The number of conformally equivalent maximal graphs

The number of conformally equivalent maximal graphs Isabel Fern´andez ∗ Abstract We show that the number of entire maximal graphs with finitely many singular points that are conformally equivalent is a universal constant that depends only on the number of singularities, namely 2nfor graphs with n+ 1 singularities. We also give an explicit description of the family of entire maximal graphs with a finite number of singularities all of them lying on a plane orthogonal to the limit normal vector at infinity. 1 Introduction The present paper is devoted to the study of maximal graphs in the Lorentz-Minkowski space L3= (R3,h·,·i),where h(x1, x2, x3),(y1, y2, y3)i=x1y1+x2y2−x3y3. Maximal graphs appear in a natural way when considering variational problems. If u: Ω ⊂R2≡ {x3= 0} → Ris a smooth function defining a spacelike graph in L3(that is, a graph with Riemannian induced metric), then its area is given by the expression A(u) = ZΩp1− |∇u|2, (recall that |∇u|<1 since the graph is spacelike). The corresponding equation for the critical points of the area functional in L3is Div ∇u p1− |∇u|2= 0.(1) ∗Research partially supported by Spanish MEC-FEDER Grant MTM2007-64504, and Regional J. Andaluc´ıa Grants P06-FQM-01642 and FQM325. 2000 Mathematics Subject Classification. Primary 53C50; Secondary 53A10, 53A30. Key words and phrases: maximal surfaces, conelike singularities, conformal structures. 1 Spacelike graphs satisfying this (elliptic) differential equation are called maximal graphs, since they represent local maxima for the area functional. Geometrically, this condition is equivalent to the fact that the mean curvature of the surface in L3vanishes identically. Besides of their mathematical interest, these surfaces, and more generally those having constant mean curvature, have a significant importance in physics [MT]. Figure 1: Left: Lorentzian catenoid. Right: Riemann type surface. From a global point of view, it is known by Calabi’s theorem [Ca] that the only everywhere regular complete maximal surface is the plane. In particular, there are no entire maximal graphs besides the trivial one. This motivates to allow the existence of singularities, i.e., points of the surface where the metric degenerates. We will focus here our attention to the case where the singular set is the smallest possible, that is, a finite number of points. The first and most known example is the Lorentzian catenoid (Figure 1, left), an entire maximal graph with one singular point, and actually the only one as proved in [Ec], but there are examples with any arbitrary number of singularities. Among them it is worth mentioning the Riemann type maximal graphs (Figure 1, right) obtained in [LLS], with two singular points and characterized by the property of being foliated by circles and lines. Other highly symmetric examples with arbitrary number of singularities (even infinitely many) were constructed in [FL2] (Figure 2). Actually there is a huge amount of such graphs. Indeed, in [FLS] the authors study the moduli space Gnof entire maximal graphs with n+1 singularities, proving that it is an analytic manifold of dimension 3n+4. A global system of coordinates in this space is given by the position of the singular points in L3and a real number called the logarithmic growth that controls the asymptotic behavior. If u: Ω →Rdefines a maximal graph, singular points appear where |∇u|= 1. At a singular point, the PDE (1) stops being elliptic. Moreover, the tangent plane of the surface becomes lightlike, the normal vector has no well defined limit, and the surface is asymptotic to a half of the light cone of the singular point. For this reason they are called 2 Figure 2: Entire maximal graphs with isolated singularities. conelike singularities. It should be pointed out that a maximal surface with isolated conelike singularities is an entire graph if and only if it is complete (that is, divergent curves have infinite length), as proved in [FLS]. If Sis a maximal surface with singular set F⊂S, its regular part S\Fhas a natural conformal structure associated to its Riemannian metric. The conformal type of a maximal surface has