COMPLETE SUBMANIFOLDS IN EUCLIDEAN SPACES WITH CONSTANT SCALAR CURVATURE*

QING-MING CHENG

Dedicated to Professor Buchin Su for his 100th birthday

1. Introduction

It is well-known that Nash proved that every finite dimensional Riemannian manifold possesses an isometric embedding into a Euclidean space of sufficiently high dimension. On the other hand, Yamabe problem was solved affirmatively, that is, every compact n-dimensional Riemannian manifold can be deformed to a Riemannian manifold with constant scalar curvature by a conformal transformation. Hence, every compact n-dimensional Riemannian manifold can be immersed in a Euclidean space such that it has constant scalar curvature. Therefore, in order to research complete submanifolds with constant scalar curvature in a Euclidean space, we must do it under some additional conditions. In this article, we shall study and classify complete submanifolds in a Euclidean space under some adaptable conditions on curvatures and the second fundamental form. When the codimension of submanifolds is one, Yau conjecture and the theorem due to Hartman and Nirenberg, its generalizations and so on will be discussed. For the case of higher codimensions, we shall prove that the totally umbilical sphere S^{n}(r), totally geodesic Euclidean space E^{n} and generalized cylinder S^{n−1}(c) × E^{1} are the only n-dimensional (n > 2) complete submanifolds M^{n} with constant scalar curvature n(n − 1)r in the Euclidean space E^{n+p}, which satisfy S ≤ \frac{n(n−1)r}{n−2}. Where S denotes the squared norm of the second fundamental form of M^{n}

2. Preliminaries

Let E^{n+p} be an (n + p)-dimensional Euclidean space and M^{n} an ndimensional connected submanifold in E^{n+p}. We choose a local field of orthonormal frames {e_{1}, ⋯, e_{n+p}} adapted to E^{n+p} and the dual coframes {ω_{1}, ⋯, ω_{n+p}} in such a way that, restricted to the submanifold M^{n}, {e_{1}, ⋯, e_{n}} are tangent to M^{n}. Let {ω_{AB}} denote the connection forms of E^{n+p}. The canonical forms {ω_{A}} and connection forms {ω_{AB}} restricted to M^{n} are also denoted by the same symbols. We then have

* Research partially Supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science, Sports and Curvature, Japan.

2000 Mathematics Subject Classification 53C42.

ω_{α} = 0, α = n + 1, ⋯, n + p. (2.1)

We see that e_{1}, ⋯, e_{n} is a local field of orthonormal frames adapted to the induced Riemannian metric on M^{n} and ω_{1}, ⋯, ω_{n} is a local field of its dual coframes on M^{n}. It follows from (2.1) and Cartan’s Lemma that

ω_{αi} = ∑_{j=1}^{n} h_{ij}^{α}ω_{j}, h_{ij}^{α} = h_{ji}^{α}. (2.2)

The second fundamental form II and the mean curvature vector h of M^{n} are defined by

II = ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n} h_{ij}^{α}ω_{i}ω_{j}e_{α}, (2.3)

h = \frac{1}{n} ∑_{α=n+1}^{n+p}(∑_{i=1}^{n} h_{ii}^{α})e_{α}. (2.4)

The mean curvature H of M^{n} is defined by

H = \frac{1}{n}\sqrt{∑_{α=n+1}^{n+p}(∑_{i=1}^{n} h_{ii}^{α})^{2}}. (2.5)

Let S = ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2} denote the squared norm of the second fundamental form of M^{n}. The connection form of M^{n} are characterized by the structure equations

dω_{i} = − ∑_{j=1}^{n} ω_{ij} ∧ ω_{j}, ω_{ij} + ω_{ji} = 0, (2.6)

dω_{ij} = − ∑_{k=1}^{n} ω_{ik} ∧ ω_{kj} + \frac{1}{2} ∑_{k,l=1}^{n} R_{ijkl}ω_{k} ∧ ω_{l}, (2.7)

R_{ijkl} = ∑_{α=n+1}^{n+p}(h_{ik}^{α}h_{jl}^{α} − h_{il}^{α}h_{jk}^{α}). (2.8)

where R_{ijkl} is components of the curvature tensor of M^{n}. Letting R_{ij} and n(n − 1)r denote components of the Ricci curvature and the scalar curvature of M^{n}, respectively, we have from (2.8)

R_{jk} = ∑_{α=n+1}^{n+p}(∑_{i=1}^{n} h_{ii}^{α}h_{jk}^{α} − ∑_{i=1}^{n} h_{ik}^{α}h_{ji}^{α}), (2.9)

n(n − 1)r = n^{2}H^{2} − S. (2.10)

We also have

dω_{αβ} = − ∑_{γ=n+1}^{n+p} ω_{αγ} ∧ ω_{γβ} + \frac{1}{2} ∑_{i,j=1}^{n} R_{αβij}ω_{i} ∧ ω_{j}, (2.11)

