Constant mean curvature spheres in homogeneous three-manifolds

Constant mean curvature spheres in homogeneous three-manifolds

Abstract

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.

Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42

Key words and phrases: Constant mean curvature, Hopf uniqueness, homogeneous three-manifold, Cheeger constant.

1 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 be a Riemannian homogeneous three-manifold, denote its Riemannian universal cover, and let denote the Cheeger constant of . Then, any two spheres in of the same constant mean curvature differ by an isometry of . Moreover:

  1. If is not diffeomorphic to , then, for every , there exists a sphere of constant mean curvature in .

  2. If is diffeomorphic to , then the values   for which there exists a sphere of constant mean curvature in are exactly those with .

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 be a Riemannian homogeneous three-manifold, denote its Riemannian universal cover and let be a sphere of constant mean curvature in . Then:

  1. is maximally symmetric; that is, there exists a point , which we call the center of symmetry of , with the property that any isometry of that fixes also satisfies . In particular, constant mean curvature spheres are totally umbilical if has constant sectional curvature, and are spheres of revolution if is rotationally symmetric.

  2. If and is a Riemannian product , where is a sphere of constant curvature , then it is well known that is totally geodesic, stable and has nullity for its stability operator. Otherwise, has index and nullity for its stability operator and the immersion of into extends as the boundary of an isometric immersion of a Riemannian 3-ball which is mean convex1.

Moreover, if denotes the space of spheres of nonnegative constant mean curvature in that have a base point as a center of symmetry, then the map that assigns to each sphere its mean curvature is a homeomorphism between: (i) and if is not diffeomorphic to , or (ii) and if is diffeomorphic to . This homeomorphism is real analytic except at when .

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 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 and then used it to prove that constant mean curvature spheres in are round. His proof also worked in the other simply-connected three-dimensional 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 and , where (resp. ) denotes the two-dimensional sphere (resp. the hyperbolic plane) of constant Gaussian curvature , 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 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 .

In 2013, Daniel and Mira [9] introduced a new method for studying constant mean curvature spheres in the homogeneous manifold with its usual Thurston geometry. Using this method, they classified constant mean curvature spheres in for values of the mean curvature greater than , 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 : For any there is a unique constant mean curvature sphere in with mean curvature ; moreover, 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 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 , since the proof by Meeks uses in an essential way special properties of that are not shared by a general .

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 , and we proved Theorems 1.1 and 1.2 in the case that 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 diffeomorphic to admits a foliation by leaves of constant mean curvature of value . 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 of the same constant mean curvature in a metric Lie group 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 is a simply connected homogeneous three-manifold.

Definition 1.3.

Given a nowhere zero Killing field on , we say that an immersed surface is a Killing graph with respect to if whenever intersects an integral curve of , this intersection is transversal and consists of a single point. If additionally intersects every integral curve of , then we say that is an entire Killing graph.

By classification, any homogeneous manifold diffeomorphic to is isometric to a metric Lie group. In this way, any nonzero vector field on that is right invariant for such Lie group structure is a nowhere-zero Killing field on . We note that any integral curve of is a properly embedded curve in diffeomorphic to .

Theorem 1.4.

Let be a homogeneous manifold diffeomorphic to , and let be a sequence of constant mean curvature spheres in with . Then, there exist isometries of and compact domains with the property that a subsequence of the converges uniformly on compact subsets of to an entire Killing graph with respect to some nowhere zero Killing vector field on , which in fact is right invariant with respect to some Lie group structure on .

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 , we proved that when is diffeomorphic to , Theorems 1.1 and 1.2 hold for . In that same paper we proved that if is a metric Lie group not diffeomorphic to (and hence, diffeomorphic to ), then there exists a constant such that: (i) for any there is an index-one sphere of constant mean curvature in ; (ii) any sphere of constant mean curvature is a left translation of , and (iii) if , then the areas of the spheres diverge to as ; furthermore, the spheres bound isometrically immersed mean convex Riemannian three-balls .

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 in a metric Lie group that are invariant, i.e., such that is everywhere tangent to a nonzero right invariant Killing vector field on (equivalently, is invariant under the flow of ), and we explain in more detail the geometry of metric Lie groups diffeomorphic to . For this analysis, we divide these metric Lie groups into two categories: metric semidirect products and those of the form , where is the universal cover of the special linear group and is a left invariant metric; here the metric semidirect products under consideration are the semidirect product of a normal subgroup with and we refer to the cosets in of as being horizontal planes.

