COMPLETE HYPERSURFACES IN A EUCLIDEAN

SPACE R^{n+1} WITH CONSTANT SCALAR CURVATURE*

Qing-Ming Cheng

Dedicated to professor Katsuhiro Shiohama for his 60th birthday

Abstract. In this paper, we consider n-dimensional oriented complete hypersurfaces with constant scalar curvature of a Euclidean space R^{n+1}. We characterize the hypersurface S^{k}(c) × R^{n−k} in a Euclidean space R^{n+1} and show that Yau’s conjecture is true for the class of oriented compact locally conformally flat hypersurfaces with constant scalar curvature in a Euclidean space R^{n+1}

1. Introduction

The study of hypersurfaces in Euclidean spaces has a long and interesting history. It is obvious that the mean curvature and scalar curvature of a given hypersurface in Euclidean spaces are the basic invariants with important geometric meaning.

In 1900, Liebmann [10] proved that a strictly convex, compact surface of constant mean curvature in a Euclidean space R^{3} is necessarily a standard round sphere. In 1951, H. Hopf [8] proved a much stronger theorem, namely, the only possible differentiable immersions of sphere S^{2}(c) into R^{3} with constant mean curvature are exactly those round spheres. And he proposed the following:

Hopf’s conjecture. Let M be an oriented compact hypersurface with constant mean curvature in R^{n+1}. Must M be the standard round sphere?

To this conjecture, A. D. Alexandrov [1] obtained an important result. That is, he showed that the conjecture is true if M is an embedded hypersurface in R^{n+1}. The basic problem of whether there exist any other oriented compact hypersurfaces in R^{n+1} with constant mean curvature was open until W.Y. Hsiang [9] and Wente [17] discovered the examples which are not standard round sphere. By making use of the study of generalized rotational hypersurfaces of O(k) × O(l)-type, W. Y. Hsiang [9] constructed a family of important examples of constant mean curvature immersions

Typeset by AMS-TEX

Key words and phrases. constant scalar curvature, principal curvature, geodesic and locally conformally flat hypersurface.

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

1991 Mathematics Subject Classification 53C20, 53C42

of n (n > 2) dimensional sphere S^{n}(c) into R^{n+1}. In 1986, Wente [17] constructed an immersion of a 2-dimensional oriented compact surface of genus one in R^{3} with constant mean curvature. Hence, the assumption of embedding in the theorem of Alexandrov is a necessity which can not be replaced by immersion.

On the other hand, S. Y. Cheng and S. T. Yau [6] investigated oriented complete hypersurfaces with constant scalar curvature in Euclidean spaces. They proved that if M is an oriented complete hypersurface in R^{n+1} with constant scalar curvature and the sectional curvature is non-negative, then M is isometric to a standard round sphere or a generalized cylinder S^{n−k}(c) × R^{k}. In 1982, Yau [18] proposed to study oriented compact hypersurfaces with constant scalar curvature in R^{n+1} and conjectured the following:

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

In 1977, Reilly [13] studied the Hessian of a function fϵC^{∞}(Ω̄), where Ω is a bounded domain in R^{n+1}. He obtained a so-called Reilly’s formula. As an application of this formula, he gave a much simpler proof of the theorem of Alexandrov. By following a modification of Reilly’s proof for the Alexandrov theorem, A. Ros [14] showed that Yau’s conjecture is also true if M is an embedded hypersurface. On the other hand, the topological types of compact locally conformally flat hypersurfaces in R^{n+1} (n > 3) were determined by do Carmo, Dajczer and Mercuri [7]. They also described the extrinsic geometry of compact locally conformally flat hypersurfaces in R^{n+1}. Finally, Pinkall [12] determined their intrinsic conformal geometry. He proved that every compact locally conformally flat hypersurface in R^{n+1} (n > 3) is conformally equivalent to a classical Schottky manifold. Since Yamabe’s problem was solved, that is, any metric on a compact manifold can be conformally deformed to a metric of constant scalar curvature, it is natural to ask whether there exist classical Schottky manifolds with constant scalar curvature such that they can be immersed into R^{n+1} as locally conformally flat hypersurfaces? Namely, is Yau’s conjecture true for the class of locally conformally flat hypersurfaces. In this paper, we shall consider oriented complete n-dimensional hypersurfaces with constant scalar curvature in R^{n+1} and give an affirmative answer for the above problem in the Section 3, that is, Yau’s conjecture is true for the class of locally conformally flat hypersurfaces. We also characterize hypersurfaces S^{n−1}(c) × R in the Section 4.

