ARCHIVUM MATHEMATICUM (BRNO)
Tomus 47 (2011), 119–131
A NEW CHARACTERIZATION OF r-STABLE
HYPERSURFACES IN SPACE FORMS
H. F. de Lima∗ and M. A. Velásquez
Abstract. In this paper we study the r-stability of closed hypersurfaces with
constant r-th mean curvature in Riemannian manifolds of constant sectional
curvature. In this setting, we obtain a characterization of the r-stable ones
through of the analysis of the first eigenvalue of an operator naturally attached
to the r-th mean curvature.
1. Introduction
The notion of stability concerning hypersurfaces of constant mean curvature of
Riemannian ambient spaces was first studied by Barbosa and do Carmo in [3], and
Barbosa, do Carmo and Eschenburg in [4], where they proved that spheres are the
only stable critical points of the area functional for volume-preserving variations.
In [1], Alencar, do Carmo and Colares extended to hypersurfaces with constant
scalar curvature the above stability result on constant mean curvature. In order to
do that, they assumed that the Riemannian ambient space had positive constant
sectional curvature.
The natural generalization of mean and scalar curvatures for an n-dimensional
hypersurface are the r-th mean curvatures Hr , for r = 1, . . . , n. In fact, H1 is just
the mean curvature and H2 defines a geometric quantity which is related to the
scalar curvature.
In [2], Barbosa and Colares studied closed hypersurfaces immersed in space forms
with constant r-th mean curvature. The authors showed that such hypersurfaces
are r-stable if and only if they are geodesic spheres, thus generalizing the previous
results on constant mean curvature hypersurfaces. More recently, Yijun He and
Haizhong Li [9] treated the case of compact hypersurfaces without boundary
Hr+1
immersed in space forms with
constant. They proved that such hypersurfaces
H1
are r-stable if and only if they are totally umbilical.
Motivated by these works, here we consider closed hypersurfaces with constant
r-th mean curvature in a space form in order to obtain a relation between r-stability
2010 Mathematics Subject Classification: primary 53C42; secondary 53B30, 53C50, 53Z05,
83C99.
Key words and phrases: space forms, r-th mean curvatures, r-stability.
∗ Corresponding author.
Received December 17, 2010, revised December 2010. Editor O. Kowalski.
120
H. F. DE LIMA AND M. A. VELÁSQUEZ
and the spectrum of a certain elliptic operator naturally attached to such r-th mean
curvature of the hypersurfaces. Our approach is based on the use of the Newton
transformations Pr and the associated second order linear differential operators Lr
(cf. Section 2).
More precisely, we will prove the following characterization of r-stable hypersurfaces (cf. Theorem 3.5):
n+1
Let M c
be either the Euclidean space Rn+1 (c = 0), an open hemisphere of
n+1
the sphere S
(c = 1), or the hyperbolic space Hn+1 (c = −1). Let r be an integer
n+1
satisfying the inequality 0 ≤ r ≤ n−2, and x : M n → M c be a closed hypersurface
with positive constant (r + 1)-th mean curvature Hr+1 . Suppose that
n
n
n
λ = c(n − r)
Hr + nH1
Hr+1 − (r + 2)
Hr+2
r
r+1
r+2
is constant. Then x is r-stable if and only if λ is the first eigenvalue of Lr on M n .
2. Preliminaries
n+1
Mc
Let
be an orientable simply connected Riemannian manifold with constant
sectional curvature c, Riemannian metric g = h , i, volume element dM and
Levi-Civita connection ∇. In this context, we consider hypersurfaces x : M n →
n+1
M c , namely, isometric immersions from a connected, n−dimensional orientable
n+1
Riemannian manifold M n into M c . We also let ∇ denote the Levi-Civita connection of M n .
Since M n is orientable, one can choose a globally defined unit normal vector
field N on M n . Let A denote the shape operator with respect to N , so that, at
each p ∈ M n , A restricts to a self-adjoint linear map Ap : Tp M → Tp M .
