Hypersurfaces with constant mean curvature in a hyperbolic space

Acta Mathematica Scientia 2011,31B(3):1091–1102 http://actams.wipm.ac.cn

HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN A HYPERBOLIC SPACE∗

 )

Su Bianping (

Department of Science, Xi’an University of Architecture and Technology, Xi’an 710055, China E-mail: [email protected]


)†

Shu Shichang (

Department of Mathematics, Xianyang Normal University, Xianyang 712000, China E-mail: [email protected]

Yi Annie Han Department of Mathematics, Borough of Manhattan Community College, City University of New York, New York 10007, USA E-mail: [email protected]

Abstract Let M n be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H n+1 (c) with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that (1) if the multiplicities of the two distinct principal curvatures are greater than 1, then M n is isometric to the Riemannian product S k (r) × H n−k (−1/(r2 + ρ2 )), where r > 0 and 1 < k < n − 1; (2) if H 2 > −c and one of the two distinct principal curvatures is simple, then M n is isometric to the Riemannian product S n−1 (r) × H 1 (−1/(r2 + ρ2 )) or S 1 (r) × H n−1 (−1/(r2 + ρ2 )), r > 0, if one of the n 2 2 −2 following conditions is satisfied (i) S ≤ (n − 1)t22 + c2 t−2 2 on M or (ii) S ≥ (n − 1)t1 + c t1 n 2 2 −2 2 2 −2 n on M or (iii) (n − 1)t2 + c t2 ≤ S ≤ (n − 1)t1 + c t1 on M , where t1 and t2 are the positive real roots of (1.5). Key words hypersurface; hyperbolic space; scalar curvature; mean curvature; principal curvature 2000 MR Subject Classification

1

53C42; 53A10

Introduction

Let N n+1 (c) be an (n + 1)-dimensional connected Riemannian manifold with constant sectional curvature c. According as c > 0, c = 0 and c < 0, it is called spherical space, Euclidean space or hyperbolic space, and denoted by S n+1 (c), Rn+1 or H n+1 (c), respectively. As it is well known that there were many rigidity results for hypersurfaces with constant mean curvature or ∗ Received

December 16, 2008. Project supported by NSF of Shaanxi Province (SJ08A31), NSF of Shaanxi Educational Committee (2008JK484; 2010JK642) and Talent Fund of Xi’an University of Architecture and Technology. † Corresponding author: Shu Shichang.

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constant scalar curvature in S n+1 (c) or Rn+1 , for example, see [1–4], [7], [11] and [16] etc., but less were obtained for hypersurfaces immersed into a hyperbolic space. S.Y.Cheng and Yau [2] proved that an n-dimensional (n ≥ 2) complete hypersurface M n with constant scalar curvature in Rn+1 is isometric to a sphere, a hyperplane or a generalized cylinder S k (c) × Rn−k , 1 ≤ k ≤ n − 1, if the sectional curvature of M n is nonnegative. They also proved that an n-dimensional compact hypersurface M n with constant scalar curvature n(n − 1)R satisfying R ≥ 1 in the unit sphere S n+1 (1) is isometric to a sphere, or a Riemannian product S k (c1 ) × S n−k (c2 ), 1 ≤ k ≤ n − 1, if the sectional curvature of M n is nonnegative. In [7], Li extended the results due to S.Y.Cheng and Yau [2] in terms of the squared norm of the second fundamental form of M n . Cheng [3] and [4] characterized the hypersurface S k (c) × Rn−k in a Euclidean space Rn+1 and the hypersurface S k (c1 ) × S n−k (c2 ) in a unit sphere S n+1 (1), respectively. On the other hand, Morvan-Wu [10], Wu [15] proved some rigidity theorems for complete hypersurfaces M n in a hyperbolic space H n+1 (c) under the assumption that the mean curvature is constant and the Ricci curvature is non-negative. To our best knowledge, there were almost no intrinsic rigidity results for the hypersurfaces with constant scalar curvature in a hyperbolic space until Liu and Su [9], Hu and Zhai [6] and the author obtained some interesting results, in [13], he proved that, let M n be an n-dimensional complete hypersurface with constant scalar curvature n(n − 1)R in a hyperbolic space H n+1 (−1), if R + 1 ≥ 0 and the sectional curvature of M n is nonnegative, then M n is a totally umbilical hypersurface; or M n is isometric to S n−1 (r) × H 1 (−1/(r2 + 1)) or S 1 (r) × H n−1 (−1/(r2 + 1)), where r > 0. In this paper, we investigate the complete hypersurfaces with constant mean curvature H and with two distinct principal curvatures. In order to represent our theorems, we need some notations, for details see Lawson [8], Ryan [12] or Liu [9]. First, we give a description of the real hyperbolic space H n+1 (c) of constant curvature c (< 0). For any two vectors x and y in Rn+2 , we set g(x, y) = x1 y1 + · · · + xn+1 yn+1 − xn+2 yn+2 , (Rn+2 , g) is the so-called Minkowski space-time. Denote ρ =

