Servicios Personalizados
Revista
Articulo
Inglés (pdf)
Articulo en XML
Referencias del artículo
Traducción automática
Enviar articulo por emailIndicadores
-
Citado por SciELO
Links relacionados
-
Similares en
SciELO
Compartir
Permalink
Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.2 Bahía Blanca jul./dic. 2006
On the stability index of minimal and constant mean curvature hypersurfaces in spheres
Luis J. Alías
Abstract: The study of minimal and, more generally, constant mean curvature hypersurfaces in Riemannian space forms is a classical topic in differential geometry. As is well known, minimal hypersurfaces are critical points of the variational problem of minimizing area. Similarly, hypersurfaces with constant mean curvature are also solutions to that variational problem, when restricted to volume-preserving variations. In this paper we review about the stability index of both minimal and constant mean curvature hypersurfaces in Euclidean spheres, including some recent progress by the author, jointly with some of his collaborators. One of our main objectives on writing this paper has been to make it comprehensible for a wide audience, trying to be as self-contained as possible.
2000 Mathematics Subject Classification. Primary 53C42; Secondary 53A10
The author was partially supported by MEC/FEDER project no. MTM2004-04934-C04-02, F. Séneca project no. 00625/PI/04, and F. Séneca grant 01798/EE/05, Spain
1. Stability and index of minimal hypersurfaces
Let us consider 
an orientable hypersurface immersed into the unit Euclidean sphere 
. We will denote by 
the shape operator of 
with respect to a globally defined normal unit vector field 
. That is, 
is the endomorphism determined by
where 
and 
denote, respectively, the Levi-Civita connections on 
and 
. As is well known, 
defines a symmetric endomorphism on 
whose eigenvalues 
are usually referred to as the principal curvatures of the hypersurface. The mean curvature of 
is then defined as
Throughout this paper, we will assume that 
is compact. Every smooth function 
induces a normal variation 
of the original immersion 
, given by
where Exp denotes the exponential map in 
. Since 
is compact and 
is an immersion, there exists 
such that every 
is also an immersion, for 
. Then we can consider the area function 
which assigns to each 
the 
-dimensional area of 
with respect to the metric induced on 
by the immersion 
. That is,
where 
stands for the manifold 
endowed with the metric induced by 
from the Euclidean metric on 
, and 
is the 
-dimensional area element of that metric on 
. The first variation formula for the area [16, Chapter I, Theorem 4] establishes that
|
| (1.1) |
As a consequence, 
is a minimal hypersurface (that is, 
on 
) if and only if 
for every smooth function 
. In other words, minimal hypersurfaces are characterized as critical points of the area functional.
The stability operator of this variational problem is given by the second variation formula for the area [16, Chapter I, Theorem 32], which in our case is written as follows
|
| (1.2) |
Here 
, where 
stands for the Laplacian operator of 
and 
is the square of the norm of the shape operator. The operator 
is called the Jacobi (or stability) operator of the minimal hypersurface 
. The Jacobi operator 
belongs to a class of operators which are usually referred to as Schrödinger operators, that is, operators of the form 
, where 
is any continuous function on 
. As is well known, the spectrum of 
consists of an increasing sequence of eigenvalues 
with finite multiplicities 
and such that 
. Moreover, the first eigenvalue is simple (
) and it satisfies the following min-max characterization
|
| (1.3) |
Observe that with our criterion, a real number 
is an eigenvalue of 
if and only if 
for some smooth function 
, 
.
The Jacobi operator induces the quadratic form 
acting on the space of smooth functions on 
by
and the index of a minimal hypersurface 
, denoted by 
, is defined as the maximum dimension of any subspace 
of 
on which 
is negative definite. That is,
Equivalently, 
is the number of negative eigenvalues of 
(counted with multiplicity), which is necessarily finite and it is given by
A minimal hypersurface would be said to be stable if 
for every 
. Equivalently, in terms of the index, stability would mean that 
.
Intuitively, 
measures the number of independent directions in which the hypersurface fails to minimize area. To see it, observe that if 
for some 
, then 
and therefore 
for small values of 
, in the normal variation of 
induced by 
. That means that the minimal hypersurface 
, while a critical point of the area functional, is not a local minimum. For minimal hypersurfaces in 
this is always the case. In fact, taking the constant function 
one has
|
| (1.4) |
In particular, every compact minimal hypersurface in 
is unstable.
2. Minimal hypersurfaces with low index
We have just seen that there exists no compact stable minimal hypersurface in 
. Equivalently, 
for every compact minimal hypersurface in the sphere. In [22, Theorem 5.1.1], Simons characterized the totally geodesic equators 
as the only compact minimal hypersurfaces in 
with 
. Later on, Urbano [23] when 
, and El Soufi [11] for general 
(see also [20, Lemma 3.1]), proved that if 
is not a totally geodesic equator, then not only must be 
but in fact it must hold
Therefore, we have the following result.
