PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 67(81) (2000), 112–120

RIGIDITY THEOREMS OF HYPERSURFACES IN A SPHERE

Li Haizhong

Communicated by Mileva Prvanović

Abstract. By the study of Cheng-Yau’s self-adjoint operator □, we prove two rigidity theorems for a class of n-dimensional hypersurfaces in the (n + 1)- dimensional unit sphere S^{n+1}.

1. Introduction and Theorems

Let S^{n+1} be an (n + 1)-dimensional unit sphere with constant sectional curvature 1, let M be an n-dimensional compact hypersurface in S^{n+1}, and e_{1}, …, e_{n} a local orthonormal frame field on M, ω_{1}, …, ω_{n} its dual coframe field. Then the second fundamental form of M is

B = ∑_{i,j} h_{ij}ω_{i} ⊗ ω_{j}.

Further, near any given point p ∈ M, we can choose a local frame field e_{1}, …, e_{n} so that at p, ∑_{ij} h_{ij}ω_{i} ⊗ ω_{j}= ∑_{i} k_{i}ω_{i} ⊗ ω_{i}, then the Gauss equations say

R_{ijij} = 1 + k_{i}k_{j},  i ≠ j,  (1.1)
n(n − 1)(R − 1) = n^{2}H^{2} − |B|^{2},  (1.2)

where R is the normalized scalar curvature, H = \frac{1}{n} ∑_{i} k_{i} is the mean curvature and

|B|^{2} = ∑_{i} k_{i}^{2} the norm square of the second fundamental form of M.

As it is well known, there are many rigidity results for minimal hypersurfaces or hypersurfaces with constant mean curvature H in S^{n+1} by use of J. Simons’ method, for example, see [1], [4], [6], [7] etc. In [2], Cheng and Yau introduced a self-adjoint operator □ they proved some rigidity theorems for n-dimensional hypersurfaces with constant scalar curvature in an (n + 1)-dimensional unit sphere S^{n+1}. In [3], the author also established some rigidity results by the study of Cheng-Yau’s operator and some new estimates. In this paper, we will prove the following results

1991 Mathematics Subject Classification. Primary 53C42; Secondary 53A10.

Supported by the NNSFC No˙19701017 and Basic Science Research Foundation of Tsinghua University.

Theorem 0.1. Let M be an n-dimensional (n ≥ 3) compact hypersurface in an (n + 1)-dimensional unit sphere S^{n+1}. If

|∇B|^{2} ≥ n^{2}|∇H|^{2},  (1.3)

and

0 ≤ |B|^{2} ≤ 2\sqrt{n − 1},  (1.4)

then either |B|^{2} ≡ 0 and M is a totally umbilical hypersurface; or |B|^{2} ≡ 2\sqrt{n − 1} and M = S^{1}(r_{1}) × S^{n−1}(r_{2}), where

r_{1}^{2} = \frac{1}{1 + \sqrt{n − 1}}, r_{2}^{2} = \frac{\sqrt{n − 1}}{1 + \sqrt{n − 1}}.  (1.5)

Corollary 0.1. Let M be an n-dimensional (n ≥ 2) compact hypersurface with constant mean curvature in an (n + 1)-dimensional unit sphere S^{n+1}. If (1.4) holds, then either |B|^{2} ≡ 0 and M is a totally umbilical hypersurface; or |B|^{2} ≡ 2\sqrt{n − 1} and M = S^{1}(r_{1}) × S^{n−1}(r_{2}), where r_{1} and r_{2} are defined by (1.5).

Corollary 0.2. Let M be an n-dimensional (n ≥ 2) compact hypersurface with constant normalized scalar curvature R in an (n + 1)-dimensional unit sphere S^{n+1}. If (1.4) holds and R ≥ 1, then M is either |B|^{2} ≡ 0 and is a totally umbilical hypersurface; or |B|^{2} ≡ 2\sqrt{n − 1} and M = S^{1}(r_{1}) × S^{n−1}(r_{2}), where r_{1} and r_{2} are defined by (1.5).