 −1/c. We define

H n+1 (c) = {x ∈ Rn+2 |g(x, x) = −ρ2 , xn+2 > 0}. Then H n+1 (c) is a simply-connected hypersurface of Rn+2 . Hence, we obtain a model of a real hyperbolic space. We define M1 = {x ∈ H n+1 (c)|x1 = 0}, M2 = {x ∈ H n+1 (c)|x1 = r > 0}, M3 = {x ∈ H n+1 (c)|xn+2 = xn+1 + ρ}, M4 = {x ∈ H n+1 (c)|x21 + · · · + x2n+1 = r2 > 0}, M5 = {x ∈ H n+1 (c)|x21 + · · · + x2k+1 = r2 > 0, x2k+2 + · · · + x2n+1 − x2n+2 = −ρ2 − r2 }, M1 , · · · , M5 are often called the standard examples of complete hypersurfaces in H n+1 (c) with at most two distinct constant principal curvatures. It is obvious that M1 , · · · , M4 are totally

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umbilical. In the sense of Chen [5], they are called the hyperspheres of H n+1 (c). M3 is called the horosphere and M4 the geodesic distance sphere of H n+1 (c). Ryan [12] obtained the following. Proposition 1.1 Let M n be a complete hypersurface in H n+1 (c) Suppose that, under a suitable choice of a local orthonormal tangent frame field of T M n. The shape operator over T M n is expressed as a matrix A. If M n has at most two distinct constant principal curvatures, then it is congruent to one of the following: (1) M1 . In this case, A = 0, and M1 is totally geodesic. Hence M1 is isometric to H n (c); 2 (2) M2 . In this case, A = √ 1/ρ I , where In denotes the identity matrix of order n, 2 2 n 1/ρ +1/r

and M2 is isometric to H n (−1/(r2 + ρ2 )); (3) M3 . In this case, A = 1ρ In , and M3 is isometric to an Euclidean space Rn ;  (4) M4 . In this case, A = 1/r2 + 1/ρ2 In , M4 is isometric to a round sphere S n (r) of radius r;  2 (5) M5 . In this case, A = λIk ⊕ μIn−k , where λ = 1/ρ2 + 1/r2 , and μ = √ 1/ρ , M5 2 2 1/r +1/ρ

is isometric to S k (r) × H n−k (−1/(r2 + ρ2 )). We know that S k (r)×H n−k (−1/(r2 +ρ2 )) in H n+1 (c) has two distinct principal curvatures  2 λ = 1/ρ2 + 1/r2 with multiplicity k and μ = √ 1/ρ with multiplicity n−k; Therefore, we 2 2 1/r +1/ρ

also know that the Riemannian product S 1 (r) × H n−1 (−1/(r2 + ρ2 )) or S n−1 (r) × H 1 (−1/(r2 + ρ2 )) in H n+1 (c) has two distinct principal curvatures one of which is simple. Without lose of generality, we can denote the two distinct principal curvatures by λ and μ, and say λ with multiplicity n − 1 and μ with multiplicity 1. Therefore, we have (n − 1)λ + μ = nH,

(1.1)

where H is the mean curvature of S 1 (r) × H n−1 (−1/(r2 + ρ2 )) or S n−1 (r) × H 1 (−1/(r2 + ρ2 )). Note that λ = 0, μ = 0 and λ and μ satisfy λμ = −c.

(1.2)

μ = nH − (n − 1)λ.