For 1 ≤ r ≤ n, if we let Sr (p) denote the r-th elementary symmetric function
on the eigenvalues of Ap , we get n smooth functions Sr : M n → R such that
det(tI − A) =
n
X
(−1)k Sk tn−k ,
k=0
n
where S0 = 1 by definition. If p ∈ M and {e1 , . . . , en } is an orthonormal basis of
Tp M formed by eigenvectors of Ap , with corresponding eigenvalues λ1 , . . . , λn , one
immediately sees that
Sr = σr (λ1 , . . . , λn ) ,
where σr ∈ R[X1 , . . . , Xn ] is the r-th elementary symmetric polynomial on the
indeterminates X1 , . . . , Xn .
For 1 ≤ r ≤ n, one defines the r-th mean curvature Hr of x by
n
Hr = Sr (λ1 , . . . , λn ) .
r
In particular, for r = 1,
n
H1 =
1X
λk = H
n
k=1
A NEW CHARACTERIZATION OF r-STABLE HYPERSURFACES IN SPACE FORMS 121
is the mean curvature of M n , which is the main extrinsic curvature of the hypersurface. When r = 2, H2 defines a geometric quantity which is related to the (intrinsic)
normalized scalar curvature R of the hypersurface. More precisely, it follows from
the Gauss equation that
R = c + H2 .
(2.1)
On the other hand, with a straightforward computation we verify that
(2.2)
|A|2 = n2 H 2 − n(n − 1)H2 ,
where |A|2 denotes the squared norm of the shape operator of M n .
We also define, for 0 ≤ r ≤ n, the r-th Newton transformation Pr on M n by
setting P0 = I (the identity operator) and, for 1 ≤ r ≤ n, via the recurrence
relation
Pr = Sr I − APr−1 .
(2.3)
A trivial induction shows that
Pr = (Sr I − Sr−1 A + Sr−2 A2 − · · · + rAr ) ,
so that Cayley-Hamilton theorem gives Pn = 0. Moreover, since Pr is a polynomial
in A for every r, it is also self-adjoint and commutes with A. Therefore, all bases of
Tp M diagonalizing A at p ∈ M n also diagonalize all of the Pr at p. Let {e1 , . . . , en }
be such a basis. Denoting by Ai the restriction of A to hei i⊥ ⊂ Tp Σ, it is easy to
see that
n−1
X
det(tI − Ai ) =
(−1)k Sk (Ai )tn−1−k ,
k=0
where
Sk (Ai ) =
X
λj1 · · · λjk .
1≤j1 <...<jk ≤n
j1 ,...,jk 6=i
With the above notations, it is also immediate to check that Pr ei = Sr (Ai )ei , and
hence (cf. Lemma 2.1 of [2])
tr(Pr ) = (−1)r (n − r)Sr = br Hr ;
(2.4)
tr(APr ) = (−1)r (r + 1)Sr+1 = −br Hr+1 ;
tr(A2 Pr ) = (−1)r (S1 Sr+1 − (r + 2)Sr+2 ) ,
where br = (n − r) nr .
Associated to each Newton transformation Pr one has the second order linear
differential operator Lr : C ∞ (M ) → C ∞ (M ), given by
Lr (f ) = tr(Pr Hess f ) .
Note that L0 = ∆, the Laplacian operator on M .
n+1
According to [11], Pr is a divergence-free whenever M c is of constant sectional
curvature; consequently,
Lr (f ) = div(Pr ∇f ) .
122
H. F. DE LIMA AND M. A. VELÁSQUEZ
For future use, we recall Lemma 2.6 of [6]: if (aij ) denotes the matrix of A with
respect to a certain orthonormal basis β = {e1 , . . . , en } of Tp M , then the matrix
(arij ) of Pr with respect to the same basis is given by
arij
(2.5)
n
1 X j1 ...jr j
=
aj i . . . ajr ir ,
r! i ,j =1 i1 ...ir i 1 1
k
k
where
...jr
ji11...i
r
=


sgn(σ) ,


0,
n+1
For x : M n → M c
if the ik are pairwise distinct and
σ = (jk ) is a permutation of them;
else .
as above, a variation of it is a smooth mapping
n+1
X : M n × (−, ) → M c
satisfying the following conditions:
n+1
(1) For t ∈ (−, ), the map Xt : M n → M c
immersion such that X0 = x.