In Section 5 we prove that if is a sequence of index-one spheres with constant mean curvatures of values in a metric Lie group diffeomorphic to , then, after taking limits of an appropriately chosen subsequence of left translations of these spheres, there exists an invariant surface of constant mean curvature that is a limit of compact domains of the spheres . Moreover, is complete, stable, has the topology of a plane or a cylinder, and if is not a coset of a two-dimensional subgroup of , then the closure of its left invariant Gauss map image in (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 is tangent to some coset of a two-dimensional subgroup of , then 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 can be chosen so that its left invariant Gauss map image is a point or a closed embedded regular curve. In the case is a point, Theorem 1.4 follows easily from the fact that is a coset of a two-dimensional subgroup of . In the case that is a closed curve, one can choose so that one of the following exclusive possibilities occurs:

  1. is diffeomorphic to an annulus.

  2. is diffeomorphic to a plane, and is an immersed annulus in .

  3. is diffeomorphic to a plane and there exists an element such that the left translation by in leaves invariant, but this left translation does not lie in the -parameter subgroup of isometries generated by the nonzero right invariant Killing vector field that is everywhere tangent to .

In Section 7 we prove Theorem 1.4 in the case that is a metric semidirect product. For that, it suffices to prove that is an entire Killing graph for some right invariant Killing vector field in . This is proved as follows. First, we show that if case 3 above holds, then the Killing field which is everywhere tangent to is horizontal with respect to the semidirect product structure of , and we prove that is an entire graph with respect to any other horizontal Killing field linearly independent from . 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 in the abstract Riemannian three-balls that the Alexandrov-embedded index-one spheres bound, and whose volumes tend to infinity as . 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 that submerses isometrically into with boundary , and then proving that a certain CMC flux of in this abstract cylinder is different from zero. This gives a contradiction with the homological invariance of the CMC flux and the fact that is a limit of the (homologically trivial) Alexandrov-embedded constant mean curvature spheres .

In Section 8 we prove Theorem 1.4 in the remaining case where is isomorphic to . 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 .

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

In order to show that Theorems 1.1 and 1.2 are implied by Theorem 1.4, we will prove the next two assertions in Sections 3.1 and 3.2, respectively.

Assertion 3.1.

Assume that Theorem 1.4 holds. Then Theorems 1.1 and 1.2 hold for the particular case that the homogeneous three-manifold is simply connected.

Assertion 3.2.

Theorems 1.1 and 1.2 hold provided they hold for the particular case that the homogeneous three-manifold is simply connected.

In what follows, will denote a homogeneous three-manifold and will denote its universal Riemannian cover; if is simply connected, we will identify . By an -surface (or -sphere) in or we will mean an immersed surface (resp. sphere) of constant mean curvature of value immersed in or .

The stability operator of an -surface is the Schrödinger operator , where stands for the Laplacian with respect to the induced metric in , and is the smooth function given by

where denotes the squared norm of the second fundamental form of , and is the unit normal vector field to . The immersion is said to be stable if is a nonnegative operator. When is closed, the index of is defined as the number of negative eigenvalues of , and the nullity of is the dimension of the kernel of . We say that a function is a Jacobi function if . Functions of the type where is a Killing vector field on the ambient space are always Jacobi functions on .

3.1 Proof of Assertion 3.1

Let denote a simply connected homogeneous three-manifold. It is well known that if is not isometric to a Riemannian product space , where denotes a sphere of constant Gaussian curvature , then is diffeomorphic to or to .

When is isometric to , the statements contained in Theorems 1.1 and 1.2 follow from the Abresch-Rosenberg theorem that states that constant mean curvature spheres in are rotational spheres, and from an ODE analysis of the profile curves of these spheres. More specifically, an explicit expression of the rotational -spheres in can be found in Pedrosa and Ritoré [35, Lemma 1.3], or in [1]. From this expression it follows that if , then each of these rotational -spheres is embedded, and bounds a unique compact subdomain in which is a mean convex ball that is invariant under all the isometries of 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 , the corresponding rotational sphere is a slice , that is trivially stable, and hence has index zero and nullity one. Moreover, any point of can be defined as the center of symmetry of the sphere in this case. That the rotational -spheres in for have index one and nullity three is shown in Souam [38, proof of Theorem 2.2]. Also, the rotational -spheres for with a fixed center of symmetry converge as to a double cover of the slice that contains . So, from all this information and the Abresch-Rosenberg theorem, it follows that Theorems 1.1 and 1.2 hold when is isometric to .

When is diffeomorphic to , 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, will denote a homogeneous manifold diffeomorphic to . It is well known that is isometric to a metric Lie group, i.e., a three-dimensional Lie group endowed with a left invariant Riemannian metric. In the sequel, will be regarded as a metric Lie group, and will denote its identity element. Given , we will denote by the left and right translations by , respectively given by , for all . Thus, is an isometry of for every .