(1.3)

From (1.1), we have From (1.2) and (1.3), we know that λ satisfies (n − 1)λ2 − nHλ − c = 0. Putting t = λ, we have (n − 1)t2 − nHt − c = 0,

(1.4)

and the squared norm S of the second fundamental form of the Riemannian product S (r) × H n−1 (−1/(r2 +ρ2 )) or S n−1 (r)×H 1 (−1/(r2 +ρ2 )) is S = (n−1)λ2 +(− λc )2 = (n−1)t2 +c2 t−2 , where t satisfies (1.4). Denote by PH (t) the following function 1

PH (t) = (n − 1)t2 − nHt − c (H 2 > −c).

(1.5)

From H 2 > −c, we have n2 H 2 > −4(n − 1)c. Therefore, we know that (1.5) has two positive real roots   nH − n2 H 2 + 4(n − 1)c nH + n2 H 2 + 4(n − 1)c t1 = , t2 = , (1.6) 2(n − 1) 2(n − 1)

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and t1 < t2 . By a direct calculation, we have 2 2 −2 (n − 1)t22 + c2 t−2 2 ≤ (n − 1)t1 + c t1 .

(1.7)

We shall prove the following result: Theorem 1.1 Let M n be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H n+1 (c) with non-zero constant mean curvature H and with two distinct principal curvatures. If both the multiplicities of these two distinct principal curvatures are greater than 1, then M n is isometric to the Riemannian product S k (r)×H n−k (−1/(r2 +ρ2 )), where r > 0 and 1 < k < n − 1. Theorem 1.2 Let M n be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H n+1 (c) with non-zero constant mean curvature H and with two distinct principal curvatures, one of which is simple. If H 2 > −c, then M n is isometric to the Riemannian product S 1 (r) × H n−1 (−1/(r2 + ρ2 )) or S n−1 (r) × H 1 (−1/(r2 + ρ2 )), if one of the following conditions is satisfied: n (1) S ≤ (n − 1)t22 + c2 t−2 2 on M , n (2) S ≥ (n − 1)t21 + c2 t−2 1 on M , 2 2 −2 n (3) (n − 1)t2 + c t2 ≤ S ≤ (n − 1)t21 + c2 t−2 1 on M , where t1 and t2 are the positive real roots of (1.5).

2

Preliminaries

Let M n be an n-dimensional hypersurface in a hyperbolic space H n+1 (c) with c < 0. We choose a local orthonormal frame e1 , · · · , en+1 in H n+1 (c) such that e1 , · · · , en are tangent to M n . Let ω1 , · · · , ωn+1 be the dual coframe. We use the following convention on the range of indices: 1 ≤ A, B, C, · · · ≤ n + 1; 1 ≤ i, j, k, · · · ≤ n. The structure equations of H n+1 (c) are given by  dωA = ωAB ∧ ωB , B



ωAB + ωBA = 0,

(2.1)

ωAC ∧ ωCB + ΩAB ,

(2.2)

1 KABCD ωC ∧ ωD , 2

(2.3)

KABCD = c(δAC δBD − δAD δBC ).

(2.4)

dωAB =

C

where ΩAB = −

C,D

Restricting ourselves to M n , ωn+1i

ωn+1 = 0.  = hij ωj , hij = hji .

(2.5) (2.6)

j

The structure equations of M n are dωi =

 j

ωij ∧ ωj ,

ωij + ωji = 0,

(2.7)

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dωij =



ωik ∧ ωkj −

k

1 Rijkl ωk ∧ ωl , 2

1095 (2.8)

k,l

Rijkl = c(δik δjl − δil δjk ) + (hik hjl − hil hjk ),  hik hkj , Rij = (n − 1)cδij + nHhij −

(2.9) (2.10)

k

n(n − 1)(R − c) = n2 H 2 − S 2 ,

(2.11)

where n(n − 1)R is the scalar curvature, H is the mean curvature and S is the squared norm of the second fundamental form of M n . The Codazzi equation and the Ricci identity are

hijkl − hijlk

hijk = hikj ,   = hmj Rmikl + him Rmjkl , m

(2.12) (2.13)

m

where hijk and hijkl denote the first and the second covariant derivatives of hij , respectively. We choose e1 , · · · , en such that hij = λi δij . From (2.6) we have ωn+1i = λi ωi ,

i = 1, 2, · · · , n.

(2.14)

Hence, we have, from the structure equations of M n , dωn+1i = dλi ∧ ωi + λi dωi = dλi ∧ ωi + λi



ωij ∧ ωj .

