A CHARACTERIZATION OF THE CLIFFORD TORUS

Qing-Ming Cheng* and Susumu Ishikawa**

Abstract. In this paper, we prove that an n-dimensional closed minimal hypersurface M with Ricci curvature Ric(M) ≥ \frac{n}{2} of a unit sphere S^{n+1}(1) is isometric to a Clifford torus if n ≤ S ≤ n + \frac{14(n+4)}{9n+30} , where S is the squared norm of the second fundamental form of M.

1. Introduction

Let M be an n-dimensional closed minimal hypersurface in a unit sphere S^{n+1}(1) of dimension n+1. Let S denote the squared norm of the second fundamental form of M. From the Gauss equation (see section 2), we know that S, which is exitrinsic by definition, is actually an intrinsic quantity. It is well-known that Chern, do Carmo and Kobayashi [3] and Lawson [4] obtained independently that Clifford tori are the only closed minimal hypersurfaces of the unit sphere with S = n. When the scalar curvature of M is constant, Yang and the first named author in [6] and [7] proved that

if n ≤ S ≤ n + \frac{n}{3}, then M is isometric to a Clifford torus S^{m}(\sqrt{\frac{m}{n}}) × S^{n−m}(\sqrt{\frac{n−m}{n}}). A natural problem is that for a closed minimal hypersurface M of a unit sphere, whether there exists a constant ϵ(n) > 0 such that if n ≤ S ≤ n + ϵ(n), then S = n

and M is isometric to a Clifford torus S^{m}(\sqrt{\frac{m}{n}}) × S^{n−m}(\sqrt{\frac{n−m}{n}}). The first named author [2] gave a positive answer under additional condition that M has only two distinct principal curvatures. In general, it still remains open and it is a very hard problem. On the other hand, the Clifford torus S^{m}(\sqrt{\frac{m}{n}})×S^{n−m}(\sqrt{\frac{n−m}{n}}) is a closed minimal hypersurface in S^{n+1}(1) with S = n and its Ricci curvature varies between

\frac{n(m − 1)}{m} and \frac{n(n − m − 1)}{n − m}. If 2 ≤ m ≤ n − 2, then Ric(M) ≥ \frac{n}{2}. Hence it is

natural to ask whether there exists a constant ϵ(n) > 0 such that if M is a closed minimal hypersurface with Ric(M) ≥ \frac{n}{2} and n ≤ S ≤ n + ϵ(n), then S = n and M

Typeset by AMS-TEX

Key words and phrases. Minimal hypersurfaces, scalar curvature, Ricci curvature and Clifford torus.

*Research partially supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture and by a Grant-in-Aid for Scientific Research from Josai University.

**Research partially supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture.

1991 Mathematics Subject Classification 53C20, 53C42

is isometric to a Clifford torus S^{m}(\sqrt{\frac{m}{n}}) × S^{n−m}(\sqrt{\frac{n−m}{n}}) (1 < m < n − 1). In this paper, we give an affirmative answer for the above problem.

Theorem 1. Let M be an n-dimensional closed minimal hypersurface of a unit sphere S^{n+1}(1) with Ricci curvature Ric(M) ≥ \frac{n}{2}. If

n ≤ S ≤ n + \frac{14(n + 4)}{9n + 30},

then S = n and M is isometric to a Clifford torus S^{m}(\sqrt{\frac{m}{n}}) × S^{n−m}(\sqrt{\frac{n−m}{n}}) (1 < m < n − 1).

