Constant mean curvature spheres in homogeneous three-spheres
We give a complete classification of the immersed constant mean curvature spheres in a three-sphere with an arbitrary homogenous metric, by proving that for each , there exists a constant mean curvature sphere in the space that is unique up to an ambient isometry.
Key words and phrases:Minimal surface, constant mean curvature, -potential, stability, index of stability, nullity of stability, curvature estimates, Hopf uniqueness, metric Lie group, homogeneous three-manifold, left invariant metric, left invariant Gauss map.
1991 Mathematics Subject Classification:Primary 53A10; Secondary 49Q05, 53C42
July 22, 2019
The classification and geometric description of constant mean curvature spheres is a fundamental problem in classical surface theory. There are two highly influential results on this problem, see [1, 2, 11]:
Theorem 1.1 (Hopf Theorem).
An immersed sphere of constant mean curvature in a complete, simply connected three-dimensional manifold of constant sectional curvature is a round sphere.
Theorem 1.2 (Abresch-Rosenberg Theorem).
An immersed sphere of constant mean curvature in a simply connected homogeneous three-manifold with a four-dimensional isometry group is a rotational sphere.
When the ambient space is an arbitrary homogeneous three-manifold , the type of description of constant mean curvature spheres given by the above theorems is no longer possible due to the lack of ambient rotations in . Because of this, the natural way to describe constant mean curvature spheres in that general setting is to parameterize explicitly the moduli space of these spheres, and to determine their most important geometric properties.
In this manuscript we develop a theoretical framework for studying constant mean curvature surfaces in any simply connected homogeneous three-manifold that is not diffeomorphic to (the classification of constant mean curvature spheres in follows from Theorem 1.2). We will apply this general theory to the classification and geometric study of constant mean curvature spheres when is compact. Specifically, Theorem 1.3 below gives a classification of constant mean curvature spheres in any homogeneous three-manifold diffeomorphic to , by explicitly determining the essential properties of such spheres with respect to their existence, uniqueness, moduli space, symmetries, embeddedness and stability.
Let be a compact, simply connected homogenous three-manifold. Then:
For every , there exists an immersed oriented sphere in of constant mean curvature .
Up to ambient isometry, is the unique immersed sphere in with constant mean curvature .
There exists a well-defined point in called the center of symmetry of such that the isometries of that fix this point also leave invariant.
is Alexandrov embedded, in the sense that the immersion of in can be extended to an isometric immersion of a Riemannian three-ball such that is mean convex.
has index one and nullity three for the Jacobi operator.
Moreover, let be the set of oriented immersed spheres of constant mean curvature in whose center of symmetry is a given point . Then, is an analytic family parameterized by the mean curvature value of .
Every compact, simply connected homogeneous three-manifold is isometric to the Lie group endowed with a left invariant metric. There exists a three-dimensional family of such homogeneous manifolds, which includes the three-spheres of constant sectional curvature and the two-dimensional family of rotationally symmetric Berger spheres, which have a four-dimensional isometry group. Apart from these two more symmetric families, any other left invariant metric on has a three-dimensional isometry group, with the isotropy group of any point being isomorphic to . Item (3) in Theorem 1.3 provides the natural generalization of the theorems by Hopf and Abresch-Rosenberg to this more general context, since it implies that any constant mean curvature sphere in such a space inherits all the ambient isometries fixing some point; in particular, is round (resp. rotationally invariant) in (resp. in Berger spheres).
Items (1) and (2) together with the last statement of Theorem 1.3 provide an explicit description of the moduli space of constant mean curvatures spheres in any compact, simply connected homogeneous three-manifold . Items (4) and (5) in Theorem 1.3 describe general embeddedness and stability type properties of constant mean curvature spheres in which are essentially sharp, as we explain next. In the manifolds , the constant mean curvature spheres are round; hence, they are embedded and weakly stable (see Section 4 for the definition of weak stability). However, for a general homogeneous diffeomorphic to , the constant mean curvature spheres need not be embedded (Torralbo  for certain ambient Berger spheres) or weakly stable (Torralbo and Urbano  for certain ambient Berger spheres, see also Souam ), and they are not geodesic spheres if is not isometric to some . Nonetheless, item (4) in Theorem 1.3 shows that any constant mean curvature sphere in a general satisfies a weaker notion of embeddedness (which is usually called Alexandrov embeddedness), while item (5) describes the index and the dimension of the kernel of the Jacobi operator of a constant mean curvature sphere.
