Submanifolds with constant scalar curvature in a space form

J. Math. Anal. Appl. 447 (2017) 488–498

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Submanifolds with constant scalar curvature in a space form Jogli G. Araújo, Henrique F. de Lima ∗ , Fábio R. dos Santos, Marco Antonio L. Velásquez Departamento de Matemática, Universidade Federal de Campina Grande, Campina Grande 58.429-970, Brazil

a r t i c l e

i n f o

Article history: Received 27 July 2016 Available online 21 October 2016 Submitted by H.R. Parks Keywords: Riemannian space forms Complete submanifolds Parallel normalized mean curvature vector field Constant normalized scalar curvature Clifford torus Circular and hyperbolic cylinders

a b s t r a c t We deal with complete submanifolds M n having constant positive scalar curvature and immersed with parallel normalized mean curvature vector field in a Riemannian space form Qn+p of constant sectional curvature c ∈ {1, 0, −1}. In this setting, we c show that such a √ submanifold M n must be either totally umbilical or isometric to a  Clifford torus S1 1 − r2 × Sn−1 (r), when c = 1, a circular cylinder R × Sn−1 (r),  √  when c = 0, or a hyperbolic cylinder H1 − 1 + r2 × Sn−1 (r), when c = −1. This characterization theorem corresponds to a natural improvement of previous ones due to Alías, García-Martínez and Rigoli [2], Cheng [4] and Guo and Li [6]. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Many authors have approached the problem of characterizing hypersurfaces immersed with constant mean curvature or with constant scalar curvature in a Riemannian space form Qn+1 of constant sectional c curvature c. For instance, in the seminal work [5], Cheng and Yau introduced a new self-adjoint differential operator  acting on smooth functions defined on Riemannian manifolds. As a by-product of such approach they were able to classify closed hypersurfaces M n with constant normalized scalar curvature R satisfying R ≥ c and nonnegative sectional curvature immersed in Qn+1 . Later on, Li [8] extended the results due to c Cheng and Yau [5] in terms of the squared norm of the second fundamental form of the hypersurface M n . In [3], Brasil Jr., Colares and Palmas used the generalized maximum principle of Omori [9] and Yau [13] to characterize complete hypersurfaces with constant scalar curvature in Sn+1. In [1], by applying a weak Omori–Yau maximum principle due to Pigola, Rigoli, Setti [10], Alías and García-Martínez studied the behavior of the scalar curvature R of a complete hypersurface immersed with constant mean curvature into * Corresponding author. E-mail addresses: [email protected] (J.G. Araújo), [email protected] (H.F. de Lima), [email protected] (F.R. dos Santos), [email protected] (M.A.L. Velásquez). http://dx.doi.org/10.1016/j.jmaa.2016.10.033 0022-247X/© 2016 Elsevier Inc. All rights reserved.

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a real space form Qn+1 , deriving a sharp estimate for the infimum of R. Afterwards, Alías, García-Martínez c and Rigoli [2] obtained another suitable weak maximum principle for complete hypersurfaces with constant scalar curvature in Qn+1 , and gave some applications of it in order to estimate the norm of the traceless c part of its second fundamental form. In particular, they extended the main theorem of [3] for the context of Qn+1 . c Considering higher codimension, Cheng [4] showed that the totally umbilical sphere Sn (r), totally geodesic Euclidean space Rn and generalized cylinder R × Sn−1 (r) are the only n-dimensional complete submanifolds with constant scalar curvature and parallel normalized mean curvature vector field (that is, the normalized mean curvature vector field is parallel as a section of the normal bundle) in the Euclidean space Rn+p, which satisfy a suitable constrain on the norm of the second fundamental form. Later on, Guo and Li [6] generalized the results of [8] showing that the only closed submanifolds in the unit sphere Sn+p with constant scalar curvature, parallel normalized mean curvature vector field and whose second fundamental form satisfies some appropriate boundedness are the totally umbilical sphere Sn(r) and the Clifford torus √ S1 ( 1 − r2 ) × Sn−1 (r). Motivated by these works, we deal with complete submanifolds M n having constant positive scalar curvature and immersed with parallel normalized mean curvature vector field in a Riemannian space form Qn+p of constant sectional curvature c ∈ {1, 0, −1}. In this setting, we establish a suitable Simons type c formula (cf. Proposition 3.1) and an Omori type maximum principle for the square operator (cf. Lemma 4.5) in order to obtain the following characterization result: Theorem 1.1. Let M n be a complete submanifold immersed with parallel normalized mean curvature vector field in a Riemannian space form Qn+p (c ∈ {1, 0, −1} and n ≥ 4), with constant normalized scalar curvature c R ≥ 1, when c = 1, and R > 0, when c ∈ {0, −1}. Then i. either |Φ| ≡ 0 and M n is totally umbilical, ii. or sup |Φ|2 ≥ αn,c (R) = M

