Stable Hypersurfaces with Constant Scalar Curvature

STABLE HYPERSURFACES WITH CONSTANT SCALAR CURVATURE

HILARIO´ ALENCAR, WALCY SANTOS AND DETANG ZHOU

Abstract. We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there 4 is no complete noncompact strongly stable hypersurface M in R with zero scalar curvature S2, nonzero Gauss-Kronecker curvature R 3 and ﬁnite total curvature (i.e. M |A| < +∞).

2000 Mathematics Subject Classiﬁcation: Primary: 53042.

1. Introduction In this paper we study the complete noncompact stable hypersurfaces with constant scalar curvature in Euclidean spaces. It has been proved by Cheng and Yau [CY] that any complete noncompact hypersurfaces in the Euclidean space with constant scalar curvature and nonnegative sectional curvature must be a generalized cylinder. Note that the as- sumption of nonnegative sectional curvature is a strong condition for hypersurfaces in the Euclidean space with zero scalar curvature since the hypersurface has to be ﬂat in this case. Let M n be a complete ori- n n+1 entable Riemannian manifold and let x : M → R be an isometric n+1 immersion into the Euclidean space R with constant scalar curvature. We can choose a a global unit normal vector ﬁeld N and the Riemannian n+1 connections ∇ and ∇e of M and R , respectively, are related by

∇e X Y = ∇X Y + hA(X),Y iN, where A is the second fundamental form of the immersion, deﬁned by

A(X) = −∇e X N.

The authors were partially supported by CNPq and FAPERJ, Brazil. 1 2 Hil´arioAlencar, Walcy Santos and Detang Zhou

Let λ1, ..., λn be the eigenvalues of A. The r-mean curvature of the immersion in a point p is deﬁned by 1 X 1 Hr = n λi1 ...λir = nSr, r i1<...

Theorem A. (see Theorem 3.1) There is no complete noncompact sta- n+1 ble hypersurface M in R with zero scalar curvature S2 and 3-mean curvature S3 6= 0 satisfying R 3 S1 (1.2) lim BR = 0, R→+∞ R2 where BR is the geodesic ball in M. 2 2 When S2 = 0, S1 = |A| we have Stable hypersurfaces with constant scalar curvature 3

Corollary B. There is no complete noncompact stable hypersurface M 4 in R with zero scalar curvature S2, nonzero Gauss-Kronecker curvature R 3 and ﬁnite total curvature (i.e. M |A| < +∞). We remark that Shen-Zhu (see [SZ]) proved that a complete stable mini- n+1 mal n-dimensional hypersurface in R with ﬁnite total curvature must be a hyperplane. The above Corollary can be seen as a similar result in dimension 3 for hypersurfaces with zero scalar curvature. We also prove the following result for hypersurfaces with positive constant scalar curvature in Euclidean space. Theorem C. (see Theorem 3.2) There is no complete immersed strongly n n+1 stable hypersurface M → R , n ≥ 3, with positive constant scalar curvature and polynomial growth of 1-volume, that is R S1dM lim BR < ∞, R→∞ Rn

n where BR is a geodesic ball of radius R of M .

As a consequence of the properties of a graph with constant scalar curvature, we have the following corollary: n Corollary D. (see Corollary 4.1) Any entire graph on R with nonnegative constant scalar curvature must have zero scalar curvature. This can be compared with a result of Chern [Ch] which says any en- n tire graph on R with constant mean curvature must be minimal. It has been be known by a result of X. Cheng in [Che] (see also [ENR]) n+1 that any complete noncompact stable hypersurface in R with constant mean curvature must be minimal if n < 5. It is natural to ask that n+1 any complete noncompact stable hypersurface in R with nonnegative constant scalar curvature must have zero scalar curvature. It should be remarked that Chern[Ch] proved that there is no entire n graph on R with Ricci curvature less than a negative constant. We n don’t know whether there exists an entire graph on R with constant negative scalar curvature. The rest of this paper is organized as follows: we include some results and deﬁnitions which will be used in the proof of our theorems in Section 2. The proof of main results are given in Section 3 and Section 4 is an appendix in which we prove some stability properties for graphs with constant scalar curvature in the Eucildean space. 4 Hil´arioAlencar, Walcy Santos and Detang Zhou