R_{αβij} = ∑_{l=1}^{n}(h_{il}^{α}h_{lj}^{β} − h_{jl}^{α}h_{li}^{β}). (2.12)

By taking the exterior differentiation of (2.3) and defining h_{ijk}^{α} by

∑_{k=1}^{n} h_{ijk}^{α}ω_{k} = dh_{ij}^{α} − ∑_{k=1}^{n} h_{ik}^{α}ω_{kj} − ∑_{k=1}^{n} h_{jk}^{α}ω_{ki} − ∑_{β=n+1}^{n+p} h_{ij}^{β}ω_{βα}, (2.13)

we obtain the Codazzi equation by straightforward computations h_{ijk}^{α} = h_{ikj}^{α} = h_{jik}^{α}. (2.14) We take the exterior differentiation of (2.13) and define h_{ijkl}^{α} by

∑_{l=1}^{n} h_{ijkl}^{α}ω_{l} = dh_{ijk}^{α} − ∑_{l=1}^{n} h_{ljk}^{α}ω_{li} − ∑_{l=1}^{n} h_{ilk}^{α}ω_{lj} − ∑_{l=1}^{n} h_{ijl}^{α}ω_{lk} − ∑_{β=n+1}^{n+p} h_{ijk}^{β}ω_{βα}. (2.15)

Then the Ricci formula for the second fundamental form is given by

h_{ijkl}^{α} − h_{ijlk}^{α} = ∑_{m=1}^{n} h_{mj}^{α}R_{mikl} + ∑_{m=1}^{n} h_{im}^{α}R_{mjkl} + ∑_{β=n+1}^{n+p} h_{ij}^{β}R_{βαkl}. (2.16)

The Laplacian Δh_{ij}^{α} of h_{ij}^{α} is defined by

Δh_{ij}^{α} = ∑_{k=1}^{n} h_{ijkk}^{α}.

From the Codazzi equation (2.14) and the Ricci formula (2.16), we obtain for any α, n + 1 ≤ α ≤ n + p,

Δh_{ij}^{α} = ∑_{k=1}^{n} h_{kijk}^{α} (2.17)

= ∑_{k=1}^{n} h_{kikj}^{α} + ∑_{k,m=1}^{n} h_{km}^{α}R_{mijk}

+ ∑_{k,m=1}^{n} h_{mi}^{α}R_{mkjk} + ∑_{k=1}^{n} ∑_{β=n+1}^{n+p} h_{ki}^{β}R_{βαjk}

= ∑_{k=1}^{n} h_{kkij}^{α} + ∑_{k,m=1}^{n} h_{km}^{α}R_{mijk}

+ ∑_{k,m=1}^{n} h_{mi}^{α}R_{mkjk} + ∑_{k=1}^{n} ∑_{β=n+1}^{n+p} h_{ki}^{β}R_{βαjk}.

3. Complete hypersurfaces

A classical theorem of Hilbert says that a complete surface with the non-zero constant curvature in the 3-dimensional Euclidean space E^{3} must be isometric to the sphere. On the other hand, Hartman and Nirenberg [16] proved that a complete surface with zero curvature in the 3-dimensional Euclidean space E^{3} must be isometric to the plane or the cylinder. Hence, these two results give a complete classification of complete surfaces with constant curvature in the 3-dimensional Euclidean space E^{3}. As a generalization of these theorems, Nomizu and Smyth [26] proved

Theorem 3.1 (Nomizu and Smyth [26] ). An n-dimensional (n > 2) complete hypersurface M^{n} with constant scalar curvature in E^{n+1} is isometric to a sphere, a hyperplane or a generalized cylinder S^{k}(c) × E^{n−k}, 1 ≤ k ≤ n − 1, if the sectional curvature of M^{n} is nonnegative and the mean curvature is constant.

Proof. By making use of a simple computation, we obtain

\frac{1}{2}ΔS = ∑_{i,j,k=1}^{n} h_{ijk}^{2} + ∑_{i<j}(λ_{i} − λ_{j})^{2}K_{ij} (3.1)

since the mean curvature is constant. Where λ_{i}’s are the principal curvatures of M^{n} and K_{ij} is the sectional curvature of the section spanned by {e_{i}, e_{j}}. From (2.10), we know that S is constant because the scalar curvature is constant. Hence, we infer that M^{n} is isoparametric, that is, Theorem 3.1 is true.

In 1977, S.Y. Cheng and Yau [9] studied n-dimensional complete hypersurfaces with constant scalar curvature in the Euclidean space E^{n+1}. They proved the result due to Nomizu and Smyth [26] is true without the assumption that the mean curvature is constant. Namely, they proved:

Theorem 3.2 (S. Y. Cheng and Yau [9]). An n-dimensional (n > 2) complete hypersurface M^{n} with constant scalar curvature in E^{n+1} is isometric to a sphere, a hyperplane or a generalized cylinder S^{k}(c) × E^{n−k}, 1 ≤ k ≤ n − 1, if the sectional curvature of M^{n} is nonnegative.