Acknowledgement. The main result of this work was done while the author was visiting the Department of Mathematics of Saga University in 1999. I would like to thank Professor K. Shiohama and Professor S. Ishikawa for their interesting in this problem. I owe special thanks to Professor K. Shiohama for his valuable discussion and suggestion. I also express my appreciation to my colleague Professor M. Yamasaki for many valuable conversations.

2. Preliminaries

Let M be an n-dimensional hypersurface in a Euclidean space R^{n+1} with constant scalar curvature n(n−1)r. We take a local orthonormal frame field {e_{1}, …, e_{n+1}} in R^{n+1}, restricted to M, so that e_{1}, …, e_{n} are tangent to M. Let ω_{1}, ⋯, ω_{n+1} denote the dual coframe fields in R^{n+1}. We shall make use of the following convention on the ranges of indices:

1 ≤ A, B, C, ⋯, ≤ n + 1; 1 ≤ i, j, k, ⋯, ≤ n.

Then the structure equations of R^{n+1} are given by

dω_{A} = ∑_{B=1}^{n+1} ω_{AB} ∧ ω_{B}, ω_{AB} + ω_{BA} = 0,

dω_{AB} = ∑_{C=1}^{n+1} ω_{AC} ∧ ω_{CB}.

Then, in M,

ω_{n+1} = 0.

It follows from Cartan’s Lemma that

(2.1)  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

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

respectively. M is said to be totally umbilical if the h_{ij} can be expressed as h_{ij} = Hδ_{ij}. The structure equations of M are given by

dω_{i} = ∑_{k=1}^{n} ω_{ik} ∧ ω_{k}, ω_{ij} + ω_{ji} = 0,

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

(2.2)  R_{ijkl} = (h_{ik}h_{jl} − h_{il}h_{jk}),

where Ω_{ij} (resp. R_{ijkl}) denotes the Riemannian curvature form (resp. the components of the Riemannian curvature tensor) of M. From the above equation, we have

(2.3)  n(n − 1)r = n^{2}H^{2} − S

where n(n − 1)r is the scalar curvature of M and S = ∑_{ij=1}^{n} h_{ij}^{2} is the squared norm of the second fundamental form of M.

We choose e_{1}, …, e_{n} such that h_{ij} = λ_{i}δ_{ij}. Then we have from (2.1)

(2.4)  ω_{n+1i} = λ_{i}ω_{i}, i = 1, 2, ⋯, n.

Hence, we have from the structure equations of M

dω_{n+1i} = dλ_{i} ∧ ω_{i} + λ_{i}dω_{i}

= dλ_{i} ∧ ω_{i} + λ_{i} ∑_{j=1}^{n} ω_{ij} ∧ ω_{j}.

Therefore, from the structure equations of R^{n+1}, we obtain

dω_{n+1i} = ∑_{j=1}^{n} ω_{n+1j} ∧ ω_{ji}

= ∑_{j=1}^{n} λ_{j}ω_{ij} ∧ ω_{j}.

We infer

(2.5)  dλ_{i} ∧ ω_{i} + ∑_{j=1}^{n}(λ_{i} − λ_{j})ω_{ij} ∧ ω_{j} = 0.

We define

(2.6)  ϕ_{ij} = (λ_{i} − λ_{j})ω_{ij}.

Then ϕ_{ij} = ϕ_{ji}. From (2.5) and (2.6), we obtain

(2.7)  (ϕ_{ij} + δ_{ij}dλ_{j}) ∧ ω_{j} = 0.

By E. Cartan’s lemma, we conclude

(2.8)  ϕ_{ij} + δ_{ij}dλ_{j} = ∑_{k} T_{ijk}ω_{k},

where T_{ijk} are uniquely determined functions such that

(2.9)  T_{ijk} = T_{ikj}.