2. Some stability and index properties for hypersurfaces with S2 = const.

We introduce the r’th Newton transformation, Pr : TpM → TpM, which are deﬁned inductively by

P0 = I, Pr = SrI − A ◦ Pr−1, r ≥ 1. The following formulas are useful in the proof (see, [Re], Lemma 2.1).

(2.1) trace(Pr) = (n − r)Sr,

(2.2) trace(A ◦ Pr) = (r + 1)Sr+1, 2 (2.3) trace(A ◦ Pr) = S1Sr+1 − (r + 2)Sr+2. From [AdCC] we have the second variation formula for hypersurfaces n+1 in a space form of constant curvature c, Qc , with constant 2-mean curvature:

2 d A1 R R 2 ∞ (2.4) |t=0= hP1(∇f),∇fidM− (S1S2−3S3+c(n−1)S1)f dM, ∀f∈C (D). dt2 D D c

Definition 2.1. When S2 = 0 and c = 0, M is stable if and only if Z Z 2 (2.5) hP1(∇f), ∇fidM ≥ −3 S3f dM, M M ∞ for any f ∈ Cc (M). One can see that if P1 ≡ 0, then S3 = 0 and M is stable. When S2 = const. 6= 0, M is stable if and only if Z Z 2 hP1(∇f), ∇fidM ≥ (S1S2 − 3S3 + c(n − 1)S1)f dM, D D ∞ R for all f ∈ Cc (M) and M fdM = 0. We say that M is strongly stable ∞ if and only if the above inequality holds for all f ∈ Cc (M). Similar to minimal hypersurface we can also define the index I for hypersurfaces with constant scalar curvature. Given a relatively compact domain Ω ⊂ M, we denote by Ind 1(Ω) the number of linearly independent normal deformations with support on Ω that decrease A1. The index of the immersion are defined as 1 1 (2.6) Ind (M) := sup{Ind (Ω) Ω ⊂ M, Ωrelatively compact}. M is strongly stable if Ind 1(M) = 0. The following result has been known in [El]. Stable hypersurfaces with constant scalar curvature 5

n n+1 Lemma 2.1. Let M → Qc be a noncompact hypersurface with S2 = const. > 0. If M has finite index then there exist a compact set K ⊂ M such that M \ K is strongly stable. For hypersurfaces with constant mean curvature, do Carmo and Zhou [dCZ] proved that n+1 Theorem 2.1. Let x : M n → M be an isometric immersion with constant mean curvature H. Assume M has subexponential volume growth and finite index. Then there exist a constant R0 such that H ≤ −Ric (N), M\BR0 where N is a smooth normal vector field along M and Ric(N) is the Ricci curvature of M in the normal vector N. The technique in [dCZ] was generalized by Elbert [El] to prove the following result: Theorem 2.2. Let x : M n → Q(c)n+1 be an isometric immersion with 1 S2 = constant > 0. Assume that Ind M < ∞ and that the 1-volume of M is infinite and has polynomial growth. Then c is negative and

S2 ≤ −c. In particular, it implies that when c = 0 the hypersurfaces in the above theorem must have nonpositive scalar curvature.

