in Ω(v0),where Ujis the open disc of radius ǫ > 0 centered at zj, j = 1,...,s. Put U=Qs j=1 Umj j⊂Ω(v0)n= Ω(v0)×n ... ×Ω(v0). Then pn(U) is a neighborhood of D0in Divn(Ω(v0)),where pn: Ω(v0)n→Divn(Ω(v0)) is the projection associated to the action of the group of permutations of order non Ω(v0)n. A coordinate chart for Divn(Ω(v0)) around D0is given by ξ:pn(U)→Cn, ξ( s Y j=1 Q1,mj·...·Qmj,mj) = ((t1,mj,...,tmj,mj)j=1,...,s), where th,mj=Pmj l=1(zj(Ql,mj))h, h = 1,...,mj, j = 1,...,s. For more details see [6]. Label p:Cn→ J (v0) = Cn/L(v0) as the natural projection and consider a neighborhood W′of H(D0) = 0 such that p:W:= p−1(W)→W′is a diffeomorphism and H0(U)⊂W′. Write ηi(v0)(wj) = fi,j(wj)dwjon Wj:= wj(Uj) for i= 1, . . . , n, j = 1, . . . s. The local expression ˆ H0of H0around D0,ˆ H0=p−1◦H0◦ξ−1,is given by ˆ H0:ξ(pn(U)) →W ˆ H0(t) = s X j=1 mj X h=1 Zwh,mj 0 fj(wj)dwj where fj=T(f1,j,...,fn,j), wh,mj≡wj, tl,mj=Pmj h=1 wl h,mj, h = 1,...,mj, t = (t1,mj,...,tmj,mj)j=1,...,s. Put fj(wj) = P∞ l=0 bj,lwl j, bj,l ∈Cn, j = 1 . . . s. Then the Taylor series for the holomorphic map wh,mj7→ Rwh,mj 0fj(wj)dwjis Rwh,mj 0fj(wj)dwj=P∞ l=1 aj,lwl h,mj,where aj,l = 1 lbj,l−1, l ≥1, j = 1,...,s. It is not hard to check that ˆ H0(t)Ps j=1 Pmj l=1 aj,ltl,mj+R(t), where the first derivatives of Rwith respect to tl,mjvanish at t= 0,and so the column vectors of the Jacobian matrix of ˆ H0are {al,j, l = 1,...,mj, j = 1 ...,s}. Reasoning by contradiction, suppose that the rows of that matrix are linearly dependent, which is equivalent to saying that there exists a holomorphic 1-form ω0in N(v0) having a zero at zj∈Ω(v0)⊂N(v0) of order at least mj, j = 1,...,s.A direct application of Riemann-Roch theorem gives the existence of a non-constant meromorphic function fon N(v0) having poles at z1, . . . , zswith order at most m1,...,ms,respectively. In particular, fhas degree less than or equal to n. As Jv0(0) is not a pole of f, up to adding a constant we can suppose that f(Jv0(0)) = 0. 17 On the other hand, since ϕv0(D0) + ϕv0(0) = Ei(v0) and Iv0(Ei(v0)) = Ei(v0), i ∈ {1,...,22n},then we get ϕv0(D0·0) −ϕv0(Jv0(D0·0)) = 0.Therefore, a direct application of Abel’s theorem gives the existence of a meromorphic function gof degree n+1 on N(v0) whose principal divisor coincides with D0·0 Jv0(D0·0).As Jv0is an antiholomorphic involution with fixed points, it is not hard to check that g◦Jv0=r/g, r > 0.Hence, up to multiplying g by the factor r−1/2,we can suppose that g◦Jv0= 1/g. Note that g∈ Fv0,where for any v∈ Tn,Fvdenotes the family of meromorphic functions hof degree n+ 1 in N(v) with zeroes in Ω(v) and satisfying h◦Jv= 1/h. Claim: Let fλdenote the meromorphic function 1+λf 1+λ(f◦Jv0), λ ∈C.Then, fλis not constant, for any λ∈C∗.Moreover, gλ:= gfλ∈ Fv0for any λ∈C. Assume fλ=c, where c, λ ∈C∗.Then, we infer that 1 + λf =c(1 + λ(f◦Jv0)) and so the polar divisor of f, which is contained in D0,is invariant under Jv0.This is absurd because D0∈Divn(Ω(v0)) and Ω(v0)∩Jv0(Ω(v0)) = ∅. For the second part of the claim, first note that the principal divisor of gλis (gλ) = Dλ·∞ Jv0(Dλ)·Jv0(∞),where Dλis an integral