Genus Zero Complete Maximal Maps and Maxfaces with an Arbitrary Number of Ends

We prove the existence of a genus-zero complete maximal map with a prescribed singularity set and an arbitrary number of simple and complete ends. We also discuss the conditions under which this maximal map can be made into a complete maxface.


Introduction
Similar to minimal surfaces in R 3 , maximal surfaces are immersions with zero mean curvature in the Lorentz Minkowski space E 3  1 .Maximal surfaces share many similarities with minimal surfaces, such as being critical points of the area functional and admitting the Weierstrass-Enneper representation.However, the theory of complete maximal immersions is limited compared to the theory of complete minimal surfaces.It has been proven [10] that the plane is the only complete maximal immersion.
Maximal immersions naturally appear with singularities.Following the terminology from [4], [7], [10], etc., we use the term "maximal map" to refer to generalized maximal immersions.Maxfaces are a subclass of generalized maximal immersions where the singularities occur only at points where the limiting tangent plane contains a light-like vector [10].
The object of the discussion in this article is genus-zero maximal maps and maxfaces.We summarize the existing results and examples for genus-zero complete maximal maps/maxfaces as follows: (1) The space-like plane is a maximal immersion.
(2) The Lorentzian catenoid is a maxface with two complete, embedded ends.
(4) In [2], the authors provide an example of a genus-zero maxface with two complete ends, with swallowtails, cone-like features, cuspidal edge, and cuspidal crosscaps.These ends are not embedded.(5) Imaizumi and Kato [4] discuss different types of simple ends and classify maximal surfaces of genus zero with at most three embedded ends.
In [4], authors discuss the classifications of ends of the maximal map, and in [8], we see the regularity results about the ends of zero mean curvature surfaces in the Lorentz-Minkowski space.
The question addressed in this article is as follows: Question 1.1.Given a prescribed singular set and the prescribed nature of singularity, can we construct a genus-zero maximal map or maxface with an arbitrary number of simple ends and complete ends?
One possible approach to constructing a maximal map or maxface is by starting with a minimal immersion.For a minimal immersion, there exists a corresponding maximal immersion (known as a companion [10]).However, this method does not always yield a complete maxface.An example of this is the Jorge-Meeks minimal surface [5].
To construct a complete maximal map or maxface, the Björling problem and its solution are not always sufficient.In some cases, starting from the Björling data leads to a non-complete maxface.Examples in [6] (for instance, see Example 3.11) do not provide the Weierstrass data for complete maxfaces.
To construct examples of complete maxfaces, we need to find a suitable meromorphic function g and a holomorphic 1-form ω on C ∪ {∞} \ {p 1 , ..., p n } that satisfies the period condition (as in (3) of Theorem 2.1).However, unlike in the case of minimal surfaces, when solving the balance equation of the period problem, we have to ensure that the meromorphic function g is chosen in a way that |g(p i )| = 1 after extension to the ends.Due to this restriction, most methods and techniques used for minimal surfaces can not be directly applied.
In this article, we prove the existence of maximal maps or maxfaces with complete ends, and since we are not specifically looking for embedded ends, the method for solving the period problem can be simplified.Here, we explain the method we will use.
For any r and the subset {p 1 , ..., p r } ⊂ C, if we take then it is well-known (see [11]) that there exist constants a i 's and b j 's such that (g, f (z)dz) gives a complete branched minimal surface on C ∪ {∞} \ {p 1 , ..., p r }.
The same data and solution can be used to obtain a maximal map with any number of ends.However, this solution does not guarantee that |g(p i )| = 1 at the ends, and thus we do not obtain complete ends.Moreover, the a i 's and b j 's are not explicitly determined, so we do not have control over the exact singularity set for the maxface.
In this article, we will modify this method to solve the period problem and find the desired maxface or maximal map.
In Section 3, we explain the period problem and prove the following results: (1) Let C be a singular curve in C as defined in the definition 3.1, and let m, n ∈ N * .Then: (a) (Theorem 3.4) there exists a maximal map with m complete ends and C as its singularity set.(b) (Theorem 3.6) there is a maximal map with m complete ends, n simple ends, and C as its singularity set.All simple ends, in this case, are embedded.
