MINIMAL HYPERSURFACES IN SPHERE

Qing-Ming Cheng*

Department of Mathematics, Josai University, Saitama, Sakado 350-02, Japan

1. INTRODUCTION

An important problem in Differential Geometry is to investigate minimal hypersurfaces in space forms. The problem involves analysis on manifolds equipped with the moving frame introduced by E.Cartan. A classical result due to Myers in [10] stated that if a closed Riemannian manifold M is minimally immersed into a closed hemi-sphere of the unit sphere, then it is contained entirely in the great sphere. This may be viewed as a sphere version of the Berntein theorem on complete minimal surface into Euclidean 3-space. The idea of the proof is that the distance from every point on M to the boundary of the closed hemi-sphere is a superharmonic function on M and the maximum principle then implies that the distance is identically zero on M. By making use of this idea, an important contribution in the investigation of closed minimal hypersurfaces in the sphere was established by Simons in [14] who derived the linear elliptic second order differential equation satisfied by the second fundamental form of minimal hypersurfaces. He obtained an important estimate of a lower bound for the index and the nullity of a non-totally geodesic minimal hypersurface in sphere and proved the following:

Theorem 1. (Simons [14]). Let M be an n-dimensional closed minimal hypersurface in unit sphere S^{n+1}(1). Then

∫_{M}(S − n)Sdv ≥ 0,

where S is the squared norm of the second fundamental form of M.

It is follows that if S ≤ n, then we have S = 0 (e.g. M is totally geodesic) or else S = n. The complete determination of the latter case was obtained by Chern-do Carmo-Kobayashi in [8] and Lawson in [9], respectively, as follows. For positive

integers m and n with m < n let M_{m,n−m} := S^{m}(\sqrt{\frac{m}{n}}) × S^{n−m}(\sqrt{\frac{n−m}{n}}) be the Clifford torus.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 53C20, 53C42.

Key words and phrases. minimal hypersurface, second fundamental form, scalar curvature.

*Supported by a Grant-in-Aid for Scientific Rersearch from the Japanese Ministry of Education, Science and Culture

Theorem 2. (Chern-do Carmo-Kobayashi [8] and Lawson [9]). The Clifford torus M_{m,n−m} are the only closed minimal hypersurfaces in S^{n+1}(1) satisfying S = n.

Notice that the second fundamental form of M_{m,n−m} has exactly two eigenvalues with multiplicities m and n − m. Conversely, if M is an n-dimensional closed minimal hypersurface in S^{n+1}(1) having two distinct principal curvatures with multiplicities m > 1 and n − m > 1, then Otsuki in [11] proved that it is the Clifford torus M_{m,n−m}. Otsuki also proved that minimal hypersurfaces with two distinct principal curvatures one of which has multiplicity 1 are determined by a nonlinear second order ordinary differential equation. They have the same topological types as S^{n−1} × R (when M is noncompact), or S^{n−1} × S^{1} (when M is closed). There are infinitely many isometrically distinct such minimal hypersurfaces in S^{n+1}(1) on which the scalar curvatures are not constant. Every minimal hypersurface of constant scalar curvature in S^{n+1}(1) has the property that S is constant on it. Chern conjectured as follows:

Chern’s Conjecture. For n-dimensional closed minimal hypersurfaces in the unit sphere S^{n+1}(1) with constant scalar curvature, the values {S} of the squared norm of the second fundamental form should be discrete.

Simons [14], Chern-do Carmo-Kobayashi [8] and Lawson [9] proved that the first and the second value are 0 and n, respectively. Since there are isoparametric hypersurfaces of E.Cartan with S=2n, the following Theorem 3 suggests the third value of S should be 2n.

Theorem 3. (Yang and Cheng [15], [16] and [17]). Let M be a closed minimal hypersurface in S^{n+1}(1) with constant scalar curvature. If S ≤ n + \frac{n}{3}, then S = 0 (e.g. M is totally geodesic) or else S = n (e.g. M is the Clifford torus).

In particular, when n = 3, The Chern’s conjecture was solved in [2]. In fact, Cheng and Wan [7] obtained the following more general result:

Theorem 4.(Cheng and Wan [7]). The totally geodesic sphere S^{3}(1), the Clifford torus M_{1,2} and the tube of the Veronese surface P^{2}R are the only complete minimal hypersurfaces in S^{4}(1) with constant scalar curvature.

