Properness of associated minimal surfaces

We prove that for any open Riemann surface $N$ and finite subset $Z\subset \mathbb{S}^1=\{z\in\mathbb{C}\,|\;|z|=1\},$ there exist an infinite closed set $Z_N \subset \mathbb{S}^1$ containing $Z$ and a null holomorphic curve $F=(F_j)_{j=1,2,3}:N\to\mathbb{C}^3$ such that the map $Y:Z_N\times N\to \mathbb{R}^2,$ $Y(v,P)=Re(v(F_1,F_2)(P)),$ is proper. In particular, $Re(vF):N \to\mathbb{R}^3$ is a proper conformal minimal immersion properly projecting into $\mathbb{R}^2=\mathbb{R}^2\times\{0\}\subset\mathbb{R}^3,$ for all $v \in Z_N.$


Introduction
Given an open Riemann surface N , a conformal minimal immersion X : N → R 3 is said to be flux-vanishing if the conjugate immersion X * : N → R 3 is well defined, or equivalently, if X is the real part of a null holomorphic curve F : N → C 3 (see Definition 2.3).In this case, the family of isometric associated minimal immersions X v ≡ Re(vF ) : N → R 3 , v ∈ S 1 = {z ∈ C | |z| = 1}, is well defined.Notice that X = X 1 and recall that X * = X −ı , ı = √ −1.
The aim of this paper is to study the interplay between topological properness and associated minimal surfaces.Not so many years ago, it was a general thought that properness strongly influences the underlying conformal structure of minimal surfaces in R 3 .In this line, Schoen and Yau asked whether there exist hyperbolic minimal surfaces in R 3 properly projecting into R 2 ≡ R 2 × {0} ⊂ R 3 [SY,p. 18].A complete answer to this question can be found in [AL1], where examples with arbitrary conformal structure and flux map are shown.
On the other hand, any flux-vanishing minimal surface all whose associated surfaces uniformly properly project into R 2 is parabolic, see Proposition 4.3.This suggests a correlation between properness of associated surfaces and conformal structure of minimal surfaces.The following questions arise: (Q1) Do there exist hyperbolic flux-vanishing minimal surfaces S such that both S and its conjugate surface S * properly project into R 2 ?(Q2) More generally, how many associated surfaces of a hyperbolic flux-vanishing minimal surface can properly project into R 2 ?
Motivated by the above questions, this paper deals with those subsets Z ⊂ S 1 allowing proper projections in a uniform way, accordingly to the following Definition 1.1.A closed subset Z ⊂ S 1 is said to be a projector set for an open Riemann surface N if there exists a null holomorphic curve F = (F j ) j=1,2,3 : N → C 3 such that the map Moreover, Z is said to be a universal projector set if it is a projector set for any open Riemann surface.
If Z is a projector set for N and F is as in Definition 1.1, then Re(vF ) : N → R 3 is a proper conformal minimal immersion in R 3 which properly projects into R 2 , for all v ∈ Z.
One can easily check that if Z ⊂ S 1 is a projector set for N , then so are vZ for all v ∈ S 1 , Z ∪ (−Z), and any closed subset of Z.
Pirola's results [Pi] imply that S 1 is a projector set for any parabolic Riemann surface of finite topology (see also [Lo]).On the other hand, {1} is a universal projector set [AL1], whereas Proposition 4.3 in this paper shows that S 1 is not.In this line we have obtained the following Main Theorem.For any finite subset Z ⊂ S 1 and any open Riemann surface N , there exists an infinite projector set Z N for N containing Z.
In particular, Z is a universal projector set.
As a corollary, for any open Riemann surface N and finite set Z ⊂ S 1 , there exist an infinite subset Z N ⊂ S 1 containing Z and a flux-vanishing conformal minimal immersion X : N → R 3 such that X v properly projects into R 2 for all v ∈ Z N .This particularly answers (Q1) in the positive and enlightens about (Q2).
It is not hard to check that if S 1 is a projector set for an open Riemann surface N , then N is parabolic (see Proposition 4.3).Furthermore, if N is of finite topology and F : N → C 3 is a null holomorphic curve such that the map Y : , is proper, then F has finite total curvature (see Corollary 4.4).Connecting with a classical Sullivan's conjecture for properly immersed minimal surfaces in R 3 , see [Mo], the following question remains open: (Q3) Let N be an open Riemann surface of finite topology, and assume there exists a null holomorphic curve F : N → C 3 such that the map is proper.Must N be of parabolic conformal type?Even more, must F be of finite total curvature?
Our main tools come from approximation results for minimal surfaces and null holomorphic curves developed by the authors in [AL1,AL2].