Proof. (1) When M^{n} is compact, we consider □(nH). We then obtain, by a simple computation,

□(nH) = ∑_{i,j,k=1}^{n} h_{ijk}^{2} − n^{2}|gradH|^{2} + ∑_{i<j}(λ_{i} − λ_{j})^{2}K_{ij} (3.2)

since the scalar curvature is constant. From (2.10), we can prove

∑_{i,j,k=1}^{n} h_{ijk}^{2} − n^{2}|gradH|^{2} ≥ 0. (3.3)

Hence, from Stokes formula, we infer that M^{n} is isoparametric, that is, M^{n} is isometric to a standard round sphere because M^{n} is compact.

(2) When M^{n} is complete and noncompact, we consider □⟨N, a⃗⟩, where N is the unit normal vector field and a⃗ is a fixed unit vector. Note that, since M^{n} is of nonnegative sectional curvature, M^{n} is convex. Hence there exists a unit vector a⃗ in the Euclidean space so that ⟨N, a⃗⟩ ≥ 0 on M^{n} (see Wu [32]). By the same curvature assumption, we also know that the scalar curvature n(n − 1)r = ∑_{i≠j} K_{ij} ≥ 0. Since the scalar curvature is constant, r = 0 at some point if and only if the sectional curvature is zero. Hence, in the case that the scalar curvature is zero, Theorem 3.2 is true from a theorem due to Hartman and Nirenberg [16]. If the scalar curvature is positive, we can prove that the operator □ is elliptic. By a direct computation, we infer that

□⟨N, a⃗⟩ = − ∑_{k,l=1}^{n}(nHδ_{kl} − h_{kl}) ∑_{i=1}^{n}(h_{ki}h_{il})⟨N, a⃗⟩. (3.4)

From Maximum Principle, we obtain that ⟨N, a⃗⟩ is constant. Therefore, ⟨N, a⃗⟩ > 0 or ⟨N, a⃗⟩ ≡ 0. From this assertion, we infer that the Theorem 3.2 holds.

In particular, when n = 3, the author and Wan [8] proved that the assertion due to Nomizu and Smyth [26] is true without the assumption on the sectional curvature. That is, we proved the following:

Theorem 3.3 (Cheng and Wan [8]). Let M^{n} be an oriented complete hypersurface in E^{4} with constant scalar curvature. If the mean curvature is constant, then M^{n} is isometric to a standard round sphere, the Euclidean three-space E^{3} or the Riemannian product S^{k}(c) × E^{3−k} for k = 1, 2.

In 1982, Yau [34] proposed to study oriented compact hypersurfaces with constant scalar curvature in E^{n+1} and conjectured the following

Yau’s conjecture . The standard round spheres are the only possible oriented compact hypersurfaces in E^{n+1} with constant scalar curvature.

By making use of the so-called Reilly’s formula, Ros [29] showed that Yau’s conjecture is true if M^{n} is an embedded hypersurface. Namely,

Theorem 3.4 (Ros [29]). If M^{n} is an embedded compact hypersurface with constant scalar curvature in E^{n+1}, then M^{n} is isometric to a standard round sphere.

Proof. From (2.10), we have

n(n − 1)r = n^{2}H^{2} − S ≤ n(n − 1)H^{2}, (3.5)

since S = ∑_{i=1}^{2} λ_{i}^{2} ≥ nH^{2} holds, where λ_{i} are the principal curvatures of M^{n}. Since M^{n} is a compact hypersurface in E^{n+1}. Let ϕ denote the position vector field of E^{n+1} relative to the origin. With a straightforward calculation we then have on M^{n}

□⟨ϕ, ϕ⟩ = n(n − 1)H + n(n − 1)r⟨ϕ, N⟩. (3.6)

According to the Stokes formula, we have

∫_{M^{n}}(n(n − 1)H + n(n − 1)r⟨ϕ, N⟩)dM^{n} = 0, (3.7)

where dM^{n} denotes the volume element on M^{n}. Since M^{n} is a compact embedded hypersurface in E^{n+1}. Then, there exists a compact domain Ω ⊂ E^{n+1} such that the boundary ∂Ω = M^{n} holds. Let Δ̄ denote the Euclidean Laplacian of E^{n+1}. We then have

Δ̄⟨ϕ, ϕ⟩ = 2(n + 1) (3.8)

on E^{n+1}. Hence, from the divergence theorem, we obtain

− ∫_{M^{n}}⟨ϕ, N⟩dM^{n} = (n + 1)volΩ, (3.9)

where N is the inner unit normal field of Ω. Since the mean curvature H of M^{n} with respect to the inner normal N is positive everywhere on M^{n}, then we have

∫_{M^{n}} \frac{1}{H} dM^{n} ≥ (n + 1)volΩ, (3.10)

and the equality holds if and only if M^{n} is isometric to a standard round sphere. This assertion can be proved by the so-called Reilly’s formula (see Ros [29]) and it can also be proved by using the idea of Heintze and Karcher [11] (see Montiel and Ros [25]). From (3.5), we conclude

∫_{M} n(n − 1)HdM = ∫_{M} \frac{n(n − 1)H^{2}}{H} dM ≥ ∫_{M} \frac{n(n − 1)r}{H}dM. (3.11)

Hence, we have

∫_{M} \frac{1}{H} dM = (n + 1)volΩ

from (3.7), (3.10) and the above inequality (3.11). We infer that Theorem 3.2 is true.

