Free Boundary Minimal Annuli Immersed in the Unit Ball

We construct a family of compact free boundary minimal annuli immersed in the unit ball B3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {B}^3$$\end{document} of R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^3$$\end{document}, the first such examples other than the critical catenoid. This solves a problem formulated by Nitsche in 1985. These annuli are symmetric with respect to two orthogonal planes and a finite group of rotations around an axis, and are foliated by spherical curvature lines. We show that the only free boundary minimal annulus embedded in B3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {B}^3$$\end{document} foliated by spherical curvature lines is the critical catenoid; in particular, the minimal annuli that we construct are not embedded. On the other hand, we also construct families of non-rotational compact embedded capillary minimal annuli in B3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {B}^3$$\end{document}. Their existence solves in the negative a problem proposed by Wente in 1995.


Introduction
Amid the general theory of free boundary minimal surfaces, the case where the ambient space is the unit ball B 3 of R 3 is of special significance [13,25].Here, we say that a compact minimal surface Σ is free boundary in B 3 if it intersects ∂B 3 orthogonally along ∂Σ.These surfaces appear as critical points of the area functional among all surfaces in B 3 whose boundaries lie on ∂B 3 .After the seminal work of Fraser and Schoen [12,13], the last decade has seen a great success in the construction of embedded free boundary minimal surfaces in B 3 of different topological types, by using different methods; see [3,4,14,13,15,16,17,18].
A trivial example of such a free boundary surface is the flat equatorial disk of B 3 .In 1985, Nistche [25] proved its topological uniqueness: any free boundary minimal disk immersed in B 3 must be an equatorial disk.
The simplest non-trivial example of a free boundary minimal surface in B 3 is the critical catenoid, i.e., the only compact piece of a catenoid that intersects ∂B 3 orthogonally along its boundary.This surface is rotational and has the topology of an annulus.The problem of the topological uniqueness of the critical catenoid among free boundary minimal annuli in B 3 was already formulated by Nitsche [25] in 1985, and it has been a relevant open problem of the theory for years, see e.g.[21,9,31,6,20,26,25,37].Our aim in this paper is to give a negative answer to this question.Specifically, we prove: Theorem 1.1.There exists an infinite, countable, family of non-rotational free boundary minimal annuli immersed in B 3 .These annuli have one family of spherical curvature lines, and are not embedded.They are invariant under reflection through the planes x 2 = 0 and x 3 = 0, and under a finite group of rotations around the x 3 -axis.The reflection with respect to x 3 = 0 interchanges the boundary components of the annulus.See  Two views from above of a free boundary minimal annulus immersed in B 3 .The example is symmetric with respect to the x 2 = 0 and x 3 = 0 planes, and also with respect to the rotations with angles 6πk/5, k ∈ {1, . . ., 5}, around the x 3 -axis.Its symmetry group is isomorphic to the dihedral group D 10 .The Gauss map along its central planar geodesic is a 3-folded covering map of the great circle S 2 ∩ {x 3 = 0}.See Figure 1.2 for two more views of the annulus.
A fundamental problem of the theory of free boundary minimal surfaces in B 3 is the conjecture according to which the critical catenoid should be the only free boundary minimal annulus embedded in B 3 , see [10], and [21] for a detailed discussion.Theorem 1.1 shows that the embeddedness assumption in the conjecture cannot be removed.Some partial affirmative answers to this conjecture have been recently obtained by several authors in [2,7,13,19,23,28,32].
The diversity of immersed minimal annuli provided by Theorem 1.1 suggests to look for a potential counterexample to the critical catenoid conjecture with the same geometric structure, i.e., so that both boundary curves are elements of a foliation by spherical curvature lines of the annulus.However, we can actually show the following uniqueness result: Theorem 1.2.The only free boundary minimal annulus embedded in B 3 and foliated by spherical curvature lines is the critical catenoid.
Theorems 1.1 and 1.2 give relevant insight not only towards the critical catenoid conjecture, but also about the general geometry of free boundary minimal annuli.One may regard our examples as free boundary versions of the constant mean curvature tori in space forms with planar or spherical curvature lines constructed in the 1980s by Wente [35] and then Abresch [1] and Walter [33,34].In this sense, Theorem 1.1 suggests in a natural way the interesting problem of classifying all free boundary minimal annuli immersed in B 3 .See Section 7.3.The geometry of the examples in Theorem 1.1 also has several formal similarities with the classical Riemann minimal examples foliated by circles in parallel planes; see Shiffman [29] and Meeks-Perez-Ros [24] for the fundamental uniqueness theorems of these Riemann examples.
The previous discussion can also be formulated in the more general capillary context of compact constant mean curvature surfaces that meet ∂B 3 at a constant angle along their boundary.By Nitsche's theorem [25], any capillary CMC disk immersed in B 3 is an equatorial disk; see also Ros and Souam [27].Regarding the topological uniqueness of capillary annuli, Wente [37] constructed for the case of non-zero mean curvature H = 0 examples of immersed, nonembedded, capillary and free boundary annuli in B 3 .For that, he used special properties of Abresch's solutions to the sinh-Gordon equation in [1] that are not available in the minimal case that we treat here.Also in [37], Wente asked whether any embedded capillary CMC annulus in B 3 should be rotational.This can be seen as a capillary version of the critical catenoid conjecture.See also [31,26].
In this paper we give a negative answer to Wente's problem; see Theorem 8.1 for a more precise statement.See also Figure 1.3.Theorem 1.3.There exist compact, embedded non-rotational minimal annuli in B 3 , with boundary contained in ∂B 3 , that are foliated by spherical curvature lines.In particular, they are embedded capillary minimal annuli in B 3 .
The general family of minimal surfaces in R 3 with spherical curvature lines was described by Dobriner [8] in the 19th century in terms of elliptic theta functions.Much more recently, in 1992, Wente [36] recovered and reformulated Dobriner's classification using the solutions to a certain Hamiltonian planar system.Wente described these minimal surfaces as catenoids, perhaps covered infinitely often, from which a number of flat ends have been extruded.Since these flat ends are placed along the planar curvature lines of the surface, this picture seems to forbid that such a surface could contain a compact free boundary minimal annulus in B 3 .In general, if a minimal annulus with spherical curvature lines has its two boundary curves on the same sphere, one would expect following Wente's description that it will have one or more flat ends in the middle.
The surprising realization that started this work is that, for some minimal surfaces with spherical curvature lines, one can perform a phase shift of a half period in its Weierstrass data to avoid such flat ends.This allows the construction of compact minimal annuli with a central planar geodesic, similar to a catenoidal neck, but perhaps with an immersed dihedral flower structure as the one depicted in Figure 1.1.Still, in order to prove Theorem 1.1, one needs to control several aspects simultaneously: compactness, periods, center of the spheres that contain the curvature lines, and orthogonal intersection along the boundary.
Our construction also produces interesting non-compact examples.
Corollary 1.4.There exist complete, non-compact minimal strips Σ with free boundary in B 3 , foliated by spherical curvature lines.That is, ∂Σ has two non-compact connected components, both of them contained in ∂B 3 , and Σ intersects ∂B 3 orthogonally along ∂Σ.
We next outline the paper.In Section 2 we review, following Wente [36], the geometry of minimal surfaces in R 3 with spherical curvature lines, and the integration of their associated Hamiltonian system.In Section 3 we analyze the phase space of this system in order to control later on the free boundary condition.In Section 4 we solve explicitly the Hamiltonian system in a degenerate case that appears as a limit of the examples that we will construct.
In Section 5 we provide Weierstrass data in terms of elliptic functions for a family of complete minimal surfaces in R 3 foliated by spherical curvature lines, having the additional property that one of such curvature lines is a bounded planar geodesic.This possibility was not considered in [36].The phase space analysis in Section 3 ensures that, for some of these minimal surfaces Σ, one of their curvature lines intersects some sphere S(p, R) of R 3 orthogonally.
In Section 6 we study the period map that indicates when are the spherical curvature lines of Σ periodic.We view it as an R-valued map from the 2-dimensional space of rectangular lattices in C where the Weierstrass data of Σ are defined, and show that its level sets are connected, regular, analytic curves.
In Section 7 we show that there exists an analytic 1-parameter family of the examples Σ constructed in Section 5 for which the sphere of orthogonal intersection B(p, R) has its center in the plane where the bounded planar geodesic of Σ lies; so, by symmetry, Σ has free boundary in some sphere of R 3 .For that, we use our study of the degenerate Hamiltonian system of Section 4. Our analysis of the period map in Section 6 implies that there is a dense family of examples within that 1-parameter family that, after homothety and translation, are compact free boundary minimal annuli in the unit ball B 3 .This proves Theorem 1.1.
In Section 8 we prove Theorem 1.3 on the existence of embedded capillary minimal annuli in B 3 , using the results of Sections 5, 6 and 7. Finally, in Section 9 we prove the uniqueness result stated in Theorem 1.2.