Definition 3.3.

Given an immersed oriented surface with unit normal vector field (here refers to the tangent bundle of ), we define the left invariant Gauss map of to be the map that assigns to each , the unit tangent vector to at the identity element given by .

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 be any metric Lie group diffeomorphic to . Then, there exists a number such that:

  1. For every , there exists an -sphere in with index one and nullity three for its stability operator. Furthermore, is Alexandrov embedded, i.e., is the boundary of a (unique) mean convex immersed ball in .

  2. If is an -sphere in for some , then is a left translation of .

  3. The left invariant Gauss map of is an orientation-preserving diffeomorphism.

  4. Each lies inside a real-analytic family of index-one spheres in for some , where and has constant mean curvature of value . In particular, there exists a unique component of the space of index-one spheres with constant mean curvature in such that for all .

  5. For every there exists some positive constant such that if , then the norm of the second fundamental form of is at most .

  6. For every there exists some positive constant such that if , then the area of is at most .

  7. If is a sequence of constant mean curvature spheres in whose mean curvatures satisfy , then .

Proof.

After a change of orientation if necessary, it clearly suffices to prove all statements for the case where . By [25, item 3 of Theorem 4.1], there exists a unique component of the space of index-one spheres with constant mean curvature in such that the values of the mean curvatures of the spheres in are not bounded from above. As is diffeomorphic to , then [25, item 6 of Theorem 4.1] ensures that the map that assigns to each sphere in its mean curvature is not surjective. By item 5 in the same theorem, there exists such that for every , there is an index-one sphere , and the following property holds:

  1. The areas of any sequence with satisfy .

Since the sphere has index one, then Cheng [5, Theorem 3.4] gives that the nullity of is three. Alexandrov embeddedness of 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 stated in items 1,2,4, and of the fact also proved in [25] that, for large enough, the spheres bound small isoperimetric regions in . 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 .

Definition 3.5.

Let be a homogeneous manifold diffeomorphic to , let be the collection of all closed, orientable immersed surfaces in , and given a surface , let stand for the absolute mean curvature function of . The critical mean curvature of is defined as

Item 2 of Theorem 1.4 in [26] gives that Ch. By definition of , every compact -surface in satisfies . Observe that Proposition 3.4 gives that for every there is an -sphere in . Thus,

(3.1)

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 . If are constant mean curvature spheres satisfying item 7 of Proposition 3.4, then by Theorem 1.4 we obtain the existence of a surface in of constant mean curvature that is an entire Killing graph with respect to some nonzero right invariant Killing field . Let be the -parameter subgroup of given by , . As left translations are isometries of , it follows that defines a foliation of by congruent surfaces of constant mean curvature , 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 in with (otherwise, we left translate until is contained in the region of on the mean convex side of , and then start left translating towards its mean convex side until it reaches a first contact point with ; this provides a contradiction with the mean curvature comparison principle). Therefore, we conclude that and that, by item 1 of Proposition 3.4, the values for which there exists a sphere of constant mean curvature in are exactly those with . 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 is diffeomorphic to (assuming that Theorem 1.4 holds).

Regarding the proof of Theorem 1.2 when is diffeomorphic to , 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 in 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 , and the analyticity properties in item 4 of Proposition 3.4.

In the case that the isometry group of is of dimension 6, all constant mean curvature spheres in 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 be a metric Lie group diffeomorphic to whose isometry group has dimension or . Let denote the group of isometries of with , and let denote the index-two subgroup of orientation-preserving isometries in . The next proposition gives some basic properties of the elements of that will be needed for proving the maximal symmetry of -spheres in . Its proof follows from the analysis of metric Lie groups in [27].

Proposition 3.6.

There exists a -parameter subgroup of such that:

  1. There exists an isometry of order two in that is a group automorphism of , and whose fixed point set is , i.e., .

  2. Every leaves invariant, in the sense that .

Moreover, if there exists another -parameter subgroup of that satisfies the previous two properties, then has a three-dimensional isometry group, and , where is given with respect to by item (1) above.

Proof.