(2.15)

j

On the other hand, we have, on the curvature forms of H n+1 (c), 1 Kn+1iCD ωC ∧ ωD 2 C,D 1 =− c(δn+1C δiD − δn+1D δiC )ωC ∧ ωD 2

Ωn+1i = −

C,D

= −cωn+1 ∧ ωi = 0.

(2.16)

Therefore, from the structure equations of H n+1 (c), we have   ωn+1j ∧ ωji + ωn+1n+1 ∧ ωn+1i + Ωn+1i = λj ωij ∧ ωj . dωn+1i = j

(2.17)

j

From (2.15) and (2.17), we obtain dλi ∧ ωi +



(λi − λj )ωij ∧ ωj = 0.

(2.18)

j

Putting ψij = (λi − λj )ωij . Then ψij = ψji . (2.18) can be written as  (ψij + δij dλj ) ∧ ωj = 0. j

(2.19)

(2.20)

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By E.Cartan’s lemma, we get ψij + δij dλj =



Qijk ωk ,

(2.21)

k

where Qijk are functions uniquely determined such that Qijk = Qikj .

2

(2.22)

Proof of Theorems

We first have the following Proposition 3.1, whose proof can be given by making use a methed due to Otsuki [11]. Proposition 3.1 [11] Let M n be a hypersurface in a hyperbolic space H n+1 (c) such that the multiplicities of the principal curvatures are constant. Then the distribution of the space of the principal vectors corresponding to each principal curvature is completely integrable. In particular, if the multiplicity of a principal curvature is greater than 1, then this principal curvature is constant on each integral submanifold of the corresponding distribution of the space of the principal vectors. Proof of Theorem 1.1 Let λ, μ be the principal curvatures of multiplicities k and n−k, respectively, where 1 < k < n − 1. We have kλ + (n − k)μ = nH.

(3.1)

Denote by Dλ and Dμ the integral submanifolds of the corresponding distribution of the space of principal vectors corresponding to the principal curvature λ and μ, respectively. From Proposition 3.1, we know that λ is constant on Dλ . Since the mean curvature is constant, (3.1) implies that μ is constant on Dλ . By making use of Proposition 3.1 again, we have μ is constant on Dμ . Therefore, we know that μ is constant on M n . By the same assertion we know that λ is constant on M n . Therefore M n is isoparametric. By Proposition 1.1, we know that M n is isometric to S k (r) × H n−k (−1/(r2 + ρ2 ))), where r > 0 and 1 < k < n − 1. This completes the proof of Theorem 1.1. From now on, we consider the complete connected and oriented hypersurfaces in H n+1 (c) with non-zero constant mean curvature and with two distinct principal curvatures, one of which is simple. We can choose an orientation for M n such that H > 0. Without lose of generality, we may assume λn = μ, λ1 = λ2 = · · · = λn−1 = λ, where λi for i = 1, 2, · · · , n are the principal curvatures of M n . Therefore, we know that (n − 1)λ + μ = nH, S = (n − 1)λ2 + μ2 .

(3.2)

μ = nH − (n − 1)λ.

(3.3)

We have

From λ − μ = n(λ − H) = 0,

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we know λ − H = 0. 1 Let  = |λ − H|− n . We denote the integral submanifold through x ∈ M n corresponding to λ by M1n−1 (x). Putting n 

dλ =

λ,k ωk ,

dμ =

k=1

n 

μ,k ωk .

(3.4)

k=1

From Proposition 3.1, we have λ,1 = λ,2 = · · · = λ,n−1 = 0 on M1n−1 (x).

(3.5)

dμ = −(n − 1)dλ.

(3.6)

From (3.3), we have

Hence, we also have μ,1 = μ,2 = · · · = μ,n−1 = 0

on M1n−1 (x).

(3.7)

In this case, we may consider locally λ to be a function of the arc length s of the integral curve of the principal vector field en corresponding to the principal curvature μ. From (2.21) and (3.5), we have for 1 ≤ j ≤ n − 1, dλ = dλj =

n 

Qjjk ωk =

k=1

n−1 

Qjjk ωk + Qjjn ωn = λ,n ωn .

(3.8)

k=1

Therefore, we have Qjjk = 0, 1 ≤ k ≤ n − 1, and Qjjn = λ,n .