It is well known that the theorem due to Hartman and Nirenberg [16] states that complete hypersurfaces with zero curvature in E^{n+1} are isometric to the hyperplane E^{n} or the cylinder E^{n−1} × S^{1}(c). The condition of zero curvature yields that the distinct principal curvatures are at most two and one of principal curvatures with at least n − 1 multiplicities is zero. Furthermore, the scalar curvature is zero. In [4], the author studied the more general case, that is, we studied oriented complete hypersurfaces with constant scalar curvature and with two distinct principal curvatures in E^{n+1} and proved that Yau’s conjecture is true for the class of locally conformally flat hypersurfaces (see Corollary 3.1).

Theorem 3.5 (Cheng [4]). If M^{n} is an oriented complete hypersurface in E^{n+1}, n > 2, with constant scalar curvature n(n − 1)r and with two distinct principal curvatures, then M^{n} is isometric to one of the following hypersurfaces:

(1) E^{n−k} × S^{k}(c), for 1 ≤ k ≤ n − 1,

(2) a complete non-compact hypersurface of revolution S^{n−1}(c(s)) × M^{1}, where S^{n−1}(c(s)) is of constant curvature [\frac{d}{ds}\{log |λ^{2}−r|^{1/n}\}]^{2}+ λ^{2} and M^{1} is a plane curve. Moreover, w = {λ^{2} − r}^{−1/n} satisfies the following ordinary differential equation of order 2:

\frac{d^{2}w}{ds^{2}} − w(\frac{n − 2}{2} \frac{1}{w^{n}} − r) = 0,

where λ is the principal curvature with multiplicity n − 1.

Theorem 3.6 (Cheng [4]). If M^{n} is an oriented complete locally conformally flat hypersurface in E^{n+1}, n > 3, with constant scalar curvature n(n−1)r, then M^{n} is isometric to one of the following hypersurfaces:

(1) a standard round sphere,

(2) E^{n},

(3) E^{n−1} × S^{1}(c),

(4) E^{1} × S^{n−1}(c),

(5) a complete non-compact hypersurface of revolution S^{n−1}(c(s)) × M^{1}, where S^{n−1}(c(s)) is of constant curvature [\frac{d}{ds}\{log |λ^{2}−r|^{1/n}\}]^{2}+ λ^{2} and M^{1} is a plane curve. Moreover, w = {λ^{2} − r}^{−1/n} satisfies the ordinary differential equation of order 2 in Theorem 3.5.

Corollary 3.1 (Cheng [4]). If M^{n} is an oriented compact locally conformally flat hypersurface in E^{n+1}, n > 3, with constant scalar curvature n(n − 1)r, then M^{n} is isometric to a standard round sphere S^{n}(r).

From Theorem 3.5, we know that there are many complete hypersurfaces, which have two distinct principal curvatures, one of which is simple, but are not isometric to S^{n−1}(c) × E and S^{1}(c) × E^{n−1}. The following theorem gives a characterization of the Riemannian products S^{n−1}(c) × E and S^{1}(c) × E^{n−1}.

Theorem 3.7 (Cheng [4]). Let M^{n} be an n-dimensional oriented complete hypersurface in E^{n+1} with constant scalar curvature n(n−1)r and with two distinct principal curvatures one of which is simple. If S ≥ \frac{n(n−1)r}{n−2}, then M^{n} is isometric to the Riemannian product S^{n−1}(c) × E or S^{1}(c) × E^{n−1}.

Corollary 3.2 (Cheng [4]). Let M^{n} be an n-dimensional (n > 3) oriented complete locally conformally flat hypersurface with constant scalar curvature n(n − 1)r in E^{n+1}. If S ≥ \frac{n(n−1)r}{n−2}, then M^{n} is isometric to one of the following hypersurfaces:

(1) E^{n},

(2) S^{1}(c) × E^{n−1},

(3) S^{n−1}(c) × E.

Proof. Since M^{n} is a locally conformally flat hypersurface in E^{n+1} and n > 3, from a result of Cartan-Schouten, we know that M^{n} is of at most two distinct principal curvatures and one of them is of at least n − 1 multiplicities. Without loss of the generality, we assume λ_{1} = λ_{2} = ⋯ = λ_{n−1} = λ and λ_{n} = μ.

(a). In the case r = 0.

From (2.10), we have λ = 0 on M^{n} or μ = −\frac{(n−2)λ}{2} on M^{n}.