First suppose that has dimension 4. By [27, item 2 of Proposition 2.21], there exists a unique principal Ricci eigendirection whose associated eigenvalue is simple, and Stab contains an -subgroup , all whose elements have differentials that fix . Let be the 1-parameter subgroup of generated by , and let be the orientation-preserving automorphism of whose differential satisfies , (see [27, proof of Proposition 2.21] for the construction of ). Clearly satisfies item 1 of Proposition 3.6. Regarding item 2, observe that Stab, where is the set of -rotations about the geodesics of such that and is orthogonal to . Therefore, every satisfies . If Stab, then item 2 of Proposition 3.6 holds. Otherwise, Stab and in this case, item 2 of Proposition 2.24 of [27] ensures that is isomorphic and homothetic to endowed with its standard product metric (hence is a nonzero vertical vector and is the vertical line passing through ), and every is either a reflectional symmetry with respect to a vertical plane or it is the composition of a reflectional symmetry with respect to with a rotation about . Thus, leaves invariant and item 2 of Proposition 3.6 holds. Note that the moreover part of the proposition cannot hold in this case of being four-dimensional, because given any 1-parameter subgroup of different from , there exists such that .

Next suppose that has dimension 3. First suppose that the underlying Lie group structure of is not unimodular2. Hence is isomorphic to some semidirect product for some matrix with trace and since the dimension of , see Section 4.2 for this notion of nonunimodular metric Lie group. Then [27, item 4 of Proposition 2.21] gives that where . Therefore, item 1 of Proposition 3.6 holds with the choice and the above . Regarding item 2, we divide the argument into two cases in this nonunimodular case for . Given , consider the matrix

(3.2)
  1. If is not isomorphic and homothetic to any with given by (3.2) with , then [27, item 3 of Proposition 2.24] gives that .

  2. If is isomorphic and homothetic to for some , then [27, item (3a) of Proposition 2.24] implies that then where and . In the case that , then , which is not the case presently under consideration as the isometry group of for has dimension four.

In both cases (A1) and (A2), item 2 of Proposition 3.6 holds with being the -axis. The moreover part of Proposition 3.6 also holds, because item 1 only holds for .

If dim and is unimodular, then there exists a frame of left invariant vector fields on that are eigenfields of the Ricci tensor of , independently of the left invariant metric on , see [27, Section 2.6]. By the proof of Proposition 2.21 of [27], for each there exists an orientation-preserving automorphism whose differential satisfies and whenever . Furthermore, is an isometry of every left invariant metric on (in particular, of the metric of ). Also, item 1 of Proposition 2.24 in [27] gives that is the dihedral group . Therefore, item 1 of Proposition 3.6 holds with any of the choices and , for , where is the -parameter subgroup of generated by . Regarding item 2, we again divide the argument into two cases in this unimodular case for .

  1. If is not isomorphic and homothetic to (recall that is Sol with its standard metric), then [27, item 3 of Proposition 2.24] ensures that . Thus item 2 of Proposition 3.6 holds for every choice of the form and with . Once here, and since it is immediate that for any , we conclude that the moreover part of Proposition 3.6 holds in this case.

  2. If is isomorphic and homothetic to , then [27, item 3b of Proposition 2.24] gives that , where

    Therefore, item 2 of Proposition 3.6 only holds for the choice and (in particular, the moreover part of Proposition 3.6 also holds).

Now the proof is complete. ∎

We are now ready to prove the existence of the center of symmetry of any -sphere in . Let denote an -sphere, with left invariant Gauss map . By item 3 of Proposition 3.4 and , we know is an orientation preserving diffeomorphism. Without loss of generality we can assume that , where is the unique point such that and is an arc length parameterization of the 1-parameter subgroup of that appears in Proposition 3.6. Let be an isometry associated to in the conditions of item 1 of Proposition 3.6. Clearly , hence and are two -spheres in passing through with the same left invariant Gauss map image at . By the uniqueness of -spheres up to left translations in item 2 of Proposition 3.4, we conclude that . Let be the unique point in such that . Since , we see that the Gauss map image of at is also . Since , this implies (by uniqueness of the point ) that is a fixed point of , which by item 1 of Proposition 3.6 shows that .

Let be the midpoint of the connected arc of that connects with , and define . As the family is real analytic with respect to (by items 2 and 4 of Proposition 3.4), then the family is also real analytic in terms of . Also, note that the points and both lie in , and are equidistant from along . This property and item 2 of Proposition 3.6 imply that every satisfies , and hence either and , or alternatively and .

Take . We claim that leaves invariant. To see this, we will distinguish four cases.

  1. Suppose that preserves orientation on and satisfies , . The first condition implies that is an -sphere in . So, by items 2 and 3 of Proposition 3.4 we have for some , and that corresponds to the unique point of where its Gauss map takes the value . But now, from the second condition we obtain that restricts to as the identity map, which implies that the value of the left invariant Gauss map image of at is . Thus, , hence and , that is, leaves invariant.