(2) (Theorem 3.5) Let X : C ∪ {∞} \ {p 1 , ..., p n } → E 3 1 be a complete maximal map with the Gauss map g.Then, there exists a finite set E and a family of complete maxfaces on M \ E with the Gauss map g, such that g is very close to g away from the ends.It is important to note that these results are trivially solved in the case of minimal immersions (such as the Jorge-Meeks surface).However, for maximal surfaces, due to the presence of non-isolated singularities, finding such results requires a slightly different approach.

Preliminary
We denote the Lorentz Minkowski space as E 3  1 , which is the vector space R 3 with the metric dx 2 + dy 2 − dz 2 .Similar to the case of minimal immersions, Esutidillo and Romero [1] have proved the Weierstrass-Enneper representation for generalized maximal immersions (maximal maps).We mention it here.
Theorem 2.1 (Weierstrass-Enneper representation for maximal maps [1]).Let M be a Riemann surface, g be a meromorphic function, and ω = f (z)dz be a holomorphic 1-form on M such that: (1) If p ∈ M is a pole of g of order m, then f has a zero at p of order at least 2m.
Then, the map X : M → E 3 1 given by (2.1) is a maximal map with base point 0 ∈ M.
Moreover, any maximal map can be expressed in this form.
The third condition in the above theorem 2.1 is called the period condition.For the maximal map X : M → E 3  1 with the Weierstrass data (g, ω), the pullback metric is given by ds The singularities of the maximal map X : M → E For the maxface, there are no branch points, and we have a singularity where |g|= 1 and ω = 0. Therefore, if (g, ω) is the Weierstrass data for the maxface, we must have We refer to the above equation as the divisor condition.
The completeness of a maxface can be understood using the following equivalent criteria: Fact 2.2.A maxface is complete (see [10], corollary 4.8) if and only if: ( (3) The induced metric ds 2 is complete at the ends.
In this case, the points p i 's are called ends for the maxface, and g and ω can be extended as meromorphic function and meromorphic 1-form respectively at the ends.
Similarly, for the maximal map defined on M \ {p 1 , ..., p n }, we say it is a complete maximal map if and only if |g(p i )| = 1, 1 ≤ i ≤ n, and the induced metric ds 2 is complete at the ends.
In the next section, we will prove the existence of a maximal map with an arbitrary number of complete ends and a prescribed singularity set.We will start the discussion by finding a complete maximal map with the prescribed Gauss map.

Genus zero complete maxface with prescribed singularities
We will start this section by constructing a maximal map with a given Gauss map.
3.1.Maximal map with an arbitrary number of complete ends and a prescribed Gauss map.Let g be a meromorphic function on C ∪ {∞} such that Without loss of generality, we assume p n = ∞.
Next, we will find a suitable holomorphic 1-form ω such that (g, ω) becomes the Weierstrass data for a maximal map X defined on C ∪ {∞} \ {p 1 , ..., p n }, and all the ends are complete.
Consider the following expression for f (z): where a i , b j ∈ C (we will find suitable values for a i 's and b j 's).Set ω 0 := f (z)dz.
We need to show that there exist constants (a i ; b j ) ∈ C 3n−2 such that the period condition, as in the theorem 2.1, holds for all loops on C ∪ {∞} \ {p 1 , ..., p n }.Since ω 0 is an exact form, and if g 2 ω 0 and gω 0 have no residues on C ∪ {∞}, the pair (g, ω 0 ) satisfies the period condition.Therefore, we need to find values of (a i ; b j ) such that the following equations hold: Equation 3.3 gives a homogeneous system of linear equations in the variables a i and b j .Moreover, since the total residues of gω 0 and g 2 ω 0 are zero on C ∪ {∞}, we have a homogeneous system of at most 2n − 2 linear equations in 3n − 2 variables a i and b j .Therefore, the solution space (denoted by Sol ⊂ C 3n−2 ) for the system of equations 3.3 has a dimension of at least n.This proves that we can choose suitable (a i ; b j ) ∈ Sol \ {0} such that (g, ω 0 ) satisfies the period condition.
Since ω 0 has poles at p i 's, the divisor condition for (g, ω 0 ), as stated in the theorem 2.1, holds trivially on C ∪ {∞} \ {p 1 , ..., p n }.Moreover, since the solution space is of dimension n, for (a i ; b j ) ∈ Sol \ {0}, at least one of the b j 's is non-zero.Therefore, p n is a pole of both ω 0 and g, making p n a complete end for the maximal map.