been widely studied, for example in [FL1, AA] parabolicity criteria for maximal surfaces are given, but there also exist hyperbolic examples, [Al1, Al2, MUY]. In the case of entire graphs with n+1 singularities, it turns out that S\Fis conformally equivalent to a n-connected circular domain of the complex plane, that is, the plane with n+ 1 discs removed. Each one of these boundary circles corresponds to a singular point of the graph. Our aim in this paper is to study the space of entire maximal graphs with the same conformal structure, that is Problem. Given a n-connected circular domain Ωof the complex plane, how many entire maximal graphs with n+1 singularities are there whose conformal structure is biholomorphic to Ω? We will answer this question by proving that the number of (non congruent) maximal graphs supported by a fixed circular domain is finite and does not depend on the circular domain, but only on the number of connected component of the boundary, that is, the number of singularities. This will be the aim of Section 3. Thus, our problem reduces to compute the number of graphs for a fixed conformal structure. In Section 4 we will fix an specific n-connected circular domain (Definition 4.1) and we will find out how many entire graphs are there with this conformal structure, obtaining that there are exactly 2nnon- congruent surfaces. Moreover, the graphs constructed in Section 4 can be characterized by the property of having all their singularities in a plane orthogonal to the limit normal vector at infinity (Theorem 5.1). Let us point out that our main result contrast with the analogous problem in the 3 related theory of solutions to the Monge-Amp`ere equation Hess(u) = 1.(2) Specifically, in [GMM] it is proved that any solution to (2) globally defined on R2with finitely many isolated singularities is uniquely determined by its associated conformal structure, which is also a circular domain of the complex plane. 2 Preliminaries 2.1 Maximal surfaces A differentiable immersion X:M→L3from a surface Mto L3is said to be spacelike if the tangent plane at any point is spacelike, that is to say, the induced metric on M is Riemannian. The Gauss map of a spacelike surface in L3takes values in the sphere of radius −1, H2={p∈L3:hp, pi=−1}. Since H2has two connected components, H2 +=H2∩ {x3>0}and H2 −=H2∩ {x3<0}, spacelike surfaces are always orientable. A maximal immersion is a spacelike immersion whose mean curvature vanishes. A remarkable property of maximal surfaces in L3is the existence of a Weierstrass-type representation for maximal surfaces, similar to the one of minimal surfaces. Roughly speaking, the Weierstrass representation of a conformal maximal immersion X:M→L3 is a pair (g, φ3) of a meromorphic function and a holomorphic 1-form defined on Msuch that, up to translation, the immersion can be recovered as X(p) := Real Zp p0i 2(1 g−g)φ3,−1 2(1 g+g)φ3, φ3,(3) where p0∈Mis an arbitrary point. It is worth mentioning that gagrees with the stereographic projection of the Gauss map of the surface. We refer to [Ko, Ec] and Theorem 2.1 below for more details. We will focus our attention to entire maximal graphs, that is, maximal graphs defined on the whole plane {x3= 0}. As we explained in Section 1, the only everywhere regular example is the plane [Ca], and so singularities (i.e., points where the induced metric converges to zero) appear in a natural way in this setting. The following theorem condense the information regarding the global structure of entire maximal graphs with isolated singularities (also called conelike singularities). 4 Proposition 2.1 (Global behavior, [FLS]) Let Sbe a surface with isolated singularities in L3. Then the following two statements are equivalents: (i) Sis a complete embedded maximal surface, (ii) Sis an entire graph over any spacelike plane. In this case Sis asymptotic at infinity to either a half-catenoid or a plane. If we label F⊂Sas the singular set, S\Fis conformally equivalent to Ω0:= C\ ∪p∈FDp,where Dp are pairwise disjoint closed discs. Moreover, the associated conformal reparameterization X: Ω0→L3extends analytically to Ω := C\ ∪p∈FInt(Dp)by putting X(∂(Dp)) = X(p). The point p∞=∞is called the end of the surface. 