If λ = 0 on M^{n}, we have that the sectional curvature of M^{n} is non-negative. The result of S.Y. Cheng and Yau [9] yields that M^{n} is isometric to E^{n} or E^{n−1} × S^{1}(c).

If μ = −\frac{(n−2)λ}{2} on M^{n}, we can obtain that M^{n} is isometric to S^{1}(c)× E^{n−1}. This is impossible because λ ≠ 0.

(b). In the case r ≠ 0.

If at some point p, these principal curvatures are equal with each other, then S = nH^{2} at this point p. From (2.10), we have S = nH^{2} = nr > 0. Since S ≥ \frac{n(n−1)r}{n−2}, we have nr ≥ \frac{n(n−1)r}{n−2}. This is impossible. Hence M^{n} is of two distinct principal curvatures and one of them is of n − 1 multiplicities. From the Theorem 3.7, we know that M^{n} is isometric to S^{n−1}(c) × E. That is, Corollary 3.2 is proved.

Proof of Theorem 3.7. If w is constant, then λ is constant because of w^{−n} = λ^{2} − r. Hence, the Theorem 3.7 is true. Next, we shall prove that w must be constant. If w is not constant, we shall obtain a contradiction. In fact, since w^{−n} = λ^{2} − r satisfies the equation

\frac{d^{2}w}{ds^{2}} − w(\frac{n − 2}{2} \frac{1}{w^{n}} − r) = 0, (3.12)

we know that w(s) is a function defined in (−∞, +∞) because M^{n} is complete and the integral curve of the principal vector field e_{n} corresponding to the principal curvature μ is a geodesic, which can be proved by a simply way. If r ≤ 0, then from (3.12), we know that \frac{dw}{ds} is an increasing function. If r > 0, then S ≥ \frac{n(n−1)r}{n−2} holds if and only if w^{−n} = λ^{2} − r ≥ \frac{2r}{n−2}. In fact, we consider a function

f(t) = \frac{n^{2}}{4} t + \frac{n^{2}r^{2}}{4t} − \frac{n(n − 2)}{2}r

for t > 0. Since

\frac{df(t)}{dt} = \frac{n^{2}}{4} − \frac{n^{2}r^{2}}{4t^{2}} ,

we know that t^{2} ≥ r^{2} holds if and only if f(t) is an increasing function and t^{2} ≤ r^{2} holds if and only if f(t) is a decreasing function. And f(t) obtains its minimum at t^{2} = r^{2}.

It is not difficult to prove λ^{2} − r > 0. Hence, we can assume λ > 0. Since

(3.13)  S = (n − 1)λ^{2} + μ^{2}

= (n − 1)λ^{2} + (\frac{nr}{2λ} − \frac{n − 2}{2}λ)^{2}

= \frac{n^{2}}{4} λ^{2} + \frac{n^{2}r^{2}}{4λ^{2}} − \frac{n(n − 2)}{2}r,

we conclude that S is an increasing function of λ^{2}. When λ^{2} −r = \frac{2r}{n−2}, we know S = \frac{n(n−1)r}{n−2}. Hence, we obtain that S ≥ \frac{n(n−1)r}{n−2} holds if and only if w^{−n} = λ^{2} − r ≥ \frac{2r}{n−2} holds because S is an increasing function of λ^{2}.

From (3.12), we obtain \frac{d^{2}w}{ds^{2}} ≥ 0. Hence, \frac{dw}{ds} is an increasing function of s if w is not constant.

Finally, we infer that it is impossible. Integrating the equation (3.12), we have

(\frac{dw}{ds})^{2} = C − rw^{2} − \frac{1}{w^{n−2}} , (3.14)

where C is an integral constant. Hence, we must have

C ≥ min_{w>0}(rw^{2} + \frac{1}{w^{n−2}} ) = \frac{n}{2}(\frac{2r}{n − 2})^{\frac{n−2}{n}} ,

since g(w) = rw^{2} + \frac{1}{w^{n−2}} is an increasing function if w^{−n} ≥ \frac{2r}{n−2} and g(w) = rw^{2} + \frac{1}{w^{n−2}} is a decreasing function if 0 < w^{−n} < \frac{2r}{n−2} and g(w) = rw^{2} + \frac{1}{w^{n−2}} obtains its minimum \frac{n}{2}(\frac{2r}{n−2})^{\frac{n−2}{n}} at w^{−n} = \frac{2r}{n−2}. And C = \frac{n}{2}(\frac{2r}{n−2})^{\frac{n−2}{n}} if and only if w is constant. Thus we infer that, for any non-constant solution w of (3.12),

C > \frac{n}{2}(\frac{2r}{n − 2})^{\frac{n−2}{n}} . (3.15)

Since g(w) = rw^{2}+ \frac{1}{w^{n−2}} is an increasing function if w^{−n} ≥ \frac{2r}{n−2}, g(w) = rw^{2}+ \frac{1}{w^{n−2}} is a decreasing function if 0 < w^{−n} < \frac{2r}{n−2} and C > min g(w), we obtain that the following equation has two distinct positive roots w_{1}, w_{2}:

rw^{2} + \frac{1}{w^{n−2}} − C = 0.