Corollary 0.3. Let M be an n-dimensional (n ≥ 2) compact hypersurface in an (n + 1)-dimensional unit sphere S^{n+1}. Suppose that the normalized scalar curvature R is proportional to the mean curvature H of M, that is, there exists a constant a satisfying

R = aH,  a^{2} > 4n/(n − 1).  (1.6)

If (1.4) holds, then either |B|^{2} ≡ 0 and M is a totally umbilical hypersurface; or |B|^{2} ≡ 2\sqrt{n − 1} and M = S^{1}(r_{1}) × S^{n−1}(r_{2}), where r_{1} and r_{2} are defined by (1.5).

Theorem 0.2. Let M be an n-dimensional (n ≥ 4) compact hypersurface in an (n + 1)-dimensional unit sphere S^{n+1}. If

|∇B|^{2} ≥ n^{2}|∇H|^{2},

and

Ric(M) ≥ n − 2,  (1.7)

then either Ric(M) = n−1 and M is a totally umbilical hypersurface; or Ric(M) = n − 2 and M = S^{m}(r_{1}) × S^{n−m}(r_{2}) for some m with 1 ≤ m ≤ n − 1, where

r_{1}^{2} = \frac{m − 1}{n}, r_{2}^{2} = \frac{n − m − 1}{n}.  (1.8)

Corollary 0.4. Let M be an n-dimensional (n ≥ 4) compact hypersurface with constant mean curvature in an (n + 1)-dimensional unit sphere S^{n+1}. If (1.7) holds, then either Ric(M) = n − 1 and M is a totally umbilical hypersurface; or Ric(M) = n − 2 and M = S^{m}(r_{1}) × S^{n−m}(r_{2}) for some m with 1 ≤ m ≤ n − 1, where r_{1} and r_{2} are defined by (1.8).

Corollary 0.5. Let M be an n-dimensional (n ≥ 4) compact hypersurface with constant normalized scalar curvature R in an (n + 1)-dimensional unit sphere S^{n+1}. If (1.7) holds and R ≥ 1, then either Ric(M) = n − 1 and M is a totally umbilical hypersurface; or Ric(M) = n − 2 and M = S^{m}(r_{1}) × S^{n−m}(r_{2}) for some m with 1 ≤ m ≤ n − 1, where r_{1} and r_{2} are defined by (1.8).

Corollary 0.6. Let M be an n-dimensional (n ≥ 4) compact hypersurface in (n+1)-dimensional unit sphere S^{n+1}. Suppose that the normalized scalar curvature R is proportional to the mean curvature H of M, that is, there exists a constant a satisfying

R = aH, a^{2} > 4n/(n − 1).

If (1.7) holds, then either Ric(M) = n−1 and M is a totally umbilical hypersurface; or Ric(M) = n − 2 and M = S^{m}(r_{1}) × S^{n−m}(r_{2}) for some m with 1 ≤ m ≤ n − 1, where r_{1} and r_{2} are defined by (1.8).

2. Preliminaries

Let M be an n-dimensional compact hypersurface in an (n + 1)-dimensional unit sphere S^{n+1}. For any p ∈ M, we choose a local orthonormal frame e_{1}, …, e_{n}, e_{n+1} in S^{n+1} around p, so that e_{1}, …, e_{n} are tangent to M. Take the corresponding dual coframe {ω_{1}, …, ω_{n}, ω_{n+1}}. In this paper, we make the following convention on the range of indices:

1 ≤ A, B, C ≤ n + 1; 1 ≤ i, j, k ≤ n.

The structure equations of S^{n+1} are

dω_{A} = ∑_{B} ω_{AB} ∧ ω_{B}, ω_{AB} = −ω_{BA},

dω_{AB} = ∑_{C} ω_{AC} ∧ ω_{CB} − ω_{A} ∧ ω_{B}.