2.2 Double surface and representation theorem As showed in the previous section, the underlying conformal structure of an entire maximal graph with an isolated set of singularities is conformally equivalent to a circular domain in the complex plane. We now go into this aspect in depth to obtain a representation theorem for entire maximal graphs with a finite number of singularities that will be crucial in our study. For any finitely connected circular domain Ω = C\ ∪k j=1Int(Dj), let Ω∗be its mirror surface and Nthe double surface obtained by gluing Ω and Ω∗along their common boundaries as in Figure 3 (see [FK] for an explicit description of this construction). It is clear that Nis a compact Riemann surface of genus k−1 minus two points. We denote by Nthe compactification of Nby adding these two points. Finally, we label J:N → N as the mirror involution mapping a point in Ω into its mirror image and viceversa. Notice that Jextends to an antiholomorphic involution on N, and its fixed point set of Jcoincides with ∂Ω≡∂Ω∗. This double surface is used in [FLS] to give a characterization of complete maximal surfaces with a finite number of singularities in terms of their Weierstrass data: Theorem 2.1 (Representation) Let X: Ω →L3be a conformal immersion of an entire maximal graph with n+ 1 conelike singularities, where Ω = C\ ∪n+1 j=1 Int(Dj), Dj pairwise disjoint closed discs. Label Nas the compactification of the double surface of Ω. Then the Weierstrass data of X, (g, φ3), satisfy: (i) gis a meromorphic function on Nof degree n+ 1,|g|<1on Ω, and g◦J=1 g, 5 Figure 3: The double surface associated to a maximal surface with singularities (ii) φ3is a holomorphic 1-form on N \ {p∞, J(p∞)},where p∞=∞ ∈ Ω,with poles of order at most two at p∞and J(p∞),and satisfying J∗(φ3) = −φ3, (iii) the zeros of φ3in N \ {p∞, J(p∞)}coincide (with the same multiplicity) with the zeros and poles of g. Conversely, let Nbe a compact genus nRiemann surface. Suppose that there exists an antiholomorphic involution J:N → N such that the fixed point set of Jconsists of n+ 1 pairwise disjoint analytic Jordan curves γj, j = 0,1,...,n, and that N \ Sn j=0 γj= Ω0∪J(Ω0),where Ω0is topologically equivalent (and so conformally) to Cminus a finite number of pairwise disjoint open discs. Then, for any (g, φ3)satisfying (i),(ii)and (iii)the map X:Ω0\ {p∞} → L3given by Equation (3) is well defined and S=X(Ω0\ {p∞})is an entire maximal graph with conelike singularities corresponding to the points qj:= X(γj), j = 0,1,...,n. 2.3 Divisors on a Riemann surface. An important part of our work in this paper deals with classical properties of divisors on compact Riemann surfaces. We recall here the notation and basics results that will be used in the sequel (see [FK] for more details). 6 Let Σ be a Riemann surface. A (multiplicative) divisor on Σ is a formal symbol D=pk1 1·...pkh h,where pkj∈Σ and kj∈Z.We can also write the divisor Das D=Y p∈Σ pkp, where kp6= 0 only for finitely many. We call Div(Σ) to the multiplicative group of divisors on Σ. We can define an order in Div(Σ), indeed, given D1=Qp∈Σpk1 pand D2=Qp∈Σpk2 p∈Div(Σ), we say that D1≥ D2if k1 p≥k2 pfor all p∈Σ. The degree of the divisor Dis defined as the integer deg(D) = Pp∈Σkp.D ∈ Div(Σ) is an integral divisor if kp≥0 for any p∈Σ.We denote by Divk(Σ) the set of integral divisors of degree k. Let fbe a meromorphic function on Σ.The associated divisor of fis defined as (f) = Qp∈Σpkp,where for any zero (resp. pole) pof fof order αwe have kp=α > 0 (resp. kp=−α < 0), and kp= 0 in other case. Likewise we define the associated divisor of a meromorphic 1-form. Classical theory of Riemann surfaces give that both functions and 1-forms are determined by their divisors up to a multiplying constant. Moreover, the degree of a meromorphic function on a compact Riemann surface is 0, whereas the associated divisor of a 1-form has degree 2n−2, where nis the genus of the surface. 