Constant mean curvature spheres in homogeneous three-manifolds

We prove that two spheres of the same constant mean curvature in an arbitrary homogeneous three-manifold only differ by an ambient isometry, and we determine the values of the mean curvature for which such spheres exist. This gives a complete classification of immersed constant mean curvature spheres in three-dimensional homogeneous manifolds.


Introduction
In this manuscript we solve the fundamental problem of classifying constant mean curvature spheres in an arbitrary homogeneous three-manifold, where by a sphere, we mean a closed immersed surface of genus zero: Theorem 1.1. Let M be a Riemannian homogeneous three-manifold, X denote its Riemannian universal cover, and let Ch(X) denote the Cheeger constant of X. Then, any two spheres in M of the same constant mean curvature differ by an isometry of M . Moreover: (1) If X is not diffeomorphic to R 3 , then, for every H ∈ R, there exists a sphere of constant mean curvature H in M .
(2) If X is diffeomorphic to R 3 , then the values H ∈ R for which there exists a sphere of constant mean curvature H in M are exactly those with |H| > Ch(X)/2.
Our study of constant mean curvature spheres in homogeneous three-manifolds provides a natural parameterization of their moduli space and fundamental information about their geometry and symmetry, as explained in the next theorem: Theorem 1.2. Let M be a Riemannian homogeneous three-manifold, X denote its Riemannian universal cover and let S H be a sphere of constant mean curvature H ∈ R in M . Then: (1) S H is maximally symmetric; that is, there exists a point p ∈ M , which we call the center of symmetry of S H , with the property that any isometry Φ of M that fixes p also satisfies Φ(S H ) = S H . In particular, constant mean curvature spheres are totally umbilical if M has constant sectional curvature, and are spheres of revolution if M is rotationally symmetric.
(2) If H = 0 and X is a Riemannian product S 2 (κ) × R, where S 2 (κ) is a sphere of constant curvature κ > 0, then it is well known that S H is totally geodesic, stable and has nullity 1 for its stability operator. Otherwise, S H has index 1 and nullity 3 for its stability operator and the immersion of S H into M extends as the boundary of an isometric immersion F : B → M of a Riemannian 3-ball B which is mean convex 1 .
Moreover, if M M (p) denotes the space of spheres of nonnegative constant mean curvature in M that have a base point p ∈ M as a center of symmetry, then the map S H ∈ M M (p) → H ∈ R that assigns to each sphere S H its mean curvature is a homeomorphism between: (i) M M (p) and [0, ∞) if X is not diffeomorphic to R 3 , or (ii) M M (p) and (Ch(X)/2, ∞) if X is diffeomorphic to R 3 . This homeomorphism is real analytic except at H = 0 when X = S 2 (κ) × R.
The classification of constant mean curvature spheres is an old problem. In the 19th century, Jellet [16] proved that any constant mean curvature sphere in R 3 that is star-shaped with respect to some point is a round sphere. In 1951, Hopf [14] introduced a holomorphic quadratic differential for any constant mean curvature surface in R 3 and then used it to prove that constant mean curvature spheres in R 3 are round. His proof also worked in the other simply-connected threedimensional spaces of constant sectional curvature, which shows that these spheres are, again, totally umbilical (hence, they are the boundary spheres of geodesic balls of the space); see e.g. [6,7].
In 2004, Abresch and Rosenberg [1,2] proved that any constant mean curvature sphere in the product spaces H 2 (κ) × R and S 2 (κ) × R, where S 2 (κ) (resp. H 2 (κ)) denotes the two-dimensional sphere (resp. the hyperbolic plane) of constant Gaussian curvature κ = 0, and more generally, in any simply connected Riemannian homogeneous three-manifold with an isometry group of dimension four, is a rotational sphere. Their theorem settled an old problem posed by Hsiang-Hsiang [15], and reduced the classification of constant mean curvature spheres in these homogeneous spaces to an ODE analysis. For their proof, Abresch and Rosenberg introduced a perturbed Hopf differential, which turned out to be holomorphic for constant mean curvature surfaces in these spaces. The Abresch-Rosenberg theorem was the starting point for the development of the theory of constant mean curvature surfaces in rotationally symmetric homogeneous three-manifolds; see e.g., [8,10] for a survey on the beginnings of this theory.
It is important to observe here that a generic homogeneous three-manifold M has an isometry group of dimension three, and that the techniques used by Abresch and Rosenberg in the rotationally symmetric case do not work to classify constant mean curvature spheres in such an M .
In 2013, Daniel and Mira [9] introduced a new method for studying constant mean curvature spheres in the homogeneous manifold Sol 3 with its usual Thurston geometry. Using this method, they classified constant mean curvature spheres in Sol 3 for values H of the mean curvature greater than 1 √ 3 , and reduced the general classification problem to the obtention of area estimates for the family of spheres of constant mean curvature greater than any given positive number. These crucial area estimates were subsequently proved by Meeks [22], which completed the classification of constant mean curvature spheres in Sol 3 : For any H > 0 there is a unique constant mean curvature sphere S H in Sol 3 with mean curvature H; moreover, S H is maximally symmetric, embedded, and has index one [9,22].
In [25] we extended the Daniel-Mira theory in [9] to arbitrary simply connected homogeneous three-manifolds X that cannot be expressed as a Riemannian product of a two-sphere of constant curvature with a line. However, the area estimates problem in this general setting cannot be solved following Meeks' approach in Sol 3 , since the proof by Meeks uses in an essential way special properties of Sol 3 that are not shared by a general X.
In order to solve the area estimates problem in any homogeneous three-manifold, the authors have developed in previous works an extensive theoretical background for the study of constant mean curvature surfaces in homogeneous three-manifolds, see [27,25,26,24,23]. Specifically, in [27] there is a detailed presentation of the geometry of metric Lie groups, i.e., simply connected homogeneous three-manifolds given by a Lie group endowed with a left invariant metric. In [25,27] we described the basic theory of constant mean curvature surfaces in metric Lie groups. Of special importance for our study here is [25], where we extended the Daniel-Mira theory [9] to arbitrary metric Lie groups X, and we proved Theorems 1.1 and 1.2 in the case that M is a homogeneous three-sphere. In [23] we studied the geometry of spheres in metric Lie groups whose left invariant Gauss maps are diffeomorphisms. In [26] we proved that any metric Lie group X diffeomorphic to R 3 admits a foliation by leaves of constant mean curvature of value Ch(X)/2. In [24] we established uniform radius estimates for certain stable minimal surfaces in metric Lie groups. All these works are used in the present manuscript to solve the area estimates problem.
We note that the uniqueness statement in Theorem 1.1 does not follow from the recent Gálvez-Mira uniqueness theory for immersed spheres in three-manifolds in [11]. Indeed, their results imply in our context (see Step 2 in the proof of Theorem 4.1 in [25]) that two spheres Σ 1 , Σ 2 of the same constant mean curvature in a metric Lie group X are congruent provided that the left invariant Gauss map of one of them is a diffeomorphism (see Definition 3.3 for the notion of left invariant Gauss map); Theorem 1.1 does not assume this additional hypothesis.
The main results of this paper, Theorems 1.1 and 1.2, can be reduced to demonstrating Theorem 1.4 below; this reduction is explained in Section 3. To state Theorem 1.4, we need the notion of entire Killing graph given in the next definition, where X is a simply connected homogeneous three-manifold. Definition 1.3. Given a nowhere zero Killing field K on X, we say that an immersed surface Σ ⊂ X is a Killing graph with respect to K if whenever Σ intersects an integral curve of K, this intersection is transversal and consists of a single point. If additionally Σ intersects every integral curve of K, then we say that Σ is an entire Killing graph.
By classification, any homogeneous manifold X diffeomorphic to R 3 is isometric to a metric Lie group. In this way, any nonzero vector field K on X that is right invariant for such Lie group structure is a nowhere-zero Killing field on X. We note that any integral curve of K is a properly embedded curve in X diffeomorphic to R. Theorem 1.4. Let X be a homogeneous manifold diffeomorphic to R 3 , and let S n be a sequence of constant mean curvature spheres in X with Area(S n ) → ∞. Then, there exist isometries Φ n of X and compact domains Ω n ⊂ Φ n (S n ) with the property that a subsequence of the Ω n converges uniformly on compact subsets of X to an entire Killing graph with respect to some nowhere zero Killing vector field on X, which in fact is right invariant with respect to some Lie group structure on X.
2 Organization of the paper.
We next explain the organization of the paper and outline of the strategy of the proof of the main theorems. In our study in [25] of the space of constant mean curvature spheres of index one in metric Lie groups X, we proved that when X is diffeomorphic to S 3 , Theorems 1.1 and 1.2 hold for M = X. In that same paper we proved that if X is a metric Lie group not diffeomorphic to S 3 (and hence, diffeomorphic to R 3 ), then there exists a constant h 0 (X) ≥ 0 such that: (i) for any H > h 0 (X) there is an index-one sphere S H of constant mean curvature H in X; (ii) any sphere of constant mean curvature H > h 0 (X) is a left translation of S H , and (iii) if H n h 0 (X), then the areas of the spheres S Hn diverge to ∞ as n → ∞; furthermore, the spheres S n bound isometrically immersed mean convex Riemannian three-balls f n : B n → X.
In Section 3 we explain how Theorems 1.1 and 1.2 can be deduced from Theorem 1.4 and from the result stated in the last paragraph. Section 4 is an introductory section; in it we prove some basic properties of constant mean curvature surfaces f : Σ X in a metric Lie group X that are invariant, i.e., such that f (Σ) is everywhere tangent to a nonzero right invariant Killing vector field K Σ on X (equivalently, f (Σ) is invariant under the flow of K Σ ), and we explain in more detail the geometry of metric Lie groups diffeomorphic to R 3 . For this analysis, we divide these metric Lie groups into two categories: metric semidirect products and those of the form ( SL(2, R), , ), where SL(2, R) is the universal cover of the special linear group SL(2, R) and , is a left invariant metric; here the metric semidirect products X under consideration are the semidirect product of a normal subgroup R 2 with R and we refer to the cosets in X of R 2 as being horizontal planes.
In Section 5 we prove that if {S n } n is a sequence of index-one spheres with constant mean curvatures of values H n h 0 (X) in a metric Lie group X diffeomorphic to R 3 , then, after taking limits of an appropriately chosen subsequence of left translations of these spheres, there exists an invariant surface f : Σ X of constant mean curvature h 0 (X) that is a limit of compact domains of the spheres S n . Moreover, f is complete, stable, Σ has the topology of a plane or a cylinder, and if f (Σ) is not a coset of a two-dimensional subgroup of X, then the closure γ f of its left invariant Gauss map image γ f in S 2 (see Definition 3.3 for this notion) has the structure of a one-dimensional lamination. Another key property proved in Section 5 is that whenever f (Σ) is tangent to some coset of a two-dimensional subgroup of X, then f (Σ) lies in one of the two closed half-spaces bounded by this coset; see Corollary 5.7. In Section 6 we prove in Theorem 6.1 that the invariant limit surface f (Σ) can be chosen so that its left invariant Gauss map image γ f ⊂ S 2 is a point or a closed embedded regular curve.
In the case γ f is a point, Theorem 1.4 follows easily from the fact that f (Σ) is a coset of a twodimensional subgroup of X. In the case that γ f is a closed curve, one can choose f so that one of the following exclusive possibilities occurs: 1. Σ is diffeomorphic to an annulus.
2. Σ is diffeomorphic to a plane, and f (Σ) is an immersed annulus in X.
3. Σ is diffeomorphic to a plane and there exists an element a ∈ X such that the left translation by a in X leaves f (Σ) invariant, but this left translation does not lie in the 1-parameter subgroup of isometries generated by the nonzero right invariant Killing vector field K Σ that is everywhere tangent to f (Σ).
In Section 7 we prove Theorem 1.4 in the case that X is a metric semidirect product. For that, it suffices to prove that f (Σ) is an entire Killing graph for some right invariant Killing vector field V in X. This is proved as follows. First, we show that if case 3 above holds, then the Killing field K Σ which is everywhere tangent to f (Σ) is horizontal with respect to the semidirect product structure of X, and we prove that f (Σ) is an entire graph with respect to any other horizontal Killing field V linearly independent from K Σ . So, it suffices to rule out cases 1 and 2 above. We prove that case 2 is impossible by constructing in that situation geodesic balls of a certain fixed radius R * > 0 in the abstract Riemannian three-balls B n that the Alexandrov-embedded index-one spheres S n bound, and whose volumes tend to infinity as n → ∞. This unbounded volume result eventually provides a contradiction with Bishop's theorem. Finally, we show that case 1 is impossible by constructing an abstract three-dimensional cylinder bounded by Σ that submerses isometrically into X with boundary f (Σ), and then proving that a certain CMC flux of Σ in this abstract cylinder is different from zero. This gives a contradiction with the homological invariance of the CMC flux and the fact that f (Σ) is a limit of the (homologically trivial) Alexandrov-embedded constant mean curvature spheres S n .
In Section 8 we prove Theorem 1.4 in the remaining case where X is isomorphic to SL(2, R). The arguments and the basic strategy of the proof in this situation follow closely those from the previous Section 7. However, several of these arguments are by necessity different from those in Section 7, as many geometric properties of metric semidirect products do not have analogous counterparts in SL(2, R).
The paper finishes with an Appendix that can be read independently of the rest of the manuscript. In it, we prove a general nonvanishing result for the CMC flux that is used in Sections 7 and 8 for ruling out case 1 above, as previously explained.
3 Reduction of Theorems 1.1 and 1.2 to Theorem 1.4 3

.1 Proof of Assertion 3.1
Let M = X denote a simply connected homogeneous three-manifold. It is well known that if X is not isometric to a Riemannian product space S 2 (κ) × R, where S 2 (κ) denotes a sphere of constant Gaussian curvature κ > 0, then X is diffeomorphic to R 3 or to S 3 .
When X is isometric to S 2 (κ) × R, the statements contained in Theorems 1.1 and 1.2 follow from the Abresch-Rosenberg theorem that states that constant mean curvature spheres in S 2 (κ)×R are rotational spheres, and from an ODE analysis of the profile curves of these spheres. More specifically, an explicit expression of the rotational H-spheres in S 2 (κ)×R can be found in Pedrosa and Ritoré [35,Lemma 1.3], or in [1]. From this expression it follows that if H = 0, then each of these rotational H-spheres is embedded, and bounds a unique compact subdomain in X which is a mean convex ball that is invariant under all the isometries of X that fix the mid point in the ball of the revolution axis of the sphere (this point can be defined as the center of symmetry of the sphere). If H = 0, the corresponding rotational sphere is a slice S 2 (κ) × {t 0 }, that is trivially stable, and hence has index zero and nullity one. Moreover, any point of S 2 (κ) × {t 0 } can be defined as the center of symmetry of the sphere in this case. That the rotational H-spheres in S 2 (κ) × R for H = 0 have index one and nullity three is shown in Souam [38, proof of Theorem 2.2]. Also, the rotational H-spheres for H > 0 with a fixed center of symmetry p 0 converge as H → 0 to a double cover of the slice S 2 (κ) × {t 0 } that contains p 0 . So, from all this information and the Abresch-Rosenberg theorem, it follows that Theorems 1.1 and 1.2 hold when X is isometric to S 2 (κ) × R.
When X is diffeomorphic to S 3 , the statements in Theorem 1.1 and 1.2 were proven in our previous work [25].
So, from now on in this Section 3.1, X will denote a homogeneous manifold diffeomorphic to R 3 . It is well known that X is isometric to a metric Lie group, i.e., a three-dimensional Lie group endowed with a left invariant Riemannian metric. In the sequel, X will be regarded as a metric Lie group, and e will denote its identity element. Given x ∈ X, we will denote by l x , r x : X → X the left and right translations by x, respectively given by l x (y) = xy, r x (y) = yx for all y ∈ X. Thus, l x is an isometry of X for every x ∈ X. Definition 3.3. Given an immersed oriented surface f : Σ X with unit normal vector field N : Σ → T X (here T X refers to the tangent bundle of X), we define the left invariant Gauss map of f to be the map G : Σ → S 2 ⊂ T e X that assigns to each p ∈ Σ, the unit tangent vector G(p) to X at the identity element e given by (dl f (p) ) e (G(p)) = N p .
The following proposition follows from a rearrangement of ideas taken from [25], but for the sake of clarity, we will include a proof here.
Proposition 3.4. Let X be any metric Lie group X diffeomorphic to R 3 . Then, there exists a number h 0 (X) ≥ 0 such that: (1) For every |H| > h 0 (X), there exists an H-sphere S H in X with index one and nullity three for its stability operator. Furthermore, S H is Alexandrov embedded, i.e., S H is the boundary of a (unique) mean convex immersed ball f n : B n → X in X.
(2) If Σ is an H-sphere in X for some |H| > h 0 (X), then Σ is a left translation of S H .
(3) The left invariant Gauss map G : S H → S 2 of S H is an orientation-preserving diffeomorphism.
(4) Each S H lies inside a real-analytic family {Σ H | H ∈ (H − ε, H + ε)} of index-one spheres in X for some ε > 0, where Σ H = S H and Σ H has constant mean curvature of value H . In particular, there exists a unique component C of the space of index-one spheres with constant mean curvature in X such that S H ∈ C for all H > h 0 (X).
, then the norm of the second fundamental form of S H is at most C. (7) If S n is a sequence of constant mean curvature spheres in X whose mean curvatures H n > h 0 (X) satisfy H n → h 0 (X), then Area(S n ) → ∞.
Proof. After a change of orientation if necessary, it clearly suffices to prove all statements for the case where H ≥ 0. By [25, item 3 of Theorem 4.1], there exists a unique component C of the space of index-one spheres with constant mean curvature in X such that the values of the mean curvatures of the spheres in C are not bounded from above. As X is diffeomorphic to R 3 , then [25, item 6 of Theorem 4.1] ensures that the map H : C → R that assigns to each sphere in C its mean curvature is not surjective. By item 5 in the same theorem, there exists h 0 (X) ≥ 0 such that for every H > h 0 (X), there is an index-one sphere S H ∈ C, and the following property holds: (7)' The areas of any sequence {S Hn } n ⊂ C with H → h 0 (X) satisfy lim n→∞ Area(S Hn ) = ∞.
Since the sphere S H has index one, then Cheng [5,Theorem 3.4] gives that the nullity of S H is three. Alexandrov embeddedness of S H follows from [25,Corollary 4.4]. Now item 1 of Proposition 3.4 is proved.
Once this item 1 has been proved, items 2, 3, 4, 5 of Proposition 3.4 are proved in items 1, 2, 3, 4 of [25,Theorem 4.1]. Item 6 of Proposition 3.4 is a direct consequence of the existence, uniqueness and analyticity properties of the spheres S H stated in items 1,2,4, and of the fact also proved in [25] that, for H large enough, the spheres S H bound small isoperimetric regions in X. Finally, item 7 of Proposition 3.4 follows from property (7)' above and from the uniqueness given by the already proven item 2 of Proposition 3.4. Now the proof is complete.
A key notion in what follows is the critical mean curvature of X.
Definition 3.5. Let X be a homogeneous manifold diffeomorphic to R 3 , let A(X) be the collection of all closed, orientable immersed surfaces in X, and given a surface Σ ∈ A(X), let |H Σ | : Σ → [0, ∞) stand for the absolute mean curvature function of Σ. The critical mean curvature of X is defined as Item 2 of Theorem 1.4 in [26] gives that Ch(X) = 2H(X). By definition of H(X), every compact H-surface in X satisfies |H| ≥ H(X). Observe that Proposition 3.4 gives that for every H > h 0 (X) there is an H-sphere S H in X. Thus, In order to prove Assertion 3.1, in the remainder of this section we will assume that Theorem 1.4 holds.
We will prove next that h 0 (X) = H(X). If S n are constant mean curvature spheres satisfying item 7 of Proposition 3.4, then by Theorem 1.4 we obtain the existence of a surface Σ 0 in X of constant mean curvature h 0 (X) that is an entire Killing graph with respect to some nonzero right invariant Killing field K. Let Γ = {Γ(t) | t ∈ R} be the 1-parameter subgroup of X given by Γ(0) = e, Γ (0) = K(e). As left translations are isometries of X, it follows that {l a (Σ 0 ) | a ∈ Γ} defines a foliation of X by congruent surfaces of constant mean curvature h 0 (X), and this foliation is topologically a product foliation. A standard application of the mean curvature comparison principle shows then that there are no closed surfaces Σ in X with max Σ |H| < h 0 (X) (otherwise, we left translate Σ 0 until Σ is contained in the region of X on the mean convex side of Σ 0 , and then start left translating Σ 0 towards its mean convex side until it reaches a first contact point with Σ; this provides a contradiction with the mean curvature comparison principle). Therefore, we conclude that h 0 (X) = H(X) and that, by item 1 of Proposition 3.4, the values H ∈ R for which there exists a sphere of constant mean curvature H in X are exactly those with |H| > H(X). This fact together with item 2 of Proposition 3.4 proves item 2 of Theorem 1.1, and thus it completes the proof of Theorem 1.1 in the case X is diffeomorphic to R 3 (assuming that Theorem 1.4 holds).
Regarding the proof of Theorem 1.2 when X is diffeomorphic to R 3 , similar arguments as the one in the preceding paragraph prove item 2 of Theorem 1.2. To conclude the proof of Theorem 1.2 (and hence of Assertion 3.1) we need to show that all spheres S H in X are maximally symmetric, something that we will prove next. We note that the 'Moreover' part in the statement of Theorem 1.2 follows directly from the existence of a center of symmetry of each sphere S H , and the analyticity properties in item 4 of Proposition 3.4.
In the case that the isometry group I(X) of X is of dimension 6, all constant mean curvature spheres in X are totally umbilical; in particular they are maximally symmetric, and the definition of the center of symmetry of the sphere is clear. So, in the remainder of this section we will let X be a metric Lie group diffeomorphic to R 3 whose isometry group I(X) has dimension 3 or 4. Let Stab e (X) denote the group of isometries ψ of X with ψ(e) = e, and let Stab + e (X) denote the index-two subgroup of orientation-preserving isometries in Stab e (X). The next proposition gives some basic properties of the elements of Stab e (X) that will be needed for proving the maximal symmetry of H-spheres in X. Its proof follows from the analysis of metric Lie groups in [27]. Proposition 3.6. There exists a 1-parameter subgroup Γ of X such that: (1) There exists an isometry φ ∈ Stab + e (X) of order two in X that is a group automorphism of X, and whose fixed point set is Γ, i.e., Γ = {x ∈ X : φ(x) = x}.
Moreover, if there exists another 1-parameter subgroup Γ = Γ of X that satisfies the previous two properties, then X has a three-dimensional isometry group, and φ( Γ) = Γ, where φ is given with respect to Γ by item (1) above.
Proof. First suppose that I(X) has dimension 4. By [27, item 2 of Proposition 2.21], there exists a unique principal Ricci eigendirection w ∈ T e X whose associated eigenvalue is simple, and Stab + e (X) contains an S 1 -subgroup A, all whose elements have differentials that fix w. Let Γ be the 1-parameter subgroup of X generated by w, and let φ be the orientation-preserving automorphism of X whose differential satisfies dφ e (w) = w, (dφ e )| w ⊥ = −1 w ⊥ (see [27, proof of Proposition 2.21] for the construction of φ). Clearly φ satisfies item 1 of Proposition 3.6. Regarding item 2, observe that Stab + e (X) = A ∪ B, where B is the set of π-rotations about the geodesics γ of X such that γ(0) = e and γ (0) is orthogonal to w. Therefore, every ψ ∈ Stab + e (X) satisfies ψ(Γ) = Γ. If Stab + e (X) = Stab e (X), then item 2 of Proposition 3.6 holds. Otherwise, Stab e (X)−Stab + e (X) = Ø and in this case, item 2 of Proposition 2.24 of [27] ensures that X is isomorphic and homothetic to H 2 ×R endowed with its standard product metric (hence w is a nonzero vertical vector and Γ is the vertical line passing through e), and every ψ ∈ Stab e (X) − Stab + e (X) is either a reflectional symmetry with respect to a vertical plane or it is the composition of a reflectional symmetry with respect to H 2 × {0} with a rotation about Γ. Thus, ψ leaves Γ invariant and item 2 of Proposition 3.6 holds. Note that the moreover part of the proposition cannot hold in this case of I(X) being four-dimensional, because given any 1-parameter subgroup of X different from Γ, there exists ψ ∈ B such that ψ(Γ) = Γ. Next suppose that I(X) has dimension 3. First suppose that the underlying Lie group structure Y of X is not unimodular 2 . Hence Y is isomorphic to some semidirect product R 2 A R for some matrix A ∈ M 2 (R) with trace(A) = 0 and det(A) = 0 since the dimension of I(X) = 3, see Section 4.2 for this notion of nonunimodular metric Lie group. Then [27, item 4 of Proposition 2.21] gives that Stab + e (X) = {1 X , φ} where φ(x, y, z) = (−x, −y, z). Therefore, item 1 of Proposition 3.6 holds with the choice Γ = {(0, 0, z) | z ∈ R} and the above φ. Regarding item 2, we divide the argument into two cases in this nonunimodular case for Y . Given b ∈ R − {0}, consider the matrix y, z) and ψ 2 (x, y, z) = (x, −y, z). In the case that b = 0, then det(A(b)) = 0, which is not the case presently under consideration as the isometry group of X for b = 0 has dimension four.
If dim(I(X)) = 3 and Y is unimodular, then there exists a frame of left invariant vector fields E 1 , E 2 , E 3 on Y that are eigenfields of the Ricci tensor of X, independently of the left invariant metric on X, see [27,Section 2.6]. By the proof of Proposition 2.21 of [27], for each i = 1, 2, 3 there exists an orientation-preserving automorphism φ i : X → X whose differential satisfies (φ i ) * (E i ) = E i and (φ i ) * (E j ) = −E j whenever j = i. Furthermore, φ is an isometry of every left invariant metric on Y (in particular, of the metric of X). Also, item 1 of Proposition 2.24 in [27] gives that Stab + e (X) is the dihedral group {1 X , φ 1 , φ 2 , φ 3 }. Therefore, item 1 of Proposition 3.6 holds with any of the choices φ = φ i and Γ = Γ i , for i = 1, 2, 3, where Γ i is the 1-parameter subgroup of X generated by (E i ) e . Regarding item 2, we again divide the argument into two cases in this unimodular case for Y .
(A1)' If X is not isomorphic and homothetic to R 2 A(−1) R (recall that R 2 A(−1) R is Sol 3 with its standard metric), then [27, item 3 of Proposition 2.24] ensures that Stab e (X) = Stab + e (X). Thus item 2 of Proposition 3.6 holds for every choice of the form Γ = Γ i and φ = φ i with i = 1, 2, 3. Once here, and since it is immediate that φ i (Γ j ) = Γ j for any i, j ∈ {1, 2, 3}, we conclude that the moreover part of Proposition 3.6 holds in this case.
We are now ready to prove the existence of the center of symmetry of any H-sphere in X. Let f H : S H X denote an H-sphere, with left invariant Gauss map G : S H → S 2 . By item 3 of Proposition 3.4 and h 0 (X) = H(X), we know G is an orientation preserving diffeomorphism. Without loss of generality we can assume that f H (q H ) = e, where q H ∈ S H is the unique point such that G(q H ) = v := Γ (0) and Γ : R → X is an arc length parameterization of the 1-parameter subgroup of X that appears in Proposition 3.6. Let φ ∈ Stab + e (X) be an isometry associated to Γ in the conditions of item 1 of Proposition 3.6. Clearly dφ are two H-spheres in X passing through e with the same left invariant Gauss map image at e. By the uniqueness of H-spheres up to left translations in item 2 of Proposition 3.4, we conclude (B1) Suppose that ψ preserves orientation on X and satisfies ψ( is an H-sphere in X. So, by items 2 and 3 of Proposition 3.4 we have l a ( f H (S H )) = (ψ • f H )(S H ) for some a ∈ X, and that l a (p H ) corresponds to the unique point of (ψ • f H )(S H ) where its Gauss map takes the value v. But now, from the second condition we obtain that ψ restricts to Γ as the identity map, which implies that the value of the left invariant Gauss map image of (ψ (B2) Suppose that ψ reverses orientation on X and satisfies ψ(p H ) = p * H , ψ(p * H ) = p H . The first condition implies that after changing the orientation of (ψ • f H )(S H ), this last surface is an H-sphere in X. Since by the second condition ψ restricts to Γ as minus the identity map, then the left invariant Gauss map image of (ψ • f H )(S H ) (with the reversed orientation) at p H is v. Now we deduce as in case (B1) that ψ leaves f H (S H ) invariant.
(B3) Suppose that ψ reverses orientation on X and satisfies ψ(p H ) = p H , ψ(p * H ) = p * H . As in case (B2), we change the orientation on (ψ • f H )(S H ) so it becomes an H-sphere in X. Item 2 of Proposition 3.4 then gives that there exists a ∈ X such that (ψ . This is only possible if a = a −1 , which, in our setting that X is diffeomorphic to R 3 , implies that a = e (note that this does not happen when X is diffeomorphic to the three-sphere, because then X is isomorphic to SU(2), which has one element of order 2). The claim then holds in this case.
(B4) Finally, assume that ψ preserves the orientation on X and satisfies ψ( Since ψ restricts to Γ as minus the identity map, then the value of the left invariant Gauss map image of (ψ H . Now we finish as in case (B3).
To sum up, we have proved that any ψ ∈ Stab e (X) leaves f H (S H ) invariant; this proves that the H-sphere f H (S H ) is maximally symmetric with respect to the identity element e, in the sense of item 1 of Theorem 1.2. So, we may define e to be the center of symmetry of f H (S H ). We next show that the above definition of center of symmetry does not depend on the choice of the 1-parameter subgroup Γ of X that appears in Proposition 3.6. Suppose Γ is another 1-parameter subgroup of X satisfying the conclusions of Proposition 3.6. As we explained in the paragraph just after Proposition 3.6, Γ determines two points q, q * ∈ S H with f H ( q), f H ( q * ) ∈ Γ, such that G( q) = Γ (0), G( q * ) = − Γ (0), where G is the left invariant Gauss map of f H . To show that the above definition of center of symmetry of f H (S H ) does not depend on Γ, we must prove that the mid point of the arc of Γ between f H ( q) an f H ( q * ) is e. To do this, let φ ∈ Stab + e (X) be an order-two isometry that satisfies item 1 of Proposition 3.6 with respect to Γ. By Proposition 3.6, φ( Γ) = Γ. As φ Since the set of fixed points of φ is Γ and f H ( q), f H ( q * ) / ∈ Γ (because two different 1-parameter subgroups in X only intersect at e, here we are using again that X is diffeomorphic to R 3 ), then φ( f H ( q)) = f H ( q * ). As φ is an isometry with φ(e) = e and φ( Γ) = Γ, we deduce that the lengths of the arcs of Γ that join e to f H ( q), and e to f H ( q * ), coincide. This implies that the mid point of the arc of Γ between f H ( q)) an f H ( q * ) is e, as desired. As every H-sphere in X is a left translation of f H (S H ), we conclude that all H-spheres in X are maximally symmetric with respect to some point p ∈ X, which is obtained by the corresponding left translation of the identity element e, and thus can be defined as the center of symmetry of the sphere. This completes the proof of Theorem 1.2 (assuming that Theorem 1.4 holds) for the case that X is diffeomorphic to R 3 . Thus, the proof of Assertion 3.1 is complete.
The next remark gives a direct definition of the center of symmetry of any H-sphere in X: Remark 3.7 (Definition of the center of symmetry). Let f : S H X be an H-sphere in a homogeneous manifold X diffeomorphic to R 3 with an isometry group of dimension three or four. Let q, q * ∈ S H be given by G(q) = v, G(q * ) = −v, where G : S H → S 2 denotes the left invariant Gauss map of S H and v := Γ (0), with Γ(t) a 1-parameter subgroup satisfying the conditions in Proposition 3.6. Then, by our previous discussion, the left coset α := l f (q) (Γ) passes through both f (q) and f (q * ), and we define the center of symmetry of S H as the midpoint p ∈ α of the subarc of α that joins f (q) with f (q * ).
It is worth mentioning that this definition gives a nonambiguous definition of the center of symmetry of an H-sphere; however, when X has an isometry group of dimension three and is nonunimodular, there are many points p ∈ X besides this center of symmetry such that the Hsphere is invariant under all isometries of X that fix p. Nevertheless, in order to make sense of the analyticity properties of the family of constant mean curvature spheres with a fixed center of symmetry (see Theorem 1.2), we need to make the definition of center of symmetry nonambiguous.