In particular, if n ≤ 5, we obtain the following

Theorem 2. Let M be an n (n ≤ 5)-dimensional closed minimal hypersurface of a unit sphere S^{n+1}(1). If

n ≤ S ≤ n + ϵ(n),

then S = n and M is isometric to a Clifford torus S^{m}(\sqrt{\frac{m}{n}})×S^{n−m}(\sqrt{\frac{n−m}{n}}), where

ϵ(3) = \frac{42}{85}, ϵ(4) = \frac{8}{31} and ϵ(5) = \frac{3(21−5\sqrt{17})}{28+3\sqrt{17}}

Remark. For n ≤ 5, Peng-Terng [5] proved that Let M be an n (n ≤ 5)-dimensional closed minimal hypersurface of a unit sphere S^{n+1}(1). If

n ≤ S ≤ n + ϵ_{1}(n),

then S = n, where ϵ_{1}(n) = \frac{6 − 1.13n}{5 + \sqrt{17}}. It is obvious that our pinching constant in

Theorem 2 is larger than their one.

2. Local formulae.

Let M be an n-dimensional hypersurface in a unit sphere S^{n+1}(1). We choose a local orthonormal frame field {e_{1}, …, e_{n+1}} in S^{n+1}(1), restricted to M, so that e_{1}, …, e_{n} are tangent to M. Let ω_{1}, ⋯, ω_{n+1} denote the dual coframe field in S^{n+1}(1). Then, in M,

ω_{n+1} = 0.

It follows from Cartan’s Lemma that

(2.0)  ω_{n+1i} = ∑_{j} h_{ij}ω_{j}, h_{ij} = h_{ji}.

The second fundamental form α and the mean curvature of M are defined by

(2.1)  α = ∑_{i,j} h_{ij}ω_{i}ω_{j}e_{n+1} and nH = ∑_{i} h_{ii},

respectively. We recall that M is by definition a minimal hypersurface if the mean curvature of M is identically zero. The connection form ω_{ij} are characterized by the structure equations

(2.2)  \{ \begin{matrix}dω_{i} + ∑_{j} ω_{ij} ∧ ω_{j} = 0, ω_{ij} + ω_{ji} = 0, \\ dω_{ij} + ∑_{k} ω_{ik} ∧ ω_{kj} = Ω_{ij}, \\ Ω_{ij} = \frac{1}{2} ∑_{k,l} R_{ijkl}ω_{k} ∧ ω_{l}, \end{matrix}

where Ω_{ij} (resp. R_{ijkl}) denotes the curvature form (resp. the components of the curvature tensor) of M. The Gauss equation is given by

(2.3)  R_{ijkl} = (δ_{ik}δ_{jl} − δ_{il}δ_{jk}) + (h_{ik}h_{jl} − h_{il}h_{jk}).

The covariant derivative ∇α of the second fundamental form α of M with components h_{ijk} is given by

∑_{k} h_{ijk}ω_{k} = dh_{ij} + ∑_{k} h_{jk}ω_{ik} + ∑_{k} h_{ik}ω_{jk}.

Then the exterior derivative of (2.0) together with the structure equation yields the following Codazzi equation:

(2.4)  h_{ijk} = h_{ikj} = h_{jik}.

From Codazzi equation, we know that h_{ijk} is symmetric in the indices i, j and k. Similarly, we have the covariant derivative ∇^{2}α of ∇α with components h_{ijkl} as follows

∑_{l} h_{ijkl}ω_{l} = dh_{ijk} + ∑_{l} h_{ljk}ω_{il} + ∑_{l} h_{ilk}ω_{jl} + ∑_{l} h_{ijl}ω_{kl}.

And it is easy to get the following Ricci formula:

(2.5)  h_{ijkl} − h_{ijlk} = ∑_{m} h_{im}R_{mjkl} + ∑_{m} h_{mj}R_{mikl}.

Similarly, we also have

(2.6)  h_{ijklm} − h_{ijkml} = ∑_{r} h_{rjk}R_{rilm} + ∑_{r} h_{irk}R_{rjlm} + ∑_{r} h_{ijr}R_{rklm},

where h_{ijklm}’s are the components of the covariant derivative ∇^{3}α of ∇^{2}α. We should remark that h_{ijkl} and h_{ijklm} are symmetric in the first three indices i, j and k and not symmetric in the other ones generally. The components of the Ricci curvature and the scalar curvature are given by

(2.7)  R_{ij} = (n − 1)δ_{ij} − ∑_{k} h_{ik}h_{jk},

(2.8)  R = n(n − 1) − ∑_{i,j} h_{ij}^{2}.