3 A first approach to the problem Let Gbe an entire maximal graph with n+ 1 conelike singularities. When n= 0, Ecker [Ec] characterized the Lorentzian catenoid (Figure 1, left) as the unique entire maximal graph with 1 singular point, so we will assume from now on that n≥1. As showed in Section 2.1, the underlying conformal structure of a maximal graph is conformally equivalent to a circular domain Ω ⊂Cwith n+ 1 boundary components. Moreover, if we rotate the surface so that the end is horizontal, as a consequence of Theorem 2.1 the divisors of the Weierstrass data (g, φ3) of Gmust be of the form (g) = D·p∞ D∗·p∗ ∞ ,(φ3) = D·D∗ p∞·p∗ ∞ ,(4) where p∞=∞ ∈ Ω is the end of the surface, D∈Divn(Ω), and ∗denotes the mirror involution. Notice that the divisor Ddetermines uniquely the Weierstrass data (g, φ3) up to replacing by (eiθ g, A φ3), for any θ, A ∈R. 7 Conversely, for any integral divisor Dof degree non Ω such that there exist a meromorphic function gand 1-form φ3satisfying (4), it is immediate to check that (g, φ3) fulfill conditions (i) to (iii) in Theorem 2.1. Thus by means of Equation (3) we can obtain an entire maximal graph with n+ 1 conelike singularities, horizontal end, and conformal structure Ω. Moreover, this graph is unique up to homotheties and vertical rotations. The problem of finding out whether exists a pair (g, φ3) satisfying (4) for a given divisor Dis closely related with the Abel-Jacobi map of the corresponding compact Riemann surface N,ϕ: Div(N)→ J (N), where J(N) denotes the Jacobian bundle of N(see [FK] for its definition). Abel Theorem states that D ∈ Div(N) is the divisor associated to a meromorphic function (resp. 1-form) on Nif and only if ϕ(D) = 0 (resp. ϕ(D) = T, where T∈ J (N) is a fixed element in the Jacobian bundle). Thus, in our case the divisors Dcoming from Weierstrass data are precisely those satisfying: ϕ(D) + ϕ(p∞)−ϕ(D∗)−ϕ(p∗ ∞) = 0, ϕ(D) + ϕ(D∗)−ϕ(p∞)−ϕ(p∗ ∞) = T. This set of divisors is deeply studied in [FLS], proving that the previous two equations are equivalent to 2ϕ(D)−2ϕ(p∗ ∞) = T. (5) Before going into the properties of this set, let us fix some notation. Let Ω be a n-connected circular domain and write ∂Ω = ∪n j=0γcj(rj), with γcj(rj) = {z∈C,|z−cj|=rj}. Up to a M¨obius transformations we can assume that c0= 0, r0= 1 and c1∈R+. Thus, we can parameterize the space Tnof marked (i.e., with an ordering in the boundary components) n-connected circular domains (up to biholomorphisms) by their corresponding uplas v= (c1, r1,...,cn, r1,...,rn)∈R+×Cn−1×(R+)n,of centers and radii, with the convention c0= 0 and r0= 1. By this identification, Tncan be considered as an open subset of R+×Cn−1×(R+)n, and therefore it inherits a natural analytic structure of manifold of dimension 3n−1. We label as Ω(v) the circular domain defined by v∈ Tn. Now define the spinorial bundle Sn={(v, D) : v∈ Tn,2ϕv(D)−2ϕv(p∗ ∞) = Tv}, where the subscript vrefers to the double surface of Ω(v), then Theorem 3.1 ([FLS]) The spinorial bundle Sndefined above is an analytical manifold of dimension 3n−1. Moreover, the map ν:Sn→ Tn 8 ν(v, D) = v is a finitely sheeted covering. Thus, the number of divisors D∈Divn(Ω(v)) satisfying Equation (5) is a universal constant that depends not on the conformal structure Ω(v), but only on the number of boundary components (equivalently, the number of singularities of the maximal graph). As explained above, each divisor corresponds to a unique congruence class of entire maximal graphs with n+ 1 singularities and conformal structure Ω(v). Thus we have the following Corollary 3.1 For each n∈Nthere exists a constant C(n)∈Nsuch that, for any n-connected circular domain Ω, the number of non-congruent entire maximal graph with conformal structure biholomorphic to Ωis exactly C(n). Remark 3.1 Since the space Tnis simply-connected, it follows from Corollary 3.1 that the number of connected components of Snis C(n). In particular, the number of connected components of the moduli space of entire maximal graphs with n+ 1 singularities is also C(n). Indeed, label Gnas the space of marked entire maximal graph with horizontal end and n+1 singularities, where a mark means an ordering m= (q0,...,qn)of the singular points of the graph. As we commented in Section 1, Gncan be endowed with a differentiable structure of manifold of dimension 3n+ 4 with coordinates given by (G, m)7→ (m, c), being cthe logarithmic growth at the end. On