(2) Xt = x , for all t ∈ (−, ).
∂M
given by Xt (p) = X(t, p) is an
∂M
In all that follows, we let dMt denote the volume element of the metric induced
on M by Xt and Nt the unit normal vector field along Xt .
The variational field associated to the variation X is the vector field ∂X
.
∂t t=0
Letting f = h ∂X
∂t , Nt i, we get
∂X >
∂X
= f Nt +
,
∂t
∂t
where > stands for tangential components.
The balance of volume of the variation X is the function V : (−, ) → R given
by
Z
V(t) =
X ∗ (dM ) ,
(2.6)
M ×[0,t]
and we say X is volume-preserving if V is constant.
n+1
From now on, we will consider only closed hypersurfaces x : M n → M c . The
following lemma is enough known and can be found in (cf. [12]).
n+1
Lemma 2.1. Let M c
be an orientable simply connected Riemannian manifold
n+1
with constant sectional curvatures c and x : M n → M c
a closed hypersurface. If
n+1
X : M n × (−, ) → M c
is a variation of x, then
Z
dV
=
f dMt .
dt
M
R
In particular, X is volume-preserving if and only if M f dMt = 0 for all t.
A NEW CHARACTERIZATION OF r-STABLE HYPERSURFACES IN SPACE FORMS 123
We remark
R that Lemma 2.2 of [4] shows that if f0 : M → R is a smooth function
such that M f0 dM = 0, then there exists a volume-preserving variation of M
whose variational field is f0 N .
Following [2], we define the r-area functional Ar : (−, ) → R associated to the
variation X be given by
Z
Ar (t) =
Fr (S1 , S2 , . . . , Sr )dMt ,
M
where Sr = Sr (t) and Fr is recursively defined by setting F0 = 1, F1 = S1 and, for
2 ≤ r ≤ n − 1,
Fr = Sr +
c(n − r + 1)
Fr−2 .
r−1
We notice that if r = 0, the functional A0 is the classical area functional.
The following result follows from Proposition 4.1 of [2]. Since it seems to us that
their proof only works on a neighborhood free of umbilics, and in order to keep
this work self-contained, we present an alternative one here.
n+1
be a closed hypersurface of the orientable
Lemma 2.2. Let x : M n → M c
n+1
simply connected Riemannian manifold M c
with constant curvature c, and let
n+1
n
X : M × (−, ) → M c
be a variation of x. Then,
(2.7)
D ∂X >
E
∂Sr+1
= Lr f + ctr(Pr )f + tr(A2 Pr )f +
, ∇Sr+1 .
∂t
∂t
Proof. Formula (2.5) gives
(r + 1)Sr+1 = tr(APr ) =
X
aji arij =
i,j
1 X j1 ...jr j
aj i . . . ajr ir aji ,
r! i,j,i ,j i1 ...ir i 1 1
k
k
where the functions Sr+1 are seen as functions of t. Hence, differentiation with
respect to t gives
0
(r + 1)Sr+1
=
1 X j1 ...jr j 0
[a
. . . ajr ir aji + · · · + aj1 i1 . . . ajr ir a0ji ]
r! i,j,i ,j i1 ...ir i j1 i1
k
k
(r + 1) X j1 ...jr j 0
=
a aj i . . . ajr ir
r! i,j,i ,j i1 ...ir i ji 1 1
k k
X
= (r + 1)
a0ji arij
i,j
= (r + 1)tr
∂A
∂t
Pr .