3. Proof of the theorems 2 2 When S2 = 0 we know that |S1| = |A| . Thus, if S3 6= 0, we have 2 that |A| > 0. Hence S1 6= 0 and we can choose an orientation such that P1 is semi-positive deﬁnite. Since p 2 2 | P1A| = trace(A ◦ P1)

= −3S3, then, when c = 0, M is stable if Z Z p 2 2 (3.1) hP1(∇f), ∇fidM ≥ | P1A| f dM, M M ∞ for any f ∈ Cc (M). When S2 = 0, we have that the folowing inequality which is essentially due to Cheng and Yau[CY](see also Lemma 4.1 in [AdCC] or Lemma 3.2 in [Li]). 2 2 (3.2) |∇A| − |∇S1| ≥ 0. 6 Hil´arioAlencar, Walcy Santos and Detang Zhou

In the following lemma, we characterize the equality case in some special case. Lemma 3.1. Let M n(n ≥ 3) be a non-ﬂat connected immersed 1- n+1 2 2 minimal hypersurface in R . If |∇A| = |∇S1| holds on all non- vanishing point of |A| in M, then each component of M with |A|= 6 0 must be a cylinder over a curve. Proof. To prove our lemma we recall here the computations in [SSY]. Choose a frame at p so that the second fundamental form is diagonalized, 2 P 2 we have |A| = i hii, and (3.3)   X 2 2 X 2 X 2 X X 2 hijk − |∇|A|| = ( hij)( hstk) − ( hijhijk)  i,j,k i,j s,t,k k i,j

−1 P 2 i,j hij

1 X = (h h − h h )2|A|−2 2 ij stk st ijk i,j,k,s,t  1 X 2 X 2 = 2  (hiihstk − hsthiik) + hss i,k,s,t s

  X X 2 −2  hijk |A| k i6=j

  1 X 2 −2 X 2 = 2  (hiihssk − hsshiik)  |A| + 2 hiij i,k,s i6=j

X 2 + hijk. i6=j,j6=k,i6=k The right hand side of above equation is nonnegative and is zero if and only if all terms on the right hand side of the last equation are zero. n+1 Suppose x : M → R is the 1-minimal immersion. Since M is not a hyperplane, then |A| is a nonnegative continuous function which does Stable hypersurfaces with constant scalar curvature 7 not vanish identically. Let p be such a point such that |A|(p) > 0. Then |A| > 0 in a connected open set U containing p. The equality in (3.2) implies

hjji = 0, for all j 6= i,

hijk = 0, for all j 6= i, j 6= k, k 6= i

hiihssk = hsshiik, for all i, s, k.

So we have hjij = 0, for all j 6= i, and from the last equation we claim that at most one i such that hiii 6= 0. Otherwise, without the loss of generality we assume h111 6= 0, and h222 6= 0, we have h11h22k = h22h11k for all k. This implies h11 = h22 = 0 by choosing k = 1, 2. Using again the third formula we have hjjh111 = h11hjj1 for j = 3, ··· , n. Hence hjj = 0 for all j = 3, ··· , n, which contradicts to |A|= 6 0. We now assume h111 6= 0 by continuity we can also assume h11 6= 0. From the last equation of above equation, we have h11hss1 = hssh111 for s 6= 1. Hence hss = 0 for all s 6= 1. This implies that M is a cylinder over a curve.

We are now ready to prove

Theorem 3.1. There is no complete noncompact stable hypersurfaces n+1 in R with S2 = 0 and S3 6= 0 satisfying

R 3 S1 lim BR = 0. R→+∞ R2

Proof. Assume for the sake of contradiction that there were such a hypersurface M. From Lemma 3.7 in [AdCC], we have

2 2 (3.4) L1S1 = |∇A| − |∇S1| + 3S1S3. 8 Hil´arioAlencar, Walcy Santos and Detang Zhou

∞ Since for any φ ∈ Cc (M),

Z Z hP1(∇(φS1)), ∇(φS1)idM = hP1((∇φ)S1 + φ∇S1), (∇φ)S1) M M

+φ∇S1idM Z 2 = hP1(∇φ), ∇φiS1 dM M Z +2 hP1(∇φ), ∇S1iφS1dM M Z 2 + φ hP1(∇S1), ∇S1idM, M Stable hypersurfaces with constant scalar curvature 9 then using (3.4) we have