An important property of constant mean curvature spheres in not listed in the statement of Theorem 1.3 is that, after identifying with the Lie group endowed with a left invariant metric, the left invariant Gauss map of every constant mean curvature sphere in is a diffeomorphism to ; see Theorem 4.1 for this result. This diffeomorphism property will be crucial in the proof of Theorem 1.3.
As an application of Theorem 1.3, we provide in Theorem 7.1 a more detailed description of the special geometry of minimal spheres in a general compact . In particular, it follows from Theorem 7.1 that minimal spheres in are embedded; this follows from the existence of an embedded minimal sphere in endowed with an arbitrary Riemannian metric as proved by  (see also Remark 7.2), and the fact that any two such minimal spheres in are congruent by item (2) of Theorem 1.3.
The theorems of Hopf and Abresch-Rosenberg rely on the existence of a holomorphic quadratic differential for constant mean curvature surfaces in homogeneous three-manifolds with isometry group of dimension at least four. This approach using holomorphic quadratic differentials does not seem to work when the isometry group of the homogeneous three-manifold has dimension three. Instead, our approach to proving Theorem 1.3 is inspired by two recent works on constant mean curvature spheres in the Thurston geometry , i.e., in the solvable Lie group Sol with any of its most symmetric left invariant metrics. One of these works is the local parameterization by Daniel and Mira  of the space of index-one, constant mean curvature spheres in equipped with its standard metric via the left invariant Gauss map and the uniqueness of such spheres. The other one is the obtention by Meeks  of area estimates for the subfamily of spheres in whose mean curvatures are bounded from below by any fixed positive constant; these two results lead to a complete description of the spheres of constant mean curvature in endowed with its standard metric. However, the proof of Theorem 1.3 for a general compact, simply connected homogeneous three-manifold requires the development of new techniques and theory. These new techniques and theory are needed to prove that the left invariant Gauss map of an index-one sphere of constant mean curvature in is a diffeomorphism, that constant mean curvature spheres in are Alexandrov embedded and have a center of symmetry, and that there exist a priori area estimates for the family of index-one spheres of constant mean curvature in . On the other hand, some of the arguments in the proofs of Theorem 3.7 and Theorem 4.1 below are generalizations of ideas in Daniel and Mira , and are therefore merely sketched here. For more specific details on these computations, we refer the reader to the announcement of the results of the present paper in Chapter 3 of the Lecture Notes  by the first and third authors.
Here is a brief outline of the proof of Theorem 1.3. First we identify the compact, simply connected homogenous three-manifold isometrically with endowed with a left invariant metric. Next we show that any index-one sphere of constant mean curvature in has the property that any other immersed sphere of constant mean curvature in is a left translation of . The next step in the proof is to show that the set of values for which there exists an index-one sphere of constant mean curvature in is non-empty, open and closed in (hence, ). That is non-empty follows from the existence of small isoperimetric spheres in . Openness of follows from an argument using the implicit function theorem, which also proves that the space of index-one spheres with constant mean curvature in modulo left translations is an analytic one-dimensional manifold. By elliptic theory, closedness of can be reduced to obtaining a priori area and curvature estimates for index-one, constant mean curvature spheres with any fixed upper bound on their mean curvatures. The existence of these curvature estimates is obtained by a rescaling argument. The most delicate part of the proof of Theorem 1.3 is obtaining a priori area estimates. For this, we first show that the non-existence of an upper bound on the areas of all constant mean curvature spheres in implies the existence of a complete, stable, constant mean curvature surface in which can be seen to be the lift via a certain fibration of an immersed curve in , and then we prove that such a surface cannot be stable to obtain a contradiction. The Alexandrov embeddedness of constant mean curvature spheres follows from a deformation argument, using the smoothness of the family of constant mean curvature spheres in . Finally, the existence of a center of symmetry for any constant mean curvature sphere in is deduced from the Alexandrov embeddedness and the uniqueness up to left translations of the sphere.