By investiating the examples of Otsuki in detail, we find the value of S satisfies that inf S < n and sup S > n. On the other hand, Cheng [5] and Chen [3] proved the following:

Theorem 5. (Cheng [5] and Chen [3]). The Clifford torus M_{m,n−m} are the only closed minimal hypersurfaces in S^{n+1}(1) with two distinct principal curvatures and satisfying S ≥ n.

From these above theorems, the following problem is asked naturally:

Problem. For a closed minimal hypersurface of the unit sphere S^{n+1}(1), whether there exists a constant ϵ(n) > 0 such that if n ≤ S ≤ n + ϵ(n), then S = n and M is the Clifford torus M_{m,n−m}.

Remark 1. From Theorem 3, we have that if the scalar curvature of M is constant, then there exists ϵ(n) = \frac{n}{3}.

Remark 2. From Theorem 5, we have that if the distinct principal curvatures of M are two, then there exists ϵ(n) = ∞.

For the Problem, Cheng and Ishikawa in [6] proved the following:

Theorem 6. (Cheng and Ishikawa [6]). Let M be an n-dimensional closed minimal hypersurface of the 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 the Clifford torus M_{m,n−m}.

Theorem 7. (Cheng and Ishikawa [6]). Let M be an n(n ≤ 5) -dimensional closed minimal hypersurface of the unit sphere S^{n+1}(1). If

n ≤ S ≤ n + ϵ(n)

then S = n and M is the Clifford torus M_{m,n−m}, 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 3. For n ≤ 5, Peng-Terng [13] proved that for an n (n ≤ 5)-dimensional closed minimal hypersurface M 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 7 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 gives 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 some local formulae. For any fixed point p in M, we can choose a local 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 gotten by direct computations (cf. [4]). 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).

\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} (2.11)

+ 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

From (2.10) and Stokes formula, we know that Thorem 1 is true obviously.

The following Lemmas will be used in the proofs of Theorem 6 and Theorem 7, which can be proved by making use of the method of the Lagrange multipliers.

Lemma 1. (Cheng and Ishikawa [6]). Let a_{i} (i = 1, 2, 3, 4) be real number and they satisfy ∑_{i} a_{i} = 0 and ∑_{i} a_{i}^{2} = a. Then ∑_{i} a_{i}^{4} ≤ \frac{7}{12}a^{2}.

Lemma 2. (Cheng and Ishikawa [6]). 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)} .

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

(3.1)  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.

Proposition 1. (Cheng and Ishikawa [6]). 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)} .

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

h_{iijj} − h_{jjii} = (λ_{i} − λ_{j})(1 + λ_{i}λ_{j}). (3.2)

We define

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

Since h_{ijkl} is symmetric for indices i, j, k, from formula (3.2), we get

(3.4)  ∑_{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 get

∑_{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 Lemma 2. From the definition of u_{ijkl}, we know that u_{ijkl} is symmetric for indices i, j, k, l. From Lemma 2, we infer

(3.5)  ∑_{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 (3.4) and (3.5), we know that Proposition 1 is true.

Proposition 2. (Cheng and Ishikawa [6]). 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.

Proof. The following integral formula (3.6) can be proved, by a direct computation,

(3.6)  ∫_{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

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

Formulas (3.6) and (3.7) yield

(3.8)  ∫_{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 get

∫_{M} ∑_{i,j,k,l} h_{ijkl}^{2}dM (3.9)

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

From Proposition 1, (3.6) and (3.9), we infer

(3.10) ∫_{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.

(3.8)+2 × (3.10) 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 Proposition 2 is valid. Proof of Theorem 6. According to (2.10) and Stokes Theorem, we obtain

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

and

(3.12)  − ∫_{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 Theorem 6, we have

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

Therefore

(3.13)  f_{4} ≤ \frac{n − 2}{2}S.

From Proposition 2 and (3.13), 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.

By (3.11), (3.12) and the above inequality, we get

∫_{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 from the

assumption of Theorem , M is a Clifford torus S^{m}(\sqrt{\frac{m}{n}})×S^{n−m}(\sqrt{\frac{n−m}{n}}) according to a result due to Chern, do Carmo and Kobayashi [8] or Lawson [9] . We complete the proof of Theorem 6.

Proof of Theorem 7. 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 (3.12) 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.

By making use of the same proof of Theorem 6, we know Theorem 7 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 get, 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 get that Theorem 7 is valid in the case n = 4.