Theorem 1. Let 
be a compact orientable minimal hypersurface immersed into the Euclidean sphere 
. Then
- either
(and 
is a totally geodesic equator 
), - or
.
On the other hand, apart from the totally geodesic equators, which are obtained as intersections of 
with linear hyperplanes of 
, the easiest minimal hypersurfaces in 
are the minimal Clifford tori. They are obtained by considering the standard immersions 
and 
, for a given integer 
, and taking the product immersion 
. As we will see in the next section, all minimal Clifford tori have 
. For that reason, it has been conjectured for a long time that minimal Clifford tori are the only compact minimal hypersurfaces in 
with 
, changing Theorem 1 into the following conjecture.
Conjecture 2. Let 
be a compact orientable minimal hypersurface immersed into the Euclidean sphere 
. Then
- either
(and 
is a totally geodesic equator 
), - or
, with equality if and only if 
is a minimal Clifford torus 
.
In [23], Urbano showed that the conjecture is true when 
. See also next section for some other partial answers to Conjecture 2 in the general 
-dimensional case.
On the other hand, using also the constant function 
as a test function in (1.3) to estimate 
, from (1.4) one finds that
|
| (2.1) |
Moreover, equality 
holds if and only if 
on 
, that is, if and only if 
is a totally geodesic equator 
. In [22, Lemma 6.1.7] Simons proved that when 
is not a totally geodesic equator, then not only must be 
but in fact it must hold
Later on, Wu [24] was able to characterize the case in which equality 
holds, by showing that equality holds if and only if 
is a minimal Clifford torus. More recently, Perdomo [21] has considered again this problem, giving a new proof of that spectral characterization of minimal Clifford tori by the first stability eigenvalue. Summing up, we have the following result.
Theorem 3. Let 
be a compact orientable minimal hypersurface immersed into the Euclidean sphere 
, and let 
stand for the first eigenvalue of its Jacobi operator. Then
- either
(and 
is a totally geodesic equator 
), - or
, with equality if and only if 
is a minimal Clifford torus 
.
3. Proof of Theorem 1 and some partial answers to Conjecture 2
Before giving the proof of Theorem 1, we will fix some notation and establish some basic formulae which will be useful throughout this paper. Let us consider 
an orientable hypersurface immersed into the unit Euclidean sphere 
, with normal unit vector field 
. If 
denotes the Levi-Civita connection on 
, then the Gauss and Weingarten formulae for the immersion 
are given by
|
| (3.1) |
and
|
| (3.2) |
for every tangent vector fields 
.
As is well known, the curvature tensor 
of the hypersurface 
is described in terms of 
by the Gauss equation of 
, which can be written as
|
| (3.3) |
for 
. Observe that our criterion for the definition of the curvature is
![R (X, Y )Z = ∇ [X,Y]Z - [∇X ,∇Y ]Z.](/img/revistas/ruma/v47n2/2a04171x.png)
As a consequence of (3.3), the Ricci curvature of 
is given by
|
| (3.4) |
It follows from here that the scalar curvature of every minimal hypersurface in 
satisfies
|
| (3.5) |
with equality only at points where 
is totally geodesic. As a consequence, the only minimal hypersurfaces in 
which are isometric to a unit round sphere are the totally geodesic equators. On the other hand, the Codazzi equation of 
is given by
|
| (3.6) |
for every tangent vector fields 
, where 
denotes the covariant differential of 
,
For a fixed arbitrary vector 
, we will consider the functions 
and 
defined on 
. Observe that 
and 
are, respectively, the coordinates of the immersion 
and the Gauss map 
. A standard computation, using Gauss (3.1) and Weingarten (3.2) formulae, shows that the gradient and the hessian of the functions 
and 
are given by
for every tangent vector fields 
. Here 
denotes the tangential component of 
along the immersion 
, that is,
|
| (3.11) |
Equation (3.8) directly yields
|
| (3.12) |
On the other hand, using Codazzi equation (3.6) in (3.10) we also get
Here we are using the fact that trace commutes with the covariant derivative, which yields
Now we are ready to prove Theorem 1.
Proof of Theorem 1. We already know from our previous discussions that 
for every compact minimal hypersurface in 
. Moreover, if 
is a totally geodesic equator in 
, then the Jacobi operator reduces to 
, where 
is the Laplacian operator on the unit sphere 
. In particular, the eigenvalues of 
are given by 
, where 
denotes the 
-th eigenvalue of the Laplacian on 
, with the same multiplicity. Then, 
with multiplicity 1 and 
. In particular, 
for a totally geodesic equator of the sphere.