Now we compute certain local formulae. For any fixed point p in M, we can choose a local orthonormal frame field e_{1}, ⋯, e_{n} such that

(2.9)  h_{ij} = \{ \begin{matrix}0 if i ≠ j \\ λ_{i} if i = j. \end{matrix}

The following formulas can be obtained by a direct computation (cf. [1]). Let

S := ∑_{i,j} h_{ij}^{2} = ∑_{i} λ_{i}^{2}.

(2.10)  \frac{1}{2}ΔS = ∑_{i,j,k} h_{ijk}^{2} − S(S − n).

(2.11)  \frac{1}{2}Δ ∑_{i,j,k} h_{ijk}^{2} = ∑_{i,j,k,l} h_{ijkl}^{2} + (2n + 3 − S) ∑_{i,j,k} h_{ijk}^{2}

+ 3(2B − A) − \frac{3}{2}|∇S|^{2},

where A = ∑_{i,j,k} λ_{i}^{2}h_{ijk}^{2} and B = ∑_{i,j,k} λ_{i}λ_{j}h_{ijk}^{2}.

3. Proofs of Theorems

At first we give two algebra Lemmas which will play a crucial role in the proofs of our theorems.

Lemma 1. Let a_{i} (i = 1, 2, 3, 4) be real number satisfying ∑_{i} a_{i} = 0 and ∑_{i} a_{i}^{2} = a. Then ∑_{i} a_{i}^{4} ≤ \frac{7}{12}a^{2}.

Proof. We maximize the function ∑_{i} a_{i}^{4} with the constraints ∑_{i} a_{i} = 0 and ∑_{i} a_{i}^{2} = a. By means of the method of the Lagrange multiplier, we solve the following problem:

f = ∑_{i} a_{i}^{4} + λ ∑_{i} a_{i} + μ(∑_{i} a_{i}^{2} − a),

where λ and μ are the Lagrange multipliers. The maximum point of ∑_{i} a_{i}^{4} is a critical point of f. Taking derivative of f with respect to a_{i}, we have

f_{a_{i}} = 4a_{i}^{3} + λ + 2μa_{i} = 0.

Hence, at most three of the a_{i}’s are distinct with each other at critical point of f. We consider the following three cases.

(1) Three of the a_{i}’s are distinct with each other. Without loss of generality, we denote them by a_{1}, a_{2}, a_{3} and assume a_{1} = a_{4}, then

2a_{1} + a_{2} + a_{3} = 0, 2a_{1}^{2} + a_{2}^{2} + a_{3}^{2} = a.

Hence,

∑_{i} a_{i}^{4} = 2a_{1}^{4} + a_{2}^{4} + a_{3}^{4} = \frac{(a + 2a_{1}^{2})^{2}}{2} − 14a_{1}^{4}

= \frac{a^{2}}{2} + 2aa_{1}^{2} − 12a_{1}^{4} ≤ \frac{7}{12}a^{2},

i.e.,

∑_{i} a_{i}^{4} ≤ \frac{7}{12}a^{2}.

(2) Two of the a_{i}’s are distinct with each other. Without loss of generality, we denote them by a_{1}, a_{2} and assume a_{1} = a_{4} and a_{2} = a_{3} or a_{1} = a_{3} = a_{4}, then ∑_{i} a_{i}^{4} ≤ \frac{7}{12}a^{2}.

(3) If all of the a_{i}’s are same, then ∑_{i} a_{i}^{4} = 0. Therefore, we conclude

∑_{i} a_{i}^{4} ≤ \frac{7}{12}a^{2}.