If we denote by the same letters the restrictions of ω_{A}, ω_{AB} to M, then we have

dω_{i} = ∑_{j} ω_{ij} ∧ ω_{j},  ω_{ij} = −ω_{ji},  (2.1)

dω_{ij} = ∑_{k} ω_{ik} ∧ ω_{kj} − \frac{1}{2} ∑_{k,l} R_{ijkl}ω_{k} ∧ ω_{l},  (2.2)

where R_{ijkl} is the curvature tensor of the induced metric on M.

Restricted to M, we have ω_{n+1} = 0, thus

0 = dω_{n+1} = ∑_{i} ω_{n+1i} ∧ ω_{i},  (2.3)

and from Cartan’s lemma we can write

ω_{in+1} = ∑_{j} h_{ij}ω_{j}, h_{ij} = h_{ji}.

The quadratic form B = ∑_{i,j} h_{ij}ω_{i} ⊗ ω_{j} is the second fundamental form of M. The Gauss equations are

R_{ijkl} = (δ_{ik}δ_{jl} − δ_{il}δ_{jk}) + h_{ik}h_{jl} − h_{il}h_{jk},  (2.4)

R_{ii} = n − 1 + nHh_{ii} − ∑_{k} h_{ik}h_{ki}

n(n − 1)(R − 1) = n^{2}H^{2} − |B|^{2},  (2.5)

where R is the normalized scalar curvature, H = \frac{1}{n} ∑_{i} h_{ii} the mean curvature and |B|^{2} = ∑_{i,j} h_{ij}^{2} the norm square of the second fundamental form of M, respectively.

The Codazzi equation is

h_{ijk} = h_{ikj},  (2.6)

where the covariant derivative of the second fundamental form is defined by

∑_{k} h_{ijk}ω_{k} = dh_{ij} + ∑_{k} h_{kj}ω_{ki} + ∑_{k} h_{ik}ω_{kj}.  (2.7)

The second covariant derivative of h_{ij} is defined by

∑_{l} h_{ijkl}ω_{l} = dh_{ijk} + ∑_{m} h_{mjk}ω_{mi} + ∑_{m} h_{imk}ω_{mj} + ∑_{m} h_{ijm}ω_{mk}.  (2.8)

By exterior differentiation of (2.7), we can see that the following Ricci identities hold

h_{ijkl} − h_{ijlk} = ∑_{m} h_{mj}R_{mikl} + ∑_{m} h_{im}R_{mjkl}.  (2.9)

For a C^{2}-function f defined on M, the gradient and the Hessian (f_{ij}) are defined by

df = ∑_{i} f_{i}ω_{i},  ∑_{j} f_{ij}ω_{j} = df_{i} + ∑_{j} f_{j}ω_{ji}.  (2.10)

The Laplacian of f is defined by Δf = ∑_{i} f_{ii}.

Let T = ∑_{ij} T_{ij}ω_{i} ⊗ ω_{j} be a symmetric tensor defined on M, where
T_{ij} = nHδ_{ij} − h_{ij}.  (2.11)

Following Cheng-Yau [2], we introduce an operator □ associated to T acting on any C^{2}-function f by

□f = ∑_{i,j} T_{ij}f_{ij} = ∑_{i,j}(nHδ_{ij} − h_{ij})f_{ij},  (2.12)

since T_{ij} is divergence-free, it follows [2] that the operator □ is self-adjoint relative to the L^{2} inner product of M, i.e.,

∫_{M} f□g = ∫_{M} g□f.  (2.13)