And

w_{1} < (\frac{2r}{n − 2})^{− \frac{1}{n}} < w_{2}. (3.16)

Since

(\frac{dw}{ds})^{2} = C − rw^{2} − \frac{1}{w^{n−2}} ,

and \frac{dw}{ds} is an increasing function, we know that \frac{dw}{ds} has at most one zero point in (−∞, +∞).

(a). If \frac{dw}{ds} has no zero point in (−∞, +∞), then w(s) is a monotone function of s in (−∞, +∞).

(b). If \frac{dw}{ds} has only one zero point in (−∞, +∞), denoting it by s_{0}, then w(s) is a monotone function of s both in (−∞, s_{0}] and in [s_{0}, +∞).

In any case, w(s) is a monotone function of s both in (−∞, s_{0}] and in [s_{0}, +∞). Since

(\frac{dw}{ds})^{2} = C − rw^{2} − \frac{1}{w^{n−2}} ≥ 0,

we know that w(s) is bounded, that is, w_{1} ≤ w(s) ≤ w_{2}. Then for some s_{0}, from the mean value theorem, we have that for any s ∈ [s_{0}, +∞), there exists a t ∈ [s_{0}, s] such that

\frac{dw(t)}{ds} = \frac{w(s) − w(s_{0})}{s − s_{0}}.

Hence,

lim_{s→∞} \frac{dw(s)}{ds} = 0

holds because \frac{dw(s)}{ds} is a monotone increasing function and w(s) is bounded. Thus, we infer \frac{dw(s)}{ds} < 0 for any s ∈ (−∞, +∞) and sup \frac{dw(s)}{ds} = 0. By making use of the same argument, we consider \frac{dw(s)}{ds} on s ∈ (−∞, s_{0}] and we can conclude inf \frac{dw(s)}{ds} = 0. This is impossible because \frac{dw(s)}{ds} < 0 for any s ∈ (−∞, +∞) and \frac{dw(s)}{ds} is a monotone increasing function. Hence, we infer that w(s) is constant. Thus, the proof of Theorem 3.7 is completed.

4. Complete submanifolds

In the section 3, we consider complete hypersurfaces with constant scalar curvature in the Euclidean space E^{n+1}. It is natural and very important to study n-dimensional submanifolds with constant scalar curvature and higher codimension p in the Euclidean space E^{n+p}. But there are few results about it. In this section, we shall propose to study complete submanifolds with constant scalar curvature in the Euclidean space E^{n+p}. As a generalization of the result due to Normizu and Smyth [26], under the stronger conditions, Yano and Ishihara [33] proved the following:

Theorem 4.1 (Yano and Ishihara [33]). Let M^{n} be an n-dimensional (n > 2) complete submanifold with constant scalar curvature n(n − 1)r in the Euclidean space E^{n+p}. If the following conditions are satisfied, then M^{n} is isometric to the totally umbilical sphere S^{n}(r), the totally geodesic Euclidean space E^{n} or the generalized cylinder S^{n−k}(c) × E^{k} for 1 ≤ k ≤ n − 1.

(1) The sectional curvatures of M^{n} are non-negative,

(2) the mean curvature vector is parallel in the normal bundle,

(3) the normal bundle of M^{n} is flat.

Proof. Since the normal bundle is flat we can choose a local orthonormal frame field {e_{1}, ⋯, e_{n}} such that h_{ij}^{α} = λ_{i}^{α}δ_{ij} for any α. Since the mean curvature vector is parallel, from (2.17), we obtain

\frac{1}{2}ΔS = ∑_{i,j,k,α}(h_{ijk}^{α})^{2} + ∑_{α, i<j}(λ_{i}^{α} − λ_{j}^{α})^{2}K_{ij}.

By using the same assertion as in Theorem 3.1, we know that Theorem 4.1 is true.

In [6], we successfully prove the following:

Theorem 4.2 (Cheng [6]). Let M^{n} be an n-dimensional (n > 2) complete submanifold with constant scalar curvature n(n − 1)r in the Euclidean space E^{n+p}. If S ≤ \frac{n(n−1)r}{n−2} is satisfied, then M^{n} is isometric to the totally umbilical sphere S^{n}(r), the totally geodesic Euclidean space E^{n} or the generalized cylinder S^{n−1}(c) × E^{1}. Where S denote the squared norm of the second fundamental form of M^{n}.