Therefore, it remains to show that, if 
is not a totally geodesic equator, then 
. When 
is not totally geodesic, we already know from the estimate (2.1) that 
with multiplicity 
. Therefore, we will prove that 
by showing that 
is also another negative eigenvalue of 
with multiplicity at least 
. Since 
, equation (3.13) implies that 
, and then
for every 
. Thus, whenever 
, the functions 
are eigenfunctions of 
with negative eigenvalue 
. We claim that if 
is not totally geodesic in 
, then the dimension of the linear subspace 
is 
. If our claim is true, then the multiplicity of 
as an eigenvalue of 
will be at least 
and this will finish the proof of the theorem.
To show our claim, we will follow the ideas of Urbano in [23] for the case 
. Obviously, 
. If 
, then there exists a unit vector 
such that 
on 
. From (3.8), that implies that 
. Moreover, by (3.11) we also have
which in particular implies that the function 
cannot be constant on 
. Then a classical result by Obata [18, Theorem A] implies that 
is isometric to a unit round sphere. But we have already seen as a consequence of Gauss equation (3.3) that the only minimal hypersurfaces in 
which are isometric to a unit round sphere are the totally geodesic equators. Thus, if 
is not totally geodesic we have 
, as claimed. We also refer the reader to [20, Lemma 3.1] for an alternative proof of our claim, using a more geometric argument. □
Let us consider now a minimal Clifford torus 
in 
. At a point 
, the vector field
defines a normal unit vector at the point 
. With respect to this orientation, its principal curvatures are given by
Then, every minimal Clifford torus has 
. In particular, its Jacobi operator reduces to 
, where 
is the Laplacian on the product manifold 
, and the eigenvalues of 
are given by 
, where 
are the eigenvalues of 
. Therefore, the index of 
reduces to the number of eigenvalues of 
(counted with multiplicity) which are strictly less than 
.
To compute it, simply recall that if 
is an eigenvalue of the Laplacian on a Riemannian manifold 
with multiplicity 
and 
is an eigenvalue of the Laplacian on a Riemannian manifold 
with multiplicity 
, then 
is an eigenvalue of the Laplacian on the product manifold 
, and the multiplicity of 
is the sum of the products 
for all the possible values of 
and 
satisfying 
[8]. In our case, the eigenvalues of the Laplacian on 
are given by
with multiplicities
and
and the eigenvalues of the Laplacian on 
are given by
with multiplicities
and
It easily follows from here that 
with multiplicity 1, 
with multiplicity 
and 
. Therefore, all minimal Clifford tori in 
have 
, which supports Conjecture 2.
In [23] Urbano obtained the following characterization of minimal Clifford tori in 
, solving Conjecture 2 when 
.
Theorem 4. Let 
be a compact orientable minimal surface immersed into 
, which is not a totally geodesic equator. Then 
, with equality if and only if 
is a minimal Clifford torus 
.
Later on, Guadalupe, Brasil Jr. and Delgado [14] showed that the conjecture is true for every dimension 
, under the additional hypothesis that 
has constant scalar curvature, obtaining the following result.
Theorem 5. Let 
be a compact orientable minimal hypersurface immersed into 
, which is not a totally geodesic equator. Assume that 
has constant scalar curvature. Then 
, with equality if and only if 
is a minimal Clifford torus 
.
More recently, Perdomo [20] has showed that the conjecture is also true for every dimension 
with an additional assumption about the symmetries of 
, and, in particular, the conjecture is true for minimal hypersurfaces with antipodal symmetry.
The proof of Theorem 3 makes use of a celebrated formula for the Laplacian of the function 
on 
, which was established by Simons in [22]. Specifically, for the case of minimal hypersurfaces in 
, Simons formula reads as follows,
|
| (4.1) |
To give a proof of (4.1), let us introduce the following standard notation. Let 
be two self-adjoint operators. Then
and
where 
is a local orthonormal frame on 
. Recall that, in our notation,
for 
. On the other hand, the rough Laplacian of an operator 
is defined as the operator 
given by
Recall again that in our notation, 
.
Consider now 
an orientable hypersurface immersed in 
, which in principle we do not assume to be minimal. A standard tensor computation implies
|
| (4.2) |
By the Codazzi equation (3.6), we know that 
is symmetric and, hence, 
is also symmetric in its two first variables,
Regarding to the symmetries of 
in the other variables, it is not difficult to see that
Thus, using the Gauss equation (3.3) we conclude from here that
where we have used the facts that trace commutes with 
and that 
. In particular, if 
is a minimal hypersurface in 
we conclude that
which jointly with (4.2) implies (4.1).
As a first application of Simons formula (4.1), we have the following result.
Theorem 6. Let 
be a compact orientable minimal hypersurface immersed into the Euclidean sphere 
, and assume that 
on 
. Then
- either
(and 
is a totally geodesic equator 
), - or
and 
is a minimal Clifford torus.