n(n − 1)R2 . (n − 2)(nR − (n − 2)c)

Moreover, assuming in addition that R > 1 when c = 1, the equality supM |Φ|2 = αn,c (R) holds and this supremum is attained at some point of M n if, and only if, M n is isometric to a √  (a) Clifford torus S1 1 − r2 × Sn−1 (r) → Sn+1 → Sn+p , when c = 1, (b) circular cylinder R × Sn−1 (r) → Rn+1 → Rn+p , when c = 0,  √  (c) hyperbolic cylinder H1 − 1 + r2 × Sn−1 (r) → Hn+1 → Hn+p , when c = −1,  n−2 . where r = nR Here, Φ stands for the traceless part of the second fundamental form of the submanifold M n. We point out that Theorem 1.1 is a natural extension of Theorems 1 and 2 in [2] for higher codimension and, as well as, it can be regarded as a suitable improvement of the main results of [4] and [6]. The proof of Theorem 1.1 is given in Section 5. 2. Preliminaries Let M n be an n-dimensional connected submanifold immersed in a space form Qn+p , with constant c sectional curvature c. We will make use of the following convention on the range of indices: 1 ≤ A, B, C, . . . ≤ n + p,

1 ≤ i, j, k, . . . ≤ n

and n + 1 ≤ α, β, γ, . . . ≤ n + p.

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We choose a local orthonormal frame field {e1 , . . . , en , en+1 , . . . , en+p } along M n , where {ei }i=1,...,n are tangent to M n and {eα }α=n+1,...,n+p are normal to M n . Let {ωB } be the corresponding dual coframe, and {ωBC } the connection 1-forms on Qn+p . With restricting on M n , the second fundamental form A, the c curvature tensor R and the normal curvature tensor R⊥ of M n can be given by ωiα =



hα ij ωj ,



A=

j

dωij =

i,j,α



1 Rijkl ωk ∧ ωl , 2

ωik ∧ ωkj −

k

dωαβ =

hα ij ωi ⊗ ωj ⊗ eα ,

k,l



ωαγ ∧ ωγα −

γ

1 ⊥ Rαβkl ωk ∧ ωl . 2 k,l

Moreover, the components hα ijk of the covariant derivative ∇A satisfy 

α hα ijk ωk = dhij +

k



hα ki ωkj +



k

hα kj ωki +

k



hβij ωβα .

(2.1)

β

The Gauss equation is Rijkl = c(δik δjl − δil δjk ) +

 α α α (hα ik hjl − hil hjk ).

(2.2)

α

In particular, the components of the Ricci tensor Rik are given by Rik = c(n − 1)δik + n



H α hα ik −

α



α hα ij hjk ,

(2.3)

α,j

  α where H α = n1 i hα ii are the components of the mean curvature vector field H = α H eα . Moreover, the normalized scalar curvature R is given by R=

 1 Rii . (n − 1) i

(2.4)

From (2.3) and (2.4), we get the following relation n(n − 1)R = n(n − 1)c + n2 H 2 − |A|2 ,

(2.5)

 2 where |A|2 = α,i,j (hα ij ) is the squared norm of the second fundamental form and H = |H| is the mean n curvature function of M . By exterior differentiation of (2.1), we have the following Ricci identity α hα ijkl − hijlk =



hα mj Rmikl +

m



hα im Rmjkl +

m



⊥ hβij Rβαkl .