Z Z 2 2 2 2 φ S1(|∇A| − |∇S1| )dM = (L1S1 − 3S1S3)φ S1dM M M Z 2 = − hP1(∇S1), ∇(φ S1)idM M Z 2 2 − 3S3φ S1 dM M Z 2 = − φ hP1(∇S1), ∇S1idM M Z −2 hP1(∇φ), ∇S1iφS1dM M Z 2 2 − 3S3φ S1 dM ZM = − hP1(∇(φS1)), ∇(φS1)idM M Z 2 + hP1(∇φ), ∇(φ)iS1 dM M Z 2 2 − 3S3φ S1 dM M Z 2 ≤ hP1(∇φ), ∇φiS1 dM M Z 2 3 ≤ (n − 1) |∇φ| S1 dM, M

∞ for any φ ∈ Cc (M). Here we have used the stability inequality (2.5) in the ﬁfth line and use the following consequence of (2.1) in the last inequality:

2 (3.5) (n − 1)S1|∇φ| ≥ hP1(∇φ), ∇φi. 10 Hil´arioAlencar, Walcy Santos and Detang Zhou

We can choose φ as  2R−r(x)  R , on B2R \ BR; φ(x) = 1, on BR;  0, on M \ B2R. 2 2 Thus from the choice of φ we have S1(|∇A| − |∇S1| ) ≡ 0. Therefore the elipticity of L1 implies L1S1 = 3S1S3. From Lemma 3.1, M must be a cylinder over a curve which contradicts S 6= 0. The proof is complete. 3 The following Lemma is of some independent interest and we include here since its second part is useful in the proof of Theorem 3.2.

n+1 Lemma 3.2. Let M be a complete immersed hypersurface in Qc with n(n−1) positive constant scalar curvature S2 > − 2 c and S1 6= 0. 1)If M is strongly stable outside a compact subset, then either M has ﬁnite 1-volume, or 1 Z lim S1 = +∞. R→+∞ 2 R BR 2)If M is strongly stable, then 1 Z lim S1 = +∞. R→+∞ 2 R BR In particular M has inﬁnite 1-volume.

Proof. We can assume that there exists a geodesic ball BR0 ⊂ M such that M \ BR0 is strongly stable. That is, Z Z 2 (3.6) (S1S2 − 3S3 + c(n − 1)S1)f dM ≤ hP1(∇f), ∇fidM, M M ∞ for all f ∈ Cc (M \ BR0 ). Now, since S2 > 0, we have (see [AdCR], p. 392)

H1H2 ≥ H3, and 1/2 H1 ≥ H2 . n(n − 1) n(n − 1(n − 2) By using that S = nH , S = H and S = H , 1 1 2 2 2 3 6 3 it follows that (n − 2) S S ≥ 3S , n 1 2 3 Stable hypersurfaces with constant scalar curvature 11 that is, (n − 2) (3.7) − 3S ≥ − S S . 3 n 1 2 We also have that S 2S 1/2 1 ≥ 2 , n n(n − 1) which implies 2n 1/2 (3.8) S ≥ S1/2. 1 n − 1 2 By using inequality (3.7) in (3.6), it follows that Z Z n − 2 2 S1S2 − S1S2 + c(n − 1)S1 f dM ≤ hP1(∇f), ∇fidM, M n M that is, Z Z n(n − 1)c 2 n S2 + S1f dM ≤ hP1(∇f), ∇fidM. M 2 2 M By using (3.5), we obtain that Z Z 2 (n − 1) S1|∇f| dM ≥ hP1(∇f), ∇fidM M M Therefore, there exists a constant C > 0 such that Z Z 2 2 (3.9) S1|∇f| dM ≥ C S1f dM. M M

1) When M is strongly stable outside BR0 . We can choose f as  r(x) − R0, on BR +1 \ BR ;  0 0  1, on BR+R0+1 \ BR0+1; f(x) = 2R+R0+1−r(x)  R , on B2R+R0+1 \ BR+R0+1;  0, on M \ B2R+R0+1, where r(x) is the distance function to a ﬁxed point. Then

