Compact hypersurfaces in a unit sphere

(1)

Compact hypersurfaces in a unit sphere

Qing-Ming Cheng

Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan ([email protected]) Shichang Shu^∗

Department of Mathematics, Weinan Teachers’ College, Weinan 714000, Shaanxi, People’s Republic of China ([email protected])

Young Jin Suh

Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea ([email protected])

(MS received 31 August 2004; accepted 9 February 2005)

We study curvature structures of compact hypersurfaces in the unit sphereSⁿ⁺¹(1) with two distinct principal curvatures. First of all, we prove that the Riemannian productS¹(√

1−c²)×Sⁿ⁻¹(c) is the only compact hypersurface inSⁿ⁺¹(1) with two distinct principal curvatures, one of which is simple and satisﬁes

r >1−2

n, r=n−2

n−1 and S(n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2, wheren(n−1)ris the scalar curvature of hypersurfaces andc²= (n−2)/nr. This generalized the result of Cheng, where the scalar curvature is constant is assumed.

Secondly, we prove that the Riemannian productS¹(√

1−c²)×Sⁿ⁻¹(c) is the only compact hypersurface with non-zero mean curvature inSⁿ⁺¹(1) with two distinct principal curvatures, one of which is simple and satisﬁes

r >1−2

n and S(n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2. This gives a partial answer for the problem proposed by Cheng.

1. Introduction

LetMbe ann-dimensional hypersurface in a unit sphereSⁿ⁺¹(1) of dimensionn+1.

It is well known that a compact minimal hypersurface with S = n in Sⁿ⁺¹(1) is isometric to a Cliﬀord torus

S¹

1

n

×Sⁿ⁻¹

n−1

n

,

whereS is the squared norm of the second fundamental form of the hypersurface (cf. [6,7,10]). Furthermore, Otsuki [12] investigated the converse problem. He proved

∗Present address: Department of Mathematics, Faculty of Science and Engineering, Saga Uni- versity, Saga 840-8502, Japan.

1129

c 2005 The Royal Society of Edinburgh

(2)

that Riemannian product S^k

k

n

×Sⁿ⁻^k

n−k

n

is the only compact minimal hypersurface with two distinct principal curvatures whose multiplicities are greater than 1. For compact minimal hypersurfaces with two distinct principal curvatures, one of which is simple, Otsuki constructed many examples that are not congruent to each other. However, they have the same topo- logical type asSⁿ⁻¹×S¹ when they are compact. Cheng [3] investigated in detail compact minimal hypersurfaces with two distinct principal curvatures, one of which is simple, and proved that the Cliﬀord torus

S¹

1

n

×Sⁿ⁻¹

n−1

n

is the only compact minimal hypersurface inSⁿ⁺¹(1) with nS C(n), where C(n) is a constant depending only on n. As a generalization of the result of Cheng [3], Hasanis and Vlachos [9] deleted the condition of the above bounding ofS, that is, they proved that the Cliﬀord torus

S¹

1

n

×Sⁿ⁻¹

n−1

n

is the only compact minimal hypersurface in Sⁿ⁺¹(1) with two distinct principal curvatures, one of which is simple, ifSn.

Recently, Cheng [4] studied n-dimensional complete hypersurfaces in Sⁿ⁺¹(1) with constant scalar curvature and proved the following theorem.

Theorem 1.1(Cheng [4]). LetM be ann-dimensional complete hypersurface with constant scalar curvaturen(n−1)rinSⁿ⁺¹(1). IfM has only two distinct principal curvatures, one of which is simple, thenr >1−(2/n)holds andM is isometric to S¹(√

1−c²)×Sⁿ⁻¹(c) ifr= (n−2)/(n−1) and S (n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2, wherec²= (n−2)/nr.

Further, he constructed inﬁnitely many isometrically distinct compact hypersurfaces in Sⁿ⁺¹(1) which have constant scalar curvature but are not congruent to each other.

Alternatively, it is natural to ask whether the condition of constant scalar curvature in Cheng [4, theorem 1.1] is necessary. In this paper, we shall ﬁrst of all solve the problem and prove the following theorem.