divisor of degree ≤nand so the degree of gλis ≤n+ 1, λ ∈C.Moreover, gλis not constant for any λ(otherwise, Jv0(0) would be a zero of 1 + λf, contradicting f(Jv0(0)) = 0). Let Abe the set {λ∈C:gλ∈ Fv0},and observe that 0 ∈A. It suffices to see that A is open and closed. The openness of Ais an elementary consequence of Hurwitz theorem (we are using the fact that the degree of gλis at most n+ 1). Finally, let us prove that Ais closed. Let λ0∈A, and take {λn}n∈N→λ0,where {λn:n∈N} ⊂ A. The sequence {gn:= gλn}n∈Nconverges to g0:= gλ0uniformly on N(v0).We know that gn◦Jv0= 1/gnand so the zeros of gnlie in Ω(v0),therefore, gnis holomorphic on Ω(v0), n ∈Nand so the same holds for g0.Moreover, since |g0|= 1 on ∂Ω(v0) and it is non constant, the maximum principle implies that |g0|<1 on Ω(v0) and we infer that g0has no critical points on ∂Ωv0.As ∂Ωv0consists of n+ 1 disjoint circles, this means that g0takes on any complex number θ∈S1at least n+ 1 times. Hence the degree of g0must be n+ 1 and g0∈ Fv0.This concludes the proof of the claim. To get the desired contradiction take P∈∂Ω(v0) such that f(P)6= 0,∞,and choose λ′=−1 f(P).Since Jv0(P) = P, the meromorphic function gλ′has degree less than n+ 1,and so, λ′/∈A=C,which is absurd. Summarizing, we have proved that H|Sn:Sn→0, H(v, D) = (v, 0),is a local diffeomorphism, where 0={(v, 0) : v∈ Tn} ⊂ Jnis the null section in the Jacobian bundle. Consequently, the projection v:Sn→ Tn,v(v, D) = v, is a local diffeomorphism too. To finish, it suffices to check that vis also proper. Indeed, take a sequence {(vk, Dk)}k∈N⊂ Sn such that {vk}k∈Nconverges to a point v∞∈ Tn.We can assume that ϕvk(Dk·0) = Ei(vk) for any k∈N.Since Ivk(Ei(vk)) = Ei(vk),we get ϕvk(Dk·0) −ϕvk(Jvk(Dk·0)) = 0.By Abel’s theorem there is a meromorphic function gk∈ Fvkwith canonical divisor Dk·0 Jvk(Dk·0). 18 Let us see that, up to taking a subsequence, {gk}k∈N→g∞∈ Fv∞.Reflecting about all the components of ∂Ω(vk),we can meromorphically extend gkto a planar open neighborhood Wkof Ω(vk)), k ∈N.By continuity and for k0large enough, the set W=∩k≥k0Wk is a planar neighborhood of Ω(v∞).Classical normality criteria show that, up to taking a subsequence, {gk}k∈Nconverges uniformly on Ω(v∞) to a function g∞which is meromorphic beyond Ω(v∞).It is clear that |g∞|= 1 on ∂Ω(v∞),|g∞|<1 on Ω(v∞) and g∞(0) = 0.This proves that g∞is non constant and can be extended to N(v∞) by the Schwarz reflection g∞◦J∞= 1/g∞.Since deg(gk) = n+ 1,then Hurwitz theorem implies that deg(g∞)≤n+ 1.On the other hand, |g∞|= 1 only on ∂Ω(v∞),and so g∞is injective on every boundary component of Ω(v∞).Therefore, the degree of g∞must be exactly n+1 and g∞∈ Fv∞. Finally, note that D∞·0 Jv∞(D∞·0) where D∞∈Divnand use Hurwitz theorem to infer that {Dk}k∈N→D∞∈Divn.Since Snis a closed subset of Divn,we get D∞∈ Sn(i),which proves the properness of v:Sn→ Tnand so the theorem. ✷ 3.3 Proof of the Main Theorem In the preceding section, we have endowed Mnof a differentiable structure. It is natural to ask whether s2:Mn→R3n+4 is a smooth map or not. In order to do this, we have to show that the Weierstrass data of elements in Mn depends