Proof. If r = 0, then we have S = 0 on M^{n}. Namely, M^{n} is totally geodesic. Hence, M^{n} is isometric to the hyperplane E^{n} since M^{n} is complete. Next, we consider the case of r > 0. From Gauss equation, we know the mean curvature H > 0. Hence, the mean curvature vector is not vanish on M^{n}. Thus, we know that e_{n+1} = \frac{h}{H} is a normal vector field defined globally on M^{n}. We define S_{1} and S_{2} by

S_{1} := ∑_{i,j=1}^{n}(h_{ij}^{n+1} − Hδ_{ij})^{2}, S_{2} := ∑_{α=n+2}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2}, (4.1)

respectively. Then, S_{1} and S_{2} are functions defined on M^{n} globally, which do not depend on the choice of the orthonormal frame {e_{1}, ⋯, e_{n}}. And

S − nH^{2} = S_{1} + S_{2}. (4.2)

From the definition of the mean curvature vector h, we know nH = ∑_{i=1}^{n} h_{ii}^{n+1} and ∑_{i=1}^{n} h_{ii}^{α} = 0 for n+2 ≤ α ≤ n+p on M^{n}. From (2.17), the Gauss equation (2.8) and (2.12), we can infer

\frac{1}{2}ΔS_{2} = ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} (4.3)

+ nH ∑_{α=n+2}^{n+p} trace(H_{n+1}H_{α}^{2}) − ∑_{α=n+2}^{n+p}[trace(H_{n+1}H_{α})]^{2}

− ∑_{α,β=n+2}^{n+p} N(H_{α}H_{β} − H_{β}H_{α}) − ∑_{α,β=n+2}^{n+p}[trace(H_{α}H_{β})]^{2}

+ ∑_{α=n+2}^{n+p} trace(H_{n+1}H_{α})^{2} − ∑_{α=n+2}^{n+p} trace(H_{n+1}^{2}H_{α}^{2}).

From the following Lemma 4.1 and Lemma 4.2, we have

\frac{1}{2}ΔS_{2} ≥ ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} (4.4)

+ (nH^{2} − \sqrt{\frac{n}{n − 1}}(n − 2)H\sqrt{S_{1}} − S_{1} − \frac{3}{2}S_{2})S_{2}.

Lemma 4.1 (Cheng [6]). (1). Let a_{1}, ⋯, a_{n} and b_{ij} for i, j = 1, ⋯, n be real numbers satisfying ∑_{i=1}^{n} a_{i} = 0, ∑_{i=1}^{n} b_{ii} = 0, ∑_{i,j=1}^{n} b_{ij}^{2} = b and b_{ij} = b_{ji} for i, j = 1, ⋯, n. Then

−(∑_{i=1}^{n} b_{ii}a_{i})^{2} + ∑_{i,j=1}^{n} b_{ij}^{2}a_{i}a_{j} − ∑_{i,j=1}^{n} b_{ij}^{2}a_{i}^{2} ≥ − ∑_{i=1}^{n} a_{i}^{2}b.

(2). Let b_{i} for i = 1, ⋯, n be real numbers satisfying ∑_{i=1}^{n} b_{i} = 0 and ∑_{i=1}^{n} b_{i}^{2} = B. Then

∑_{i=1}^{n} b_{i}^{4} − \frac{B^{2}}{n} ≤ \frac{(n − 2)^{2}}{n(n − 1)}B^{2}.

(3). Let a_{i} and b_{i} for i = 1, ⋯, n be real numbers satisfying ∑_{i=1}^{n} a_{i} = 0 and ∑_{i=1}^{n} a_{i}^{2} = a. Then

∑_{i=1}^{n} a_{i}b_{i}^{2} ≥ −\sqrt{∑_{i=1}^{n} b_{i}^{4} − \frac{(∑_{i=1}^{n} b_{i}^{2})^{2}}{n}} \sqrt{a}.

Lemma 4.2 (see [23]). For symmetric matrices H_{1}, ⋯, H_{p} (p ≥ 2), put S_{αβ} = trace(H_{α}H_{β}), S = ∑_{α=1}^{p} S_{αα} and N(H_{α}) = trace(^{t}H_{α}H_{α}). Then

∑_{α,β=1}^{p} N(H_{α}H_{β} − H_{β}H_{α}) + ∑_{α,β=1}^{p} S_{αβ}^{2} ≤ \frac{3}{2}S^{2}.

Hence, we infer

\frac{1}{2}ΔS_{2} ≥ ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} (4.5)

+ (nH^{2} − \sqrt{\frac{n}{n − 1}}(n − 2)H\sqrt{S_{1}} − S_{1} − \frac{3}{2}S_{2})S_{2}

≥ ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2}

+ (nH^{2} − \frac{n(n − 2)}{2(n − 1)}H^{2} − \frac{n − 2}{2}S_{1} − S_{1} − \frac{3}{2}S_{2})S_{2}

= ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2}

+ (nH^{2} − \frac{n(n − 2)}{2(n − 1)}H^{2} + \frac{n^{2}H^{2}}{2} − \frac{n}{2}S + \frac{(n − 3)}{2}S_{2})S_{2}

= ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} + \{\frac{n}{2}(\frac{n^{2}H^{2}}{n − 1} − S) + \frac{(n − 3)}{2}S_{2}\}S_{2}

≥ ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} + \{\frac{(n − 3)}{2}S_{2}\}S_{2} ≥ 0.