124
H. F. DE LIMA AND M. A. VELÁSQUEZ
∂A
∂t Pr
We see from the above that it is enough to compute tr
we do next:
, and this is what
∂A X D ∂A
E
0
Sr+1
= tr
Pr =
Pr ek , ek
∂t
∂t
k
X
=
Sr (Ak ) (∇ ∂X A)ek , ek
∂t
k
=
X
Sr (Ak ) h∇ ∂X Aek , ek i − hA∇ ∂X ek , ek i
∂t
∂t
k
=−
X
Sr (Ak )h∇ ∂X ∇ek N, ek i −
X
∂t
Sr (Ak )hA∇ek ∂X/∂t, ek i ,
k
k
where we used that [∂X/∂t, ek ] = 0 in the last summand. If R denotes the curvature
tensor of M , we have
R(ek , ∂X/∂t)N = ∇ ∂X ∇ek N − ∇ek ∇ ∂X N + ∇[ek , ∂X ] N .
∂t
∂t
∂t
Thus, by using also (2.6),
0
Sr+1
= −
X
−
X
Sr (Ak ) hR(ek , ∂X/∂t)N, ek i + h∇ek ∇ ∂X N, ek i
∂t
k
Sr (Ak )h∇ek (f N + (∂X/∂t)> ), Aek i .
k
Since M is of constant sectional curvature, we get
hR(X, Y )W, Zi = c{hX, W ihY, Zi − hX, ZihY, W i} .
Therefore,
0
Sr+1
=−
X
Sr (Ak )c {hek , N ih∂X/∂t, ek i − hek , ek ih∂X/∂t, N i}
k
−
X
Sr (Ak )h∇ek ∇ ∂X N, ek i −
X
∂t
−
Sr (Ak )hAek , ∇ek f N i
k
k
X
Sr (Ak )h∇ek (∂X/∂t)> , Aek i
k
=c
X
Sr (Ak )f −
k
−
X
+
X
k
k
X
Sr (Ak )ek h∇ ∂X N, ek i +
X
∂t
k
Sr (Ak )hAek , f ∇ek N i −
Sr (Ak )h∇ ∂X N, ∇ek ek i
∂t
k
X
Sr (Ak )ek hAek , (∂X/∂t)> i
k
Sr (Ak )h∇ek Aek , (∂X/∂t)> i .
A NEW CHARACTERIZATION OF r-STABLE HYPERSURFACES IN SPACE FORMS 125
AskingPfurther that ∇ei ej = 0 at p (which is always possible) and writing
f N + l αl el , we have
X
X
Sr (Ak )h∇ ∂X N, ∇ek ek i =
Sr (Ak )h∇ ∂X N, λk δkk N i
∂t
∂X
∂t
=
∂t
k
k
=
X
Sr (Ak )λk h∇f N +P
l
k
=
X
αl el N, N i
o
n
X
αl hAel , N i
Sr (Ak )λk f h∇N N, N i −
l
k
= 0.
By now applying expression (2.4) for the trace of Pr , we get
X
X
0
Sr+1
= ctr(Pr )f +
Pr ek hN, ∇ ∂X ek i + f
Sr (Ak )hAek , Aek i
∂t
k
−
X
k
>
Pr ek hAek , (∂X/∂t) i +
X
k
Sr (Ak )h∇ek Aek , (∂X/∂t)> i
k
= ctr(Pr )f +
X
Pr ek hN, ∇ek ∂X/∂ti + f
X
hAPr ek , Aek i
k
k
−
X
Pr ek hAek , ∂X/∂ti +
X
k
h∇Pr ek Aek , (∂X/∂t)> i
k
= ctr(Pr )f +
X
Pr ek ek (f ) − Pr ek h∇ek N, ∂X/∂ti
k
2
+ tr(A Pr )f −
X
Pr ek hAek , ∂X/∂ti + h
X
k
∇Pr ek Aek , (∂X/∂t)> i .
k
Finally, by applying Codazzi’s equation, we arrive at
X
X
∇Pr ek Aek =
(∇ek APr ek + A[Pr ek , ek ])
k
k
=
X
∇ek (Sr+1 I − Pr+1 )ek +
k
=
X
X
A (∇Pr ek ek − ∇ek Pr ek )
k
ek (Sr+1 )ek +
k
X
∇ek Pr+1 ek −
k
X
A∇ek Pr ek
k
= ∇Sr+1 + div(Pr+1 ) − A(divPr )
= ∇Sr+1 ,
since Pr is divergence-free. Hence,
0
Sr+1
= ctr(Pr )f + Lr f + tr(A2 Pr )f + h∇Sr+1 , (∂X/∂t)> i .
