The number of conformally equivalent maximal graphs
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 for graphs with 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.
The present paper is devoted to the study of maximal graphs in the Lorentz-Minkowski space where . Maximal graphs appear in a natural way when considering variational problems. If is a smooth function defining a spacelike graph in (that is, a graph with Riemannian induced metric), then its area is given by the expression
(recall that since the graph is spacelike). The corresponding equation for the critical points of the area functional in is
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 vanishes identically.
Besides of their mathematical interest, these surfaces, and
more generally those having
constant mean curvature, have a significant importance in physics [MT].
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 of entire maximal graphs with singularities,
proving that it is an analytic manifold of dimension . A global system of coordinates in this space is given
by the position of the singular points in and a real number called the logarithmic growth that controls
the asymptotic behavior.
If defines a maximal graph, singular points appear where .
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 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 is a maximal surface with singular set , its regular part has 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 singularities, it turns out that is conformally equivalent to a -connected circular domain of the complex plane, that is, the plane with 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 -connected circular domain of the complex plane, how many entire maximal graphs with singularities are there whose conformal structure is biholomorphic to ?
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 -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 non-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 related theory of solutions to the Monge-Ampère equation
Specifically, in [GMM] it is proved that any solution to (2) globally defined on with finitely many isolated singularities is uniquely determined by its associated conformal structure, which is also a circular domain of the complex plane.
2.1 Maximal surfaces
A differentiable immersion from a surface to
is said to be spacelike if the tangent plane at any point
is spacelike, that is to say, the induced metric on is
Riemannian. The Gauss map of a spacelike surface in takes values in the sphere of radius , .
Since has two connected components, and , spacelike surfaces are always orientable.
A maximal immersion is a spacelike immersion whose mean curvature vanishes. A remarkable property of maximal surfaces in is 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 is a pair of a meromorphic function and a holomorphic -form defined on such that, up to translation, the immersion can be recovered as
We will focus our attention to entire maximal graphs, that is, maximal graphs defined on the whole plane . 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).
Proposition 2.1 (Global behavior, [Fls])
Let be a surface with isolated singularities in . Then the following two statements are equivalents:
is a complete embedded maximal surface,
is an entire graph over any spacelike plane.
In this case is asymptotic at infinity to either a half-catenoid or a plane. If we label as the singular set, is conformally equivalent to where are pairwise disjoint closed discs. Moreover, the associated conformal reparameterization extends analytically to by putting The point 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 , let be its mirror surface and the double surface obtained by gluing and along their common boundaries as in Figure 3 (see [FK] for an explicit description of this construction). It is clear that is a compact Riemann surface of genus minus two points. We denote by the compactification of by adding these two points.
Finally, we label as the mirror involution mapping a point in into its mirror image and viceversa. Notice that extends to an antiholomorphic involution on , and its fixed point set of coincides with
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 be a conformal immersion of an entire maximal graph with conelike singularities, where pairwise disjoint closed discs. Label as the compactification of the double surface of Then the Weierstrass data of , satisfy:
is a meromorphic function on of degree on , and ,
is a holomorphic 1-form on where with poles of order at most two at and and satisfying
the zeros of in coincide (with the same multiplicity) with the zeros and poles of
Conversely, let be a compact genus Riemann surface. Suppose that there exists an antiholomorphic involution such that the fixed point set of consists of pairwise disjoint analytic Jordan curves and that where is topologically equivalent (and so conformally) to minus a finite number of pairwise disjoint open discs.
Then, for any satisfying and the map given by Equation (3) is well defined and is an entire maximal graph with conelike singularities corresponding to the points
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).
Let be a Riemann surface. A (multiplicative) divisor on is a formal symbol where and We can also write the divisor as
where only for finitely many. We call to the multiplicative group of divisors on . We can define an order in , indeed, given and , we say that if for all
The degree of the divisor is defined as the integer
is an integral divisor if
for any We denote by the set of
integral divisors of degree
Let be a meromorphic function on The associated divisor of is defined as where for any zero (resp. pole) of of order we have (resp. ), and in other case. Likewise we define the associated divisor of a meromorphic 1-form. Classical theory of Riemann surfaces give that both functions and -forms are determined by their divisors up to a multiplying constant. Moreover, the degree of a meromorphic function on a compact Riemann surface is , whereas the associated divisor of a -form has degree , where is the genus of the surface.
3 A first approach to the problem
Let be an entire maximal graph with conelike singularities.
When , Ecker [Ec] characterized the Lorentzian catenoid (Figure 1, left) as the unique entire maximal graph with
singular point, so we will assume from now on that .
As showed in Section 2.1, the underlying conformal structure of a maximal graph is conformally equivalent to a circular domain with 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 of must be of the form
where is the end of the surface, , and denotes the mirror involution. Notice that the divisor determines uniquely the Weierstrass data up to replacing by , for any .
Conversely, for any integral divisor of degree on such that there exist a meromorphic function and -form satisfying (4), it is immediate to check that fulfill conditions to in Theorem 2.1. Thus by means of Equation (3) we can obtain an entire maximal graph with conelike singularities, horizontal end, and conformal structure . Moreover, this graph is unique up to homotheties and vertical rotations.