(3.9)

By (2.21) and (3.7), we have dμ = dλn =

n 

Qnnk ωk =

k=1

n−1 

Qnnk ωk + Qnnn ωn =

k=1

n 

μ,i ωi = μ,n ωn .

(3.10)

i=1

Hence, we obtain Qnnk = 0,

1 ≤ k ≤ n − 1, and Qnnn = μ,n .

(3.11)

From (3.6), we get Qnnn = μ,n = −(n − 1)λ,n .

(3.12)

From the definition of ψij , if i = j, we have ψij = 0 for 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ n − 1. Therefore, from (2.21), if i = j and 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ n − 1 we have Qijk = 0 for any k.

(3.13)

By (2.21), (3.9), (3.11), (3.12) and (3.13), we get ψjn =

n 

Qjnk ωk = Qjjn ωj + Qjnn ωn = λ,n ωj .

(3.14)

k=1

From (2.21), (3.3) and (3.14) we have ωjn =

λ,n ψjn λ,n = ωj = ωj . λ−μ λ−μ n(λ − H)

(3.15)

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Therefore, from the structure equations of M n we have dωn =

n−1 

ωk ∧ ωkn + ωnn ∧ ωn = 0.

k=1

Therefore, we may put ωn = ds. By (3.8) and (3.10), we get dλ = λ,n ds,

λ,n =

dλ ds

dμ = μ,n ds,

μ,n =

dμ . ds

and

Then we have ωjn =

dλ ds

1

n(λ − H)

ωj =

d{log |λ − H| n } ωj . ds

(3.16)

From (3.16) and the structure equations of H n+1 (c), we have dωjn =

n−1 

ωjk ∧ ωkn + ωjn ∧ ωnn + ωjn+1 ∧ ωn+1n + Ωjn

k=1

=

n−1 

ωjk ∧ ωkn + ωjn+1 ∧ ωn+1n − cωj ∧ ωn

k=1

=

1 n−1 d{log |λ − H| n }  ωjk ∧ ωk − (c + λμ)ωj ∧ ds. ds

k=1

From (3.16), we have 1

dωjn

1

d2 {log |λ − H| n } d{log |λ − H| n } dωj = ds ∧ ωj + ds2 ds 1 1 n d2 {log |λ − H| n } d{log |λ − H| n }  = ds ∧ ω + ωjk ∧ ωk j ds2 ds k=1 2    1 1 d2 {log |λ − H| n } d{log |λ − H| n } ωj ∧ ds = − + ds2 ds 1 n−1 d{log |λ − H| n }  ωjk ∧ ωk . + ds

k=1

From the above two equalities, we have 2  1 1 d2 {log |λ − H| n } d{log |λ − H| n } − − (c + λμ) = 0. ds2 ds

(3.17)

From (3.3), we get 1

d2 {log |λ − H| n } − ds2



1

d{log |λ − H| n } ds

2 − {c + nHλ − (n − 1)λ2 } = 0.

(3.18)

1

Since we have defined  = |λ − H|− n , we obtain from the above equation d2  + {c + nHλ − (n − 1)λ2 } = 0. ds2

(3.19)

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We can prove the following lemmas. Lemma 3.1 Let PH (t) = (n − 1)t2 − nHt − c, where c < 0 and H 2 > −c. Then PH (t) has two positive real roots t1 and t2 and (i) if t ≥ H, then t ≥ t2 holds if and only if PH (t) ≥ 0 and t ≤ t2 holds if and only if PH (t) ≤ 0; (ii) if t ≤ H, then t ≤ t1 holds if and only if PH (t) ≥ 0 and if t ∈ [t1 , t0 ), then t ≥ t1 holds nH if and only if PH (t) ≤ 0; if t ∈ [t0 , H], then PH (t) < 0, where t0 = 2(n−1) . Proof We have dPH (t) = 2(n − 1)t − nH, dt nH H (t) = 0 is t0 = 2(n−1) > 0. Therefore, we know that t ≤ t0 if it follows that the solution of dPdt and only if PH (t) is a decreasing function, t ≥ t0 if and only if PH (t) is an increasing function and PH (t) obtain its minimum at t = t0 . n2 H 2 ] < 0. This implies Since PH (t) is continuous and H 2 > −c, we have PH (t0 ) = −[c + 4(n−1) that PH (t) has two distinct real roots t1 , t2 and t1 < t0 < t2 . From PH (0) = −c > 0, we infer that t1 > 0, t2 > 0 and t1 and t2 satisfy (1.6). Since t0 < H and PH (H) = −(c + H 2 ) < 0, we know that H < t2 . In fact, if H ≥ t2 , from the increasing property of PH (t), we have PH (H) ≥ PH (t2 ) = 0, which is a contraction. Now, we prove the second part of Lemma 3.1. If t ≥ H, from the increasing property of PH (t), we obtain that t ≥ t2 holds if and only if PH (t) ≥ PH (t2 ) = 0 and t ≤ t2 holds if and only if PH (t) ≤ PH (t2 ) = 0. If t ≤ H, from the decreasing property of PH (t), we directly obtain that t ≤ t1 holds if and only if PH (t) ≥ PH (t1 ) = 0. Now, we consider the case t ≤ H and t ≥ t1 . In this case, we have t ∈ [t1 , t0 ) or t ∈ [t0 , H]. If t ∈ [t1 , t0 ), from the decreasing property of PH (t), we infer that t ≥ t1 holds if and only if PH (t) ≤ PH (t1 ) = 0. If t ∈ [t0 , H], from the increasing property of PH (t), we infer that PH (t) ≤ PH (H) < 0. This completes the proof of Lemma 3.1. Lemma 3.2 Let