In [11], Otsuki proved the following:

Theorem ( Otsuki [11]). Let M be a hypersurface in a Euclidean space R^{n+1} such that the multiplicities of the principal curvatures are constant. Then the distribution of the space of the principal vector fields correspending to each principal curvature is completely integrable. In particular, if the multiplicity of a principal curvature is greater than 1, then this principal curvature is constant on each integral submanifold of the corresponding to distribution of the space of the principal vector fields.

From the above theorem of Otsuki, we have

Corollary 2.1. Let M be an n-dimensional oriented complete hypersurface in a Euclidean space R^{n+1} with constant scalar curvature and with two distinct principal curvatures. If the multiplicities of these two distinct principal curvatures are greater than 1, then M is isometric to Riemannian product R^{k} × S^{n−k}(c), 1 < k < n − 1.

Proof. Since M has only two distinct principal curvatures and the scalar curvature is constant, from the Theorem of Otsuki, we obtain that these two distinct principal curvatures are constant on M. Hence, Corollary 2.1 is true from a result of E. Cartan.

3. Complete locally conformally flat hypersurfaces in R^{n+1}

From now on, we assume that M is an n-dimensional oriented complete hypersurface with constant scalar curvature and with two distinct principal curvatures in R^{n+1}. If multiplicities of these two distinct principal curvatures are all greater than 1, from Corollary 2.1 we know that M is isometric to Riemannian product R^{k} × S^{n−k}(c), 1 < k < n − 1. Hence, we shall assume that one of these two distinct principal curvatures is simple, that is, we assume

(3.1)  λ_{1} = λ_{2} = ⋯ = λ_{n−1} = λ and λ_{n} = μ.

Since the scalar curvature n(n − 1)r is constant, from Gauss equation (2.3), we get

(3.2)  n(n − 1)r = (n − 1)(n − 2)λ^{2} + 2(n − 1)λμ.

Theorem 3.1. If M is an n-dimensional oriented complete hypersurface (n > 2) in R^{n+1} with constant scalar curvature n(n − 1)r and with two distinct principal curvatures one of which is simple, then M is isometric to one of the following hypersurfaces:

(1) R^{n−1} × S^{1}(c),

(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\{log |λ^{2}−r|^{\frac{1}{n}} \}}{ds}}\}^{2} + λ^{2} and M^{1} is a plane curve. And w = \{λ^{2} − r\}^{− \frac{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.

In order to prove the Theorem 3.1, we prepare a lemma.

Let us denote the integral submanifold through x ∈ M corresponding to λ by M_{1}^{n−1}(x).

Lemma 3.1. If λ ≠ 0, then

(1) the integral submanifold M_{1}^{n−1}(x) through x ∈ M corresponding to λ is umbilical in M and R^{n+1},

(2) the integral curve of the principal vector field e_{n} corresponding to the principal curvature μ is a geodesic.

Proof. Because of λ ≠ 0 on M, (3.2) yields

(3.3)  μ = \frac{nr}{2λ} − \frac{n − 2}{2}λ.

Put

(3.4)  dλ = ∑_{k=1}^{n} λ_{,k}ω_{k}, dμ = ∑_{k=1}^{n} μ_{,k}ω_{k}.

From the Theorem of Otsuki in the Section 2, we imply

(3.5)  λ_{,1} = λ_{,2} = ⋯ = λ_{,n−1} = 0 on M_{1}^{n−1}(x).

From (3.3),

(3.6)  dμ = [−\frac{nr}{2λ^{2}} − \frac{n − 2}{2}]dλ

holds. Hence, we also have

(3.7)  μ_{,1} = μ_{,2} = ⋯ = μ_{,n−1} = 0 on M_{1}^{n−1}(x).

In this case, we may consider locally λ is a function of the arc length s of the integral curve of the principal vector field e_{n} corresponding to the principal curvature μ. From (2.8) and (3.5), we have, for 1 ≤ j ≤ n − 1,

dλ = dλ_{j} = ∑_{k=1}^{n} T_{jjk}ω_{k} (3.8)

= ∑_{k=1}^{n−1} T_{jjk}ω_{k} + T_{jjn}ω_{n}

= λ_{,n}ω_{n}.

Hence, we obtain

(3.9)  T_{jjk} = 0, for 1 ≤ k ≤ n − 1, and T_{jjn} = λ_{,n}.