Proof of Assertion 3.2
Let M be a homogeneous three-manifold with universal covering space Π : X → M . To prove Assertion 3.2, we will assume in this Section 3.2 that Theorems 1.1 and 1.2 hold for X.
We first prove Theorem 1.1 in M . Since every constant mean curvature sphere f : S M is the projection via Π of some lift f : S X of f with the same mean curvature, then clearly the first and second items in Theorem 1.1 hold in M , since they are true by hypothesis in X. In order to prove the uniqueness statement in Theorem 1.1, let f 1 : S 1 M , f 2 : S 2 M be two spheres with the same constant mean curvature in M . For i = 1, 2, choose f i respective lifts of f i , let p i be the centers of symmetry of f i , denote p i = Π( p i ), and let I : M → M be an isometry with I(p 1 ) = p 2 . Let I : X → X be the lift of I that takes p 1 to p 2 . It follows that the H-spheres I • f 1 and f 2 have the same center of symmetry in X and so these immersions have the same images. In particular, it follows that I • f 1 and f 2 also have the same images, which completes the proof of Theorem 1.1 in M .
We next prove Theorem 1.2 in M . Suppose f : S M is an oriented H-sphere, and let f : S X denote some lift of f . Since the stability operators of f and f are the same, the index and nullity of f and f agree. Also, note that if the immersion f extends to an isometric immersion F : B → X of a mean convex Riemannian three-ball B into X, then Π • F is an isometric immersion of B into M that extends f . These two trivial observations prove that item 2 of Theorem 1.2 holds in M .
If M is covered by S 2 (κ) × R and f (S) is totally geodesic, then any point of the image sphere satisfies the property of being a center of symmetry of f . This observation follows by the classification of homogeneous three-manifolds covered by S 2 (κ) × R as being those spaces isometric to is a circle of circumference R for some R > 0 and P 2 (κ) is the projective plane of constant Gaussian curvature κ > 0.
Suppose now that M is not covered by S 2 (κ) × R with f (S) being totally geodesic. In order to complete the proof of item 1 of Theorem 1.2, we next define the center of symmetry of f in M as p := Π( p), where p is the center of symmetry of a lift f in X. Since the center of symmetry of the lift f is uniquely defined by f and any other lift is equal to the composition of f with an isometry σ : X → X that is a covering transformation, then by uniqueness of the center of symmetry in X, σ maps the center of symmetry of f to the center of symmetry of the composed oriented immersion σ • f ; hence p is independent of the choice of the lift of f to X. We now show that f (S) is invariant under all isometries of M that fix p. Let I : M → M be any such isometry, and let I : X → X be the lift of I such that I( p) = p. As p is a center of symmetry of f , then, I induces an isometry of f : S X. It follows that I induces an isometry of the H-sphere f , which proves item 1 of Theorem 1.2 in M . Once here, the homeomorphism and analyticity properties in the last statement of Theorem 1.2 follow trivially from the validity of the corresponding statement in X. This completes the proof of Theorem 1.2 in M , and thus also the proof of Assertion 3.2.

Constant mean curvature spheres in complete locally homogeneous threemanifolds
As a consequence of Theorems 1.1 and 1.2 and of the discussion in Sections 3.1 and 3.2 we can provide a description of the space of constant mean curvature spheres in complete, locally homogeneous three-manifolds. Specifically, let Y be a complete, connected, locally homogeneous Riemannian three-manifold. Then, the universal Riemannian covering space Π : X → Y is a simply connected homogeneous three-manifold. Let Ch(X) be the Cheeger constant of X. Remark 3.7, the proof of Proposition 3.6 and the results on the existence and uniqueness of the center of symmetry for any H-sphere in X imply that associated to any H-sphere S H in Y with H > 0, there is a unique point p(S H ) ∈ Y that is the image by Π of the center of symmetry of any lift of S H to X; that p(S H ) does not depend on the lift of S H can be proved following the arguments used in Section 3.2. The following properties of spheres with nonzero constant mean curvature in Y are then easy to check, and provide a parameterization of the space of such constant mean curvature spheres.
We observe that item 6 above is the natural generalization to complete locally homogeneous three-manifolds of the uniqueness result in Theorem 1.1 that any two spheres of the same constant mean curvature in a homogeneous three-manifold differ by an ambient isometry.

Background on constant mean curvature surface theory in metric Lie groups
In this section we will collect some basic material about metric Lie groups diffeomorphic to R 3 and immersed surfaces of constant mean curvature in these spaces, that will be needed in later sections. For the purposes of this paper, it is worth dividing metric Lie groups X diffeomorphic to R 3 into two classes depending on whether the underlying group structure of X is that of a semidirect product, or it is the one of SL(2, R), that can be naturally identified as the universal cover of the group PSL(2, R) of orientation-preserving isometries of the hyperbolic plane. We divide this section into three parts: in Section 4.1 we describe some geometric aspects of constant mean curvature surfaces in a metric Lie group X that are invariant under the flow of a nonzero right invariant vector field on X. Sections 4.2 and 4.3 describe the ambient geometry and particular aspects of surface theory in a metric Lie group isomorphic to a semidirect product or to SL(2, R), respectively. We will use the contents of Section 4.1 in Section 5, but Sections 4.2 and 4.3 will not be needed until Section 6. The basic references for this material are Milnor [33], the notes [27] and the paper [25].

Invariant constant mean curvature surfaces
Given a unit vector v ∈ T e X in the Lie algebra of X, we can extend v by left translation to a leftinvariant unit vector field V globally defined on X. By using right translations in X in a similar way, we can extend v to a right-invariant vector field K in X. The vector field K will not be, in general, of unit length anymore, but it is a nowhere-zero Killing field on X.
By [27,Corollary 3.17], left cosets of two-dimensional subgroups of X are characterized by their left invariant Gauss map being constant, and they all have constant mean curvature with absolute value at most H(X). As every right coset ∆x of a two-dimensional subgroup ∆ of X is the left coset of a conjugate subgroup of ∆ (namely ∆x = x∆ 1 where ∆ 1 = x −1 ∆x), then we deduce: (C) The left invariant Gauss map of a left or right coset of a two-dimensional subgroup of X is constant.
The next proposition is a reformulation of Corollary 3.8 in [25]. We remark that Corollary 3.8 in [25] was written in terms of the H-potential of X, a concept that we will not introduce here, but one can easily translate that formulation into the one below by using [27,Corollary 3.21].
Proposition 4.1. Let f : Σ X be a complete immersed H-surface in X and let G : Σ → S 2 ⊂ T e X be its left invariant Gauss map. Assume that there are no two-dimensional subgroups in X with mean curvature H whose (constant) left invariant Gauss map lies in the Gauss map image G(Σ). Then: (1) The differential dG of G has rank at most 1 everywhere on Σ if and only if f is invariant under the flow of a nonzero right invariant vector field K on X.
(2) If f is invariant under the flow of a nonzero right invariant vector field of X, then rank(dG) = 1 everywhere on Σ, and G(Σ) is a regular curve in S 2 .
Remark 4.2. If K is a nonzero right invariant vector field on a metric Lie group X diffeomorphic to R 3 , then each integral curve of K is diffeomorphic to R and the quotient space X/K of integral curves of K inherits a two-dimensional differentiable structure that makes the natural projection Therefore, when f : Σ X satisfies the hypothesis in item 2 of Proposition 4.1, we can consider f (Σ) as the surface in X obtained by pulling back via Π K Σ an immersed curve β contained in X/K Σ . Moreover, after identifying in the standard way X/K Σ with any entire Killing graph S 0 ⊂ X with respect to K Σ , the invariant surface f : Σ X can be parameterized locally (and even globally if Σ is simply connected) as where β(s) : I ⊂ R → X/K Σ ≡ S 0 is the immersed curve above, and Γ = Γ(t) is the 1-parameter subgroup of X given by Γ (0) = K Σ (e).
Let f : Σ X be an immersed H-surface in X whose left invariant Gauss map G : Σ → S 2 has rank one at every point. It is shown in [25, proof of Corollary 3.8] that around any z 0 ∈ Σ there exist conformal parameters (s, t) on Σ such that G does not depend on t, so we can write G = G(s). With respect to these coordinates, the Gauss map G(s) satisfies a second order autonomous ODE of the form where Ψ : S 2 × R 3 → R 3 is real analytic; this follows directly from the ODE (4.3) in [25]. The special form of this ODE has the following trivial consequence: is a solution of the ODE (4.3) in [25], then g(δ 1 s + δ 2 ), δ 1 = 0, is also a solution of the same ODE. Consequently, the same property holds for a solution G(s) of (4.1).
Proof. By the previous comments, we can view f 1 , f 2 locally around p 1 , p 2 as conformal immersions f i (s, t) into X, and G 1 , G 2 as regular parameterized curves G i (s) : J i → S 2 , both of them satisfying the ODE (4.1) on an open interval J i ⊂ R. The conditions in the statement of the lemma imply that G 1 (s 1 ) = G 2 (s 2 ) and (G 1 ) (s 1 ) = ±(G 2 ) (s 2 ) for some s i ∈ J i . But once here, property (D) above and the uniqueness of solution to the Cauchy problem for ODEs imply that G 1 (s) = G 2 (δ 1 s + δ 2 ) for adequate constants δ 1 = 0 and δ 2 . In particular, the two Gauss maps coincide after conformal reparameterization of f 1 or f 2 . Recall now that, by [25,Theorem 3.7], the Gauss map of a conformally parameterized H-surface in X determines the surface up to left translation in X. Since f 1 (p 1 ) = f 2 (p 2 ), this left translation is trivial in our case, and so we conclude that f 1 (Σ 1 ) = f 2 (Σ 2 ).

Metric semidirect products
Then, a left invariant frame {E 1 , E 2 , E 3 } of X is given by Observe that {E 1 , E 2 , E 3 } is the left invariant extension of the canonical basis (∂ x ) e , (∂ y ) e , (∂ z ) e of the tangent space T e X at the identity element e = (0, 0, 0). The right invariant extensions on X of the same vectors of T e X define the frame F 1 , F 2 , F 3 where In terms of A, the Lie bracket relations are: Definition 4.4. We define the canonical left invariant metric on R 2 A R to be that one for which the left invariant frame {E 1 , E 2 , E 3 } given by (4.3) is orthonormal. Equivalently, it is the left invariant extension to X = R 2 A R of the inner product on T e X that makes (∂ x ) e , (∂ y ) e , (∂ z ) e an orthonormal basis.
Some basic properties of the canonical left invariant metric , on R 2 A R are the following ones.
(E2) The mean curvature of each leaf of the foliation F = {R 2 A {z} | z ∈ R} with respect to the unit normal vector field E 3 is the constant H = trace(A)/2. All the leaves of the foliation F are intrinsically flat. Moreover, the Gauss map image of each leaf of F with respect to the orientation given by E 3 (resp. −E 3 ) is the North (resp. South) pole of S 2 with respect to the left invariant frame {E 1 , E 2 , E 3 }.
(E3) The change (4.3) from the orthonormal basis {E 1 , E 2 , E 3 } to the basis {∂ x , ∂ y , ∂ z } produces the following expression of , in the x, y, z coordinates of X: The π-rotation about any of the integral curves of ∂ z = E 3 (vertical lines in the (x, y, z)coordinates) is an order-two orientation preserving isometry.
Proof. If A = 0, then X is the standard R 3 . If A = 0 and trace(A) = 0, then [27,Theorem 2.15] gives that either det(A) = 0 (which contradicts our hypothesis) or X is isometric to the Heisenberg space Nil 3 (recall that Nil 3 admits a 1-parameter family of homogeneous metrics, all of which are homothetic). Finally, if trace(A) = 0 then X is isometric to some E(κ, τ ) with κ < 0 and τ ∈ R as explained in [27,Section 2.8].
A simple consequence of equations (4.4) and (4.5) is that every horizontal right invariant vector field (i.e., every linear combination of ∂ x , ∂ y with constant coefficients) is bounded in R 2 A [z 1 , z 2 ], for all z 1 , z 2 ∈ R with z 1 ≤ z 2 . The following lemma guarantees the existence of at least one nonzero horizontal right invariant vector field in R 2 A R which is bounded in a horizontal halfspace.
Lemma 4.6. Let X be a semidirect product R 2 A R endowed with its canonical metric, where A ∈ M 2 (R). Then, there exists a nonzero horizontal right invariant vector field which is bounded Proof. If trace(A) = 0, the lemma follows from [24, item 2 of Proposition 6.1]. If trace(A) = 0, then [27, item 2 of Theorem 2.15] implies that there are three possible cases for A up to rescaling the metric. We will prove the lemma by studying each case separately.
• A = 0 −c 1/c 0 for some 1 ≤ c < ∞. This case corresponds to the group structure of E(2), the universal cover of the group of orientation-preserving rigid motions of the Euclidean plane. A direct computation using (4.4) and (4.5) gives |F 1 | 2 = cos 2 z + c −2 sin 2 z and |F 2 | 2 = cos 2 z + c 2 sin 2 z, both bounded.
• A = 0 c 1/c 0 for some 1 ≤ c < ∞. In this case, X is isomorphic to Sol 3 . A similar reasoning as in the preceding case gives that the basis of horizontal right invariant vector fields • A = 0 1 0 0 and X is isomorphic to the Heisenberg space Nil 3 . Using again (4.4) and It is worth recalling that PSL(2, R) has several isomorphic models, that we will use in the sequel: (F1) The group of orientation-preserving isometries of the hyperbolic plane, (F2) The group of conformal automorphisms of the unit disc D := {z ∈ C : |z| < 1}; these automorphisms are the Möbius transformations of the type φ(z) = ξ z+a 1+az , with a ∈ D and |ξ| = 1. The universal cover map θ → ξ := e iθ : R → S 1 gives SL(2, R) a structure of an R-bundle over D.
(F3) The unit tangent bundle of H 2 . This interpretation of PSL(2, R) as an S 1 -bundle over H 2 (and of SL(2, R) as an R-bundle over H 2 ) produces a submersion from SL(2, R) to H 2 .
There are three types of 1-parameter subgroups of SL(2, R): • Elliptic subgroups. Elements of any of these subgroups correspond to liftings to SL(2, R) of rotations around any fixed point in H 2 . These groups of rotations fix no points in the boundary at infinity ∂ ∞ H 2 . Any two elliptic 1-parameter subgroups are conjugate.
• Hyperbolic subgroups. These subgroups correspond to liftings to SL(2, R) of translations along any fixed geodesic in H 2 . In the Poincaré disk model of H 2 , the hyperbolic subgroup associated to a geodesic Γ fixes the two points at infinity given by the end points of Γ. As in the elliptic case, any two 1-parameter hyperbolic subgroups are conjugate.
• Parabolic subgroups. They correspond to liftings to SL(2, R) of rotations about any fixed point θ ∈ ∂ ∞ H 2 . Any of these rotation subgroups only fixes the point θ at infinity, and leaves invariant the 1-parameter family of horocycles based at θ. As in the previous cases, 1-parameter parabolic subgroups are all conjugate by elliptic rotations of the Poincaré disk.
The character of a 1-parameter subgroup Γ of SL(2, R) refers to the property of Γ being elliptic, parabolic or hyperbolic. The character is invariant by conjugation, i.e., if Γ is a 1-parameter subgroup of SL(2, R) and we define Γ a := aΓa −1 = l a (r a −1 (Γ)) for some a ∈ SL(2, R), then Γ a has the same character as Γ.
Nonzero right invariant vector fields K are in one-to-one correspondence with 1-parameter subgroups Γ and with tangent vectors at the identity element e of SL(2, R) by the formula K e = Γ (0). We will say that a nonzero right invariant vector field K on SL(2, R) is elliptic (resp. hyperbolic, parabolic) when the related 1-parameter subgroup Γ in SL(2, R) is elliptic (resp. hyperbolic, parabolic).
The matrices in the Lie algebra sl(2, R) given by The fields E 1 , E 2 , E 3 are respectively generated by the 1parameter subgroups of SL(2, R) given by where the first two are hyperbolic subgroups and the third one is an elliptic subgroup. The center Z of SL(2, R) is an infinite cyclic subgroup (hence isomorphic to Z) whose elements correspond to liftings to SL(2, R) of rotations by angles that are multiples of 2π; if Λ : SL(2, R) → PSL(2, R) denotes the universal covering map, then Z = Λ −1 (e) where e is the identity element of PSL(2, R). In this way, Z is contained in the integral curve Γ 3 ⊂ SL(2, R) of the left invariant vector field E 3 given by (4.7) (indeed, Z is contained in every elliptic subgroup of SL(2, R)).
By declaring the left invariant frame {E 1 , E 2 , E 3 } to be orthonormal, we define a left invariant metric , on SL(2, R) such that the metric Lie group ( SL(2, R), , ) is isometric to the E(κ, τ )space with base curvature κ = −4 and bundle curvature τ 2 = 1. Let be the associated Riemannian submersion onto the hyperbolic plane endowed with the metric of constant curvature −4 (this submersion can be naturally identified with any of the submersions described in properties (F2) and (F3) above). Let φ : defines a submersion which is Riemannian with respect to the particular left invariant metric , defined above. The next result follows from [27, Section 2.7] and describes the geometry of each 1-parameter subgroup of SL(2, R) in terms of the coordinates of its velocity vector at the identity element e with respect to the basis {E 1 , E 2 , E 3 }. Lemma 4.7. Consider a vector Γ (0) ∈ T e SL(2, R) with coordinates (a, b, c) with respect to the basis given in (4.6), and let Γ denote the 1-parameter subgroup generated by Γ (0). Then: is a constant geodesic curvature circle passing through Π(e) and completely contained in H 2 (−4).
2. Γ is hyperbolic if and only if a 2 +b 2 > c 2 . In this case, Π(Γ) is a constant geodesic curvature arc passing through Π(e) with two end points in the boundary at infinity of H 2 (−4).