Near a given point p ∈ M, we choose an orthonormal frame field {e_{1}, …, e_{n}} and their dual frame field {ω_{1}, …, ω_{n}}, so that h_{ij} = k_{i}δ_{ij} at p, we have the following computation by use of (2.12) and (2.5)

\begin{aligned}□(nH) & = nHΔ(nH) − ∑_{i} k_{i}(nH)_{ii} \\ & = \frac{1}{2}Δ(nH)^{2} − ∑_{i}(nH)_{i}^{2} − ∑_{i} k_{i}(nH)_{ii} \\ & = \frac{1}{2}n(n − 1)ΔR + \frac{1}{2}Δ|B|^{2} − n^{2}|∇H|^{2} − ∑_{i} k_{i}(nH)_{ii}. \end{aligned}  (2.14)

On the other hand, we have through a standard calculation by use of (2.6) and (2.9) (also see (2.8) of [2])

\frac{1}{2}Δ|B|^{2} = ∑_{i,j,k} h_{ijk}^{2} + ∑_{i} k_{i}(nH)_{ii} + \frac{1}{2} ∑_{i,j} R_{ijij}(k_{i} − k_{j})^{2}.  (2.15)

Putting (2.15) into (2.14), we have

□(nH) = \frac{1}{2}n(n − 1)ΔR + |∇B|^{2} − n^{2}|∇H|^{2} + \frac{1}{2} ∑_{i,j} R_{ijij}(k_{i} − k_{j})^{2}. (2.16)

Now we assume that M is compact (without boundary) and we obtain the following key formula by integrating (2.16) and by noting ∫_{M} ΔR dv = 0 and ∫_{M} □(nH) dv = 0

0 = ∫_{M}[|∇B|^{2} − n^{2}|∇H|^{2} + \frac{1}{2} ∑_{i,j} R_{ijij}(k_{i} − k_{j})^{2}] dv.  (2.17)

3. An algebraic Lemma

From (2.4), we have R_{ijij} = 1 + k_{i}k_{j}, i ≠ j, and by putting this into (2.17), we obtain

0 = ∫_{M}[|∇B|^{2} − n^{2}|∇H|^{2} + n|B|^{2} − n^{2}H^{2} − |B|^{4} + +nH ∑_{i} k_{i}^{3}]dv. (3.1)

Let μ_{i} = k_{i} − H and |Z|^{2} = ∑_{i} μ_{i}^{2}, we have

∑_{i} μ_{i} = 0,  |Z|^{2} = |B|^{2} − nH^{2},  (3.2)

∑_{i} k_{i}^{3} = ∑_{i} μ_{i}^{3} + 3H|Z|^{2} + nH^{3}.  (3.3)

From (3.1)–(3.3), we get

0 = ∫_{M}[|∇B|^{2} − n^{2}|∇H|^{2} + |Z|^{2}(n + nH^{2} − |Z|^{2}) + nH ∑_{i} μ_{i}^{3}]dv.  (3.4)

We need the following algebraic lemma due to Okumura (see [5])

Lemma 0.1. [5]. With the same notations as above, for n ≥ 3, we have

−\frac{n − 2}{\sqrt{n(n − 1)}}|Z|^{3} ≤ ∑_{i} μ_{i}^{3} ≤ \frac{n − 2}{\sqrt{n(n − 1)}}|Z|^{3},  (3.5)

and equality holds in (3.5) if and only if at least (n − 1) of the μ_{i} are equal.

Proof. We can get Lemma 3.1 by using the method of Lagrange’s multipliers to find the critical points of ∑_{i} μ_{i}^{3} subject to the conditions: ∑_{i} μ_{i} = 0, ∑_{i} μ_{i}^{2} = |Z|^{2}. We omit it here. □

Combining (3.4) with (3.5), we obtain

0 ≥ ∫_{M}[|∇B|^{2} − n^{2}|∇H|^{2} + |Z|^{2}(n + nH^{2} − |Z|^{2} − \frac{n(n − 2)}{\sqrt{n(n − 1)}}|H||Z|)] dv. (3.6)

4. Proof of Theorem 1

By a well-known inequality, we have for an arbitrary real number a > 0

2|H||Z| ≤ aH^{2} + \frac{1}{a}|Z|^{2}.  (4.1)