Minimal surfaces foliated by spherical curvature lines
We say that a minimal surface in R 3 has spherical curvature lines if one its two families of curvature lines has the property that each of its elements lies in some sphere S of R 3 , and hence, intersects this sphere at a constant angle.Here, we allow that the sphere S has infinite radius, i.e., that it is a plane, for some of these curvature lines.In this section we review some aspects of their geometry, following Wente [36].
Let Σ be a minimal surface with spherical curvature lines.Then, around each point of Σ there exists a local conformal parameter z = u + iv on a domain D ⊂ C with the following properties: (1) The second fundamental form of Σ is II = −du 2 + dv 2 , and its Hopf differential is ψ zz , N = −1/2.Here, ψ : D → R 3 is a conformal parametrization of Σ, and N is the unit normal.
(3) The metric of the surface is ds 2 = e 2ω |dz| 2 , where ω is a solution to the Liouville equation ∆ω − e −2ω = 0.The principal curvature associated to the v-curves is κ 2 = e −2ω .(4) There exist functions α(u), β(u) such that (2.1) Equation (2.1) characterizes the property that the v-curves of ψ are spherical curvature lines.The local function ω can actually be extended analytically to a global solution to the Liouville equation, defined on C minus a discrete set of points Z, at which ω → ∞.This extended function, that will also be denoted by ω, satisfies (2.1) globally on C \ Z for adequate functions α, β : R → R, and defines a conformal minimal immersion which extends our original local immersion, and that has flat ends at the points of Z.The values of α(u), β(u) are finite for every u ∈ R, and they describe the radius R(u) of the sphere S(u) ⊂ R 3 where ψ(u, v) lies, and the intersection angle θ(u) of Σ with S(u) along ψ(u, v) by the equations As for the center c(u) of S(u), it is given (when α(u) = 0) by An important property is that all the centers c(u) of the spheres S(u where, for each u ∈ R, p(u, X) is the (at most) fourth degree polynomial defined by 2.1.A Hamiltonian system.The functions (α(u), β(u)) above satisfy the autonomous system (2.7) with respect to some constant δ ∈ R. The system (2.7) has a Hamiltonian nature, and in particular has some preserved quantities.More specifically, any solution (α(u), β(u)) to (2.7) is defined on R, and associated to it there exist constants h, k ∈ R such that hold for every u ∈ R. The constants h, k, δ let us define, associated to any minimal surface foliated by spherical curvature lines, the polynomial (2.10) q(x) := −x 3 + δx 2 + hx + k that will play an important role.This polynomial is strongly related to the one introduced in (2.6).
For instance, a computation using (2.8) and (2.9) gives the relation for every u ∈ R. In particular, the discriminant of p(u, X) is actually independent from u, and described by the constants h, k, δ.
The system (2.7) can be integrated by separated variables following a procedure by Jacobi.Consider the change of coordinates (2.11) αβ = s + t, α 2 = −st.
Conversely, given a solution (α, β) to (2.7) with respect to some constant δ ∈ R, one can seek to integrate (2.1) to find a solution ω to the Liouville equation, and then obtain from ω a minimal surface Σ that satisfies (2.1) with respect to (α, β), and in particular has spherical curvature lines.In order to do this, in the view of (2.5), it is also necessary to impose the additional condition that p(u, X) > 0 for some u ∈ R and some X > 0.
It turns out that this positivity condition is also sufficient for the existence, as explained in the next lemma, that is contained in [36,Theorem 2.3 then there exists a conformal minimal immersion ψ(u, v) with spherical curvature lines that satisifies (2.1) with respect to α(u), β(u).
Remark 2.2.Assume that α(u 0 ) = 0.By (2.3), the curvature line ψ(u 0 , v) intersects a plane at a constant angle, and so it can be unbounded.If ψ(u 0 , v) is unbounded, the surface has a flat end at some point in the (u 0 , v) line, and e ω is unbounded along that line.However, assume that additionally to α(u 0 ) = 0 we also have α (u 0 ) > 0. In that situation, the polynomial p(u 0 , X) in (2.6) has degree three and negative leading coefficient.Thus, by (2.5), e ω must be bounded, i.e., the planar curvature line ψ(u 0 , v) is bounded.

Weierstrass representation.
Minimal surfaces with spherical curvature lines can be described by elliptic Weierstrass data.Specifically, given g 2 , g 3 ∈ R, let ℘(z) = ℘(z; g 2 , g 3 ) be the (possibly degenerate) Weierstrass P -function associated to g 2 , g 3 , so that it satisfies its standard differential equation ℘ 2 = 4℘ 3 − g 2 ℘ − g 3 .There are three cases: (1) g 2 = g 3 = 0.In that degenerate case, ℘(z) = 1/z 2 , which is holomorphic in C \ Λ, with Λ = {0}.(2) The modular discriminant ∆ mod := g 3 2 − 27g 2 3 is zero, with g 2 g 3 = 0.In that case, ℘ is a degenerate Weierstrass P-function that is singly periodic with respecto to either a real period or a purely imaginary period.See [5].The function ℘(z) is holomorphic in C \ Λ, where Λ is the set of multiples of this fundamental period, and has double poles at the points of Λ.
(3) ∆ mod = 0.Then, ℘(z) is doubly periodic with respect to a lattice Λ ⊂ C. The map ℘(z) is holomorphic in C \ Λ, and has double poles at the points of Λ.
These Weierstrass data Φ define a complete, conformally immersed minimal surface ψ : Note that φ has poles at the points of Λ.At each z 0 ∈ Λ, ψ has an embedded flat end.By construction, the second fundamental form of ψ is II = −du 2 + dv 2 , where z = u + iv, and so the v-curves ψ(u 0 , v) are curvature lines of the surface associated to the positive principal curvature.It can be checked that, by our choice of Φ, these are actually spherical curvature lines, and the centers c(u) of the spheres S(c(u), R(u)) that contain the curves ψ(u, v) all lie in a common vertical line L of R 3 .If a v-curve contains a point z 0 = u 0 + iv 0 of the lattice Λ, then the curvature line ψ(u 0 , v) is unbounded (the surface ψ has an end at z 0 ), and in particular it is contained in a plane, that is, Conversely, it is proved by Wente, see Theorems 4.1, 4.2 and 4.3 in [36], that up to an isometry and a homothety of R 3 , and except for some degenerate cases, any minimal surface in R 3 with spherical curvature lines can be constructed from the above Weierstrass data.Here, the degenerate cases correspond to the surfaces with planar curvature lines (including the catenoid), which appear when α(u) ≡ 0.