Γ is parabolic if and only if
Every left invariant metric in SL(2, R) can be obtained by choosing numbers λ 1 , λ 2 , λ 3 > 0 and declaring the length of the left invariant vector field E i to be λ i , 1, 2, 3, while keeping them orthogonal; see [26,Proposition 4.2]. For the remainder of this section, we will suppose that the metric Lie group X is isometric and isomorphic to SL(2, R) endowed with the left invariant metric given by an (arbitrary but fixed) choice of λ i > 0, i = 1, 2, 3.
(2) Σ(γ) is ruled by geodesics of X which are integral curves of E 3 , and each integral curve Γ of E 3 contained in Σ(γ) is the fixed point set of the isometry R Γ : X → X, which restricts to an order-two, orientation-reversing isometry on Σ(γ).
(5) The critical mean curvature of X (see Definition 3.5) satisfies H(X) ≥ δ, where δ is any positive lower bound for the mean curvature function of Σ(γ r ) for any given r > 0.
To prove item 4, take a curve γ r at distance r > 0 from a geodesic γ ⊂ H 2 (−4). Given a point q ∈ γ r , there exists a geodesic γ ⊂ H 2 (−4) passing through q which is tangent to γ r at q, and γ lies entirely on the side of γ r that does not contain γ. By item 3, the surface Σ( γ) = Π −1 ( γ) is minimal. Since Σ( γ) is tangent to Σ(γ r ) and lies at one side of Σ(γ r ), then the usual mean curvature comparison principle implies that the mean curvature of Σ(γ r ) with respect to the unit normal vector field that points towards Σ(γ), is nonnegative along Γ = Π −1 ({q}). In fact, the mean curvature of Σ(γ r ) is strictly positive along Γ as we explain next: otherwise the second fundamental forms II of Σ(γ r ) and II of Σ( γ) would have the same trace and the same value in the direction given by the tangent vector Γ to Γ; choosing Γ as the first vector of an orthonormal basis B of the common tangent plane of both surfaces at any point of Γ, the matrix of the difference II − II with respect to B is symmetric with zero diagonal entries; therefore, II − II has opposite eigenvalues a, −a. If a = 0, then we contradict that Σ( γ) lies at one side of Σ(γ r ); hence II = II along Γ, which in turn implies that γ r would have a point of second order contact with γ, which is false. Therefore, the mean curvature function of Σ(γ r ) is strictly positive.
To see that the mean curvature function of Σ(γ r ) is bounded away from zero, consider the 1-parameter subgroup G of hyperbolic translations of SL(2, R) associated to the translations of H 2 (−4) along γ. Then, Σ(γ r ) is invariant under every element in G. Also, Σ(γ r ) is invariant under left translation by every element in the center of SL(2, R) (recall that elements in this center correspond to liftings to SL(2, R) of rotations by multiples of 2π around a point in H 2 (−4)). Hence Σ(γ r ) can be considered to be a doubly periodic surface. This property, together with the positivity of the mean curvature function of Σ(γ r ), imply that item 4 holds.
Finally, item 5 is a direct consequence of item 4 and of the mean curvature comparison principle by the following argument. Let ∆ ⊂ X be any closed surface and take r > 0. Since the mean convex side of Σ(γ r ) contains arbitrarily large balls of X in its interior, then after a left translation of ∆, we can assume that ∆ lies on the mean convex side of Σ(γ r ). After further continuous left translations inside the mean convex side of Σ(γ r ) applied to ∆, we can assume that ∆ is also tangent to Σ(γ r ) at some common point p. By the mean curvature comparison principle, the absolute mean curvature function of ∆ at p is at least as large as any lower positive bound δ of the positive mean curvature function of Σ(γ r ). Hence, H(X) ≥ δ, which completes the proof. Lemma 4.9. The push-forward by any left translation in SL(2, R) of a right invariant elliptic (resp. hyperbolic, parabolic) vector field is again right invariant elliptic (resp. hyperbolic, parabolic) vector field. Furthermore, given a nonzero elliptic or parabolic vector field F on SL(2, R), the inner product of F and the left invariant vector field E 3 given by (4.6), has constant nonzero sign on SL(2, R), independently of the left invariant metric on SL(2, R).
Proof. The fact that the push-forward by a left translation of a right invariant vector field is again right invariant holds in any Lie group, and is well known.
Consider now F a := (l a ) * (F ), where F is a right invariant vector field and a ∈ SL(2, R). If Γ denotes the 1-parameter subgroup with Γ (0) = F (e), then the corresponding 1-parameter subgroup Γ a that generates the right invariant vector field F a is given by Γ a = aΓa −1 = l a (r a −1 (Γ)).
In particular, Γ a and Γ are conjugate subgroups, and so they have the same character. Thus, F a and F also have the same character, as claimed.
Finally, observe that from the paragraph just before the statement of Lemma 4.8 we deduce that the sign of the inner product of a pair of tangent vectors v, w ∈ T x SL(2, R) at any x ∈ X does not depend on the left invariant metric on SL(2, R). Therefore, in order to prove the last sentence of the lemma, it suffices to choose a nonzero elliptic or parabolic vector field F and prove that the function F, E 3 : SL(2, R) → R has no zeros, where , is the metric defined just before (4.8).
Given x ∈ SL(2, R), let F x be the vector field on SL(2, R) defined by (l x ) * (F x ) = F . By the first sentence of Lemma 4.9, F x is a nonzero right invariant vector field of the same character as F (i.e., elliptic or parabolic), and where we have used that E 3 is left invariant and that l x is an isometry of , . As F x is elliptic or parabolic, by Lemma 4.7, the coordinates (a, b, c) of F x e satisfy a 2 + b 2 ≤ c 2 , which implies that F, E 3 (x) cannot vanish, as desired. Now the proof of Lemma 4.9 is complete.
We will finish this section with some considerations about two-dimensional subgroups of SL(2, R) and their constant left invariant Gauss map images.
Given θ ∈ ∂ ∞ H 2 , let H 2 θ ⊂ SL(2, R) be the set of liftings to SL(2, R) of the orientationpreserving isometries of H 2 that fix θ. Elements in H 2 θ are of two types: liftings to SL(2, R) of hyperbolic translations along geodesics one of whose end points is θ, and liftings to SL(2, R) of parabolic rotations around θ. H 2 θ is a noncommutative, two-dimensional simply connected subgroup of SL(2, R), and H 2 θ contains a unique 1-parameter parabolic subgroup, namely the liftings to SL(2, R) of parabolic rotations around θ. Every two-dimensional subgroup of SL(2, R) is of the form H 2 θ for some θ ∈ ∂ ∞ H 2 , see the paragraph around equation (2.30) in [27] for more details. Given a hyperbolic 1-parameter subgroup Γ of SL(2, R), by Lemma 4.7, the image of Γ through the map Π defined in (4.9) is an arc in H 2 (−4) of constant geodesic curvature that has two extrema θ 1 , θ 2 ∈ ∂ ∞ H 2 . These two end-point values θ 1 , θ 2 determine the two two-dimensional subgroups in coordinates with respect to the orthonormal frame {λ Figure 1.
with (a, b, c) = (cos θ, sin θ, 0)) produces a parabolic (resp. hyperbolic) 1-parameter subgroup Γ P (resp. Γ H ) of SL(2, R). Furthermore, the projection of Γ P (resp. of Γ H ) through the map Π defined in (4.9) is the horocycle that passes through the origin of the Poincaré disk and through the point at infinity e iθ ∈ ∂ ∞ H 2 (resp. the segment with extrema ±e iθ ). It follows that both subgroups Γ P , Γ H are contained in the two-dimensional subgroup H 2 θ of SL(2, R).
The coordinates of v P , v H with respect to the frame {λ . After normalizing the cross product of these 3-tuples, we deduce that the (constant) value of the left invariant Gauss map of H 2 θ in the left invariant metric on SL(2, R) determined by λ 1 , λ 2 , λ 3 > 0 is the one given by equation (4.10). The remaining conclusions of the lemma are direct consequences of (4.10).