Even though the geometry of constant mean curvature surfaces in homogeneous three-manifolds with an isometry group of dimension at least four has been deeply studied, the case where the ambient space is an arbitrary homogeneous three-manifold remains largely unexplored. The results in this paper and the Lecture Notes  seem to constitute the first systematic study of constant mean curvature surfaces in general homogeneous three-manifolds, as well as a starting point for further development of this area. It is worth mentioning that many of the results in this paper are proven for any homogeneous, simply connected three-manifold not diffeomorphic to . In our forthcoming paper  these results provide the foundation for understanding the geometry and classification of constant mean curvature spheres in any homogeneous three-manifold diffeomorphic to .
2. Background material on three-dimensional metric Lie groups.
We next state some basic properties of three-dimensional Lie groups endowed with a left invariant metric that will be used freely in later sections. For details of these basic properties, see Chapter 2 of the general reference .
Let denote a simply connected, homogeneous Riemannian three-manifold, and assume that it is not isometric to the Riemannian product , where is the Gaussian curvature of . Then is isometric to a simply connected, three-dimensional Lie group equipped with a left invariant metric , i.e., for every , the left translation , , is an isometry of . We will call such a space a metric Lie group, and denote it by . For the underlying Lie group structure of such an , there are two possibilities, unimodular and non-unimodular.
2.1. is unimodular.
Among all simply connected, three-dimensional Lie groups, the cases , (universal cover of ), (universal cover of the Euclidean group of orientation preserving rigid motions of the plane), (Sol geometry, or the universal cover of the group of orientation preserving rigid motions of the Lorentzian plane), Nil (Heisenberg group of upper triangular real matrices) and comprise the unimodular Lie groups.
Suppose that is a simply connected, three-dimensional unimodular Lie group equipped with a left invariant metric . It is always possible to find an orthonormal left invariant frame such that
for certain constants , among which at most one is negative. The triple of numbers and are called the structure constants and the canonical basis of the unimodular metric Lie group , respectively. The following associated constants are also useful in describing the geometry of a simply connected, three-dimensional unimodular metric Lie group:
For instance, the Levi-Civita connection associated to is given by
where denotes the cross product associated to and to the orientation on defined by declaring to be a positively oriented basis. From here it is easy to check that the symmetric Ricci tensor associated to diagonalizes in the basis with eigenvalues (see Milnor  for details)
Depending on the signs of the structure constants , we obtain six possible different Lie group structures, which are listed in the table below together with the possible dimension of the isometry group for a given left invariant metric on the Lie group .
|+, +, +|
|+, +, –||Ø|
|+, +, 0||Ø|
|+, –, 0||Sol||Ø||Ø|
|+, 0, 0||Ø||Nil||Ø|
|0, 0, 0||Ø||Ø|
Table 1. Three-dimensional, simply connected unimodular metric Lie groups. Each horizontal line corresponds to a unique Lie group structure; when all the structure constants are different, the isometry group of is three-dimensional. If two or more constants agree, then the isometry group of has dimension or . We have used the standard notation for the total space of the Riemannian submersion with bundle curvature over a complete simply connected surface of constant curvature .
Fix a unimodular metric Lie group with underlying Lie group and left invariant metric . Let be a (-orthonormal) canonical basis of . Now pick numbers and declare the length of to be , while keeping them orthogonal. This defines uniquely a left invariant metric on , and every left invariant metric on can be described with this procedure (see e.g., the discussion following Corollary 4.4 in ). It turns out that are Ricci eigendirections for every , although the associated Ricci eigenvalues depend on the and so, on .
For , the integral curve of passing through the identity element is a 1-parameter subgroup of .
In the above situation, each is the fixed point set of an order-two, orientation preserving isomorphism which we will call the rotation of angle about . Moreover, the following properties hold:
leaves invariant each of the collections of left cosets of , .