In the case n = 5, from Proposition 1, (3.6) and (3.9), we have

(3.14) ∫_{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 (3.14), the similar proof as in the case n = 3 yields Theorem 7 is also valid in this case. We complete the proof of Theorem 7.

In order to prove Theorem 3, the following Lemma 3 will be used

Lemma 3. (see Yang and Cheng [16] and [17]). For any fixed c ∈ R, there exists a point x_{c} ∈ M^{n} such that at the point x_{c},

(3.15)  \begin{matrix}(S − n) [c (∑_{i} λ_{i}^{3})^{2} − S ∑_{i} λ_{i}^{4}] = \\ = & c (∑_{i,j,k} 2λ_{i}h_{ijk}^{2}) ∑_{i} λ_{i}^{3} − (2A + B)S + 3c ∑_{j} (∑_{i} λ_{i}^{2}h_{iij})^{2} , \end{matrix}

(3.16)  f ≥ \frac{(λ_{1} − λ_{2})^{2}}{λ_{1}^{2} + λ_{2}^{2}} (\frac{S}{n} + λ_{1}λ_{2})^{2} ,

(3.17)  A − B ≤ \frac{1}{3}(λ_{1} − λ_{2})^{2}tS^{2}(1 − α)

where the nonnegative constant α satisfies α < 1 and αtS^{2} = ∑_{i} h_{iii}^{2} and

(3.18)  S − n = tS,

(3.19)  ∑_{k} (∑_{i} λ_{i}^{2}h_{iik})^{2} ≤ \frac{1 + 2α}{3}tS^{2}f,

where f = f_{0} − t^{2}S^{2}/n,

(3.20) (∑_{i,j,k} 2λ_{i}h_{ijk}^{2})^{2} ≤ [\frac{4}{3}(A + 2B) − \frac{16}{3} ∑_{k} \frac{1}{S + 2λ_{k}^{2}} (∑_{i} λ_{i}^{2}h_{iik})^{2}] tS^{2}.

The outline of proof of Theorem 3. In order to prove the Theorem 3, we only need to prove that if S > n, then S > n + \frac{n}{3}.

First of all, we give another estimate of ∑_{i,j,k,l} u_{ijkl}^{2} for the proof of Theorem 3. Define

(3.21)  f_{ij} = ∑_{p} h_{ip}h_{pj} − xh_{ij} − \frac{S}{n} δ_{ij}, i, j = 1, …, n.

where

x = \frac{1}{S} ∑_{i} λ_{i}^{3}, δ_{ij} = \{ \begin{matrix}1, if i = j, \\ 0, if i ≠ j. \end{matrix}

It is easy to see that

\begin{matrix}∑_{i,j} h_{ij}f_{ij} = ∑_{i} λ_{i} (λ_{i}^{2} − xλ_{i} − \frac{S}{n} ) = 0, \\ ∑_{i,j} δ_{ij}f_{ij} = ∑_{i} (λ_{i}^{2} − xλ_{i} − \frac{S}{n} ) = 0, \end{matrix}

(3.22)  ∑_{i,j} f_{ij}^{2} = ∑_{i} λ_{i}^{4} − \frac{1}{S} (∑_{i} λ_{i}^{3})^{2} − \frac{S^{2}}{n} = f_{0} − \frac{t^{2}S^{2}}{n} ≡ f.

Since ∑_{i,j} h_{ij}^{2} = ∑_{i} λ_{i}^{2} = S, ∑_{i,j,k} h_{ijk}^{2} = tS^{2} , we have

(3.23)  ∑_{i,j} h_{ij}h_{ijkl} + ∑_{i,j} h_{ijk}h_{ijl} = 0,  ∑_{i,j,k} h_{ijk}h_{ijkl} = 0.