The geometry of invariant limit surfaces of index-one spheres
In what follows, we will denote by M 1 X the moduli space of all index-one constant mean curvature spheres in a metric Lie group X diffeomorphic to R 3 . Inside M 1 X we have the component C described in item 4 of Proposition 3.4. We will also use the number h 0 (X) ≥ 0 defined in Proposition 3.4. Definition 5.1. We say that a complete, noncompact, connected H-surface f : Σ X is a limit surface of C with base point p ∈ Σ (also called a pointed limit immersion and denoted by f : (Σ, p) (X, e)) if e = f (p) and there exists a sequence { f n : S n X} n ⊂ C, compact domains Ω n ⊂ S n and points p n ∈ Ω n such that the following two conditions hold: Here, the convergence is the uniform convergence in the C k -topology for every k ≥ 1, when we view the surfaces as local graphs in the normal bundle of the limit immersion.
(G2) The area of f n is greater than n, for all n ∈ N.
We should observe that, by item 6 of Proposition 3.4, any such pointed limit immersion has constant mean curvature equal to h 0 (X). Moreover, by [25, Lemma 5.2], the space of pointed limit immersions of C is nonempty.
(X, e) be a pointed limit immersion of C, and let G : Σ → S 2 ⊂ T e X be its left invariant Gauss map. Then, the rank of the differential dG is constant on Σ. Furthermore: (1) If rank(dG) = 0, then f is injective and f (Σ) is a two-dimensional subgroup of X.
(2) If rank(dG) = 1, then there exists a unique (up to scaling) right invariant vector field on X which is everywhere tangent to f (Σ).
Proof. Suppose first that dG has rank zero at some point q ∈ Σ. By [27, equation (3.9) and Corollary 3.21], there exists a two-dimensional subgroup ∆ of X which has the same mean curvature as f and (constant) left invariant Gauss map of value G(q) ∈ S 2 . We are going to prove next that Consider the foliation F = {l a (∆) = a∆ | a ∈ X} of left cosets of ∆. Then, we may write the leaves of F as the level sets of a Morse function without critical points h : X → R, so that {h = 0} = f (q)∆. Also, we can consider local coordinates ( Arguing by contradiction, assume that f (Σ) and f (q) ∆ do not coincide in a neighborhood of f (q). Then, ϕ is not identically zero around the origin. Since the graphs x 3 = ϕ(x 1 , x 2 ) and x 3 = 0 have the same constant mean curvature in X (and thus define solutions to the same quasilinear elliptic PDE), then ϕ − 0 = ϕ satisfies a second order linear homogeneous elliptic equation with smooth coefficients. So, by Bers' theorem [4], we have ϕ( is a homogeneous polynomial of the form p k = h k • Φ, with h k a harmonic homogeneous polynomial of degree k ≥ 2 in R 2 , and Φ a linear transformation of the (x 1 , x 2 )coordinates. In particular, the level set structure of ϕ(x 1 , x 2 ) around the origin is C 1 -diffeomorphic to the level set structure of the harmonic function Re(z k ) around the origin in the complex plane; see e.g., Kuo [19] or Cheng [5]. Note that the level sets of ϕ describe the intersection of the foliation F with V around f (q).
As f is a pointed limit immersion of C, there exist a sequence { f n : S n X} n ⊂ C, compact domains Ω n ⊂ S n and points p n ∈ Ω n such that the conditions (G1) and (G2) hold. To find the desired contradiction, we will show that for n large enough, the foliation F intersects tangentially f n (S n ) at some point f n (q n ) with q n ∈ Ω n , so that f n (S n ) has points at both sides of f n (q n )∆, which contradicts the Transversality Lemma [23, Lemma 3.1].
To start, note that it follows from the above discussion on the asymptotic behavior of ϕ around the origin that f (Σ) intersects F nearby f (q) in a hyperbolic manner, which is to say that the intersection of F with V produces a 1-dimensional foliation of V with an isolated singularity of index −k at f (q), and the leaves of F are transverse to V except at f (q). For a > 0 sufficiently close to zero, V ∩ {x 3 = a} separates V into of k + 1 components, of which k lie in {x 3 > a} while just one lies in {x 3 < a}. Analogously, for a < 0 sufficiently close to zero, For n large, let V (n) ⊂ f n (Ω n ) be subdomains that are expressed as small normal graphs over V and converge to V as n → ∞. Since the leaves of F are transverse to V except at f (q), for a = 0 fixed and sufficiently close to zero, this description of components of V − {x 3 = a} persists on V (n) − {x 3 = a} for n sufficiently large. Also observe that x 3 | ∂V (n) has no critical points in ∂V (n) − {|x 3 | > a} (since this property holds for x 3 | ∂V ). By Morse theory, we then conclude that for n large enough, there exists a tangency point f n (q n ) ∈ V (n) of f n (S n ) with some leaf {x 3 = a n } such that f n (q n ) → f (q) and a n → 0 as n → ∞. This last property implies that for n sufficiently large, there exist points r n , s n ∈ ∂V (n) ⊂ f n (S n ) such that x 3 (r n ) < x 3 ( f n (q n )) and x 3 (s n ) > x 3 ( f n (q n )), which is the desired contradiction with the Transversality Lemma [23, Lemma 3.1].
Suppose now that the rank of dG is constant one on Σ. By item 1 of Proposition 4.1, f (Σ) is invariant under the flow of a nonzero right invariant vector field K on X. If K is a right invariant vector field on X linearly independent from K, and K is everywhere tangent to f (Σ), then the Lie bracket [K, K] is also everywhere tangent to f (Σ). This implies that K, K generate an integrable two-dimensional subalgebra of the algebra of right invariant vector fields on X, whose integral submanifold passing through e is a two-dimensional subgroup ∆ of X. By uniqueness of integral submanifolds of an integrable distribution, we have f (Σ) = ∆, which contradicts that the rank of dG is constant one on Σ. Hence item 2 of the proposition is proved.
We next prove that if dG has rank one at a some point q ∈ Σ, then the rank of dG is one everywhere on Σ, which together with the previous paragraphs implies the constancy of the rank of dG in any case and finishes the proof of Proposition 5.2. Suppose then that rank(dG q ) = 1 at some q ∈ Σ. By the argument in Step 1 of the proof of [25,Theorem 4.1], there is a nonzero right invariant vector field V on X that has a contact at f (q) with f (Σ) of order at least two. Suppose for the moment that V is everywhere tangent to f (Σ). In this case, the differential of G has rank at most one at every point of Σ by item 1 of Proposition 4.1. But by the previous arguments of this proof, the rank of dG cannot be zero at any point of Σ, and so, dG has constant rank one.
To finish the proof, we assume that V is not everywhere tangent to f (Σ) and we will obtain a contradiction. This contradiction will follow from the existence of two disjoint compact domains in the index-one H n -spheres f n : S n X that give rise to the limit immersion f (in the sense of conditions (G1) and (G2) above), such that each of these compact domains is unstable. To create these compact subdomains, we proceed as follows. Consider the Jacobi function J = V, N on Σ, where N stands for the unit normal vector field along f . By [5,Theorem 2.5], in a small compact neighborhood E q of q in Σ, the set J −1 (0) has the appearance of a set of k embedded arcs α 1 , . . . , α k crossing at equal angles at q with very small geodesic curvatures (their geodesic curvatures all vanish at the common point q), where k ≥ 2 is the degree of vanishing of J at q; see Figure 2. Let E(n) ⊂ Ω n be compact disks that converge to E q as n → ∞, where Ω n ⊂ S n is defined in condition (G1) above. As S n has index one, for n large, the zero set of the Jacobi function J n = V, N n (here N n is the unit normal vector of f n ) is a regular analytic Jordan curve that decomposes S n into two nodal domains D 1 (n), D 2 (n) and the zero sets J −1 n (0) ∩ Ω n converge as n → ∞ to the zero set J −1 (0) of J. Then one of the two nodal domains, say D 1 (n), intersects E(n) in a connected set, see Figure 2.
Let U 1 , U 2 ⊂ Σ be two small compact disjoint geodesic disks of radius ε 0 > 0 centered at be points in the zero set of J n that converge to q 1 , q 2 , respectively, and let U 1 (n), (H2) {x ∈ S n | d Sn (x, β n ) ≤ 1 n } n converges to {q} as n → ∞ and dist Sn (β n , U 1 (n) ∪ U 2 (n)) > δ for some small δ > 0 independent of n, where dist Sn denotes intrinsic distance in S n .
Note that β n decomposes D 1 (n) into two disk components D 1 (n) + , D 1 (n) − , each one being disjoint from exactly one of the disks U 1 (n), U 2 (n). Without loss of generality, we can assume that U 1 (n) ∩ D 1 (n) − = Ø (and thus, U 2 (n) ∩ D 1 (n) + = Ø), see Figure 2. Let φ n : S n → [0, 1] be a piecewise smooth cut-off function with the following properties: Note that φ n J n is a piecewise smooth function that vanishes along ∂D 1 (n) + .
We claim that where L n = ∆ + P n is the stability operator of S n , i.e., P n = |σ n | 2 + Ric(N n ), |σ n | is the norm of the second fundamental form of S n and Ric(N n ) denotes the Ricci curvature of X in the direction of the unit normal to S n . To prove (5.1), observe that since L n J n = 0 on S n , then Multiplying by φ 2 n in (5.2) and integrating on D 1 (n) + we find , hence the Divergence Theorem and the fact that φ n J n = 0 along ∂D 1 (n) + give that .
can be made smaller than C 1 /n 2 for n large, for some C 1 > 0 independent of n. Therefore, the last right-hand-side is bounded from above by which tends to zero as n → ∞ by the first property in (H2) above, because J(q) = 0. This proves our claim (5.1). Next we will define a test function for the stability operator on the domain D 1 (n) + ∪ U 1 (n).
To start, let us consider for ε 0 > 0 sufficiently small, local coordinates (x, y) around q 1 in Σ, so that: corresponds in these coordinates to a compact neighborhood V 1 ⊂ U 1 of q 1 .
By noting the convergence of the disks U 1 (n) to U 1 as n → ∞, it is clear that we can choose coordinates (x (n) , y (n) ) in S n around q 1 (n) for each n large enough on domains V 1 (n) converging to V 1 , given by a slight deformation of the coordinates (x, y), and with similar properties. More specifically, for n ≥ n 0 (ε) large enough there exist local coordinates around q 1 (n) in S n , that will be denoted for simplicity also by (x, y), so that: • The curves {y = constant} are geodesic arcs orthogonal to {x = 0} ⊂ ∂D 1 (n) + .
For n large, v 1 (n) can be extended to a piecewise smooth map on S n (also denoted by v 1 (n)), with the following properties for λ, ε sufficiently small: Observe that (J1)', (J2)' respectively follow from (J1), (J2) and the fact that φ n = 1 in U 1 (n) for n large. Let us explain next why properties (J3)' and (J4)' hold for v 1 (n).
To start, note that, in terms of the (x, y) coordinates in V 1 (n), the Jacobi function J n satisfies J n (0, y) = 0 and |∇J n |(0, y) = 0 for every y ∈ [−ε, ε]. Thus, for ε small enough, J n can be arbitrarily well approximated in D(ε) by a linear function of the type ax for some a = 0. This shows that the equality in (J3)' holds for ε, λ small enough. The inequality in (J3)' is immediate from the definition of v 1 (n).
Similarly, by the previous comments and the definition by interpolation of v 1 (n) on R(n), we see that v 1 (n) is arbitrarily well-approximated in R(n) ⊂ D(ε) for ε small enough by the piecewise linear function a 2 (x − λ|y| + λε), where a = 0 is the previous constant associated to J n . Thus, the (almost constant) norm of the gradient of v 1 (n) in R(n) ⊂ D(ε) is approximately √ 1 + λ 2 /2 times the (almost constant) norm of the gradient of J n in D(ε). By taking λ small enough, we obtain condition (J4)'.
Observe that v 1 (n) vanishes along the boundary of D 1 (n) + ∪ R(n). We will prove that, for ε, λ sufficiently small, there is n 0 = n 0 (ε, λ) such that, whenever n ≥ n 0 , the following inequality holds: Assuming (5.5) holds, we conclude that D 1 (n) + ∪ R(n) is a strictly unstable domain in S n , and thus, D 1 (n) + ∪ U 1 (n) is strictly unstable as well. Arguing in a similar way we also conclude that we contradict that the index of S n is one. Hence it only remains to prove (5.5) in order to find the desired contradiction in the case that the differential dG has rank one at some point but the right invariant Killing vector field V is not everywhere tangent to Σ. We next show (5.5). Since Using property (I2) above, we split the first integral in the right-hand-side of (5.6) for n large as By the previous claim in equation (5.1), the first integral in (5.7) tends to zero as n → ∞. Hence, to prove (5.5) we just need to show that if ε, λ are sufficiently small and n is large enough, then for some constant C(ε, λ) < 0. First note that for ε, λ sufficiently small and n large enough (once ε, λ are fixed), we have by (J3)' Observe that the potential P n converges smoothly to the corresponding potential P = |σ| 2 +Ric(N ) for the limit surface Σ. Therefore, max R(n) |P n | can be supposed to be less than some µ > 0 independent of ε, λ. This implies that where in (A) we have used that the sequences of numbers |∇J n |} n converge to the same positive limit |(∇J)(q 1 )| when we make ε > 0 decrease to zero, and take n large enough with respect to each such choice of ε.
On the other hand, where (B) holds for ε small and n large enough; here, we are using that by taking ε small enough, the metric space structure on S n induced by f n can be assumed to be arbitrarily close to the flat one of the (x, y)-coordinates. By property (J4)', we obtain from the above inequality for ε, λ small and n large that Hence, (5.9), (5.10) and (5.11) give that which implies directly that inequality (5.8) holds for ε, λ small enough and n large. It then follows that for n large, there exist two disjoint unstable regions on S n which contradicts that S n has index one. This completes the proof of the proposition. (X, e) is a pointed limit immersion of C. We define ∆(f ) as the set of pointed immersions f : (Σ, p) (X, e) where Σ is a complete, noncompact connected surface, p ∈ Σ, f (p) = e and f is obtained as a limit of f under an (intrinsically) divergent sequence of left translations. In other words, there exist compact domains Ω n ⊂ Σ and points q n ∈ Ω n diverging to infinity in Σ such that the sequence of left translated immersions {(l f (qn) −1 • f )| Ω n } n converges on compact sets of Σ to f as n → ∞. (1) f is a limit surface of C, thus of constant mean curvature h 0 (X).
(3) There exists a nonzero right invariant vector field on X which is everywhere tangent to f (Σ).
(4) f (Σ) is topologically an immersed plane or annulus in X. We next prove (K2). Arguing by contradiction, suppose that f (Σ) is not a two-dimensional subgroup and there exist a two-dimensional subgroup ∆ of X with mean curvature h 0 (X), and points x ∈ X, y ∈ f (Σ) ∩ (x∆) such that f (Σ) and x∆ = l x (∆) are tangent at y. Therefore, f (Σ), x∆ are different surfaces tangent at y with the same unit normal at this point and the same constant mean curvature. Since f is a limit surface of C by the already proven item 1 of this proposition, we can easily adapt the arguments in the first part of the proof of Proposition 5.2 to deduce that f (Σ) = x∆, which is impossible since e ∈ f (Σ), and so f (Σ) = ∆, a subgroup. Therefore, property (K2) holds and the proof of item 3 of Proposition 5.4 is complete.
Once item 3 is proved, the proofs of items 4,5 of the proposition are the same as the proofs of the related items 2,3 of [25, Corollary 5.4], respectively. It remains to prove the 'Moreover part' of the proposition, so assume that f (Σ) is not a two-dimensional subgroup. By properties (K1), (K2) above, we conclude that there are no twodimensional subgroups in X with mean curvature h 0 (X), whose (constant) left invariant Gauss map lies in G(Σ). In this setting, the already proven item 3 of this proposition implies that we can apply Proposition 4.1 to deduce that f (Σ) is a regular curve. This completes the proof.
The next technical lemma will be used in the proof of Corollary 5.7.
Lemma 5.5 (Unique Limit Surface Lemma). Let f 1 : (Σ 1 , p) (X, e) be a pointed limit immersion of C, with associated pointed immersions f n : (S n , p n ) (X, e), S n ∈ C, so that each H n -sphere S n contains a compact subdomain Ω 1 n with p n ∈ Ω 1 n and f n | Ω 1 n converges to f 1 as n → ∞, and Area( f n | Ω 1 n ) > n. Suppose that the rank of the differential dG 1 of the left invariant Gauss map G 1 of f 1 is one. Let G n be the left invariant Gauss map of f n and let q n ∈ S n be the unique point with G n (q n ) = v = G 1 (p). Then, after choosing a subsequence, there exist compact domains Ω 2 n ⊂ S n with q n ∈ Ω 2 n such that: by left translating f n | Ω 2 n by the inverse of f n (q n ) in X, converges to a pointed immersion f 2 : (Σ 2 , q) (X, e) that is a limit of C, and which has unit normal vector v at q ∈ Σ 2 .
(2) The immersion f 2 has the same image as f 1 .
Remark 5.6. Lemma 5.5 remains true if we drop the hypothesis that the rank of dG 1 is one, but we will not need this more general version in this paper.
Proof. The existence of the compact subdomains Ω 2 n ⊂ S n (after passing to a subsequence) and item 1 follow from standard arguments in elliptic theory, as we have uniform curvature estimates for the H n -immersions l fn(qn) −1 • f n : (S n , q n ) (X, e) by item 5 of Proposition 3.4. By construction, the h 0 (X)-immersions f 1 , f 2 have an oriented contact of order at least one at e. By Proposition 5.2, the differentials of the respective Gauss maps G 1 and G 2 of f 1 and f 2 have constant ranks, which are possibly different. We now distinguish cases depending on the value of rank(dG 2 ).
(L1) Suppose rank(dG 2 ) = 0. In this case, item 1 of Proposition 5.2 gives that f 2 is injective and is not a two-dimensional subgroup (because rank(dG 1 ) = 1), and both f 1 (Σ 1 ), f 2 (Σ 2 ) are tangent at e with the same constant mean curvature, we can apply the arguments in the first part of the proof of Proposition 5.2 (see also the proof of property (K2) above) to find a contradiction. Therefore, this case cannot occur.
(L3) Suppose rank(dG 2 ) = 2. By the Inverse Function Theorem applied to G 2 around q, we can find ε > 0 small such that if B ε denotes the intrinsic ball in Σ 2 of radius ε centered at q, then G 2 | Bε is a diffeomorphism onto its image. Choosing ε > 0 small enough, we can assume that for n large, G 2 (B ε ) ⊂ G n (B Sn (q n , 2ε)), where B Sn (q n , 2ε) is the intrinsic ball of radius 2ε around q n in S n . As G n (p n ) → v, we have G n (p n ) ∈ G n (B Sn (q n , 2ε)), and as G n is a diffeomorphism, this implies that p n ∈ B Sn (q n , 2ε) for n large enough. As ε is arbitrarily small, we conclude that d Sn (p n , q n ) → 0 as n → ∞. This implies that the limit surfaces f 1 (Σ 1 ), f 2 (Σ 2 ) are the same, so in particular rank(dG 2 ) = 1, a contradiction. Therefore, this case does not occur, and this completes the proof of Lemma 5.5.
As a consequence of Lemma 5.5 and the Transversality Lemma [23, Lemma 3.1], we have the following corollary. Proof. Suppose the image immersed surface f (Σ) is tangent to a left or right coset E of some two-dimensional subgroup of X at a point f (y) ∈ f (Σ) ∩ E, where y ∈ Σ. By the discussion just before property (C) in Section 4.1, we can assume that E = f (y)∆ for some two-dimensional subgroup ∆ of X. After a left translation of f by f (y) −1 , we may assume that p = y, and hence f (Σ) is tangent to ∆ at e.
If f (Σ) is a two-dimensional subgroup of X, then f (Σ) = ∆ and there is nothing to prove. Hence in the sequel we will assume that f (Σ) is not a two-dimensional subgroup of X. In this case, as f ∈ ∆(f ) and f is a limit immersion of C, then Proposition 5.4 gives that G(Σ) is a regular curve in S 2 , where G : Σ → S 2 ⊂ T e X is the left invariant Gauss map of f . In particular, rank(dG) = 1.
Applying Lemma 5.5 to f 1 = f , we conclude that f (Σ) coincides with the image set f 2 (Σ 2 ) of the limit f 2 : (Σ 2 , p 2 ) (X, e) of a sequence of pointed immersions f 2 n : (Ω 2 n , q n ) (X, e), where each Ω 2 n is a compact subdomain of an H n -sphere S n in C and such that the left invariant Gauss map of f 2 n at q n ∈ Ω 2 n is v = G(p). By the Transversality Lemma, the spheres f 2 n (S n ) all lie on one side of ∆. Thus, f (Σ) = f 2 (Σ 2 ) = lim n→∞ f 2 n (Ω 2 n ) must lie on one side of ∆. This completes the proof of the corollary.
Assume that ∆(f ) contains no elements with constant left invariant Gauss map. Then: (2) The image γ f = G(Σ) of the left invariant Gauss map G : Σ → S 2 ⊂ T e X of f is a complete embedded regular curve in S 2 . Moreover, its closure γ f ⊂ S 2 admits the structure of a lamination of S 2 , whose leaves correspond to the Gauss map images γ f of elements f ∈ ∆(f ).
In (b) If γ f is not a simple closed curve, then the lamination γ f has uncountably many leaves and γ f has the following recurrency property: given any compact arc I of γ f , there exists a sequence of intrinsically divergent, pairwise disjoint arcs in γ f that converge to I in the C 1 -topology.
Proof. Item 1 follows from a standard diagonal argument.
In the sequel, we will assume that ∆(f ) contains no elements with constant left invariant Gauss map. In particular, f (Σ) is not a two-dimensional subgroup of X.
We next prove item 2. By Proposition 5.4, there exists a nonzero, right invariant vector field on X which is everywhere tangent to f (Σ) and the Gauss map image γ f is a regular curve in S 2 . Next we show that the curve γ f is complete, or equivalently, it has no end points in S 2 . If such an end point x ∈ S 2 of γ f exists, then we can consider a divergent sequence q n ∈ Σ such that G(q n ) → x. After passing to a subsequence, there exist compact subdomains Ω n ⊂ Σ with q n ∈ Ω n such that the restrictions f qn | Ωn converge to an element f ∈ ∆(f ) (see the first paragraph in the proof of Lemma 5.5 for a similar argument). Clearly f has constant left invariant Gauss map, which contradicts our hypothesis. Therefore, γ f is complete.
By the arguments in the proof of Lemma 4.3, γ f has no tangential self-intersections. Transversal self-intersections of γ f can also be ruled out by using a straightforward modification of the arguments in case (L2) of the proof of Lemma 5.5. Therefore, the curve γ f is embedded. Also, note that the surface f (Σ) can be viewed locally as the graph of a solution to a quasilinear elliptic PDE, and recall that f has uniformly bounded second fundamental form. By standard elliptic estimates, this provides a priori C 3 estimates for f (Σ), and this shows in particular that γ f has bounded geometry. These properties for γ f imply that its closure γ f has the structure of a lamination of S 2 , all whose leaves have bounded geometry as well. Hence item 3 of the lemma is proved.
If f ∈ ∆(f ), then the arguments above apply to f to give that the Gauss map image γ f of f is a complete embedded regular curve in S 2 . By definition, there exists a sequence {q n } n ⊂ Σ such that the immersions f q n converge smoothly to f , and thus the corresponding Gauss map images γ f q n converge to γ f in S 2 . But clearly γ f q n = γ f as sets, from where we deduce that the leaves of the lamination γ f correspond to the Gauss map images of elements of ∆(f ). This proves item 2 of the lemma. We finish by proving item 4. Consider the set S of sublaminations of the lamination γ f , which is partially ordered by the inclusion. We want to apply Zorn's lemma to S in order to find a minimal element in S, i.e., a sublamination of γ f with no proper sublaminations. To do this, we must check that every totally ordered subset S 1 of S has a lower bound. This is clear provided that the intersection of all sublaminations in S 1 is nonempty. Since S 1 is a collection of closed sets of the compact topological space S 2 and S 1 clearly satisfies the finite intersection property, then the intersection of all sublaminations in S 1 is nonempty. By Zorn's lemma, there exists a sublamination L of γ f with no proper sublaminations. Take a leaf Γ of L. As Γ is a leaf of γ f , then item 2 implies that there exists an immersion f ∈ ∆(f ) whose Gauss map image γ f is equal to Γ. Therefore, γ f is a sublamination of L, and by minimality of L in S implies that γ f = L. This proves the first sentence of item 4 of the lemma.
In order to prove items 4a and 4b, it is worth adapting some known facts about laminations to our setting. Take a sublamination γ f of γ f with no proper sublaminations, corresponding to an element f ∈ ∆(f ). A point x ∈ γ f is called a limit point of γ f if x is the limit in S 2 of an intrinsically divergent sequence {x n } n ⊂ γ f . If x ∈ γ f is a limit point of γ f , then the leaf component L of the lamination γ f that contains x consists entirely of limit points of γ f (and L is called a limit leaf). The set Lim(γ f ) of limit points of γ f is a (closed) sublamination of γ f , possibly empty.
If γ f is a closed curve, then clearly Lim(γ f ) is empty and so, γ f consists of the single leaf γ f , which is item 4a of the lemma. Next suppose that γ f is not a simple closed curve. Thus, Lim(γ f ) = Ø. Since γ f contains no proper sublaminations, then Lim(γ f ) = γ f . We next show that γ f contains an uncountable number of leaves. Consider a small compact arc α ⊂ S 2 cutting γ f transversally. Since complete one-manifolds are second countable in their intrinsic topology (each leaf of the lamination γ f has this property), to prove that γ f has an uncountable number of leaves, as a lamination has a local product structure it suffices to prove that W = α ∩ γ f is uncountable. We can consider W to be a complete metric space (note that α is a compact arc in S 2 with its usual topology, hence α has a natural structure of a complete metric space, and W is a closed subset of α). W has no isolated points, since Lim(γ f ) = γ f . In this setting, the uncountability of W is a consequence of the following well known elementary application of the Baire category theorem: any complete metric space without isolated points is uncountable.
Finally, the recurrency property in item 4b of the lemma follows easily from the fact that γ f ⊂ γ f = Lim(γ f ). Now Lemma 5.8 is proved.
For the remainder of this section we will use the notation introduced in Lemma 5.8, and denote by f : (Σ, p) (X, e) an element of ∆(f ) such that ∆(f ) contains no elements with constant left invariant Gauss map. In particular, f (Σ) is not a two-dimensional subgroup of X.
By Proposition 5.4, γ f is a regular curve in S 2 and thus, the right invariant vector field K Σ given by item 2 of Proposition 5.2 is unique up to scaling. Once we pick K Σ and an orientation on X, the curve γ f ⊂ S 2 has a natural orientation as follows. γ f σ σ n Figure 3: If γ f is not a simple closed curve, then it contains compact arcs σ n converging to a compact arc σ, with the orientations of the σ n being opposite to the one of σ.
Definition 5.9. Suppose that q ∈ Σ and N (q) ∈ T f (q) X is the unit normal vector to Σ at q. Consider a short arc β in Σ transverse to the integral curves of the vector field K Σ induced by K Σ on Σ, parameterized so that β(0) = q and {df q (β (0)), K Σ (f (q)), N (q)} is a positively oriented basis for T f (q) X. Now, the orientation on γ f is defined to be the one given by G • β where G is the Gauss map of f . Lemma 5. 10. Assume that γ f is not a simple closed curve in S 2 . Then, there exists a compact interval σ ⊂ γ f with G(p) ∈ σ (here G is the left invariant Gauss map of f and p is the base point of f ) and there exists a sequence of pairwise disjoint arcs σ n ⊂ γ f that are small normal graphs over σ, which converge to σ in the C 1 -topology and, as graphs oriented by γ f , satisfy that their induced orientations are opposite of the orientation of σ, see Figure 3.
Proof. By item 4b of Lemma 5.8, given a small arc σ ⊂ γ f containing G(p), there exists a sequence of pairwise disjoint intervals σ n ⊂ γ f which converge C 1 in S 2 to σ. After extracting a subsequence, we can assume that all the arcs σ n lie on the same side of σ; more specifically, we may assume that there exists a one-sided neighborhood U of σ diffeomorphic to a square [0, 1] × [0, 1], so that σ corresponds to {0} × [0, 1] and for each n ∈ N, σ n corresponds to {1/n} × [0, 1] under this diffeomorphism. Also note that the lamination structure on the closure of γ f ensures that we can take U so that every component of γ f ∩ U is a small normal graph over σ. Therefore, in the model of U as [0, 1] × [0, 1], γ f ∩ U can be represented as A × [0, 1] where A ⊂ [0, 1] is a compact infinite subset, 1 ∈ A and 0 is an accumulation point of A.
Without loss of generality, we can also assume that the orientation on σ induced by the one of γ f is pointing upwards in this model of σ as {0} × [0, 1]. If Lemma 5.10 fails, then we have that after choosing U small enough, every component of γ f ∩ U ≡ A × [0, 1] is oriented by pointing upwards. We can assume that γ f is parameterized by the real line R, and its orientation is the induced one by this parameterization. Then, each of the segments σ, σ n corresponds respectively to a closed interval I, I n ⊂ R by this parameterization, and these intervals form a pairwise disjoint collection. Each point or closed interval J in R defines a "future" (the component of R − J that limits to +∞) and a "past" (the component of R − J that limits to −∞), and these definitions can be translated to γ f via the parameterization. After passing to a subsequence, we can assume that one of the two following possibilities holds: (M2) I n is contained in the past of I, for all n ∈ N.
If case (M1) holds, then consider the end point t 1 = max(I) and the first t 2 > t 1 whose image by the parameterization lies in the closed topological square U ⊂ S 2 ; the image of the interval Let D ⊂ S 2 be the topological disk bounded by α whose interior is disjoint from the square U . Consider the embedded open arc (I ) + of γ f given by the future of I . Note that (I ) + must eventually intersect the interior of D (to see this, observe that for n sufficiently large, the segment σ n ≡ {1/n}×[0, 1] lies in (I ) + and intersects ∂D transversely at its initial point). Since γ f has no self-intersections, we deduce that (I ) + must cross ∂D along the arc (0, r) × {0}. Moreover, at the points of (I ) + ∩ ∂D along the arc (0, r) × {0} where (I ) + enters D, the induced orientation of (I ) + points down, which contradicts our assumption that every component of γ f ∩ U ≡ A × [0, 1] is oriented by pointing up. If possibility (M2) above occurs, then a similar argument leads to a contradiction. This finishes the proof of Lemma 5.10.
Lemma 5.11. Suppose that γ f is not a closed curve. Let σ, σ n ⊂ γ f , n ∈ N, be the arcs given by Lemma 5.10, and assume that G(p) ∈ σ (here p is the base point of f and G is its left invariant Gauss map). Let D, D n ⊂ Σ be given by D := G −1 (σ), D n := G −1 (σ n ). Take points p n ∈ D n so that {G(p n )} n converges to G(p). Then, there exist diffeomorphisms Φ n : D → D n such that the immersions l f (pn) −1 • f • Φ n converge to f uniformly on compact sets of D.
Proof. As γ f is not a closed curve, Σ cannot be diffeomorphic to an annulus, and so by Proposition 5.4 it is simply connected. Thus, by the arguments described after Proposition 4.1 and the uniformization theorem, we can parameterize f as a conformal immersion f : U ⊂ C X where U is either C or a vertical strip {s + it | a 1 < s < a 2 }, and the left invariant Gauss map G : U → S 2 of f depends solely on s. From now on, we will identify Σ with U and assume that the imaginary part of the base point p ∈ U of f is zero. We will also consider the stereographically projected Gauss map g := π • G : U → C, where π : S 2 → C is the stereographic projection from the South pole of S 2 defined in terms of a previously chosen left invariant orthonormal frame is the value of the left invariant Gauss map of f at p n . By hypothesis on the points p n , we have that We next translate property (N2) above into the desired convergence of l f (pn) −1 • f • Φ n to f . To do this, we observe that given z ∈ U, we have where A 1 , A 2 , A 3 : U → C are given in terms of g by [25, equation (3.4)], and {E 1 , E 2 , E 3 } is the orthonormal basis of left invariant vector fields on X that we prescribed previously. Note that the immersion f • Φ n satisfies the same complex linear differential equation (5.12) with respect to g n . Using properties (N1), (N2) above and the smooth dependence of the solution of an ODE with respect to initial conditions, we deduce that l f (pn) −1 • f • Φ n converges uniformly on compact sets Finally we prove the last sentence in the statement of Lemma 5.11. Let us consider the pointed immersions l f (pn) −1 • f : (Σ, p n ) (X, e), which have uniform curvature estimates by item 5 of Proposition 3.4. By elliptic theory, these immersions converge (up to subsequence) to some other complete pointed immersion of constant mean curvature. Now, the uniform convergence property that we obtained in the previous paragraph, and the unique continuation principle for surfaces of constant mean curvature (see e.g., Aronszajn [3]) concludes the proof of Lemma 5.11.

