A UNIQUENESS THEOREM FOR PROPERLY EMBEDDED MINIMAL SURFACES BOUNDED BY

STRAIGHT LINES

FRANCISCO J. LOPEZ and FRANCISCO MARTIN

(March 3, 2000)

Abstract

In this paper we prove a uniqueness theorem for minimal discs in ℝ^{3} spanning a polygonal boundary.

1991 Mathematics Subject Classification (Amer. Math. Soc.): primary 53A10; secondary 53C42. Key words and phrases: properly embedded minimal surfaces, boundary, straight lines.

1. Introduction

A classical problem considered by Schwarz, Weierstrass and Riemann was to determine minimal surfaces bounded by straight lines. These authors obtained existence results for minimal surfaces with boundary a given polygon, where the sides of the polygon could be of finite or infinite length.

Jenkins and Serrin [6] proved an existence and uniqueness theorem for minimal graphs bounded by straight lines. They obtained simple, necessary and sufficient conditions to solve the Dirichlet problem in a compact convex domain bounded by a polygon assuming values +∞, −∞ and continuous data on different straight segments in the boundary.

In [9], the authors construct a deformation of some particular Jenkins-Serrin graphs (see Figure 1) which consists of properly embedded minimal discs bounded by straight lines, contained in a wedge of a slab (see Figure

RESEARCH PARTIALLY SUPPORTED BY DGICYT GRANT NUMBER PB97-0785.

\begin{matrix}c \\ ◯ \end{matrix} 0 Australian Mathematical Society 0263-6115/0 $A2.00 + 0.00

Figure 1. A Jenkins-Serrin graph.

3). These examples arise as a solution to Plateau’s problem for a polygonal noncompact boundary consisting of a twice U shaped contour. These surfaces can be used as a new type of barrier for the maximum principle application. As a consequence, in [9], the authors have proved that any properly immersed minimal surface in a slab wedge of angle less than π lies in the convex hull of its boundary. Furthermore, they have proved non existence results for minimal surfaces with planar boundaries which generalize some classical results by Nitsche [12] (see also Rossman’s work [15] for a good survey).

One of the most interesting questions as regards to Plateau’s problem is to determine how many minimal surfaces can be spanned in a given boundary. The answer to this question is not generally known, not even if we fix the topological type of the solutions. The difficulty increases if, in addition, the boundary is not compact.

An existence and uniqueness theorem for properly immersed minimal discs whose boundaries consist of two disjoint straight lines and a segment which joins them orthogonally has been recently obtained in [10]. In [4], Fang have considered the non compact Plateau’s problem for surfaces of annular type bounded by a pair of non-compact convex curves in parallel planes, obtaining existence results for certain symmetric boundaries.

The aim of this paper is to obtain a uniqueness theorem for the surfaces in the family of examples exhibited in [9]. To be more precise, we deal with the uniqueness of properly embedded minimal surfaces whose boundary Γ_{θ d} consists of the following configuration of straight lines:

Fix θ ∈ [0, π[ and d ≥ 0, and consider two half-lines r_{1}^{+} and r_{1}^{−} in ℝ^{3}, meeting at an angle of θ. If θ = 0 this means that the straight lines are parallel and distinct. Let q_{1}^{+} and q_{1}^{−} be two points in r_{1}^{+} and r_{1}^{−}, respectively, such that they are symmetric with respect to the inner bisector of these halflines. If and only if θ ≠ 0, we allow q_{1}^{+} = q_{1}^{−}. We choose q_{1}^{+} and q_{1}^{−} in such a way that either q_{1}^{+} = q_{1}^{−} or the half-lines ℓ_{1}^{+} and ℓ_{1}^{−} on r_{1}^{+} and r_{1}^{−} starting at q_{1}^{+} and q_{1}^{−}, respectively, do not intersect. Write d = dist(q_{1}^{+}, q_{1}^{−}).

Let π_{1} be the plane determined by ℓ_{1}^{+} and ℓ_{1}^{−} and let π_{2} denote a plane parallel and distinct to π_{1}. Let ℓ_{2}^{+} and ℓ_{2}^{−} be the orthogonal projections to π_{2} of ℓ_{1}^{+} and ℓ_{1}^{−}, respectively. Denote q_{2}^{+} (q_{2}^{−}) as the orthogonal projection to π_{2} of q_{1}^{+} (q_{1}^{−}), and label ℓ_{0}^{+} ( ℓ_{0}^{−}) as the segment [q_{1}^{+}, q_{2}^{+}] ([q_{1}^{−}, q_{2}^{−}]). Finally, we write

Γ_{θ d}^{+} = ⋃_{i=0}^{2} (ℓ_{i}^{+}) , Γ_{θ d}^{−} = ⋃_{i=0}^{2} (ℓ_{i}^{−})

and define Γ_{θ d} = Γ_{θ d}^{+} ∪ Γ_{θ d}^{−}.

Figure 2. The curve Γ_{θ d}.

We consider the following generalized Plateau’s problem: PROBLEM Determine a properly immersed minimal surface X : M → ℝ^{3} satisfying:

(1) M is homeomorphic to the closed unit disc \overline{𝔻} minus two boundary points E_{1} and E_{2}, that we call the ends of M.

(2) X(∂(M)) = Γ_{θ d}.

(3) If d > 0, X is an embedding.

(4) In the limit case ℓ_{0}^{+} = ℓ_{0}^{−} (that is, d = 0), the maps X|_{M−γ^{+}} and X|_{M−γ^{−}} are injective, where γ^{+} and γ^{−} are the two connected components of ∂(M).

(5) X(M) lies in the convex hull, E(Γ_{θ d}), of Γ_{θ d}.

In the limit case θ = 0, it is known that 0 < d < ∥v⃗∥, where v⃗ is the vector with origin q_{1}^{+} and end q_{2}^{+}, and the example is uniquely determined as one of the above mentioned Jenkins-Serrin graphs. See the remark in page 9.

Concerning to the general case θ > 0, we have proved the following: MAIN THEOREM There exists d_{θ} > 0 such that:

(A) • If d ∈]0, d_{θ}[, the above Plateau’s problem has only two solutions.

• If d = 0 or d = d_{θ}, the above Plateau’s problem has a unique solution.

(B) For d > d_{θ}, there are no solutions.

Figure 3. The two solutions in case θ = π/3, d = 0.56 ∥v⃗∥. Figure (a) corresponds to an unstable example, and Figure (b) corresponds to a stable one.

As we mentioned above, the existence part of this result can be found in [9]. Anyway, in Section 2 we briefly describe this family of surfaces. The statement of this theorem also involves the uniqueness of the solutions, which is the matter of this paper. This result is an elaborate consequence of Theorem 3.8, Theorem 3.12, Theorem 3.17, Theorem 3.24 and Theorem 3.32 in this paper.

Finally, note that the above assumption (5) cannot be derived from the ones (1)-(4). As a matter of fact, in Figure 4 we illustrate two examples satisfying (1), (2), (3) and (4), but none of them are contained in the convexhull of its boundary.

This paper is organized as follows. In Section 2, we give several results we need in this paper. We also describe the family of minimal surfaces

Figure 4. (a) A minimal disc bounded by Γ_{θ d}, for θ = π/3 and d = 0.56∥v⃗∥. (b) A minimal disc bounded by Γ_{θ d}, for θ = 0 and d = 0.40∥v⃗∥.

arising as solutions of Plateau’s problem above, and recall its geometrical properties. In Section 3, we obtain the main uniqueness results for the space of solutions. In Subsection 3.1 we prove that M is conformally equivalent to a twice punctured closed disc with piecewise analytic boundary (see Paragraph 3.1.1), and in Paragraph 3.1.2 we prove that the meromorphic data of X extend to the closed disc. In case θ/π ∈ ℚ, it is not hard to see that these results are a consequence of the Cone Lemma, proved by Hoffman and Meeks in [5]. Our approach also satisfactorily answers the case θ/π ∉ ℚ. In Subsection 3.2, we prove that any solution of Plateau’s problem above inherits the symmetries of its boundary. This fact is a consequence of the reflection method of Alexandrov. In this sense, we generalize the ideas of Schoen to our particular case of non compact boundary with unbounded slope. Finally, in Subsection 3.3 we make use of the results in the preceding subsections to determine a model of the complex structure and Weierstrass representation of any solution of our Plateau problem.

2. Background and Notation

The aim of this section is to fix the principal notation used in this paper, and to summarize some results about complete minimal surfaces.

Let X : M → ℝ^{3} be a proper conformal minimal immersion, where M is a Riemann surface with piecewise analytic boundary homeomorphic to the closed unit disc 𝔻 minus two boundary points E_{1} and E_{2}, that we call the ends of M.

Remark. We say that M is a Riemann surface with piecewise analytic boundary if and only if M is a subset of an open Riemann surface M′, the conformal structure of M − ∂(M) is that induced by M′ and ∂(M) consists of a set of piecewise analytic curves. Meromorphic (respectively holomorphic) functions and 1-forms on M are, by definition, the restriction of meromorphic (respectively holomorphic) functions and 1-forms on M′.

The Weierstrass representation of X is denoted by (g, η). Recall that g is a meromorphic function and η a holomorphic 1-form on M. Both of them determine the minimal immersion X as follows:

X(P) = Re (∫ ^{P} (ϕ_{1}, ϕ_{2}, ϕ_{3})) ,

where

ϕ_{1} = \frac{1}{2}(1 − g^{2})η, ϕ_{2} = \frac{i}{2}(1 + g^{2})η, ϕ_{3} = gη  (1)

are holomorphic 1-forms on M satisfying:

∑_{j=1}^{3} | ϕ_{j} |^{2}≠ 0.  (2)

Furthermore, g is the stereographic projection of the Gauss map N : M → 𝕊^{2}.

In this paper we often use the following classical results about minimal surfaces with common points.

Let M_{1} and M_{2} be two minimal surfaces in ℝ^{3}. Assume p_{0} is an interior point of both M_{1} and M_{2} and suppose T_{p_{0}}M_{1} = T_{p_{0}}M_{2}. Up to a rigid motion, assume that p_{0} = (0, 0, 0) and T_{p_{0}}M_{1} = {x_{3} = 0} so that both M_{1} and M_{2} are given near p_{0} as the graphs of two real analytic functions u_{1}(x_{1}, x_{2}) and u_{2}(x_{1}, x_{2}), respectively.

The difference u(x_{1}, x_{2}) = u_{1}(x_{1}, x_{2}) − u_{2}(x_{1}, x_{2}) can be expanded as a power series u(x_{1}, x_{2}) = ∑_{k=1}^{∞} f_{k}(x_{1}, x_{2}) in a neighbourhood of the origin, where each f_{k}(x_{1}, x_{2}) is an homogeneous polynomial of degree k in x_{1} and x_{2}. If f_{k}(x_{1}, x_{2}) ≡ 0, 1 ≤ k ≤ ν − 1, and f_{ν}(x_{1}, x_{2}) does not vanish identically, then we say that M_{1} and M_{2} have a contact of (ν − 1)^{th} order at p_{0}. If ν − 1 = 1, we say that they have an ordinary contact. In case ν = 1, the surfaces are transverse at p_{0}.

Theorem 2.1 (Order of contact). Two minimal surfaces M_{1} and M_{2} have a contact of order ν − 1 at p_{0} if and only if M_{1} intersects M_{2} along ν curves C_{1}, …, C_{ν} in a neighbourhood of p_{0}. If ν − 1 > 0, these curves intersect each other at p_{0} at an angle π/ν. They divide a neighbourhood of p_{0} in M_{1} into 2ν open sectors, such that M_{1} lies on one side of M_{2} in one sector and on the other side in the next sector.

Furthermore, if m_{i} is the multiplicity of the Gauss map of M_{i} at p_{0}, i = 1, 2, then

ν − 1 ≥ Minimum{m_{1}, m_{2}}.

In particular, when M_{2} is the tangent plane T_{p_{0}}M_{1} at p_{0}, the multiplicity of the Gauss map g of M_{1} at p_{0} is ν − 1 if, and only if, T_{p_{0}}M_{1} intersects the surface along ν curves C_{1}, …, C_{ν} in a neighbourhood of p_{0}. As above, these curves intersect each other at p_{0} forming angles different to 0 and π. They divide a neighbourhood of p_{0} into 2ν open sectors, such that M_{1} lies on one side of the tangent plane in one sector and on the other side in the next sector.

We say that p_{0} is an ordinary point of contact if the multiplicity of the Gauss map at p_{0} is 1. Obviously, at an ordinary point of contact the tangent plane to a minimal surface intersects the surface along two orthogonal curves.

Minimal surfaces containing straight lines have special properties. These were studied by Schwarz, who obtained the following result:

Theorem 2.2 (Schwarz’s Reflection Principle). If a minimal surface contains a piece of a straight line ℓ in its boundary, then the union of the surface and its 180 degrees rotation about ℓ is a minimal surface which has ℓ as an axis of symmetry.

Two good references about these results are [13, Section 437] and [3, Section 3.4].

Subsection 3.2 is devoted to proving that the minimal surface X(M) inherits the symmetries of its boundary. The proof of these facts is a suitable version of the reflection method of Alexandrov [1] and, as such, is based essentially on the Hopf maximum principle. We state two theorems which summarize the well known versions of the maximum principle which we require.

Theorem 2.3 (Interior maximum principle). Suppose M_{1}, M_{2} are minimal surfaces in ℝ^{3}. Suppose p is an interior point of both M_{1} and M_{2}, and suppose T_{p}M_{1} = T_{p}M_{2}. Assume that T_{p}M_{1} = {x_{3} = 0} so that both M_{1}, M_{2} are given near p as the graphs of two real analytic functions u_{1} and u_{2}, respectively. If u_{1} ≥ u_{2} in a neighbourhood of p, then M_{1} = M_{2}.

An elementary consequence of this result is the nonexistence of compact nonplanar minimal surfaces with boundary contained in a plane Π. This result remains true in the non compact case if in addition the surface lies in the slab determined by Π and a plane Π′ parallel to Π. See [11, Lemma 2.1] for details.

Theorem 2.4 (Maximum principle at the boundary). Suppose M_{1}, M_{2} are minimal surfaces in ℝ^{3} with boundaries β_{1} and β_{2}. Suppose p is an interior point of both β_{1} and β_{2}, and suppose T_{p}M_{1} = T_{p}M_{2}, T_{p}β_{1} = T_{p}β_{2}. Assume that T_{p}M_{1} = {x_{3} = 0} so that both M_{1}, M_{2} are given near p as the graphs of two real analytic functions u_{1} and u_{2}, respectively. If u_{1} ≥ u_{2} in a neighbourhood of p, then M_{1} = M_{2}.

Figure 5. (a) The domain z(M). (b) The surface M.

We conclude these preliminaries describing the family of minimal surfaces constructed by the authors in [9]. We also include a summary of the main geometrical properties of these examples.