the other hand, we can consider the map ǫ:Gn→ Sn×L3×S1×R ǫ((G, m)) = ((v, D), q0, g(1), h(1)) where, if (g, φ3)denote the Weierstrass data of the graph, then •(v, D)∈ Snis given by the conformal structure of G(with the order in v∈ Tngiven by the order in m), and the divisor Ddefined as in Equation (5), •q0is the first singular point in m, •h:= φ3 dz (here zmeans the natural conformal parameter in Ω(v)⊂C, recall that 1∈∂Ω(v)for all v∈ Tn). Then, it is clear from the above explanation that ǫis bijective. Moreover, the induced topology in Gnby ǫagree with the one given by its before mentioned differentiable structure, as proved in [FLS]. Thus, the number of connected components of Gnis C(n). 9 Observe that replacing the set τby its complementary {a1,...,a2n+2}\τgives congruent surfaces (more specifically, (g, φ3) are transform into (−g, −φ3)). So, we can assume without loss of generality that b1=a1. To avoid congruences, we will also normalize so that g(a1) = h(a1) = 1, where h=φ3 dz w . Looking at the expressions for gand φ3this means that θ= 0, A = 1. Thus, the number of non-congruent maximal graphs defined on Ω0is the number of possible choices of bj∈ {a2j−1, a2j+1}, j = 2,...,n+ 1, which is 2n.✷ Taking into account our previous discussion in Section 3, we can conclude that: Theorem 4.2 The number of non-congruent entire maximal graphs with the same conformal structure is 2n, where n+ 1 is the number of (conelike) singularities. Equivalently, the number of connected components of the space Gnof entire marked maximal graphs with n+ 1 singularities and horizontal end is 2n. 5 Maximal graphs with coplanar singularities We will prove now that the surfaces constructed in the previous section are characterized by the property of having all its singularities on a plane orthogonal to the limit normal vector at infinity. In particular, for n= 1, surfaces obtained in Section 4 describe the whole moduli space of the entire maximal graphs with two singular points. Theorem 5.1 Let G⊂L3be an entire maximal graph with n+ 1 conelike singularities. Then Ghas all its singularities lying on a timelike plane in L3orthogonal (in the Lorentzian sense) to the normal vector at the end if and only if Gis congruent to one of the examples given in Definition 4.2. Proof : Assume that Ghas all its singularities in an orthogonal plane to the normal vector at the end. Up to a rigid motion in L3we can assume that the end is horizontal and the singularities lie in the plane {x1= 0}. Let X: Ω →G⊂L3a conformal reparameterization of G. By the uniqueness result in [Kly] (see also [FLS] Remark 2.5), the surface is symmetric with respect to the plane {x1= 0}.This symmetry induces an antiholomorphic involution T: Ω →Ω leaving ∂Ω globally fixed. It follows that Textends to an antiholomorphic involution T:N → N, where Nis the mirror surface, by putting T◦J=J◦T(Jis the mirror involution). Moreover, if (g, φ3) are the Weierstrass data of 16 the immersion, g◦T=gand T∗(φ3) = φ3.It is straightforward that Tmust have exactly two fixed points on every connected component of the circular domain ∂Ω.We call these points p1,...,p2n+2.Observe that the end p∞∈Ω is also fixed by T. Consider the holomorphic involution F=J◦T, whose fixed points are exactly p1,...,p2n+2.Therefore, Nis a compact genus nRiemann surface with 2n+2 fixed points, this means that Nis hyperelliptic with Weiersrtass points p1,...,p2n+2 (see [FK]), N ≡ {(z, w)∈C2:w2= 2n+2 Y i=1 (z−aj)}, where (aj,0) corresponds to pjfor any j(and so aj6=akfor k6=j). With this identification we have F(z, w) = (z, −w).Up to a M¨obius transformation we can suppose that z(p2n+1) = 1, z(p2n+2) = −1,and z(p∞) = ∞. In what follows we will identify aj= (aj,0) ∈N.To prove aj∈Rnotice that the divisor associated to the meromorphic 1-form d(z◦J) coincides with the one of dz and therefore z◦J=k z +λ, for some k, λ ∈R.Since a2n+1 = 1 and a2n+2 =−1 are fixed by Jit follows that z◦J=zwhich implies that aj∈R.Moreover, since w2◦J=w2,then w◦J=±w. Taking into account that Jinterchanges the two points with z=∞,namely p∞and p∗ ∞=J(p∞),then w◦J=−w. Therefore J(z, w) = (z, −w) and T(z, w) = (z, w). 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