Phase space analysis
In this section we analyze the systems (2.7), (2.12) analytically.To start, we fix the polynomial q(x) in (2.10).While the next discussion can be carried out more generally, we will directly work in the conditions of our construction and make the assumption that q(x) can be factorized as where r 1 < r 2 < 0 < r 3 .
In this way, the trajectory (s(λ), t(λ)) is contained in the rectangle R := [0, r 3 ] × [r 2 , 0] of the (s, t)-plane.If s(λ) or t(λ) is constant, this trajectory is contained in one of the edges of R. In any other case, the trajectory meets the interior of R. Thus, for any connected arc of this trajectory that lies in the interior of R, there exist While this gives four different autonomous systems in normal form, the change λ → −λ reverses both signs of ε 1 , ε 2 , and so the corresponding systems have the same orbits, with opposite orientation.So, we can regard the rectangle R as two different phase spaces R + , R − , associated respectively to the systems where ε = ±1, s > 0 and t < 0. Note that while each of these systems is formally decoupled, and can be integrated as two independent first order autonomous ODEs in normal form, the common parameter λ establishes a link between both solutions.We describe next the behavior of the orbits of R + and R − .Any trajectory (s(λ), t(λ)) of R + has negative slope at every point of int(R).Assume that ε = 1 for definiteness.Then, s, t are defined (up to a translation in the λ parameter) for every λ ≤ 0, so that s(λ), t(λ) → 0 with the order of exp(−|λ|) as λ → −∞, and either s(0) = r 3 or t(0) = r 2 at the other end.So, the orbits Γ of R + that touch the interior of R start (in infinite λ-time) at the equilibrium (0, 0), and end up (in finite λ-time) at a point in the outer rim R ∩ ({s = r 3 } ∪ {t = r 2 }) of the rectangle R.Moreover, Γ is tangent to either s = r 3 or t = r 2 at this point in the outer rim, unless Γ is the special orbit Γ 1 of R + that joins the origin to the opposite vertex (r 2 , r 3 ) of R, that is also an equilibrium of (3.3).
One can discuss similarly the trajectories of R − .This time, any such trajectory (s(λ), t(λ)) that meets the interior of R has positive slope in int(R), and joins in finite λ-time a point (r 3 , t), t ∈ (r 2 , 0) with a point (s, r 2 ), s ∈ (0, r 3 ).The orbits in R − foliate the interior of R, and none of them passes through the vertex (r 3 , r 2 ).
The orbits of R + and R− can be joined in a tangential, analytic way at any point of the outer rim of R (with the vertex (r 3 , r 2 ) excluded), to create a trajectory of system (2.12).In this way, we obtain trajectories Γ(λ) = (s(λ), t(λ)) of (2.12) that bounce at the walls s = r 3 and t = r 2 of R following the orbits of R + , R − .See Figure 3.1.These trajectories Γ(λ) are thus defined for every λ ∈ R, with Γ(λ) → (0, 0) as λ → ±∞.
We are specially interested in one particular orbit of (2.12).Note that by the previous discussion, we do not need to fix the sign of α (0) in the definition below.Definition 3.2.In the conditions above, we denote by Γ 0 the orbit of (2.12) that corresponds to the choice α(0) = 0, β(0) = 0. Remark 3.3.In the case that the solution (α(u), β(u)) to (2.7) defines a minimal surface Σ, the condition α(0) = β(0) = 0 is equivalent, by (2.3), to the property that Σ intersects orthogonally a plane along the curvature line ψ(0, v).
Consider next the following assumptions on the polynomial q(x) in (2.10) (written as (3.1)): The first condition had already been imposed on q(x).The second assumption is just a normalization that can be attained after a homothety and a conformal reparametrization of the surface.The third one will be fundamental to control the free boundary condition for minimal surfaces (see Corollary 3.7 below).By a basic algebraic manipulation we have: Lemma 3.4.For (r 1 , r 2 , r 3 ) ∈ R 3 , the following two claims are equivalent: i) (r 1 , r 2 , r 3 ) satisfy conditions (1)-(3) above.ii) r 3 = 1/(r 1 r 2 ), where (r 1 , r 2 ) satisfy Along the rest of the paper, we will consider the open sets Ω 0 ⊂ Ω ⊂ W ⊂ R 2 given by the following relations for (r 1 , r 2 ) (see Figure 3.2): We remark that the condition r 1 r 2 2 < −1 for Ω 0 is equivalent to r 2 + r 3 < 0. Proposition 3.5.Let q(x) be the polynomial (3.1),where (r 1 , r 2 ) ∈ Ω and r 3 = 1/(r 1 r 2 ).Then: (1) The orbit Γ 0 intersects the line s + t = 0 in at most one point before hitting the horizontal segment R ∩ {t = r 2 }. (2) If (r 1 , r 2 ) ∈ Ω 0 , this intersection point always exists.
For this, we look at the slope m(s 0 ) of the orbit of (3.3)-(a) that passes through a point (s 0 , −s 0 ) in the diagonal s + t = 0.By (3.3)-(a), it is given by m(s 0 ) = − q(−s 0 ) q(s 0 ) .
On the other hand, a simple computation from (2.10) shows that the function q(−x) − q(x) is negative in [0, √ h) and positive in ( √ h, ∞), where h = q (0) > 0. Thus, for small positive values of s, we see that m(s) > −1, and we have two possibilities: In that situation, m(s) > −1 for every s ∈ (0, min{r 3 , −r 2 }).
Case 2: Recall that the orbit Γ 0 starts above the diagonal s + t = 0. So, as long as Γ 0 stays in the phase space (3.3)-(a), it can only cross s + t = 0 at a point (s 0 , −s 0 ) at which m(s 0 ) ≤ −1.Thus, by the previous dichotomy, Γ 0 can only cross once s + t = 0 while in the phase space (3.3)-(a).From here and the monotonicity of the orbits of (3.3)-(b), we see that item (1) holds.This completes the proof.

Solution of the system in the degenerate case
We now solve system (2.12) in that case that q(x) is given by (3.1) for the (degenerate) case that r 1 = r 2 = r < 0. Our aim in doing so is to have an exact control of the orbit Γ 0 in this situation.