Theorem 1.2. Let M be ann-dimensional compact hypersurface in Sⁿ⁺¹(1) with two distinct principal curvatures, one of which is simple. Then,M is isometric to S¹(√

1−c²)×Sⁿ⁻¹(c) if r >1−2

n, r=n−2

n−1 and S(n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2, wherec²= (n−2)/nr, andn(n−1)r denotes the scalar curvature.

(3)

Remark1.3. In our theorem above, we do not know whether the condition r = (n−2)/(n−1) is necessary, which was also assumed in theorem 1.1. From the examples constructed by Cheng [4], we know that the condition

S(n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2 is essential. It is well known that the Cliﬀord torus

S¹

1

n

×Sⁿ⁻¹

n−1

n

is a compact minimal hypersurface inSⁿ⁺¹(1) satisfyingr= (n−2)/(n−1).

On the other hand, Li [11] proved the following. Let M be an n-dimensional compact hypersurface with constant scalar curvaturen(n−1)r. Ifr1 and

S (n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2,

thenM is isometric to either the totally umbilical hypersurface or the Riemannian product

S¹(

1−c²)×Sⁿ⁻¹(c), c²=n−2

nr n−2 n .

For any 0 < c < 1, by considering the standard immersions Sⁿ⁻¹(c)⊂Rⁿ, S¹(√

1−c²)⊂R² and taking the Riemannian product immersion S¹(√

1−c²)× Sⁿ⁻¹(c)→R²×Rⁿ, we obtain a compact hypersurfaceS¹(√

1−c²)×Sⁿ⁻¹(c) in Sⁿ⁺¹(1) with constant scalar curvaturen(n−1)r, wherer= (n−2)/nc²>1−2/n.

Hence, some of the Riemannian productsS¹(√

1−c²)×Sⁿ⁻¹(c) do not appear in the results in [11].

From theorem 1.1 above (see Cheng [4]), it is natural and interesting to generalize Li’s results [11] to the caser >1−2/n. That is, it is interesting to prove the following problem.

Problem1.4 (Cheng [4]). LetM be ann-dimensional compact hypersurface with constant scalar curvaturen(n−1)rinSⁿ⁺¹(1). If

r >1−2

n and S(n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2, can we classify the hypersurfaceM?

In order to solve this problem, Cheng [5] proved the following theorem.

Theorem 1.5. Let M be an n-dimensional compact hypersurface with an inﬁnite fundamental group inSⁿ⁺¹(1). If

r n−2

n−1 and S(n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2, then M is isometric to the Riemannian product S¹(√

1−c²)×Sⁿ⁻¹(c), where n(n−1)ris the scalar curvature ofM andc²= (n−2)/nr.

(4)

In the following theorem, we solve the problem 1.4 partially.

Theorem 1.6. Let M be ann-dimensional compact hypersurface in Sⁿ⁺¹(1) with non-zero mean curvature and with two distinct principal curvatures, one of which is simple. Then,M is isometric toS¹(√

1−c²)×Sⁿ⁻¹(c)if r >1−2

n and S(n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2, wherec²= (n−2)/nr andn(n−1)rdenotes the scalar curvature.

2. Preliminaries

Throughout this paper, all manifolds are assumed to be smooth and connected with- out boundary. Letϕ:M →Sⁿ⁺¹(1) be an isometric immersion of then-dimensional Riemannian manifoldM into a unit sphereSⁿ⁺¹(1). We choose a local orthonormal frame {e1, . . . ,en,en+1} and the dual co-frame {ω1, . . . , ωn, ωn+1} in such a way that{e1, . . . ,en} is a local orthonormal frame onM. Hence, we have

ω_n+1= 0 onM.

From Cartan’s lemma, we have ω_n+1,i=

n

j=1

h_ijω_j, h_ij =h_ji. (2.1)

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

H = 1 n

n

i=1

hii, α=hijωi⊗ωjen+1.

When the mean curvature H of M is identically zero, we recall that M is by deﬁnition a minimal hypersurface.