According to (2.8) and (3.7), we have

dμ = dλ_{n} = ∑_{k=1}^{n} T_{nnk}ω_{k} (3.10)

= ∑_{k=1}^{n−1} T_{nnk}ω_{k} + T_{nnn}ω_{n}

= ∑_{i=1}^{n} μ_{,i}ω_{i} = μ_{,n}ω_{n}.

Hence, we obtain

(3.11)  T_{nnk} = 0, for 1 ≤ k ≤ n − 1, and T_{nnn} = μ_{,n}.

From (3.6), we infer

(3.12)  T_{nnn} = μ_{,n} = [−\frac{nr}{2λ^{2}} − \frac{n − 2}{2}]λ_{,n}.

From the definition of ϕ_{ij}, if i ≠ j we have ϕ_{ij} = 0 for 1 ≤ i ≤ n−1 and 1 ≤ j ≤ n−1. Therefore, from (2.8), if i ≠ j and 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ n − 1 we have

(3.13)  T_{ijk} = 0, for any k.

According to (2.8), (3.9), (3.11), (3.12) and (3.13), we obtain

ϕ_{jn} = ∑_{k=1}^{n} T_{jnk}ω_{k} (3.14)

= T_{jjn}ω_{j} + T_{jnn}ω_{n} = λ_{,n}ω_{j}.

Since the λ and the μ are two distinct principal curvatures of M, we know

λ − μ = n\frac{λ^{2} − r}{2λ} ≠ 0.

The formula (2.6) , (3.3) and (3.14) yield

(3.15)  ω_{jn} = \frac{ϕ_{jn}}{λ − μ} = \frac{λ_{,n}}{λ − μ}ω_{j}

= \frac{λ_{,n}}{λ − \{\frac{nr}{2λ} − \frac{n−2}{2}λ\}}ω_{j}

= \frac{2λλ_{,n}}{n\{λ^{2} − r\}}ω_{j}.

Hence, from the structure equations of M we conclude

dω_{n} = ∑_{k=1}^{n−1} ω_{k} ∧ ω_{kn} = 0.

Thus, we may put ω_{n} = ds. According to (3.8) and (3.10), we have

dλ = λ_{,n}ds and λ_{,n} = \frac{dλ}{ds}

and

dμ = μ_{,n}ds and μ_{,n} = \frac{dμ}{ds}.

Hence,

(3.16)  ω_{jn} = \frac{2λλ_{,n}}{n\{λ^{2} − r\}}ω_{j}

= \frac{2λ\frac{dλ}{ds}}{n\{λ^{2} − r\}}ω_{j}

= \frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}ω_{j}

holds. This means that the integral submanifold M_{1}^{n−1}(x) corresponding to λ and s is umbilical in M and R^{n+1}.

On the other hand, from (3.16) we have

∇_{e_{n}}e_{n} = ∑_{k=1}^{n} ω_{ni}(e_{n})e_{i} = 0.

According to the definition of geodesic, we know that the integral curve of the principal vector field e_{n} corresponding to the principal curvature μ is a geodesic. This finished the proof of Lemma 3.1

Proof of Theorem 3.1. If λ = 0 at some point, then r = 0 at this point from (3.2). Since r is constant, we know r = 0 on M. Since these principal curvatures λ and μ are continuous on M, from (3.2) and r = 0 we obtain λ = 0 on M. Hence, the sectional curvature of M is not less than 0 from Gauss equation (2.2). From the result of S. Y. Cheng and Yau in [6], we know that M is isometric to a totally geodesic hypersurface R^{n} or the Riemannian product R^{n−1} × S^{1}(c) . Since M has two distinct principal curvatures, we have that M is isometric to the Riemannian product R^{n−1} × S^{1}(c).

Next, we assume λ ≠ 0 on M. From the structure equations of R^{n+1}, we have

dω_{jn} = ∑_{k=1}^{n−1} ω_{jk} ∧ ω_{kn} + ω_{jn+1} ∧ ω_{n+1n}

= \frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds} ∑_{k=1}^{n−1} ω_{jk} ∧ ω_{k} − λμω_{j} ∧ ds.