The function −ln(H(x)
) is a primitive of 1/(x q(x)).Then, the general solution (s(λ), t(λ)) to system (a) in (3.3) (assume ε = 1 for definiteness) is implicitly given by where In our present degenerate case, we can still carry out a qualitative analysis of the system following our study of Section 3.There is, however, an important difference in that, due to the existence of the double root of q(x) at x = r < 0, the function t(λ) can only approach the value t(λ) = r as λ → ∞.This means that the trajectories (s(λ), t(λ)) of system (2.12) start (for λ = −∞) at (0, 0), they reach in finite time the wall {(1/r 2 , t) : t ∈ (r, 0)}, and end (for λ = ∞) at the point (0, r).See Figure 4.1.In particular, any orbit of (2.12) that starts at the origin above the diagonal s + t = 0 must eventually intersect again this diagonal, as it must end at (0, r).Trajectories of (2.12) in the (s, t)-plane when r 1 = r 2 .If they start initially above the diagonal s + t = 0 they eventually cross it, maybe after bouncing at the right wall.
From now on, we assume the conditions (4.4) which mean that (r, r) ∈ ∂Ω 0 .
both of them defined on (0, 1/r 2 ]. Also, let r ≈ −1.155867 be the unique solution to θ(x) = −1, where . ( Proof.Initially, i.e. for λ ≈ −∞, Γ 0 is an orbit of system (3.3)-(a), with ε = 1.By the general solution (4.2) to this system we have the first integral along any trajectory.We now look at the orbits of the phase space R + associated to (3.3)-(a) that intersect the diagonal s + t = 0.For any such point (s 0 , −s 0 ) in the diagonal, let c(s 0 ) be its corresponding constant c(s 0 ) := H(s 0 )/H(−s 0 ) = G(s 0 ) for (4.7).Let c 0 ∈ R denote the integration constant for Γ 0 associated to (4.7).Clearly, c 0 is the limit as s 0 → 0 of the constants c(s 0 ).It follows directly from (4.1) that So, one has H(s) = −H(t) along the orbit Γ 0 , while we stay in the phase space R + of (3.3)-(a), i.e., as long as the orbit Γ 0 does not hit the wall s = 1/r 2 .
The function F(x) is strictly increasing, with F(x) → −∞ as x → 0, and so its maximum value is F(1/r 2 ) = θ(r), where θ(x) is given by (4.6).The function G(x) has a unique critical point, a minimum, at and so the maximum of G in [0, 1/r 2 ] is the largest value between .
The function . Thus, in this Case 1 we have θ(r) > −1, and so G(x) < −1 for every x ∈ [0, 1/r 2 ].In particular H(x) = −H(−x).Since the relation H(s) = −H(t) holds along the orbit Γ 0 until it leaves the phase space R + , we deduce then that Γ 0 cannot intersect the diagonal s + t = 0 while in R + .
By the analysis in Section 3, the orbit Γ 0 extends analytically beyond (1/r 2 , t 0 ) by passing to the phase space R − associated to system (3.3)-(b).Since Γ 0 must intersect s + t = 0 and did not do it while in R + , there must exist some point ( α, − α) ∈ R − met by Γ 0 .But now, the general solution The proof is analogous, so we merely sketch it.This time θ(r) < −1, and so F(x) < −1 for every x.This is used to show by means of the previous arguments that Γ 0 cannot intersect s + t = 0 after reaching the wall, i.e., it must intersect it while still in the phase space R + .At that intersection point ( α, − α) ∈ R + , using that H(s) = −H(t) along Γ 0 when in R + , we must have G( α) = −1, as stated.This finishes the proof of Case 2.
Finally, when r = r , the orbit Γ 0 intersects s + t = 0 exactly at the point 5. Compact minimal annuli with spherical curvature lines 5.1.Weierstrass data.Fix (r 1 , r 2 ) ∈ W , where W ⊂ R 2 is given by (3.5), and define It follows that ∆ mod := g 3 2 − 27g 2 3 > 0, and so these values define a rectangular lattice Λ in C. Let ℘ be the Weierstrass P -function associated to Λ.It satisfies For all the properties of ℘ and other associated Weierstrass elliptic functions that will be used in this paper, see e.g.[38,5].
Define the doubly periodic meromorphic function Note that φ is real along both R and iR, and it has no poles in the strip |Re(z)| < ω 1 .Also, φ has poles at the points of the lattice Z := Λ − ω 1 for which it is periodic.The values of φ along iR are contained in the interval [r 1 , r 2 ], with φ(0) = r 1 and φ(ω 2 ) = r 2 .In particular, φ(z) < 0 for every z ∈ iR.
Define next where g 0 > 0 is a positive constant.So, g is the unique solution to g/g = φ with g(0) = g 0 .
From our discussion in Section 2 we have: Lemma 5.3.The surface Σ given by (5.9) has the following properties: (1) Each curve ψ(u 0 , v) is a curvature line contained in some sphere S(c(u 0 ), R(u 0 )).
(2) All the centers c(u) lie in a common vertical line L of R 3 .
Proof.Items (1) and (2) hold by Wente [36], see our discussion in Section 2. Note that we have made a translation z → z + ω 1 in the conformal parameter z = u + iv with respect to the formulas presented in Section 2.2, but this does not affect the properties of Σ detailed there.Since φ is real along iR, it follows by (2.16) that ψ(0, v) lies in the x 3 = 0 plane.Since the imaginary part of φ is never identically zero along u 0 + iR for any u 0 ∈ (−ω 1 , ω 1 ) with u 0 = 0, it follows that ψ(u 0 , v) does not lie in a horizontal plane, and so R(u 0 ) is finite.This proves item (3).
In Lemma 5.4 below we give an explicit expression of the Gauss map g in terms of the Weierstrass zeta and sigma functions.Recall that these classical functions satisfy ζ (z) = −℘(z) and σ (z)/σ(z) = ζ(z).
It is a classical property of Weierstrass functions that ℘(z) is real and injective along the boundary of the rectangle generated by the half-periods ω 1 , ω 2 .Thus, there exists exactly one value µ in the boundary of that rectangle where 4℘(µ) = b.By (5.1), and taking into account the values of ℘ at the half-periods, see (5.4), we have So, µ lies in the horizontal segment between ω 2 and ω 1 + ω 2 .The function ℘(z) is increasing along that segment, and hence, from (5.3) and (2.15) we have ℘ (µ) = 1/4.

Period and symmetries of minimal annuli.
We now consider the period problem for Σ along iR, i.e., we discuss when are the spherical curvature lines ψ(u 0 , v) of Σ periodic.We show in Lemma 5.5 below that this is controlled by the number κ ∈ R given by (5.16) Note that, by (5.8), κ measures the variation of g(z) as z varies from 0 to 2ω 2 ∈ iR: recall that |g(z)| = g 0 , constant, if z ∈ iR.Also, we observe that if ψ(u, v) is periodic in the v-direction, this period must be a multiple of Im(2ω 2 ), by the periodicity of φ and (2.16).
(3) There is some m ∈ Z such that κ = m/n ∈ Q.
Remark 5.6.The value κ in (5.16) does not depend on the choice of g 0 .
The quantity κ in (5.16) allows to control not only the period problem for ψ(u, v) in the vdirection as explained in Lemma 5.5, but also the symmetries of the resulting minimal annuli when the period closes.
(2) The plane The rotations of angles 2πk/n, with k ∈ {0, . . ., n − 1}, around the vertical line L of R 3 that contains the centers of the spheres S(c(u), R(u)).
Therefore, we have: Corollary 5.7.The symmetry group of the minimal annulus Σ * is isomorphic to the dihedral group D n if g 0 = 1, and to Proof.It suffices to show that Σ * is only invariant with respect to the group of isometries of R 3 generated by those listed above.Let γ = Σ * ∩ {x 3 = 0} be the central planar curvature line of Σ * .Since the u-curvature lines of Σ * are not closed, and γ is the only v-curvature line that is planar, we deduce that any isometry of R 3 that leaves Σ * invariant also leaves γ invariant.Since γ has a line of symmetry in the x 3 = 0 plane and it is not a circle, its (planar) symmetry group is a dihedral group D n .Thus, n must be a divisor of n and by the argument in Lemma 5.5, we have that κ = m /n for some m ∈ Z.But since κ = m/n irreducible, we obtain n = n and the result follows.
In the case g 0 = 1, Σ * is invariant with respecto to a prismatic group of order 2n; see [4] for a description of such groups.
In addition, the numerator m of κ = m n describes the number of times that the planar curvature line γ = Σ * ∩ {x 3 = 0} wraps around.This can be seen as follows.First, note that the Gauss map g of Σ * maps γ ≡ ψ(0, v) into the circle |z| = g 0 of C, and that g (iv) = 0 for any v, since φ(iv) = ∞.Thus, g defines a regular covering map of this circle.The value m describes the degree of this map.This implies that m gives the rotation index of the unit normal of the (locally convex) planar curve γ.
It is important to observe that, for any u ∈ (0, ω 1 ), all the fundamental quantities α(u), β(u), c 3 (u) can be explicitly computed in terms of the Weierstrass elliptic functions ℘, ζ, σ.We explain this next.