It follows from (3.23) that

(3.24)  \begin{matrix}∑_{i,j,k,l} h_{ijkl}h_{ij}h_{kl} = ∑_{i,j} λ_{i}h_{iijj}λ_{j} = − ∑_{i,j,k} λ_{i}h_{ijk}^{2}. \\ ∑_{i,j,k,l,p} h_{ijkl}h_{ij}h_{kp}h_{pl} = ∑_{i,j} λ_{i}h_{iijj}λ_{j}^{2} = − ∑_{i,j,k} λ_{i}^{2}h_{ijk}^{2} = −A. \end{matrix}

By (3.2), we also have

(3.25)  \begin{matrix}∑_{i,j} λ_{i}^{2}h_{iijj}λ_{j} = −A + ∑_{i,j} λ_{i}^{2}λ_{j}(λ_{i} − λ_{j})(1 + λ_{i}λ_{j}) \\ = −A + S ∑_{i} λ_{i}^{4} − (∑_{i} λ_{i}^{3})^{2} − S^{2} = −A + Sf + \frac{S^{2}t}{1 − t}. \end{matrix}

And then, by the symmetry of u_{ijkl},

(3.26)  \begin{matrix}∑_{i,j,k,l} u_{ijkl}h_{ij}f_{kl} = −A + \frac{1}{2}Sf + \frac{S^{2}t}{1 − t} + \frac{1}{S} ∑_{i,j,k} λ_{i}h_{ijk}^{2} ∑_{i} λ_{i}^{3} =: F, \\ ∑_{i,j,k} u_{ijkl}h_{ijk} = −\frac{3}{2} ∑_{i} λ_{i}^{2}h_{iil}λ_{l},  ∑_{i,j,k} f_{ij}h_{kl}h_{ijk} = ∑_{i,j} λ_{i}^{2}h_{iij}h_{jl}, \\ ∑_{i,j,k,l} u_{ijkl}h_{ij}δ_{kl} = −\frac{1}{2}S^{2}t. \end{matrix}

From (3.24), (3.25) and (3.26), we can get

\begin{matrix}∑_{i,j,k,l} u_{ijkl}^{2} ≥ \frac{1}{2}Sf + \frac{4}{tS^{2}} ∑_{i} λ_{i}^{2} (∑_{j} λ_{j}^{2}h_{jji})^{2} − 2A \\ + \frac{1}{S} ∑_{i} λ_{i}^{3} ∑_{i,j,k} 2λ_{i}h_{ijk}^{2} + \frac{1}{1 − t}t (2 + \frac{1}{2}t) S^{2} + \frac{1}{S^{2}} (∑_{i,j,k} λ_{i}h_{ijk}^{2})^{2} . \end{matrix}

Then (2.11) and (3.4) give

(3.27)  \begin{matrix}2Sf + \frac{1}{S} ∑_{i} λ_{i}^{3} ∑_{i,j,k} 2λ_{i}h_{ijk}^{2} + \frac{1}{4S^{2}} (∑_{i,j,k} 2λ_{i}h_{ijk}^{2})^{2} \\ + \frac{4}{tS^{2}} ∑_{i} λ_{i}^{2} (∑_{j} λ_{j}^{2}h_{jji})^{2} \\ ≤ tS^{2} (2tS − S − 5 − \frac{4t}{1 − t}) + 5A − 6B. \end{matrix}

Now we apply Lemma 3 to the case that c = 1 + ^{\frac{22+3t+\sqrt{(22+3t)^{2}−400}}{26}} ≡ 1 + c_{0}. It is easy to see that c_{0} satisfies the following equation:

(3.28)  \frac{1}{3c_{0}} (1 + c_{0} − \frac{3}{13})^{2} = \frac{14 + t}{13}.

By Lemma 3, there is a point x_{c} ∈ M such that at x_{c},

− tS ∑_{i} λ_{i}^{4} + t(1 + c_{0}) (∑_{i} λ_{i}^{3})^{2}

= \frac{1 + c_{0}}{S} ∑_{i} λ_{i}^{3} ∑_{i,j,k} 2λ_{i}h_{ijk}^{2} − (2A + B)

+ \frac{3(1 + c_{0})}{S} ∑_{j} (∑_{i} λ_{i}^{2}h_{iij})^{2} .

From now on we only consider the formulas and the estimate at the point x_{c}. We define

G ≡ −\frac{13}{3}tSf − \frac{1}{S} ∑_{i} λ_{i}^{3} ∑_{i,j,k} 2λ_{i}h_{ijk}^{2} (3.29)

− \frac{1}{4S^{2}} (∑_{i,j,k} 2λ_{i}h_{ijk}^{2})^{2} − \frac{4}{tS^{2}} ∑_{i} λ_{i}^{2} (∑_{j} λ_{j}^{2}h_{jji})^{2} .