Take n ∈ [1, 2] and r ∈] − 1, 1[. Put r = −cos(x_{0}), x_{0} ∈]0, π[. Consider the Riemann surface

N = {(z, w) ∈ ℂ^{∗} × ℂ / w^{2} = (z − e^{ix_{0}/n})(z − e^{−ix_{0}/n})},

and define in the z-plane:

s_{1}^{+} = \{λe^{− \frac{πi}{n}} : λ ∈]0, 1]\}, s_{1}^{−} = \{λe^{\frac{πi}{n}} : λ ∈]0, 1]\},
s_{2}^{+} = \{λe^{− \frac{πi}{n}} : λ ∈ [1, +∞[\}, s_{2}^{−} = \{λe^{\frac{πi}{n}} : λ ∈ [1, +∞[\},
s_{0}^{+} = {e^{−it} : t ∈ [x_{0}/n, π/n]}, s_{0}^{−} = {e^{it} : t ∈ [x_{0}/n, π/n]}.

Then, label C ⊂ N as the connected component of z^{−1}(ℂ − (∪_{i=0}^{2}(s_{i}^{+} ∪s_{i}^{−}))) containing the point

P_{0} = (1, +\sqrt{2(1 − cos(x_{0}/n))}).

Define

M = \overline{C}  (3)

where \overline{C} means the closure of C in N.

Finally, label γ_{i}^{+} = z^{−1}(s_{i}^{+}), γ_{i}^{−} = z^{−1}(s_{i}^{−}), i = 0, 1, 2. Denote

γ^{+} = ⋃_{i=0}^{2} γ_{i}^{+}, γ^{−} = ⋃_{i=0}^{2} γ_{i}^{−}.

It is clear that ∂(M) = γ^{+} ∪ γ^{−}. Furthermore, note that z|_{γ_{i}^{+}} and z|_{γ_{i}^{−}} are bijective maps onto s_{i}^{+} and s_{i}^{−}, respectively, i = 1, 2. However, γ_{0}^{+} and γ_{0}^{−} consist of two copies of s_{0}^{+} and s_{0}^{−}, respectively. See Figure 5 for more details.

Since z(M) is simply connected and 0 ∉ z(M) , then the function z^{n} + z^{−n} +2r is well defined on M. We choose the branch of z^{n} satisfying 1^{n} = 1.

This choice of the branch of z^{n} implies that the function z^{n} + z^{−n} + 2r has neither zeroes nor poles on M − ∂(M).

Hence, the function φ((z, w)) = \sqrt{z^{n} + z^{−n} + 2r} has a well defined branch on the (simply connected) domain M − ∂(M), that can be extended continuously to M. For convenience, we choose the branch verifying φ(P_{0}) < 0. Moreover, note that given z_{0} ∈ (s_{0}^{+} ∪ s_{0}^{−}) and denoting {P^{+}, P^{−}} = z^{−1}(z_{0}), we have φ(P^{+}) = −φ(P^{−}).

We consider the meromorphic data on M

g = iz, ϕ_{3} = B \frac{dz}{zφ},  (4)

where B > 0.

For simplicity, we write

τ_{r} = \frac{dz}{zφ},

and as usual, we denote

(ϕ_{1}, ϕ_{2}, ϕ_{3}) = \frac{B}{2} (−i(1/z + z)τ_{r}, (1/z − z)τ_{r}, 2 τ_{r}) .

As M is homeomorphic to a closed disc minus two boundary points, then

X : M ⟶ ℝ^{3}

X(P) = Re ∫_{P_{0}}^{P} (ϕ_{1}, ϕ_{2}, ϕ_{3})  (5)

is a well defined conformal minimal immersion of M in ℝ^{3}.

Remark. If n = 2 and r ∈] − 1, 1[, the immersions associated to the Weierstrass data (4) are known and correspond to some Jenkins-Serrin graphs (see Figure 1). Following the notation of [6], these minimal graphs are the only ones with boundary values +∞, 0, +∞, 0 on a rectangle. See Karcher’s work [7] for a good setting.

For the reasons explained in last remark, we are going to restrict our attention to the case n < 2, and in what follows we suppose n ∈ [1, 2[.

Concerning to its symmetries, let S_{h}, S_{v} denote the antiholomorphic transformations on M

S_{h}((z, w)) = (1/\overline{z}, \overline{w}/\overline{z}), S_{v}((z, w)) = (\overline{z}, \overline{w}).

Notice that S_{h}(P_{0}) = S_{v}(P_{0}), and

g ∘ S_{h} = 1/\overline{g}, g ∘ S_{v} = −\overline{g},
S_{h}^{∗}(ϕ_{3}) = −\overline{ϕ_{3}}, S_{v}^{∗}(ϕ_{3}) = \overline{ϕ_{3}};  (6)

so elementary arguments imply that S_{h} (respectively S_{v}) induces on X(M) a symmetry with respect to the plane x_{3} = 0 (respectively x_{1} = 0).

It has been proved in [9] the following theorem.

Theorem 2.5. Let θ ∈]0, π] and r ∈] − 1, 1[.

Take n = 2π/(θ+π) and consider the Weierstrass data (M, g, ϕ_{3}), where M is described in Figure 5-(b), and

g = iz, ϕ_{3} = B \frac{dz}{z\sqrt{z^{n} + z^{−n} + 2r}} , B > 0.

Then, the minimal immersion

X = Re ∫_{1} (\frac{1}{2}(g^{−1} − g)ϕ_{3}, \frac{i}{2}(g^{−1} + g)ϕ_{3}, ϕ_{3})

satisfies:

(i) X(M) is a properly immersed minimal disk with two boundary ends in ℝ^{3}.

(ii) X(∂(M)) = Γ = ⋃_{i=0}^{2}(ℓ_{i}^{+} ∪ ℓ_{i}^{−}), where

1. The curves ℓ_{1}^{+} and ℓ_{1}^{−} are half-lines contained in a plane x_{3} = k, k > 0, they are symmetric with respect to the plane x_{1} = 0, and the straight lines containing them meet at an angle θ.

2. The curve ℓ_{2}^{+} (respectively ℓ_{2}^{−}) is the image under the symmetry with respect to the plane x_{3} = 0 of ℓ_{1}^{+} (resp. ℓ_{1}^{−}).

3. The curve ℓ_{0}^{+} (resp. ℓ_{0}^{−}) is the vertical segment joining the end points of ℓ_{1}^{+} and ℓ_{2}^{+} (respectively ℓ_{1}^{−} and ℓ_{2}^{−}).

(iii) If θ ∈]0, π[, X(M) is contained in the convex-hull E(Γ) of Γ. If θ = π, X(M) is contained in the intersection of the slab −k ≤ x_{3} ≤ k and one of the two halfspaces determined by the plane containing ℓ_{i}^{+}, ℓ_{i}^{−}, i = 0, 1, 2.

(iv) X(M) is invariant under the symmetries with respect to the planes x_{1} = 0 and x_{3} = 0.

(v) Up to a suitable choice of B (rescaling), we can suppose k = 1/2. Then, denote by d oriented distance between ℓ_{0}^{+} and ℓ_{0}^{−}. This means that

1. |d| = dist(ℓ_{0}^{+}, ℓ_{0}^{−}); 2. d ≤ 0 if and only if ℓ_{i}^{+} ∩ ℓ_{i}^{−} ≠ ∅, i = 1, 2, and d > 0 otherwise;

3. d = 0 if and only if ℓ_{0}^{+} = ℓ_{0}^{−}.

Obviously, the function d depends on r and θ. Moreover, for each θ ∈]0, π], there exists an only r_{θ} ∈]0, 1[ such that d(r_{θ}) = 0. The function d(r) is negative in ]r_{θ}, 1[ and positive in ] − 1, r_{θ}[. Furthermore, lim_{r→−1} d(r) = 0 and d(r) has an unique critical point r′_{θ} ∈] − 1, r_{θ}[. The number d_{θ} = d(r′_{θ}) is the maximum of the distance function d(r) on ] − 1, r_{θ}] .

(vi) X is an embedding if and only if d > 0. If d = 0 (that is, r = r_{θ}), then X|_{M−γ^{+}} and X|_{M−γ^{−}} are injective, where γ^{+} and γ^{−} are the two connected components of ∂(M).

Up to rescaling, and in what follows, we suppose that k = 1/2, and so, X(M) lies in the slab −1/2 ≤ x_{3} ≤ 1/2.

The numbers r ∈] − 1, 1[ and θ ∈]0, π] are analytical parameters of our family of surfaces. For being more precise and when it is necessary, we can refer to X_{θ r} as the immersion arising in above theorem for the values r and θ. Analogously we write M_{θ r} ≡ M as the underlying complex structure of the disc.

Then we can write the family M of surfaces arising from Theorem 2.5 as follows:

M = {X_{θ r} : M_{θ r} → ℝ^{3} / r ∈] − 1, 1[, θ ∈]0, π]}.  (7)

Note that θ has a clear geometrical meaning: the angle that ℓ_{i}^{+} makes with ℓ_{i}^{−}, i = 1, 2. The meaning of parameter r concerns to the underlying complex structure of the surface.

3. The uniqueness results

In this section we prove a uniqueness theorem for the surfaces X_{θ r}, r ∈] − 1, r_{θ}], θ ∈]0, π[. For the uniqueness in case θ = 0, see the remark in page 9.

From a topological point of view, embeddedness is a very natural assumption that plays a fundamental role in some geometrical arguments. For this reason, we restrict our attention to the embedded case. From a technical point of view, some arguments in Subsection 3.1 (see Lemma 3.4) only work under this topological assumption.

Moreover, we also exclude case θ = π. These surfaces are not contained in the convex hull of their boundaries, and in order to obtain the same results as in Subsection 3.1 and Subsection 3.2, this case needs to be treated slightly differently. Anyway, it can be shown that the surfaces X_{π r} admit a similar uniqueness theorem. Furthermore, this particular family of surfaces has a very natural interpretation in the Lorentz-Minkowski three dimensional space (see [8] for details).

For the remainder of this paper, we assume that

θ ∈]0, π[.

For θ ∈]0, π[, we consider two half-lines r_{1}^{+} and r_{1}^{−} in ℝ^{3}, meeting at an angle of θ. Let q_{1}^{+} and q_{1}^{−} be two points in r_{1}^{+} and r_{1}^{−}, respectively, such that they are symmetric with respect the inner bisector of this half-lines. We allow q_{1}^{+} = q_{1}^{−}. Label ℓ_{1}^{+} (ℓ_{1}^{−}) as the half-line on r_{1}^{+} (r_{1}^{−}), starting at q_{1}^{+} (q_{1}^{−}), and choose q_{1}^{+} and q_{1}^{−} in such a way that either q_{1}^{+} = q_{1}^{−} or ℓ_{1}^{+}∩ℓ_{1}^{−} = ∅.

Let π_{1} be the plane determined by ℓ_{1}^{+} and ℓ_{1}^{−} and label π_{2} a plane parallel to π_{1} and distinct. Let ℓ_{2}^{+} and ℓ_{2}^{−} be the orthogonal projections to π_{2} of ℓ_{1}^{+} and ℓ_{1}^{−}, respectively. The orthogonal projection of q_{1}^{+} and q_{1}^{−} are denoted by q_{2}^{+} and q_{2}^{−}. We label ℓ_{0}^{+} as the segment [q_{1}^{+}, q_{2}^{+}] and ℓ_{0}^{−} as [q_{1}^{−}, q_{2}^{−}]. Finally, we write

Γ^{+} = ⋃_{i=0}^{2} (ℓ_{i}^{+}) , Γ^{−} = ⋃_{i=0}^{2} (ℓ_{i}^{−}) and Γ = Γ^{+} ∪ Γ^{−}.

Up to a homothetic transformation, we can suppose that the distance be-

Figure 6. The curve Γ.

tween π_{1} and π_{2} is 1. We label d as the distance between ℓ_{0}^{+} and ℓ_{0}^{−}. Notice that the possibility ℓ_{0}^{+} = ℓ_{0}^{−} is allowed, and so d could be zero.

Along the remainder of this paper, we use a set of Cartesian coordinates, such that

• ℓ_{0}^{+} and ℓ_{0}^{−} have the direction of x_{3}-axis;

• the planes π_{1} and π_{2} coincide with the planes x_{3} = 1/2 and x_{3} = −1/2, respectively; • the origin is the middle point between (q_{1}^{+} + q_{2}^{+})/2 and (q_{1}^{−} + q_{2}^{−})/2

and;

• the x_{2}-axis is the inner bisector of the orthogonal projection of ℓ_{i}^{+} and ℓ_{i}^{−} to the plane x_{3} = 0, i = 1, 2.

Throughout this section, X : M → ℝ^{3} is a proper conformal minimal immersion satisfying:

1. M is homeomorphic to the closed unit disc \overline{𝔻} minus two boundary points E_{1} and E_{2}, that we call the ends of M. The boundary of M has two connected components that we have called γ^{+} and γ^{−}.

2. X(∂(M)) = Γ.

3. If d > 0, X is an embedding.

4. In the limit case ℓ_{0}^{+} = ℓ_{0}^{−}, X|_{M−γ^{+}} and X|_{M−γ^{−}} are injective. As a consequence, the points on ℓ_{0} = ℓ_{0}^{+} = ℓ_{0}^{−} are double. (For this reason, throughout the paper, we will distinguish between M and X(M).)

5. X(M) lies in the convex hull, E(Γ), of Γ.

3.1. The underlying complex structure and Weierstrass data This subsection aims at showing what is the complex structure of M and the behaviour of the Weierstrass data. We prove that M is conformally equivalent to a closed disc with piecewise analytic boundary and twice punctured at the boundary. As a consequence of this, the Gauss map and Weierstrass data will admit a meromorphic extension to the ends. The principal ideas of this subsection are inspired in Osserman’s works [14].

3.1.1. The underlying complex structure. This paragraph is devoted to study the conformal type of the disk M, and the main result corresponds to Theorem 3.8. To obtain this theorem, we need some information about the Gauss map of the immersion X. This information is summarized in Theorem 3.1 below.

Let a⃗_{2} = (0, 1, 0). Observe that, a⃗_{2} is the unitary vector in the direction of the inner bisector of ℓ_{2}^{+} and ℓ_{2}^{−} pointing to E(Γ) (see Figure 6).

As a first step, we want to prove the following:

Theorem 3.1. There exists a neighbourhood of a⃗_{2}, U(a⃗_{2}), in 𝕊^{2} such that, counting multiplicities,

♯(N^{−1}({a⃗, −a⃗})) = 1, ∀a⃗ ∈ U(a⃗_{2}),

where N : M → 𝕊^{2} is the Gauss map of X.

To obtain this theorem, we need additional results.

The plane x_{2} = t is denoted by Πt ; a⃗_{2}, t ∈ ℝ. Notice that the fact X(M) ⊂ E(Γ) and the interior maximum principle imply Π0 ; a⃗_{2} ∩ X(M) = ℓ_{0}^{+} ∪ ℓ_{0}^{−}.