For every left invariant metric of , is a geodesic and is an isometry.
In Proposition 2.21 of  it is proved that, in the above conditions, there exist order two, orientation-preserving isomorphisms , , such that , and such that the pulled-back metric equals for every left invariant metric on . Equation (2.3) implies that the integral curve of is a geodesic of (for every left invariant metric ). Since , we conclude from uniqueness of geodesics that consists entirely of fixed points of . This proves Proposition 2.1. ∎
2.2. is non-unimodular.
The simply connected, three-dimensional, non-unimodular metric Lie groups correspond to semi-direct products with , as we explain next. A semi-direct product is the Lie group , where the group operation is expressed in terms of some real matrix as
here denotes the usual exponentiation of a matrix . Let
Then, a left invariant frame of is given by
In terms of , the Lie bracket relations are:
We define the canonical left invariant metric on the semidirect product to be that one for which the left invariant basis given by (2.7) is orthonormal. Equivalently, it is the left invariant extension to of the inner product on the tangent space at the identity element that makes an orthonormal basis.
We next emphasize some other metric properties of the canonical left invariant metric on :
The mean curvature of each leaf of the foliation with respect to the unit normal vector field is the constant . All the leaves of the foliation are intrinsically flat.
The change from the orthonormal basis to the basis given by (2.7) produces the following expression for the metric in the coordinates of :
The Levi-Civita connection associated to the canonical left invariant metric is easily deduced from (2.9) as follows:
A simply connected, three-dimensional Lie group is non-unimodular if and only if it is isomorphic to some semi-direct product with . If is a non-unimodular metric Lie group, then up to the rescaling of the metric of , we may assume that . This normalization in the non-unimodular case will be assumed throughout the paper. After an orthogonal change of the left invariant frame, we may express the matrix uniquely as
The canonical basis of the non-unimodular metric Lie group is, by definition, the left invariant orthonormal frame given in (2.7) by the matrix in (2.12). In other words, every simply connected, three-dimensional non-unimodular metric Lie group is isomorphic and isometric (up to possibly rescaling the metric) to with its canonical metric, where is given by (2.12).
We give two examples. If where is the identity matrix, we get a metric Lie group that we denote by , which is isometric to the hyperbolic three-space with its standard constant metric and where the underlying Lie group structure is isomorphic to that of the group of similarities of . If , we get the product space , where has constant negative curvature .
Under the assumption that , the determinant of determines uniquely the Lie group structure. This number is the Milnor -invariant of :
Every non-unimodular group admits a -rotation about , with similar properties to the three -rotations that appeared in Proposition 2.1 for the unimodular case. The following statement is an elementary consequence of equations (2.10) and (2.11).
Let be a non-unimodular metric Lie group endowed with its canonical metric. Then, the -axis is a geodesic and it is the fixed point set of the order-two isomorphism, orientation preserving isometry , . In particular, given any point , the subgroup of orientation-preserving isometries of that fix contains the subgroup .
In the case that the isometry group of a metric Lie group has dimension three, then the orientation-preserving isometries that fix reduce to the ones that appear in Propositions 2.1 and 2.3. In other words, the group of orientation-preserving isometries of that fix a given point is either isomorphic to (when is unimodular), or to (when is non-unimodular); see [17, Proposition 2.21].
3. The left invariant Gauss map of constant mean curvature surfaces in metric Lie groups.
Throughout this paper, by an -surface we will mean an immersed, oriented surface of constant mean curvature surface . By an -sphere we will mean an -surface diffeomorphic to .
It is classically known that the Gauss map of an -surface in with is a harmonic map into , and that is determined up to translations by its Gauss map. We next give an extension of this result to the case of an arbitrary metric Lie group , after exchanging the classical Gauss map by the left invariant Gauss map which we define next.
Given an oriented immersed surface with unit normal vector field (here refers to the tangent bundle of ), we define the left invariant Gauss map of the immersed surface to be the map that assigns to each the unit tangent vector to at the identity element given by .