The period map
We now vary the values (r 1 , r 2 ) ∈ W that we fixed at the start of Section 5. Below, we follow the notations in that section.Definition 6.1.We define the period map Per(r 1 , r 2 ) : W → (0, 1) as (6.1) .16).By construction, Per(r 1 , r 2 ) depends analytically on (r 1 , r 2 ).
Our next objective is to control the level sets of the map Per.We first prove: Theorem 6.2.For any (r 1 , r 2 ) ∈ W we have In particular, Per(r 1 , r 2 ) ∈ (0, 1), and Per(r 1 , r 2 ) extends C 1 -smoothly to the diagonal D := {(r, r) : r < 0}, with values Proof.Recall that the function φ(z) in (5.5) is real on iR, with φ(0) = φ(2ω 2 ) = r 1 and φ(ω 2 ) = r 2 .Moreover, φ is a strictly increasing (resp.decreasing) diffeomorphism from the segment [0, . Then, we can make in the integral expression of Per in (6.1) the change of variable t = φ(ν) on each of these two segments, and we have by Lemma 5.2 (6.4) where q(x) = q(x) is given by (3.1).Writing t = r 1 + (r 2 − r 1 )s we obtain the more convenient expression (6.5) where (6.6) The function Q(s, r 1 , r 2 ) is positive.Also, (6.7) , since it can be easily checked that the function in the middle of these inequalities is increasing.So, by (6.5), (6.6) and the first inequality in (6.7), (6.8) The function ϕ(s) := 2 arctan r 2 s r 1 (1 − s) is a primitive of the above integral.Thus, from (6.8), we obtain the left inequality of expression (6.2).Operating in the same way using the second inequality of (6.7), we obtain (6.2).
Finally, by differentiating (6.6) with respect to r 1 , we have that as (r 1 , r 2 ) → (r, r), By an analogous computation, the derivative of Per(r 1 , r 2 ) with respect to r 2 converges to the same value as (r 1 , r 2 ) → (r, r).Thus, Per(r 1 , r 2 ) extends C 1 to the diagonal D.
Note that, by the implicit function theorem, from each (r, r) ∈ D starts a unique regular, real analytic level curve of Per, that intersects D orthogonally at (r, r).Moreover, (6.9) ∂Per(r 1 , r 2 ) ∂(r 1 + r 2 ) > 0 holds for every (r 1 , r 2 ) ∈ W .In particular, the level curves of Per intersect transversely the lines r 1 − r 2 = const, and Per(r 1 , r 2 ) is strictly increasing along them.
Proof.It follows from Theorem 6.2 that, for any c ∈ (0, 1), the level set Per −1 (c) contains a regular, real analytic curve that intersects D orthogonally at (r c , r c ).We wish to show that any level set Per −1 (c) is one of such regular, real analytic curves starting at D; in particular, these level sets are connected.
To start we will prove (6.9).First, by differentiation of Q(s, r 1 , r 2 ) in (6.6) we have (6.10) ) and The numerator L(s; r 1 , r 2 ) is linear in s, and is always negative at s = 1, since r 1 < r 2 < 0. As Υ(s; Assume first of all that and so it is non-negative at every point.By (6.5) we obtain directly that (6.9) holds.
Hence, the gradient of Per(r 1 , r 2 ) does not vanish in W , and so the level sets of Per are regular, real analytic curves, at first maybe not connected, that intersect transversely the lines r 1 − r 2 = const, and Per(r 1 , r 2 ) increases along any such line as r 1 (or r 2 ) increases.
Finally, we can see that the union of all the level curves of Per(r 1 , r 2 ) that start at the diagonal D is equal to W , by a connectedness argument.Indeed, such set is trivially closed, and it is also open due to the regularity of all the level curves of Per. is the open set defined by (3.5), and let Σ τ = Σ τ (r 1 , r 2 ) denote the minimal surface in R 3 of Definition 5.9.In this way, Σ τ is parametrized as map ψ(u, v) : [−τ, τ ] × R → R 3 , and the curve ψ(0, v) is a horizontal planar geodesic contained in the x 3 = 0-plane.
The next proposition follows from our analysis in Section 5, and collects some of the most important properties of Σ τ proved there: Proposition 7.1.Σ τ is an immersed minimal strip in R 3 that satisfies the following properties: (1) The curvature lines v → ψ(•, v) of Σ τ are spherical curvature lines.
Assume also that Per(r 1 , r 2 ) = m n ∈ Q, where m, n have no common divisors.Then: i) The quotient of Σ τ by the conformal projection (5.18) defines a compact minimal annulus Σ * τ in R 3 , with all the above properties.ii) The Gauss map of Σ * τ defines a regular m-fold covering of the great circle S 2 ∩ {x 3 = 0} along the horizontal planar geodesic ψ(0, v) of Σ * τ .
iii) Σ * τ has a prismatic symmetry group D n × Z 2 of order 4n.
In this section we show that some of the compact minimal annuli Σ * τ of Proposition 7.1 are actually free boundary in the unit ball (after a homothety and a translation).For that, by item (4) of Proposition 7.1, we need to control the height c 3 (τ ) of the boundary curve ψ(τ, v) of ∂Σ τ .This will be done by studying the nodal set of the height map h(r 1 , r 2 ) that we introduce below.