This completes the proof of the Lemma 1. Lemma 2. Let a_{ij} and b_{i} (i, j =1, ⋯, n) be real numbers satisfying ∑_{i} b_{i} = 0 and ∑_{i} b_{i}^{2} = b > 0, ∑_{i,j} b_{i}a_{ij} = \frac{1}{2}b(n − b) and ∑_{i,j} b_{j}a_{ij} = \frac{1}{2}b(n − b). Then

∑_{i} a_{ii}^{2} + 3 ∑_{i≠j} a_{ij}^{2} ≥ \frac{3b(n − b)^{2}}{2(n + 4)} .

Proof. We consider F = ∑_{i} a_{ii}^{2} + 3 ∑_{i≠j} a_{ij}^{2} as a function of a_{ij} with constraint ∑_{i,j} b_{i}a_{ij} = \frac{1}{2}b(n − b) and ∑_{i,j} b_{j}a_{ij} = \frac{1}{2}b(n − b). Let

f := ∑_{i} a_{ii}^{2} + 3 ∑_{i≠j} a_{ij}^{2} + λ[∑_{i,j} b_{i}a_{ij} − \frac{1}{2}b(n − b)] + μ[∑_{i,j} b_{j}a_{ij} − \frac{1}{2}b(n − b)],

where λ and μ are the Lagrange multipliers. It is obvious that the minimum point of F is a critical point of f. Taking derivative of f with respect to a_{ij}, we get

(3.1)  f_{a_{ii}} = 2a_{ii} + λb_{i} + μb_{i} = 0, for i,
(3.2)  f_{a_{ij}} = 6a_{ij} + λb_{i} + μb_{j} = 0, for i ≠ j.

Hence

∑_{i} a_{ii}f_{a_{ii}} = 2 ∑_{i} a_{ii}^{2} + λ ∑_{i} a_{ii}b_{i} + μ ∑_{i} a_{ii}b_{i} = 0

and

∑_{i≠j} a_{ij}f_{a_{ij}} = 6 ∑_{i≠j} a_{ij}^{2} + λ ∑_{i≠j} a_{ij}b_{i} + μ ∑_{i≠j} a_{ij}b_{j} = 0.

Therefore,

(3.3)  2[∑_{i} a_{ii}^{2} + 3 ∑_{i≠j} a_{ij}^{2}] = λ\frac{1}{2}b(b − n) + μ\frac{1}{2}b(b − n).

From (3.1) and (3.2), we have

(3.4)  2 ∑_{i} b_{i}a_{ii} + (λ + μ) ∑_{i} b_{i}^{2} = 0,

6 ∑_{i≠j} b_{i}a_{ij} + λ ∑_{i≠j} b_{i}^{2} + μ ∑_{i≠j} b_{i}b_{j} = 0

and

6 ∑_{i≠j} b_{j}a_{ij} + λ ∑_{i≠j} b_{i}b_{j} + μ ∑_{i≠j} b_{j}^{2} = 0.

From (3.4) and these two equalities above, we get

−4 ∑_{i} b_{i}a_{ii} + 3b(n − b) + λnb = 0

and

−4 ∑_{i} b_{i}a_{ii} + 3b(n − b) + μnb = 0.

λ + μ = \frac{6(b − n)}{(n + 4)}.

According to (3.3), we obtain

f_{min} = \frac{3b(n − b)^{2}}{2(n + 4)} .

Thus we have finished the proof of the Lemma 2.

For any fixed point p in M, we can choose a local frame field e_{1}, ⋯, e_{n} such that

(3.5)  h_{ij} = λ_{i}δ_{ij}.

Defining f_{3} = ∑_{i} λ_{i}^{3} and f_{4} = ∑_{i} λ_{i}^{4}, then f_{3} and f_{4} are functions defined globally on M.

Proposition 1. Let M be a minimal hypersurface in S^{n+1}(1). Then

∑_{i,j,k,l} h_{ijkl}^{2} ≥ \frac{3}{2}(Sf_{4} − f_{3}^{2} − 2S^{2} + nS) + \frac{3S(S − n)^{2}}{2(n + 4)}

holds.