smoothly on its associated divisor in Sn.This requires a convenient concept of differentiability for maps from the bundle of divisors to the space of meromorphic functions or 1-forms. The first part of this subsection is devoted to present these concepts. For any v∈ Tn,call M(v) the family of meromorphic functions on N(v).The corresponding bundle over Tnis denoted by Mn=∪v∈TnM(v). Likewise, we call H(v) the space of meromorphic 1-forms on N(v) and denote by Hn=∪v∈TnH(v) the associated bundle over Tn. Given v∈ Tnand k1, k2∈Nwe denote by Divk1,k2(v) the product manifold, Divk1,k2(v) = Divk1(v)×Divk2(v), and by Divk1,k2its associated bundle over Tn,Divk1,k2=∪v∈TnDivk1,k2(v).Like in the case of Divk,Divk1,k2has a natural structure of analytical manifold. We use the convention Divk,0=Divkand Div0,0=Tn. Definition 3.3 (smoothness with k-regularity) Let Mjbe a real manifold of dimension mj, j = 1,2,3,and let f:M1×M2→M3be a Ckmap. The map fis said to be differentiable (or smooth) with k-regularity in M1if, for any charts U1×U2,x≡ (x1,...,xm1), y ≡(y1, . . . , ym2)in M1×M2and (U3, z ≡(z1,...,zm3)) in M3,the local 19 expression of f, f(x, y) : x(U1)×y(U2)→z(U3),satisfies that f(·, y)is smooth in x(U1) for any y∈y(U2),and all the partial derivatives of f(x, y)with respect to variables in x are Ckin x(U1)×y(U2). Definition 3.4 (smooth deformation of the double of a circular domain) Let v0∈ Tnand ǫ > 0small enough. Denote by V(ǫ)the Euclidean ball of radius ǫin Tncentered at v0.Since V(ǫ)is simply connected, standard homotopy arguments in differential topology show the existence of a family of diffeomorphisms {Fv:N(v0)→N(v) : v∈V(ǫ)}such that Fv0=Id, Fv(∞) = ∞, Jv◦Fv◦Jv0=Fv,for any v∈V(ǫ),and F:V(ǫ)×Ω(v0)→C, F(v, z) := Fv(z),is smooth. By definition, we say that {Fv:N(v0)→N(v) : v∈V(ǫ)}is a smooth deformation of N(v0).Moreover note that, for ǫsmall enough, ∂F ∂z 6= 0 in V(ǫ)×Ω(v0). Let W⊂Divk1,k2be a submanifold, and let h:W→ Mnbe a map preserving the fibers, that is to say, hv,D1,D2:= h(v, D1, D2)∈ M(v) for any (v, D1, D2)∈W. We are going to define the notion of differentiability with k-regularity of h. Take Vany coordinate neighborhood in Divk1,k2meeting W. Denote by Vthe associated neighborhood to Vin Tn and call v0∈Van interior point. Take a smooth deformation of N(v0),{Fv:N(v0)→ N(v) : v∈V(ǫ)}.We say that his differentiable with k-regularity in V ∩ Wif the map ˆ h: (V ∩ W)×N(v0)→C, ˆ h((v, D1, D2), x) = hv,D1,D2(Fv(x)) is smooth with k-regularity in V ∩ W. The map his said to be differentiable with kregularity on Wif it does in V ∩ W, for any coordinate neighborhood Vmeeting W. It is easy to check that this definition does not depend on choice of neither v0nor the smooth deformation of N(v0). Likewise, for a map ω:W→ Hnpreserving the fibers, define ˆω:V(ǫ)∩W→ H(v0) by ˆω(v, D1, D2) = F∗ v(ωv,D1,D2)(1,0) where the superscript (1,0) means the (1,0) part of the 1-form (by definition (f dz + g dz)(1,0) =f dz). We say that ωis differentiable with k-regularity in V(ǫ)∩Wif for any local chart (U, z) in N(v0),the map ˆ f: (V(ǫ)∩W)×U→C,given by ˆ f((v, D1, D2), z) = ˆω(v, D1, D2)(z)/dz is