The existence of periodic invariant limit surfaces
In this section we will apply the results of Section 5 to prove Theorem 6.1 below, as well as its consequence, Corollary 6.7. In later sections we will also quote some of the steps in the proof of Theorem 6.1.
We will use the following notation along this section. X is a metric Lie group diffeomorphic to R 3 ; h 0 (X) ≥ 0, C are respectively the number and the component of the space M 1 X of index-one spheres with constant mean curvature in X that appear in Proposition 3.4. f : (Σ, p) (X, e) is a pointed limit immersion of C produced by Proposition 5.4; in particular, f is stable with constant mean curvature h 0 (X), and there exists a nonzero, right invariant vector field K Σ on X which is everywhere tangent to f (Σ). Γ Σ ⊂ X will denote the 1-parameter subgroup of X generated by K Σ .
Observe that if the space ∆(f ) defined in Lemma 5.8 contains an element f 1 : (Σ 1 , p 1 ) (X, e) with constant left invariant Gauss map (equivalently, f 1 (Σ 1 ) is a two-dimensional subgroup of X), then f 1 is injective and f 1 (Σ 1 ) is an entire Killing graph in X with respect to any nonzero right invariant vector field in X transverse to f 1 (Σ 1 ). In particular, Theorem 1.4 holds trivially in this case (note that by item 1 of Proposition 5.4, f 1 is a limit surface of C, and so it has the convergence properties in the statement of Theorem 1.4). Therefore, in the sequel we will assume that no element in ∆(f ) has constant left invariant Gauss map; in particular, f (Σ) is not a two-dimensional subgroup of X.
By Proposition 5.4, Σ is either simply connected (with f (Σ) being a plane or an annulus) or Σ is diffeomorphic to an annulus (so f (Σ) is an annulus as well). By Lemma 5.8, the left invariant Gauss map image γ f = G(Σ) of f is an embedded regular curve in S 2 ⊂ T e X, and we can choose f so that the closure γ f is a lamination of S 2 with no proper sublaminations. Theorem 6.1. In the above situation, γ f is a closed curve in S 2 .
Proof. As γ f is not a point, then we can consider on γ f the orientation given by Definition 5.9. We will divide the proof of the theorem into three assertions for the sake of clarity. Assertion 6.2. γ f is a closed curve if one of the following two conditions hold: (1) X is isomorphic to R 2 A R for some matrix A ∈ M 2 (R) and K Σ / ∈ Span{F 1 , F 2 } where F 1 , F 2 are given by equation (4.4).
(2) X is isomorphic to SL(2, R) and K Σ is either elliptic or parabolic.
Proof of Assertion 6.2. Arguing by contradiction, suppose that γ f is not a simple closed curve. Let σ and {σ n } n be the arcs given in Lemma 5.10, where q := G(p) lies in σ. Let p n be points of Σ such that the sequence G(p n ) ∈ σ n converges to q as n → ∞. By Lemma 5.11, the pointed immersions l f (pn) −1 • f : (Σ, p n ) (X, e) converge uniformly on compact sets to f : (Σ, p) (X, e). As K Σ is everywhere tangent to f (Σ), then the nonzero vector field (l f (pn) −1 ) * (K Σ ) is everywhere tangent to (l f (pn) −1 • f )(Σ). Observe that (l f (pn) −1 ) * (K Σ ) is right invariant by the first paragraph of the proof of Lemma 4.9. After normalizing, we deduce that the sequence of right invariant vector fields converges to a nonzero right invariant vector field K 1 on X which is everywhere tangent to f (Σ).
Since f (Σ) is not a two-dimensional subgroup of X, then K 1 must be a (constant) nonzero multiple of K Σ . As the orientations of the arcs σ n are all opposite to the orientation of the arc σ, by definition of the orientation of γ f we deduce that K 1 is a negative multiple of K Σ . We now check that the last property cannot happen under the hypotheses of Assertion 6.2. First suppose that we are in case 1 of this assertion. Since K Σ / ∈ Span{F 1 , F 2 }, then we may assume that for some constants c 1 , c 2 ∈ R; here F 3 is given by (4.4). Plugging (4.4) into this last equation we get K Σ = ∂ z + h 1 (x, y)∂ x + h 2 (x, y)∂ y for some functions h 1 , h 2 . Since ∂ z is left invariant and (l a ) * preserves Span{∂ x , ∂ y } for every a ∈ X, then we conclude that K(n) cannot converge to a negative multiple of K Σ , which is the desired contradiction if case 1 holds.
In case 2, we can use similar arguments to find a contradiction, using the last sentence of Lemma 4.9. This finishes the proof of Assertion 6.2. Assertion 6.3. If X = R 2 A R and K Σ ∈ Span{F 1 , F 2 }, then γ f is a closed curve. Moreover, if γ f does not pass through the North or South pole, then f (Σ) is an entire Killing graph with respect to any horizontal right invariant vector field linearly independent from K Σ .
Proof of Assertion 6.3. As K Σ ∈ Span{F 1 , F 2 }, equation (4.4) implies that K Σ is a linear combination λ∂ x + µ∂ y for some λ, µ ∈ R. Since f (Σ) is foliated by integral curves of K Σ , then f (Σ) is ruled by straight lines in the direction of λ∂ x + µ∂ y . There are two cases to consider: (O1) The angles between the immersed surface f (Σ) and the planes R 2 A {z}, z ∈ R, are bounded away from zero. Equivalently, γ f does not contain the North or South poles of S 2 .
(O2) There exists a sequence {q n } n ⊂ Σ such that the angle between f (Σ) and R 2 A {z(f (q n ))} at f (q n ) tends to zero as n → ∞. Equivalently, the North pole or the South pole in S 2 is in γ f . Suppose case (O1) holds. Note that the length of the pullback ∇(z • f ) by f of the tangential component of ∂ z to Σ is bounded away from zero. Without loss of generality, we can assume that K Σ = ∂ x (this is just a linear change of the coordinates x, y, which does not affect to our hypotheses). Hence, the image through f of the integral curve of ∇(z • f ) passing through the base point p is a curve in R 2 A R that passes through the origin and intersects every horizontal plane R 2 A {t} at a single point Q(t) = (x(t), y(t), t). Furthermore, the intersection of f (Σ) with R 2 A {t} is the straight line passing through Q(t) in the direction of ∂ x . Hence, f (Σ) is an entire Killing graph with respect to any horizontal right invariant vector field V = a∂ x + b∂ y linearly independent from ∂ x .
We now prove that γ f is a simple closed curve by using arguments that are similar to the ones in the proof of the first item of Assertion 6.2. Consider the cross product K Σ × N , which defines a nowhere zero tangent vector field to f (Σ) (here N is the unit normal of f ). Thus, the tangent vector field W to Σ given by pulling back K Σ × N through f is never tangent to the foliation {R 2 A {z} | z ∈ R}. By connectedness of Σ, the function W, ∇(z • f ) has constant (nonzero) sign on Σ. Observe that this sign does not change when we compose f with a left translation in X and do the same process with the translated surface. Assume γ f is not a simple closed curve and consider for each n ∈ N the right invariant vector field K(n) given by (6.1), which in this case lies in Span{∂ x , ∂ y } (because (l a ) * preserves Span{∂ x , ∂ y } for every a ∈ X). As before, the K(n) converge to a negative multiple of K Σ , and so the vector fields W n obtained by pulling back K(n) × N n through l f (pn) −1 • f converge to a negative multiple of W (here, N n denotes the unit normal of l f (pn) −1 • f ). This contradicts the previously explained invariance of the sign of the function W, ∇(z • f ) under left translations. Hence, γ f is a simple closed curve, what proves Assertion 6.3 if case (O1) holds.
In order to analyze case (O2), we will prove the following general property to be used later on.

Claim 6.4 (Slab Property in
be the Killing field on Σ induced from the right invariant vector field K Σ . If there exists a sequence {q n } n ⊂ Σ such that the angle between f (Σ) and R 2 A {z(f (q n ))} at f (q n ) tends to zero as n → ∞ (i.e., case (O2) holds), then: (2) If q ∈ Σ satisfies that f (Σ) is tangent to R 2 A {z(f (q))}, then z(f (q)) ∈ {a, b}.
(3) If Σ is simply connected, then the points q n can be taken to lie on distinct integral curves of K Σ such that z(f (q n )) = a if n is odd, b if n is even.
(4) γ f is a simple closed curve that passes through the North and South poles of S 2 .
Proof of Claim 6.4. By the uniform curvature estimates in item 5 of Proposition 3.4, after passing to a subsequence, the sequence of left translated pointed immersions l f (qn) −1 • f : (Σ, p) (X, e) converges to a pointed immersion [ f : ( Σ, p) (X, e)] ∈ ∆(f ). Since the angle between f (Σ) and . From now on, we will assume that f ( Σ) ⊂ R 2 A [0, ∞) (this does not affect the arguments that follow). Note that f is invariant under a nonzero right invariant vector field K Σ , which is the limit as n → ∞ of the right invariant vector fields K(n) given by equation (6.1) after replacing p n by q n ; in particular, both K(n) and K Σ lie in Span{∂ x , ∂ y }. Without loss of generality, we can assume that K Σ = ∂ x .
Therefore, f ( Σ) is ruled by straight lines in the direction of ∂ x . If f ( Σ) = R 2 A {0}, then f has constant left invariant Gauss map, which contradicts our assumption stated before Theorem 6.1.

Since f ( Σ) is analytic and f ( Σ) = R 2
A {0}, then there exists ε > 0 such that if B Σ ( p, ε) denotes the intrinsic closed disk in Σ centered at the base point p with radius ε, then D = f B Σ ( p, ε) is a small graphical disk over its vertical projection to R 2 A {0}, D is foliated by line segments in the direction of ∂ x , exactly one of these line segments lies in the x-axis, and all the other ones lie in R 2 A (0, ∞). As l f (qn) −1 • f converges to f , then the sequence of disks D n = (l f (qn) −1 • f )(B Σ (q n , ε)) converges smoothly to D, where B Σ (q n , ε) stands for the intrinsic closed disk in Σ centered at q n with radius ε. Consider the integral curves of K(n) (which are horizontal straight lines not necessarily parallel to the x-axis) which intersect D n . For each n, this set of parallel lines forms a smooth surface D n contained in (l f (qn) −1 •f )(Σ), with boundary consisting of two horizontal lines, and the D n converge smoothly on compact sets to the ruled surface D ⊂ f ( Σ) consisting of the integral curves of ∂ x that intersect D; note that D contains the x-axis and its boundary consists of two lines that both lie in R 2 A (0, ∞). In particular, for n large, the boundary curves of D n lie strictly above the lowest straight line in D n . Therefore, along this lowest straight line in D n , the tangent plane to this surface is horizontal. Hence, f (Σ) is also tangent to some horizontal plane and by Corollary 5.7 we see that f (Σ) lies above this plane.
After possibly replacing f by a left translation of it and changing the base point p, we may assume that f (Σ) is tangent to the plane P 0 = R 2 A {0} at f (p) = (0, 0, 0) and it lies above P 0 . By analyticity of f , we can choose an open strip S ⊂ Σ containing p and such that: • ∇(z • f ) only vanishes in S along the integral curve β 1 of K Σ that passes through p.
• z • f attains the same value at the two boundary components of S.
Let C 1 ⊂ Σ be one of the (at most two) components of Σ − S, and let P be the horizontal plane that contains f (∂C 1 ). We claim that the angle that f (C 1 ) makes with the foliation by horizontal planes is not bounded away from zero: otherwise there exists a sequence of points r n ∈ Σ such that |(z • f )(r n )| → ∞ as n → ∞. After left translating f by f (r n ) −1 and extracting a subsequence, we find a new limit f ∈ ∆(f ) such that its image makes angles bounded away from zero with the foliation of horizontal planes {R 2 A {t} | t ∈ R}. Since γ f is a leaf of the lamination γ f and γ f = γ f (recall that we were assuming that γ f has no proper sublaminations), then item 2 of Lemma 5.8 implies that f is also a limit of f under left translations. This clearly contradicts that f (Σ) is tangent to P 0 . Hence, our claim is proved.
By repeating the above arguments, there is an integral curve β 2 ⊂ C 1 of K Σ such that ∇(z • f ) vanishes along β 2 and being the closest such integral curve to β 1 in C 1 ; let b := (z •f )(β 2 ) > 0. As before, Corollary 5.7 implies that f (Σ) is contained on one side of the plane P b = R 2 A {b}; by connectedness of f (Σ), this side must be R 2 A (−∞, b]. This implies that the immersed surface f (Σ) is contained in the slab between the planes P 0 and P b ; also, the strip Y 1 ⊂ Σ between β 1 and β 2 has an embedded image contained in that slab (because z • f has no critical points in the interior of Y 1 ). Repeating these arguments in the domain C 2 := C 1 − Y 1 , one finds an integral curve β 3 of K Σ in C 2 closest to β 2 where z • f has a critical value and f (Σ) lies on one side of the corresponding horizontal plane. Again by connectedness of f (Σ), this critical value must be 0.
Continuing inductively with this argument, in the case that Σ is simply connected, one produces the desired sequence of points {q n } n∈N , where q n ∈ Γ n and for n odd (resp. even) one has the related integral curves β n contained in R 2 A {0} (resp. R 2 A {b}). The remainder of the statements listed in Claim 6.4 follow immediately from this discussion.
Finally, Assertion 6.3 follows from the Slab Property (Claim 6.4) and our previous discussion of case (O1).
In the last step of the proof of Theorem 6.1 we will study the case in which X is isomorphic to SL(2, R) and K Σ is hyperbolic, or equivalently, the 1-parameter subgroup Γ Σ of SL(2, R) generated by K Σ is hyperbolic. Recall from Section 4.3 that the image of Γ Σ through the projection Π : SL(2, R) → H 2 defined by (4.9) is an arc of constant geodesic curvature (possibly zero) in the standard metric of H 2 , that passes through the origin of the Poincaré disk and its two extrema θ 1 , θ 2 ∈ ∂ ∞ H 2 determine the two two-dimensional subgroups H 2 θ 1 , H 2 θ 2 of SL(2, R) that contain Γ Σ . We will also use in the next statement the pair of antipodal, simple closed curves Υ ∪ (−Υ) ⊂ S 2 defined in Lemma 4.10.
Assertion 6.5. If X is isomorphic to SL(2, R) and K Σ is hyperbolic, then γ f is a simple closed curve. Moreover, if γ f does not intersect Υ ∪ (−Υ), and H is either is an entire Killing graph with respect to the parabolic right invariant vector field which is tangent at e to H.
Proof of Assertion 6.5. We denote by Γ 3 the 1-parameter elliptic subgroup of X defined by Γ 3 (0) = (E 3 ) e (see equation 4.6)). Let H be one of the subgroups H 2 θ 1 or H 2 θ 2 . Let F = {r t (H) = H t | t ∈ SL(2, R)} be the foliation of X whose leaves are the right cosets of H. Since Γ 3 intersects H only at the identity element, it is easily seen that every element of X lies in a unique right coset of H of the form H t, for some t ∈ Γ 3 .
Consider the oriented distance function d : X → R to the closed set H, which is positive on H t, for t > 0, where we have naturally parameterized Γ 3 by R. As each right coset of a codimension one connected subgroup of a metric Lie group is at a constant distance from the subgroup (see [27,Lemma 3.9]) and Γ 3 is isomorphic to R, then the level sets of d are the leaves of F. Hence, we have an induced function d : F → R so that the absolute value of d at a given element H t of F is the constant distance from H to H t; thus F can be parameterized by Γ 3 , i.e., F = {H t | t ∈ Γ 3 }, and we may assume that ∇ d, E 3 > 0, where here ∇ stands for gradient in X.
The approach to prove Assertion 6.5 is to follow, when possible, the arguments in the proof of Assertion 6.3 by exchanging the former foliation {R 2 A {t} | t ∈ R} by the current one F. Notice that the ambient function z in Assertion 6.3 is now replaced by the oriented distance function d : X → R. For the proof, it is also important to take into account the following list of elementary observations: (P1) Each of the right cosets H t with t ∈ R ≡ Γ 3 is a left coset of some two-dimensional subgroup H t which is a conjugate subgroup of H; see Section 4.1. If t ∈ Γ 3 lies in the center of X (which is isomorphic to Z), then H t = t H. In this sense, F is a periodic foliation invariant under the left action of Z = center(X) ⊂ Γ 3 .
(P2) Limits of f after left translations by the inverses of points on the image surface f (Σ) give rise to limit surfaces f ( Σ) that are invariant under some other hyperbolic right invariant vector field and such that γ f is a leaf of γ f . Otherwise, Lemma 4.9 implies that f ( Σ) would be tangent to a right invariant parabolic vector field (because the limit of a normalized sequence of hyperbolic right invariant vector fields is either hyperbolic or parabolic). As the set of parabolic vector fields normalized to have length 1 at e is compact, then every element in ∆( f ) is tangent to a parabolic right invariant vector field. As γ f is a leaf of the lamination γ f = γ f (here we are using γ f has no proper sublaminations), then f is itself a limit of f , i.e., f ∈ ∆( f ), and so f is tangent to a right invariant parabolic vector field. This is a contradiction since K Σ is hyperbolic and f (Σ) is not a two-dimensional subgroup of X.
(P3) Given a ∈ X, the family a F = {a L = l a (L) | L ∈ F} is a foliation of X by the right cosets of some conjugate subgroup of H.
(P4) As the foliation F is periodic in the sense of item (P1), then for every sequence {a n } n ⊂ X, the sequence of foliations {a n F} n∈N defined as in (P3) has a convergent subsequence to another foliation of X by the right cosets of some conjugate subgroup of H.
(P5) Let Γ P be the unique 1-parameter parabolic subgroup contained in H. As Γ Σ H = H and Γ P H = H, we deduce that the left actions of Γ Σ , Γ P on X leave invariant each of the right cosets of H. In particular, K Σ and the parabolic right invariant vector field P on X generated by Γ P are everywhere tangent to each of the leaves of F.
(P6) Given a leaf H t of F, each integral curve of K Σ restricted to H t intersects each integral curve of P restricted to H t in exactly one point.
To prove Assertion 6.5, let us start by assuming that the Gaussian image γ f is at a positive distance from the set Υ ∪ (−Υ). In particular, the angle between f (Σ) and the leaves of F is bounded away from zero. Using observations (P5) and (P6) above, it follows from simple modifications of the arguments in the proof of case (O1) of Assertion 6.3 that in these conditions, f (Σ) is an entire Killing graph with respect to P , and that γ f is closed. Details are left to the reader. Claim 6.6 (Slab Property in SL(2, R)). Suppose X = SL(2, R) and K Σ is hyperbolic. Let K Σ = f * (K Σ ) be the Killing field on Σ induced by the right invariant vector field K Σ . If there exists a sequence {q n } n ⊂ Σ such that the angle between f (Σ) and the leaf of F passing through f (q n ) tends to zero as n → ∞, then: (1) f (Σ) is contained in a smallest topological slab bounded by two different leaves L a , L b of F, and f (Σ) is tangent to both leaves.
(2) If f (Σ) is tangent to a leaf L of F, then L = L a or L = L b .
(3) γ f is a simple closed curve that intersects both Υ and −Υ.
Proof of Claim 6.6. Consider for each n ∈ N the left translated pointed immersion f n := l f (qn) −1 • f : (Σ, q n ) (X, e), which is everywhere tangent to the hyperbolic right invariant vector field K n := (l f (qn) −1 ) * (K Σ ); see observation (P2). Note that K n is everywhere tangent to the foliation by right cosets F n := {H n t | t ∈ R}, where H n is the two-dimensional subgroup of X given by H n = f (q n ) −1 H f (q n ) . After extracting a subsequence, the f n converge on compact sets to a pointed limit immersion f : ( Σ, p) (X, e) and by observation (P4), F n converges to some foliation F of X by the right cosets of some two-dimensional subgroup H. Observe that by hypothesis, the angle that f n (Σ) makes with F n at f n (q n ) = e tends to zero as n → ∞. Thus, f ( Σ) is tangent to H at f ( p) = e. In particular, the left invariant Gauss map of f at p lies in Υ ∪ (−Υ). Furthermore, f is everywhere tangent to some right invariant hyperbolic Killing field K, by observation (P2) ( K is the limit of appropriate rescalings of the K n after extracting a subsequence, as we did in (6.1)). In particular, K is everywhere tangent to H, which implies that K is generated by a 1-parameter subgroup of H. Let d : X → R stand for the oriented distance function associated to the foliation F. Note that d is constant along each of the integral curves of K. By Corollary 5.7, f ( Σ) is contained in one side of H, i.e., the restriction of d to f ( Σ) is nonpositive or nonnegative. Moreover, arguing as in the proof of Claim 6.4 there exists a small compact topological disk D = f B Σ ( p, ε) which is foliated by compact arcs of integral curves of K, such that: (Q) Exactly one of these compact arcs of integral curves of K in D lies in the zero set of d, while all the other ones lie in d −1 (0, ∞) (or in d −1 (−∞, 0)).
Since f n converges to f , then the sequence of disks D n := f n (B Σ (q n , ε)) converges smoothly to D. As {H n } n → H, then we can assume that the distance function d n : SL(2, R) → R associated to the foliation F n converges on compact sets of SL(2, R) to d. As K n is everywhere tangent to the leaves of F n , we conclude that d n is constant along the integral curves of K n . Furthermore, item (Q) and the previous convergence properties ensure that for n large enough, D n contains an integral curve segment of K n along which d n has a local extremum. Hence, there are points r n ∈ B Σ (q n , ε) such that f n is tangent at f n (r n ) to some leaf of F n . Therefore, f is tangent at f (r n ) to some leaf L a of F. By Corollary 5.7, we deduce that f (Σ) lies on one side of L a . Once here, and using similar ideas, we can adapt the last four paragraphs of the proof of Claim 6.4 to our SL(2, R) setting, and conclude that f (Σ) is tangent to another leaf L b of F, and lies in the topological slab of SL(2, R) bounded by L a ∪ L b . This proves item 1 of Claim 6.6, and item 2 is a direct consequence of item 1, Corollary 5.7 and the connectedness of Σ that we leave to the reader.
As for item 3 of Claim 6.6, since the leaves of F are minimal, the mean curvature comparison principle applied to f (Σ) and L a , L b implies that the mean curvature vector of f (Σ) points towards the interior of the previously defined topological slab at the tangency points between f (Σ) and L a ∪ L b . From here, a straightforward continuity argument applied to the normal field to F gives that γ f intersects both Υ and −Υ. If Σ is an annulus, then γ f is clearly a closed curve. Otherwise, Σ is simply connected and f (Σ) is tangent to L a along at least two (in fact, infinitely many) distinct integral curves of K Σ = f * (K Σ ). This implies that γ f takes the same value at these integral curves of K Σ . As γ is embedded, we now conclude that γ f is a closed curve. Assertion 6.5 follows from the Slab Property (Claim 6.6) and our discussion in the paragraph just before Claim 6.6.
We now complete the proof of Theorem 6.1. Since X is not isomorphic to SU(2), it is either isomorphic to a semidirect product or it is isomorphic to SL(2, R). By Assertions 6.2, 6.3 and 6.5, γ f is a simple closed curve, which completes the proof of Theorem 6.1.
Corollary 6.7. Let f : (Σ, p) (X, e) be a pointed limit immersion that satisfies the hypotheses stated just before Theorem 6.1.
(1) If Σ is diffeomorphic to an annulus, then it has linear area growth.
(2) If Σ is simply connected, then one of the two following possibilities holds: (a) f factors through a pointed immersion f : ( Σ, p) (X, e) of an annulus. In this case, Σ has quadratic area growth and the constant mean curvature immersion f is stable.
(b) f is periodic, in the sense that there exists an element a Σ ∈ X − Γ Σ such that the left translation by a Σ leaves f (Σ) invariant.
Proof. Recall that f (Σ) is foliated by integral curves of K Σ . Since X is a simply connected Lie group diffeomorphic to R 3 , then the integral curves of K Σ are proper Jordan arcs. Let K Σ = f * (K Σ ) denote the Killing field induced on Σ by K Σ after pulling back K Σ through f . Assume that Σ is diffeomorphic to an annulus and we will show that item 1 holds. Consider the natural projection Π Γ Σ : X → X/Γ Σ , where X/Γ Σ denotes the quotient surface whose points are the integral curves of K Σ in X. Since Σ is an annulus, then (Π Γ Σ • f )(Σ) is an immersed closed curve β ⊂ X/Γ Σ and f : Σ → β is a trivial R-bundle over β, whose fibers are the integral curves of K Σ . Take a global section of this trivial bundle. The image set of this global section is an embedded closed curve α ⊂ Σ which intersects transversely each of the integral curves of K Σ . Since K Σ is a Killing vector field, then the image ψ t (α) of α through the flow {ψ t | t ∈ R} by isometries of Σ associated to K Σ produces a foliation of Σ by curves isometric to α, so in particular all of these curves ψ t (α) have the same length. From here it is straightforward to show that Σ has linear area growth, so item 1 is proved.
Next suppose that Σ is simply connected. Consider the quotient space Σ/ K Σ of integral curves of K Σ . Thus, Σ/ K Σ is a 1-dimensional manifold diffeomorphic to R, and there is a natural projection Π : Σ → Σ/ K Σ . We can identify Σ/ K Σ = R, so that Π(p) = 0 for the base point p of Σ satisfying f (p) = e. Let G : R = Σ/ K Σ → γ f ⊂ S 2 be the mapping such that G = G • Π, where G stands for the left invariant Gauss map of f . Since γ f is a simple closed regular curve by Theorem 6.1, it is easy to check that the map G is the universal cover of γ f . Let τ : Σ/ K Σ → Σ/ K Σ be one of the two generators of the group of automorphisms of the covering G : Since the left invariant Gauss map determines the surface up to left translations by [25,Theorem 3.7], given a point q ∈ Π −1 (τ (0)), the left translation l f (q) : X → X induces a nontrivial isometry l q : Σ → Σ that is a lift of τ to Σ, i.e., τ • Π = Π • l q , and satisfies l q (p) = q. Note that l q acts freely on Σ and the quotient Σ = Σ/ l q is an annulus. We now have two possibilities.
In any of the two possibilities above, the isometry group Iso(Σ) of the induced metric on Σ by f contains an R-type subgroup (which corresponds to left translations by any element in Γ Σ ) and a Ztype subgroup (which corresponds to the subgroup generated by l q ), such that these two subgroups generate a subgroup ∆ of Iso(Σ), although ∆ is not necessarily isomorphic to R × Z, because elements in the R-type subgroup do not necessarily commute with those in the Z-subgroup; also note that Σ would have at most quadratic area growth provided that ∆ is isomorphic to R × Z.
If case (R1) holds, then the quotient metric on Σ (which is the induced metric by f ) has linear area growth, and so Σ has quadratic area growth. Since f is stable and the cover Σ → Σ is cyclic, then the quotient immersion f : Σ X is also stable (see for instance [28,Proposition 2.5]), which proves that item 2a in the statement of Corollary 6.7 holds. Finally, if case (R2) holds, then we define a Σ = f (q), which lies in X − Γ Σ since f (p) = e ∈ Γ Σ and the integral curves f (Π −1 (0)), f (Π −1 (τ (0))) of K Σ are disjoint. Then, the left translation by a Σ leaves f (Σ) invariant and we have item 2b of the corollary. Now the proof is complete.