Preliminaries
Denote by • the Euclidean norm in K n , where K = R or C. For any compact topological space K and continuous map f : K → K n , denote by Given an n-dimensional topological manifold M, we denote by ∂M the (n − 1)dimensional topological manifold determined by its boundary points.For any A ⊂ M, A • and A will denote the interior and the closure of A in M, respectively.Open connected subsets of M − ∂M will be called domains, and those proper topological subspaces of M being n-dimensional manifolds with boundary are said to be regions.If M is a topological surface, M is said to be open if it is non-compact and ∂M = ∅.

Riemann surfaces
Remark 2.1.Throughout this paper N will denote a fixed but arbitrary open Riemann surface, and σ 2 N a conformal Riemannian metric on N .
The key tool in this paper is a Mergelyan's type approximation result by null holomorphic curves in C 3 (see Lemma 2.6 below and [AL1,AL2]).This subsection and the next one are devoted to introduce the necessary notations for a good understanding of this result.
A Jordan arc in N is said to be analytical if it is contained in an open analytical Jordan arc in N .

Classically, a compact region
, where H 1 (•, Z) means first homology group with integer coefficients.More generally, an arbitrary subset Given an open subset W ⊂ N , we denote by • F h (W ) the space of holomorphic functions on W, and • Ω h (W ) the space of holomorphic 1-forms on W.
The following definition is crucial in our arguments, see  A (complex) 1-form θ on S is said to be of type (1, 0) if for any conformal chart (U, z) in N , θ| U ∩S = h(z)dz for some function h : U ∩ S → C.An ntuple Λ = (θ 1 , . . ., θ n ), where θ j is a (1, 0)-type 1-form for all j, is said to be an n-dimensional vectorial (1, 0)-form on S. The space of continuous n-dimensional (1,0)-forms on S will be endowed with the C 0 topology induced by the norm (see Remark 2.1).

We denote by
• F h (S) the space of continuous functions f : S → C which are holomorphic on an open neighborhood of M S in N , and • Ω h (S) the space of 1-forms θ of type (1, 0) on S such that θ/ϑ ∈ F h (S) for any nowhere-vanishing holomorphic 1-form ϑ on N (the existence of such a θ is well known, see for instance [AFL]).
Smoothness of functions and 1-forms on admissible sets is defined as follows: is smooth, for any nowhere-vanishing holomorphic 1-form ϑ on N .
Given a smooth function f ∈ F h (S), the differential df of f is given by . Notice that df ∈ Ω h (S) and is smooth as well.
Finally, the C 1 -norm on S of a smooth f ∈ F h (S) is defined by In a similar way, one can define the notions of smoothness, (vectorial) differential and C 1 -norm for functions f : S → C k , k ∈ N.

Null curves in C 3
Throughout this paper we adopt column notation for both vectors and matrices of linear transformations in C 3 .As usual, (•) T means transpose matrix.The following operators are strongly related to the geometry of C 3 and null curves.We denote by We also set , and the equality holds if and only if u = 0 := (0, 0, 0) T .
We denote by O(3, C) the complex orthogonal group {A ∈ M 3 (C) | A T A = I 3 }, that is to say: the group of matrices whose column vectors determine a ≺ •, • ≻conjugate basis of C 3 .We also denote by A : Let M be an open Riemann surface.
Definition 2.3.A holomorphic map F : M → C 3 is said to be a null curve if ≺dF, dF ≻= 0 and ≪dF, dF ≫ never vanishes on M.
Conversely, given an exact holomorphic vectorial 1-form Φ on M satisfying that ≺ Φ, Φ ≻= 0 and ≪ Φ, Φ ≫ never vanishes on M, then the map F : M → C 3 , F (P ) = P Φ, defines a null curve in C 3 .In this case Φ = dF is said to be the Weierstrass representation of F. The following definition deals with the notion for null curve on admissible subsets.
Definition 2.5.Let S ⊂ N be an admissible subset.A smooth map F ∈ F h (S) 3 is said to be a generalized null curve in C 3 if it satisfies the following properties: • ≺dF, dF ≻= 0 and ≪dF, dF ≫ never vanishes on S.
If F is a null curve and A ∈ O(3, C), then A • F is a null curve as well.The same holds for generalized null curves.
The following Mergelyan's type result for null curves is a key tool in this paper.It will be used to approximate generalized null curves by null curves which are defined on larger domains.
Lemma 2.6 ([AL1, AL2]).Let S ⊂ N be admissible and connected, let F = (F j ) j=1,2,3 ∈ F h (S) 3 be a generalized null curve in C 3 , and let W ⊂ N be a domain of finite topology containing S such that (i S ) * : H 1 (S, Z) → H 1 (W, Z) is an isomorphism, where i S : S → W denotes the inclusion map.