The height map.
Let Ω 0 ⊂ R 2 be the open set defined by (3.5).By Proposition 5.8, the height c 3 (τ ) of the center of the sphere that contains the boundary curve ψ(τ, v) of Σ τ is given by (5.23).
In order to control the nodal set h −1 (0) ⊂ Ω 0 , we will study how h(r 1 , r 2 ) behaves when (r 1 , r 2 ) approaches the segment For this, we will use the analysis of the degenerate case of system (2.12) in Section 4.
These are the Weierstrass data of the universal covering of a catenoid with necksize r 2 and vertical axis, and so, the surfaces Σ τ = Σ τ (r 1 , r 2 ) converge uniformly on compact sets to some translation of it.
Unluckily, the catenoid is a very degenerate situation with respect to our study.Indeed, on the catenoid, the fundamental equation (2.1), and in particular the pair (α(u), β(u)), does not carry meaningful information on it, due to the fact that the metric e 2ω only depends on u.More specifically, for any horizontal curvature line γ = ψ(u 0 , v) of any catenoid Σ c , there exists a sphere S(p, R) with center in the axis of Σ c that intersects Σ c orthogonally along γ.This means in our situation that we cannot read the limit values of the height map h(r 1 , r 2 ) directly from the geometry of the limit catenoid, and we need to follow a more indirect argument.
Coming now back to (5.23), we have for its last term So, by (7.3) and the first equality in (5.28), we deduce that h(r 1 , r 2 ) in (7.1) satisfies This proves in particular that h(r 1 , r 2 ) extends continuously to the segment L 0 .We now prove properties (1) and (2) in the statement of the theorem.
To start, let us assume that r ∈ (− 3 √ 2, r ], where r is the value in Theorem 4.2.We prove next that h(r, r) = 0 in that interval.
By item (ii) of Theorem 4.2, we have G(r) = −1, where However, one can check that G(x) does not take the value −1 in the interval [− 3 √ 2, r ]; indeed, the only solution to G(x) = −1 happens near the value r ≈ −1.0584, outside the interval.This contradiction shows that h(r, r) = 0 in (− 3 √ 2, r ].As a matter of fact, we easily show that h(r, r) < 0 there, since by Theorem 4.2 we can compute explicitly We now can make a similar argument on the interval [r , −1].This time, if h(r, r) = 0 for some r ∈ [r , −1], we can use item (i) of Theorem 4.2 to obtain that H(r) = −1, where Thus, there exists a unique value We also have h(r, r) > 0 for r ∈ (r * , 0) by direct evaluation at r = −1.This proves items (1) and ( 2) of the theorem.Item (3) is a direct consequence of items ( 1), ( 2) and the analyticity of h(r 1 , r 2 ) in Ω 0 .
To start, let us recall that, by Theorem 6.3, the level sets of Per(r 1 , r 2 ) are regular, connected, real analytic curves that start from the diagonal r 1 = r 2 and intersect at most once every line of the form r 1 − r 2 = const, since Per(r 1 , r 2 ) is strictly increasing along any such line.

Proof of Theorem 1.1:
Let γ * ⊂ Ω 0 be the real analytic curve constructed in Theorem 7.4.For each (r 1 , r 2 ) ∈ γ * such that Per(r 1 , r 2 ) ∈ Q, we can consider the compact minimal annulus Σ * τ constructed in Proposition 7.1.Note that Σ * τ has a prismatic symmetry group, as detailed in item (iii) of Proposition 7.1 or in Corollary 5.7.Now, the curve γ * lies in the nodal set h −1 (0) of the height map h(r 1 , r 2 ).Thus, by item (4) of Proposition 7.1, a homothety and translation of Σ * τ defines a compact minimal annulus with free boundary in B 3 , and all the desired properties.

Examples, discussion and open problems.
The most interesting examples of free boundary minimal annuli of our family are those associated to periods m n ∈ (0, 1) where both m, n are as small as possible.Indeed, m gives the rotation index of the Gauss map along the orthogonal intersection of the minimal annulus with the plane x 3 = 0, while n gives the number of periods that are necessary for the annulus to close, and determines its symmetry group.See Corollary 5.7.
The value of the period for the free boundary minimal annuli Σ * τ in B 3 obtained in Theorem 1.1 cannot be equal to 1/2, or more generally, to 1/n, n ∈ N. We do not detail the complete argument.We merely indicate that if the period of Σ * τ was 1/n, then its Gauss map would be a diffeomorphism onto its image in S 2 (observe that it is a diffeomorphism along its central planar geodesic, by item (ii) of Proposition 7.1), and in these conditions, Σ * τ would be embedded (Σ * τ would actually be a radial graph, as explained in [31]).Since Σ * τ is symmetric with respect to the three coordinate planes of R 3 , this contradicts McGrath's theorem in [23].We note that the main theorem in [23] has been extended by  to the case of antipodal symmetry, and by Seo [28] to the case of two arbitrary planes of reflective symmetry.These results are based in part on a characterization by Fraser and Schoen [13] of the critical catenoid in terms of Steklov eigenvalues, and on the two-piece property of embedded free boundary minimal surfaces by Lima and Menezes [22].
We have not been able to find any free boundary minimal annulus in B 3 within our family with a period ≥ 2/3.When we approach the diagonal r 1 = r 2 along the curve Per −1 (2/3), the value h(r 1 , r 2 ) of the height map is very small, but always positive.In this sense, let us observe that if r * is the value at which γ * intersects r 1 = r 2 (see Theorem 7.4), then We believe that there should exist examples of free boundary minimal annuli in B 3 for all values of the period in the interval (1/2, Per(r * , r * )).
The example in Figures 1.1 and 1.2 corresponds to a period equal to 3/5.We next prove the existence of this example.By Theorem 7.4, we have h(r, r) < 0 at the point (r, r) of intersection of Per −1 (3/5) with r 1 = r 2 .Thus, by monotonicity of c 3 (u), we have at (r, r) that the value τ at which β(τ ) = 0 and the value u * at which c 3 (u * ) = 0 satisfy τ < u * .In this way, β(u * ) < 0 at (r, r), by our orbit analysis in Section 4.
Similarly, one can prove existence of free boundary minimal annuli in B 3 with other low periods, like 4/7, 5/8 or 5/9.
Note that there obviously exist different points in h −1 (0) whose associated periods are rational numbers with the same (maybe large) denominator, say n.This provides examples of noncongruent free boundary minimal annuli in B 3 with the same symmetry group (the prismatic group of order 4n).The authors thank Mario B. Schulz for this remark.This is related to a recent topological non-uniqueness theorem by Carlotto, Schulz and Wiygul [4], who constructed examples of embedded non-congruent free boundary minimal surfaces in B 3 with the same topology and symmetry group.
In any case, the classification of all free boundary minimal annuli in B 3 with spherical curvature lines is far from complete.For instance, there seem to exist such free boundary minimal annuli that are not symmetric with respect to a central planar geodesic, and so, their isometry group would not be prismatic (it would be isomorphic to D n ).
The critical catenoid is stable (since it is a radial graph), and is characterized as the unique free boundary minimal annulus in B 3 with index 4 (as a free boundary minimal surface in B 3 ), see [8,30,32].It is an interesting problem to study the related stability properties of the minimal annuli in B 3 constructed in Theorem 1.1.
Remark 7.5.The minimal annuli of Theorem 1.1 are, to the authors' knowledge, the first examples of non-embedded free boundary minimal surfaces in B 3 (excluding trivial coverings of embedded ones).They are also the first non-trivial examples of free boundary minimal surfaces in B 3 whose first Steklov eigenvalue σ 1 is known to be smaller than 1; see [13,21] for a discussion on the meaning and importance of this eigenvalue condition.We remark that Fraser and Schoen proved that σ 1 ≤ 1, and that if σ 1 = 1 holds for an immersed free boundary minimal annulus in B 3 , then this annulus is the critical catenoid; thus, σ 1 < 1 holds for our minimal annuli here.
A challenging but very interesting open problem is to classify all the free boundary minimal annuli in B 3 .A more specific problem is the following intriguing question: Open problem: Are there free boundary minimal Möbius bands in B 3 ?By topological reasons, such examples will never be embedded.We note that Fraser and Schoen found in [13] an embedded minimal Möbius band with free boundary in the unit ball B 4 of R 4 , the so-called critical Möbius band; see also Fraser and Sargent [11].(1) A n (µ) intersects ∂B 3 at a constant angle along ∂A n (µ) ⊂ ∂B 3 , i.e., A n (µ) is a capillary minimal annulus in B 3 .(2) Each A n (µ) is foliated by spherical curvature lines.
Proof.By Theorem 6.3, the level sets of Per are regular, connected, real analytic curves, and for any n ∈ N the level curve Per −1 (1/n) intersects the diagonal r 1 = r 2 at the point (r n , rn ), By Corollary 5.7, every (r 1 , r 2 ) ∈ Per −1 (1/n) together with the choice of initial condition g 0 = 1 for the Gauss map g determines a minimal annulus Σ * n = Σ * n (r 1 , r 2 ) in R 3 foliated by spherical curvature lines, with the symmetries specified in item (3) above, and so, in particular, symmetric with respect to the x 3 = 0 plane.The height with respect to x 3 = 0 of the centers c(u) of the spheres where the spherical curves ψ(u 0 , v) of Σ * n lie decreases from ∞ to −∞ as we move u 0 from 0 to the half-period ω 1 ; see Proposition 5.12.Thus, there exists some intermediate value u * n ∈ (0, ω 1 ) such c 3 (u * n ) = 0, and therefore the restriction of Σ * n to [−u * n , u * n ] × R produces (via the quotient (5.18)) a compact minimal annulus A n in a ball B of R 3 , with center at some point of the plane x 3 = 0.
Consider next the planar geodesic Σ * n ∩ {x 3 = 0} of this example.It is a closed, locally convex planar curve that has rotation index equal to 1, since the period is 1/n.Thus, it is globally convex, and the Gauss map along it defines a diffeomorphism onto the horizontal great circle S 2 ∩{x 3 = 0}.We note, however, that the larger, non-compact annulus Σ * n loses its embeddedness as we approach u → ω 1 , due to the apparition of a flat end at (ω 1 , 0) (note that φ has a pole at ω 1 ).We want to understand next the embeddedness of the compact piece A n of Σ * n when (r 1 , r 2 ) is close to the diagonal.
As (r 1 , r 2 ) → (r n , rn ) along Per −1 (1/n), the meromorphic functions φ in (5.6) that define the minimal annuli Σ * n converge to the constant rn , see the beginning of the proof of Theorem 7.4.Thus, taking into account (7.2), the annuli Σ * n converge uniformly on compact sets to a subset of the minimal annulus with Weierstrass data on the conformal cylinder C/(2nω 2 Z) given by (8.1) where ω 2 is as in (7.2), i.e. ω 2 = −iπr n /n.These are the Weierstrass data of a (singly-covered, embedded) catenoid C n with vertical axis and necksize r2 n .Assume for one moment that the limit of the compact annuli A n = A n (r 1 , r 2 ) is a compact set of this limit catenoid C n , as (r 1 , r 2 ) → (r n , rn ).Then, for (r 1 , r 2 ) ∈ Per −1 (1/n) close enough to (r n , rn ), all the annuli A n are embedded.Thus, they are compact embedded minimal annuli foliated by spherical curvature lines, whose boundary lies in a ball of R 3 .By Theorem 7.1, they have a prismatic symmetry group D n × Z 2 , see Corollary 5.7.
Therefore, after a homothety and a translation, we would obtain a real analytic family {A n (µ)}, µ ≥ 0, of minimal annuli in B 3 with all the properties stated in Theorem 8.1.Here the parameter µ of this family is just a parametrization of the level curve Per −1 (1/n), so that µ = 0 corresponds to the point (r n , rn ).
Thus, it only remains to show that the limit of the annuli A n is a compact set of C n .For that, it suffices to show that the values u * n ∈ (0, ω 1 ) that define A n as a subset of Σ * n are bounded in (0, ∞); here, one should recall that ω 1 → ∞ as (r 1 , r 2 ) → (r n , rn ), see (7.2).
By the monotonicity of c 3 (u), see Proposition 5.12, we have that c 3 ( k ) > 0 for every k, since k < u * n (r k 1 , r k 2 ).Thus, by (8.2), h(ū) ≥ 0, a contradiction.This contradiction shows that the u * n must be bounded.As discussed previously, this finishes the proof of Theorem 8.1.