S(t) = (n − 1)t2 + [nH − (n − 1)t]2 , where c < 0 and H 2 > −c. Then (i) if t ≥ H, then t ≥ t2 holds if and only if S(t) ≥ S(t2 ) and t ≤ t2 holds if and only if S(t) ≤ S(t2 ); (ii) if t ≤ H, then t ≥ t1 holds if and only if S(t) ≤ S(t1 ) and t ≤ t1 holds if and only if S(t) ≥ S(t1 ), where t1 , t2 are the two distinct real roots of PH (t) and t1 < t2 . Proof We have dS(t) = 2n(n − 1)(t − H), dt it follows that the solution of dS(t) dt = 0 is t = H. Therefore, we know that if t ≤ H if and only if S(t) is a decreasing function, t ≥ H if and only if S(t) is an increasing function and S(t) reaches its minimum at t = H.

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From the proof of Lemma 3.1, we know that t1 < H < t2 . Since t ≥ H if and only if S(t) is an increasing function, we infer that if t ≥ H, then t ≥ t2 holds if and only if S(t) ≥ S(t2 ) and t ≤ t2 holds if and only if S(t) ≤ S(t2 ). If t ≤ H, from the decreasing property of S(t), we directly have that t ≥ t1 holds if and only if S(t) ≤ S(t1 ) and t ≤ t1 holds if and only if S(t) ≥ S(t1 ). This completes the proof of Lemma 3.2. 2 Lemma 3.3 If H 2 > −c, then the positive function  is bounded. Proof From the definition of  and (3.19), we have d2  + [c + H 2 + (2 − n)H−n + (1 − n)−2n ] = 0 ds2

(3.20)

d2  + [c + H 2 + (n − 2)H−n + (1 − n)−2n ] = 0 ds2

(3.21)

for λ − H > 0, or

for λ − H < 0. We easily know that equation (3.20) or (3.21) is equivalent to its first order integral 

for λ − H > 0, or



d ds

d ds

2 + (c + H 2 )2 + 2H2−n + 2−2n = C

(3.22)

+ (c + H 2 )2 − 2H2−n + 2−2n = C

(3.23)

2

for λ − H < 0, where C is a constant. From (3.22) and (3.23), we have (c + H 2 )2 + 2H2−n + 2−2n ≤ C

(3.24)

(c + H 2 )2 − 2H2−n + 2−2n ≤ C

(3.25)

for λ − H > 0, or for λ − H < 0. Since H 2 + c > 0, from (3.24) and (3.25), we infer that the positive function  is bounded from above. This completes the proof of Lemma 3.3. Proof of Theorem 1.2 Putting t = λ, from (3.19), we have d2  − PH (t) = 0. ds2 Since λ − μ = n(t − H) = 0, we have t − H = 0. (1) If S ≤ (n − 1)t22 + c2 t−2 2 , we consider two cases t > H or t < H. If t > H, we have S(t2 ) = (n − 1)t22 + [nH − (n − 1)t2 ]2

(3.26)