Then from (3.27), (3.29) and Lemma 3, we get

G ≤ −4A + 5B − \frac{1}{S} [8\sqrt{\frac{7}{3t}} − \frac{2}{t} − 13(1 + c_{0})] ∑_{i} (∑_{j} λ_{j}^{2}h_{jji})^{2} + \frac{13t}{3(1 − t)}S^{2}.

Denote

(3.30)  \begin{matrix}g(t) ≡ \frac{1}{13t}(8\sqrt{21t} − 6) − 3(1 + c_{0}) \\ = \frac{1}{13t}(8\sqrt{21t} − 6) − 3 \{1 + \frac{1}{26} [22 + 3t − \sqrt{(22 + 3t)^{2} − 400}]\} . \end{matrix}

Since \frac{dg(t)}{dt} = 0 has only three real roots: t_{1} = 0.101786, t_{2} = 0.109566 and t_{3} = 1.61358. Hence g(t) is a decreasing function when \frac{1}{9} < t < \frac{1}{4} because \frac{dg(t)}{dt} < 0. Hence g(t) ≥ g(\frac{1}{4}) when t ∈ (\frac{1}{9}, \frac{1}{4}]. S − n ≥ \frac{1}{8}n was proved in [16], i.e. t ≥ \frac{1}{9}. On the other hand, if Theorem 3 is not true, we can assume that S − n ≤ \frac{1}{3}n, or t ≤ \frac{1}{4}. Therefore, we can assume that t ∈ [\frac{1}{9}, \frac{1}{4}]. Then, by a simple calculation, we have

(3.31)  g(t) ≥ g (\frac{1}{4}) ≥ −\frac{3}{5}.

By Lemma 3, combining (3.31) with (3.29) of the expression of G, we have

\begin{matrix}− \frac{13}{3} [1 + \frac{1}{5}(1 + 2α)] tSf − \frac{1}{S} ∑_{i} λ_{i}^{3} ∑_{i,j,k} 2λ_{i}h_{ijk}^{2} − \frac{1}{4S^{2}} (∑_{i,j,k} 2λ_{i}h_{ijk}^{2})^{2} \\ − \frac{4}{tS^{2}} ∑_{i} λ_{i}^{2} (∑_{j} λ_{j}^{2}h_{jji})^{2} ≤ −4A + 5B + \frac{13}{3(1 − t)}tS^{2}, \end{matrix}

or

− \frac{13}{10} (1 + \frac{1}{3}α) Sf − \frac{1}{S} ∑_{i} λ_{i}^{3} ∑_{i,j,k} 2λ_{i}h_{ijk}^{2}

− \frac{4}{tS^{2}} ∑_{i} λ_{i}^{2} (∑_{j} λ_{j}^{2}h_{jji})^{2} − \frac{1}{4S^{2}} (∑_{i,j,k} 2λ_{i}h_{ijk}^{2})^{2}

≤ −4A + 5B + \frac{13tS^{2}}{3(1 − t)}.

Then

(3.32)  \begin{matrix}− \frac{13}{10}Sf − \frac{1}{S} ∑_{i} λ_{i}^{3} ∑_{i,j,k} 2λ_{i}h_{ijk}^{2} − \frac{1}{4S^{2}} (∑_{i,j,k} 2λ_{i}h_{ijk}^{2})^{2} \\ − \frac{4}{tS^{2}} ∑_{i} λ_{i}^{2} (∑_{j} λ_{j}^{2}h_{jji})^{2} \\ ≤ \frac{1}{1 + (1/3)α} [−\frac{13}{3}(A − B) + \frac{1}{3}(A + 2B)] + \frac{13tS^{2}}{3(1 − t)}. \end{matrix}

Since A + 2B = \frac{1}{3} ∑_{i,j,k}(λ_{i} + λ_{j} + λ_{k})^{2}h_{ijk}^{2} ≥ 0, adding (3.32) up to (3.27), we get

(3.33) 0 ≤ −\frac{7}{10}Sf + \frac{1}{3} [16 − \frac{13}{1 + (1/3)α}] (A−B)+[2tS − S − \frac{2}{3} + \frac{t}{3(1 − t)}] tS^{2}.