The problem of finding out whether exists a pair satisfying (4) for a given divisor is closely related with the Abel-Jacobi map of the corresponding compact Riemann surface , , where denotes the Jacobian bundle of (see [FK] for its definition). Abel Theorem states that is the divisor associated to a meromorphic function (resp. 1-form) on if and only if (resp. , where is a fixed element in the Jacobian bundle). Thus, in our case the divisors coming from Weierstrass data are precisely those satisfying:
This set of divisors is deeply studied in [FLS], proving that the previous two equations are equivalent to
Before going into the properties of this set, let us fix some notation. Let be a -connected circular domain and write , with . Up to a Möbius transformations we can assume that and . Thus, we can parameterize the space of marked (i.e., with an ordering in the boundary components) -connected circular domains (up to biholomorphisms) by their corresponding uplas of centers and radii, with the convention and . By this identification, can be considered as an open subset of , and therefore it inherits a natural analytic structure of manifold of dimension . We label as the circular domain defined by . Now define the spinorial bundle
where the subscript refers to the double surface of , then
Theorem 3.1 ([Fls])
The spinorial bundle defined above is an analytical manifold of dimension . Moreover, the map
is a finitely sheeted covering.
Thus, the number of divisors satisfying Equation (5) is a universal constant that depends not on the conformal structure , 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 singularities and conformal structure . Thus we have the following
For each there exists a constant such that, for any -connected circular domain , the number of non-congruent entire maximal graph with conformal structure biholomorphic to is exactly .
Since the space is simply-connected, it follows from Corollary 3.1 that the number of connected components of is . In particular, the number of connected components of the moduli space of entire maximal graphs with singularities is also .
Indeed, label as the space of marked entire maximal graph with horizontal end and singularities, where a mark means an ordering of the singular points of the graph. As we commented in Section 1, can be endowed with a differentiable structure of manifold of dimension with coordinates given by , being the logarithmic growth at the end. On the other hand, we can consider the map
where, if denote the Weierstrass data of the graph, then
is given by the conformal structure of (with the order in given by the order in ), and the divisor defined as in Equation (5),
is the first singular point in ,
(here means the natural conformal parameter in , recall that for all ).
Then, it is clear from the above explanation that is bijective. Moreover, the induced topology in by agree with the one given by its before mentioned differentiable structure, as proved in [FLS]. Thus, the number of connected components of is .
4 Counting maximal graphs on a given circular domain
As it was showed in the previous section, the number of maximal
graphs that share the same underlying conformal structure only depends on the number of boundary components of the conformal support.
Thus, in this section we will fix an specific circular domain and we
will find out how many non-congruent maximal graphs are defined on that
Let and Throughout this section, will denote the (hyperelliptic) compact genus Riemann surface associated to the function that is,
And we will also define
The surface can be realized as a two sheeted covering of the Riemann sphere. Indeed, consider two copies of Following [FK], we label these copies as sheet and sheet We ”cut” each copy along curves joining with for any We assume that these cuts does not intersect each others (see Figure 4). Each cut has two banks: a N-bank and a S-bank. We recover the surface by identifying the N-bank (resp. S-bank) of a cut in the sheet with the corresponding S-bank (resp. N-bank) in the sheet
We denote by the two canonical projections, whose associated divisors are
We will label as the one where the coefficient of degree of the Laurent series of is .
Finally we define as the antiholomorphic involution given by The fixed points of are the Jordan curves Moreover, has two connected components, each one of them corresponding to a single-valued branch of , and biholomorphic to a -connected circular domain.
Let and Consider the above defined compact Riemann surface
with the antiholomorphic involution . Label as the set of fixed points of . We will define as the closure of the connected component of containing , and will denote the circular domain .
Let be Weierstrass data on of an entire maximal graph with singularities and horizontal end. Then there exists distinct points , such that
where , , and .
By Theorem 2.1, the associated divisors to are given by
where Here, denotes the point in , and .
We will denote by the holomorphic involution given by .
In the above conditions there exist distint points , such that for two meromorphic functions , on satisfying
Since has degree and is hyperelliptic, the two meromorphic functions and satisfy a relation where is a polynomial in two variables with algebraic degree two in the first one and in the second (see [FK]). We can rewrite this relation as with polynomials whose maximum algebraic degree is Solving this equation we obtain
Consider the meromorphic function Let us check that for some constant Indeed, any meromorphic function on the hyperelliptic surface can be expressed as with rational functions (see [FK]). In our case, is a polynomial function in , and so it follows that either or The last case would imply that is a rational function of which is impossible from Equation (7) so Now observe that has poles only at and with order at most , which implies that is a holomorphic function on and therefore constant. Thus, for some . Up to replace by we can suppose that
We will also assume that the leading coefficient of is one. Since and are meromorphic functions of degree that only depend on , it is not hard to realize that (7) implies that
where . Thus, the meromorphic function
satisfies that and therefore up to a multiplying constant On the other hand, , and reasoning as before we can deduce that for some polynomial functions in with algebraic degree less than or equal to Since we infer that and so, setting we can write
Looking at the divisor of is immediate to realize that there exists an integral divisor with such that:
Since points in
are zeros of both and , they must be distinct
(recall that only has simple zeroes) points of
Setting and the claim is proved.
Up to multiplicative constants, the functions and in Claim 4.1 are given by and being .
Call to the integral divisor given by . By Riemann-Roch Theorem, the dimension of the linear space of meromorphic functions on satisfying condition (resp. ) in Claim 4.1 is where is the dimension of the linear space of meromorphic 1-forms on satisfying (resp. ). Let us see that
Indeed, observe first that by the residues theorem, both spaces agree with the space of holomorphic -forms with . But since is a basis for the space of holomorphic 1-forms on any must be of the form where is a polynomial with algebraic degree less than Thus, if a Weierstrass point is a zero of then its order is at least two. It follows that the number of zeroes of the holomorphic 1-form is at least which is impossible because has genus
Therefore the dimension of the linear space of meromorphic functions satisfying condition (resp. ) in the Claim 4.1 is It is easy to show that the function (resp. ) is a basis for this space, so Claim 4.2 is proved.
As a consequence of the previous claims, we can write:
for a suitable constant . As we infer that for some .
To finish observe that the divisor of coincides with the divisor for the 1-form and as a consequence
since we get This concludes the proof.