Definition 3.1. For t ≥ 0, we define σ_{t} =^{def} X^{−1}(X(M) ∩ Πt ; a⃗_{2}).

Taking into account that X is proper and X(M) ⊂ E(Γ), we obtain that σ_{t} is compact, for every t ≥ 0. Furthermore, σ_{t} is a set of properly immersed analytic lines, because it is the nodal set of an harmonic function.

Claim 3.2. Π0 ; a⃗_{2} and X(M) are transverse along ℓ_{0}^{+} ∪ ℓ_{0}^{−}. In particular, the set σ_{0} consists of two disjoint regular curves on ∂(M).

To see this, it suffices to observe that Π0 ; a⃗_{2} and X(M) are transverse along ℓ_{0}^{+} ∪ ℓ_{0}^{−}. If not, using the order of contact theorem (or the maximum principle at the boundary), we obtain that X(M) has points in both two sides of Π0 ; a⃗_{2}, which is absurd because X(M) ⊂ E(Γ).

On the other hand:

Claim 3.3. The plane Πt ; a⃗_{2} and the surface X(M) are transverse on ℓ_{1}^{+}∪ ℓ_{1}^{−} ∪ ℓ_{2}^{+} ∪ ℓ_{2}^{−}, ∀t > 0.

Indeed, given a point in P ∈ (ℓ_{1}^{+} ∪ ℓ_{1}^{−} ∪ ℓ_{2}^{+} ∪ ℓ_{2}^{−})∩Πt ; a⃗_{2}, the normal vector at P lies either in the orthogonal plane to ℓ_{1}^{+} or in the orthogonal plane to ℓ_{1}^{−}, but a⃗_{2} does not belong to any of both two planes, which implies that Πt ; a⃗_{2} is transverse to X(M) at P.

Remark. Using Claim 3.3 we deduce that one, and only one, curve lying in σ_{t}, t > 0, approaches to each one of the four points in σ_{t} ∩ ∂(M).

On the other hand, taking into account that σ_{t} is compact and the interior maximum principle, we deduce there are no regions in M bounded by curves in σ_{t}.

Since M is simply connected, then σ_{t} is the union of two regular simple curves, σ_{t}^{1} and σ_{t}^{2}, with at most one point in common. Furthermore, if σ_{t}^{1} and σ_{t}^{2} intersect at P_{0} ∈ M, then the order of contact theorem implies that X(P_{0}) is an ordinary point of tangential contact between X(M) and Πt ; a⃗_{2}. The three topological possibilities that could occur for σ_{t} have been illustrated in Figure 7.

Lemma 3.4 (Boundary behaviour). Up to relabellings, the immersion X satisfies

1. X(γ^{+}) = Γ^{+} and X(γ^{−}) = Γ^{−};

Figure 7. The topological possibilities for σ_{t}. (a) The ends E_{1}, E_{2} are in the same connected component of M − σ_{t}. (b) σ_{t}^{1} and σ_{t}^{2} intersect at P_{0} ∈ M, then X(P_{0}) is an ordinary point of tangential contact between X(M) and Πt ; a⃗_{2}. (c) The curves σ_{t}^{1} and σ_{t}^{2} are disjoint and E_{1}, E_{2} are not in the same connected component of M − σ_{t}.

2. X^{−1}(ℓ_{1}^{+}) ∪ X^{−1}(ℓ_{1}^{−}) diverges to E_{1} and X^{−1}(ℓ_{2}^{+}) ∪ X^{−1}(ℓ_{2}^{−}) diverges to E_{2}.

Proof. If d = dist(ℓ_{0}^{+}, ℓ_{0}^{−}) > 0, then Statement 1 of Lemma 3.4 holds by an easy connectedness argument.

In the case θ > 0 and d = 0, suppose that X(γ^{+}) ≠ Γ^{+}. As X(γ^{+}) is a connected divergent curve over X(∂M), then X(γ^{+}) must be, up to relabellings, ℓ_{1}^{+} ∪ ℓ_{0} ∪ ℓ_{2}^{−}, and so X(γ^{−}) = ℓ_{1}^{−} ∪ ℓ_{0} ∪ ℓ_{2}^{+}.

Label γ_{0}^{+}, γ_{0}^{−} as the two connected components of X^{−1}(ℓ_{0}). From Claim 3.2, both components are two regular arcs in ∂(M) and Π0 ; a⃗_{2} is transverse along ℓ_{0}. Thus, there exist two open neighbourhoods U(γ_{0}^{+}), U(γ_{0}^{−}) in M of γ_{0}^{+}, γ_{0}^{−} , respectively, such that X (U(γ_{0}^{+})), X (U(γ_{0}^{−})) are graphs over Π0 ; a⃗_{2}. In particular, there exists t′ > 0 small enough such that, for each t < t′, X(σ_{t}) consists of two curves, one of them starting at ℓ_{1}^{+} and ending at ℓ_{2}^{−}, and the other one starting at ℓ_{1}^{−} and ending at ℓ_{2}^{+}. As both curves lie in Πt ; a⃗_{2}, then they meet at least one point. This means that X(M)−ℓ_{0} presents self-intersections, which is absurd.

In order to prove Assertion 2, recall that M is homeomorphic to a closed disc minus two boundary points, and so, Assertion 1 in the lemma implies that X^{−1}(ℓ_{1}^{+}) and X^{−1}(ℓ_{2}^{+}) diverge each other to different ends, and the same happens for X^{−1}(ℓ_{1}^{−}) and X^{−1}(ℓ_{2}^{−}) .

If ℓ_{1}^{+} and ℓ_{2}^{−} diverge to the same end, then, without loss of generality, we can assume that ℓ_{1}^{+}, ℓ_{2}^{−} diverge to E_{1}, and so ℓ_{1}^{−}, ℓ_{2}^{+} diverge to E_{2}. Let U(E_{i}), i = 1, 2, be two open disjoint neighbourhoods of the ends. As X is proper, then there is a t′′ > 0 large enough such that, for every t > t′′ σ_{t} ⊂ U(E_{1}) ∪ U(E_{2}).

Therefore, the remark in page 14 says that σ_{t} consists of two disjoint regular arcs, σ_{t}^{1} and σ_{t}^{2}, where σ_{t}^{i} ⊂ U(E_{i}), i = 1, 2. It is clear that X(σ_{t}^{1}) is a curve starting at ℓ_{1}^{+} and ending at ℓ_{2}^{−}, and X(σ_{t}^{2}) starts at ℓ_{1}^{−} and ends at ℓ_{2}^{+}. As both curves lie in Πt ; a⃗_{2} ∩ E(Γ), then they intersect, which is contrary to our assumptions.

In what follows, we label γ_{i}^{+} = X^{−1}(ℓ_{i}^{+}), γ_{i}^{−} = X^{−1}(ℓ_{i}^{−}), i = 1, 2. We define also γ_{0}^{+} = X^{−1}(ℓ_{0}^{+} ∪ ℓ_{0}^{−}) ∩ γ^{+} and γ_{0}^{−} = X^{−1}(ℓ_{0}^{+} ∪ ℓ_{0}^{−}) ∩ γ^{−}.

Observe that

γ^{+} = γ_{0}^{+} ∪ γ_{1}^{+} ∪ γ_{2}^{+}, γ^{−} = γ_{0}^{−} ∪ γ_{1}^{−} ∪ γ_{2}^{−}.

The following lemma is a consequence of the remark in page 14 and basic facts in Morse Theory. We include the proof for completeness.

Lemma 3.5. Counting multiplicities, ♯ [N^{−1} ({−a⃗_{2}, a⃗_{2}})] = 1.

Proof. Let us call A = {t ∈ ℝ^{+} / E_{1}, E_{2}lie in the closure of the same connected component of (M − σ_{t})}, and B = {t ∈ ℝ^{+} / γ_{0}^{+}, γ_{0}^{−} are in the same connected component of (M − σ_{t})}.

Observe that A and B are not void. For, note that Claim 3.2 yields 0 ∈ A. Moreover, during the proof of Lemma 3.4 we have seen that for t > 0 large enough, σ_{t} consists of two disjoint arcs belonging to two disjoint small neighbourhoods of the ends. As we can choose these neighbourhoods not intersecting to γ_{0}^{+} ∪ γ_{0}^{−}, then t ∈ B.

From the remark in page 14, it is not hard to see that A ∩ B = ∅. Furthermore, t ∈ [0, +∞[− (A ∪ B) if and only if σ_{t} is connected and contains a unique point of ordinary contact with normal vector in {−a⃗_{2}, a⃗_{2}}.

The set {σ_{t}, t ∈ [0, +∞[} consists of the level curves of the proper Morse function ⟨X, a⃗_{2}⟩. Since M is a closed disc minus two boundary points, basic Morse theory (see Claim 3.2, Claim 3.3 and the remark in page 14) gives that A = [0, t_{0}[ and B =]t_{0}, +∞[, t_{0} > 0. Moreover, σ_{t_{0}} contains a unique point of ordinary contact and σ_{t}, t ≠ t_{0}, do not. This concludes the proof.

Proof of Theorem 3.1.

Let β :]0, 1[→ M − ∂(M) be a continuous curve which diverges to both ends E_{1} and E_{2}. Define A and B as in the proof of Lemma 3.5.

Let t be a point in A, such that σ_{t} ∩ β(]0, 1[) = ∅. Then, as we have obtained during the proof of Lemma 3.5, there exists ϵ > 0 such that t−ϵ > 0 and [0, t + ϵ[⊂ A. As X is proper, we can choose ϵ small enough such that L ∩ X(β(]0, 1[)) = ∅, where L is the open slab in ℝ^{3} bounded by Πt−ϵ ; a⃗_{2} and

Πt+ϵ ; a⃗_{2}.

Let P be the set of planes Π ⊂ ℝ^{3} such that Π ∩ Γ ⊂ L ∩ Γ, and label V = {a⃗(Π) / Π ∈ P} ⊂ 𝕊^{2}, where a⃗(Π) is the unit normal vector to Π satisfying ⟨a⃗(Π), a⃗_{2}⟩ ≥ 0. It is clear that V is an open set in 𝕊^{2}, containing a⃗_{2} = a⃗(Π0 ; a⃗_{2}). From Claim 3.2, Claim 3.3 and shrinking L if necessary, we can assume that any plane in P is transverse to X(M) along Γ.

On the other hand, let H be the closed halfspace bounded by Πt+ϵ ; a⃗_{2} with inner normal vector −a⃗_{2}. As X is proper, then the set C = X^{−1}(H) is compact. Moreover, since t ∈ A, then the proof of Lemma 3.5 gives ±a⃗_{2} ∉ N(C).

As N(C) is a compact subset of 𝕊^{2}, then there exists an open neighbourhood V′ of a⃗_{2} such that (±V′) ∩ N(C) = ∅.

Thus, U(a⃗_{2}) =^{def} V ∩ V′ is the neighbourhood that we are looking for. Indeed, given a unitary vector a⃗ ∈ U(a⃗_{2}), let Π0 ; a⃗ be a plane orthogonal to a⃗ and contained in P, and for any t ≥ 0, define Πt ; a⃗ as the plane parallel to Π0 ; a⃗ whose height with respect to a⃗ is t.

Then, like in the proof of Lemma 3.5, we define A(a⃗) = {t ∈ ℝ^{+} / E_{1}, E_{2} lie in the closure of the same connected component of (M − X^{−1}(Πt ; a⃗))}, and B(a⃗) = {t ∈ ℝ^{+} / γ_{0}^{+}, γ_{0}^{−} are in the same connected component of (M − X^{−1}(Πt ; a⃗))}. Reasoning like in that proof, it is not hard to get A(a⃗) = [0, t_{0}[ and B(a⃗) =]t_{0}, +∞[, where t_{0} > 0. Therefore, there exists a unique point of ordinary contact in Πt_{0} ; a⃗, and there are no such points in Πt ; a⃗, t ≠ t_{0}. □

Consider B as in the proof of Lemma 3.5.

Definition 3.2. For t ∈ B, we define D_{t}^{i} as the closure of the connected component of M −σ_{t} containing E_{i}, i = 1, 2, where σ_{t} = X^{−1}(X(M)∩Πt ; a⃗_{2}).

Observe that D_{t}^{i} is homeomorphic to \overline{𝔻} − \{1\}, and its boundary is the union of three analytic curves ”laid end to end”. One of them is, up to relabelling, σ_{t}^{i} (see the remark in page 14), and the other two ones, a_{t}^{i} and b_{t}^{i}, are divergent curves lying in ∂(M).

Figure 8. The topology of D_{t}^{i}.

For the remainder of this paragraph, we use a set of coordinates such that a⃗_{2} = (0, 0, 1).

As Πt ; a⃗_{2} is transverse to X(M), for every t ∈ B, then we deduce that:

Remark. For any t ∈ B, the map g has neither zeroes nor poles on D_{t}^{i}, i = 1, 2 and so ηg has no zeroes in these sets.

Let X_{3} : M → ℝ be the third coordinate function, and let X_{3}^{∗} its conjugate harmonic function. Thus, h = X_{3} + iX_{3}^{∗} is a holomorphic function satisfying h′(z) dz = ηg, where z is a conformal parameter in D_{t}^{i}. In particular, from the remark in page 17 one has

h′(z) ≠ 0, for every z ∈ D_{t}^{i}, i = 1, 2.  (8)

Lemma 3.6. For any t ∈ B, the map h : D_{t}^{i} → h(D_{t}^{i}) is bijective.

Proof. Since (X_{3})|_{D_{t}^{i}}^{−1}(s) = σ_{s}^{i}, it suffices to prove that (X_{3}^{∗})|_{σ_{s}^{i}} is injective, for every s ≥ t.

On the other hand, as σ_{s}^{i} is a regular curve and h′ never vanishes, then X_{3}^{∗} is monotone on σ_{s}^{i}. This concludes the proof.

Lemma 3.6 implies that h is a biholomorphism from D_{t}^{i} − ∂(D_{t}^{i}) onto h(D_{t}^{i} − ∂(D_{t}^{i})), and (8) implies that it extends to the boundary in a regular way.

Now, we can prove the following:

Lemma 3.7. For t ∈ B large enough, we have

R < |g(P)| < \frac{1}{R}, for every P ∈ D_{t}^{i}, i = 1, 2,

where R ∈]0, 1[ is a suitable constant depending on t.

Proof. Let U be the stereographic projection of U(a⃗_{2}) ∪ (−U(a⃗_{2})) over the Riemann sphere. Taking into account the choice of the frame, U is a neighbourhood of \{0, ∞\} in \overline{ℂ} = ℂ ∪ \{∞\}.

Let P_{0} be the unique point of M, such that g(P_{0}) ∈ {0, ∞} (see Theorem 3.1). As g is continuous, then there is a compact neighbourhood \overline{V} of P_{0} satisfying

g(\overline{V}) ∪ (− \frac{1}{\overline{g}} ) (\overline{V}) ⊂ U.