1 Z Z Z S dM+ S dM ≥ C S dM. R2 1 1 1 B2R+R0+1\BR+R0+1 BR0+1\BR0 BR+R0+1\BR0+1 If the 1-volume is infinite, we can choose R large such that Z Z C S1dM > S1dM, BR+R0+1\BR0+1 BR0+1\BR0 12 HilárioAlencar, Walcy Santos and Detang Zhou hence 1 Z lim S1dM = +∞. R→+∞ R2 B2R+R0+1\BR+R0+1 2) When M is strongly stable we can choose a simpler test function f as  1, on B ;  R f(x) = 2R−r(x) R , on B2R \ BR;  0, on M \ B2R, which implies that when S1 6= 0, 1 Z lim S1dM = +∞. R→+∞ 2 R B2R The proof is complete. Theorem 3.2. There is no complete immersed strongly stable hyper- n n+1 surface M → R , n ≥ 3, with positive constant scalar curvature and polynomial growth of 1-volume, that is R S1dM lim BR < ∞, R→∞ Rn n where BR is a geodesic ball of radius R of M . Proof. Suppose that M is a complete immersed strongly stable hy- n n+1 persurface M → R , n ≥ 3, with positive constant scalar curvature. R From Theorem 2.2, it suffices to show that the 1-volume M S1dM is infinite which is the part (2) of Lemma 3.2.

4. Graphs with S2 = const in Euclidean space In this section we include some stability properties and estimates for n entire graphs on R which may be known to experts and not easy to ﬁnd a reference. Using these facts we give the proof of Corollary 4.1. Let n n+1 n M a hypersurface of R given by a graph of a function u : R → R ∞ n of class C (R ). For such hypersurfaces we have: n n Proposition 4.1. Let M a graph of a function u : R → R of class ∞ n C (R ). Then n (1) If S2 = 0 and S1 does not change sign on M, then M is a stable hypersurface. n (2) If M has S2 = C > 0, then M is strongly stable. Stable hypersurfaces with constant scalar curvature 13

∞ Proof. Considerer and f : M → R a C function with compact p 2 support and let W = 1 + |∇u| . In order to calculate hP1(∇f), ∇fi, write g = fW . Thus g g hP (∇f), ∇fi = hP (∇( )), ∇( )i 1 1 W W 1 1 1 1 = hP (g∇ + ∇g ), g∇( ) + ∇gi 1 W W W W 1 1 1 1 = hgP (∇ ) + P (∇g), g∇( ) + ∇gi 1 W W 1 W W 1 1 g 1 = g2hP (∇ ), ∇ i + hP (∇ ), ∇gi 1 W W W 1 W g 1 1 + hP (∇g), ∇ i + hP (∇g), ∇gi. W 1 W W 2 1

By using that P1 is selfadjoint, we have: (4.1) 1 1 g 1 1 hP (∇f), ∇fi = g2hP (∇ ), ∇ i+2 hP (∇ ), ∇gi+ hP (∇g), ∇gi. 1 1 W W W 1 W W 2 1