Proof of Theorem 1.4 when X is a semidirect product
In this section we will prove Theorem 1.4 in the case that X is a semidirect product R 2 A R endowed with its canonical metric.
Let S n be a sequence of constant mean curvature spheres in X with Area(S n ) > n for all n. In order to prove Theorem 1.4 in X it suffices to consider the case where all the spheres S n lie in the connected component C of the space of index-one spheres in X with H > h 0 (X) given in Proposition 3.4. Indeed, if Theorem 1.4 holds for any such sequence of spheres (S n ) n ⊂ C, the discussion after Definition 3.5 ensures that any H-sphere Σ in X satisfies that H > h 0 (X), and so Theorem 1.4 holds in full generality.
So, assuming these conditions (in particular, that S n ∈ C for every n), Proposition 5.4, Theorem 6.1 and Corollary 6.7 imply that there exists a pointed limit immersion f : (Σ, p) (X, e) of the spheres S n with the following properties: (S1) f is complete, has constant mean curvature h 0 (X), it is stable and its left invariant Gauss map image γ f is either a point or a regular, simple closed curve in S 2 ⊂ T e X.
(S2) f is everywhere tangent to a nonzero, right invariant Killing vector field K Σ in X. Moreover, exactly one of the following three situations happens: (S2.1) Σ is diffeomorphic to an annulus, and it has linear area growth.
(S2.2) Σ is simply connected and f : Σ X factors through an immersion f : Σ X of an annulus. In this case, Σ has quadratic area growth and f is a stable immersion with constant mean curvature.
(S2.3) There exists a ∈ X − Γ Σ such that the left translation by a leaves f (Σ) invariant.
Here, Γ Σ is the 1-parameter subgroup of X that generates K Σ .
Recall that if γ f is a point, then f (Σ) is a two-dimensional subgroup of X. In particular f (Σ) is an entire Killing graph with respect to some right invariant vector field in X, and so Theorem 1.4 holds. Hence, from now on we will assume that f (Σ) is not a two-dimensional subgroup; in particular, γ f is a regular Jordan curve in S 2 , and K Σ is unique (up to scaling) among right invariant vector fields on X everywhere tangent to f (Σ).
First assume that the matrix A is singular. By Lemma 4.5, X is isometric to R 3 or to some E(κ, τ ) space with κ ≤ 0. By the classification theorems of Hopf, Chern and Abresch-Rosenberg cited in the Introduction, constant mean curvature spheres in these spaces are embedded and rotational, and the moduli space of such spheres can be parameterized analytically by the constant values of their mean curvatures, which take all possible values in (H(X), ∞). In particular, h 0 (X) = H(X), which is the mean curvature of the subgroup R 2 A {0} with respect to its upward-pointing unit normal vector. If γ f passes through the North or South pole of S 2 , then Claim 6.4 ensures that f (Σ) in contained in a smallest horizontal slab which is a contradiction. Hence, γ f does not pass through the North or South poles of S 2 . As γ f is a closed curve, then γ f is at positive distance from the North and South poles. In this situation, Assertion 6.3 implies that f (Σ) is an entire Killing graph with respect to some nonzero Killing vector field in X. Thus, Theorem 1.4 holds in this situation.
So, from now on, we will assume that the matrix A is regular. By the previous discussion, the next result directly implies that Theorem 1.4 holds for any metric semidirect product X, as desired.
Theorem 7.1. In the above conditions, K Σ is horizontal, i.e., K Σ ∈ Span{F 1 , F 2 }, and f (Σ) is an entire Killing graph in X with respect to any nonzero right invariant Killing vector field V ∈ Span{F 1 , F 2 } that is linearly independent from K Σ (with the notation of (4.4)).
Observe that if Theorem 7.1 holds, then case (S2.3) occurs for f (because in cases (S2.1) and (S2.2) the image surface f (Σ) is an immersed annulus in X, which contradicts that f (Σ) is an entire graph).
We will divide the proof of Theorem 7.1 into four steps. In Section 7.1 we will prove that Theorem 7.1 holds when K Σ ∈ Span{F 1 , F 2 }; thus, Theorem 7.1 will be proved provided that we find a contradiction whenever K Σ is not horizontal. Assuming that K Σ / ∈ Span{F 1 , F 2 }, in Sections 7.2, 7.3, 7.4 we will respectively show that cases (S2.3), (S2.2), (S2.1) above cannot occur. This will complete the proof of Theorem 7.1 (and thus of Theorem 1.4 for X a metric semidirect product).
7.1 Proof of Theorem 7.1 when K Σ is horizontal Lemma 7.2. If K Σ ∈ Span{F 1 , F 2 }, then f is an entire Killing graph with respect to any right invariant vector field V ∈ Span{F 1 , F 2 } that is linearly independent from K Σ .
Proof. In this case, K Σ = λ∂ x + µ∂ y for some λ, µ ∈ R. Since f (Σ) is foliated by integral curves of K Σ , then f (Σ) is ruled by straight lines in the direction of K Σ . By Assertion 6.3, we conclude that the lemma will hold provided that γ f does not pass through the North or South pole of S 2 . Arguing by contradiction, assume that γ f passes through the North or South pole of S 2 . Then, f is in the conditions of Claim 6.4, which ensures that f (Σ) is contained in a smallest slab-type region A {b} in nonempty sets whose components are integral curves of K Σ , and f (Σ) transversely intersects every intermediate plane a, b]. Thus, we can consider f : Σ Y to be a complete flat immersion, which is ruled by straight lines parallel to K Σ . In particular, Σ has quadratic area growth with respect to the pullback metric of Y through f . Since the coefficients of the canonical metric on R 2 A R with respect to the (x, y, z)-coordinates only depend on z by equation (4.5), we conclude that the identity map ] is a quasi-isometry. From here, we deduce that Σ has at most quadratic area growth with respect to the pullback metric of X through f .
Let V be any horizontal right invariant vector field on X that is linearly independent from K Σ . As the coefficients of the canonical metric are bounded in every horizontal slab of finite width by equation (4.5), then V is bounded in R 2 A [a, b]. Denoting by N the unit normal vector to f , we deduce that J := N, V is a bounded Jacobi function on Σ that changes sign (changes of sign of J occur at points of f −1 (R 2 A {a, b})). As f : Σ X is a complete stable h 0 (X)-surface with at most quadratic area growth, then the conformal structure of Σ is parabolic, and Corollary 1 in Manzano, Pérez and Rodríguez [20] implies that J cannot change sign, which is a contradiction. This contradiction completes the proof of the lemma.

Case (S2.3) is impossible when K Σ is not horizontal
Assume that K Σ / ∈ Span{F 1 , F 2 }. The next lemma reduces this case to the specific nonhorizontal, right invariant vector field K Σ = F 3 given by (4.4), which is generated by the 1-parameter subgroup of X corresponding to the z-axis. In proving the next lemma, we will be making use of the assumption that A is a regular matrix, as discussed just before the statement of Theorem 7.1. Lemma 7.3. If K Σ / ∈ Span{F 1 , F 2 }, then after a horizontal left translation of f and a scaling of K Σ , the following statements hold: (2) Case (S2.3) does not happen.
Proof. Since K Σ / ∈ Span{F 1 , F 2 }, then after scaling and using (4.2) and (4.4), for some s, t ∈ R, where p = (x, y), q = (s, t). Given P 0 = (p 0 , 0) ∈ R 2 A {0}, the right invariant vector field (l P 0 ) * (K Σ ) is everywhere tangent to the left translation of f by P 0 . Hence, to prove item 1 of the lemma it suffices to find p 0 ∈ R 2 such that (l P 0 ) * (K Σ ) = F 3 . By direct computation using the group operation in R 2 A R, see Section 4.2, we have that l P 0 (p, z) = (p 0 + p, z) is an Euclidean translation, hence its differential is the identity after identifying the tangent spaces of X at both points (p, z), (p 0 + p, z) with R 3 in the basis ∂ x , ∂ y , ∂ z . Therefore, the value of (l P 0 ) * (K Σ ) at l P 0 (p, z) is given by hence we only need to define p 0 such that Ap 0 = q, which can be done since A is regular. This proves the first statement in the lemma.
To prove the second assertion, assume that case (S2.3) holds with K Σ = F 3 . Hence, there exists a ∈ X − Γ Σ = X − {z-axis} such that the left translation by a leaves f (Σ) invariant. Writing a = (p 0 , z 0 ) ∈ R 2 A R as the product (p 0 , 0) * (0, z 0 ) and using that (0, z 0 ) ∈ Γ Σ , we conclude that a can be chosen so that z 0 = 0. As l a leaves f (Σ) invariant and F 3 is unique up to rescaling among right invariant vector fields which are nonzero and everywhere tangent to f (Σ), then (l a ) * (F 3 ) = λ F 3 for some λ ∈ R. But the value at l a (p, z) = (p 0 + p, z) of (l a ) * (F 3 ) is is given by (7.1). Since A is a regular matrix, then we deduce that (l a ) * (F 3 ) cannot be a multiple of F 3 . This completes the proof.

Case (S2.2) is impossible when K Σ is not horizontal
Assume that K Σ / ∈ Span{F 1 , F 2 } and that case (S2.2) above holds. By item 1 of Lemma 7.3, we can assume after a horizontal left translation that K Σ = F 3 . This of course may produce a change of base point for f , of the form f (p) := Q ∈ R 2 A {0}. Recall from the beginning of Section 7 that f : (Σ, p) (X, e) is obtained as a limit of pointed limit immersions f n : (S n , p n ) (X, e) of spheres S n ∈ C. Applying to the f n the same horizontal left translation that we applied to f in order to have K Σ = F 3 , we have f n (p n ) = Q for all n.
By Proposition 3.4, each sphere S n ∈ C is Alexandrov embedded. In this way, for each n ∈ N there exists a three-dimensional Riemannian manifold (B n , g n ) which is topologically a closed ball and a Riemannian submersion f n : (B n , g n ) → X such that ∂B n = S n , f n | Sn = f n and ∂B n is mean convex. Lemma 7.4. In the conditions above, there is some R * > 0 and points q n ∈ S n such that the volumes of the Riemannian metric balls B gn (q n , R * ) in (B n , g n ) of radius R * centered at q n tend to ∞ as n → ∞.
Proof. As explained in Remark 4.2, the surface f (Σ) can be obtained by pulling back via Π K Σ a closed immersed curve α contained in X/K Σ . Since in our case K = K Σ = F 3 and every integral curve of F 3 intersects any horizontal plane R 2 A {z 0 } transversely at a single point, we can identify X/K Σ with R 2 A {0}.
Consider the vertical geodesic of X Given r > 0, let W(Γ, r) denote the solid metric cylinder of radius r around Γ, i.e., the set of points of X whose distance to Γ is at most r. Since f (Σ) is an annulus invariant under the flow of F 3 , and since all cylinders W(Γ, r) are also invariant under this flow, it is clear that f (Σ) is contained in a solid metric cylinder W 0 := W(Γ, r 0 ) for some r 0 > 0, such that ∂W 0 is at a positive distance from f (Σ). By [24, Theorem 1.1], there exists some R = R(r 0 ) > 0 associated to W 0 with the following property: ( ) If M is any compact, stable minimal surface in X whose boundary is contained in W 0 , then the intrinsic radius 4 of M is less than R.
In particular, since the mean curvature of the complete stable surface f (Σ) is h 0 (X) and f (Σ) is contained in W 0 , it follows that f (Σ) cannot be minimal, thus, h 0 (X) > 0. Let us also consider the positive number D 0 = D 0 (r 0 ) defined in the following way. Observe that W 0 is foliated by integral curves of F 3 . We define D 0 as the maximum length of all the arcs of integral curves of F 3 that are contained in the compact piece of W 0 that lies in the slab determined by the planes R 2 As Σ is simply connected and f : Σ X is invariant under the flow of F 3 , then by the discussion in Remark 4.2 we may parameterize f as f (s, t) = l Γ(t) (α(s)), (s, t) ∈ R 2 , (7 .3) where α = α(s) : R → R 2 A {0} is a curve parameterized by arc length in R 2 A {0}, and so that under the natural identification Σ ≡ {(s, t) ∈ R 2 }, the base point p ∈ Σ for f corresponds to (0, 0). In particular, f (0, 0) = α(0) = Q. Note that since we are in the conditions of case (S2.2), the image α(R) is an immersed closed curve and α(s) is L-periodic, where L is the length of α(R).
For each s 0 ∈ R we may define numbers c − (s 0 ) < 0, c + (s 0 ) > 0 such that is the arc of the integral curve of F 3 that passes through α(s 0 ) whose endpoints lie in , and that the length of A(s 0 ) is at most D 0 . Take some a ∈ (0, L) and consider for each j ∈ N the compact region of Σ (viewed as R 2 with coordinates (s, t)) given by see Figure 5. Figure 5: Left: the compact regions K 1 ⊂ K 2 in the (s, t)-plane. The vertical segments A(0), A(L), A(2L) apply through f into the compact arc A(0), which is part of the integral curve of F 3 passing through Q. Each K j wraps j times through f around its image, due to the L-periodicity of f in the s-variable.
Since f is a limit surface of the f n , it follows that for n large enough, there exist compact simply connected domains K j n ⊂ S n such that f n (K j n ) ⊂ W 0 and the sequence {f n | K j n } n converges to f | K j uniformly as n → ∞. For ε > 0 sufficiently small and less than the injectivity radius of Σ, denote by U k the closed geodesic disk in Σ of radius ε centered at (kL, 0) with k ∈ {1, . . . , j}. After choosing ε > 0 sufficiently small, we may assume that {U 1 , . . . , U j } forms a pairwise disjoint collection, and the intrinsic distance between U i , U k for i, k ∈ {1, . . . , j}, i = k, is greater than 2δ for some δ > 0 independent of j. Also, there exist closed geodesic disks of radius ε, U k n ⊂ K j n , such that {f n | U k n } n converges to f | U k uniformly as n → ∞. Observe that for each j, there exists n 0 (j) ∈ N such that if n ≥ n 0 (j), the related compact disks U k n ⊂ K j n satisfy that the intrinsic distance between U i n and U k n inside K j n is greater than δ for every i, k ∈ {1, . . . , j}, i = k. It is important to notice that δ does not depend on j, since f (s, t) is L-periodic with respect to s.
Let I denote a small compact segment in R 2 A {0} centered at Q and transversal to f (Σ) at Q. Let α n ⊂ S n denote the set f −1 n (R 2 A {0}). Since f is a limit of the f n and f (Σ) intersects R 2 A {0} transversely, then by the Transversality Lemma (Lemma 3.1 in [23]), α n is a closed simple curve in S n . As f (s, t) is L-periodic in the variable s, the previous convergence properties imply that for n = n(j) large enough there exist j points q 1 n , . . . , q j n ∈ α n ∩ K j n such that, for every k = 1, . . . , j, we have q k n ∈ U k n and f n (q k n ) ∈ I. In particular, we note for later use the following property: ( ) For n large enough, the intrinsic compact metric disks D Sn (q k n , δ/2) in S n of radius δ/2 and center q k n , k = 1, . . . , j, are pairwise disjoint.
Let J k n be the compact connected arc of α n contained in K j n whose endpoints are q 1 n and q k n , for each k ∈ {1, . . . , j}. Let W n be the vector field on S n obtained by pulling back via f n the tangent part to f n (S n ) of the right invariant vector field K Σ = F 3 ; clearly, W n has no zeros on K j n ⊂ S n for n large enough, since K Σ has no zeros, is everywhere tangent to f (Σ) and {f n | K j n } n converges to f | K j uniformly as n → ∞. Moreover, again for n large enough, the integral curves of W n that start at any point of any of the arcs J k n satisfy that their images by f n intersect both planes R 2 A {−R − 1} and R 2 A {R + 1}. In particular, we can define the compact disks D k n ⊂ S n obtained by letting the arc J k n flow under W n between the sets f −1 In fact, D k n ⊂ K j n for all k ∈ {1, . . . , j} and for n large. Let γ k n := ∂D k n , which is a Jordan curve in S n . Observe that γ k n can be written as a union A n 1 ∪ A n 2 ∪ B n 1 ∪ B n 2 , where: (T1) A n 1 , A n 2 are compact arcs of integral curves of W n that pass through q 1 n and q k n , respectively. In particular, the endpoints of both f n (A n 1 ) and f n (A n 2 ) lie on the planes R 2 Moreover, since f n (A n 1 ) and f n (A n 2 ) converge uniformly as n → ∞ to proper subarcs of the compact arc A(0) defined by equation (7.4), then the lengths of A n 1 and A n 2 are smaller than the constant D 0 for n large enough.
(T2) The arc f n (B n 1 ) (resp. f n (B n 2 )) lies in the plane R 2 Recall that (B n , g n ) is the abstract Riemannian three-ball that S n bounds, and f n : (B n , g n ) → X is a Riemannian submersion with mean convex boundary f n | ∂Bn = f n . In particular, S n is a good barrier to solve Plateau problems in B n in the sense of Meeks-Yau [32]. More precisely, for each k ∈ {1, . . . , j} (and n large enough), there exists a compact stable embedded minimal disk M k n ⊂ B n of least area, with ∂M k n = γ k n . Furthermore, also by [32], M k n is an immersion transverse to ∂B n at the regular points of its boundary γ k n . In particular, M k n is an immersion transverse to ∂B n at the interior points of the arcs A n 1 and A n 2 . Therefore, if we denote by Λ n ⊂ B n the embedded surface {q ∈ B n | f n (q) ∈ R 2 A {0}} and β k n := M k n ∩ Λ n , then β k n ∩ ∂M k n = {q 1 n , q k n }, and the intersection of M k n with Λ n is clearly transverse at q 1 n , q k n since the arcs A n 1 , A n 2 are transverse to Λ n . Moreover, up to a small perturbation of Λ n in B n that fixes its boundary, we may assume that the intersection of M k n with Λ n is transverse. Therefore, β k n is a (possibly disconnected) compact 1-dimensional manifold with boundary, and a connected component of β k n is a compact arc in B n joining q 1 n with q k n . We will keep denoting this compact arc by β k n , see Figure 6. Take r ∈ β k n . We next give a lower bound of d X ( f n (r), f n (B n 1 ∪ B n 2 )), the distance in X from f n (r) to f n (B n 1 ∪ B n 2 ). Since f n (B n 1 ∪ B n 2 ) is contained in the union of planes can be taken arbitrarily close to R + 1. In particular, d X ( f n (r), f n (B n 1 ∪ B n 2 )) > R, for all r ∈ β k n . (7.6) As f n | M k n : M k n X is by construction a compact immersed stable minimal disk in X whose boundary is contained in the solid metric cylinder W 0 , we deduce from the radius estimate ( ) above that for any r ∈ β k n the distance d M k n (r, γ k n ) in M k n from r to γ k n = ∂M k n is not greater  Figure 6: The least area disk M k n inside the abstract Riemannian ball B n has the same boundary γ k n as the disk D k n ⊂ S n = ∂B n . γ k n consists of four consecutive arcs A n 1 , B n 1 , A n 2 , B n 2 satisfying properties (T1), (T2). than R. Note that d M k n (r, B n 1 ∪ B n 2 ) ≥ d X ( f n (r), f n (B n 1 ∪ B n 2 )), which by (7.6) is greater than R. Therefore, , for all r ∈ β k n . (7.7) Moreover, since q 1 n ∈ A n 1 and q k n ∈ A n 2 , we deduce by the continuity and connectedness of the arc β k n that there exists some midpoint r 0 ∈ β k n such that d M k n (r 0 , A n 1 ) = d M k n (r 0 , A n 2 ). Thus, we can estimate the distance in B n from r 0 to A n i , i = 1, 2, by (by equation (7.7)) ≤ R.
(inequality (7.8)) (7.9) Recall that, by item 5 of Proposition 3.4, the norms of the second fundamental forms of the spheres S n are uniformly bounded by some constant C > 0. Also, note that since we have already proved that h 0 (X) > 0 in this case, the values H n of the mean curvatures of S n are uniformly bounded away from zero (since H n > h 0 (X) for every n). Hence, by [29], there exists some ε 1 > 0 smaller than the injectivity radius of B n such that S n = ∂B n has a regular neighborhood V ε 1 n in B n of fixed size ε 1 (independent of n). In other words, we have a diffeomorphism where N n is the inward pointing unit normal in (B n , g n ) of the mean convex sphere S n = ∂B n and exp n denotes the exponential map in (B n , g n ).
By Property ( ) above, for n large enough, the solid cylinders C 1 n , . . . , C j n in V ε 1 n given by the image under Φ n of the compact cylinders D Sn (q k n , δ/2) × [0, ε 1 /2] are mutually disjoint, see Figure 6. Also, note that the volume of each C k n in (B n , g n ) is greater than some V > 0, independent of n, j and k (again this follows from the uniform bound of the second fundamental forms of the S n , as well as from the fact ε 1 does not depend on n). In particular, for n large enough, the total volume in (B n , g n ) of the (disjoint) union C 1 n ∪ · · · ∪ C j n is at least jV . Note that this volume can be made arbitrarily large, as j ∈ N was chosen fixed but arbitrary.
Also, by the triangle inequality we have for any x ∈ C k n , k ∈ {1, . . . , j}, that d Bn (q k n , x) ≤ ε 1 +δ 2 . This inequality and (7.9) imply that from where we deduce that C 1 n ∪ · · · ∪ C j n is contained in the Riemannian ball of (B n , g n ) centered at q 1 n and of radius R * := 2(D 0 + R) + ε 1 +δ 2 .
In particular, we have proved the following: given a fixed but arbitrary j ∈ N, then for all n ∈ N large enough, the metric ball B gn (q 1 n , R * ) in the Riemannian manifold (B n , g n ) centered at q 1 n ∈ ∂B n of radius R * has volume at least jV , where V, R * > 0 do not depend on j, n. This clearly proves Lemma 7.4.
We will next obtain a contradiction with Lemma 7.4, that will prove that case (S2.2) is impossible when K Σ / ∈ Span{F 1 , F 2 }, which was the objective of the present Section 7.3. First, observe that as the second fundamental forms of the f n are uniformly bounded, there exists δ > 0 such that, for all n, the following properties hold: (U1) (B n , g n ) can be extended to a compact Riemannian three-manifold with boundary ( B n , g n ), in the sense that B n is a compact subset of B n and g n | Bn = g n .
is a diffeomorphism, where exp n denotes the exponential map of ( B n , g n ) and N n is the unit normal vector to S n at x with respect to g n such that H n N n is the mean curvature vector of S n .
(U3) The metrics g n are uniformly bounded in the C 2 -topology.
Let ϕ : (−δ, 0] → (0, ∞) be a smooth positive function such that ϕ = 1 in [−δ/3, 0] and ϕ(t) = 1 t+δ in (−δ, −2δ/3]. We define the complete Riemannian metric g n on B n − ∂ B n by (we are identifying S n × [−δ, 0] with B n − Int(B n ) through the diffeomorphism that appears in property (U2)). Observe that g n = 1 (t + δ) 2 g n in S n × (−δ, −2δ/3]. (7.10) As g n | ∂ Bn is a Riemannian metric on the closed surface ∂ B n , then (7.10) implies that given ε > 0, there exists δ 1 = δ 1 (ε) ∈ (2δ/3, δ) such that the restriction of g n to S n × (−δ, −δ 1 ] is εclose in the C 2 -topology to the hyperbolic metric of constant sectional curvature −1. In particular, the Ricci curvature of g n satisfies where b(ε) can be made arbitrarily small for ε small. As the coefficients of the Ricci tensor of ϕ 2 g n are smooth expressions of the coefficients of g n , of those of the Ricci tensor of g n and of ϕ, ϕ , ϕ , then property (U3) and the smoothness of ϕ imply that the Ricci curvature of ϕ 2 g n satisfies for all n: Ric ϕ 2 gn is uniformly bounded in S n × [−δ 1 , 0]. (7.12) As g n is locally homogeneous, (7.11) and (7.12) imply that there exist c < 0 independent of n such that Ric gn ≥ 2c in B n − ∂ B n for all n. Since ( B n − ∂ B n , g n ) is a complete Riemannian manifold, then Bishop's comparison theorem gives the following upper bound for the volume of the metric ball B gn (q n , R * ) in ( B n − ∂ B n , g n ) of radius R * > 0 centered at q n : where V (c, R * ) denotes the volume of any metric ball of radius R * in the three-dimensional space form of constant sectional curvature c. This is a contradiction with Lemma 7.4, as desired.