  2. Suppose that reverses orientation on and satisfies , . The first condition implies that after changing the orientation of , this last surface is an -sphere in . Since by the second condition restricts to as minus the identity map, then the left invariant Gauss map image of (with the reversed orientation) at is . Now we deduce as in case (B1) that leaves invariant.

  3. Suppose that reverses orientation on and satisfies , . As in case (B2), we change the orientation on so it becomes an -sphere in . Item 2 of Proposition 3.4 then gives that there exists such that . As and fixes pointwise, then . Moreover, the value of the left invariant Gauss map image of (with the reversed orientation) at is . Since the only point of where its left invariant Gauss map image (with the original orientation) takes the value is , we deduce that . Similarly, . This is only possible if , which, in our setting that is diffeomorphic to , implies that (note that this does not happen when is diffeomorphic to the three-sphere, because then is isomorphic to , which has one element of order 2). The claim then holds in this case.

  4. Finally, assume that preserves the orientation on and satisfies , . As in case (B1), is an -sphere in and so there exists such that . As , then . Since restricts to as minus the identity map, then the value of the left invariant Gauss map image of at is . Since the only point of where its left invariant Gauss map image takes the value is , we deduce that . Now we finish as in case (B3).

To sum up, we have proved that any leaves invariant; this proves that the -sphere is maximally symmetric with respect to the identity element , in the sense of item 1 of Theorem 1.2. So, we may define to be the center of symmetry of . We next show that the above definition of center of symmetry does not depend on the choice of the 1-parameter subgroup of that appears in Proposition 3.6. Suppose is another -parameter subgroup of satisfying the conclusions of Proposition 3.6. As we explained in the paragraph just after Proposition 3.6, determines two points with , such that , , where is the left invariant Gauss map of . To show that the above definition of center of symmetry of does not depend on , we must prove that the mid point of the arc of between an is . To do this, let be an order-two isometry that satisfies item 1 of Proposition 3.6 with respect to . By Proposition 3.6, . As leaves invariant and is a diffeomorphism, then . Since the set of fixed points of is and (because two different 1-parameter subgroups in only intersect at , here we are using again that is diffeomorphic to ), then . As is an isometry with and , we deduce that the lengths of the arcs of that join to , and to , coincide. This implies that the mid point of the arc of between an is , as desired. As every -sphere in is a left translation of , we conclude that all -spheres in are maximally symmetric with respect to some point , which is obtained by the corresponding left translation of the identity element , 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 is diffeomorphic to . Thus, the proof of Assertion 3.1 is complete.

The next remark gives a direct definition of the center of symmetry of any -sphere in :

Remark 3.7 (Definition of the center of symmetry).

Let be an -sphere in a homogeneous manifold diffeomorphic to with an isometry group of dimension three or four. Let be given by , , where denotes the left invariant Gauss map of and , with a -parameter subgroup satisfying the conditions in Proposition 3.6. Then, by our previous discussion, the left coset passes through both and , and we define the center of symmetry of as the midpoint of the subarc of that joins with .

It is worth mentioning that this definition gives a nonambiguous definition of the center of symmetry of an -sphere; however, when has an isometry group of dimension three and is nonunimodular, there are many points besides this center of symmetry such that the -sphere is invariant under all isometries of that fix . 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.

3.2 Proof of Assertion 3.2

Let be a homogeneous three-manifold with universal covering space . To prove Assertion 3.2, we will assume in this Section 3.2 that Theorems 1.1 and 1.2 hold for .

We first prove Theorem 1.1 in . Since every constant mean curvature sphere is the projection via of some lift of with the same mean curvature, then clearly the first and second items in Theorem 1.1 hold in , since they are true by hypothesis in . In order to prove the uniqueness statement in Theorem 1.1, let , be two spheres with the same constant mean curvature in . For , choose respective lifts of , let be the centers of symmetry of , denote , and let be an isometry with . Let be the lift of that takes to . It follows that the -spheres and have the same center of symmetry in and so these immersions have the same images. In particular, it follows that and also have the same images, which completes the proof of Theorem 1.1 in .

We next prove Theorem 1.2 in . Suppose is an oriented -sphere, and let denote some lift of . Since the stability operators of and are the same, the index and nullity of and agree. Also, note that if the immersion extends to an isometric immersion of a mean convex Riemannian three-ball into , then is an isometric immersion of into that extends . These two trivial observations prove that item 2 of Theorem 1.2 holds in .

If is covered by and is totally geodesic, then any point of the image sphere satisfies the property of being a center of symmetry of . This observation follows by the classification of homogeneous three-manifolds covered by