(2.6)

β

The Codazzi equation and the Ricci equation are given, respectively, by α α hα ijk = hikj = hjik

(2.7)

and ⊥ Rαβij =

 β α β (hα ik hkj − hjk hki ). k

(2.8)

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3. A Simons type formula From now on, we will deal with submanifolds M n of Qn+p having parallel normalized mean curvature vecc tor field, which means that the mean curvature function H is positive and that the corresponding normalized H is parallel as a section of the normal bundle. mean curvature vector field H H In this context, we can choose a local orthonormal frame {e1 , . . . , en+p } such that en+1 = . Thus, H H n+1 =

1 tr(hn+1 ) = H n

1 tr(hα ) = 0, α ≥ n + 2. n

and H α =

(3.1)

We will also consider the following symmetric tensor Φ=



Φα ij ωi ⊗ ωj ⊗ eα ,

α,i,j α α where Φα ij = hij − H δij . Consequently, we have that

Φn+1 = hn+1 − Hδij ij ij

α and Φα ij = hij ,

n + 2 ≤ α ≤ n + p.

(3.2)

 2 Let |Φ|2 = α,i,j (Φα ij ) be the square of the length of Φ. From (2.5), it is not difficult to verify that Φ is traceless with |Φ|2 = |A|2 − nH 2 = n(n − 1)(c + H 2 − R).

(3.3)

Extending the ideas of [6], we obtain the following Simons type formula Proposition 3.1. Let M n be an n-dimensional (n ≥ 2) submanifold immersed with parallel normalized mean curvature vector field in a Riemannian space form Qn+p . Then, we have c   1 β β α α Δ|A|2 = |∇A|2 + nHij hij + cn|Φ|2 + n Hhn+1 ij hjk hki 2 i,j,α −







i,j,k,l

β,i,j,k

2 α hα ij hkl

α

−



⊥ (Rαβij )2 .

i,j,α,β

Proof. Taking into account that   1 α 2 Δ|A|2 = hα (hα ij Δhij + ijk ) , 2 α,i,j

(3.4)

α,i,j,k

 α α α where the Laplacian Δhα ij of hij is defined by Δhij = k hijkk , using Codazzi equation (2.7) into (3.4) we have       1 2 α α 2 2 α Δ|A| = hij hijkk + (hα hα (3.5) ijk ) = |∇A| + ij hkijk . 2 α,i,j k

α,i,j,k

α,i,j,k

Thus, from (2.6) and (3.5) we conclude that    1 α α α Δ|A|2 = |∇A|2 + (hα hα hα ij hkki )j − ijj hkki + ij hmi Rmj 2 α,i,j,m α,i,j,k

α,i,j,k

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+

α,i,j,k,m

Hence, observing that

 α,i,j,k



α hα ij hkm Rmijk +

β ⊥ hα ij hki Rβαjk .

α 2 ⊥ 2 ⊥ 2 hα ijj hkki = n |∇ H| , where |∇ H| =

   1 α Δ|A|2 = |∇A|2 − n2 |∇⊥ H|2 + hα ij hkki j + 2 α,i,j,k



+

(3.6)

β,α,i,j,k



α hα ij him Rmj +

α,i,j,m



α 2 α,i (Hi ) ,



from (3.6) we get

α hα ij hmk Rmijk

(3.7)

α,i,j,k,m

β ⊥ hα ij hik Rβαjk .

α,β,i,j,k

But, using once more Codazzi equation (2.7) we obtain that 

−n2 |∇⊥ H|2 +



α (hα ij hkki )j =

α α nHij hij .

(3.8)

i,j,α

α,i,j,k

From (2.2) and (2.3) we also conclude that 

α hα ij hmk Rmijk +



α hα ij him Rmj +

α,i,j,m

α,i,j,k,m

= c|Φ|2 −





β ⊥ hα ji hik Rβαjk

β,α,i,j,k β α β hα ij hij hmk hmk + n

α,β,i,j,k,m



−



β β α hα ij him hml hlj +

α,β,i,j,m,l

(3.9)



α H β hβmj hα ij him

α,β,i,j,m β β α hα ij hkm hjm hik +

α,β,i,j,k,m



β ⊥ hα ji hik Rβαjk .