Observe that the left invariant Gauss map of a two-dimensional subgroup of is constant.
We will prove that, after stereographically projecting the left invariant Gauss map of an -surface in from the south pole111Here, the south pole of is defined in terms of the canonical basis of . of the unit sphere , the resulting function satisfies a conformally invariant elliptic PDE that can be expressed in terms of the -potential of the space , that we define next:
Let be a non-unimodular metric Lie group. Rescale the metric on so that is isometric and isomorphic to with its canonical metric, where is given by (2.12) for certain constants . Given , we define the -potential of to be the map given by
where denotes the complex conjugate of .
Note that, has a finite limit as (in particular, blows up when ). We will say that the -potential for has a zero at if . A simple analysis of the zeros of the -potential in (3.1) and (3.2) gives the following lemma.
Let be a metric Lie group and . Then, the -potential for is everywhere non-zero if and only if:
is isomorphic to , or
is not isomorphic to , is unimodular and , or
is non-unimodular with -invariant and , or
is non-unimodular with -invariant and .
We use next Lemma 3.4 to prove the following fact, that will be used later on:
Assume that there exists a compact -surface in . Then, the -potential of is everywhere non-zero.
If is non-unimodular, then the horizontal planes produce a foliation of by leaves of constant mean curvature ; recall that the matrix is given by (2.12). Applying the usual maximum principle to and to the leaves of such foliation we deduce that . So, by Lemma 3.4, the -potential for does not vanish in this case.
If is unimodular and not isomorphic to , then is diffeomorphic to . Suppose is a compact minimal surface in . After a left translation, we can assume that . Choose so that the Lie group exponential map restricts to the ball of radius centered at as a diffeomorphism into . Let be a point in and denote by the 1-parameter subgroup of generated by the unique such that . As is a proper arc in and is compact, there exists a largest such that , where , . Applying the maximum principle for minimal surfaces to and we deduce that . By applying again, we have , which contradicts the defining property of . Therefore, there are no compact minimal surfaces in , and so . By Lemma 3.4, the -potential for is everywhere non-zero. Finally, if is isomorphic to , the same conclusion holds again by Lemma 3.4. This proves Fact 3.5. ∎
If is unimodular and not isomorphic to , then is either isometric and isomorphic to for a matrix with trace zero, or to . In both cases, it is not difficult to check that there exists a minimal two-dimensional subgroup; hence the argument in the first paragraph of the last proof could be also adapted to this situation to prove the non-existence of a compact immersed minimal surface in .
Suppose is a simply connected Riemann surface with conformal parameter , is a metric Lie group, and .
Let be a solution of the complex elliptic PDE
such that everywhere222By we mean that if and that if ., and such that the -potential of does not vanish on (for instance, this happens if satisfies the conditions of Lemma 3.4). Then, there exists an immersed -surface , unique up to left translations, whose Gauss map is .
Conversely, if is the Gauss map of an immersed -surface in a metric Lie group , and the -potential of does not vanish on , then satisfies the equation (3.3), and moreover holds everywhere.
We will only consider the case that is unimodular, since the non-unimodular situation can be treated in a similar way.
Given a solution to (3.3) as in the statement of the theorem, define the functions by
and is a conformal parameter on . Noting that has a finite limit as , we can easily deduce from (3.4) that actually take values in .
A direct computation shows that
Since satisfies (3.3), the above equation yields
Now, equation (3.2) implies that
where are given by (2.2) in terms of the structure constants of the unimodular metric Lie group .
Working analogously with , we obtain
In particular, from (2.2) we have
Therefore, if we define where and is the identity element in , we have by (3.9) that
We want to solve the first order PDE in the unknown . To do this, we will apply the classical Frobenius Theorem. By a direct computation, we see that the formal integrability condition for the local existence of such map is given precisely by (3.10). Thus, as is simply connected, the Frobenius theorem ensures that there exists a smooth map such that and is unique once we prescribe an initial condition .