On the other hand, if {e1, ..., en} is a geodesic frame along M,

n 1 X 1 div(fgP (∇ ) = h∇ (fgP (∇ )), e i 1 W ei 1 W i i=1

n X 1 1 1 = hfg P (∇ ) + f gP (∇ ) + fg∇ (P (∇ )), e i i 1 W i 1 W ei 1 W i i=1

n X 1 1 = fg hP (∇ ), e i + f ghP (∇ ), e i i 1 W i i 1 W i i=1

1 +fgh∇ (P (∇ )), e i . ei 1 W i

g Since f = , we get W

1 1 fi = gi + g , W W i 14 Hil´arioAlencar, Walcy Santos and Detang Zhou that is, 1 2 1 gfi = ggi + g W W i 2 1 = fgi + g W i Hence, n 1 X 1 1 1 1 div(fgP (∇ ) = {fg hP (∇ ), e i + (fg + g2 )hP (∇ ), e i} + fgL ( ) 1 W i 1 W i i W 1 W i 1 W i=1 i n X 1 1 1 1 = {2fg hP (∇ ), e i + g2 hP (∇ ), e i} + fgL ( ) i 1 W i W 1 W i 1 W i=1 i 1 1 1 1 = 2fhP (∇ ), ∇gi + g2hP (∇ ), ∇( )i + fgL ( ) 1 W 1 W W 1 W g 1 1 1 1 = 2 h∇ ,P (∇g)i + g2hP (∇ ), ∇( )i + f 2WL ( ). W W 1 1 W W 1 W Thus, (4.2) g 1 1 1 1 1 2 h∇ ,P (∇g)i = div(fgP (∇ ))−g2hP (∇ ), ∇( )i−f 2WL ( ). W W 1 1 W 1 W W 1 W Now, by using (4.2) into equation (4.1), we get 1 1 1 hP (∇f), ∇fi = div(fgP (∇ )) − f 2WL ( ) + hP (∇g), ∇gi. 1 1 W 1 W W 2 1 Now, the divergence theorem implies that Z Z Z 2 1 1 hP1(∇f), ∇fidM = − f WL1( )dM+ 2 hP1(∇g), ∇gidM. M M W M W 2 2 Choose the orientation of M in such way that S1 ≥ 0. Since S1 − |A| = 2 2S2 ≥ 0, we obtain that S1 ≥ |A|. Thus, hP1(∇g), ∇gi = S1|∇g| − 2 hA∇g, ∇gi ≥ (S1 − |A|)|∇g| ≥ 0, which implies that Z Z 2 1 (4.3) hP1(∇f), ∇fidM ≥ − f WL1( )dM. M M W

When S2 is constant, we will use the following formula proved by Reilly (see [Re], Proposition C). 1 1 L ( ) = L (hN, e i) = −(S S −3S )hN, e i = −(S S −3S ) , 1 W 1 n+1 1 2 3 n+1 1 2 3 W Stable hypersurfaces with constant scalar curvature 15 where N is the normal vector of M and en+1 = (0, ..., 0, ±1), according to our choice of the orientation of M. Thus, Z Z 2 hP1(∇f), ∇fidM ≥ (S1S2 − 3S3)f dM, M M for all function f with compact support. Hence M is stable if S2 = 0 and strongly stable in the case S 6= 0. 2

Remark 4.1. We would like to remark that the operator L1 need not to be elliptic in the above proof.

n n Proposition 4.2. Let M a graph of a function u : R → R of class ∞ n C (R ), with S1 ≥ 0. Let BR be a geodesic ball of radius R in M. Then Z C(n) n S1dM ≤ R , BθR 1 − θ Z where C(n) and θ are constants, with 0 < θ < 1. In particular, S1dM M has polynomial growth.

∞ Proof. Let f : M → R be a be a function in C0 (M), that is a smooth function with compact support. Observe that ∇u ∇u ∇u div f = fdiv + ∇f, , W W W ∇u where W = p1 + |∇u|2. By using the fact that S is given by S = div , 1 1 W we have that

Z Z ∇u Z ∇u (4.4) fS1dM = fdiv dM = − ∇f, dM. M M W M W

Now, choose a family of geodesic balls BR that exhausts M. Fix θ, with 0 < θ < 1 and let f : M → R be a continuous function that is one on BθR, zero outside BR and linear on BR \ BθR. Therefore, from equation (4.4) we obtain Z Z Z ∇u S1dM ≤ fS1dM ≤ , ∇f dM. BθR BR BR W 16 Hil´arioAlencar, Walcy Santos and Detang Zhou