α,β,i,j,k

On the other hand, from (3.1) we get 

α H β hβmj hα ij him =

α,β,i,j,m



β β Hhn+1 ij hjk hki ,

(3.10)

β,i,j,k

and 

β α β hα ij hij hmk hmk

=

α,β,i,j,k,m







i,j,k,l

2 α hα ij hkl

.

(3.11)

α

Furthermore, using (2.8) we have 

⊥ (Rαβjk )2 =

α,β,j,k



α β ⊥ (hβji hα ik − hji hik )Rβαjk

α,β,i,j,k

=



α,β,i,j,m,l

β β α hα ij him hml hlj −

 α,β,i,j,k,m

(3.12) β β α hα ij hkm hjm hik −



β ⊥ hα ji hik Rβαjk .

α,β,i,j,k

Therefore, considering (3.8), (3.9), (3.10), (3.11) and (3.12) in (3.7), we conclude the proof. 2 4. Key lemmas This section is devoted to quote the key lemmas which will be used to prove Theorem 1.1 in the next section. The first one is just Lemma 2.1 of [6].

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Lemma 4.1. Let M n be a submanifold immersed in a Riemannian space form Qn+p , with constant normalized c scalar curvature R ≥ c. Then |∇A|2 ≥ n2 |∇H|2 . The second auxiliary lemma is an algebraic result, whose proof can be found in [11]. Lemma 4.2. Let B, C : Rn −→ Rn be symmetric linear maps that BC − CB = 0 and trB = trC = 0, then −

n−2 n(n − 1)

|B|2 |C| ≤ tr(B 2 C) ≤

n−2 n(n − 1)

|B|2 |C|.

(4.1)

Consider the following algebraic lemma, whose proof can be found in [7]. Lemma 4.3. Let B 1 , B 2 , . . . , B p be symmetric (n×n)-matrices. Set Sαβ = tr(B α B β ), Sα = Sαα , S = then 

|B α B β − B β B α |2 +

α,β

Now, let φ =

 i,j



2 Sαβ

α,β

3 ≤ 2





 α

Sα ,

2 Sα

.

(4.2)

α

φij ωi ωj be a symmetric tensor on M n defined by φij = nHδij − hn+1 ij .

Following Cheng–Yau [5], we introduce a operator  associated to φ acting on any smooth function f by f =

 i,j

φij fij =



(nHδij − hn+1 ij )fij .

(4.3)

i,j

By taking a local orthonormal frame field e1 , . . . , en at q ∈ Σn such that hn+1 = λn+1 δij , we can reason ij i as in the proof of Lemma 5 in [2] in order to obtain the following sufficient criterion of ellipticity for the square operator. Lemma 4.4. Let M n be a submanifold immersed with parallel normalized mean curvature vector field in a Riemannian space form Qn+p , with normalized scalar curvature R > c. Then,  is elliptic. c Our last key lemma corresponds to a Omori type maximum principle for the square operator. Lemma 4.5. Let M n be a complete submanifold immersed with parallel normalized mean curvature vector field in a Riemannian space form Qn+p , with constant normalized scalar curvature R ≥ c. If the mean c curvature function H is bounded on M n , then there exists a sequence of points {qk }k∈N ⊂ M n such that lim nH(qk ) = sup nH, k

M

lim |∇nH(qk )| = 0 k

lim sup (nH(qk )) ≤ 0.

and

k

Proof. Let us choose a local orthonormal frame {e1 , . . . , en } on M n such that hn+1 = λn+1 δij . From (4.3) ij i we have (nH) =

  (nHδij − λn+1 δ )(nH) = n (nH − λn+1 )Hii . ij ij i i i,j

i

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On the other hand, since we are assuming that R ≥ c, from (2.5) we get, for all i = 1, . . . , n and n + 1 ≤ α ≤ n + p, 2 2 2 2 2 (λα i ) ≤ |A| = n H + n(n − 1)(c − R) ≤ (nH) .

Consequently, for all i = 1, . . . , n and n + 1 ≤ α ≤ n + p, we have |λα i | ≤ nH.

(4.4)

Moreover, from (2.2) we obtain 

Rijij = c +

α hα ii hjj −

 2 (hα ij ) .