Part (i) and the sharp bound given in (ii) are due to Simons [22, Corollary 5.3.2]. On the other hand, the characterization of minimal Clifford tori given in (ii), which is local, was obtained independent and simultaneously by Chern, do Carmo and Kobayashi [10] and Lawson [15].
Proof of Theorem 6. Integrating (4.1) on 
, and using Stokes' theorem and the hypothesis 
, we obtain that
Therefore 
on 
, and either 
(and 
is totally geodesic) or 
. This proves part (i) of the theorem and the first statement of part (ii). If 
, then a local argument using the fact that 
implies that 
has exactly two constant principal curvatures
with multiplicities 
and 
, respectively (for the details, see [10] or the proof of [15, Lemma 1]). In other words, 
is a minimal isoparametric hypersurface of 
with two distinct principal curvatures, and from a classical result by Cartan [9], 
must be a minimal Clifford tori. Actually, in [9] Cartan showed that an isoparametric hypersurface of 
with two distinct principal curvatures must be an open piece of a standard product 
with 
, but 
being minimal it must be 
. □
The proof of Theorem 3 below makes use also of the following auxiliary result, which can be found in [7] (see also [24, Lemma 1]).
Lemma 7. Let 
be a Riemannian manifold, and consider 
a symmetric tensor on 
such that 
and its covariant differential 
is symmetric. Then
Proof of Theorem 3. We already know from (2.1) that 
with equality if and only if 
is a totally geodesic equator. Then, assume that 
is not totally geodesic and consider, for every 
, the positive smooth function 
. We will use 
as a test function to estimate 
in (1.3). Observe that
Therefore, using Simons formula (4.1) we obtain
On the other hand, Lemma 7 applied to 
yields
so that
Then,
Therefore, using 
as a test function in (1.3), we get
Since 
is not totally geodesic, then
Now, letting 
in (4.4) we conclude from here that
Moreover, if 
then 
on 
, and Lemma 7 implies that 
is a positive constant. Thus 
, where 
is a constant, and the first eigenvalue of 
is simply the constant 
. Therefore, 
and by Theorem 6 we conclude that 
must be a minimal Clifford torus. □
In [21], Perdomo gave another proof of the characterization of minimal Clifford tori by the equality 
. His proof is based on a maximum principle, and it works as follows. Let us assume that 
. In particular, by (2.1) we know that 
is not totally geodesic. Let 
be the (non-empty) open subset of non-geodesic points of 
. The function 
is smooth on 
. Writing 
and using Simons formula (4.1), we obtain that
|
| (4.5) |
on 
. By Lemma 7 we also have
Therefore, using this into (4.5) we obtain
|
| (4.6) |
on 
. Moreover, if equality 
holds at a point 
, then 
.
As is well known, the first eigenvalue 
is simple, and its eigenspace is generated by a positive smooth function 
. Then 
or, equivalently,
Observe that
|
| (4.7) |
Consider the smooth function 
defined on 
by 
. A straightforward computation using (4.6) and (4.7) yields
|
| (4.8) |
Summing up,
|
| (4.9) |
where 
is the differential operator on 
given by 
. Let 
be a point where the function 
attains its (positive) maximum on 
, and let 
be a region around 
on which 
is greater than some positive constant. Since the maximum of 
in 
is attained in the interior of 
, by (4.9) and the maximum principle applied to 
we deduce that 
is constant on 
. Since 
is connected, we conclude that 
is a positive constant on the whole 
. Therefore, equality trivially holds in (4.9). That means that the inequality in (4.8) must be an equality, but this happens if and only if equality happens in (4.6), which implies that 
on 
. Then, by Lemma 7 we know that 
is a positive constant, and since 
, we obtain that 
. Then Theorem 6 implies that 
must be a minimal Clifford torus. This finishes Perdomo's proof.
5. Stability and index of constant mean curvature hypersurfaces
Let 
be a compact orientable hypersurface immersed into the unit Euclidean sphere. As another consequence of the first variation formula for the area (1.1), we have that 
has constant mean curvature (not necessarily zero) if and only if 
for every smooth function 
satisfying the additional condition 
. To see it, assume that 
for every 
satisfying 
, and write 
, where
Since 
, then
but this implies that 
is constant on 
.