It follows from Lemma 3 that

(3.34)  \begin{matrix}\frac{1}{3} [16 − \frac{13}{1 + (1/3)α}] (A − B) ≤ \\ ≤ \frac{1}{9} [16 − \frac{13}{1 + (1/3)α}] (1 − α)(λ_{1} − λ_{2})^{2}tS^{2} \\ = \frac{1}{3} (16 − \frac{13}{y}) (\frac{4}{3} − y) (λ_{1} − λ_{2})^{2}tS^{2} (where y = 1 + (1/3)α) \\ = \frac{1}{3} (\frac{64}{3} + 13 − \frac{52}{3y} − 16y) (λ_{1} − λ_{2})^{2}tS^{2} \\ ≤ \frac{7}{20}(λ_{1} − λ_{2})^{2}tS^{2}. \end{matrix}

Then by Lemma 3, (3.33) and (3.34), we have

0 ≤tS^{2} (2tS − S − \frac{2}{3} + \frac{1}{9}) + \frac{7}{20}(λ_{1}^{2} + λ_{2}^{2} − 2λ_{1}λ_{2})tS^{2} − \frac{7}{10} (λ_{1}λ_{2} + \frac{1}{1 − t})^{2} S

≤tS^{2} [2tS − S + \frac{7}{20}S − \frac{5}{9} + \frac{7}{10(1 − t)}] + \frac{7}{10}(zt − z^{2})S^{3}

(where zS = −λ_{1}λ_{2} − \frac{1}{1 − t})

≤tS^{2} (2tS + \frac{7}{40}tS − S + \frac{7}{20}S − \frac{5}{9} + \frac{42}{45}) = tS^{2} (\frac{87}{40}tS − \frac{13}{20}S + \frac{17}{45}) .

Therefore,

t > \frac{26}{87} − \frac{16}{87S} ,

or

S − n > \frac{26}{87}S − \frac{16}{87}.

Finally, we get

S − n > \frac{26}{61}n − \frac{16}{61} ≥ \frac{1}{3}n, (when n ≥ 4)

which is a contradiction to the assumption that S − n ≤ \frac{1}{3}n. Theorem 3 is proved.

References

1. Cartan, E., Sur des familles remarquables d’hypersurfaces isoparametriques dans les espaces spheriques, Math. Z., 45 (1939), 335-367.

2. Chang, S.P., On minimal hypersurfaces with constant scalar curvatures in S^{4}, J. Diff. Geom. 37 (1993), 523-534.

3. Chen, Q., Rigidity and non-rigidity of minimal hypersurfaces in a unit sphere with two distinct principal curvatures, preprint.

4. 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.

5. 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.

6. Cheng, Q.M. and Ishikawa, S., A characterization of the Clifford torus, preprint 1996.

7. Cheng, Q.M. and Wan, Q.R., Hypersurfaces of space forms M^{4}(c) with constant mean curvature, Geometry and Global Analysis (Sendai 1993), Tohoku Univ., Sendai, Japan 1993, 437-442.

8. Chern, S.S. , do Carmo, M. and Kobayashi, S., Minimal submanifolds of a sphere with second fundamental form of constant length, Functioal Analysis and Related Fields (1970), Springer-Verlag, Berlin, pp. 59-75.

9. Lawson, H.B. Jr., Local rigidity theorems for minimal hypersurfaces, Ann. Math. 89, 167-179(1969).

10. Myers, S.B., Curvature of closed hypersurfaces and nonexistence of closed minimal hypersurfaces, Trans. Amer. Math. Soc. 91 (1951), 211-217.

11. Otsuki, T., Minimal hypersurfaces in a Riemannian manifold of constant curvatures, Amer. J. Math. 92 (1970), 145-173.

12. Peng, C.K. and Terng, C.L., Minimal hypersurfaces of spheres with constant scalar curvature, in “Seminar on minimal submanifolds” (1983), Princeton Univ. Press, Princeton, pp. 179-198.

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

14. Simons, J., Minimal varieties in Riemannian manifolds, Ann. Math. 88, 62-105(1968).

15. Yang, H.C. and Cheng, Q.M., A note on the pingching constant of minimal hypersurfaces with constant scalar curvature in the unit sphere, Chinese Science Bull. 36, 1-6(1991).

16. 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.

17. Yang, H.C. and Cheng, Q.M., Chern’s conjecture on minimal hypersurfaces, to appear in Math. Z..

18. Yau, S.T., Submanifolds with constant mean curvature II, Amer. J. Math. 97, 76-100(1975).

19. Yau, S.T., Seminar on Diff. Geom., Princeton Univ. Press, Princeton, 1982, p. 699-706.