Since S ≤ \frac{n(n−1)r}{n−2}, from (2.9), we know that the Ricci curvature of M^{n} is bounded from below. By applying the following Generalized Maximum Principle due to Omori [24] and Yau [36] to the function S_{2}, we have that there exists a sequence {p_{k}} ⊂ M such that

(4.6)  lim_{k→∞} S_{2}(p_{k}) = sup S_{2} and lim_{k→∞} sup ΔS_{2}(p_{k}) ≤ 0.

Generalized Maximum Principle (Omori [24] and Yau [36]). Let M^{n} be a complete Riemannian manifold whose Ricci curvature is bounded from below and f ∈ C^{2}(M) a function bounded from above on M^{n}. Then for any ϵ > 0, there exists a point p ∈ M^{n} such that

f(p) ≥ sup f − ϵ, ∥gradf∥(p) < ϵ, Δf(p) < ϵ.

Since S ≤ \frac{n(n−1)r}{n−2}, we know that \{h_{ij}^{α}(p_{k})\}, for any i, j = 1, 2, ⋯, n and any α = n + 1, ⋯, n + p, is a bounded sequence. Hence, we can assume lim_{k→∞} h_{ij}^{α}(p_{k}) = h̃_{ij}^{α}, if necassary, we can take a subsequence. From (4.5) and (4.6), by taking limitation of (4.5), we know that all of inequalities is equalities. Hence, sup S_{2} = 0 for n > 3. When n = 3, if sup S_{2} ≠ 0, we know lim_{k→∞}(\frac{n^{2}H^{2}}{n−1} − S)(p_{k}) = 0 and lim_{k→∞} \sqrt{\frac{n}{n−1}}H(p_{k}) = lim_{k→∞} \sqrt{S_{1}(p_{k})}. Let lim_{k→∞} H(p_{k}) = H̃, lim_{k→∞} S(p_{k}) = S̃. lim_{k→∞} S_{1}(p_{k}) = S̃_{1}. We have \frac{n^{2}H̃^{2}}{n−1} = S̃, \frac{n}{n−1} H̃^{2} = S̃_{1} and S̃ = sup S_{2} +S̃_{1} +nH̃^{2} = S̃+sup S_{2}. This is impossible. Hence, we obtain sup S_{2} = 0. That is, S_{2} = 0 on M^{n}. From (4.5), we have

∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} = 0 (4.7)

on M^{n}. From (2.13), we have, for any α ≠ n + 1,

∑_{i,k=1}^{n} h_{iik}^{α}ω_{k} = −nHω_{αn+1}.

Hence, (4.7) yields ω_{αn+1} = 0 for any α. Thus, we know that e_{n+1} is parallel in the normal bundle T^{⊥}(M^{n}) of M^{n}. Hence, if we denote by N_{1} the normal subbundle spanned by e_{n+2}, e_{n+3}, ⋯, e_{n+p} of the normal bundle of M^{n}, then M^{n} is totally geodesic with respect to N_{1}. Since the e_{n+1} is parallel in the normal bundle, we know that the normal subbundle N_{1} is invariant under parallel translation with respect to the normal connection of M^{n}. Then from the Theorem 1 in [35], we conclude that M^{n} lies in a totally geodesic submanifold E^{n+1} of E^{n+p}. We denote by H′ the mean curvature of M^{n} in E^{n+1}. Since E^{n+1} is totally geodesic in E^{n+p}, we have H = H′, that is, the mean curvature H′ of M^{n} in E^{n+1} is the same as in E^{n+p}. We also know that the squared norm S′ of the second fundamental form of M^{n} in E^{n+1} is the same as in E^{n+p}. Hence, we imply S′ ≤ \frac{n^{2}(H′)^{2}}{n−1} and H′ ≠ 0. We choose a local orthonormal frame field {e_{1}, ⋯, e_{n}} such that h_{ij} = λ_{i}δ_{ij} for i, j = 1, 2, ⋯, n. Where h_{ij} and λ_{i} denote components of the second fundamental form and principal curvatures of M^{n} in E^{n+1}, respectively.

Thus, we obtain

∑_{i=1}^{n}(λ_{i})^{2} ≤ \frac{(∑_{i=1}^{n} λ_{i})^{2}}{n − 1} .

From the Lemma 4.1 in Chen [3, p.56], we have, for any i, j,

λ_{i}λ_{j} ≥ 0.

Hence, we have the sectional curvature of M^{n} is nonnegative. Thus, M^{n} is a complete hypersurface in E^{n+1} with constant scalar curvature and its sectional curvature is nonnegative. From the Theorem 3.2 due to S.Y. Cheng and S.T. Yau [9], we know that our Theorem 4.2 is true.

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Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan

E-mail address: cheng@ms.saga-u.ac.jp