However, it may happen that for some i (1 ≤ i ≤ n − 1), a i = 0, in which case p i may be a zero of ω 0 .Thus, while the pair (g, ω 0 ) gives a maximal map, p i may not be a complete end for this maximal map.To address this, we aim to modify ω 0 such that the resulting pair becomes the Weierstrass data for a maximal map with all p i 's as poles of ω 0 , ensuring that all p i 's are complete ends.
To achieve this, for 1 ≤ i ≤ n − 1, we consider and we want to find (α k ) ∈ C 2n with at least one α k = 0 such that (g, f i ) satisfies the period condition on C ∪ {∞}.By using the technique mentioned earlier, i.e., Res p j (gf i dz) = 0 and Res p j (g 2 f i dz) = 0 for 1 ≤ j ≤ n, we obtain 2n−2 independent linear homogeneous equations with 2n variables.Therefore, there exists (α k ) ∈ C 2n \ {0} such that (g, f i ) satisfies the period condition and p i is a pole of f i .
Let F := f + n−1 i=1 f i and ω := F (z)dz.It is clear that g and ω satisfy the divisor condition and the period condition on C ∪ {∞} \ {p 1 , ..., p n }.Moreover, all p i 's are poles of g and ω, making all ends complete.
To have a complete maximal map X : C ∪ {∞} \ {p 1 , ..., p n } → E 3 1 with the Weierstrass data (g, ω), we only need to prove that the singular set is compact.It is evident from the fact that g has poles at the ends and the manner in which we adjusted ω 0 to obtain ω, which implies that {z : |g|= 1} ∪ {z : ω(z) = 0} is compact.
To summarize, we have proved the following proposition: Proposition 3.1.Let g be a meromorphic function on the Riemann sphere such that the number of distinct poles of g is n.Then there exists a maximal map with n complete ends having the Gauss map g.
The maximal map obtained from the proposition 3.1 is not necessarily a maxface because the 1-form ω may have some zeros on C ∪ {∞} \ {p 1 , ..., p n }.In subsection 3.3, we will perturb the Gauss map to obtain the complete maxface.Furthermore, the above method takes care of the ends as complete, but in this process, we obtain higher-order poles at the ends, which prevents the possibility of embedded ends.
Moreover, if we replace F as defined above by where, for n ≥ 2, h(z) = n−1 j=1 (z−p j ) 2 and c = p j for 1 ≤ j ≤ n, and for n = 1, h(z) is any polynomial, we obtain the same result.Additionally, there are other ways to define the required ω.In all these cases, we can increase the number of variables to obtain a larger dimensional solution space, which may help us in obtaining the maxface with a prescribed nature of singularity.This aspect needs to be explored further.
Below, we will define the singular curve and discuss the possibility of the existence of a maximal map with such a curve as its singularity set.

3.2.
Maximal map with an arbitrary number of complete ends and a prescribed singularity set.For a given curve or collection of disjoint union of curves C ⊂ C, we aim to find a complete maximal map with an arbitrary number of complete ends such that C is the singularity set.So firstly, for a given C, we will be looking for the meromorphic function g on C ∪ {∞} such that for z ∈ C, |g(z)|= 1, and then applying the proposition 3.1, we will get the required maximal map.
Considering the above, we will take C as follows in the rest of the article.
Definition 3.1.The singular curve C is a curve or disjoint union of curves such that there exists a meromorphic function g on C ∪ {∞} such that g(C) is mapped to the circle |z|= R.
There are many examples of singular curves.Below, we present two examples that constitute a significant class of singular curves.
Example 3.2.We begin by recalling the definition of the Schwarzian derivative of . The Möbius transformations are the only functions for which S[f ](z) = 0 [9].
Consider a regular simple closed curve C such that the Schwarzian derivative of any conformal parameterization f (z) of C is identically zero.In this case, there exists a meromorphic function G(z) on C ∪ {∞} that maps C to the unit circle.