α

(4.5)

α

Since |A|2 ≤ (nH)2 , we have that 2 2 2 (hα ij ) ≤ |A| ≤ (nH) ,

for every α, i, j and, hence, from (4.4) we have α α α 2 |hα ii hjj | = |hii ||hjj | ≤ (nH) .

Thus, since we are supposing that H is bounded on M n , it follows that α 2 hα ii hjj ≥ −(nH) > −∞

2 2 and −(hα ij ) ≥ −(nH) > −∞.

(4.6)

Hence, from (4.5) and (4.6) we conclude that the sectional curvatures of M n are bounded from below. Thus, we can apply the well known generalized maximum principle of Omori [9] to the function nH, obtaining a sequence of points {qk }k∈N in M n such that lim nH(qk ) = sup nH, k

lim |∇nH(qk )| = 0 and

lim sup

k



k

nHii (qk ) ≤ 0.

(4.7)

i

Moreover, from (4.4) we also have 0 ≤ nH(qk ) − |λn+1 (qk )| ≤ nH(qk ) − λn+1 (qk ) i i (qk )| ≤ 2nH(qk ). ≤ nH(qk ) + |λn+1 i This previous estimate shows that the function nH(qk ) − λn+1 (qk ) is nonnegative and bounded on M n , i for all k ∈ N. Therefore, taking into account (4.7), we obtain lim sup((nH)(qk )) ≤ n k

 i

 lim sup (nH − λn+1 )(qk )Hii (qk ) ≤ 0. i k

2

5. Proof of Theorem 1.1 Now, we are in position to present the proof of Theorem 1.1. Proof. From (2.5), (4.3), Proposition 3.1 and Lemma 4.1 we obtain that (nH) ≥ cn|Φ|2 + n



β β Hhn+1 ij hjk hki −

β,i,j,k



i,j,k,l



 α

2 α hα ij hkl

−



⊥ (Rαβij )2 .

α,β,i,j

(5.1)

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From (3.1) and (3.2) we have 

β β Hhn+1 ij hjk hki =

i,j,k,β



n+p 

n+1 n+1 Hhn+1 + ij hjk hki

i,j,k



β β Hhn+1 ij Φjk Φki

(5.2)

β=n+2 i,j,k n+p 

= Htr(Φn+1 + HI)3 +



n+p 

β β HΦn+1 ij Φjk Φki +

β=n+2 i,j,k

H 2 |Φβ |2

β=n+2 n+p 

= Htr(Φn+1 )3 + 3H 2 |Φn+1 |2 + nH 4 +

H 2 |Φβ |2

β=n+2 n+p 

+



β β HΦn+1 ij Φjk Φki .

β=n+2 i,j,k

Noticing that tr Φα = 0 and Φn+1 Φβ − Φβ Φn+1 = 0, n + 2 ≤ β ≤ n + p, from Lemma 4.2 we obtain n+p 

Htr(Φn+1 )3 + 3H 2 |Φn+1 |2 + nH 4 +

H 2 |Φβ |2 +

β=n+2

≥ −

n−2 n(n − 1)

−

n+p 



β β HΦn+1 ij Φjk Φki

(5.3)

β=n+2 i,j,k

H|Φn+1 |3 + 2H 2 |Φn+1 |2 + H 2 |Φ|2 + nH 4 n+p 

n−2

n(n − 1) β=n+2

H|Φn+1 ||Φβ |2

n−2 = 2H 2 |Φn+1 |2 + H 2 |Φ|2 + nH 4 − H|Φn+1 ||Φ|2 . n(n − 1) Hence, from (5.2) and (5.3) we have 

n−2 β β 2 n+1 2 Hhn+1 | + H 2 |Φ|2 + nH 4 − H|Φn+1 ||Φ|2 . ij hjk hki ≥ 2H |Φ n(n − 1) β,i,j,k

(5.4)

From Ricci equation (2.8) we get 



i,j,k,l



2 α hα ij hkl

+

α



⊥ (Rαβij )2 =

α,β,i,j

  ⊥ (tr(Aα Aβ ))2 + (Rαβij )2 α,β

= [tr(An+1 An+1 )]2 + 2 +

(5.5)

α=n+1,β=n+1,i,j





[tr(An+1 Aβ )]2

β=n+1

(tr(Aα Aβ ))2 +

α=n+1,β=n+1



|Aα Aβ − Aβ Aα |2 .