From the structure equations ofM, the Gauss equation and Codazzi equation, respectively, are given by

Rijkl= (δikδjl−δilδjk) + (hikhjl−hilhjk), (2.2)

h_ijk=h_ikj. (2.3)

From (2.2), we have

n(n−1)r=n(n−1) +n²H²−S, (2.4) where n(n−1)r and S denote the scalar curvature and the squared norm of the second fundamental form ofM, respectively. We choose a local ﬁeld of orthonormal framese1, . . . ,en onM such that, at the point that we consider,

h_ij =

λi ifi=j,

0 ifi=j, (2.5)

where theλi are the principal curvatures ofM.

(5)

Next, we consider the parallel hypersurfacesϕ_θ:M →Sⁿ⁺¹(1) ofM inSⁿ⁺¹(1) given by the map

ϕθ= cosθϕ+ sinθN, (2.6)

whereN is the Gauss map ofϕ:M →Sⁿ⁺¹(1). It is well known that, when cotθ is not a principal curvature ofϕ,ϕθis an isometric immersion if we endowM with the pullback metric gθ via ϕθ. We take cotθ = λi for any i = 1,2, . . . , n. Thus, we know that the shape operatorAθ of the immersionϕθ with respect to the unit normal vectorNθ= cosθN −sinθϕis given by

A_θ={cotθA_N+I}{cotθI−A_N}⁻¹, (2.7) where A_N is the shape operator with respect to the unit normal vector N and Iis the identity mapping. Hence, the principal curvatures ofϕθ:M →Sⁿ⁺¹(1) are given by

λi(θ) =cotθλi+ 1 cotθ−λi

. (2.8)

3. Proofs of theorem 1.2 and theorem 1.6

Before we prove our theorems, we shall review some known results which will be used in the proof.

In [5], Cheng studied compact hypersurfaces with inﬁnite fundamental group in Sⁿ⁺¹(1) and proved the following theorem.

Theorem 3.1. LetM be ann-dimensional compact hypersurface with inﬁnite fun- damental group inSⁿ⁺¹(1). If the sectional curvatures are non-negative, thenM is isometric to the Riemannian productS¹(√

1−c²)×Sⁿ⁻¹(c).

The following theorem due to do Carmoet al. [8] will be used.

Theorem 3.2. LetM be ann-dimensional(n3)compact orientable hypersurface inSⁿ⁺¹(1)with two distinct principal curvatures, one of which is simple. Then,M is homeomorphic toS¹(√

1−c²)×Sⁿ⁻¹(c).

Remark3.3. We note that, in the statement of theorem 3.2, do Carmo et al.

considered that M is an n-dimensional (n 4) conformal ﬂat hypersurface in Sⁿ⁺¹(1). But, in their proof, they used only the condition that the hypersurface has two distinct principal curvatures, one of which is simple. Hence, their result is also true forn= 3.

Proof of theorem 1.2. Since M has two distinct principal curvatures, we denote them byλandµ. We then have

(n−1)λ+µ=nH, S= (n−1)λ²+µ². (3.1) From (3.1) and the Gauss equation (2.4), we obtain

λµ= (n−1)(r−1)−(n−2)H²±(n−2)H

H²−(r−1).

Since

S(n−1)n(r−1) + 2

n−2 + n−2

n(r−1) + 2 and r= n−2 n−1,

(6)

we know thatH = 0. In fact, ifH = 0 holds at some point, we have, at the point, S=−n(n−1)(r−1). Hence,

−n(n−1)(r−1)(n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2,

−(n−1)(n(r−1) + 2) + 2(n−1)(n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2. Since the scalar curvature is continuous, we have

r > n−2

n−1 or r < n−2

n−1 onM.

Ifr >(n−2)/(n−1) holds, thenn(r−1) + 2>(n−2)/(n−1). Ifr <(n−2)/(n−1) holds, thenn(r−1) + 2<(n−2)/(n−1). We consider a function f(t) deﬁned by

f(t) = (n−1)²

n−2 t+n−2

t −2(n−1).

We can prove that f(t) is an increasing function of t when t > (n−2)/(n−1).

Hence, we infer a contradiction because f

n−2 n−1

= 0.

Ift <(n−2)/(n−1), thenf(t) is a decreasing function. This is also impossible.

Hence, we haveH= 0.