8.2.
Discussion and open problems for the capillary case.While the geometry of free boundary minimal surfaces in B 3 has received a great number of contributions in the past decade, the more general situation of capillary minimal surfaces in B 3 has remained, in contrast, largely unexplored.The examples that we construct in Theorem 8.1 seem to indicate that there should exist an interesting bifurcation theory for capillary minimal surfaces in B 3 , analogous to the CMC case.For instance, we expect the following behavior regarding the minimal annuli of Theorem 8.1.
Conjecture.The capillary minimal annuli A n (µ) in B 3 constructed in Theorem 8.1 are embedded for every n > 1 and every µ ≥ 0.Moreover, as µ → ∞, each family A n (µ) converges to a necklace of n flat vertical disks in B 3 whose projection is a regular n-polygon inscribed in the unit circle of the x 3 = 0 plane.
In the n = 2 case, the above conjecture claims that the minimal annuli A 2 (µ) are always embedded and converge as µ → ∞ to a doubly covered vertical equator of B 3 .In particular, Each of the minimal annuli A n (µ) has at least three planes of reflective symmetry.Thus, McGrath's characterization [23] of the critical catenoid in the free boundary case does not extend to the general capillary situation.Motivated by [4], it is natural to ask if any embedded capillary minimal annulus in B 3 with prismatic symmetry group of order 4n is one of the annuli A n (µ).
In this line, J. Choe proposed to the authors the following conjecture: any embedded capillary minimal annulus in B 3 that is symmetric with respect to the three coordinate planes of R 3 is either a compact piece of a catenoid, or one of the minimal annuli A 2n (µ).