Proof. From Ricci formula (2.5) and Gauss equation (2.3), we have

h_{iijj} − h_{jjii} = h_{ijij} − h_{ijji} = ∑_{m} h_{im}R_{mjij} + ∑_{m} h_{mj}R_{miij} (3.6)

= λ_{i}R_{ijij} + λ_{j}R_{jiij} = (λ_{i} − λ_{j})R_{ijij}
= (λ_{i} − λ_{j})(1 + λ_{i}λ_{j}).

We define

(3.7)  u_{ijkl} = \frac{1}{4}(h_{ijkl} + h_{lijk} + h_{klij} + h_{jkli}).

Since h_{ijkl} is symmetric in the indices i, j, k, from formula (3.6), we obtain

(3.8)  ∑_{i,j,k,l} h_{ijkl}^{2} ≥ ∑_{i,j,k,l} u_{ijkl}^{2} + \frac{3}{2}[Sf_{4} − f_{3}^{2} − 2S^{2} + nS].

Since Δh_{ij} = (n − S)h_{ij} and ∑_{i} h_{iikl} = 0, we have

∑_{i,j} u_{iijj}λ_{i} = ∑_{i,j} u_{iijj}λ_{j} = \frac{1}{2}S(n − S).

From ∑_{i} λ_{i} = 0 and ∑_{i} λ_{i}^{2} = S and defining a_{ij} := u_{iijj} and b_{i} := λ_{i}, then a_{ij} and b_{i} satisfy the conditions in the Lemma 2. From the definition of u_{ijkl}, we know that u_{ijkl} is symmetric in the indices i, j, k, l. From Lemma 2, we infer

(3.9)  ∑_{i,j,k,l} u_{ijkl}^{2} ≥ ∑_{i} u_{iiii}^{2} + 3 ∑_{i≠j} u_{iijj}^{2} ≥ \frac{3S(S − n)^{2}}{2(n + 4)} .

Hence, from (3.8) and (3.9), we obtain

∑_{i,j,k,l} h_{ijkl}^{2} ≥ \frac{3}{2}(Sf_{4} − f_{3}^{2} − 2S^{2} + nS) + \frac{3S(S − n)^{2}}{2(n + 4)} .

This completes the proof of the Proposition 1.

Proposition 2. Let M be a closed minimal hypersurface in S^{n+1}(1). Then

∫_{M}[(S − 2n − \frac{3}{2}) ∑_{i,j,k} h_{ijk}^{2} + 2(S − n)f_{4} − \frac{3S(S − n)^{2}}{2(n + 4)} + \frac{9}{8}|∇S|^{2}]dM ≥ 0

holds Proof. The following integral formula (3.10) can be found in [2]:

(3.10)  ∫_{M}(A − 2B)dM = ∫_{M}[Sf_{4} − S^{2} − f_{3}^{2} − \frac{1}{4}|∇S|^{2}]dM.

From Ricci formula (2.5) , by a direct computation, we obtain

\frac{1}{4}Δf_{4} = (n − S)f_{4} + 2A + B.

Integrating both sides of the above equality, we have

(3.11)  ∫_{M}(S − n)f_{4}dM = ∫_{M}(2A + B)dM.

Formulas (3.10) and (3.11) yield

(3.12)  ∫_{M}[(S − 4n)f_{4} + 3f_{3}^{2} + 3S^{2} + \frac{3}{4}|∇S|^{2}]dM ≥ 0.

According to Stokes formula, we integrate the formula (2.11) and obtain

∫_{M} ∑_{i,j,k,l} h_{ijkl}^{2}dM (3.13)

= ∫_{M}[−(2n + 3 − S) ∑_{i,j,k} h_{ijk}^{2} − 3(2B − A) + \frac{3}{2}|∇S|^{2}]dM.