The previous lemma allows us to compute the first variation of the r-area
functional.
126
H. F. DE LIMA AND M. A. VELÁSQUEZ
Proposition 2.3. Under the hypotheses of Lemma 2.2, if X is a variation of x,
then
Z
0
(2.8)
Ar (t) = −(r + 1)
Sr+1 f dMt .
M
Proof. We make an inductive argument. The case r = 0 is well known, and to the
case r = 1 we use the classical formula
∂
dMt = [−S1 f + div(∂X/∂t)> ]dMt
∂t
to get
A01
Z
=
ZM
=
F10 dMt
S10 dMt
M
Z
F1
+
ZM
+
∂
dMt
∂t
S1 [−S1 f + div(∂X/∂t)> ]dMt
M
Z
[ncf + ∆f + (S12 − 2S2 )f + h(∂X/∂t)> , ∇S1 i − S12 f + S1 div(∂X/∂t)> ]dMt
Z
Z
Z
= nc
f dMt − 2
S2 f dMt +
h(∂X/∂t)> , ∇S1 idMt
M
ZM
ZM
>
+
div S1 (∂X/∂t) dMt −
h(∂X/∂t)> , ∇S1 idMt
M
M
Z
= −2
S2 f dMt ,
=
M
M
where in the last equality we used that M is closed and X is volume-preserving.
For r ≥ 2, the induction hypothesis and (2.7) give
Z
Z
∂
0
0
Ar =
Fr dMt +
Fr dMt
∂
t
M
ZM c(n − r + 1) 0
0
Fr−2 dMt
=
Sr +
r−1
M
Z c(n − r + 1)
+
Sr +
Fr−2 −S1 f + div(∂X/∂t)> dMt
r−1
Z M
Z
Z
T
=
Sr0 dMt −
S1 Sr f dMt +
Sr div (∂X/∂t) dMt
M
M
M
Z
Z
c(n − r + 1)
∂
0
+
Fr−2 dMt +
Fr−2 dMt
r−1
∂t
M
M
Z
Z
Z
=c
tr(Pr−1 )f dMt +
Lr−1 dMt +
tr(A2 Pr−1 )f dMt
M
M
M
Z
Z
Z
T
T
+
h∇Sr , (∂X/∂t) idMt −
S1 Sr f dMt +
div Sr (∂X/∂t) dMt
M
M
ZM
c(n − r + 1) 0
T
−
h∇Sr , (∂X/∂t) idMt +
Ar−2
r−1
M
A NEW CHARACTERIZATION OF r-STABLE HYPERSURFACES IN SPACE FORMS 127
Z
Z
= c(n − r + 1)
Sr−1 f dMt +
(S1 Sr − (r + 1)Sr+1 ) f dMt
M
M
Z
Z
c(n − r + 1)
(r − 1)Sr−1 f dMt
−
S1 Sr f dMt −
r−1
M
M
Z
= −(r + 1)
Sr+1 f dMt .
M
Remark 2.4. We want to point out that Lemma 2.2 and the first variation formula
(Proposition 2.3) were first proved by R. Reilly in [10].
In order to characterize hypersurfaces of constant (r + 1)−th mean curvature,
let λ be a real constant and Jr : (−, ) → R be the Jacobi functional associated to
the variation X, i.e.,
Jr (t) = Ar (t) + λV(t) .