α=n+1,β=n+1

But, using (3.2) and Lemma 4.3 we obtain 

[tr(Aα Aβ )]2 +

α=n+1,β=n+1



⎛ |Aα Aβ − Aβ Aα |2 ≤

α=n+1,β=n+1

3⎝ 2

 β=n+1

⎞2 tr(Aβ Aβ )⎠ ≤

⎛ 3⎝ 2



β=n+1

⎞2 |Φβ |2 ⎠ .

(5.6)

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Hence, from (5.5) and (5.6) we have  i,j,k,l



 α

2 α hα ij hkl

+



⊥ (Rαβij )2 ≤ [tr(An+1 An+1 )]2 + 2

α,β,i,j



⎛ [tr(An+1 Aβ )]2 +

β=n+1

= |Φn+1 |4 + 2nH 2 |Φn+1 |2 + n2 H 4 + 2



3⎝ 2

⎞2



|Φβ |2 ⎠

β=n+1

[tr(Φn+1 Φβ )]2

β=n+1

3 + (|Φ|2 − |Φn+1 |2 )2 2 5 ≤ |Φn+1 |4 + 2nH 2 |Φn+1 |2 + n2 H 4 + 2|Φn+1 |2 (|Φ|2 − |Φn+1 |2 ) 2 3 + |Φ|4 − 3|Φ|2 |Φn+1 |2 2 1 3 = |Φn+1 |4 + 2nH 2 |Φn+1 |2 + n2 H 4 − |Φ|2 |Φn+1 |2 + |Φ|4 . 2 2

(5.7)

Therefore, from (5.1), (5.4) and (5.7) we get n(n − 2) 1 3 (nH) ≥ cn|Φ|2 − H|Φn+1 ||Φ|2 + nH 2 |Φ|2 − |Φn+1 |4 + |Φ|2 |Φn+1 |2 − |Φ|4 2 2 n(n − 1)   n(n − 2) 2 2 2 = |Φ| −|Φ| − H|Φ| + n(H + c) n(n − 1)   n(n − 2) 1 n+1 2 n+1 n+1 2 |) |)(|Φ| + |Φ |) . + (|Φ| − |Φ H|Φ| − (|Φ| − |Φ 2 n(n − 1)

(5.8)

On the other hand, from (3.3) we obtain H2 =

1 |Φ|2 + (R − c) n(n − 1)

(5.9)

Thus, from (5.8) and (5.9) we get 

(nH) ≥ (|Φ| − |Φ

n+1

+

n(n − 2) 1 |) H|Φ|2 − (|Φ| − |Φn+1 |)(|Φ| + |Φn+1 |)2 2 n(n − 1)

1 |Φ|2 QR (|Φ|) , n−1



(5.10)

where QR (x) is the function introduced by Alías, García-Martínez and Rigoli in [2] and which is given by QR (x) = −(n − 2)x2 − (n − 2)x x2 + n(n − 1)(R − c) + n(n − 1)R. On the other hand, we note that holds the following algebraic inequality (3.5) of [6] (|Φ| − |Φn+1 |)(|Φ| + |Φn+1 |)2 ≤

32 3 |Φ| . 27

Moreover, since we are assuming that R ≥ c, using (2.5) we also have n2 H 2 = |A|2 + n(n − 1)(R − c) ≥ |A|2 = |Φ|2 + nH 2 ,

(5.11)

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which give us 1 H≥ |Φ|. n(n − 1)

(5.12)

Thus, from (5.11) and (5.12) we conclude that 1 n(n − 2) H|Φ|2 − (|Φ| − |Φn+1 |)(|Φ| + |Φn+1 |)2 ≥ 2 n(n − 1)



n − 2 16 − n − 1 27

 |Φ|3 .