Main Lemma
Let us start by introducing some notation.
let X be a topological space, let K ⊂ X be a compact subset, and let F = (F 1 , F 2 , F 3 ) : X → C 3 be a continuous map.We denote by The following technical result is the core of our construction.
Let Z ⊂ S 1 be a finite subset with cardinal number n, and consider F ∈ N(M ) and δ > 0 such that Then, for any ǫ > 0 and any κ > δ, there exists F ∈ N(V ) satisfying Roughly speaking, the lemma asserts that a finite family of compact associated minimal surfaces whose boundaries lie outside a cylinder in R 3 can be stretched near the boundary, in such a way that the boundaries of the new associated surfaces lie outside a larger parallel cylinder.In this process the topology and even the conformal structure of the arising family can be chosen arbitrarily large.See Figure 3.1.Firstly we split ∂M into a suitable family of small Jordan arcs α i,j (see properties (a1), (a2), and (a3) below), and assign to each of them a complex direction e i,j in C 3 (see (3.3)).The splitting is made so that deformations of F around α i,j preserving the direction e i,j , keep the boundaries of all the Z-associated minimal surfaces outside the cylinder of radius δ/n.This choice is possible by basic trigonometry, see Claim 3.2.
In a second step, we construct an admissible set S by attaching to M a family of Jordan arcs r i,j connecting α i,j and ∂V.Then, we approximate F on M by a null curve H ∈ N(V ) formally satisfying the theses of the lemma on S, see items (c1) to (c4).
Finally, we modify H hardly on S and strongly on V − S in a recursive way to obtain the null curve F ∈ N(V ) which proves the basis of the induction, see Claim 3.4.This deformation pushes the boundaries of the Z-associated surfaces of H(V ) outside the cylinder of radius κ.Furthermore, this process hardly modifies the e i,jcoordinate of H on the connected component Ω i,j of V − S with α i,j ⊂ ∂Ω i,j , see (f2).Therefore, the Z-associated surfaces of the arising null curve F (V − M • ) lie outside the cylinder of radius δ/n.
For the inductive step we reason as follows.If −χ(V − M • ) = n ∈ N, we use Lemma 2.6 as a bridge principle for null curves to obtain a region U with and a null curve H ∈ N(U ) which approximates F on M and satisfies H Z ∂M > δ.Then, we finish by applying the induction hyphotesis.