Combining (4.1) with (3.6), we get

\begin{aligned}0 ≥ ∫_{M} \{|∇B|^{2} − n^{2}|∇H|^{2} + |Z|^{2}[n & + nH^{2}(2 − \frac{(n − 2)a}{2\sqrt{n(n − 1)}} + \frac{n(n − 2)}{2\sqrt{n(n − 1)}a}) \\ & − |B|^{2}(1 + \frac{n(n − 2)}{2\sqrt{n(n − 1)}a})]\}dv. \end{aligned}

(4.2)

Now, we choose a satisfying the following equation

2 − \frac{(n − 2)a}{2\sqrt{n(n − 1)}} + \frac{n(n − 2)}{2\sqrt{n(n − 1)}a} = 0,

that is,

a = \frac{n + 2\sqrt{n − 1}}{n − 2}\sqrt{n}.  (4.3)

Substituting (4.3) into (4.2), we obtain

0 ≥ ∫_{M}\{|∇B|^{2} − n^{2}|∇H|^{2} + |Z|^{2}[n − \frac{n}{2\sqrt{n − 1}}|B|^{2}]\} dv.  (4.4)

By the assumption of Theorem 1, the right hand side of (4.4) is non-negative. Thus, either |Z|^{2} ≡ 0, that is, M is totally umbilical; or

|B|^{2} = 2\sqrt{n − 1}.  (4.5)

In the latter case, equality holds in Lemma 3.1, and it follows that (n − 1) of k_{i} are equal. After re-enumeration if necessary, we can assume that

k_{1} = k_{2} = ⋯ = k_{n−1}, k_{1} ≠ k_{n}.  (4.6)

In this case, we have from (2.17)

\frac{1}{2} ∑_{i,j} R_{ijij}(k_{i} − k_{j})^{2} = 0.  (4.7)

Combining (4.6) with (4.7), we have R_{1n1n} = 1 + k_{1}k_{n} = 0. Thus we conclude by (4.5)

k_{1} = \frac{1}{\sqrt[4]{n − 1}}, k_{n} = −\sqrt[4]{n − 1}.

Therefore M = S^{1}(r_{1}) × S^{n−1}(r_{2}), where r_{1} and r_{2} are given by (1.5). This completes the proof of Theorem 1.

5. Proofs of Corollaries 1–3

The proof of Corollary 1 is obvious. The proof of Corollary 2 follows from Theorem 1 and the following lemma.

Lemma 0.2. Let M be an n-dimensional compact hypersurface in an (n + 1)- dimensional unit sphere S^{n+1}. If the normalized scalar curvature R = constant and R − 1 ≥ 0, then (1.3) holds.

Proof. From (2.5),

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

Taking the covariant derivative of the above expression, and using the fact R = constant, we get

n^{2}HH_{k} = ∑_{i,j} h_{ij}h_{ijk}.

It follows that

∑_{k} n^{4}H^{2}(H_{k})^{2} = ∑_{k}(∑_{i,j} h_{ij}h_{ijk})^{2} ≤ (∑_{i,j} h_{ij}^{2}) ∑_{i,j,k} h_{ijk}^{2},  (5.1)

that is

n^{4}H^{2}|∇H|^{2} ≤ |B|^{2}|∇B|^{2}.  (5.2)

On the other hand, from R − 1 ≥ 0, we have n^{2}H^{2} − |B|^{2} ≥ 0. Thus

n^{2}H^{2}|∇H|^{2} ≤ H^{2}|∇B|^{2}

and Lemma 5.1 follows. □

The proof of Corollary 3 comes out from Theorem 1 and the following lemma.

Lemma 0.3. Let M be an n-dimensional compact hypersurface in an (n + 1)- dimensional unit sphere S^{n+1}. Suppose that the normalized scalar curvature R is proportional to the mean curvature H of M, that is,

R = aH,  a^{2} > \frac{4n}{n − 1},  (5.3)

where a is a constant. Then (1.3) holds.