Thus, we can assumeH >0 onM. Therefore,M is orientable. Since the principal curvaturesλandµare continuous, we have

λµ= (n−1)(r−1)−(n−2)H²+ (n−2)H

H²−(r−1) or

λµ= (n−1)(r−1)−(n−2)H²−(n−2)H

H²−(r−1) onM. From the Gauss equation (2.4), we have

λµ=2(n−1)

n (r−1)−(n−2)

n² S+(n−2) n²

S+n(n−1)(r−1)

S−n(r−1) or

λµ=2(n−1)

n (r−1)−(n−2)

n² S−(n−2) n²

S+n(n−1)(r−1)

S−n(r−1) onM.

First of all, we consider the case where λµ=2(n−1)

n (r−1)−(n−2)

n² S+(n−2) n²

S+n(n−1)(r−1)

S−n(r−1) holds onM. In this case, we can proveλµ+ 10 onM. Indeed, ifλµ+ 1<0 at some point, then

(n−2) n²

S+n(n−1)(r−1)

S−n(r−1)<−1−2(n−1)

n (r−1) +(n−2) n² S.

(7)

From

S(n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2

and a direct computation, we have a contradiction. Hence, we haveλµ+ 10 on M. From the Gauss equation (2.2), we infer that the sectional curvatures ofM are non-negative, sinceM has two distinct principal curvatures, one of which is simple.

From theorem 3.2, we ﬁnd that M is homeomorphic to S¹(√

1−c²)×Sⁿ⁻¹(c).

Thus, the fundamental group of M is inﬁnite. From theorem 3.1, we know that ϕ(M) is isometric toS¹(√

1−c²)×Sⁿ⁻¹(c).

Secondly, we consider the case where

λµ= 2(n−1)

n (r−1)−(n−2) n² S

−(n−2) n²

S+n(n−1)(r−1)

S−n(r−1) (3.2) holds onM. From

S (n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2,

we ﬁnd that λ = 0 and µ = 0. In fact, if λ = 0, we have n²H² = S. From the Gauss equation (2.4) we obtainr= 1. Hence, from formula (3.2) forλµ, we have a contradiction. Ifµ= 0, we haven²H²= (n−1)S. From the Gauss equation (2.4), we have (n−2)S =n(n−1)(r−1). It is impossible because

S(n−1)n(r−1) + 2

n−2 + n−2

n(r−1) + 2, r >1−2 n. Next we shall prove thatλµ+ 10. If

2(n−1)

n (r−1)−(n−2)

n² S+ 10,

then, from (3.2), it is obvious that the conditionλµ+ 10 holds. If 2(n−1)

n (r−1)−(n−2)

n² S+ 1>0,

then we can prove that the condition thatλµ+ 10 also holds. In fact, from S (n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2, we have

(n−2)(n(r−1) + 2)S(n−1)n²(r−1)²+ 4n(n−1)(r−1) +n², that is,

(n−2){4n(n−1)(r−1) + 2n²+ (n−2)²n(r−1)}S

{2n(n−1)(r−1) +n²}²+ (n−2)²n²(n−1)(r−1)².

(8)

Hence,

n+ 2(n−1)(r−1)−n−2 n S

2

(n−2)²

n² {n(n−1)(r−1) +S}{S−n(r−1)}. Since the condition

2(n−1)

n (r−1)−(n−2)

n² S+ 1>0 holds, we obtain

2n(n−1)(r−1) +n²−(n−2)S (n−2)

S+n(n−1)(r−1)

S−n(r−1).

From

λµ=2(n−1)

n (r−1)−(n−2)

n² S−(n−2) n²

S+n(n−1)(r−1)

S−n(r−1), we may infer thatλµ+ 10. Hence, we obtain (1/λµ) + 10.

Sinceλµ= 0, we know thatϕ_π/2is an isometric immersion fromM intoSⁿ⁺¹(1).

Therefore, from (2.8), we know that the sectional curvatures K_π/2 of ϕ_π/2 with respect to the plane spanned byej anden forj = 1, . . . , n−1, satisfy

K_π/2(e_j,e_n) = 1

λµ+10, K_π/2(e_i,e_j) = 1+ 1

λ² >1 fori, j= 1,2, . . . , n−1.