|∇u| By using Cauchy-Schwarz inequality and the fact that ≤ 1, it W follows that Z Z Z 1 1 S1dM ≤ |∇f|dM ≤ dM ≤ vol(BR). BθR BR Br\BθR (1 − θ)R (1 − θ)R n+1 We observe that since M is a graph, if ΩR = {(x1, .., xn+1) ∈ R |−R ≤ p 2 2 xn+1 ≤ R; x1 + ... + xn ≤ R}, then Z n+1 vol(BR) ≤ 1dx1...dxn+1 = C(n)R . ΩR Hence, Z 1 C(n) n S1dM ≤ vol(BR) = R . BθR (1 − θ)R 1 − θ We have the following Corollary of Theorem 3.2 n Corollary 4.1. Any entire graph on R with nonnegative constant scalar curvature must have zero scalar curvature. Proof. Suppose by sake of contradiction that there exist a entire graph with S2 = const > 0. Such graph is strongly stable and if S2 > 0, 2 2 we get that S1 = |A| + 2S2 > 0, we obtain that S1 does not change sign and we can choose the orientation in such way that S1 > 0. Thus the graph has polynomial growth of the 1-volume. Thus we have a contradiction with Theorem 3.2. Thus it follows that S2 = 0. References [AdCC] Alencar, H., do Carmo, M., Colares, A.G. - Stable hypersurfaces with constant scalar curvature, Math. Z. 213 (1993), 117–131. [AdCE] Alencar, H., do Carmo, M., Elbert, M.F. - Stablity of hypersurfaces with vanishing r-constant curvatures in euclidean spaces, J. Reine Angew. Math. 554 (2003), 201–216. [AdCR] Alencar, H., do Carmo, M., Rosenberg, H. - On the ﬁrst eigenvalue of the linearized operator of the rth mean curvature of a hypersurface. Annals of Global Analysis and Geometry 11, no. 4 (1993), 387–395. [ASZ] Alencar, H., Santos, W., Zhou, D. - Curvature integral estimates for complete hypersurfaces, arXiv:0903.2035. [Che] Cheng, X.- On constant mean curvature hypersurfeces with ﬁnite index, Arch. Math. 86 (2006), 365–374. [CY] Cheng, S.Y., Yau, S.T.- Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), 195–204. Stable hypersurfaces with constant scalar curvature 17

[Ch] Chern, S.S.- On the curvatures of a piece of hypersurface in Euclidean space, Abh. Math. Seminar der Univ. Hamburg, 29 (1965), 77–91. [dCZ] do Carmo, M.P., Zhou, D.- Eigenvalue estimate on noncompact Riemann- ian manifolds and applications, Transactions Amer. Math. Soc. 351 (1999), 1391–1401. [El] Elbert, M.F. - Constant positive 2-mean curvature hypersurface. Illinois J. of Math, 46 n.1 (2002), 247-267. [ENR] Elbert, M.F., Nelli, B., Rosenberg, H - Stable constant mean curvature hypersurfaces. Proc. Amer. Math. Soc. 135 no. 10 (2007), 3359–3366. [Li] Li, H. - Hypersurfaces with constant scalar curvature in space forms, Math. Ann., 305(1996), 665-672. [SSY] Schoen, R., Simon, L., Yau, S.T.,- Curvature estimates for stable minimal hypersurfaces, Acta Math., 134 (1975), 275–288. [SZ] Shen, Y. B., Zhu, X. H. - On stable complete minimal hypersurfaces in Rn+1. Amer. J. Math., 120 (1998), 103116. [Re] Reilly,R.C.- Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Diﬀ. Geom., 8 (1973), 465-477. [Ro] Rosenberg, H- Hypersurfaces of constant curvature in space forms. Bull. Sci. Math.,2a s´erie117 (1993), 211-239.

HilárioAlencar Universidade Federal de Alagoas, Instituto de Matemática,57072-900, Maceió-AL, Brazil. Email: [email protected]

Detang Zhou Universidade Federal Fluminense, Instituto de Matem´atica,24020-140, Niter´oi-RJ, Brazil. Email: [email protected]

Walcy Santos Universidade Federal do Rio de Janeiro, Instituto de Matem´atica, 21941-909, Rio de Janeiro-RJ, Brazil. Email: [email protected]