A straightforward computation gives that , where the conformal factor is
Since and does not vanish on , it is easy to see that on by using that has a finite limit at (which is non-zero, by hypothesis). Thus, is a conformally immersed surface in , and it is immediate from (3.4) that the stereographically projected left invariant Gauss map of is . Also, it is clear that is unique up to left translations in .
It only remains to prove that has constant mean curvature of value . Let be the mean curvature function of . By the Gauss-Weingarten formulas, we have
where is the Levi-Civita connection in and is the unit normal field to in . Thus, since , if we let then we have
Therefore, using (2.3) and that where is the cross product in the Lie algebra of , we conclude that
Comparing this equation with (3.8), we get , as desired.
Conversely, we need to show that the left invariant Gauss map of any conformally immersed -surface , where is such that the -potential of does not vanish on , satisfies the elliptic PDE (3.3) and also everywhere. But this can be done just as in the proof of Theorem 3.4 in the paper by Daniel and Mira , using now the general metric relations given by (2.3) instead of the specific ones of . We omit the details. ∎
The following application of Theorem 3.7 will be useful later on.
Let be an immersed -surface with left invariant Gauss map and assume that the -potential for does not vanish on . Then, the differential of has rank at most everywhere on if and only if is invariant under the flow of a right invariant vector field on . Furthermore, if is invariant under the flow of a right invariant vector field, then the rank of the differential of is everywhere on and the Gauss map image is a regular curve in .
First assume that has rank less than or equal to everywhere on . Since we are assuming that the -potential of does not vanish on , then the last statement in Theorem 3.7 implies . Therefore, the differential of has rank one everywhere and is a regular curve. Around any given point , there exists a conformal parameter such that with , i.e., does not depend on . In particular, can be considered to be defined on a vertical strip in the -plane, and by Theorem 3.7 the conformally immersed -surface with Gauss map can also be extended to be defined on that vertical strip in the -plane, so that differs by a left translation from for every . Hence, by analyticity, we see that the is invariant under the flow of a right invariant vector field of .
The converse implication is trivial, since any surface invariant under the flow of a right invariant vector field is locally obtained by left translating in some regular curve on the surface, and so the differential of its left invariant Gauss map has rank at most 1 everywhere. ∎
4. Index-one spheres in metric Lie groups.
The Jacobi operator of an immersed two-sided hypersurface with constant mean curvature in a Riemannian manifold is defined as
Here, is the Laplacian of in the induced metric, the square of the norm of the second fundamental form of and a unit vector field on the hypersurface.
A domain with compact closure is said to be stable if for all compactly supported smooth functions , or equivalently, for all functions in the closure of in the usual Sobolev norm. is called strictly unstable if it is not stable.
The index of a domain with compact closure is the number of negative eigenvalues of acting on functions of ; thus, is stable if and only if its index is zero. If zero is an eigenvalue of on , then the nullity of is the (finite) dimension of the eigenspace associated to this zero eigenvalue. Since the index of stability is non-decreasing with the respect to the inclusion of subdomains of with compact closure, one can define the index of stability of as the supremum of the indices over any increasing sequence of subdomains with compact closure and .
If is compact, then is called weakly stable if for all with . Every solution to the classical isoperimetric problem is weakly stable, and every compact, weakly stable constant mean curvature surface has index zero or one for the stability operator. We refer the reader to the handbook  for the basic concepts and results concerning stability properties of constant mean curvature hypersurfaces in terms of the Jacobi operator.
Since every metric Lie group is orientable, then the two-sidedness of a surface in is equivalent to its orientability. It is easy to see that the index of an -sphere in is at least one; indeed, if denotes a basis of right invariant vector fields of (that are Killing, independently of the left invariant metric on ), then the functions , , are Jacobi functions on , i.e., . Since right invariant vector fields on are identically zero or never zero and spheres do not admit a nowhere zero tangent vector field, the functions , , are linearly independent. Hence, is an eigenvalue of of multiplicity at least three. As the first eigenvalue is simple, then is not the first eigenvalue of and thus, the index of is at least one. Moreover, if the index of is exactly one, then it follows from Theorem 3.4 in Cheng  (see also [6, 25]) that has dimension three, and hence we conclude that