If we define

K =^{def} g(\overline{V}) ∪ (− \frac{1}{\overline{g}} ) (\overline{V}),

then K is a compact neighbourhood of g(P_{0}). Using Theorem 3.1, we deduce that g^{−1}(K) = \overline{V}, and so g(M − \overline{V}) ∩ K = ∅.

If we take t large enough such that D_{t}^{i} ∩ \overline{V} = ∅, then g(D_{t}^{i}) ∩ K = ∅, i = 1, 2.

Since K is a neighbourhood of {0, ∞}, we can guarantee the existence of a constant 1 > R > 0 satisfying the statement of the lemma.

For the remainder of the paragraph, we fix t ∈ B as in Lemma 3.7, and for the sake of simplicity, we write D^{i}, σ^{i}, a^{i} and b^{i} instead of D_{t}^{i}, σ_{t}^{i}, a_{t}^{i} and b_{t}^{i}, respectively.

If we consider Weierstrass data (g, η) on D^{i}, then the induced metric

ds^{2} = \frac{1}{4} (1 + |g|^{2})^{2} |η|^{2}

is complete.

We can now state the main result of this paragraph.

Theorem 3.8 (Underlying Complex Structure). The neighbourhood D^{i}− σ^{i} is conformally diffeomorphic to 𝔻^{+} − {0}, where

𝔻^{+} = {z ∈ ℂ / |z| < 1, Im(z) ≥ 0} .

In particular, the Riemann surface M is conformally diffeomorphic to a closed disc with piecewise analytic boundary \overline{M} minus two boundary points {E_{1}, E_{2}}.

Proof. We split the proof in several claims.

Claim 3.9. The bilinear form ds^{2} =^{def} |ηg|^{2} is a complete metric on D^{i}.

As g has neither zeroes nor poles in D^{i}, then ηg has no zeroes on D^{i}. Taking Lemma 3.7 into account we deduce that dŝ ≤ R ds. As ds^{2} is complete, then ds^{2} is complete too.

Claim 3.10. The map

h : (D^{i}, ds^{2}) ⟶ (h(D^{i}), ds_{0}^{2})

is an isometry, where ds_{0}^{2} represents the Euclidean metric in the complex plane. Hence, (h(D^{i}), ds_{0}^{2}) is complete.

To see this, let us consider λ(s) a differentiable curve in D^{i}, then one has

L_{ds^{2}}(λ) = ∫_{λ} |ηg| = ∫_{λ} |h′(z)dz| = L_{ds_{0}^{2}} (h ∘ λ) .

This means that h is a length-preserving map. Taking into account Lemma 3.6, it implies that h is an isometry.

Claim 3.11. h(a^{i}) and h(b^{i}) are divergent curves in ℂ.

For, it suffices to recall that X(a^{i}) and X(b^{i}) are divergent pieces of straight lines in Γ, and so our choice of the frame gives that Re(h(a^{i})) = X_{3}(a^{i}) and Re(h(b^{i})) = X_{3}(b^{i}) diverge to +∞, i = 1, 2.

From Claim 3.10 and Claim 3.11, we deduce that h(D^{i}) is a simply connected region in \overline{ℂ}, and

∂ (h(D^{i})) = h(σ^{i}) ∪ h(a^{i}) ∪ h(b^{i}), i = 1, 2.

Furthermore, these three curves ‘laid end to end’, and

• h(σ^{i}) is compact;

• h(a^{i}) and h(b^{i}) diverge to ∞.

Hence, D^{i} is conformally diffeomorphic to a closed disc with piecewise analytic boundary, and once punctured at the boundary, i = 1, 2.

Finally, let us observe that D_{i} − σ_{i} is biholomorphic to 𝔻^{+} − {0}.

Let X̃ : D̃^{i} → ℝ^{3} be the conformal minimal immersion obtained from X(D^{i} − σ^{i}) by successive Schwarz reflections about straight lines. Here, D̃^{i} is an open Riemann surface, that is ∂(D̃^{i}) = ∅.

Label J : D̃^{i} → D̃^{i} as the automorphism induced by the rotation about the x_{3}-axis by angle 2θ. Let q : D̃^{i} → D̃^{i}/⟨J⟩ be the natural projection, and observe that D̃^{i}/⟨J⟩ is conformally equivalent to 𝔻^{∗} = 𝔻 − {0}.

Represent by S̃ the antiholomorphic involution on D̃^{i} induced by the 180 degrees rotation about a straight line in X̃(D̃^{i}). Then it is easy to see that J∘S̃ = S̃∘J. This implies that S̃ induces an antiholomorphic involution S on the quotient D̃^{i}/⟨J⟩ ≡ 𝔻^{∗}, fixing 0. Then, up to a conformal transformation, S(z) = \overline{z}, and so D_{i} − σ_{i} is biholomorphic to 𝔻^{+} − \{0\}. This completes the proof.

3.1.2. The Weierstrass data. In this paragraph we prove that the Gauss map and Weierstrass data extend continuously to the ends (see Theorem 3.12 for details). As a consequence, the immersion X has finite total curvature.

At this point we return to the original frame given at the beginning of this section. In particular, if we write X(P) = (X_{1}(P), X_{2}(P), X_{3}(P)), one has X_{3}|_{γ_{2}^{+}∪γ_{2}^{−}} = −1/2 and X_{3}|_{γ_{1}^{+}∪γ_{1}^{−}} = 1/2.

Theorem 3.12. The maps X_{3} and g extend continuously to E_{i}, i = 1, 2. In particular, the total curvature of M is finite.

Furthermore the limit tangent plane to M at E_{i} is π_{i}, i = 1, 2.

Proof. The harmonic function X_{3} is bounded, and X_{3}|_{γ_{i}^{+}∪γ_{i}^{−}} is constant. From Theorem 3.8, X_{3} extends continuously in a natural way to E_{i}, i = 1, 2.

For the second part of Theorem 3.12, we need to prove several claims.

Claim 3.13. Let Q_{i} ∈ γ_{i}^{+} (respectively γ_{i}^{−}), and let Σ_{i} be the tangent plane to X(M) at X(Q_{i}), i = 1, 2.

Denote by H_{i} the closed halfspace bounded by Σ_{i} which contains planes parallel to Σ_{i} and disjoint to E(Γ) (and so, disjoint to X(M).)

Then, there are no domains Ω_{i} ⊂ M − ∂(M) such that

• ∂(Ω_{i}) ⊂ X^{−1}(Σ_{i} ∩ X(M));

• X(Ω_{i}) lies on the closed halfspace H_{i}.

Indeed, if a such domain exists, then the distance from points of Ω_{i} to Σ_{i} is bounded. So, the claim is an easy consequence of [11, Lemma 2.1].

Claim 3.14. For any P ∈ γ_{1}^{+} ∪ γ_{1}^{−} ∪ γ_{2}^{+} ∪ γ_{2}^{−}, counting multiplicities, one has

♯[g^{−1}(g(P))] < 9.

We prove this claim in the case X(P) ∈ ℓ_{1}^{+}. The proofs of the other three possibilities are similar.

Label Σ as the tangent plane to X(M) at X(P), and observe that ℓ_{1}^{+} ⊂ Σ. Let us consider Δ = X^{−1}(Σ ∩ X(M)). Since Δ is the nodal set of a harmonic function, then Δ is a set of properly immersed analytic curves.

It is straightforward to prove that Δ ∩ X^{−1}(Γ − ℓ_{1}^{+}) has at most two points, one of them in γ_{0}^{−} and the other one in γ_{2}^{−}. In particular, ∂(M) − Δ has at most four connected components.

On the other hand, the interior maximum principle gives us that there are no compact simply connected region of M bounded by curves in Δ. Since M is simply connected, this implies that:

• If δ is a curve in Δ starting at γ_{1}^{+}, then either δ diverges to an end or the end point of δ belongs to X^{−1}(Γ − ℓ_{1}^{+}) ∩ Δ.

• When two curves in Δ originates from γ_{1}^{+}, they do not intersect.

As M is simply connected, elementary topological arguments imply that M − Δ has at least as many connected components as curves in Δ approaching γ_{1}^{+} plus one. If ♯[g^{−1}(g(P))] ≥ 9 then, using the order of contact theorem, there are at least 9 curves in Δ starting at P ∈ γ_{1}^{+}. Hence, we deduce that M − Δ has at least ten connected components.

Taking into account that ∂(M) − Δ has at most four connected components, there are six components whose boundary is totally included in Δ.

Moreover, the order of contact theorem implies that at least five components of M − Δ lie above Σ, and another five below.

Therefore, we deduce the existence of a domain Ω ∈ M −∂(M) verifying

• ∂(Ω) ⊂ Δ;

• X(Ω) lies in the closed halfspace H with boundary Σ, which contains planes parallel to Σ and disjoint to E(Γ).

which is contrary to Claim 3.13. This proves Claim 3.14.

At this point we can now prove the following:

Claim 3.15. The following limits exist:

lim ; \begin{matrix}P → E_{i} \\ P ∈ γ_{+} \end{matrix} g(P), lim ; \begin{matrix}P → E_{i} \\ P ∈ γ_{−} \end{matrix} g(P), i = 1, 2.

Consider ℓ ∈ {ℓ_{1}^{+}, ℓ_{1}^{−}, ℓ_{2}^{+}, ℓ_{2}^{−}}, and let α : [0, +∞[→ M be the parametrization by arc length of X^{−1}(ℓ).

We are trying to prove that lim_{s→+∞} f(s) exists, where f = g ∘ α.

First, observe that the normal vectors along ℓ lie in a meridian of 𝕊^{2}. Then, f([0, +∞[) ⊂ w, where w is a suitable straight line in \overline{ℂ}. Suppose this limit does not exist. Then there are two sequences {p_{n}} and {q_{n}} in [0, +∞[ such that:

• lim_{n} p_{n} = lim_{n} q_{n} = +∞,
• p_{n} < q_{n}, for every n ∈ ℕ,
• lim_{n} f(p_{n}) = a, lim_{n} f(q_{n}) = b, a, b ∈ w, a ≠ b.

Let c be a point in w between a and b. As lim_{n} f(p_{n}) = a and lim_{n} f(q_{n}) = b, there exists n_{0} ∈ ℕ such that

\{c, − \frac{1}{\overline{c}} \} ∩ f(]p_{n}, q_{n}[) ≠ ∅, for n ≥ n_{0},

which is contrary to Claim 3.14.

Taking into account Theorem 3.8 and the Uniformization Theorem, we deduce that M is conformally equivalent to a closed disk minus two boundary points (or a sector {z ∈ ℂ^{∗} : arg(z) ∈ [t_{1}, t_{2}]}), where the biholomorphism extends piecewise analitycally to the boundary. At this point, we need the following technical result, by Lindelöf: Let t_{1} < t_{2}, t_{2} − t_{1} < 2π, S = \{re^{it} : r > 0, t_{1} < t < t_{2}\} and f : \overline{S} → ℂ be bounded and continuous, holomorphic in S. Suppose that for some a, b ∈ \overline{ℂ} we have

lim_{r→+∞} f(re^{it_{1}}) = a, lim_{r→+∞} f(re^{it_{2}}) = b.

Then a = b and lim_{z→∞} f(z) = a.

We refer to reference book [2, pp. 132-133].

This theorem and Claim 3.15 imply that g extends continuously to the ends.

Finally, note that θ > 0 implies that g(γ_{i}^{+}) and g(γ_{i}^{−}) lie in different great circles of 𝕊^{2}. Thus, g(γ_{i}^{+}) ∩ g(γ_{i}^{−}) ⊂ {0, ∞} and so the limit normal vectors at the ends must be vertical, that is the limit tangent plane at E_{i} coincides with π_{i}, i = 1, 2. This concludes the proof.

As a consequence of this theorem, one has the following corollary.

Corollary 3.16. For t ∈ B large enough X(D_{t}^{i}) is a graph over the plane π_{i}, i = 1, 2.

Proof. Take t ∈ B large enough and such that g(D_{t}^{i}) does not intersect the equator \{z ∈ \overline{ℂ} / |z| = 1\}.

Label 𝔭_{3} as the orthogonal projection from X(M) to the plane x_{3} = 0. Taking into account that X(M) ⊂ E(Γ) and the definition of D_{t}^{i}, it is not hard to see that 𝔭_{3}|_{X(D_{t}^{i})} is a local diffeomorphism onto Ω_{t}^{i}, where Ω_{t}^{i} is the unbounded convex domain in the plane x_{3} = 0 limited by 𝔭_{3}(ℓ_{i}^{+}), 𝔭_{3}(ℓ_{i}^{−}) and 𝔭_{3}(σ_{t}^{i}) = {(x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} / x_{3} = 0, x_{2} = t}.

As X is proper, then the same occur for the map 𝔭_{3}|_{X(D_{t}^{i})}. So, 𝔭_{3}|_{X(D_{t}^{i})} is a covering map, and taking into account that Ω_{t}^{i} is simply connected we deduce that 𝔭_{3}|_{X(D_{t}^{i})} is one-to-one. This concludes the proof.

3.2. The symmetries of the surface. The aim of this subsection is to show that X(M) posses the same planes of symmetry that Γ. So, we generalize the ideas used by Schoen in [16] to our particular case of non compact piecewise analytic boundary.

3.2.1. The vertical symmetry. In this paragraph with the frame fixed at the beginning of Section 3, we prove that X(M) is invariant under symmetry with respect to the plane x_{1} = 0.

We need to introduce some notation. For t ≥ 0 we define

1. δ_{t} = X(M) ∩ {x_{1} = t},

2. M_{+}(t) = {(x_{1}, x_{2}, x_{3}) ∈ X(M) / x_{1} ≥ t},

3. M_{−}(t) = {(x_{1}, x_{2}, x_{3}) ∈ X(M) / x_{1} ≤ t},

4. M_{+}^{∗}(t) = {(2t − x_{1}, x_{2}, x_{3}) / (x_{1}, x_{2}, x_{3}) ∈ M_{+}(t)}, the symmetric of M_{+}(t) with respect to the plane {x_{1} = t},

5. M_{−}^{∗}(t) = {(2t − x_{1}, x_{2}, x_{3}) / (x_{1}, x_{2}, x_{3}) ∈ M_{−}(t)}, the symmetric of M_{−}(t) with respect to the plane {x_{1} = t}.

In general, given a set A ⊂ ℝ^{3}, we denote by A_{+}(t) (A_{−}(t)) the elements of A on and above (on and below) {x_{1} = t}, and A_{+}^{∗}(t) (A_{−}^{∗}(t)) its reflection about the plane {x_{1} = t}.

Let 𝔭_{1} : ℝ^{3} → {x_{1} = 0} denote the orthogonal projection.

If A, B ⊂ ℝ^{3}, we say that A ≥ B provided for every x ∈ {x_{1} = 0} for which 𝔭_{1}^{−1}({x}) ∩ A ≠ ∅ and 𝔭_{1}^{−1}({x}) ∩ B ≠ ∅, we have

Infimum [x_{1} (𝔭_{1}^{−1}({x}) ∩ A)] ≥ Supremum [x_{1} (𝔭_{1}^{−1}({x}) ∩ B)] .