Uniqueness: proof of Theorem 1.2
Let Σ be a compact immersed minimal annulus in R 3 whose boundary ∂Σ is composed of two closed curvature lines of Σ.It is then well known that Σ has no umbilic points, and that there exists a foliation F of Σ by regular curvature lines, so that both connected components of ∂Σ are elements of F. We say that Σ is foliated by spherical curvature lines if each curve of the foliation F lies in some sphere of R 3 .
We will prove in Theorem 9.1 below that any free boundary minimal annulus Σ embedded in B 3 and foliated by spherical curvature lines has two planes of symmetry.Therefore, by Seo's theorem [28], Σ is the critical catenoid, and this proves Theorem 1.2.
Note that in Theorem 9.1 we do not really assume a free boundary condition.Theorem 9.1.Let Σ be a minimal annulus embedded in the unit ball B 3 , with ∂Σ ⊂ S 2 and foliated by spherical curvature lines.Then, Σ is symmetric with respect to two planes of R 3 .Proof.To start, let us consider the case that Σ is foliated by planar curvature lines.In that situation, each of the boundary curves of Σ is a planar curve contained in the unit sphere, i.e., it is a circle, and Σ intersects S 2 with constant angle along it.So, Σ is a compact piece of a catenoid, by uniqueness of the solution to Björling's problem.The result is trivial in this case.
So, from now on, we assume that Σ is not foliated by planar curvature lines.Therefore, by our discussion in Section 2, and up to an ambient isometry and a homothety, we can represent Σ (which after these normalizations is now is embedded in some unspecified ball of R 3 ) as a subset of a complete minimal surface Σ parametrized as in (2.17) in terms of Weierstrass data given by (9.1) Here, g 0 > 0, b ∈ R satisfies the cubic equation (2.15) and ℘(z) is a (possibly degenerate) Weierstrass P-function that satisifies the ODE (9.2) with respect to some g 2 , g 3 ∈ R.
More specifically, if Λ is the set of poles of ℘(z) in C, then Σ is given by a conformal parametrization ψ(u, v) : C \ Λ → R 3 as in (2.2) so that the v-curves are spherical curvature lines, contained in spheres S(c(u), R(u)).We allow the possibility that R(u 0 ) = ∞ for some u 0 , which corresponds to the situation in which ψ(u 0 , v) is a planar curvature line.All the centers c(u) lie in a common vertical line of R 3 .The minimal annulus Σ is then obtained as the quotient of an adequate closed subset of Σ by the deck transformation (u, v) → (u, v + T ) for some T > 0. In particular, ψ(u, v) is T -periodic with respect to v.
So, we view the universal cover of Σ as parametrized by ψ(u, v) : [c, d] × R → R 3 , with ψ(u, v + 2T ) = ψ(u, v) for some T > 0. By the compactness of Σ, there are no poles of φ (i.e., no points of Λ) in the vertical strip [c, d] × iR, since any such point would generate a flat end for Σ.
We divide the proof into steps.
To see this, we first note that by (2.3), the planar curvature lines of Σ correspond to the values u where α(u) = 0. Assume that α(u) = 0 for every u ∈ [c, d].The height of the centers c 3 (u) of the curvature lines ψ(u, v) satisfy (5.29) and (5.30).Thus, c 3 (u) is monotonic and finite when u ∈ [c, d].In particular, Σ cannot lie in the unit ball unless at least one of its two boundary curves is contained in two spheres of different centers along the x 3 -axis.This is only possible in the present situation if that intersection is a circle, and so Σ is a compact piece of a catenoid, again the by uniqueness of the Björling problem.Since our assumption was that Σ is not foliated by planar curvature lines, we reach a contradiction.This proves Step 1.
Step 2: The modular discriminant ∆ mod := g 3 2 − 27g 2 3 cannot be negative.Up to a horizontal translation in the (u, v)-plane, we may assume that the planar curvature line of Σ whose existence was shown in Step 1 is placed along the iR axis, i.e., ψ(0, v) is such a curvature line.After this translation, Σ is parametrized by the Weierstrass data for some a ∈ R. Now, since Σ intersects the x 3 = 0 plane at a constant angle along ψ(0, v), it follows that |g(iv)| ≡ g 0 , constant along iR.This implies by g/g = φ and b ∈ R that φ(iv) ∈ R for every v ∈ R. Therefore, ℘(a + iv) ∈ R for every v ∈ R.
But now we recall that, as explained in Section 2.2, the condition ∆ mod < 0 implies that Λ is a rhombic lattice.In such a lattice, it is well known that ℘(z) only takes real values along the horizontal and vertical lines in C that pass through points of Λ.Therefore, the line a+iR intersects the lattice Λ.So, by (9.3), φ has poles along the iR axis.Since at those points Σ would have flat ends, this contradicts the compactness of Σ.This rules out the case ∆ mod < 0.
Step 3: ∆ mod cannot be zero.
If g 2 = g 3 = 0, then ℘(z) = 1/z 2 , which only takes real vales along R and iR.Thus, the argument of Step 2 applies.If g 3 2 = 27g 2 3 = 0, the function ℘(z) is singly periodic, with either a real or a purely imaginary period.Since ψ(u, v) is T -periodic in the v-direction, we obtain from the form of the Weierstrass data (9.1) that this period is purely imaginary, i.e., ℘(z + ω 0 ) = ℘(z) for some ω 0 ∈ iR.The degenerate Weiestrass function ℘(z) is explicitly given in these conditions by (see [5], p. 46) It follows from this expression that, again, ℘(z) cannot be simultaneously real and finite along a vertical line, and the argument of Step 2 applies.This shows that ∆ mod = 0 is impossible for Σ.
In this situation, it is well known that ℘(z) is real along a vertical line a + iR if and only if a ∈ R is a multiple of the real half-period ω 1 , i.e. a is of the form jω 1 for some j ∈ Z.If j is even, the line a + iR intersects the lattice Λ, and this contradicts the compactness of Σ as in Step 2. So, j must be odd, and by periodicity we can assume j = 1, i. Consider next the planar curvature line γ(v) := ψ(0, v), which is contained in the x 3 = 0 plane.Since the principal curvatures of Σ along the v-curves are κ 2 = e −2ω > 0, see Section 2, we see that γ(v) is a locally convex planar curve.Since Σ is embedded, γ(v) is thus globally convex.Since |g(iv)| ≡ g 0 is constant, this global convexity implies that the Gauss map of Σ along γ(v) lies in a circle, and has rotation index equal to 1.In other words, the map g(iv) : [0, 2nIm(ω 2 )) → C is a regular injective parametrization of the circle |z| = g 0 in C. Now, since φ(z) is 2ω 2 -periodic, it follows from (9.4) that, in order to have g(2nω 2 ) = g(0), by the rotation index discussion above, it must happen that g(2ω 2 ) = g(0)e 2πi/n .Assume for one moment that n > 1 holds.In that situation, by the form of the Weierstrass data (9.4), the annulus Σ is invariant by the rotations around the x 3 -axis of angles 2kπ/n, with k ∈ {1, . . ., n − 1}.Also, since both φ, g are real along the real axis, it also follows from the Weierstrass representation that Σ is invariant by reflection with respect to the plane x 2 = 0. Thus, Σ is invariant by at least two reflections with respect to vertical planes of R 3 containing the x 3 -axis.This would prove Theorem 9.1.
We next prove that n > 1.For this, we will consider below two separate cases.
In order to see that n > 1, we consider the function Figures 1.1 and 1.2.

Figure 1 .
Figure 1.1.Two views from above of a free boundary minimal annulus immersed in B 3 .The example is symmetric with respect to the x 2 = 0 and x 3 = 0 planes, and also with respect to the rotations with angles 6πk/5, k ∈ {1, . . ., 5}, around the x 3 -axis.Its symmetry group is isomorphic to the dihedral group D 10 .The Gauss map along its central planar geodesic is a 3-folded covering map of the great circle S 2 ∩ {x 3 = 0}.See Figure 1.2 for two more views of the annulus.

Figure 1 . 2 .
Figure 1.2.Left: an embedded fundamental piece of the free boundary minimal annulus in Figure 1.1.It has two symmetry planes, and the whole annulus is obtained by the union of this piece and its rotations around the x 3 -axis of angles 6πk/5, k = 1, . . ., 4. Right: union of the fundamental piece and its rotation of angle 6π/5.

Figure 1 . 3 .
Figure 1.3.Two examples of embedded, capillary minimal annuli in the unit ball B 3 .They do not have free boundary but intersect ∂B 3 with a constant angle.

Figure 4 .
Figure 4.1.Trajectories of (2.12) in the (s, t)-plane when r 1 = r 2 .If they start initially above the diagonal s + t = 0 they eventually cross it, maybe after bouncing at the right wall.

Theorem 6 . 3 .
For any c ∈ (0, 1), the level set Per −1 (c) is a connected, regular, real analytic curve in W that meets D orthogonally at the point (r c , r c ), where r c = 3 1 − 1/c 2 .

8 .
Capillary minimal annuli embedded in the unit ball 8.1.Embedded minimal annuli in B 3 .The family of minimal annuli foliated by spherical curvature lines that we have constructed also has the remarkable property that it produces compact embedded minimal annuli in the unit ball, like the ones in Figure1.3.Theorem 8.1.For any n ∈ N, n > 1, there exists a real analytic family {A n (µ) : µ ≥ 0} of compact minimal annuli in the unit ball B 3 such that:

Now, we come
back to the values u * n that we wanted to show are bounded as (r 1 , r 2 ) → (r n , rn ).By their own definition, we have c 3 (u * n ) = 0.If the u * n were not bounded, we could consider a sequence {(r k 1 , r k 2 )} k∈N → (r n , rn ) inside the level curve Per −1 (1/n) and values k ∈ (0, u * n ), where here u