From Proposition 1, (3.10) and (3.13), we infer

(3.14) ∫_{M}\{(S −2n− \frac{3}{2}) ∑_{i,j,k} h_{ijk}^{2} + \frac{3}{4}|∇S|^{2}+ \frac{3}{2}[Sf_{4}−f_{3}^{2}−S^{2}]− \frac{3S(n − S)^{2}}{2(n + 4)} \}dM ≥ 0.

(3.12)+2 × (3.14) yield

∫_{M}[(S − 2n − \frac{3}{2}) ∑_{i,j,k} h_{ijk}^{2} + 2(S − n)f_{4} − \frac{3S(S − n)^{2}}{2(n + 4)} + \frac{9}{8}|∇S|^{2}]dM ≥ 0.

Thus the Proposition 2 is valid.

Proof of Theorem 1. According to (2.10) and Stokes Theorem, we obtain

(3.15)  ∫_{M} ∑_{i,j,k} h_{ijk}^{2}dM = ∫_{M}[S(S − n)]dM,

and

(3.16)  − ∫_{M} \frac{1}{2}|∇S|^{2} = ∫_{M}[S ∑_{i,j,k} h_{ijk}^{2} + (n − S)S^{2}]dM.

From formula (2.7) and the assumption in the Theorem 1, we have

R_{ii} = n − 1 − λ_{i}^{2} ≥ \frac{n}{2}.

Therefore,

λ_{i}^{2} ≤ \frac{n − 2}{2}.

∑_{i} λ_{i}^{4} ≤ \frac{n − 2}{2} ∑_{i} λ_{i}^{2},

that is,

(3.17)  f_{4} ≤ \frac{n − 2}{2}S.

From Proposition 2 and (3.17), we have

∫_{M}\{(S − 2n − \frac{3}{2}) ∑_{i,j,k} h_{ijk}^{2} + (n − 2)S(S − n) − \frac{3S(S − n)^{2}}{2(n + 4)} + \frac{9}{8}|∇S|^{2}\}dM ≥ 0.

From (3.15), (3.16) and the above inequality, we infer

∫_{M}\{(−\frac{5}{4}S − n − \frac{7}{2}) ∑_{i,j,k} h_{ijk}^{2} + [\frac{9}{4}S − \frac{3(S − n)}{2(n + 4)}]S(S − n)\}dM ≥ 0.

Since

n ≤ S ≤ n + \frac{14(n + 4)}{9n + 30},

we have

∫_{M}\{(−\frac{5}{4}S − n − \frac{7}{2}) ∑_{i,j,k} h_{ijk}^{2} + (\frac{9n}{4} + \frac{7}{2})S(S − n)\}dM ≥ 0.

Hence

∫_{M} \frac{5}{4}(S − n) ∑_{i,j,k} h_{ijk}^{2}dM = 0.

Since S and ∑_{i,j,k} h_{ijk}^{2} are continuous functions, we have S = n. Thus, M is

isometric to a Clifford torus S^{m}(\sqrt{\frac{m}{n}}) × S^{n−m}(\sqrt{\frac{n−m}{n}}) (1 < m < n − 1) from the result due to Chern, do Carmo and Kobayashi [3] or Lawson [4] . This completes the proof of the Theorem 1.

Proof of Theorem 2. In the case n = 3, because ∑_{i} λ_{i} = 0, we have f_{4} = ∑_{i} λ_{i}^{4} = \frac{S^{2}}{2} . From Proposition 2, we have

∫_{M}\{(S − 2n − \frac{3}{2}) ∑_{i,j,k} h_{ijk}^{2} + S^{2}(S − n) − \frac{3S(S − n)^{2}}{2(n + 4)} + \frac{9}{8}|∇S|^{2}\}dM ≥ 0.

From (3.16) and the above inequality, we have

∫_{M}\{(−\frac{5}{4}S − 2n − \frac{3}{2}) ∑_{i,j,k} h_{ijk}^{2} + \frac{13}{4}S^{2}(S − n) − \frac{3S(S − n)^{2}}{2(n + 4)} \}dM ≥ 0.