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2  c c = (n − 1)t22 + nH − (n − 1)t2 + − t2 t2 2  −1 c 2 2 = (n − 1)t2 + [(n − 1)t2 − nHt2 − c] − t2 t2 2  −1 c = (n − 1)t22 + PH (t2 ) − t2 t2 = (n − 1)t22 + c2 t−2 2 . From Lemma 3.1, Lemma 3.2 and (3.26), we have S(t) ≤ S(t2 ) holds if and only if t ≤ t2 , 2 d PH (t) ≤ 0 and dds 2 ≤ 0. Thus, ds is a monotonic function of s ∈ (−∞, +∞). Therefore, by the similar assertion in Wei [14], we have (s) must be monotonic when s tends to infinity. From Lemma 3.3, we know that the positive function (s) is bounded. Since (s) is bounded and monotonic when s tends to infinity, we know that both lim (s) and lim (s) exist s→−∞

s→+∞

and then we get lim

s→−∞

d(s) d(s) = lim = 0. s→+∞ ds ds

(3.27) 1

d(s) −n From the monotonicity of d(s) ds , we have ds ≡ 0 and (s) = constant. From  = |λ − H| and (3.2), we have λ and μ are constants, that is, M n is isoparametric. Therefore, by Proposition 1.1, we know that M n is isometric to the Riemannian product S n−1 (r) × H 1 (−1/(r2 + ρ2 )) or S 1 (r) × H n−1 (−1/(r2 + ρ2 )), where r > 0. If t < H, by a direct calculation, we have S(t1 ) = (n − 1)t21 + c2 t−2 1 . From (1.7), we have S(t2 ) ≤ S(t1 ). Hence, we obtain that S ≤ S(t1 ). From Lemma 3.2, we have S(t) ≤ S(t1 ) holds if and only if t ≥ t1 . Therefore, we know that t ∈ [t1 , t0 ) or t ∈ [t0 , H). 2 If t ∈ [t0 , H), from Lemma 3.1, we have PH (t) < 0. From (3.26), we have dds 2 < 0. This d(s) implies that ds is a strictly monotone decreasing function of s and thus it has at most one zero point in s ∈ (−∞, +∞). If d(s) has no zero point in (−∞, +∞), then (s) is a monotone ds d(s) function of s in (−∞, +∞). If ds has exactly one zero point s0 in (−∞, +∞), then (s) is a monotone function of s in both (−∞, s0 ] and [s0 , +∞). On the other hand, from Lemma 3.3, we know that (s) is bounded. Since (s) is bounded and monotonic when s tends to infinity, we know that both lim (s) and lim (s) exist and s→−∞

d(s) ds

s→+∞

(3.27) holds. This is impossible because is a strictly monotone decreasing function of s. Therefore, we know that the case t ∈ [t0 , H) does not occur. We conclude that t ∈ [t1 , t0 ). From Lemma 3.1, Lemma 3.2 and (3.26), we have S(t) ≤ S(t1 ) holds if and only if t ≥ t1 , PH (t) ≤ 0 2 d and dds 2 ≤ 0. Thus, ds is a monotonic function of s ∈ (−∞, +∞). By the same assertion as above, we know that M n is isometric to the Riemannian product S n−1 (r) × H 1 (−1/(r2 + ρ2 )) or S 1 (r) × H n−1 (−1/(r2 + ρ2 )), where r > 0. (2) If S ≥ (n − 1)t21 + c2 t−2 1 , we consider two cases t > H or t < H: If t > H, since S(t1 ) = (n − 1)t21 + c2 t−2 1 ≥ S(t2 ), we have S ≥ S(t2 ). From Lemma 3.1, 2 Lemma 3.2 and (3.26), we have S(t) ≥ S(t2 ) holds if and only if t ≥ t2 , PH (t) ≥ 0 and dds 2 ≥ 0. d Thus, ds is a monotonic function of s ∈ (−∞, +∞). By the same assertion as in the proof of (1), we know that (s), λ and μ are constants, that is, M n is isoparametric. By Proposition 1.1, we know that M n is isometric to the Riemannian product S n−1 (r) × H 1 (−1/(r2 + ρ2 )) or S 1 (r) × H n−1 (−1/(r2 + ρ2 )), where r > 0.