Geometrically, the additional condition 
means that the variations under consideration preserve a certain volume function. In fact, if 
is the normal variation induced by a smooth function 
, then the volume function is the function 
defined by
![∫ V (t) = Ψ *(dV ), [0,t]× Σ](/img/revistas/ruma/v47n2/2a04481x.png)
where 
denotes the 
-dimensional volume element of 
and 
is the variation of 
, 
. Then, the first variation of 
is given by
We refer the reader to [5, 6] for the details. A variation is said to be volume-preserving if 
for all 
. As shown by Barbosa, do Carmo and Eschenburg in [6, Lemma 2.2], given a smooth function 
with 
, there exists a volume-preserving normal variation whose variation vector field is 
. As a consequence, 
has constant mean curvature (not necessarily zero) if and only if 
for every volume-preserving variation of 
. In other words, constant mean curvature hypersurfaces are characterized as critical points of the area functional when restricted to volume-preserving variations.
As in the case of minimal hypersurfaces, the stability operator of this variational problem is given by the second variation formula of the area (1.2), and similarly the corresponding quadratic form is also given by
with Jacobi operator 
. However, in contrast to the case of minimal hypersurfaces, in the case of hypersurfaces with constant mean curvature one can consider two different eigenvalue problems: the usual Dirichlet problem, associated with the quadratic form 
acting on the whole space of smooth functions on 
, and the so called twisted Dirichlet problem, associated with the same quadratic form 
, but restricted to the subspace of smooth functions 
satisfying the additional condition 
.
Similarly, there are two different notions of stability and index, the strong stability and strong index, denoted by 
and associated to the usual Dirichlet problem, and the weak stability and weak index, denoted by 
and associated to the twisted Dirichlet problem. Thus, the strong index is simply
and 
is called strongly stable if and only if 
. On the other hand, the weak index is
where
and 
is called weakly stable if and only if 
. From a geometrical point of view, the weak index is more natural than the strong index. However, from an analytical point of view, the strong index is more natural and easier to use.
In [4], Barbosa and Bérard studied in depth the twisted Dirichlet problem, comparing the eigenvalues of this problem with the eigenvalues of the usual Dirichlet problem. For instance, it easily follows from the min-max principle that both spectra are interwined by
|
| (5.1) |
where
is the usual spectrum of 
and
is its twisted spectrum.
When dealing with constant mean curvature hypersurfaces, instead of the second fundamental form 
, it is more convenient to work with the so called traceless second fundamental form, which is given by 
, where 
denotes the identity operator on 
. Observe that
with equality if and only if 
is totally umbilical. For that reason 
is also called the total umbilicity tensor of 
. In terms of 
, the Jacobi operator is given by
Using again the constant function 
as a test function for estimating 
one has
In particular, 
for every constant mean curvature hypersurface in 
, which means that there is no strongly stable constant mean curvature hypersurface in 
. It also follows from here that
|
| (5.2) |
with equality 
if and only if 
is a totally umbilical round sphere 
. Observe that, in general, 
contributes to 
but not to 
because its eigenspace is generated by a positive smooth function 
which does not satisfy the additional condition 
. On the other hand, in [13, Theorem 2.1], El Soufi and Ilias derived a sharp upper bound for the second eigenvalue of a Scrödinger operator of the form 
of a compact submanifold 
of a Riemannian space form, in terms of the total mean curvature of 
and the mean value of the potential 
. In particular, for the Jacobi operator of a constant mean curvature hypersurface in 
, their estimate yields
|
| (5.3) |
with equality 
if and only if 
is totally umbilical. See also [12] for another interesting bound for the second eigenvalue of a Scrödinger operator, and its applications to the study of minimal and constant mean curvature hypersurfaces.
6. Weakly stable constant mean curvature hypersurfaces
We have just shown that there is no compact strongly stable constant mean curvature hypersurface in 
. In contrast to this, Barbosa, do Carmo and Eschenburg [6, Theorem 1.2] characterized the totally umbilical round spheres 
as the only compact weakly stable constant mean curvature hypersurfaces in 
.
Theorem 8. Let 
be a compact orientable constant mean curvature hypersurface immersed into the unit Euclidean sphere 
. Then 
is weakly stable if and only if 
is a totally umbilical round sphere 
.
Proof. If 
is a totally umbilical round sphere 
in 
with radius 
, then 
and the Jacobi operator reduces to 
. Therefore, the eigenvalues of 
are given by 
, where 
is the 
-th eigenvalue of the Laplacian operator on 
, with the same multiplicity. In particular, 
with multiplicity 1 and its associated eigenfunctions are the constant functions. Therefore, since all the other eigenfunctions of 
(for the usual Dirichlet problem) are orthogonal to the constant functions, they do satisfy the additional condition 
. Thus, in this case we have 
for every 
. Since 
, it follows from here that 
and 
is weakly stable.