(5.13)

But, taking into account our assumption that n ≥ 4, we have that n − 2 16 − > 0. n − 1 27

(5.14)

Consequently, from (5.10), (5.13) and (5.14) we get that 1 |Φ|2 QR (|Φ|) + (|Φ| − |Φn+1 |) (nH) ≥ n−1



n − 2 16 − n − 1 27

 |Φ|3 ≥

1 |Φ|2 QR (|Φ|) . n−1

(5.15)

Since we are assuming that R ≥ 1, when c = 1, and R > 0, when c ∈ {0, −1}, we have that QR (0) = n(n − 1)R > 0 and the function QR (x) is strictly decreasing for x ≥ 0, with QR (x∗ ) = 0 at  ∗

x =R

n(n − 1) = (n − 2)(nR − (n − 2)c)

 αn,c (R) > 0.

Now, suppose that M n is not totally umbilical. In this case, we can consider without loss of generality that supM |Φ| < +∞. Thus, from (3.3) we get that H is bounded on M n and we can apply Lemma 4.5 jointly with (3.3) and (5.15) to obtain a sequence of points {qk }k∈N ⊂ M n such that   1 2 sup |Φ| QR sup |Φ| . 0 ≥ lim sup (nH)(qk ) ≥ n−1 M M k

(5.16)

Consequently, from (5.16) we get that QR (supM |Φ|) ≤ 0. Hence, from the behavior of the function QR previously described, we conclude that supM |Φ|2 ≥ αn,c (R). Let us assume in addition that R > 1 when c = 1, the equality supM |Φ|2 = αn,c (R) holds and this supremum is attained at some point of M n . In this case, we note from (3.3) that H also attains its maximum on M n and, from (5.15), we have that (nH) ≥

1 |Φ|2 QR (|Φ|) ≥ 0. n−1

Since Lemma 4.4 guarantees that  is elliptic, we conclude that H is constant on M n . Hence, returning to H (5.15), we get that |Φ| = |Φn+1 | and, consequently, Φα = 0, for all n + 2 ≤ α ≤ n + p. Thus, since en+1 = H is parallel in the normal bundle of M n , we are in position to apply Theorem 1 of [12] to conclude that M n is, in fact, isometrically immersed in a (n + 1)-dimensional totally geodesic submanifold Qn+1 of Qn+p . c c Therefore, we can use Theorems 1 and 2 of [2] to finish our proof. 2

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J.G. Araújo et al. / J. Math. Anal. Appl. 447 (2017) 488–498

Acknowledgments The first author is partially supported by CAPES, Brazil. The second author is partially supported by CNPq, Brazil, grant 303977/2015-9. The fourth author is partially supported by CNPq, Brazil, grant 308757/2015-7. The authors would like to thank the referee for his/her valuable suggestions and useful comments which improved the paper. References [1] L.J. Alías, S.C. García-Martí nez, On the scalar curvature of constant mean curvature hypersurfaces in space forms, J. Math. Anal. Appl. 363 (2010) 579–587. [2] L.J. Alías, S.C. García-Martí nez, M. Rigoli, A maximum principle for hypersurfaces with constant scalar curvature and applications, Ann. Global Anal. Geom. 41 (2012) 307–320. [3] A. Brasil Jr., A.G. Colares, O. Palmas, Complete hypersurfaces with constant scalar curvature in spheres, Monatsh. Math. 161 (2010) 369–380. [4] Q.M. Cheng, Submanifolds with constant scalar curvature, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002) 1163–1183. [5] S.Y. Cheng, S.T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977) 195–204. [6] X. Guo, H. Li, Submanifolds with constant scalar curvature in a unit sphere, Tohoku Math. J. 65 (2013) 331–339. [7] A.M. Li, J.M. Li, An intrinsic rigidity theorem for minimal submanifolds in a sphere, Arch. Math. 58 (1992) 582–594. [8] H. Li, Hypersurfaces with constant scalar curvature in space forms, Math. Ann. 305 (1996) 665–672. [9] H. Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 (1967) 205–214. [10] S. Pigola, M. Rigoli, A.G. Setti, Maximum principles on Riemannian manifolds and applications, Mem. Amer. Math. Soc. 822 (2005). [11] W. Santos, Submanifolds with parallel mean curvature vector in spheres, Tohoku Math. J. 46 (1994) 403–415. [12] S.T. Yau, Submanifolds with constant mean curvature I, Amer. J. Math. 96 (1974) 346–366. [13] S.T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975) 201–228.