smooth with k-regularity in V(ǫ)∩W. The global concept of differentiability with k-regularity in Wis defined in the obvious way. The following 1-forms we will play an important role during the proof of the Main Theorem. Given D=Qs j=1 wmj j∈Divk(Ω(v)),we denote by τD(v) the unique meromorphic 1-form on N(v) satisfying: •τD(v) has simple poles at wjand Jv(wj), j = 1,...,s, and no other poles, •ResiduewjτD(v)=−ResidueJv(wj)τD(v)=−mjfor any j 20 •Rai(v)τD(v) = 0,for any i= 1,...,n. Likewise, take D1=Qs j=1 wmj j,1, D2=Qr h=1 wnh h,2∈Divk(Ω(v)) and define κD1,D2(v) as the unique meromorphic 1-form on N(v) satisfying •κD1,D2(v) has simple poles at wj,1, wh,2and Jv(wj,1), Jv(wh,2(v), j = 1, . . . , s, h = 1,...,r, and no other poles, •Residuewj,1κD1,D2(v)= ResidueJv(wj,1)κD1,D2(v)=−mj,for any j •Residuewh,2κD1,D2(v)= ResidueJv(wh,2)κD1,D2(v)=nhfor any h •Rai(v)κD1,D2(v) = 0,for any i= 1,...,n. Lemma 3.4 ([8]) The maps ηj:Tn→ Hn, v 7→ ηj(v), τ :Divk→ Hn,(v, D)7→ τD(v), and κ:Divk,k → Hn,(v, D1, D2)7→ κD1,D2(v)are differentiable with 1-regularity. As a consequence, the functions πj,k(v) := Rbj(v)ηk(v),are differentiable on Tn. The following theorem will show that Mnand Gnare analytic manifolds of dimension 3n+ 4.We first need the following lemma, proved in [8]: Lemma 3.5 ([8]) Given v∈ Tn,there exists a holomorphic 1-form ω0in N(v)having 2n−2distinct zeroes, none of them contained in ∂Ω(v),and satisfying J∗ v(ω0) = ω0. Theorem 3.2 (Main theorem) The map s2:Mn→L3/h(1,0,0)in+1×]−1,1[,s2(G, m) = (m, c), where cis the normal direction at the non-normalized end, is injective and smooth. Hence, s2(Mn)is open and so s2provides a global system of analytic coordinates on Mn. Moreover, the action of the group of permutations of order n+ 1, µ :Mn× Pn→Mn, is discontinuous. Hence the orbit space, naturally identified to Gn,has a unique analytic structure making s1:Mn→Gn,s1(G, m) = G, an analytic covering of (n+ 1)! sheets. Proof : To see that s2is one to one, suppose (Gi,mi)∈Mn, i = 1,2 satisfy s1(G1,m1) = s1(G2,m2) = (m, c).From our normalizations, one end of both of them is asymptotic to {x3= 0, x2≥0}.Since G1and G2are graphs over {x3= 0} ⊂ L3/h(1,0,0)iit follows that, for both surfaces, the other end is asymptotic to Π ∩ {x2≤0},where Π is the plane determined by the normal direction c. Therefore, G1and G2are contained in a common horizontal half space and by Theorem 2.3 we get G1=G2. To finish the first part of the theorem, it is enough to check that s2is smooth and then use the Domain Invariance Theorem. Here we have endowed Mnwith the differentiable structure induced by E:Mn→ Sn×L3/h(1,0,0)i× {1,−1}(see Definition 3.2). Let (S, m)∈Mnand label χ= (x, q0, ε0) = E(S, m).Following the notation in the proof of the proposition 3.1, call Xχthe associated maximal immersion and label as (gχ, φ3(χ)) its Weierstrass data. 21 Claim The maps Sn×L3/h(1,0,0)i× {1,−1} → Mn, and Sn×L3/h(1,0,0)i× {1,−1} → Hn, χ7→ gχχ7→ φ3(χ) are smooth with 2-regularity and 1-regularity, respectively. Consequently, the map s2is smooth. First we prove that the maps Sn→ Mn, x 7→ g0 x,and Sn→ Hn, x 7→ φ0 3(x),given by equations (2) and (3), are smooth with 2-regularity and 1-regularity, respectively. Indeed, take x0= (v0, D0)∈ Sn.From Theorem 3.1, there exists an open ball V(ǫ) in Tncentered at v0of radius ǫ > 0 and a local diffeomorphism V(ǫ)→ Sn, v 7→ (v, D(v)), where D(v0) = D0.We label V(ǫ) as the image of V(ǫ) under this map. For simplicity, we write x(v) := (v, D(v)), v ∈V(ǫ). Therefore, the map V(ǫ)→Divn+1, v →(v, D(v)·0) is smooth, and since τ:Divn+1 → Hnis also smooth with 1-regularity (see Corollary 3.4), the same holds for the map V(ǫ)→ Hn, v 7→ τv:= τD(v)·0(v). Take a smooth deformation of N(v0),{Fv:N(v0)→N(v) : v∈V(ǫ)}.Let B(v0) = {a1(v0),...,an(v0), b1(v0), . . ., bn(v0)}be the canonical homology basis on Ω(v0) defined as in Subsection 3.1. Observe that N(v0)−∪n j=1(aj(v0)∪bj(v0)) is simply connected, moreover, without loss of generality we can suppose that this domain does not contains the points in D0·0.For vclose enough to v0the curves aj(v) := Fv(aj(v0)), bj(v0) := Fv(bj(v0)) are a canonical basis of N(v) and do not pass also through the points in D(v)·0, j = 1,...,n. By Abel’s theorem, and for z∈N(v)− ∪n j=1(aj(v)∪bj(v)) we have g0 x(v)(z) = ExpZz 1 (τv+ n X j=1 mj(v)ηj(v)) In this expresion, the integration paths lie in N(v)− ∪n j=1(aj(v)∪bj(v))∪ {1},and mj(v)∈Zare integer numbers determined by the equation: fϕv(D(v)·0) −fϕv(Jv(0) ·Jv(D(v))) = n X j=1 mj(v)πj(v), where fϕvis the branch of ϕvon N(v)− ∪n j=1(aj(v)∪bj(v)) vanishing at 1. Since mj(v) depend continuously on v, then mj(v) = mj∈Zand so, by Corollary 3.4, g0 x(v)depends smoothly on vwith 2-regularity. We have to obtain the analogous result for the map V(ǫ)→ Hn, v →φ0 3(x(v)).Take the holomorphic 1-form ω0on N(v0) given in Lemma 3.5, write ν(v0) := ω0=Pn j=1 λjηj(v0), where λj∈R,and define ν(v) := Pn j=1 λjηj(v).Since the map v7→ ν(v) is smooth with 1-regularity (see Corollary 3.4) it suffices to prove that v7→ φ0 3(x(v)) ν(v)is smooth with 2-regularity. 22 By Hurwitz’s Theorem and the implicit function theorem, ν(v) satisfies also the thesis in Lemma 3.5, for v∈V(ǫ),for ǫ > 0 small enough. Moreover, as explained during the proof of Lemma 3.5, the map V(ǫ)→Div2n−2, v 7→ (v, (ν(v))) is at least C1,where as usually (ν(v)) is the canonical divisor associated to ν(v).Hence, writing (ν(v)) = Av·Jv(Av), the map V(ǫ)→Divn−1, v 7→ (v, Av),is also smooth, and therefore, the same holds for V(ǫ)→Divn,n, v 7→ (v, D(v),∞·Av).We infer from Corollary 3.4 that the map V(ǫ)→ Hn, v7→ κv:= κ∞·Av,D(v)(v),is smooth with 1-regularity. Reasoning as above, the map fx(v)(z) = ExpZz 1 (κv+ n X j=1 njηj(v)), is a well defined meromorphic function on N(v),for suitable integer numbers njnot depending on vand V(ǫ)Divn→ Mn, v 7→ fx(v),is smooth with 2-regularity. The principal divisor associated to fx(v)is given by (fx(v)) = D(v)·Jv(D(v)) ∞·Av·Jv(∞)·Jv(Av).Therefore, if we write ν(v) = hv(z)dz zon U(v) = Ω(v)− {0,∞}∪Ω(v)∗− {Jv(0), Jv(∞)}∪a0(v),we infer that φ0 3(x(v)) ν(v)=1 hv(1) fx(v),and so v7→ φ0 3(x(v)) is smooth with 1-regularity. It follows that the map X0 x:= Real Z1 Φ(x)0Φ(x)0:= i 2(1 g0 x −g0 x),−1 2(1 g0 x +g0 x),1φ0 3(x) depends smoothly on