As an immediate consequence of (2.8) we get
Z
0
Jr (t) =
[−br Hr+1 + λ]f dMt ,
M
where br = (r + 1)
n
r+1
. Therefore, if we choose λ = br H r+1 (0), where
Z
1
Hr+1 (0)dM
H r+1 (0) =
A0 (0) M
is the mean of the (r + 1)-th curvature Hr+1 (0) of M , we arrive at
Z
Jr0 (t) = br
[−Hr+1 + H r+1 (0)]f dMt .
M
Hence, a standard argument (cf. [3]) shows that M n is a critical point of Jr for all
variations of x if and only if M n has constant (r + 1)-th mean curvature.
n+1
We wish to study spacelike immersions x : M n → M c
that minimize Ar for
all volume-preserving variations X of x. The above discussion shows that M n must
have constant (r + 1)-th mean curvature and, for such an M n , leads us naturally
to compute the second variation of Ar . This motivates the following
n+1
Definition 2.5. Let M c be an orientable simply connected Riemannian manifold
n+1
with constant curvature c, and x : M n → M
be a closed hypersurface having
constant (r + 1)-th mean curvature. We say that x is r-stable if A00r (0) ≥ 0, for all
volume-preserving variation of x.
n+1
Let x : M n → M c
be a closed hypersurface with constant (r + 1)-th mean
curvature
and
denote
by
G the set of differentiable functions f : M n → R with
R
f dMt = 0. Just as [2] we can establish the following criterion of stability: x is
M
r-stable if and only if Jr00 (0) ≥ 0, for all f ∈ G.
The sought formula for the second variation of Jr is another straightforward
consequence of Proposition 2.3.
128
H. F. DE LIMA AND M. A. VELÁSQUEZ
n+1
Proposition 2.6. Let x : M n → M c
be a closed hypersurface of orientable simn+1
ply connected Riemannian manifold M c , having constant (r + 1)-mean curvature
n+1
Hr+1 . If X : M n × (−, ) → M c
is a variation of x, then Jr00 (0) is given by
Z
(2.9)
Jr00 (0)(f ) = −(r + 1)
Lr (f ) + {ctr(Pr ) + tr(A2 Pr )}f f dM .
M
3. r-Stable Hypersurfaces in Space Forms
n+1
As in the previous section, let M c
orientable simply connected Riemannian
n+1
manifold with constant curvature c. A vector field V on M c
is said to be
conformal if
LV h , i = 2ψh , i
(3.1)
∞
for some function ψ ∈ C (M ), where L stands for the Lie derivative of the
Riemann metric of M . The function ψ is called the conformal factor of V .
Since LV (X) = [V, X] for all X ∈ X(M ), it follows from the tensorial character
of LV that V ∈ X(M ) is conformal if and only if
h∇X V, Y i + hX, ∇Y V i = 2ψhX, Y i ,
(3.2)
for all X, Y ∈ X(M ). In particular, V is a Killing vector field relatively to g if and
only if ψ ≡ 0.
The following result was shown in [5].
n+1
be an orientable simply connected Riemannian manifold
Lemma 3.1. Let M c
with constant curvature c and endowed with a conformal vector field V . Let also
n+1
n+1
x: Mn → Mc
be a hypersurface of M c
and N be a Gauss map on M n . If
η = hV, N i, then
n
2
Lr (η) = {−tr(A Pr ) − c tr(Pr )}η − (n − r)
Hr N (ψ)
r
n
n
(3.3)
− (r + 1)
Hr+1 ψ −
hV, ∇Hr+1 i ,
r+1
r+1
n+1
where ψ : M
→ R is the conformal factor of V , Hj is the j-th mean curvature
of M n and ∇Hj stands for the gradient of Hj on M n .
In particular, we obtain the following
n+1
Corollary 3.2. Let M c
be an orientable simply connected Riemannian manifold
with constant curvature c and endowed with a Killing vector field W . Let also
n+1
x: Mn → Mc
be a hypersurface having constant (r + 1)-th mean curvature Hr+1 ,
N be a Gauss map on M n and η = hW, N i, then
Lr (η) + {c tr(Pr ) + tr(A2 Pr )}η = 0 .
n+1
In particular, if x : M n → M c
is a closed spacelike hypersurface with constant
(r + 1)-th mean curvature such that λ = ctr(Pr ) + tr(A2 Pr ) is constant, then λ is
an eigenvalue of the operator Lr in M n with eigenfunction η.