Case (S2.1) is impossible when K Σ is not horizontal
Assume that case (S2.1) holds and K Σ / ∈ Span{F 1 , F 2 }. We start with the same normalizations as in the previous Section 7.3. So, K Σ = F 3 , and f : (Σ, p) (X, Q) (here Q ∈ R 2 A {0}) is obtained as a limit of pointed immersions f n : (S n , p n ) (X, Q) of spheres S n ∈ C. Also, we will consider the compact Riemannian balls (B n , g n ) with ∂B n = S n , and Riemannian submersions f n : (B n , g n ) → X such that ∂B n is mean convex and f n | ∂Bn = f n . Let α n := f −1 n (R 2 A {0}) ⊂ S n , which is a simple closed curve in S n by the Transversality Lemma (Lemma 3.1 in [23]). Also, let K n Σ be the Killing field on B n that is induced by K Σ via the pullback by the submersion f n . Recall that in this case (S2.1), Σ is an annulus. Observe that f (Σ) is the immersed annulus obtained by letting the immersed closed curve is a simple closed curve in Σ, and the sequence {f n | αn } n converges uniformly to the immersion f | α∞ .
By Theorems 1 and 2 in [32], α n is the boundary of a smooth, embedded least-area disk M (n) in B n and this disk is transverse to S n = ∂B n along its boundary α n . Note that the intrinsic radii of the disks M (n) are bounded by a universal constant R > 0 by Theorem 1.1 in [24], since their compact image boundary curves f n | ∂M (n) = f n | αn converge uniformly as n → ∞ to f | α∞ , and f (α ∞ ) is contained in a solid metric cylinder around Γ = {(0, 0, t) | t ∈ R} (so the same holds for f n (∂M (n)) for all n ∈ N, for a slightly larger solid metric cylinder around Γ). It follows that a subsequence of the immersed stable minimal disks f n (M (n)), which lie locally in the mean convex side of f n (S n ) near f n (α n ), converge to an immersed stable minimal disk M (∞) in X with boundary f (α ∞ ).
We claim that M (n) lies in the piecewise smooth, closed upper half-ball To see this, first recall that for every z 0 ∈ R, the halfspace of the form R 2 A [z 0 , ∞) is mean convex in X, and the constant mean curvature of its boundary is equal to H(X). Since f n (α n ) ⊂ R 2 A {0} is the boundary of the immersed minimal disk f n (M (n)), then the maximum principle (or the mean curvature comparison principle) applied to M (n) and to the smooth surfaces f −1 n (R 2 A {z}), z ∈ R, implies that M (n) ⊂ B + n as desired. We next claim that there exists an η ∈ (0, π/4) such that the angle that M (n) makes with S n along α n is greater than η for all n ∈ N large enough. Assume that the claim does not hold; in this case, M (∞) is tangent to f (Σ) at some boundary point Q ∈ ∂M (∞). As M (∞) lies locally in the mean convex side of f (Σ) along f (α ∞ ), then the Hopf boundary maximum principle implies that f (Σ) coincides with M (∞) nearby Q . This is a contradiction since we know that h 0 (X) > 0 in this case, as it was explained just after property ( ) in the previous section. Thus, the claim at the beginning of this paragraph holds.
Let N M (n) denote the unit normal vector field to M (n) in (B n , g n ). We claim that for n sufficiently large, g n (N M (n) , K n Σ ) has no zeros along α n . Arguing by contradiction, suppose that after passing to a subsequence, K n Σ is tangent to M (n) at some point q n ∈ α n for all n. Then K n Σ (q n ) and the unit tangent vector w n to α n at q n generate the tangent space T qn M (n). By the last paragraph, the angle between T qn M (n) and T qn S n is greater than η. Since K Σ is everywhere tangent to f (Σ), then the angle that K n Σ (q n ) makes with T qn S n tends to zero as n → ∞. Since K n Σ (q n ) is tangent to M (n) at q n , then the angle that K n Σ (q n ) makes with w n becomes arbitrarily small if n is large enough (note that w n ∈ T qn S n ∩ T qn M (n)). Applying the differential of f n at q n , we deduce that the angle between (K Σ ) fn(qn) = (F 3 ) fn(qn) and (df n ) qn (w n ) tends to zero as n → ∞. Since (df n ) qn (w n ) is a nonzero horizontal vector, this only can occur provided that (F 3 ) fn(qn) becomes horizontal as n → ∞, or equivalently, the ratio between the component of (F 3 ) fn(qn) in the direction of ∂ z and its horizontal component tends to zero. By equation (4.4), this only can occur if f n (q n ) diverges in R 2 A {0}. But this contradicts the fact that f (α ∞ ) lies at a finite distance from the origin. Hence the claim follows. Therefore, after replacing by a subsequence and choosing N M (n) so that the Jacobi function J n = g n (N M (n) , K n Σ ) is positive along α n , we deduce from the stability of the minimal disk M (n) that J n is positive on M (n).
Consider Consider the mapping T n : M (n) × R → X given by T n (q, t) = l Γ(t) ( f n (q)), (7.13) where Γ(t) is given by (7.2). As J n is positive on M (n), T n is a proper submersion of M (n) × R into X (note that the projection (x, t) → t is proper). Our next goal is to take limits of these immersions T n . To do this, we use a three-point condition. Let D be the closed unit disk in R 2 and parameterize each M (n) by a conformal diffeomorphism φ n : D → M (n) in such a way that the points (1, 0), (0, 1), (−1, 0) of ∂D have as images by f n • φ n three ordered points in f n (α n ) which converge to three ordered points in the limit curve f (α ∞ ) = ∂M (∞). In this case, after passing to a subsequence, the conformal immersions f n •φ n : D X converge smoothly to a limit conformal immersion f ∞ : D X that parameterizes M (∞); for this standard type of limit result, see [34] and [31] for details.
Note that as n → ∞, the proper immersions T n • (φ n × 1 R ) : D × R → X converge to the submersion T ∞ : D × R → X given by which again is a proper submersion since (x, t) ∈ D × R → R → t ∈ R is proper (because J ∞ is positive on M (∞)). Consider the codimension-one foliation of D × R given by the family of compact disks Let K 1 be the nowhere zero, smooth vector field on D × R obtained after pulling back K Σ through T ∞ . Clearly, each of the integral curves of K 1 in D × R intersects the disk D 0 in a single point. Thus, we can view D × R in a natural manner as being the topological product W := D 0 × R, and we endow this three-manifold with boundary with the pulled-back metric by T ∞ . We will find the desired contradiction as an application of Theorem 9.1 below, as we explain next.
Observe that the boundary ∂W = (∂D 0 ) × R of W satisfies Recall that f (Σ) has constant mean curvature h 0 (X) > 0 in X with respect to the unit normal field obtained as the limit of the inward pointing unit normal fields to the spheres S n ; see the paragraph after property ( ) in Section 7.3. Also note that the inward pointing unit normal field on ∂W has positive inner product with the inner conormal to M (∞) by the Hopf boundary maximum principle, and that we can view M (∞) as an embedded minimal disk in W . Since M (∞) can be viewed as the limit of the disks M (n), each of which lies in the mean convex ball B n bounded by S n , then we conclude that ∂W has constant mean curvature h 0 (X) with respect to the inward pointing normal field to its boundary (thus, W has mean convex boundary of constant mean curvature). The vector field K 1 is clearly a Killing field on W endowed with the pulled-back metric by T ∞ . Note that K 1 arises from the proper action of Γ = Γ(t), which we can view as a 1-parameter subgroup of isometries of W .
We claim that there exists a nowhere zero Killing field K 2 in W which is tangent to the foliation F defined in (7.15), bounded on the end of W given by the end representative D 0 × [0, ∞) and such that the closure in W of every integral curve of K 2 in Int(W ) is a compact arc with end points in ∂W . To see this, first note that by Lemma 4.6 there exists a nonzero horizontal right invariant vector field V in X that is bounded in the halfspace R 2 A [0, ∞). Let K 2 be the Killing field in W defined as the pullback by T ∞ of V . Since V is everywhere horizontal and bounded in R 2 A [0, ∞), then K 2 is tangent to F and bounded on the end representative represent the same end of the solid cylinder W . Finally, since T ∞ is proper, then the closure of every integral curve of K 2 in Int(W ) is a compact arc in a disk D t for some t ∈ R, with end points in ∂W . Thus our claim in this paragraph is proved.
By Theorem 9.1, the CMC flux (see Definition 9.2) We will arrive to a contradiction by showing that the above CMC flux is actually zero. Note that α n is homologous to zero in S n , and so, the homological invariance of the CMC flux gives that Flux(∂B n , α n , K n Σ ) = 0 for all n. But this sequence of numbers converges as n → ∞ to Flux(∂W, ∂D 0 × {0}, K 1 ), which gives the desired contradiction.

Proof of Theorem 1.4 when X is isomorphic to SL(2, R)
In this section we will prove Theorem 1.4 when X is a metric Lie group isomorphic to SL(2, R). As explained at the beginning of Section 7, to prove Theorem 1.4 it suffices to prove that if S n is a sequence of constant mean curvature spheres in X that lie in the connected component C of the space of index-one spheres in X with H > h 0 (X) defined in Proposition 3.4, and for which Area(S n ) → ∞ as n → ∞, then there exists a pointed limit immersion f : (Σ, p) (X, e) of this sequence such that f (Σ) is an entire Killing graph in X with respect to a right invariant Killing vector field in X. Also by the discussion at the beginning of Section 7, we can also assume that there exists a pointed limit immersion f : (Σ, p) (X, e) of the spheres S n such that properties (S1), (S2) hold.
By item 5 of Lemma 4.8, H(X) > 0. Therefore, by (3.1) we have h 0 (X) > 0, which implies that no immersions in ∆(f ) have constant left invariant Gauss map (because two-dimensional subgroups of X are minimal, see Corollary 3.17 in [27]). In particular, f (Σ) is not a two-dimensional subgroup of X and the right invariant vector field K Σ to which f (Σ) is everywhere tangent is unique up to scaling. Theorem 1.4 when X is isomorphic to SL(2, R) will follow directly from the next result.
Theorem 8.1. In the above conditions, f (Σ) is an entire Killing graph with respect to a nonzero right invariant vector field on X. Moreover, f satisfies the periodicity condition (S2.3) stated at the beginning of Section 7.
As explained in the paragraph just after the statement of Theorem 7.1, case (S2.3) must occur for f provided that f (Σ) is an entire Killing graph. So we only need to prove that f (Σ) is an entire Killing graph with respect to some nonzero right invariant vector field. The proof of this property will be divided into four steps, depending on the character of the right invariant vector field K Σ (elliptic, hyperbolic or parabolic), and on the situations (S2.1), (S2.2) and (S2.3). In Section 8.1 we will show that K Σ cannot be elliptic. In Section 8.2 we will prove that Theorem 8.1 holds if K Σ is parabolic. In Section 8.3 we will prove that Theorem 8.1 holds if K Σ is hyperbolic and case (S2.3) occurs. In Sections 8.4 and 8.5 we will respectively show that cases (S2.2) and (S2.1) are impossible when K Σ is hyperbolic. This discussion exhausts all possible cases, and thus proves Theorem 8.1, and therefore Theorem 1.4 when X is isomorphic to SL(2, R).
8.1 K Σ cannot be elliptic Lemma 8.2. K Σ is not generated by an elliptic 1-parameter subgroup of SL(2, R).
Proof. Let Γ E be any elliptic 1-parameter subgroup of SL(2, R) and let K E be the right invariant, elliptic vector field on SL(2, R) generated by the left action of Γ E on SL(2, R). Each elliptic 1parameter subgroup of SL(2, R) consists of the set of liftings to SL(2, R) of all rotations of H 2 around a given point; see Section 4.3. If a ∈ SL(2, R) satisfies (l a ) * (K E ) = λ K E for some λ ∈ R − {0}, then aΓ E a −1 = Γ E , see e.g., the proof of Lemma 4.9. Once here, it is well known that on an elliptic subgroup this is only possible if a ∈ Γ E . Suppose now that K Σ is generated by the left action on SL(2, R) of an elliptic 1-parameter subgroup Γ Σ = Γ E of SL(2, R) and we will obtain a contradiction. If case (S2.3) holds, then there exists a ∈ X − Γ Σ such that l a (f (Σ)) = f (Σ). As the right invariant vector field (l a ) * (K Σ ) is everywhere tangent to l a (f (Σ)) and K Σ is unique up to scaling among nonzero right invariant vector fields that are everywhere tangent to f (Σ), then we conclude that (l a ) * (K Σ ) = λ K Σ for some λ ∈ R, λ = 0. Thus, the first paragraph in this proof ensures that a ∈ Γ E , a contradiction. Therefore, case (S2.3) cannot hold.
In both cases (S2.1) and (S2.2), Remark 4.2 implies that f (Σ) is obtained by the flow under the elliptic right invariant vector field K Σ of an immersed closed curve contained in SL(2, R)/K Σ . In particular, since K Σ is Killing, the distance in X of f (Σ) to Γ E is bounded, or equivalently, f (Σ) is contained in a solid metric cylinder in X of fixed radius r > 0 centered along Γ E , where d X denotes distance in X. As every elliptic 1-parameter subgroup of X contains the center Z of SL(2, R), then so does Γ E , from where we deduce that W(Γ E , r) is invariant under every left translation in Z. Also, it is clear that every right invariant vector field of SL(2, R) descends to the quotient group SL(2, R)/Z given by the action on SL(2, R) of the group of left translations by elements of Z. As W(Γ E , r)/Z is compact, the restriction to W(Γ E , r) of every right invariant vector field in SL(2, R) is bounded in W(Γ E , r).
Let V be any nonzero right invariant vector field in X tangent to f (Σ) at f (p) = e and linearly independent from K Σ . As V is not everywhere tangent to f (Σ) and V is bounded in W(Γ E , r), then the Jacobi function J = V, N is not identically zero, vanishes at p and it is bounded on Σ (here N is the unit normal vector field to f ). Note that J changes sign on Σ by the maximum principle. As f : Σ → X is a complete stable h 0 (X)-surface with at most quadratic area growth, the existence of the function J contradicts [20, Corollary 1] (see the last paragraph of the proof of Lemma 7.2 for a similar argument). This contradiction proves the lemma.

Proof of Theorem 8.1 when K Σ is parabolic
In order to prove Theorem 8.1 if K Σ is parabolic, we will use the following auxiliary result: Lemma 8.3. Let Γ P , Γ H be two 1-parameter subgroups of SL(2, R) such that Γ P is parabolic, Γ H is hyperbolic and they generate a two-dimensional subgroup of SL(2, R). Let K P , K H denote the right invariant vector fields on SL(2, R) generated by Γ P , Γ H . Then, there exists a function h ∈ C ∞ ( SL(2, R)) such that the vector field K H − h K P satisfies: (1) K H − h K P is bounded in X = ( SL(2, R), , ) (regardless of the left invariant metric on SL(2, R)).
(2) If the isometry group of X has dimension four, then K H − h K P has constant length.
Proof. First note that if we decompose K H as K H = h K P + V for some h ∈ C ∞ ( SL(2, R)) and some smooth vector field V on SL(2, R), then the boundedness of V in X = ( SL(2, R), , ) does not depend on the left invariant metric , on SL(2, R), because all such metrics are quasiisometric by the identity map in SL(2, R). Also, note that the particular bound for V on each specific X might depend on the metric , . Thus, it suffices to prove item 2 of the lemma. We next prove item 2. Let H 2 = H 2 θ be the 2-dimensional subgroup of SL(2, R) that contains Γ P , Γ H (see the description of H 2 θ before Lemma 4.10). Consider the Lie group given by the semidirect product R (1) R = {(x, y) | x, y ∈ R}, endowed with the group operation (x, y) * (x , y ) = (x + e y x , y + y ). (8.1) R be a group isomorphism, which exists since both H 2 , R (1) R are noncommutative Lie groups of dimension two. As Φ 1 is an isomorphism, the push-forward vector fields K P = (Φ 1 ) * (K P ), K H = (Φ 1 ) * (K H ) are nowhere zero and right invariant on R (1) R. Furthermore, as the unique parabolic one-dimensional subgroup of Since {∂ x , x ∂ x + ∂ y } is a basis of right invariant vector fields on R (1) R and λ = 0, there exist µ 1 , µ 2 ∈ R such that Consider the left invariant metric , on R (1) R that makes Φ 1 an isometry (recall that H 2 has the induced metric from SL(2, R)). As ∂ y is left invariant on R (1) R, then the length of ∂ y with respect to , is constant (nonzero). Hence, equation (8.2) implies that the length of Parameterize by {c(t) | t ∈ R} the 1-parameter subgroup Γ 3 of SL(2, R) generated by E 3 . As the family {r c(t) (H 2 ) | t ∈ R} of right cosets of H 2 is a foliation of SL(2, R), we can extend h to a smooth function, also called h ∈ C ∞ ( SL(2, R)), so that h is constant along the orbits of the right action by elements in Γ 3 . Suppose now that the left invariant metric on X has isometry group of dimension four. Thus, the right translations r c(t) by elements in Γ 3 are isometries of the ambient metric (because Γ 3 (0) = (E 3 ) e given by (4.6) and E 3 is a left invariant Killing vector field of every left invariant metric SL(2, R) whose isometry group has dimension four). This property and the right invariance of K H , K P imply that the length of K H − hK P is constant along the orbits of the right action by elements of Γ 3 . As (K H )| H 2 − h (K P )| H 2 has constant length on H 2 , we conclude that K H − h K P has constant length on X and item 2 of Lemma 8.3 is proved.
The next lemma implies that Theorem 8.1 holds in the case that K Σ is parabolic, which is the objective of the present Section 8.2.
Lemma 8.4. Suppose that K Σ is parabolic. Let H 2 θ be the two-dimensional subgroup of SL(2, R) that contains the 1-parameter subgroup Γ Σ generated by K Σ . Let Γ H be any hyperbolic 1parameter subgroup of H 2 θ and K H the nonzero right invariant vector field on SL(2, R) generated by Γ H . Then, f (Σ) is a smooth entire Killing K H -graph in X = ( SL(2, R), , ).
Proof. Consider the foliation F of SL(2, R) by right cosets of H 2 θ . As Γ H (resp. Γ Σ ) is a 1parameter subgroup of H 2 θ , then the integral curves of K H (resp. K Σ ) are entirely contained in right cosets of H 2 θ ; see property (P5) just before Claim 6.6. Since f (Σ) is ruled by integral curves of K Σ , we have that if f (Σ) intersects a leaf r x (H 2 θ ) of F at a point x, then the integral curve r x (Γ Σ ) of K Σ passing through x is also contained in f (Σ) ∩ r x (H 2 θ ), and the angle that f (Σ) makes with r x (H 2 θ ) along r x (Γ Σ ) is constant, since left translations by elements of Γ Σ are isometries of X.
Fix one of the leaves of F, say r x (H 2 θ ), x ∈ SL(2, R). Since each of the integral curves of K H contained in r x (H 2 θ ) intersects exactly once the unique integral curve r x (Γ Σ ) of K Σ contained in r x (H 2 θ ), then it follows that Lemma 8.4 will hold provided that we show that f (Σ) intersects each of the right cosets r x (H 2 θ ) transversely. By straightforward modifications of the arguments in Assertion 6.5 (note that in that assertion, K Σ was assumed to be hyperbolic while we are now assuming that K Σ is parabolic; nevertheless the arguments in Assertion 6.5 still hold since they will be applied to the foliation by right cosets of H 2 θ , which are tangent to both K Σ and K H ), we deduce that one of the two following possibilities holds for f : (V1) f (Σ) makes angles with the leaves of F that are bounded away from zero.
(V2) f (Σ) is contained in a smallest slab-type region between two different leaves L := r x (H 2 θ ), L := r y (H 2 θ ) of F, for some x, y ∈ SL(2, R), and f (Σ) is tangent to L (resp. L ) along an integral curve Γ x (resp. Γ y ) of K Σ (in other words, the conclusions of Claim 6.6 hold).
A straightforward modification of the arguments in the paragraph just before Claim 6.6 implies that if case (V1) holds, then Lemma 8.4 also holds. Therefore, we will assume that case (V2) holds and we will find a contradiction.
Consider the Jacobi function J = K H , N on Σ, where N stands for the unit normal vector to f . As K Σ is parabolic and everywhere tangent to f (Σ), then Lemma 8.3 applied to K P := K Σ and K H ensures that J is bounded on Σ. Also, note that J vanishes along the curves in f −1 (L ∪ L ), which exist by condition (V2). Since each nodal domain Ω ⊂ Σ of J is a smooth infinite strip invariant under a 1-parameter group of isometries (the ones generated by the flow of K Σ ), then Ω has linear area growth. Hence, the closure Ω of Ω is a parabolic Riemannian surface with boundary, in the sense that there exists a sequence {ϕ j } j ⊂ C ∞ 0 (Ω) such that 0 ≤ ϕ j ≤ 1, the ϕ −1 j (1) form an increasing exhaustion of Ω and Ω |∇ϕ j | 2 ≤ 1 j for each j. As f is stable, there exists v ∈ C ∞ (Σ) such that v > 0 and Lv = 0 in Σ, where L is the Jacobi operator of f . Since J is bounded in Σ and changes sign, then we have a contradiction with Lemma 8.5 below. This contradiction proves the lemma. Lemma 8.5. Suppose Σ is a complete Riemannian surface without boundary. Suppose that L = ∆ + P is a Schrödinger operator with potential P ∈ C ∞ (Σ). Let u, v ∈ C ∞ (Σ) satisfying Lu = Lv = 0, with u not identically zero and v > 0 in Σ. If Ω ⊂ Σ is a nodal domain of u such that Ω is parabolic and u is bounded in Ω, then u is a nonzero multiple of v and Ω = Σ.
Proof. The arguments in the proof of Theorem 1 in [20] are still valid in our current setting and we leave the details to the reader. 8.3 Proof of Theorem 8.1 when K Σ is hyperbolic and case (S2.3) holds Lemma 8.6. Suppose that K Σ is generated by a 1-parameter hyperbolic subgroup Γ H of SL(2, R), and that case (S2.3) holds. Let H 2 θ be one of the two two-dimensional subgroups of SL(2, R) containing Γ H . Let Γ P , K P be the parabolic subgroup of H 2 θ and the right invariant vector field on SL(2, R) generated by Γ P , respectively. Then, f (Σ) is a smooth entire Killing K P -graph.
Proof. We argue in a similar way as in the proof of Lemma 8.4. Consider the foliation F = {r t (H 2 θ ) | t ∈ Γ 3 } of SL(2, R) by right cosets of H 2 θ , where Γ 3 is the elliptic 1-parameter subgroup associated to E 3 given by equation (4.6). Recall that the leaves of F are equidistant surfaces to H 2 θ . Suppose first that the angles that f (Σ) makes with F are bounded away from zero. Then, f (Σ) is an entire Killing graph with respect to K P as explained just before Claim 6.6, and so Lemma 8.3 holds in this case. Otherwise, we can apply the Slab Property in SL(2, R) (Claim 6.6) to conclude that f (Σ) is contained in a smallest slab-type region S between two different leaves L, L of F. As we are in the conditions of case (S2.3), there is an element a ∈ SL(2, R) − Γ H such that l a (f (Σ)) = f (Σ). As f (Σ) is everywhere tangent to the right invariant vector field K Σ and f (Σ) is not a two-dimensional subgroup of SL(2, R), then there exists λ ∈ R − {0} such that (l a ) * (K Σ ) = λ K Σ . Observe that the 1-parameter subgroup of SL(2, R) generated by (l a ) * (K Σ ) is aΓ Σ a −1 = (l a • r a −1 )(Γ Σ ) (see e.g., the proof of Lemma 4.9), hence this last 1-parameter subgroup must be a reparameterization of Γ Σ . By Lemma 8.7 below, either a 2 lies in the center Z of SL(2, R), or a is the product of an element h ∈ Γ H with an element b ∈ Z. In the first case, a 2 = e since otherwise the left translation by a large power a 2k of a 2 sends the topological slab S into another topological slab disjoint from S, which contradicts that f (Σ) is invariant under the left translation by a 2k and f (Σ) is contained in S. But a 2 = e implies a = e, which contradicts our hypothesis that a ∈ SL(2, R) − Γ H . Therefore, a = bh with h ∈ Γ H and b ∈ Z. Observe that b = e since a / ∈ Γ H . Therefore, a large power a n = b n h n of a sends the topological slab S into another topological slab disjoint from S and we find a contradiction as before. This proves Lemma 8.6. Lemma 8.7. Suppose a ∈ SL(2, R) satisfies aΓ H a −1 = Γ H for some hyperbolic 1-parameter subgroup Γ H of SL(2, R). Let P : SL(2, R) → PSL(2, R) = SL(2, R)/Z be the quotient homomorphism where Z is the center of SL(2, R). Then P(a) ∈ P(Γ H ) or a 2 ∈ Z.
Proof. We identify P(Γ H ) with the set of hyperbolic translations of the Poincaré disk D along a geodesic γ. Since P(a)P(Γ H )P(a −1 ) = P(Γ H ), we conclude that P(a) lies in the normalizer subgroup of P(Γ H ) in PSL(2, R). In this case, it is well known that P(a) ∈ P(Γ H ) or P(a) is an elliptic isometry of order two around some point in γ. In the second case, P(a) 2 is the identity transformation of D, or equivalently a 2 lies in the kernel of P which is Z.