Proof. By use of Gauss equations (2.5) and the assumption (5.3), we have

|B|^{2} = n^{2}H^{2} + n(n − 1)(1 − aH).  (5.4)

It follows that

4|B|^{2}|∇h|^{2} ≥ 4 ∑_{k}(∑_{i,j} h_{ij}h_{ijk})^{2} = (2n^{2}H − n(n − 1)a)^{2}|∇H|^{2}.  (5.5)

By (5.3) and (5.4) we have

(2n^{2}H − n(n − 1)a)^{2} − 4n^{2}|B|^{2}

= (4n^{4}H^{2} + n^{2}(n − 1)^{2}a^{2} − 4n^{3}(n − 1)Ha) − 4n^{3}(nH^{2} + (n − 1)(1 − aH))

= n^{2}(n − 1)((n − 1)a^{2} − 4n) > 0.
(5.6)

Combining (5.5) with (5.6), we conclude that (1.3) holds. □

1. Proof of Theorem 2

Now we assume

Ric(e_{i}) = R_{ii} = n − 1 + nHk_{i} − k_{i}^{2} ≥ n − 2, 1 ≤ i ≤ n,  (6.1)

that is,

nHk_{i} − k_{i}^{2} + 1 ≥ 0.  (6.2)

We have from (6.2)

\frac{nH}{2} − \frac{1}{2}\sqrt{n^{2}H^{2} + 4} ≤ k_{i} ≤ \frac{nH}{2} + \frac{1}{2}\sqrt{n^{2}H^{2} + 4},  1 ≤ i ≤ n.  (6.3)

Therefore we get from (6.3)

R_{ijij} = 1 + k_{i}k_{j} ≥ 0, i ≠ j.  (6.4)

The assumptions of Theorem 2 imply that the right hand side of (2.17) is nonnegative, thus we have

\frac{1}{2} ∑_{i,j} R_{ijij}(k_{i} − k_{j})^{2} = 0.  (6.5)

In the same way as that of Nomizu-Smyth’s in [4], it follows that either M is totally umbilical (that is k_{1} = ⋯ = k_{n}); or M has two different principal curvatures

k_{1} = ⋯ = k_{m} ≠ k_{m+1} = ⋯ = k_{n},  R_{1n1n} = 1 + k_{1}k_{n} = 0,  (6.6)

where 1 < m < n. By the assumptions, we have

R_{aa} = (m − 1)(1 + k_{1}^{2}) ≥ (n − 2), 1 ≤ a ≤ m,

R_{αα} = (n − m − 1)(1 + k_{n}^{2}) = (n − m − 1)(1 + 1/k_{1}^{2}) ≥ (n − 2), m + 1 ≤ α ≤ n, that is,

k_{1}^{2} = \frac{n − m − 1}{m − 1}, k_{n}^{2} = \frac{m − 1}{n − m − 1},

and M = S^{m}(r_{1}) × S^{n−m}(r_{2}), where r_{1} and r_{2} are given by (1.8). This completes the proof of Theorem 2.

7. Proofs of Corollaries 4–6

The proof of Corollary 4 is obvious. The proof of Corollary 5 follows from Theorem 2 and Lemma 5.1. The proof of Corollary 6 follows from Theorem 2 and Lemma 5.2.

References

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[2] S.Y. Cheng and S.T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), 195–204.

[3] H.Z. Li, Hypersurfaces with constant scalar curvature in space forms, Math. Ann. 305 (1996), 665–672.

[4] K. Nomizu and B. Smyth, A formula of Simon’s type and hypersurfaces, J. Differential Geom. 3 (1969), 367–377.

[5] M. Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207–213.

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Department of Mathematical Science (Received 14 09 1998) Tsinghua University Beijing, 100084 People’s Republic of China

    hli@math.tsinghua.edu.cn