Conversely, assume that 
is a compact orientable hypersurface with constant mean curvature in 
which is weakly stable. This means that
for every smooth function 
with 
. As in the proof of Theorem 1, we will work with the functions 
and 
, where 
is a fixed arbitrary vector. Since 
is constant, writing 
equation (3.13) becomes
|
| (6.1) |
Let us consider the function 
. From (3.12) we have that 
, so that 
trivially satisfies the condition 
. Using (3.12) and (6.1), we easily get
and then
Therefore, we have that
for every fixed arbitrary vector 
. Let us choose 
as an element of the standard orthonormal basis of 
, 
. Then
because of
But this implies that 
on 
, and it must be totally umbilical. □
It is worth pointing out that Theorem 8 can be seen also as a consequence of the estimate (5.3), jointly with the interwining (5.1) of the two spectra of 
. In fact, if 
is weakly stable, then we have
which implies again that 
and 
must be totally umbilical.
7. Constant mean curvature hypersurfaces with low index
Apart from the totally umbilical spheres, the easiest constant mean curvature hypersurfaces in 
are the constant mean curvature Clifford tori. They are obtained by considering the standard immersions 
and 
, for a given radius 
and integer 
, and taking the product immersion 
. Its principal curvatures are given by
and its constant mean curvature 
is given by
In particular, 
precisely when 
, which corresponds to the minimal Clifford torus.
For the constant mean curvature Clifford tori, one has
and the Jacobi operator reduces to 
, where 
is the Laplacian operator on the product manifold 
. In particular, the eigenvalues of 
are given by 
, where 
are the eigenvalues of 
, and they have the same multiplicity. Thus, 
with multiplicity 1 and its associated eigenfunctions are the constant functions. Moreover, since all the rest of eigenfunctions of 
(for the usual Dirichlet problem) are orthogonal to the constant functions, they do satisfy the additional condition 
. Thus, similarly to the case of totally umbilical round spheres, we have 
for every 
, and 
reduces to the number of positive eigenvalues of the Laplacian operator (counted with multiplicity) which are strictly less than 
. This yields that 
for every constant mean curvature Clifford torus, and 
precisely when 
(for the details, see [3, Section 3]). Observe that, in particular, this happens when 
, so that, the minimal Clifford tori satisfy 
when regarded as constant mean curvature hypersurfaces.
Motivated by this value of 
, in [3] Alías, Brasil and Perdomo have recently obtained the following result, which extends Theorem 1 (under the additional hypothesis of constant scalar curvature) and Theorem 5 to the case of constant mean curvature hypersurfaces.
Theorem 9. Let 
be a compact orientable hypersurface immersed into the Euclidean sphere 
with constant mean curvature. Assume that 
has constant scalar curvature. Then
- either
(and 
is a totally umbilic sphere in 
), or 
, with equality if and only if 
is a constant mean curvature Clifford torus 
with radius 
.
As observed in [3], the value of the index of the constant mean curvature Clifford tori 
converges to infinity as 
converges either to 0 or 1. For that reason, in contrast to the case of minimal Clifford tori, it is not possible, in general, to find a characterization of all constant mean curvature Clifford tori in terms of their index.
Proof. We already know from Theorem 8 that 
for the totally umbilical round spheres, whereas 
for the rest of compact constant mean curvature hypersurfaces in 
, without any additional hypothesis. Then, first we need to show that, if the scalar curvature of 
is constant (or equivalently, 
is constant; see equation (3.5)), then 
for every compact constant mean curvature hypersurface in 
which is not totally umbilical. That is, we need to find a subspace 
of 
with 
on which 
is negative definite.
As in the proof of Theorem 8, we will consider the functions 
and 
, where 
is a fixed arbitrary vector. When 
, we can take 
. In fact, since 
and 
is a positive constant, then (6.1) implies that the functions 
are eigenfunctions of 
with negative eigenvalue 
and that they also satisfy the condition 
. Moreover, we have also seen in the proof of Theorem 1 that, when 
is minimal and not totally geodesic, then 
.
Therefore, in what follows we assume that 
is a non-zero constant. In that case, we take 
where
and 
are the two different real roots of the quadratic equation
That is,
Using (3.12) and (6.1), it is not difficult to see that 
for every 
, where
and that they also satisfy the condition 
. Therefore
It remains to estimate 
. Taking into account that 
, where 
is the linear map given by 
, we deduce that
On the other hand, using that 
we also have
which implies
and therefore 
. We refer the reader to [3, Section 4] for further details about this proof.