x= (v, D) with 2-regularity and defines a complete maximal surface in L3/hVxi,where Vx= Real2πi Res 0(Φ(x)0) Since φ0 3(x) is holomorphic at 0,the vector Vx= (wx,0) ∈C×Ris horizontal. Moreover, up to replacing φ0 3(x) by −φ0 3(x) we can suppose that the third coordinate of the flux around the curve a0(v) is positive. It is straightforward to see that θχ=¯wx |wx|and rχ=ε0 |wx|depend smoothly on χ, and so, it follows that gχ=θχg0 xand φ3(χ) = rχφ0 3(x) depend smoothly on χwith 2 and 1-regularity respectively. To conclude the proof of the claim, observe that Xχ=q0+ Real Z1i 2(1 gχ −gχ),−1 2(1 gχ +gχ),1φ0 3(χ) depends smoothly on χ= (x= (v, D), q0, ε0) with 2-regularity. Therefore qj(χ) = Xχ(aj(v)) and c(χ) = gχ(∞) are smooth functions of χ, and the same holds for s2.This proves the claim. By the injectivity of s2and the domain invariance theorem, s2(Mn) is an open domain in L3/h(1,0,0)in+1×]−1,+1[.We can then endow Mnwith the unique analytic structure making s2:Mn→s2(Mn) an analytic diffeomorphism. 23 To conclude, it remains to check that the action µis discontinuous. Indeed, let τ:Mn→Mndenote the diffeomorphism given by τ(S, m) = (S, τ(m)), τ ∈ Pn+1.Let (S0,m0)∈Mnand write m0= (q0, q1,...,qn)∈L3/h(1,0,0)in+1.Take a neighborhood Uj of qjin L3/h(1,0,0)i, j = 0,1,...,n, such that Ui∩Uj=∅, i 6=j, and call U=Qn j=0 Uj. Then, it is clear that τ(s−1 2(U × R)) ∩s−1 2(U × R) = ∅,for any τ∈ Pn+1 − {Id},which proves the discontinuity of µand concludes the proof. ✷ To finish, we prove that the underlying topology in Gncorresponds to the uniform convergence of graphs over compacts subsets of {x3= 0}. Theorem 3.3 Let {Gk}k∈Nbe a sequence in Gn,and G0∈Gn. Then {Gk)}k∈N→G0in the topology of Gnif and only if {Gk}k∈Nconverges to G0 uniformly on compact subsets of {x3= 0}. Proof : Suppose {Gk}k∈N→G0∈Gnin the topology of Gn,and choose marks in such a way that {(Gk,mk)}k∈Nconverges to (G0,m0) in Mn. Write E((Gk,mk)) = (xk, q0(k), ǫk), Xk=X(xk,q0(k),ǫk)and xk= (vk, Dk)∈ Sn, k ∈ N∪ {0}.Observe that, without loss of generality, ǫk=ǫ0,for all k∈N. Since {(xk, q0(k), ǫk)} → (x0, q0(0), ǫ0) and X(x,q0,ǫ)depends smoothly on (x, q0, ǫ) with 2-regularity (see the proof of Theorem 3.2), it is not hard to check that {Xk}k∈N diverges uniformly on k, that is to say, for any compact Win {x3= 0}there is r > 0 such that |z+ 1/z|> r implies Xk(z)/∈W×R,for all k. Let Wbe any compact domain in the cylinder {x3= 0} ⊂ L3/h(1,0,0)icontaining the singularities in m0as interior points, and let Wkdenote the compact set X−1 k(W×R)⊂ Ω(vk)− {0,∞}, k ∈N∪ {0}. As the domains Wkare uniformly contained in a compact region of C− {0},then {Wk}k∈N→W0in the Hausdorff distance and Xkconverges uniformly on W0to X0.In the last statement we have used that Xkcan be reflected analytically about the circles in ∂Ω(vk),and so all the immersions Xk, k large enough, are well defined in a universal neighborhood of W0in C.It is then obvious that the function uk:R2→Rdefining the graph Gkconverges uniformly over Wto the function u0:R2→Rdefining G0 (furthermore, {vk}k∈N→v0implies that {mk}k∈N→m0). Since Wcan be as larger as we want, {uk}k∈N→u0uniformly