A NEW CHARACTERIZATION OF r-STABLE HYPERSURFACES IN SPACE FORMS 129
Although the existence of nontrivial Killing vector fields on Riemannian space
forms is a standard fact, we will present an alternative proof of this result by using
the ideas of [8].
Lemma 3.3. There exist nontrivial Killing vector fields on Riemannian space
forms.
Proof. Given any fixed two linearly independent vectors u and v in the the
Euclidean space Rn+1 and considering the vector field W = hu, ·iv −hv, ·iu. Observe
that hW (p), pi = 0, that is, geometrically, W (p) determines an orthogonal direction
to the position vector p on the subspace spanned by u and v. Moreover, we easily
verify that
h∇X W, Y i + hX, ∇Y W i = 0 ,
for all tangent vector fields X, Y ∈ X(Rn+1 ). Therefore, W is a Killing vetor field
globally defined on the Rn+1 . Since W is orthogonal to the position vector field,
then W is also a Killing vector field on the sphere
Sn = p ∈ Rn+1 : hp, pi = 1 .
Similarly, fixed two linearly independent vectors u and v in the the Lorentz-Minkowski
space Ln+1 and considering the vector field W = hu, ·iv − hv, ·iu we get a Killing
vector field W globally defined in Ln+1 that is orthogonal to the position vector
field. Then, W is also a Killing vector field on the hyperbolic space
Hn = p ∈ Ln+1 : hp, pi = −1 .
The following result can be found in [2].
n+1
Lemma 3.4. Let x : M n → M c
be a closed orientable hypersurface, where
n+1
n+1
Mc
represent the Euclidean space R
(c = 0), an open hemisphere of the sphere
Sn+1 (c = 1) or the hyperbolic space Hn+1 (c = −1). If Hr is positive then, for
1 ≤ j ≤ r, Lj is elliptic.
We can now state and prove our main result.
n+1
Theorem 3.5. Let M c
be either the Euclidean space Rn+1 (c = 0), an open
n+1
hemisphere of the sphere S
(c = 1), or the hyperbolic space Hn+1 (c = −1). Let
n+1
r be an integer satisfying the inequality 0 ≤ r ≤ n − 2, and x : M n → M c
be a
closed hypersurface with positive constant (r + 1)-th mean curvature Hr+1 . Suppose
that
n
n
n
λ = c(n − r)
Hr + nH1
Hr+1 − (r + 2)
Hr+2
r
r+1
r+2
is constant. Then x is r-stable if and only if λ is the first eigenvalue of Lr on M n .
Proof. First we observe that Lemma 3.3 guarantees the existence of a nontrivial
n+1
Killing vector field W on M c . On the other hand, from Lemma 3.4, the operator
Lr is elliptic on M .
By using the formulas (2.4), it is easy to show that λ = ctr(Pr ) + tr(A2 Pr ).
n+1
Therefore, since that λ is constant and W is a Killing field on M c , Corollary 3.2
guarantees that λ is in the spectrum of Lr .
130
H. F. DE LIMA AND M. A. VELÁSQUEZ
Let λ1 be the first eigenvalue of Lr on M n . If λ = λ1 , then
R
− M f Lr (f )dM
R
λ = min
.
f ∈G\{0}
f 2 dM
M
It follows that, for any f ∈ G,
Z
Z
Jr00 (0)(f ) = (r + 1)
{−f Lr (f ) − λf 2 }dM ≥ (r + 1)(λ − λ)
M
f 2 dM = 0 ,
M
and hence x is r-stable.