Moreover, if 
then 
which means that 
splits as direct sum of the two subspaces 
and 
. Then, at any point 
the tangent space 
splits also as a direct sum of two subspaces
where 
and 
. Using that 
is not totally umbilical, and equations (3.7) and (3.9), we can see that, at each point 
, 
and 
are subspaces of principal directions of 
with constant principal curvatures 
and 
, respectively (see [3, Section 4] for the details). As a consequence, 
is a compact isoparametric hypersurface of 
with two distinct principal curvatures, and from the well known rigidity result by Cartan [9] we conclude that 
is a standard product of the form 
with radius 
. Finally, from our previous discussion about the values of 
for those hypersurfaces, we conclude that it must be 
. □
8. A sharp estimate for the first eigenvalue of the Jacobi operator
As we already know from (5.2), the first eigenvalue of the Jacobi operator of a compact hypersurface 
with constant mean curvature 
in 
satisfies
with equality if and only if 
is a totally umbilical round sphere. As an extension of Theorem 3 to the case of constant mean curvature, Alías, Barros and Brasil in [2] have recently proved that when 
is not totally umbilical, then not only must be 
but it fact it must hold
where 
is the norm of the total umbilicity tensor of 
. Moreover, they were also able to characterize the case where equality holds, obtaining the following extension of Theorem 1.
Theorem 10. Let 
be a compact orientable hypersurface immersed into the Euclidean sphere 
with constant mean curvature 
, and let 
stand for the first eigenvalue of its Jacobi operator. Then
- either
(and 
is a totally umbilic sphere in 
), or 
, with equality if and only if 
and 
is a minimal Clifford torus 
, with 
;
, 
, and 
is a constant mean curvature Clifford torus 
with 
, 
;
, 
, and 
is a constant mean curvature Clifford torus 
with 
.
For the proof of Theorem 10, we will need the following result due to Alencar and do Carmo, which extends Theorem 6 to the case of constant mean curvature hypersurfaces.
Theorem 11. Let 
be a compact orientable hypersurface immersed into the Euclidean sphere 
with constant mean curvature 
, and assume that 
, where
is the positive root of the polynomial
|
| (8.1) |
Then
- either
(and 
is a totally umbilic sphere in 
), or 
and 
and 
is a minimal Clifford torus 
, with 
;
, 
, and 
is a constant mean curvature Clifford torus 
with 
, 
;
, 
, and 
is a constant mean curvature Clifford torus 
with 
.
The proofs of Theorem 10 and Theorem 11 make use of an extension of Simons formula (4.1) for the case of hypersurfaces with constant mean curvature, which is due to Nomizu and Smyth [17]. To see it, consider 
the total umbilicity tensor and recall that 
. Since we are assuming now that 
is constant, we have that 
and 
, and (4.2) can be written in terms of 
as follows
|
| (8.2) |
Moreover, by (4.3) we also have
Then, taking into account that 
, (8.2) becomes
|
| (8.3) |
As a first application of equation (8.3), we may give the proof of Theorem 11. For the proof, we will also need the following auxiliary result, known as Okumura lemma, which can be found in [19] and [1, Lemma 2.6].
Lemma 12. Let 
be real numbers such that 
. Then
Moreover, equality holds in the right-hand (respectively, left-hand) side if and only if 
of the 
's are nonpositive (respectively, nonnegative) and equal.
Proof of Theorem 11. Since 
, we may use Lemma 12 to estimate 
as follows
and then
Using this in (8.3), we find 
where 
is given by (8.1). That is,
Integrating this inequality on 
, and using Stokes' theorem and the hypothesis 
, we find that
because of 
when ![x ∈ [0,αH ]](/img/revistas/ruma/v47n2/2a04805x.png)
. Therefore, 
on 
, and either 
(and 
is totally umbilical) or 
. This proves part (i) of Theorem 11 and the first statement of part (ii). If 
, then 
, 
and part (ii)(a) just follows from (ii) in Theorem 6. If 
and 
, then a local argument using the facts that 
and that equality holds in the right-hand side of Lemma 12 implies that 
has exactly two constant principal curvatures, with multiplicities 
and 1. Then, by Cartan's result on isoparametric hypersurfaces of the sphere [9] we conclude that 
must be a constant mean curvature Clifford torus of the form 
with 
and 
, since we are assuming 
. Finally,to identify which constant mean curvature Clifford tori do appear, a direct computation shows that when 
we have 
for all of them, but when 
we have
when 
, and
when 
(for the details see [1, p. 1227] or [2, p. 878]) □
Now we are ready to prove Theorem 10.
Proof of Theorem 10. We already know that 
with equality if and only if 
is totally umbilical, which proves part (i). Then we may assume that 
is not totally umbilical and consider, for every 
, the positive smooth function 
. As in the proof of Theorem 3, we will use 
as a test function to estimate 
in (1.3). We observe that
which using (8.4) yields
|
| (8.5) |
From Lemma 7 applied to 
, we also have
which jointly with (8.5) gives
Then,
Therefore, using 
as a test function in (1.3), we get
Since 
is not totally umbilical, then
and letting 
in (8.6) we conclude that
This proves the first statement of part (ii). Moreover, if
then 
on 
and by Lemma 7 we know that 
is a positive constant. Thus 
, where 
is a constant, and the first eigenvalue of 
is simply the constant
That is, 
and 
, and by Theorem 11 we know that it must hold either (a), (b) or (c). Conversely, we already know from Theorem 3 that 
for all minimal Clifford tori in 
, and it is not difficult to see that 
for the constant mean curvature Clifford tori in (b) and (c). See [2] for the details. □
Finally, it is worth pointing out that Perdomo's technique in [21] also works here to characterize the equality case. For the details about this claim, see [2, Section 4].