It is obvious that the immersionϕ_π/2has only two distinct principal curvatures and that one of them is simple. From theorem 3.2, we infer thatM is homeomorphic to S¹(√

1−c²)×Sⁿ⁻¹(c). Thus, the fundamental group of M is inﬁnite. From theorem 3.1, we know thatϕ_π/2(M) is isometric toS¹(√

1−c²)×Sⁿ⁻¹(c). Hence, the principal curvatures 1/λand 1/µofϕ_π/2are constant. Thus, we see thatλandµ are constant andλµ+1 = 0. Hence, the sectional curvatures ofM are non-negative.

From theorem 3.1, we may infer thatϕ(M) is isometric toS¹(√

1−c²)×Sⁿ⁻¹(c).

Proof of theorem 1.6. SinceM has two distinct principal curvatures, one of which is simple, from theorem 3.2 we know that M is homeomorphic toS¹(√

1−c²)× Sⁿ⁻¹(c). Hence, the fundamental group ofM is inﬁnite. Since the condition

S(n−1)n(r−1) + 2

n−2 + n−2 n(r−1) + 2 holds if and only if

n+ 2(n−1)(r−1)−n−2 n S

2

(n−2)²

n² {n(n−1)(r−1) +S}{S−n(r−1)}, from the Gauss equationn(n−1)r=n(n−1) +n²H²−S, we conclude that

(S−S₋)(S−S+)0, where

S_± =n+ n³

2(n−1)H²± n−2 2(n−1)

n⁴H⁴+ 4n²(n−1)H².

(9)

Since the mean curvature H = 0 holds on M, we have S₊ > S₋ on M. Then, sinceSis a continuous function onM, we can infer that only one of the conditions SS+ onM andSS₋ onM is satisﬁed.

If the conditionSS₋ onM holds, from the result of [1], and the fundamental group is inﬁnite, we know thatM is isometric toS¹(√

1−c²)×Sⁿ⁻¹(c). If, on the other hand, the conditionSS+ onM holds, from the result of [2], we know that M is isometric toS¹(√

1−c²)×Sⁿ⁻¹(c).

Acknowledgments

Q.-M.C. was partly supported by a Grant-in-Aid for Scientiﬁc Research from the Japan Society for the Promotion of Science. S.S. was partly supported by the Nat- ural Science Foundation of China and the NSF of Shaanxi. Y.J.S. was partly supported by Grant no. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2005.

References

1 A. C. Asperti and E. A. Costa. Vanishing of homology groups, Ricci estimate for submanifolds and applications.K¯odai Math. J.24(2001), 313–328.

2 J. N. Barbosa, A. Brasil and I. Lazaro. Hypersurfaces of the Euclidean sphere with non- negative Ricci curvature.Arch. Math.81(2003), 335–341.

3 Q.-M. Cheng. The rigidity of Cliﬀord torus S¹(

1/n)×Sⁿ⁻¹(

(n−1)/n). Comment.

Math. Helv.71(1996), 60–69.

4 Q.-M. Cheng. Hypersurfaces in a unit sphereSⁿ⁺¹(1) with constant scalar curvature.J.

Lond. Math. Soc.64(2001), 755–768.

5 Q.-M. Cheng. Compact hypersurfaces in a unit sphere with inﬁnite fundamental group.

Pac. J. Math.212(2003), 49–56.

6 Q.-M. Cheng and S. Ishikawa. A characterization of the Cliﬀord torus.Proc. Am. Math.

Soc.127(1999), 819–828.

7 S. S. Chern, M. do Carmo and S. Kobayashi. Minimal submanifolds of a sphere with second fundamental form of constant length. InFunctional Analysis and Related Fields, pp. 59–75 (Springer, 1970).

8 M. do Carmo, M. Dajczer and F. Mercuri. Compact conformally ﬂat hypersurfaces.Trans.

Am. Math. Soc.288(1985), 189–203.

9 T. Hasanis and T. Vlachos. A pinching theorem for minimal hypersurfaces in a sphere.

Arch. Math.75(2000), 469–471.

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

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

12 T. Otsuki. Minimal hypersurfaces in a Riemannian manifold of constant curvature.Am. J.

Math.92(1970), 145–173.

(Issued 16 December 2005)

(10)