Case (S2.2) is impossible if K Σ is hyperbolic
The proof that f (Σ) cannot be in the conditions of case (S2.2) if K Σ is hyperbolic will be an adaptation to the SL(2, R) setting of our arguments in Section 7.3. In order to do this, we prove first a radius estimate (Lemma 8.9) for compact stable minimal surfaces with boundary contained in some special regions of SL(2, R), similar in spirit to Theorem 1.1 in [24]. To start, we next associate to every hyperbolic 1-parameter subgroup Γ H of SL(2, R) a cylindrical-type neighborhood of Γ H in X, which depends on the left invariant metric and on a number r > 0. We will call this neighborhood of Γ H an infinite box.
Definition 8.8. Let Γ H be a hyperbolic 1-parameter subgroup of SL(2, R), and let γ ⊂ H 2 (−4) be the geodesic with the same extrema as the circle arc Π(Γ H ) ⊂ H 2 (−4) (recall that Π was defined in (4.9)). Note that Π(Γ H ) is at constant hyperbolic distance r 0 from γ, for some r 0 ≥ 0. Given r > r 0 , let γ + r , γ − r ⊂ H 2 (−4) be the two arcs at distance r from Π(Γ H ) and let P + (r) = Π −1 (γ + r ), P − (r) = Π −1 (γ − r ) be the topological planes inside SL(2, R) obtained after lifting these arcs through Π. Let H = H 2 θ be one of the two two-dimensional subgroups of SL(2, R) that contain Γ H , and call H a + r , H a − r the right cosets of H at distance r from H (here a + r , a − r are some elements in SL(2, R)). Observe that each of the surfaces P + (r), P − (r) intersects transversely both H a + r and H a − r , along integral curves of the hyperbolic right invariant vector field K H generated by the left action of Γ H on SL(2, R).
With these ingredients, we define the infinite box B(Γ H , r) to be the closure of the component ] that has portions of all four surfaces in its boundary. Observe that Γ H is contained in the interior of B(Γ H , r).  • For every r ≥ r, B(Γ H , r ) has piecewise smooth mean convex boundary (this follows from item 4 of Lemma 4.8 and the fact that every right coset H a is a left translation of some twodimensional subgroup of X, hence minimal), and the inner angles between adjacent smooth surfaces in this boundary are less than π. Furthermore, B(Γ H , r) ⊂ B(Γ H , r ).
Suppose item 3 of the lemma fails to hold. Then, there exists a sequence of compact, stable, immersed minimal surfaces h n : M n X such that h n (∂M n ) ⊂ B(Γ H , r) and points p n ∈ M n such that the intrinsic distances from p n to the boundary of M n satisfy d Mn (p n , ∂M n ) > n for all n ∈ N. Next we show that after appropriate left translations and passing to a subsequence, we can take limits of the h n . Let β be the geodesic in H 2 (−4) that passes through Π(e) and is orthogonal to γ. Note that each integral curve in SL(2, R) of the hyperbolic right invariant vector a closed ball, together with a Riemannian submersion f n : (B n , g n ) → X such that ∂B n = S n , f n | Sn = f n and ∂B n is mean convex. The key step in the proof of Proposition 8.10 will be proving that Lemma 7.4 holds in our situation. Observe that if we admit the validity of Lemma 7.4 in our current setting, then all the arguments in Section 7.4 after the proof of Lemma 7.4 remain valid in SL(2, R) and the proof of Proposition 8.10 will be finished. Therefore, we will only concentrate on the proof of the statement of Lemma 7.4 in our current SL(2, R)-setting.
Consider the hyperbolic 1-parameter subgroup Γ H of SL(2, R) generated by K Σ . Since f (Σ) is an annulus invariant under the flow of K Σ , we can find r 0 > 0 such that f (Σ) is contained in an infinite box B(Γ H , r 0 ) and ∂B(Γ H , r 0 ) is at a positive distance from f (Σ); in what follows, B(Γ H , r 0 ) will play the role of the solid cylinder W(Γ, r 0 ) of the original proof of Lemma 7.4 for metric semidirect products.
By item 3 of Lemma 8.9, there exists some R = R(r 0 ) > 0 with the following property: ( ) If M is any compact, stable immersed minimal surface in X whose boundary ∂M is contained in B(Γ H , r 0 ), then the intrinsic radius of M is less than R.
The role of the horizontal planes in the proof of the semidirect product case of Lemma 7.4 will be now played by the leaves of the foliation F = {P t = l t (Π −1 (β)) | t ∈ Γ H ≡ R}, where β is the geodesic in H 2 (−4) that passes through Π(e) and is orthogonal to γ, where both γ and β were previously defined in the proof of Lemma 8.9. Notice that F is also the foliation of X whose projection by Π to H(−4) is the foliation of H(−4) by the set of geodesics orthogonal to γ, and the leaves of F are minimal topological planes by Lemma 4.8.
As Σ is simply connected and f : Σ X is invariant under the flow of K Σ , then by the discussion in Remark 4.2 we may parameterize f as in equation (7.3) where Γ is replaced by Γ H , and α = α(s) : R → P 0 is an immersed L-periodic curve parameterized by arc length in the plane P 0 , so that f (0, 0) = α(0) = e, L > 0 being the length of α(R).
Observe that B(Γ H , r 0 ) is foliated by integral curves of K H . As B(Γ H , r 0 ) ∩ P 0 is compact, there exists the maximum length D 0 of all the arcs of integral curves of K H that are contained in the compact piece of B(Γ H , r 0 ) that lies in the topological slab of SL(2, R) determined by the leaves P −R−2 and P R+2 of F . Given s 0 ∈ R, the definition (7.4) of the compact arc A(s 0 ) ⊂ f (Σ) of the integral curve of Γ H that passes through α(s 0 ) whose endpoints lie in P −R−2 and P R+2 , remains valid after replacing Γ by Γ H , as well as the property that the length of A(s 0 ) is at most D 0 .
For each j ∈ N, consider the compact region K j ⊂ Σ defined by equation (7.5). Repeating the arguments in the proof of Lemma 7.4 verbatim, each K j produces compact simply connected domains K j n ⊂ S n for n large, such that f n (K j n ) ⊂ B(Γ H , r 0 ) and the sequence {f n | K j n } n converges to f | K j uniformly as n → ∞, and we can also find for each k ∈ {1, . . . , j} a small compact disk U k centered at (kL, 0) in K j and related compact disks U k n ⊂ K j n such that {f n | U k n } n converges to f | U k uniformly as n → ∞. Furthermore, the U k n satisfy that the intrinsic distance between U i n and U k n inside K j n is greater than δ for some δ > 0 independent of n, j, k. Notice however, that in the present setting, while the plane P 0 intersects f (Σ) transversely along the closed immersed curve α(R), we no longer know that f −1 n (P 0 ) is a simple closed curve (since P 0 is not a two dimensional subgroup). Instead, for any fixed j and for n sufficiently large, we have that f −1 n (P 0 ) ∩ K j n is a con-nected compact embedded arc which intersects each of the disks U k in B(Γ H , r 0 ) and ∂B(Γ H , r 0 ) is at a positive distance from f (Σ). Consider the related foliation F = {P t | t ∈ Γ H ≡ R} by topological minimal planes defined in the proof of Proposition 8.10. For n sufficiently large, consider the simple closed curve component α n of f −1 n (P 0 ) ⊂ S n containing p n and let K n Σ be the Killing vector field on B n obtained by pulling-back K Σ through f n . As we are assuming that case (S2.1) holds, Σ is an annulus and f (Σ) is the immersed annulus obtained by letting the immersed closed curve f (Σ) ∩ P 0 flow under K Σ . Let α ∞ = f −1 (P 0 ), which is a simple closed curve in Σ such that the sequence {f n | αn } n converges uniformly to f | α∞ .
As F is a foliation by minimal planes, then the maximum principle implies that the smooth, embedded least-area disk M (n) obtained by solving the Plateau problem in B n with boundary α n , satisfies that f n restricts to M (n) producing an immersion of M (n) into P 0 . As the mean curvatures of the S n are positive, then M (n) is transverse to ∂B n along α n .
The same argument as in the fifth paragraph of Section 7.4 proves that there exists an η ∈ (0, π/4) such that the angle that M (n) makes with S n along α n is greater than η for all n ∈ N large enough, after replacement of the intrinsic radius estimate Theorem 1.1 in [24] by item 3 of Lemma 8.9, changing Γ by Γ H , and noticing that h 0 (X) ≥ H(X) > 0.
The next step consists of proving that if N M (n) stands for the unit normal field to M (n) in (B n , g n ), then for n sufficiently large the function J n = g n (N M (n) , K n Σ ) : M (n) → R has no zeros along α n . The proof of the same property in Section 7.4 does not work now, since it uses equation (4.4) which is only valid in a semidirect product. Instead, we will argue in SL(2, R) as follows. Observe that (W) K Σ is everywhere transversal to P 0 . Now, if J n vanishes at some point q n ∈ α n , then K n Σ (q n ) is tangent to M (n) at q n . Applying the differential of f n at q n , we deduce that K Σ (f n (q n )) is tangent to P 0 at f n (q n ), which contradicts (W). Therefore, J n has no zeros along α n for all n ∈ N.
Once we know that J n has no zeros along α n for all n, we can follow the arguments in Section 7.4 with the following changes (we are using the same notation as in Section 7.4): • Replace Γ by Γ H in equations (7.13) and (7.14). In this way, we can obtain a submersion T ∞ : D × R → X.
• Replace R 2 A {t} in equation (7.15) by the horocylinder Π −1 (c t ) where c t is the horocircle in H 2 (−4) whose end point at ∂ ∞ H 2 is θ defined by H = H 2 θ , such that c t passes through Π(t), t ∈ Γ H . That is, define D t = T −1 ∞ (Π −1 (c t )), and then define the foliation F as in equation (7.15).
• Define K 1 as the nowhere zero, smooth vector field on D × R obtained after pulling K Σ by the submersion T ∞ . Observe that each of the integral curves of K 1 in D × R intersects D 0 in a single point.
• Define K 2 as the nowhere zero vector field on W := D 0 × R which is the pullback by T ∞ of a parabolic right invariant vector field V on X associated to H (V is defined up to a multiplicative nonzero constant). Observe that K 2 is tangent to the foliation F and that once we endow W with the pulled-back metric by T ∞ , K 2 is a Killing field. We claim that K 2 is bounded on one of the two ends of W : to do this, first note that to prove this boundedness property, it suffices to assume that X has isometry group of dimension four (as every left invariant metric on SL(2, R) is quasi-isometric to any fixed left invariant metric on SL(2, R)). Also notice that Π −1 (c 0 )∩H is the 1-parameter parabolic subgroup of H, and that the restriction of V to Π −1 (A 0 )∩H is bounded, where A 0 is the open horodisk in H 2 bounded by c 0 (in fact, if we identify H with the semidirect product R (1) R = {(x, y) | x, y ∈ R} as explained after (8.1), then V identifies with ∂ x up to a multiplicative constant, Π −1 (A 0 ) ∩ H is the open halfspace R (1) (0, ∞) and ∂ x is bounded in R (1) (0, ∞)). As the right translations by elements in the elliptic subgroup of SL(2, R) generated by the vector (E 3 ) e given by equation (4.6) are isometries of any E(κ, τ )-metric in SL(2, R) and V is right invariant, we conclude that V is bounded in Π −1 (A 0 ). Therefore, our claim follows.
The above changes allow us to apply Theorem 9.1 to find the same contradiction with CMC flux as in Section 7.4. This contradiction finishes the proof of Proposition 8.11.

Appendix: A nonvanishing result for the CMC flux
Let W be a Riemannian (n + 1)-manifold and M ⊂ W a two-sided n-hypersurface with constant mean curvature H ∈ R (throughout this appendix we define the mean curvature of M to be the trace of its second fundamental form divided by n). The CMC flux of M associated to a Killing vector field K on W and to a piecewise smooth (n − 1)-chain α in M which is the boundary of a piecewise smooth two-sided n-chain β in W , is the real number where η is a unit conormal vector to M along α and N is a unit normal vector field along β. In order for (9.1) not to depend on orientation choices, certain compatibility between H, η and N must be satisfied, see a more precise definition in Definition 9.2. A key property of Flux(M, α, K) is that it does not depend on the homology class of α in M . The goal of this section is to prove the following general result, in which the CMC flux of an n-hypersurface M = ∂Σ × R with respect to a certain Killing vector field K = K 1 is nonzero. Theorem 9.1 has been used in Sections 7.4 and 8.5.
Theorem 9.1. Let Σ be a smooth, compact oriented n-manifold with nonempty boundary, n ≤ 6. Let W be Σ × R equipped with a Riemannian metric such that: (1) The boundary ∂Σ × R of W is mean convex, with constant mean curvature h 0 ≥ 0.
(2) There exists a nowhere zero Killing field K 1 of W arising from a proper action 5 of an Rsubgroup of the isometry group of W . and let β, β 1 be piecewise smooth n-chains in X with boundaries ∂β = α, ∂β 1 = α 1 (which exist since [α] ∈ ker(i * )). We assume that if M (α, α 1 ) is the n-chain in M whose boundary is α 1 − α, then β − β 1 − M (α, α 1 ) is the boundary of an (n + 1)-chain Ω in X. Applying the Divergence Theorem to K in Ω, we obtain 0 = − β K, N + is independent of the choice of the representative α in a given homology class [α] ∈ ker(i * ) ⊂ H n−1 (M ) and of the n-chain β as above. This number is usually called the CMC flux of M along [α] associated to K. Observe that the mean curvature of M is computed with respect to the unit normal vector that, along α, points to the same closed component of M ∪ β as N , and that N, η ≤ 0 along α. The above argument shows the homological invariance of the CMC flux; we refer the reader to the papers [13,17,18,21] for applications of this CMC flux.
Proof of Theorem 9.1. Without loss of generality, we will assume that the Killing field K 2 is bounded on the end Σ×(−∞, 0] of W . Let {φ s | s ∈ R} be the 1-parameter subgroup of isometries of X corresponding to the flow of the Killing field K 1 . Since the action (s, x) ∈ R×W → φ s (x) is proper, then after possibly changing K 1 by −K 1 , we may suppose that φ s (Σ × {0}) lies eventually in the upper end Σ × [0, ∞) of W as s → +∞. Consider the quotient map Π : W → W/G = W , where G = {φ s | s ∈ 2πZ} (note that the discrete group of isometries G acts proper discontinuously on W ). We will denote by S 1 the related 1-parameter subgroup of isometries of W .
Let Γ be an area-minimizing oriented (n − 1)-dimensional hypersurface in ∂ W that represents [∂Σ] ∈ H n−1 (∂ W ) ( Γ is an area-minimizing integral current, which is smooth in this dimension and it exists since ∂ W is a closed Riemannian manifold). By cut-and-paste type arguments and the maximum principle for minimal hypersurfaces, any two distinct solutions to this area-minimization problem in ∂ W are disjoint or equal. It follows that ∂ W is foliated by the area-minimizing oriented (n − 1)-dimensional submanifolds φ s ( Γ), φ s ∈ S 1 . For homological reasons, Γ has a lift through Π which is a compact (n − 1)-dimensional hypersurface Γ in ∂W that is area-minimizing in ∂W , and when considered to lie in W , it is the boundary of an integral n-chain. By the properness of the R-action giving rise to K 1 and the compactness of Γ, we deduce that K 1 cannot be everywhere tangent to Γ. On the other hand, since for s = 0 it holds φ s (Γ) ∩ Γ = Ø, then the Jacobi function J Γ = K 1 , N Γ : Γ → R, is nonnegative on Γ (after choosing an appropriate orientation of Γ, or equivalently a unit normal N Γ along Γ, i.e., N Γ is tangent to ∂W , orthogonal to Γ and unitary); note that J Γ is positive at the points of Γ where K 1 is not tangent to Γ. The maximum principle (see e.g., [28, Assertion 2.2]) implies that J Γ > 0 on Γ; in particular, K 1 is transverse to Γ and each integral curve of K 1 intersects Γ in a single point.
Let ∆ ⊂ W be an area-minimizing hypersurface with boundary Γ, which in this dimension is smooth by standard interior and boundary regularity results for minimizing integral varifolds; see for example, Hardt and Simon [12] for boundary regularity results and also see [30,32] for related barrier arguments. By the mean curvature comparison principle, ∆ is never tangent to ∂W along Γ. Since J Γ > 0 by the previous paragraph, then we can choose a unit normal vector N ∆ to ∆ such that (N ∆ )| ∂Γ , N Γ is positive along Γ. As K 1 is tangent to ∂W , (N ∆ )| Γ is orthogonal to Γ and both K 1 , N Γ , (N ∆ )| ∂Γ , N Γ are positive along Γ, then we conclude that K 1 , N ∆ > 0 along Γ. As ∆ is stable and K 1 , N ∆ > 0 along ∂∆, then K 1 , N ∆ is positive on ∆ and so, ∆ intersects each of the integral curves of K 1 transversely in a single point. Therefore, the set F = {φ s (∆) | s ∈ R} is a smooth minimal foliation of W , where each leaf φ s (∆) of this foliation intersects every integral curve of K 1 transversely in a single point.
We will now deform the minimal surface ∆ by a 1-parameter family of surfaces with constant mean curvature and the same boundary as ∆. By the "blowing a bubble" technique described in [17], for δ > 0 sufficiently small there exists a unique 1-parameter family of hypersurfaces t ∈ [0, δ) → ∆(t) ⊂ W depending smoothly on t such that for all t ∈ [0, δ): (X1) ∆(t) is the graph of a smooth function f t : ∆ → R (in the direction of K 1 ), with ∆(0) = ∆ and f 0 = 0.
We now study the maximal half-open interval [0, T 0 ) of t-values in which the family ∆(t) can be defined (here T 0 ∈ (0, ∞]).
L(T 0 ) ∆ Figure 8: Left: After blowing a bubble ∆(t) with constant mean curvature −t ∈ (−T 0 , 0] (which is K 1 -graphical over ∆ = ∆(0)), we translate it with {φ s } s∈R to produce a foliation F(t), and we highlight the highest graphical leaf L(t) of F(t) lying below ∆(0) with L(t) ∩ ∆(0) = Ø. Right: Since the ∆(t) with t → T − 0 do not lie in a fixed compact set, we produce a noncompact limit L(T 0 ) = lim t→T − 0 L(t) which is a graph over a domain ∆ of ∆.