Theorem 3.17. The surface X(M) is symmetric with respect to the plane {x_{1} = 0}, and M_{+}(0) and M_{−}(0) are graphs over {x_{1} = 0}.

Proof. For any t > 0, define

Δ_{t} = M_{+}^{∗}(t) ∩ M_{−}(t).

Claim 3.18. For any t > 0, Δ_{t} ∩ (Γ_{+}^{∗}(t) ∪ Γ_{−}(t)) = δ_{t} ∩ Γ.

Since X(M) ⊂ E(Γ) and t > 0, then Γ_{−}(t) ∩ M_{+}^{∗}(t) = δ_{t} ∩ Γ.

Furthermore, by the interior maximum principle, and using once again that X(M) ⊂ E(Γ), no points of X(M) − Γ lie on the boundary of E(Γ). So, it is not hard to deduce that Γ_{+}^{∗}(t) ∩ M_{−}(t) =δ_{t} ∩ Γ.

Taking these facts into account, we deduce easily that Δ_{t} − δ_{t} cannot intersect Γ_{−}(t) ∪ Γ_{+}^{∗}(t), which proves the claim.

Define now

I_{1} = {t ∈]0, +∞[ / M_{+}(t) is a graph over the (x_{2}, x_{3}) − plane and M_{+}^{∗}(t) ≥ M_{−}(t)}.

We want to prove that I_{1} =]0, +∞[. To do this, we need several results.

Let s_{0} > 0, s_{0} ∈ B, and consider (Definition 3.2) D_{s_{0}}^{i} as the closure of the connected component of M − σ_{s_{0}} containing E_{i}, i = 1, 2, where σ_{s_{0}} = X^{−1}(X(M) ∩ {x_{2} = s_{0}}).

If s_{0} > 0 is large enough, then, by Theorem 3.12 and Corollary 3.16, X(D_{s_{0}}^{i}) is a graph over the (x_{1}, x_{2})-plane, i = 1, 2, and

x_{3}|_{X(D_{s_{0}}^{1})} > 0, x_{3}|_{X(D_{s_{0}}^{2})} < 0.  (9)

See Corollary 3.16 for details. For simplicity label D^{i} = D_{s_{0}}^{i} − σ_{s_{0}}^{i}, i = 1, 2.

Claim 3.19. If X^{−1}(Δ_{t} − δ_{t}) ⊂ D^{1} ∪ D^{2}, then Δ_{t} = δ_{t}.

Suppose that Δ_{t} − δ_{t} ≠ ∅. Since Δ_{t} is locally the intersection of distinct minimal surfaces, it follows that Δ_{t} is a one-dimensional real analytic variety, properly immersed in ℝ^{3}.

On the other hand, as X(D^{i}) are graphs over the plane x_{3} = 0 and the projection of X(D^{i}) onto this plane is a convex domain, then X^{−1}(δ_{t}) ∩ D^{i} consists of a simple arc diverging to E_{i}, i = 1, 2. Taking into account Claim 3.18, we have

X^{−1}(Δ_{t} − δ_{t})) ⊂ ((D^{1} ∪ D^{2}) − ∂(M)) .

Moreover, since any curve in Δ_{t} − δ_{t} is properly immersed in ℝ^{3}, then no such curve diverges to an interior point in σ_{s_{0}}^{i}, i = 1, 2.

On the other hand, the definition of D^{i} and (9) yield that

X^{−1}(Δ_{t}) ∩ D^{i} = X^{−1} (X(D^{i})_{+}^{∗}(t) ∩ X(D^{i})_{−}) .

So, X^{−1}(Δ_{t} − δ_{t}) ∩ D^{i} ≠ ∅ implies

X^{−1}(Δ_{t} − δ_{t}) ∩ D^{i} = X^{−1} (X(D^{i})_{+}^{∗}(t) ∩ X(D^{i})_{−}) − ∂(M) ≠ ∅.

Therefore, it is not hard to see that there exist two simply connected closed regions Ω_{1} ⊂ X(D^{i})_{−}(t) and Ω_{2} ⊂ X(D^{i})_{+}^{∗}(t) bounded by the same curve in Δ_{t} (eventually Ω_{i} could be unbounded, i = 1, 2).

Summarizing, Ω_{1} and Ω_{2} are bounded graphs over its common orthogonal projection Ω_{0} onto the (x_{1}, x_{2})-plane, and ∂(Ω_{1}) = ∂(Ω_{2}).

If Ω_{0} is compact, then the interior maximum principle gives a contradiction.

If Ω_{0} is unbounded, then observe that the functions on Ω_{0} which determine the graphs Ω_{1} and Ω_{2} are asymptotic to the same finite value at infinity. The maximum principle at infinity for minimal graphs leads to a contradiction once again. This proves the claim.

Claim 3.20. There exists t′ > 0 large enough such that [t′, +∞[⊂ I_{1}.

Let t′ > 0 such that X^{−1}(M_{+}(t)) ⊂ D^{1}∪D^{2}, and observe that X^{−1}(Δ_{t}) ⊂ D^{1} ∪ D^{2}, for t ≥ t′. Hence from Claim 3.19, Δ_{t} = δ_{t}, t ≥ t′.

It is clear that X^{−1}(M_{+}(t)) consists of two simply connected components, one of them in D^{1} and the other one in D^{2}, for t ≥ t′, and thus M_{+}(t) is the union of two disjoint graphs G_{+}^{1}(t) and G_{+}^{2}(t) over the same simply connected domain G_{+}(t) in the (x_{1}, x_{2})-plane.

From (9), we have that 𝔭_{1}(G_{+}^{1}(t′)) ∩ 𝔭_{1}(G_{+}^{2}(t′)) = ∅.

Let us see that M_{+}(t) is a graph over the plane {x_{1} = t}, t ≥ t′.

First, observe that 𝔭_{1}|_{δ_{t}} is injective, t ≥ t′. Indeed, note that δ_{t} ⊂ {x_{1} = t} is a graph over a connected piece of the straight line x_{1} − t = x_{3} = 0, and so the function x_{2}|_{each component of δ_{t}} is an increasing function.

Moreover, a similar argument gives that the set σ_{s}^{i}∩M_{+}(t′) is a connected curve, for any s > s_{0} (recall that, from Definition 3.2, D^{i} = D_{s_{0}}^{i} − σ_{s_{0}}^{i} and D_{s_{0}}^{i} is the closure of the connected component of M − σ_{s_{0}} containing to E_{i}, i = 1, 2.)

On the other hand, the functions x_{3}|_{σ_{s}∩M_{+}(t′)} are monotone, for s > s_{0}. Otherwise, there would exist some points in M_{+}(t′) whose normal vector lie in {x_{1} = 0} ∩ 𝕊^{2}. Thus, we could take t ≥ t′ in such a way that δ_{t} contains a point with normal vector in {x_{1} = 0} ∩ 𝕊^{2}. Hence, from the order of contact theorem and taking into account that X(D^{i}) are graphs over the plane x_{3} = 0, i = 1, 2, we deduce that Δ_{t} − δ_{t} ≠ ∅, which is absurd.

Figure 9. The set 𝔭_{1}(M_{+}(t)).

This proves that M_{+}(t′) is a graph over the plane {x_{1} = t′}, and so the same holds for M_{+}(t), t ≥ t′.

Taking into account that Δ_{t} = δ_{t}, t ≥ t′ and X(M) ⊂ E(Γ), we deduce that M_{+}^{∗}(t) ≥ M_{−}(t), t ≥ t′. The claim holds.

It is obvious that I_{1} is closed in [0, +∞[.

Next step demonstrates that:

Claim 3.21. The minimum of I_{1} is 0.

First note that X(M) ⊂ E(Γ) implies that {0, ∞, i, −i} ∩ g(∂(M)) = ∅, and so

N(∂(M)) ∩ {x_{1} = 0} = ∅.  (10)

Assume that t_{0} = Minimum(I_{1}) > 0.

Let K be the compact set X(M − (D^{1} ∪ D^{2})). Observe that K = X^{−1}(H_{s_{0}}), where H_{s_{0}} is the closed half space x_{2} ≤ s_{0}. In particular, M_{+}^{∗}(t) ∩ K_{−}(t) = K_{+}^{∗}(t) ∩ M_{−}(t) = Δ_{t} ∩ K, for t > 0.

Taking into account that t_{0} ∈ I_{1}, we deduce that K_{+}(t_{0}) is a graph over the (x_{2}, x_{3})-plane.

In particular, no point in K_{+}(t_{0}) − (δ_{t_{0}} ∪ Γ) has normal vector included in {x_{1} = 0}. From equation (10) we obtain also that (N(K_{+}(t_{0})∩Γ)∩{x_{1} = 0} = ∅. Furthermore, the maximum principle at the boundary yields that no point in δ_{t_{0}} ∩K_{+}(t_{0}) has normal vector lying in {x_{1} = 0}. So, we conclude that

N(K_{+}(t_{0})) ∩ {x_{1} = 0} = ∅.

Since K_{+}(t_{0}) is compact, there exists ϵ_{0} > 0 small enough such that t_{0} − ϵ_{0} > 0, N(K_{+}(t)) ∩ {x_{1} = 0} = ∅ and K_{+}(t) is a graph over the (x_{2}, x_{3})-plane , for t ∈ [t_{0} − ϵ_{0}, t_{0}].

Otherwise, we could find sequences {t_{n}} in ℝ^{+}, and {P_{n}}, {Q_{n}} in K, satisfying:

• {t_{n}} ↗ t_{0},
• P_{n}, Q_{n} ∈ K_{+}(t_{n}), P_{n} ≠ Q_{n} and 𝔭_{1}(P_{n}) = 𝔭_{1}(Q_{n}), for every n ∈ ℕ.
• {P_{n}} → P_{0} ∈ K_{+}(t_{0}), {Q_{n}} → Q_{0} ∈ K_{+}(t_{0}).

Since K_{+}(t_{0}) is a graph over {x_{1} = 0}, the cases P_{0} ≠ Q_{0} and {P_{0}, Q_{0}} ∩ (K_{+}(t_{0}) − δ_{t_{0}}) ≠ ∅ are impossible. Therefore, we would deduce that P_{0} = Q_{0} ∈ δ_{t_{0}}, and thus, N(X^{−1}(P_{0})) ∈ {x_{1} = 0}. This is absurd.

In particular, Δ_{t} ∩ K ⊂ K_{−}(t_{0} − ϵ_{0}), for t ≥ t_{0} − ϵ_{0}.

On the other hand, Δ_{t_{0}} = δ_{t_{0}}. If not, Claim 3.18 implies that any point lying in Δ_{t_{0}} −δ_{t_{0}} is an interior point of contact between M_{+}^{∗}(t_{0}) and M_{−}(t_{0}). Using the interior maximum principle, we obtain M_{+}^{∗}(t_{0}) = M_{−}(t_{0}). This is absurd because t_{0} > 0 and thus Γ is not symmetric with respect to {x_{1} = t_{0}}.

Therefore, by using that Δ_{t_{0}} = δ_{t_{0}} and that K is compact, we deduce the existence of ϵ_{1} ∈]0, ϵ_{0}] such that Δ_{t} ∩ K = δ_{t} ∩ K, t ≥ t_{0} − ϵ_{1}.

Thus, Δ_{t} − δ_{t} ⊂ X(D^{1} ∪ D^{2}), and from Claim 3.19 we get Δ_{t} = δ_{t}, for t ≥ t_{0} − ϵ_{1}.

This implies that M_{+}(t) is a graph over {x_{1} = 0}, ∀t ≥ t_{0}−ϵ_{1}. Otherwise, M_{+}(t_{0}−ϵ_{1})−K would not be a graph over {x_{1} = 0}. Thus, we could find two different points in M_{+}(t_{0} − ϵ_{1}) − K with the same orthogonal projection on {x_{1} = 0}. From the choice of s_{0}, both points would lie in the same X(D^{i}), i ∈ {1, 2}. Hence, there would exists t ≥ t_{0} − ϵ_{1} such that δ_{t} ∩ X(D^{1} ∪ D^{2}) contains points whose normal vector lie in {x_{1} = 0}. So, by the order of contact theorem and using that X(D^{1}) and X(D^{2}) are graphs over the plane x_{3} = 0, we would deduce that Δ_{t} − δ_{t} ≠ ∅, which is obviously absurd.

As X(M) ⊂ E(Γ), then a continuity argument gives M_{+}^{∗}(t) ≥ M_{−}(t), t ≥ t_{0} − ϵ_{1}, we deduce that t_{0} − ϵ_{1} ∈ I_{1}, which contradicts that t_{0} is a minimum. This proves the claim.

The last claim yields that M_{+}^{∗}(0) ≥ M_{−}(0). A symmetric argument gives M_{−}^{∗}(0) ≥ M_{+}(0), and so the theorem holds.

3.2.2. The horizontal symmetry. Once again, we need to introduce some notation. For t ≥ −1/2 we now define

1. δ_{t} = X(M) ∩ {x_{3} = t},

2. M_{+}(t) = {(x_{1}, x_{2}, x_{3}) ∈ X(M) / x_{3} ≥ t},

3. M_{−}(t) = {(x_{1}, x_{2}, x_{3}) ∈ X(M) / x_{3} ≤ t},

4. M_{+}^{∗}(t) = {(x_{1}, x_{2}, 2t − x_{3}) / (x_{1}, x_{2}, x_{3}) ∈ M_{+}(t)},

5. M_{−}^{∗}(t) = {(x_{1}, x_{2}, 2t − x_{3}) / (x_{1}, x_{2}, x_{3}) ∈ M_{−}(t)}.

Following the same notation to that of Paragraph 3.2.1, given a set A ⊂ ℝ^{3}, we denote by A_{+}(t) (A_{−}(t)) the elements of A on and above (below) {x_{3} = t}, and A_{+}^{∗}(t) (A_{−}^{∗}(t)) its reflection about the plane {x_{3} = t}.

Let 𝔭_{3} : ℝ^{3} → {x_{3} = 0} denote the orthogonal projection.

If A, B ⊂ ℝ^{3}, we say that A ≥ B provided for every x ∈ ℝ^{3} for which 𝔭_{3}^{−1}({x}) ∩ A ≠ ∅ and 𝔭_{3}^{−1}({x}) ∩ B ≠ ∅, we have

Infimum [x_{3} (𝔭_{3}^{−1}({x}) ∩ A)] ≥ Supremum [x_{3} (𝔭_{3}^{−1}({x}) ∩ B)] .