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If t < H, since S ≥ S(t1 ), from Lemma 3.1, Lemma 3.2 and (3.26), we have S(t) ≥ S(t1 ) 2 holds if and only if t ≤ t1 , PH (t) ≥ 0 and dds ≥ 0. Thus, d 2 ds is a monotonic function of s ∈ (−∞, +∞). By the same assertion as above, we know that M n is isometric to the Riemannian product S n−1 (r) × H 1 (−1/(r2 + ρ2 )) or S 1 (r) × H n−1 (−1/(r2 + ρ2 )), where r > 0. (3) If (n − 1)t22 + c2 t−2 ≤ S ≤ (n − 1)t21 + c2 t−2 2 1 , we also consider two cases t > H or t < H: If t > H, since S ≥ (n − 1)t22 + c2 t−2 2 = S(t2 ), from Lemma 3.1, Lemma 3.2 and (3.26), 2 we have S(t) ≥ S(t2 ) holds if and only if t ≥ t2 , PH (t) ≥ 0 and dds ≥ 0. Thus, d 2 ds is a monotonic function of s ∈ (−∞, +∞). Similar to the proof of (1), we know that (s), λ and μ are constants, that is, M n is isoparametric. By Proposition 1.1, we know that M n is isometric to the Riemannian product S n−1 (r) × H 1 (−1/(r2 + ρ2 )) or S 1 (r) × H n−1 (−1/(r2 + ρ2 )), where r > 0. If t < H, S ≤ (n − 1)t21 + c2 t−2 1 = S(t1 ). From Lemma 3.2, we have S(t) ≤ S(t1 ) holds if and only if t ≥ t1 . Therefore, we know that t ∈ [t1 , t0 ) or t ∈ [t0 , H). By the same assertion in the proof of (1), we know that the case t ∈ [t0 , H) does not occur. We conclude that t ∈ [t1 , t0 ). In this case, by the same assertion as in (1), we know that M n is isometric to the Riemannian product S n−1 (r) × H 1 (−1/(r2 + ρ2 )) or S 1 (r) × H n−1 (−1/(r2 + ρ2 )), where r > 0. This completes the proof of Theorem 1.2. References [1] Alencar H, do Carmo M P. Hypersurfaces with constant mean curvature in sphere. Proc Amer Math Soc, 1994, 120: 1223–1229 [2] Cheng S Y, Yau S T. Hypersurfaces with constant scalar curvature. Math Ann, 1977, 225: 195–204 [3] Cheng Q M. Complete hypersurfaces in a Euclidean space Rn+1 with constant scalar curvature. Indiana Univ Math J, 2002, 51: 53–68 [4] Cheng Q M. Hypersurfaces in a unit sphere S n+1 (1) with constant scalar curvature. J London Math Soc, 2001, 64: 755–768 [5] Chen B Y. Totally Mean Curvature and Submanifolds of Finite Type. Singapore: World Scientific, 1984 [6] Hu Z, Zhai S. Hypersurfaces of the hyperbolic space with constant scalar curvature. Result Math, 2005, 48 : 65–88 [7] Li H. Hypersurfaces with constant scalar curvature in space forms. Math Ann, 1996, 305: 665–672 [8] Lawson H B. Local rigidity theorems for minimal hypersurfaces. Ann Math, 1969, 89(2): 187–197 [9] Liu X, Su W. Hypersurfaces with constant scalar curvature in a hyperbolic space form. Balkan J Geom Appl, 2002, 7: 121–132 [10] Morvan J M, Wu B Q. Hypersurfaces with constant mean curvature in hyperbolic space form. Deom Dedicata, 1996, 59: 197–222 [11] Otsuki T. Minimal hypersurfaces in a Riemannian manifold of constant curvature. Amer J Math, 1970, 92: 145–173 [12] Ryan P J. Hypersurfaces with parallel Ricci tensor. Osaka J Math, 1971, 8: 251–259 [13] Shu S. Complete hypersurfaces with constant scalar curvature in a hyperbolic space. Balkan J of Geom and Application, 2007, 12: 107–115 [14] Wei G. Complete hypersurfaces with constant mean curvature in a unit sphere. Monatsh Math, 2006, 149: 251–258 [15] Wu B Q. Hypersurfaces with constant mean curvature in H n+1 //The Math Heritage of C.F.Gauss. Singapore: World Scientific, 1991: 862–871 [16] Zhang Y T. Rigidity theorems of Clifford torus. Acta Mathematica Scienta, 2010, 30B(3): 890–896