Now suppose that x is r−stable. Then Jr00 (0)(f ) ≥ 0, ∀f ∈ G. Let us consider
f the eigenfunction associated to the first eigenvalue λ1 of Lr . As it was already
observed, there exists a volume-preserving variation of M whose variational field is
f N . Consequently, by (2.9), we get
Z
0 ≤ Jr00 (0)(f ) = (r + 1)(λ1 − λ)
f 2 dM.
M
Therefore, since λ1 ≤ λ, we conclude that λ1 = λ.
Since L0 = ∆ is always elliptic and taking into account formula (2.2), we obtain
the following
n+1
Corollary 3.6 ([4], Proposition 2.13). Let M c
be the Euclidean space Rn+1 (c =
n+1
0), the sphere S
(c = 1), or the hyperbolic space Hn+1 (c = −1). Let x : M n →
n+1
Mc
be a closed hypersurface with constant mean curvature. Suppose that
λ = cn + |A|2
is constant, where |A|2 denotes the squared norm of the shape operator. Then x is
stable if and only if λ is the first eigenvalue of ∆ on M n .
Finally, since L1 is just the Yau’s square operator (cf. [7]), by using equation
(2.1) we get the following
n+1
Corollary 3.7. Let M c
be a Euclidean space Rn+1 (c = 0), the sphere Sn+1 (c =
n+1
1), or the hyperbolic space Hn+1 (c = −1). Let x : M n → M c
be a closed
hypersurface, with constant normalized scalar curvature R > c. If
n(n − 1)(n − 2)
n2 (n − 1)
(R − c)H1 −
H3
2
2
is constant, then x is 1-stable if and only if λ is the first eigenvalue of L1 on M n .
λ = cn(n − 1)H1 +
Acknowledgement. The first author was partially supported by CNPq/FAPESQ/PPP, Brazil. The second author was partially supported by CAPES, Brazil.
References
[1] Alencar, H., Carmo, M. do, Colares, A. G., Stable hypersurfaces with constant scalar curvature,
Math. Z. 213 (1993), 117–131.
[2] Barbosa, J. L. M., Colares, A. G., Stability of hypersurfaces with constant r-mean curvature,
Ann. Global Anal. Geom. 15 (1997), 277–297.
A NEW CHARACTERIZATION OF r-STABLE HYPERSURFACES IN SPACE FORMS 131
[3] Barbosa, J. L. M., Carmo, M. do, Stability of hypersurfaces with constant mean curvature,
Math. Z. 185 (3) (1984), 339–353.
[4] Barbosa, J. L. M., Carmo, M. do, Eschenburg, J., Stability of hypersurfaces of constant mean
curvature in Riemannian manifolds, Math. Z. 197 (1) (1988), 123–138.
[5] Barros, A., Sousa, P., Compact graphs over a sphere of constant second order mean curvature,
Proc. Amer. Math. Soc. 137 (2009), 3105–3114.
[6] Caminha, A., On spacelike hypersurfaces of constant sectional curvature Lorentz manifolds,
J. Geom. Phys. 56 (2006), 1144–1174.
[7] Cheng, S. Y., Yau, S. T., Hypersurfaces with constant scalar curvature, Math. Ann. 225
(1977), 195–204.
[8] de Lima, H. F., Spacelike hypersurfaces with constant higher order mean curvature in de
Sitter space, J. Geom. Phys. 57 (2007), 967–975.
[9] He, Y., Li, H., Stability of area-preserving variations in space forms, Ann. Global Anal.
Geom. 34 (2008), 55–68.
[10] Reilly, R., Variational properties of functions of the mean curvatures for hypersurfaces in
space forms, J. Differential Equations 8 (1973), 465–477.
[11] Rosenberg, H., Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117
(1993), 217–239.
[12] Xin, Y., Minimal submanifolds and related topics, World Scientific Publishing co., Singapore,
2003.
1 Departamento
de Matemática e Estatística,
Universidade Federal de Campina Grande,
Campina Grande, Paraíba, Brazil 58109-970
E-mail: henrique@dme.ufcg.edu.br marcolazarovelasquez@gmail.com