Example 3.3.Another example of a singular curve is the subset C = |G| −1 (1), where G is defined as follows: This curve and the meromorphic function G are taken from the proof of the theorem A [2], where the authors investigated various types of singularities on maxfaces.For this particular G, the singular curve C consists of three disjoint (topological) circles on the Riemann sphere, including the imaginary axis.
Moreover, any meromorphic function G that maps a circle of radius R to the unit circle is a Blaschke product, as discussed by Gronwall in [3], and is given by , where a i , 1 ≤ i ≤ m, and b j , 1 ≤ j ≤ n, are points inside the circle |z|= R, and For a i = 0, the factor R(z−a i ) a i (z−a ′ i ) in the equation 3.5 is replaced by − z R , and for b j = 0, the factor Note that a i 's and b j 's are repeated according to the order of zeros and poles.
For the given singular curve C, as defined in the definition 3.1, there is a meromorphic map f : C ∪ {∞} → C ∪ {∞} such that f (C) is the circle of radius R. We choose G as described in the equation 3.5 and compose it with the meromorphic map f .This composite map, denoted as g, satisfies |g(z)|= 1 on C. Additionally, by choosing G appropriately, we have the flexibility to increase the number and order of poles of g.
Furthermore, after the change of coordinates, we assume that one of the poles of g is at ∞.We denote the poles of g as p i , where 1 ≤ i ≤ n, and write (g) ∞ = n i=1 x i p i , as in the equation 3.1.Now, if we start with the meromorphic function g as described in the previous paragraph, we can use the proposition 3.1 to establish the following theorem: Theorem 3.4.Given a singular curve C as in the definition 3.1, and m ∈ N * , there exists a maximal map with m complete ends and C as its singularity.
Theorem 3.4 establishes the existence of the desired maximal map.However, if we want a maxface, we can get it by slightly modifying the normal vector (the Gauss map g).Specifically, in the following subsection, given a complete maximal map with Weierstrass data (g, ω), we will find a suitable meromorphic function g 0 such that (g = g +g 0 , ω) yields a complete maxface.Moreover, g can be chosen arbitrarily close to g, away from the ends.
3.3.Perturbation of the Weierstrass data for the complete maxface.Let (g, ω = f dz), as defined in the equations 3.1 and 3.4 respectively, be the Weierstrass data for the maximal map X : C ∪ {∞} \ {p 1 , . . ., p n } → E 3 1 with complete ends, as stated in the theorem 3.4.
Let {z j : 1 ≤ j ≤ l} be the zeros of ω and Ord z j ω = m j > 0. With α i ∈ C, we define We take g = g + g 0 .In the following, we will determine (α i ) ∈ C 3(l+n) such that (g, ω) serves as the Weierstrass data for the complete maxface on C ∪ {∞} \ {p 1 , . . ., p n , z 1 , . . ., z l }, with g being close to g away from the ends.
The divisor condition, as given in the equation 2.2, trivially holds for it to be a maxface.
If the period condition holds for (g, ω), then the corresponding maxface will be complete.This follows from the construction, since we observe that |g| = 1 at the ends {p i ; z j : 1 ≤ i ≤ n, 1 ≤ j ≤ l}.Additionally, at the end, the metric ds 2 is complete.Hence, all ends are complete.
Thus, we need to find (α i ) such that the period condition for (g, ω) is satisfied.Since gω and g 2 ω have no residues on C ∪ {∞} and ω is exact, we need to find (α i ) such that g 0 ω, g 2 0 ω, and gg 0 ω have no residues on C ∪ {∞}.This can be achieved by employing the same technique as in subsection 3.1.
Moreover, for any ǫ > 0, the vector ǫα = (ǫα i ) is also a solution.Therefore, g 0 can be suitably chosen such that away from the ends z j , we have |g 0 |< ǫ.This proves that |g − g|= |g 0 | is very small away from the ends.
Combining all the above, we establish the following result: Theorem 3.5.Let X : M = C ∪ {∞} \ {p 1 , . . ., p n } → E 3 1 be a complete maximal map with the Gauss map g.Then, there exists a finite set E and a family of complete maxfaces on M \ E with the Gauss map g such that g is very close to g away from the ends.Similar questions can be posed regarding the existence of complete maxfaces with various prescribed types of singularities occurring at different points.As the solution space for the required constants is vast, it is likely possible to choose suitable constants.However, this requires an improved method for solving the period problem.