For the sake of simplicity and in the particular case A = M_{+}^{∗}(t)−(ℓ_{0}^{+}∪ℓ_{0}^{−}) and B = M_{−}(t) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−}) (respectively A = M_{+}(t) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−}) and B = M_{−}^{∗}(t)−(ℓ_{0}^{+}∪ℓ_{0}^{−})), we write that M_{+}^{∗}(t) ⪰ M_{−}(t) (respectively M_{+}(t) ⪰ M_{−}^{∗}(t)) instead of A ≥ B, t ∈ [0, 1/2] (respectively t ∈ [−1/2, 0]).

Lemma 3.22. The set δ_{t} is compact, and consists of a simple arc, for every t ∈] − 1/2, 1/2[. In particular, the Gauss map g of X omits the points 0 and ∞.

Proof. From Theorem 3.12, we infer that δ_{t} is compact. Moreover, δ_{t} ∩ Γ has two points, one of them in ℓ_{0}^{+} and the other one in ℓ_{0}^{−}. In case ℓ_{0}^{+} = ℓ_{0}^{−}, the only point in δ_{t} ∩ Γ is, in a natural way, double.

Note that X^{−1}(δ_{t}) is the nodal set of the harmonic function x_{3}−t. Then, X^{−1}(δ_{t}) is an one-dimensional proper real analytic sub variety of M.

Using that {x_{3} = t} and X(M) are transverse along ℓ_{0}^{+} ∪ ℓ_{0}^{−} we deduce that one, and only one, curve lying in X^{−1}(δ_{t}), −1/2 < t < 1/2, approaches to each one of the two points in δ_{t} ∩ ∂(M).

On the other hand, taking into account that X^{−1}(δ_{t}) is compact and the interior maximum principle, we deduce there are no regions in M bounded by curves in X^{−1}(δ_{t}).

Figure 10. The curve δ_{t}.

Therefore, elementary topological arguments give that X^{−1}(δ_{t}) is a regular simple curve in M.

Finally, the order of contact theorem gives that there are no points in M with vertical normal vector.

Define E(Γ)_{0} = {x_{3} = 0} ∩ E(Γ), and observe that 𝔭_{3}(X(M)) ⊂ E(Γ)_{0}.

Lemma 3.23. The map 𝔭_{3}|_{X(M)} : X(M) ⟶ E(Γ)_{0} is not surjective. Moreover, there exists ϵ > 0 such that

𝔭_{3}(X^{−1}({x_{1} = 0} ∩ (X(M) − Γ))) = {(0, t, 0) / t ∈ [ϵ, +∞[}.

Proof. Consider the plane {x_{1} = 0} and denote α_{0} = X^{−1}({x_{1} = 0} ∩ (X(M) − Γ)). Let us see that α_{0} consists of a single regular analytic curve diverging to the two ends of M.

We know that α_{0} is a proper one dimensional real analytic subvariety of M −∂(M). Furthermore, from the order of contact theorem, the singularities of α_{0} occur at the points with normal vector orthogonal to {x_{1} = 0}, where a finite set of curves of α_{0} meet at equal angles.

From Corollary 3.16, only one curve of α_{0} approaches to the end E_{i}, i = 1, 2.

Note also that α_{0} does not contain any curves approaching to a point in ∂(M). In case d > 0, that is, ℓ_{0}^{+} ≠ ℓ_{0}^{−}, it is obvious. Suppose that ℓ_{0}^{+} = ℓ_{0}^{−}. First, observe that clearly no point in γ_{i}^{+} ∪ γ_{i}^{−}, i = 1, 2, is a accumulation point of α_{0}. So, we have to prove only that there are no accumulation points of α_{0} in γ_{0}^{+} ∪ γ_{0}^{−}.

To do this, consider S_{v} : M ⟶ M the antiholomorphic involution induced by the symmetry of X(M) with respect to the plane {x_{1} = 0} (see Theorem 3.17). The normal vector of any fixed point of S_{v} lies in {x_{1} = 0}. From the order of contact theorem and taking into account that X(M) ⊂ E(Γ), it is not hard to prove that the normal vector of any point in γ_{0}^{+} ∪ γ_{0}^{−} does not lie in {x_{1} = 0}. In particular, no point of γ_{0}^{+} ∪ γ_{0}^{−} is fixed by S_{v}. Therefore, if a curve α ⊂ α_{0} approaches to a point lying in γ_{0}^{+} ∪ γ_{0}^{−}, no point of α is fixed by S_{v}. Hence, the curves α and S_{v}(α) are distinct and mapped under X on the same curve in ℝ^{3}. This contradicts that X|_{M−∂(M)} is injective.

Summarizing, any divergent curve in α_{0} approaches only to either E_{1} or E_{2} and only one such curve diverges to the ends. Since there are no compact domains in M − ∂(M) bounded by curves in α_{0} (use the interior maximum principle), we deduce that α_{0} consists of a single regular arc in M − ∂(M) joining E_{1} and E_{2}. Furthermore, α_{0} is the fixed point set of S_{v}.

To finish the proof, observe that 𝔭_{3}(X(α_{0})) is a closed connected piece of the x_{2}-axis. Since the interior maximum principle implies X(M − ∂(M)) ⊂ (E(Γ) − ∂(E(Γ)), it is not hard to prove that

𝔭_{3}|_{X(α_{0})} : X(α_{0}) ⟶ {(0, t, 0) / t > 0}

is not onto. This concludes the proof.

Our purpose is to prove the following result.

Theorem 3.24. The surface X(M) is symmetric with respect to the plane {x_{3} = 0}. Furthermore, M_{+}(0) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−}) and M_{−}(0) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−}) are graphs over {x_{3} = 0}.

Proof. Let G = {P ∈ M / |g(P)| = 1}, and observe that G is a one dimensional real analytic variety containing γ_{0}^{+} ∪ γ_{0}^{−}. This variety consists of the critical points of the projection on the plane x_{3} = 0.

We label Q_{i}^{+} = γ_{0}^{+} ∩γ_{i}^{+} and Q_{i}^{−} = γ_{0}^{−} ∩γ_{i}^{−}, i = 1, 2. Taking into account that X(M) ⊂ E(Γ) and the order of contact theorem, Q_{i}^{+} and Q_{i}^{−} are not branch points of g, i = 1, 2.

Then, γ_{0}^{+} (γ_{0}^{−}) is the only curve of G which approaches to Q_{i}^{+} (Q_{i}^{−}); i = 1, 2.

On the other hand, using once again that X(M) ⊂ E(Γ) and the order of contact theorem, we check |g(P)| ≠ 1, for P ∈ γ_{i}^{+} ∪ γ_{i}^{−}, i = 1, 2.

These facts and Theorem 3.12 imply the existence of t′ > 0 satisfying M_{+}(t′)∩X(G) = M_{+}(t′)∩(ℓ_{0}^{+} ∪ℓ_{0}^{−}). Hence, by using a symmetric argument, we deduce that G is compact.

Thus, if t′ is taken as above, then 𝔭_{3}|_{M_{+}(t′)−(ℓ_{0}^{+}∪ℓ_{0}^{−})} has injective differential at any point.

Claim 3.25. Let t′ ∈]0, 1/2] such that

(M_{+}(t′) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−})) ∩ X(G) = ∅.

Then M_{+}(t′) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−}) is a graph over the (x_{1}, x_{2})-plane.

The claim is trivial for t′ = 1/2. Suppose t′ < 1/2.

First note that 𝔭_{3} is injective along (ℓ_{1}^{+} ∪ ℓ_{1}^{−} ∪ δ_{t′}) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−})).

Define υ = 𝔭_{3}(δ_{t′}) and E(Γ)_{0} = 𝔭_{3}(E(Γ)). As υ − (ℓ_{0}^{+} ∪ ℓ_{0}^{−}) has no selfintersections, then E(Γ)_{0} − υ has two connected components Υ_{1} and Υ_{2}. Up to relabeling, we suppose that Υ_{1} is the bounded component, and so Υ_{2} is the unbounded one. See Figure 11.

Figure 11. E(Γ)_{0} − υ has two connected components Υ_{1} and Υ_{2}.

Since X(∂(M)) = Γ, X(M − ∂(M)) ∩ ∂(E(Γ)) = ∅, X(M) ⊂ E(Γ) and (M_{+}(t′) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−})) ∩ X(G) = ∅ , then 𝔭_{3}|_{M_{+}(t′)−(ℓ_{0}^{+}∪ℓ_{0}^{−}∪δ_{t′})} is a local diffeomorphism onto its image in E(Γ)_{0}. Hence, Υ = 𝔭_{3}(M_{+}(t′) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−} ∪ δ_{t′})) is an open subset of E(Γ)_{0}. Moreover, 𝔭_{3}|_{X(M)} is proper, and so Υ ∩ Υ_{i} is a closed subset of Υ_{i}, i = 1, 2. Therefore, either Υ_{i} ⊂ Υ or Υ ∩ Υ_{i} = ∅, i = 1, 2.

From Theorem 3.17 and Lemma 3.23, Υ∩Υ_{1} = ∅, and so Υ_{2} ⊂ Υ, which easily implies Υ = Υ_{2}.

All these arguments give easily that 𝔭_{3}|_{M_{+}(t′)−(ℓ_{0}^{+}∪ℓ_{0}^{−})} is a proper local homeomorphism onto Υ ∪ υ, that is 𝔭_{3}|_{M_{+}(t′)−(ℓ_{0}^{+}∪ℓ_{0}^{−})} is a covering map.

As Υ∪υ is simply connected, then 𝔭_{3}|_{M_{+}(t′)−(ℓ_{0}^{+}∪ℓ_{0}^{−})} is a homeomorphism, that is, M_{+}(t′) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−}) is a graph over {x_{3} = 0}. This proves the claim.

Define now the set

I_{2} = {t ∈ [0, 1/2[ / M_{+}(t) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−}) is a graph
over the (x_{1}, x_{2})-plane and M_{+}^{∗}(t) ⪰ M_{−}(t)}.

Claim 3.26. I_{2} is not empty.

Let t′ be satisfying (M_{+}(t′) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−})) ∩ X(G) = ∅, and take t ∈ [(2t′ + 1)/4, 1/2[. From Claim 3.25, M_{+}^{∗}(t) is a graph over the plane x_{3} = 0, and so it is obvious that M_{+}^{∗}(t) ⪰ M_{−}(t). Thus, t ∈ I_{2}, as we want to prove.

For any t ∈]0, 1/2[, label Δ_{t} = M_{+}^{∗}(t) ∩ M_{−}(t). Observe that Theorem 3.12 implies that no curve in Δ_{t} diverges to E_{i}, i = 1, 2, that is, Δ_{t} is compact, t ∈]0, 1/2[.

Furthermore, for each t ∈]0, 1/2[, we write P_{1}^{t}, P_{2}^{t} as the only two points in the subset X^{−1} (δ_{t} ∩ (ℓ_{0}^{+} ∪ ℓ_{0}^{−})). With this notation we can assert:

Claim 3.27. If t ∈ I_{2}, t > 0, then P_{i}^{t} is not a branch point of g, i = 1, 2.

Consider t ∈ I_{2}, t > 0. As M_{+}^{∗}(t) ⪰ M_{−}(t), (X(M) − Γ) ⊂ (E(Γ) − ∂(E(Γ)) and t > 0, then Δ_{t} ⊂ δ_{t} ∪ ℓ_{0}^{+} ∪ ℓ_{0}^{−}. Otherwise, there would be an interior contact point between M_{+}^{∗}(t) and M_{−}(t), and so, by the interior maximum principle, M_{+}^{∗}(t) = M_{−}(t), which is absurd.

If P_{i}^{t} is a branch point of g, then the order of contact theorem implies that Δ_{t} contains at least three curves which intersect each other at this point and forming equal angles. On the other hand, Lemma 3.22 implies that X(M) and {x_{3} = t} are transverse.

So, at least one of these curves in Δ_{t} approaching P_{i}^{t} is not contained in δ_{t} ∪ ℓ_{0}^{+} ∪ ℓ_{0}^{−}, which is absurd. This contradiction proves the Claim.

Clearly I_{2} is a closed subset of [0, 1/2[. Moreover,

Claim 3.28. The minimum of I_{2} is 0.

Let t_{0} ∈ I_{2} and suppose that t_{0} > 0. Since M_{+}(t_{0}) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−}) is a graph over the plane {x_{3} = 0} and M_{+}^{∗}(t_{0}) ⪰ M_{−}(t_{0}), then M_{+}(t_{0}) ∩ X(G) ⊂ ℓ_{0}^{+} ∪ ℓ_{0}^{−}. Otherwise, δ_{t_{0}} − {X(P_{1}^{t_{0}}), X(P_{2}^{t_{0}})} would contain a point of X(G), and so the maximum principle at the boundary would imply that M_{+}^{∗}(t_{0}) = M_{−}(t_{0}), which is absurd.

Hence, by Claim 3.27, the points P_{i}^{t_{0}} are not ramification points of g. Using that X(G) is compact, there exists ϵ > 0 small enough such that t_{0} − ϵ > 0 and M_{+}(t_{0} − ϵ) ∩ X(G) ⊂ (ℓ_{0}^{+} ∪ ℓ_{0}^{−}). From Claim 3.25, M_{+}(t_{0} − ϵ) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−}) is a graph over {x_{3} = 0}.

Therefore,

Δ_{t} − (ℓ_{0}^{+} ∪ ℓ_{0}^{−} ∪ δ_{t}) ⊂ M_{−}(t_{0} − ϵ), ∀t ∈ [t_{0} − ϵ, 1/2[.  (11)

We are going to prove that Δ_{t} ⊂ (δ_{t} ∪ ℓ_{0}^{+} ∪ ℓ_{0}^{−}), t ∈ [t_{0} − ϵ, t_{0}].

As we have said before, t_{0} ∈ I_{2} yields Δ_{t_{0}} − (δ_{t_{0}} ∪ ℓ_{0}^{+} ∪ ℓ_{0}^{−}) = ∅.

Let t′ ∈ [t_{0}−ϵ, t_{0}[ and write t_{1} = Infimum{t ∈ [t′, t_{0}] / Δ_{t}−(δ_{t}∪ℓ_{0}^{+}∪ℓ_{0}^{−}) = ∅}.

Notice that Δ_{t_{1}} −(δ_{t_{1}} ∪ℓ_{0}^{+} ∪ℓ_{0}^{−}) = ∅. Otherwise, and taking into account that (X(M) − Γ) ⊂ (E(Γ) − ∂(E(Γ)), we could see that M_{+}^{∗}(t_{1}) and M_{−}(t_{1}) would have an interior point of contact in M_{−}(t_{0} − ϵ). Thus, the interior maximum principle would imply M_{+}^{∗}(t_{1}) = M_{−}(t_{1}), which is a contradiction (recall that t_{1} > 0).