From (3.16) we obtain

dω_{jn} = \frac{d^{2}\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds^{2}}ds ∧ ω_{j} + \frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}dω_{j}

= \frac{d^{2}\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds^{2}}ds ∧ ω_{j} + \frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds} ∑_{k=1}^{n} ω_{jk} ∧ ω_{k}

= [−\frac{d^{2}\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds^{2}} + \{\frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}\}^{2}]ω_{j} ∧ ds

+ \frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds} ∑_{k=1}^{n−1} ω_{jk} ∧ ω_{k}

The above two equalities yield

(3.17)  \frac{d^{2}\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds^{2}} − \{\frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}\}^{2} − λμ = 0.

From (3.3), we obtain

\frac{d^{2}\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds^{2}} (3.18)

− \{\frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}\}^{2} + \frac{n − 2}{2}\{λ^{2} − r\} − r = 0.

From the formula

λ − μ = n\frac{λ^{2} − r}{2λ} ≠ 0,

we know λ^{2} − r ≠ 0. If λ^{2} − r < 0, from the above formula we know

λ^{2} − μλ = \frac{n}{2}\{λ^{2} − r\} < 0,

and r > 0. Hence,

μλ > λ^{2}.

According to the Gauss equation (2.2), we know that the sectional curvature of M is not less than λ^{2} > 0 and r > 0. Thus, M is a totally umbilical hypersurface. This is impossible because we assumed that M is of two distinct principal curvatures. Hence, λ^{2} − r > 0. Defining w = \{λ^{2} − r\}^{− \frac{1}{n}}, we obtain from the above equation

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

Integrating the equation (3.19), we have

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

where C is the integral constant.

We consider the frame {e_{1}, e_{2}, ⋯, e_{n}, e_{n+1}} in the Euclidean space R^{n+1}. Then, from ( 2.4), (3.16) and (3.17), we have

de_{i} = ∑_{j=1}^{n−1} ω_{ij}e_{j} + ω_{in}e_{n} + ω_{in+1}e_{n+1}

= ∑_{j=1}^{n−1} ω_{ij}e_{j} + \frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}ω_{i}e_{n} − λω_{i}e_{n+1}

= ∑_{j=1}^{n−1} ω_{ij}e_{j} + [\frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}e_{n} − λe_{n+1}]ω_{i},

d\{\frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}e_{n} − λe_{n+1}\}

= \{\frac{d^{2}\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds^{2}}e_{n} − \frac{dλ}{ds}e_{n+1}\}ds

+ \frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}(∑_{j=1}^{n−1} ω_{nj}e_{j} + ω_{nn+1}e_{n+1})

− λ(∑_{j=1}^{n−1} ω_{n+1j}e_{j} + ω_{n+1n}e_{n})

= \{\frac{d^{2}\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds^{2}}e_{n} − \frac{dλ}{ds}e_{n+1}\}ds

+ \frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}(∑_{j=1}^{n−1} ω_{nj}e_{j} − μω_{n}e_{n+1})

− λ(λ ∑_{j=1}^{n−1} ω_{j}e_{j} + μω_{n}e_{n})

≡ [\frac{d^{2}\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds^{2}} − λμ]e_{n}ω_{n}

− \{\frac{dλ}{ds} + \frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}μ\}e_{n+1}ω_{n} (mod\{e_{1}, ⋯, e_{n−1}\})

= \frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}\{\frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}e_{n} − λe_{n+1}\}ds.

Putting

Z = e_{1} ∧ ⋯ ∧ e_{n−1} ∧ (\frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}e_{n} − λe_{n+1}),

these above formulas yield

dZ = \frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds}Zds.

This shows that n-vector Z in R^{n+1} is constant along M_{1}^{n−1}(x). Hence, there exists an n-dimensional linear subspace E^{n}(s) in R^{n+1} containing M_{1}^{n−1}(x). The n-vector Z only depends on s and by integrating it, we obtain

Z = \{\frac{λ^{2}(s) − r}{λ^{2}(s_{0}) − r} \}^{\frac{1}{n}} Z(s_{0}).

Therefore, we have that E^{n}(s) is parallel to E^{n}(s_{0}) in R^{n+1}. Hence, the planes of the integral curves of the principal vector field e_{n} corresponding to the principal curvature μ intersect in a line. Thus, M is a hypersurface of revolution.