Basis of the induction
Let us show that Lemma 3.1 holds in the particular instance χ(V − M • ) = 0.
Up to slightly deforming F (use Lemma 2.6), we can suppose that F is non-flat.
If U i,j = U x i,j for x i,j ∈ ∆, for simplicity we write L i,j = L x i,j for all (i, j) ∈ {1, . . ., k} × Z m .
Let {r i,j | j ∈ Z m } be a collection of pairwise disjoint analytical Jordan arcs in A i such that r i,j has initial point Q i,j ∈ α i , final point P i,j ∈ β i , and r i,j is otherwise disjoint from ∂A i for all i and j.Without loss of generality, assume that S = M ∪ (∪ i,j r i,j ) is admissible.See Figure 3.3.
Proof.Write F = (F 1 , F 2 , F 3 ) and for each i, j set and so (d i,j (t) v + L i,h ) ∩ B(δ/n) = ∅ for all t ≥ 1, h ∈ {j, j + 1} and v ∈ Z. Furthermore, we can take t 0 > 1 so that (d i,j (t 0 ) v + L i,h ) ∩ B(κ) = ∅ for all v ∈ Z and h ∈ {j, j + 1}.Up to a slightly smoothing around the points Q i,j for all i, j, it suffices to set G(r i,j ) = d i,j ([1, t 0 ]) for all i, j and G| M = F.
Then Lemma 2.6 applied to G straightforwardly provides a non-flat H ∈ N(V ) satisfying that ∀v ∈ Z, ∀h ∈ {j, j + 1}, ∀i, j, and (c4) H Z (α i,j ) ⊂ U i,j ∈ U for all i and j.
Consider now a Jordan arc , and otherwise disjoint from K n ∪(∂Ω η(n) ) (see Figure 3.4).Without loss of generality, assume that K n and γ n are chosen so that the compact set S n := (V − Ω η(n) ) ∪ K n ∪ γ n is admissible and (dG n,3 )| γn never vanishes (recall that H n−1 is non-flat and therefore so is G n ).
To check (d5 n ) we distinguish two cases.If a < n (and so n > 1), then we finish by using (d5 n−1 ) and (f1) for a small enough ǫ 0 .In case a = n we argue as follows.Assume first that P ∈ Ω η(n) − K n .Then (f2) gives that H n (P ) − H n−1 (P ) ∈≪ e n ≫ ⊥ , and so, by By (e2) we infer that H Z n (P ) > δ/n and we are done.Assume now that P ∈ K n .In this case, (f3) directly gives that H Z n (P ) > κ > δ/n as well.
The proof of (d6 n ) is analogous to that of (d5 n ).In case a < n, we use (d6 n−1 ) and (f1) for small enough ǫ 0 .In case a = n, we argue as in the proof of (d5 n ) but using (e3) instead of (e2).In this case we get that H Z n (P ) > κ for all P ∈ β η(n) − K n .Finally, (f3) shows that H Z n (P ) > κ for all P ∈ K n .The proof of Claim 3.4 is done.