In particular, the order of contact theorem gives that M_{+}^{∗}(t_{1}) and M_{−}(t_{1}) are transverse along Δ_{t_{1}} − {X(P_{1}^{t_{1}}), X(P_{2}^{t_{1}})} ⊂ δ_{t_{1}} ∪ ℓ_{0}^{+} ∪ ℓ_{0}^{−}, and {X(P_{1}^{t_{1}}), X(P_{2}^{t_{1}})} are points of ordinary contact. In case ℓ_{0}^{+} = ℓ_{0}^{−}, this means that the four sheets of M_{−}(t_{1})∪M_{+}^{∗}(t_{1}) are transverse at any point of ℓ_{0}^{+} ∩Δ_{t_{1}} = ℓ_{0}^{−} ∩ Δ_{t_{1}}, except at the point X(P_{1}^{t_{1}}) = X(P_{2}^{t_{1}}), where just two couples of sheets are tangent.

If t_{1} > t′, then the definition of t_{1} and (11) imply the existence of sequences {t_{n}}_{n∈ℕ} in ]t′, t_{1}[, {p_{n}}_{n∈ℕ} in M_{+}(t_{n}) − (δ_{t_{n}} ∪ ℓ_{0}^{+} ∪ ℓ_{0}^{−}) and {q_{n}}_{n∈ℕ} in M_{−}(t_{0} − ϵ) − (δ_{t_{n}} ∪ ℓ_{0}^{+} ∪ ℓ_{0}^{−}) such that: lim_{n→∞} t_{n} = t_{1} and q_{n} = p_{n}^{∗}(t_{n}). Without loss of generality, we can suppose that {p_{n}}_{n∈ℕ} and {q_{n}}_{n∈ℕ} converge to p_{0} ∈ M_{+}(t_{1}) and q_{0} ∈ M_{−}(t_{0} − ϵ). Since q_{0} = p_{0}^{∗}(t_{1}), we have q_{0} ∈ Δ_{t_{1}} ∩ M_{−}(t_{0} − ϵ). Therefore, and taking into account that t_{0} − ϵ < t_{1}, we get q_{0} ∈ ((ℓ_{0}^{+} ∪ ℓ_{0}^{−}) − {X(P_{1}^{t_{1}}), X(P_{2}^{t_{1}})}). This obviously contradicts that M_{+}^{∗}(t_{1}) and M_{−}(t_{1}) are transverse at q_{0}.

Hence, we deduce that t_{1} = t′, and since t′ is an arbitrary point of [t_{0} − ϵ, t_{0}[, we have

Δ_{t} ⊂ (δ_{t} ∪ ℓ_{0}^{+} ∪ ℓ_{0}^{−}), t ∈]t_{0} − ϵ, t_{0}].

Since M_{+}^{∗}(t_{0}) ⪰ M_{−}(t_{0}) and X(M) ⊂ E(Γ), a continuity argument gives M_{+}^{∗}(t) ⪰ M_{−}(t), t ∈ [t_{0} − ϵ, t_{0}]. This proves that [t_{0} − ϵ, t_{0}] ⊂ I_{2}, and so the claim.

Last claim implies that M_{+}(0) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−}) is a graph over {x_{3} = 0} and M_{+}^{∗}(0) ⪰ M_{−}(0).

If we repeat symmetrically the same arguments from below, we get that M_{−}(0) − (ℓ_{0}^{+} ∪ ℓ_{0}^{−}) is a graph over {x_{3} = 0} and M_{−}^{∗}(0) ⪰ M_{+}(0).

This concludes the proof.

This theorem has two interesting consequences.

Corollary 3.29. There are only two branch points R_{0}^{+} ∈ γ^{+} and R_{0}^{−} ∈ γ^{−} of g along γ_{0}^{+} ∪ γ_{0}^{−}. Furthermore, g has multiplicity two at these points and X(R_{0}^{+}),X(R_{0}^{−}) ∈ δ_{0}.

Proof. Theorem 3.24 implies that G = X^{−1}(δ_{0}) ∪ γ_{0}^{+} ∪ γ_{0}^{−}. If we label R_{0}^{+} = γ_{0}^{+} ∩ X^{−1}(δ_{0}) and R_{0}^{−} = γ_{0}^{−} ∩ X^{−1}(δ_{0}), then elementary properties of meromorphic functions give that R_{0}^{+} and R_{0}^{−} are branch points of g with multiplicity two.

Corollary 3.30. The limit normal vectors at the ends are opposite.

In what follows and without loss of generality, we assume that

g(E_{1}) = 0, g(E_{2}) = ∞.  (12)

3.3. The Geometrical Uniqueness In this subsection, we determine the conformal structure and Weierstrass data associated to the minimal immersion X : M → ℝ^{3} described at the beginning of this section.

Define

r_{1}^{+} = {λe^{−θi/2} : λ ∈]0, 1]}, r_{1}^{−} = {−λe^{θi/2} : λ ∈]0, 1]},

r_{2}^{+} = {λe^{−θi/2} : λ ∈ [1, +∞[}, r_{2}^{−} = {−λe^{θi/2} : λ ∈ [1, +∞[}.

Recall that g(E_{1}) = 0 = (1/g)(E_{2}), and as before, Q_{j}^{+} = γ_{0}^{+}∩γ_{j}^{+}, Q_{j}^{−} = γ_{0}^{−}∩ γ_{j}^{−}, j = 1, 2. Since X(M) ⊂ E(Γ), it is not hard to see that g(P) ≠ g(Q_{j}^{+}) (resp. g(P) ≠ g(Q_{j}^{−})), for P ∈ (γ_{j}^{+} − {Q_{j}^{+}}) (for P ∈ (γ_{j}^{−} − {Q_{j}^{−}})). Then, it is clear now that g(γ_{j}^{+}) = r_{j}^{+}, g(γ_{j}^{−}) = r_{j}^{−}, j = 1, 2.

On the other hand, from Corollary 3.29, we have that g|_{[Q_{j}^{+},R_{0}^{+}]} and

g|_{[Q_{j}^{−},R_{0}^{−}]} are injective, j = 1, 2, and Theorem 3.24 yields g(γ_{0}^{+}) = g([Q_{j}^{+}, R_{0}^{+}])

and g(γ_{0}^{−}) = g([Q_{j}^{−}, R_{0}^{−}]), j = 1, 2. Note that g(Q_{j}^{+}) = e^{−iθ/2} and g(Q_{j}^{−}) = e^{iθ/2}. Moreover, X(M) ⊂ E(Γ) implies that:

• g(R_{0}^{+}) = e^{−t_{0}i}, g(R_{0}^{−}) = −e^{t_{0}i}, where t_{0} ∈] − π/2, θ/2[.

• g(P) ≠ ±i, for P ∈ γ_{0}^{+} ∪ γ_{0}^{−}.

Then it is easy to conclude that g(γ_{0}^{+}) = r_{0}^{+} and g(γ_{0}^{−}) = r_{0}^{−}, where r_{0}^{+} and r_{0}^{−} are given as follows

r_{0}^{+} = {e^{it} : t ∈ [−θ/2, −t_{0}]}, r_{0}^{+} = {−e^{it} : t ∈ [t_{0}, θ/2]}.

Summarizing, we deduce that g(∂(M)) = ⋃_{j=0}^{2}(r_{j}^{+} ∪ r_{j}^{−}).

Let Λ denote the connected component of ℂ−(g(∂(M))∪{0}) containing i = \sqrt{−1}. Note that (r_{0}^{+} ∪ r_{0}^{−}) ⊂ ∂(Λ). See Figure 12.

The following lemma is a consequence of the information included in Subsection 3.1 and Subsection 3.2.

Lemma 3.31. The map g : M ⟶ ℂ satisfies:

1. g(M − ∂(M)) = Λ and g|_{M−∂(M)} : M − ∂(M) → Λ is a biholomorphism.

2. g(γ_{j}^{+}) = r_{j}^{+}, g(γ_{j}^{−}) = r_{j}^{−}, j = 1, 2. Furthermore, g|_{γ_{j}^{+}}, g|_{γ_{j}^{−}} are injective, j = 1, 2.

Figure 12. The region Λ and the curve g(∂(M)).

3. g(γ_{0}^{+}) = r_{0}^{+} and g(γ_{0}^{−}) = r_{0}^{−}. Moreover, g|_{[Q_{j}^{+},R_{0}^{+}]} , g|_{[Q_{j}^{−},R_{0}^{−}]} are injective, j = 1, 2.

Proof. To see that g(M −∂(M)) = Λ, let us observe that g(M −∂(M)) is a connected component of ℂ − (g(∂(M)) ∪ {0}). From Lemma 3.22, it is obvious that g(M − ∂(M)) is an open subset of ℂ^{∗} (holomorphic non constant functions are open). Moreover, as \overline{M} = M ∪ \{E_{1}, E_{2}\} is compact, then Theorem 3.12 gives that g(\overline{M}) is a closed subset of \overline{ℂ}. In particular, the set W = g(M −∂(M))∩(ℂ−(g(∂(M))∪\{0\}) = g(\overline{M})∩(ℂ−(g(∂(M))∪\{0\}) is a closed subset of ℂ−(g(∂(M))∪{0}). Therefore, either W = ℂ−(g(∂(M))∪ {0}) or W is a connected component of ℂ − (g(∂(M)) ∪ {0}). Since there are points ω ∈ g(M − ∂(M)) such that −ω ∉ g(M − ∂(M)) (see Lemma 3.5), then W is a connected component of ℂ − (g(∂(M)) ∪ {0}). Taking into account that the points in r_{0}^{+}, r_{0}^{−} lie in the boundary of W, we easily deduce that W = Λ. So, g(M − ∂(M)) = Λ.

To finish the proof of 1, define γ = ∂(\overline{M}). Since \overline{M} is conformally a closed disk with piecewise analytic boundary (see Theorem 3.8), then γ is a piecewise analytic curve homeomorphic to 𝕊^{1}. Moreover, as g|_{M} has no poles and g extends continously to \overline{M}, then we know that for any ω ∈ Λ = g(M − ∂(M)),

♯ (g^{−1}(ω)) = \frac{1}{2πi} ∫_{γ} \frac{dg}{g − ω} ∈ ℤ.

Thus, if we define f : Λ ⟶ ℤ by f(ω) = ♯(g^{−1}(ω)), the function f is continuous on Λ, and so it is constant. Since there exists ω ∈ Λ such that f(ω) = 1 (see Lemma 3.5), we deduce that f = 1, i. e., g : M − ∂(M) ⟶ Λ is a biholomorphism, which proves 1.

To prove 2, use 1 to check that g|_{γ_{1}^{+}} has no branch points, and the same occurs for γ_{1}^{−}, γ_{2}^{+} and γ_{2}^{−}. The remaining of 2, and 3 were proved at the beginning of this subsection.

At this point we are able to prove the principal result of this section.

Theorem 3.32. The immersion X : M → ℝ^{3} is, up to a horizontal translation, one of the minimal surfaces X_{θ r} described in (7), for a suitable r ∈] − 1, r_{θ}].

In particular, ℓ_{0}^{+} = ℓ_{0}^{−} if and only if r = r_{θ}.

Proof. Let M′ denote the Riemann surface obtained from M by successive Schwarz reflections about γ_{j}^{+} and γ_{j}^{−}, j = 1, 2. Note that ∂(M′) ≠ ∅.

Let X′ : M′ → ℝ^{3} be the corresponding minimal immersion, and label g′ as its Gauss map. Obviously, g′|_{M} = g. Observe that X′(M′) is invariant under the translation T defined as the composition of the 180 degrees rotation about ℓ_{1}^{+} and ℓ_{2}^{+}. The translation T induces a holomorphic transformation without fixed points T′ : M′ → M′. We denote M_{0} = M′/⟨T′⟩ and p_{0} : M′ → M_{0} as the natural projection. Write X_{0} : M_{0} → ℝ^{3}/⟨T ⟩ as the minimal immersion satisfying X_{0} ∘ p_{0} = X′. It is clear that ∂(M_{0}) ≠ ∅.

Clearly, p_{0}|_{M} : M → p_{0}(M) is a biholomorphism. In what follows, we identify M and p_{0}(M), and so, we consider M as a subset of M_{0}.

The symmetries of X(M) deduced in Subsection 3.2 together with the 180 degrees rotations about the horizontal lines, generate a group of symmetries of X′(M′) which contains a rotation about a straight line parallel to the x_{3}-axis by angle θ, followed by a symmetry with respect the plane x_{3} = 1/2. We call J as this symmetry.

We denote S_{v} and S_{h} as the antiholomorphic involutions on M_{0} induced by the symmetries with respect to the planes x_{1} = 0 and x_{3} = 0, respectively. Write L^{+} and L^{−} as the antiholomorphic involutions on M_{0} induced by the 180 degrees rotations about the straight lines containing ℓ_{2}^{+} and ℓ_{2}^{−}, respectively.

Note that J is the composition of the 180 degrees rotation about the straight line containing ℓ_{2}^{+} and the symmetry with respect to the plane x_{1} = 0, and so it induces on M_{0} the holomorphic transformation J_{0} = S_{v} ∘ L^{+}.

We denote g_{0} : M_{0} → \overline{ℂ} as the Gauss map of X_{0}, and observe that, up to natural identifications, g_{0}|_{M} = g.

As M is simply connected, log(g) has a well defined branch on M, and so the same occurs for g^{n}, where

n = \frac{2π}{π + θ} .  (13)

We choose the branch of log(g) in such a way that arg(log(g(P))) = 0 whenever g(P) ∈ ℝ^{+}. This choice is possible as consequence of Lemma 3.31. Furthermore, if P ∈ ∂(M) − (γ_{0}^{+} ∪ γ_{0}^{−}), then taking into account that g_{0} ∘ J_{0} = e^{2πi/n}g_{0} and Lemma 3.31, we get g^{n}(P) = g^{n}(J_{0}(P)).

On the other hand, it is clear that M_{0} = ∪_{j=1}^{∞}J_{0}^{j}(M), and recall that g_{0}|_{M} = g. As above, we can define suitable branches h_{j} of (g_{0}|_{J_{0}^{j}(M)})^{n}, j ∈ ℕ, in such a way that

• h_{0} = g^{n} is the above fixed branch .
• If P ∈ J_{0}^{j} (∂(M) − (γ_{0}^{+} ∪ γ_{0}^{−})), then h_{j}(P) = h_{j}(J_{0}(P)).
• If P ∈ J_{0}^{j}(M) ∩ J_{0}^{j+1}(M), then h_{j}(P) = h_{j+1}(P).

We easily deduce that g_{0}^{n} has a well defined branch on M_{0} satisfying g_{0}^{n}∘J_{0} = g_{0}^{n} and g_{0}^{n}|_{M} = g^{n}.

Hence, if we label ζ : M_{0} → M_{0}/⟨J_{0}⟩ as the natural projection, then there exists a holomorphic transformation f_{0} : M_{0}/⟨J_{0}⟩ → ℂ^{∗}, satisfying g_{0}^{n} = f_{0} ∘ ζ.

Taking into account our choice of the branches, we have:

• g_{0}^{n}(γ_{1}^{+} ∪ γ_{2}^{+}) = g_{0}^{n}(γ_{1}^{−} ∪ γ_{2}^{−}) = \{−λe^{\frac{nπi}{2}} / λ > 0\}.