This work was written while the author was visiting the Institut des Hautes Études Scientifiques (IHÉS) in Bures-sur-Yvette, France. He thanks IHÉS for its hospitality and support.
1. H. Alencar and M. do Carmo, Hypersurfaces with constant mean curvature in spheres, Proc. Amer. Math. Soc. 120 (1994), 1223-1229. [ Links ]
2. L.J. Alías, A. Barros and A. Brasil Jr., A spectral characterization of the H(r)-torus by the first stability eigenvalue, Proc. Amer. Math. Soc. 133 (2005), no. 3, 875-884. [ Links ]
3. L.J. Alías, A. Brasil Jr. and O. Perdomo, On the stability index of hypersurfaces with constant mean curvature in spheres, to appear in Proc. Amer. Math. Soc.. [ Links ]
4. J.L. Barbosa and P. Bérard, Eigenvalue and "twisted" eigenvalue problems. Applications to CMC surfaces, J. Math. Pures Appl. (9) 79 (2000), 427-450. [ Links ]
5. J.L. Barbosa and M. do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984), 339-353. [ Links ]
6. J.L. Barbosa, M. do Carmo and J. Eschenburg, Stability of hypersurfaces with constant mean curvature in Riemannian manifolds, Math. Z. 197 (1988), 123-138. [ Links ]
7. A. Barros, A. Brasil Jr. and L.A.M. Sousa Jr., A new characterization of submanifolds with parallel mean curvature vector in 
, Kodai. Math. J. 27 (2004), 45-56. [ Links ]
8. M. Berger, P. Gauduchon, and E. Mazet, Le spectre d'une variété riemannienne. Lecture Notes in Mathematics, Vol. 194 Springer-Verlag, Berlin-New York 1971. [ Links ]
9. E. Cartan, Familles de surfaces isoparamétriques dans les espaces à courbure constante, Annali di Mat. 17 (1938), 177-191. [ Links ]
10. S.S. Chern,M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length. 1970 Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) pp. 59-75 Springer, New York. [ Links ]
11. A. El Soufi, Applications harmoniques, immersions minimales et transformations conformes de la sphére, Compositio Math. 85 (1993), 281-298. [ Links ]
12. A. El Soufi and S. Ilias, Majoration de la seconde valeur propre d'un opérateur de Schrödinger sur une variété compacte et applications, J. Funct. Anal. 103 (1992), 294-316. [ Links ]
13. A. El Soufi and S. Ilias, Second eigenvalue of Schrödinger operators and mean curvature, Comm. Math. Phys. 208 (2000), 761-770. [ Links ]
14. I. Guadalupe, A. Brasil Jr., and J.A. Delgado, A characterization of the Clifford torus. Rend. Circ. Mat. Palermo (2) 48 (1999), 537-540. [ Links ]
15. H.B. Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187-197. [ Links ]
16. H.B. Lawson Jr., Lectures on minimal submanifolds. Vol. I. Second edition. Mathematics Lecture Series, 9. Publish or Perish, Inc., Wilmington, Del., 1980. [ Links ]
17. K. Nomizu and B. Smyth, A formula of Simons' type and hypersurfaces with constant mean curvature, J. Differential Geometry 3 (1969), 367-377. [ Links ]
18. M. Obata, Certain conditions for a Riemannian manifold to be iosometric with a sphere, J. Math. Soc. Japan 14 (1962), 333-340. [ Links ]
19. M. Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207-213. [ Links ]
20. O. Perdomo, Low index minimal hypersurfaces of spheres, Asian J. Math. 5 (2001), 741-749. [ Links ]
21. O. Perdomo, First stability eigenvalue characterization of Clifford hypersurfaces, Proc. Amer. Math. Soc. 130 (2002), 3379-3384 [ Links ]
22. J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2), 88 (1968) 62-105. [ Links ]
23. F. Urbano, Minimal surfaces with low index in the three-dimensional sphere, Proc. Amer. Math. Soc. 108 (1990), 989-992. [ Links ]
24. C. Wu, New characterizations of the Clifford tori and the Veronese surface, Arch. Math. (Basel), 61 (1993), 277-284. [ Links ]
Luis J. Alías
Departamento de Matemáticas,
Universidad de Murcia,
E-30100 Espinardo, Murcia, Spain
ljalias@um.es
Recibido: 14 de diciembre de 2005 Aceptado: 29 de agosto de 2006