Inductive step
Let n ∈ N, assume that Lemma 3.1 holds when −χ(V − M • ) < n, and let us show that it also holds when −χ(V − M Recall that M is admissible, and so a Jordan arc γ with endpoints P, Q ∈ ∂M and otherwise disjoint from ∂M, and such that γ ∈ H 1 (V, Z) − H 1 (M, Z).Consequently, since V is admissible then γ can be chosen so that S := M ∪ γ is admissible as well.
At this point, we need the following Denote by S 3 (R) the 3-dimensional Euclidean sphere of radius R > 0 in R 4 ≡ C 2 and write S 3 ≡ S 3 (1).
For each v ∈ Z, let γ v ⊂ S 3 ≡ S 3 (1) denote the spherical geodesic H v ∩ S 3 , where in the topology associated to the Hausdorff distance.Since S 3 − Γ is path-connected and contains u 1 / u 1 and u 2 / u 2 , then these two points lie in the same connected component of For the second part, since Σ is open and path-connected, then there exists a polygonal arc ĉ : [0, 1] → Σ × C connecting v and w and with ĉ′ (t) ∈ Θ at any regular point.To finish, choose c as a suitable smoothing of ĉ.By Claim 3.5 and equation (3.1), one can construct a generalized null curve G : S → C 3 satisfying G| M = F and G Z γ > δ.From Lemma 2.6 applied to G and (notice that dX k never vanishes on The sequence is obtained in a recursive way.The couple (X 0 , Z 0 ) trivially satisfies (a 0 ) and (e 0 ), whereas (b 0 ), (c 0 ), and (d 0 ) make no sense.Let n ≥ 1, assume we already have a couple (X n−1 , Z n−1 ) satisfying the corresponding properties, and let us construct (X n , Z n ).For ε n small enough, the null curve X n ∈ N(M n ) given by Lemma 3.1 applied to the data For the first assertion in (4.2), use items (c n ) and (d n−1 ) and a continuity argument for k ∈ {1, . . ., n − 1}, and Lemma 3.1-(L2) for k = n.
To close the induction, choose any . ., n, and ∂M n are compact, the existence of such a v near Z 0 is guaranteed by a continuity argument and (4.2).
By items (a n ) and (c n ), {X n } n∈N uniformly converges on compact subsets of N to a holomorphic map Y : and let us show that the couple (Y, Z N ) satisfies the theses of the theorem.
From (b n ), Z N is a closed infinite set.Item (A) is obvious.
To prove that Y is an immersion, hence a null curve, it suffices to check that dY /σ N (P ) > 0 ∀P ∈ N .Indeed, let P ∈ N and choose j ∈ N so that P ∈ M j .Then (c n ) implies that dY /σ N (P ) ≥ dX j /σ N (P ) − From (c n ) one has ε k < ε for all j ≥ 0, proving in particular (B).
To prove (C), take k ∈ N. From (4.3) and (d j ), j ≥ k, one infers Y Thus Y −1 (B(k − 2ε)) is compact in Z N × N for all k, proving (C).
Remark 4.2.Theorem 4.1 shows that Z 0 is a universal projector set and Z N ⊂ S 1 is a projector set for the fixed N .Since obviously Z N depends on N , one can not infer that it is a universal projector set.
On the other hand, up to an elementary refinement of the proof of Theorem 4.1, one can construct Z ∞ to be closed, and even with accumulation set in Z 0 .In this case, Z N = Z ∞ is countably infinite.
The following proposition shows that S 1 is a projector set for no hyperbolic Riemann surface.Then M is parabolic.As a consequence, if M has finite topology then M is biholomorphic to a finitely punctured compact Riemann surface and F 1 and F 2 extend meromorphically to the compactification of M.
Proof.To show that M is parabolic, it suffices to check that F 1 : M → C (and likewise F 2 ) is a proper holomorphic function.Reason by contradiction and take a divergent sequence {P n } n∈N ⊂ M such that {F 1 (P n )} n∈N is bounded.For each n ∈ N choose v n ∈ S 1 such that Re(v n F 2 (P n )) = 0. Then {Ψ(v n , P n )} n∈N is bounded as well, which is absurd.
For the second part of the proposition, assume that M has finite topology.The parabolicity implies that M = M − {Q 1 , . . ., Q k }, where M is a compact Riemann surface and Q 1 , . . ., Q k ∈ M .Since F 1 , F 2 : M → C are proper holomorphic functions then they have no essential singularities at the ends, and so, they extend meromorphically to M .Proof.Just write F = (F j ) j=1,2,3 , take into account that dF 1 and dF 2 , hence dF 3 , extend meromorphically to the natural compactification of M, and Osserman's classical results [Os].
Figure 2.1.Definition 2.2.A compact subset S ⊂ N is said to be admissible if and only if • M S := S • is a finite collection of pairwise disjoint compact regions in N with C 0 boundary, • C S := S − M S consists of a finite collection of pairwise disjoint analytical Jordan arcs, • any component α of C S with an endpoint P ∈ M S admits an analytical extension β in N such that the unique component of β − α with endpoint P lies in M S , and • S is Runge.

Figure 2 . 1 .
Figure 2.1.An admissible subset S of N Let S be an admissible subset of N .
Definition 2.4.Given a proper subset M ⊂ N , we denote by N(M ) the space of maps F : M → C 3 extending as a null curve to an open neighborhood of M in N .

Figure 3
Figure 3.1.Lemma 3.1 where A 1 , . . ., A k are pairwise disjoint compact annuli.Write ∂A i = α i ∪β i , where α i ⊂ ∂M and β i ⊂ ∂V for all i.Denote by B(r) the 2-dimensional Euclidean ball {p ∈ R 2 | p < r} for any r > 0.Label ∆ = (R 2 − B(δ)) n ⊂ R 2n , and for any x ∈ ∆ choose a vectorial line L x ⊂ R 2 and an open neighborhood U x of x in ∆ such that Figure 3.2.The vectorial line L x ⊂ R 2

Figure 3 . 3 .
Figure 3.3.The annulus A i ⊂ V − M • The first deformation stage starts with the following Claim 3.3.There exists a generalized null curve G : S → C 3 such that
Corollary 4.4.Let M be an open Riemann surface of finite topology and let F :M → C 3 be a null curve.Assume that the map Y : S 1 × M → R 2 , Y(v, P ) = F v (P ), is proper.Then F has finite total curvature.
and for any component α of C S and any open analytical Jordan arc β in N containing α, f admits a smooth extension f β to β satisfying that f β