• g_{0}^{n}|_{[Q_{j}^{+},R_{0}^{+}]} and g_{0}^{n}|_{[Q_{j}^{−},R_{0}^{−}]} are injective, j = 1, 2. Furthermore, g_{0}^{n}([Q_{j}^{+},

R_{0}^{+}]) = l′ and g_{0}^{n}([Q_{j}^{−}, R_{0}^{−}]) = l′′, j = 1, 2, where

l′ = {e^{−ti} / t ∈ [nπ/2 − π, −nt_{0}]},
l′′ = {e^{−ti} / t ∈ [−2π + nπ + nt_{0}, nπ/2 − π]}.

• g_{0}^{n}(M_{0} − ∂(M_{0})) = f_{0}((M_{0}/⟨J_{0}⟩ − ∂(M_{0}/⟨J_{0}⟩)) = ℂ^{∗} − l, where l = l′ ∪ l′′.

Furthermore, from Lemma 3.31, it is not hard to prove that

f_{0}|_{M_{0}/⟨J_{0}⟩−∂(M_{0}/⟨J_{0}⟩)} : M_{0}/⟨J_{0}⟩ − ∂(M_{0}/⟨J_{0}⟩) ⟶ ℂ^{∗} − l

is a biholomorphism.

Let S be the Riemann surface

S = {(y, v) ∈ ℂ^{∗} × ℂ / v^{2} = (y − e^{−nt_{0}i})(y − e^{n(t_{0}+π)i})}.

The set y^{−1}(ℂ^{∗} − l) has two connected components in S, and both of them are biholomorphic to ℂ^{∗} − l. We label Λ_{0} as the closure of one of these connected components.

From above arguments, and in a natural way, we can define a map

f̂_{0} : M_{0}/⟨J_{0}⟩ ⟶ Λ_{0}

such that y ∘ f̂_{0} = f_{0}. It is not hard to see that f̂_{0} is a biholomorphism. In what follows and for simplicity, we put f_{0} = f̂_{0}.

Note that f_{0}(∂(M_{0}/⟨J_{0}⟩)) = ∂(Λ_{0}) = l^{+} ∪ l^{−}, where l^{+} and l^{−} are the two lifts of l to S.

It is clear that

ξ = f_{0} ∘ ζ : M_{0} ⟶ Λ_{0},

is an unbranched cyclic covering.

We want to describe in a more precise way the Riemann surface M_{0} and the Weierstrass data of X_{0}. This step is fundamental in order to obtain a model of the complex structure of M and Weierstrass data of X.

To do this, we are going to determine the covering ξ.

Remember that J_{0} is the conformal transformation on M_{0} induced by the counterclockwise rotation about a straight line parallel to the x_{3}-axis by an angle of θ followed by a symmetry with respect to the plane x_{3} = 1/2 on X′(M′). Let c_{1} and c_{2} be counterclockwise circuits in Λ_{0} around zero and z^{−1}(l), respectively, and c̃_{i}, i = 1, 2, their lifts to M_{0}. The end points of c̃_{i} will differ by a deck transformation of the form J_{0}^{k_{i}}, k_{i} ∈ ℤ, i = 1, 2. The choice of J_{0} implies that k_{1} = 1 and k_{2} = 0. The numbers k_{1} and k_{2} determine the induced map from Π_{1}(Λ_{0}) onto ℤ_{ord(J_{0})} whose kernel corresponds to ξ_{∗}(Π_{1}(M_{0})), where ord(J_{0}) is the order of the automorphism J_{0} (that is, the number of sheets of the cyclic covering ξ). As usual, ℤ_{∞} = ℤ.

Any cyclic covering of Λ_{0} is equivalent to ξ if the associated representation has the same kernel.

At this point, we distinguish two cases: ord(J_{0}) < ∞ and ord(J_{0}) = ∞.

Case 1: ord(J_{0}) < ∞. Note that ord(J_{0}) < ∞ if and only if n ∈ ]1, 2[∩ℚ. Label m = π/θ = n/(2 − n) and observe that m ∈ [1, +∞[∩ℚ. Write m = p/q, where p, q ∈ ℕ and gcd(p, q) = 1. Since J_{0} is a rotation by angle θ followed by a symmetry, then ord(J_{0}) = 2p.

Let S̃ be the covering Riemann surface of S where the function u = y^{1/(2p)} is well defined and label

𝔭 : S̃ ⟶ S

as the natural projection. This covering is cyclic with group of deck transformations of order 2p and generated by

J̃_{0}((y, v), u) = ((y, v), e^{\frac{πi}{p}} u) .

If we define M_{0} = 𝔭^{−1}(Λ_{0}) and take into account the above arguments, we infer that the cyclic covering

𝔭|_{M_{0}} : M_{0} ⟶ Λ_{0},

is conformally equivalent to ξ.

Up to this biholomorphism, we may assume M_{0} = M_{0}, J̃_{0} = J_{0}. The functions y and v, up to composition with the covering map 𝔭, are meromorphic functions on M_{0}. Taking into account that y = g_{0}^{n}, we deduce also that, up to a suitable choice of the initial condition of the biholomorphism, g_{0} = u^{p+q}.

Summarizing, we have fixed the conformal structure of M_{0} and the meromorphic function g_{0}.

Label (ϕ_{1}^{0}, ϕ_{2}^{0}, ϕ_{3}^{0}) as the holomorphic 1-forms of the Weierstrass data of X_{0}. Let us determine ϕ_{3}^{0}.

It is clear that J_{0}^{∗}(ϕ_{3}^{0}) = −ϕ_{3}^{0}, and so the meromorphic function f_{3} = ϕ_{3}^{0}/dy satisfies f_{3} ∘ J_{0} = −f_{3}. This implies that f_{3} = u^{p}h_{3}, where h_{3} is the lift to M_{0} of a meromorphic function on Λ_{0}.

As X_{0}(∂(M_{0})) consists of the projection to ℝ^{3}/⟨T ⟩ of a set of straight lines in ℝ^{3} parallel to the x_{3}-axis, it is easy to check that if_{3}(P)y(P) ∈ ℝ, for any P ∈ ∂(M_{0}). Furthermore, the antiholomorphic involution S_{h} induced by the horizontal symmetry satisfies g_{0} ∘ S_{h} = 1/\overline{g_{0}} and S_{h}^{∗}(ϕ_{3}^{0}) = −\overline{ϕ_{3}^{0}}. This implies that y ∘ S_{h} = 1/\overline{y}, v ∘ S_{h} = e^{nπi/2}\overline{v}/\overline{y} and (yf_{3}) ∘ S_{h} = \overline{yf_{3}}.

Thus, given P ∈ ∂(M_{0}), we have (yf_{3})(S_{h}(P)) = −(yf_{3})(P).

Any connected component l̃of ∂(M_{0}) is a homeomorphic copy of l^{+} ∪l^{−}, and since v(S_{h}(P)) = −v(P), ∀P ∈ ∂(M_{0}), the above analysis yield that the meromorphic function yvf_{3} : M_{0} → ℂ takes the same values on the points of l̃ which are symmetric with respect to S_{h}. Furthermore, as S_{h} has fixed points, u ∘ S_{h} = 1/\overline{u}, and so the function R = yvh_{3} takes the same values on the points of l̃ which are symmetric with respect to S_{h} too.

Summarizing, we may define in a natural way a meromorphic function R(y) in the y-plane ℂ^{∗}, in such a way that

ϕ_{3}^{0} = \frac{R(y)}{\sqrt{y(y − e^{−nt_{0}i})(y − e^{n(t_{0}+π)i})}}dy.

Here, \sqrt{y(y − e^{−nt_{0}i})(y − e^{n(t_{0}+π)i})} represents the well defined meromorphic function on M_{0} given by vu^{p}.

Case 2: ord(J_{0}) = ∞. In this case, n ∈ [1, 2[−ℚ.

Let S̃ be the covering Riemann surface of S where the function u = log(y) is well defined and label

𝔭 : S̃ ⟶ S

as the natural projection. The map 𝔭 is an infinitely many sheeted cyclic covering, and its group of deck transformations is generated by

J̃_{0}((y, v), u) = ((y, v), u + 2πi) .

Defining M_{0} = 𝔭^{−1}(Λ_{0}) and reasoning as in the 1^{st} case, we infer that the cyclic covering

𝔭|_{M_{0}} : M_{0} ⟶ Λ_{0},

is conformally equivalent to ξ.

Up to this biholomorphism, we may assume M_{0} = M_{0} and J̃_{0} = J_{0}. The functions y and v, up to composition with the covering map 𝔭, are meromorphic functions on M_{0}. Taking into account that y = g_{0}^{n}, we deduce also that, up to a suitable choice of the initial condition, g_{0} = e^{u/n}.

As in the 1^{st} case, (ϕ_{1}^{0}, ϕ_{2}^{0}, ϕ_{3}^{0}) are the holomorphic 1-forms of the Weierstrass data of X_{0}.

It is obvious that J_{0}^{∗}(ϕ_{3}^{0}) = −ϕ_{3}^{0}, and so the meromorphic function f_{3} = ϕ_{3}^{0}/dy satisfies f_{3} ∘ J_{0} = −f_{3}. This implies that f_{3} = e^{u/2}h_{3}, where h_{3} is the lift to M_{0} of a meromorphic function on Λ_{0}.

Reasoning as in the 1^{st} case, we have y ∘ S_{h} = 1/\overline{y}, v ∘ S_{h} = e^{nπi/2}\overline{v}/\overline{y}, u ∘ S_{h} = −\overline{u}. Hence h_{3} = R(y)/(yv), where R(y) is a meromorphic function in the y-plane ℂ^{∗}, and so

ϕ_{3}^{0} = \frac{R(y)}{\sqrt{y(y − e^{−nt_{0}i})(y − e^{n(t_{0}+π)i})}}dy,

where \sqrt{y(y − e^{−nt_{0}i})(y − e^{n(t_{0}+π)i})} is the well defined function on M_{0} defined by ve^{u/2}.

In both cases, we have concluded that

y = g_{0}^{n}, ϕ_{3}^{0} = \frac{R(y)}{\sqrt{y(y − e^{−nt_{0}i})(y − e^{n(t_{0}+π)i})}}dy  (14)

but M_{0} is different in 1^{st} and 2^{nd} cases.

However, and in both cases, the function R(y) : ℂ^{∗} → ℂ is constant. Indeed, if we consider D′ = ζ^{−1}(D), where D is a small enough neighbourhood of either 0 or ∞, then any connected subset D′′ of M′ whose projection onto M_{0} is D′ can be identified in a natural way with D′. Hence, the immersion X_{0}|_{D′} can be identified with X′|_{D′′}. Furthermore, Theorem 3.12 yields that the third coordinate function

X′_{3} = Re ∫ \frac{R(y)}{\sqrt{y(y − e^{−nt_{0}i})(y − e^{n(t_{0}+π)i})}}dy

of X′ is bounded at D′′. The Great Picard Theorem implies easily that R(y) has not essential singularities at 0 and ∞, and thus, R(y) is a quotient of polynomial functions. As g′ has no zeroes neither poles in M′ (see Lemma 3.22), the same holds for g_{0}, and so we deduce that R(y) = Cy^{k}, C ∈ ℂ^{∗}, k ∈ ℤ. Using once again that X_{3} is bounded around any end, we deduce that k = 0, and thus R(y) = C is constant.

Taking into account that \sqrt{y} ∘ S_{h} = 1/\overline{\sqrt{y}}, v ∘ S_{h} = e^{nπi/2}\overline{v}/\overline{y} and S_{h}^{∗}(ϕ_{3}^{0}) = −\overline{ϕ_{3}^{0}}, it is not hard to check that, up to rigid motions,

C = \frac{B}{n} e^{\frac{nπi}{4}} , B ∈ ℝ^{∗}.

Thus, (14) becomes:

y = g_{0}^{n}, ϕ_{3}^{0} = \frac{B}{n} \frac{e^{\frac{nπi}{4}}}{\sqrt{y(y − e^{−nt_{0}i})(y − e^{n(t_{0}+π)i})}}dy  (15)

Consider the Riemann surface {(x, ν) ∈ ℂ^{∗} × ℂ / ν^{2} = (x + e^{t_{0}i})(x − e^{−t_{0}i})}, and label Λ_{1} as the closure of one of the two connected components of x^{−1}(Λ) (see Figure 12).

From Lemma 3.31, it is straightforward to check that, for a suitable choice of the branch of \sqrt{(x + e^{t_{0}i})(x − e^{−t_{0}i})}, the map

F : M → Λ_{1}
F (P) = (g(P), \sqrt{(g(P) + e^{t_{0}i})(g(P) − e^{−t_{0}i})})

is a conformal diffeomorphism. So, Λ_{1} and the holomorphic 1-forms ϕ_{j} = (F^{−1})^{∗}(ϕ_{j}^{0}|_{M}), j = 1, 2, 3 are Weierstrass data for the immersion X.

Therefore, up to this biholomorphism, we have

M ≡ Λ_{1}, g = x, ϕ_{3} = B \frac{e^{\frac{nπi}{4}} x^{n−1}}{\sqrt{x^{n}(x^{n} − e^{−nt_{0}i})(x^{n} − e^{n(t_{0}+π)i})}}dx,

where the branch of x^{n} is determined by the initial condition 1^{n} = 1.

After the change z = −ix, M becomes the closure of z^{−1}(−iΛ) in the Riemann surface N = {(z, w) ∈ ℂ^{∗} × ℂ / w^{2} = (z − e^{ix_{0}/n})(z − e^{−ix_{0}/n})}, where x_{0} = n(t_{0} + π/2). Since t_{0} ∈] − π/2, θ/2[, then (13) gives x_{0} ∈]0, π[.

Up to this change, the Weierstrass data of X : M → ℝ^{3} is given by:

M = z^{−1}(−iΛ), g = i z, ϕ_{3} = B \frac{dz}{z\sqrt{z^{n} + z^{−n} + 2r}} .  (16)

where r = − cos(x_{0}).

These are the Weierstrass data exhibited in Section 2.

As we have assumed that the distance between the planes π_{1} and π_{2} is 1, then the value of B is univocally determined.

Moreover, Theorem 2.5 says us that the immersion X : M → ℝ^{3} is, up to a horizontal translation, one of the surfaces X_{θ r} described in (7), where r ∈] − 1, r_{θ}[. This concludes the proof.

References

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[14] R. Osserman, A survey of minimal surfaces. (Dover Publications, New York, second edition, 1986).

[15] W. Rossman, ‘Minimal surfaces with planar boundary curves’. Kyushu J. Math. 52 (1998), 209-225.

[16] R. Schoen, ‘Uniqueness, symmetry and embeddedness of minimal surfaces’. J. Differential Geom. 18 (1983), 791-809.

Departamento de Geometría y Topología Universidad de Granada 18071 Granada, Spain

e-mail:fjlopez@goliat.ugr.es, fmartin@goliat.ugr.es.