on compact subsets of R2. Assume now that the functions uk:{x3= 0} → Rdefining Gkconverge, as k→ ∞,to the function u0:{x3= 0} → Rdefining G0uniformly on compact subsets of {x3= 0}. Let us show that singular points of G0are limits of sequences of singular points of graphs Gk, k ∈N.Indeed, let p0= (y0, u0(y0)) ∈G0be a singular point, and without loss of generality, suppose that p0is a downward pointing conelike singularity. By Lemma 2.1, there exists ǫ > 0 small enough such that u−1 0({x3≤u0(y0) + ǫ}) contains a compact component C0(ǫ) with regular boundary and containing y0as the unique (interior) singular point. Since {uk}k∈N→u0uniformly on compact subsets, u−1 k({x3≤u0(y0) + ǫ}) must contain a compact component Ck(ǫ) containing y0as well, klarge enough. Furthermore, 24 {Ck(ǫ)} → C0(ǫ) in the Hausdorff sense, and by the maximum principle Ck(ǫ) must contain at least an interior singular point ykof uk, k large enough. Since C0(ǫ) converges to {y0} as ǫ→0,we deduce that {pk:= (yk, uk(yk))}k→∞ →p0. As a consequence, there exist marked graphs (Gk,mk)∈Mn, k ∈N∪ {0},such that {mk}k∈N→m0. Call ckthe stereographic projection of the normal vector at E2of Gk, k ∈N∪ {0},and let us see that {ck} → c0.Indeed, take C < 0 small enough such that {x3= 0, x2≤C}does not contain any singular point of uk, k ∈N∪ {0},let γbe a circle in {x3= 0, x2≤C}and let Adenote a closed tubular neighborhood of γin {x3= 0}not containing any singular point of u0.It is well known that uk−u0is solution of a uniformly elliptic linear equation Lk(uk−u0) = 0 over A, k large enough. Moreover, the fact that the functions 1 1−|∇uk|, k∈N,are uniformly bounded on A(see [1]) guarantee that the coefficients of operators Lk, k ∈N,are uniformly bounded too. Therefore, since {uk}k∈N→u0uniformly on A, the classical Schauder estimates ([9] p. 93) imply that {uk}k∈N→u0in the C2norm on A. In particular, {nk:= Zγ νk(sk)dsk}k∈N→n0:= Zγ ν0(s0)ds0, where νkand skare the conormal vector and the arc-length parameter along γin Gk, respectively, for any k∈N∪ {0}.Since the normal vector at E2of Gklies in {x1= 0}and is orthogonal to nk, k ∈N∪ {0},we infer that {ck} → c0. Since s2:Mn→s2(Mn)⊂R3n+4 is an homeomorphism, {(Gk,mk)}k∈N→(G0,m0) in the manifold Mn,and so, {Gk}k∈N→G0in the manifold Gn.This concludes the proof. ✷ References [1] R. Bartnik and L. Simon: Spacelike hypersurfaceswith prescribed boundary values and mean curvature. Comm. Math. Phys.,Vol. 87(1982/83), 131-152. [2] E. Calabi.: Examples of the Bernstein problem for some nonlinear equations. Proc. Symp. Pure Math., Vol. 15, (1970), 223-230. [3] S. Y. Cheng and S. T. Yau.: Maximal space-like hypersurfaces in the Lorentz- Minkowski spaces. Ann. of Math. (2), Vol. 104 (1976), 407-419. [4] K. Ecker.: Area maximizing hypersurfaces in Minkowski space having an isolated singularity . Manuscripta Math., Vol. 56 (1986), 375-397. [5] F. J. M. Estudillo and A. Romero.: Generalized maximal surfaces in the Lorentz- Minkowski space L3Math. Proc. Camb. Phil. Soc. 111, (1992), 515-524. [6] H. M. Farkas, I. Kra.: Riemann surfaces. Graduate Texts in Math., 72, Springer Verlag, Berlin, 1980. 25