Since

n ≤ S ≤ n + \frac{42}{85},

we have

∫_{M}\{−\frac{5}{4}(S − n) ∑_{i,j,k} h_{ijk}^{2}\}dM ≥ 0.

Hence

∫_{M} \frac{5}{4}(S − n) ∑_{i,j,k} h_{ijk}^{2}dM = 0.

By making use of the same proof as in Theorem 1, we know that the Theorem 2 is true in the case n = 3.

In the case n = 4, from Lemma 1, we have f_{4} ≤ \frac{7}{12}S^{2}. By using this inequality, we obtain, from Proposition 2,

∫_{M}\{(S − 2n − \frac{3}{2}) ∑_{i,j,k} h_{ijk}^{2} + \frac{7}{6}S^{2}(S − n) − \frac{3S(S − n)^{2}}{2(n + 4)} + \frac{9}{8}|∇S|^{2}\}dM ≥ 0.

By the same proof as in the case n = 3, we know that the Theorem 2 is also valid in the case n = 4.

In the case n = 5, from Proposition 1, (3.10) and (3.13), we have

(3.18) ∫_{M}[(S − 2n − \frac{3}{2}) ∑_{i,j,k} h_{ijk}^{2} + \frac{3}{2}(A − 2B) − \frac{3S(S − n)^{2}}{2(n + 4)} + \frac{9}{8}|∇S|^{2}\}dM ≥ 0.

Since

3(A − 2B) = ∑_{i,j,k}(λ_{i}^{2} + λ_{j}^{2} + λ_{k}^{2} − 2λ_{i}λ_{j} − 2λ_{j}λ_{k} − 2λ_{k}λ_{i})h_{ijk}^{2}

= ∑_{i≠j≠k≠i}[2(λ_{i}^{2} + λ_{j}^{2} + λ_{k}^{2}) − (λ_{i} + λ_{j} + λ_{k})^{2}]h_{ijk}^{2}

+ 3 ∑_{i≠k}(λ_{k}^{2} − 4λ_{i}λ_{k})h_{iik}^{2} − 3 ∑_{i} λ_{i}^{2}h_{iii}^{2}

≤ \frac{\sqrt{17} + 1}{2}S ∑_{i,j,k} h_{ijk}^{2},

by making use of this inequality and (3.18), the similar proof as in the case n = 3 yields that the Theorem 2 is also valid in this case. We have finished the proof of the Theorem 2.

Acknowledgement. The first named author wishes to thank professor K. Shiohama for his constant encouragement and help.

References

1. Cheng, Q.M., The classification of complete hypersurfaces with constant mean curvature of space form of dimension 4, Mem. Fac. Sci. Kyushu Univ. 47 (1993), 79-102.

2. Cheng, Q.M., The rigidity of Clifford torus S^{1}(\sqrt{\frac{1}{n}}) × S^{n−1}(\sqrt{\frac{(n−1)}{n}}), Comment. Math. Helvetici 71 (1996), 60-69.

3. Chern, S.S., do Carmo, M. and Kobayashi, S., Minimal submanifolds of a sphere with second fundamental form of constant length, Functional analysis and related fields, 1970, pp. 59-75.

4. Lawson, H.B., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. 89 (1969), 179-185.

5. Peng, C.K. and Terng, C.L., The scalar curvature of minimal hypersurfaces in spheres, Math. Ann. 266 (1983), 105-113.

6. Yang, H.C. and Cheng, Q.M., An estimate of the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit spheres, Manuscripta Math., 84 (1994), 89-100.

7. Yang, H.C. and Cheng, Q.M., Chern’s conjecture on minimal hypersurfaces, Math. Z. 227 (1998), 377-390.

Qing-Ming Cheng Department of Mathematics, Faculty of Science, Josai University, Sakado, Saitama 350-0295, Japan

Susumu Ishikawa Department of Mathematics, Saga University, Saga 840-0027, Japan