From

dω_{ij} − ∑_{k=1}^{n−1} ω_{ik} ∧ ω_{kj}

= ω_{in} ∧ ω_{nj} + ω_{in+1} ∧ ω_{n+1j}

= −\{(\frac{d\{log |λ^{2} − r|^{\frac{1}{n}} \}}{ds})^{2} + λ^{2}\}ω_{i} ∧ ω_{j},

we know that the curvature of M_{1}^{n−1}(x) is (^{\frac{d\{log |λ^{2}−r|^{\frac{1}{n}} \}}{ds}})^{2} + λ^{2} and M_{1}^{n−1}(x) is locally isometric to S^{n−1}(c(s)). Therefore, M is isometric to a complete hypersurface of revolution S^{n−1}(c(s)) × M^{1}.

If M is compact, then M^{1} is a closed plane curve. Since M is a hypersurface of revolution and M has two distinct principal curvatures, then the closed plane curve M^{1} can not intersect in the rotating axis which is a line. In this case, there must exsit some points such that λ = 0. This is impossible. Hence M is complete and noncompact. This completed the proof of Theorem 3.1.

Theorem 3.2. If M is an n-dimensional oriented complete locally conformally flat hypersurface (n > 3) in R^{n+1} with constant scalar curvature n(n − 1)r then M is isometric to one of the following hypersurfaces:

(1) a standard round sphere,

(2) R^{n},

(3) R^{n−1} × S^{1}(c),

(4) 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\{log |λ^{2}−r|^{\frac{1}{n}} \}}{ds}}\}^{2} + λ^{2} and M^{1} is a plane curve. And w = \{λ^{2} − r\}^{− \frac{1}{n}} satisfies the ordinary differential equation of order 2 (3.19).

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

Proof. From the Theorem 3.2, the assertion is true obviously.

Proof of Theorem 3.2. Since M is a locally conformally flat hypersurface in R^{n+1} and n > 3, from a result of Cartan-Schouten, [2], [15], we know that M is of at most two distinct principal curvatures and one of them has at least n−1 multiplicities. If M is of two distinct principal curvatures at each point, from the Theorem 3.1 we know that M is isometric to R^{n−1}×S^{1}(c) or 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\{log |λ^{2}−r|^{\frac{1}{n}} \}}{ds}}\}^{2} + λ^{2} and M^{1} is a plane curve, and w = \{λ^{2} − r\}^{− \frac{1}{n}} satisfies the ordinary differential equation of order 2 (3.19).

Next, we shall prove that if at some point, these principal curvatures are same, then M is isometric to a totally umbilical hypersurface, namely, M is isometric to a standard round sphere or a Euclidean space R^{n}. In fact, if at some points these principal curvatures of M are same and M is not totally umblical, then there exist an open set U ⊂ M at which M is of two distinct principal curvatures and one of them is simple. From the Lemma 3.1 (2), on U the integral curve C(s) of the principal vector field e_{n} corresponding to the principal curvature μ is a geodesic. Since M is complete, M is geodesicly complete. Hence, the geodesic C(s) can be expanded to infnite for both sides. If the geodesic C(s) does not go through an umbilical point, we know that M is a hypersurface of revolution from the proof of Theorem 3.1. Hence, M has no umbilical points. It is a contradiction. Hence, the geodesic C(s) must go through umbilical points. Let p = C(s_{0}) is the first umbilical point which the geodesic C(s) goes through. Then p = C(s_{0}) ∉ U and p = C(s_{0}) ∈ Ū. In U, w satisfies the differential equation (3.19). By integrating it, we have

(3.21)  (\frac{dw}{ds})^{2} = C − w^{2}(r + \frac{1}{w^{n}}) ≥ 0.

At the point p, we obtain λ^{2} = r from a simple calculation. According to the definition of w, we have that w^{−n} = λ^{2} − r and lim_{s→s_{0}} w(s) = +∞. Therefore, lim_{s→s_{0}}\{C − w^{2}(r + \frac{1}{w^{n}})\} = −∞ which is contradiction with (3.21). Hence, M is a totally umbilical hypersurface. This finished the proof of Theorem 3.2.

4. A characterization of Riemannian product S^{n−1}(c) × R

Let M be an n-dimensional hypersurface in R^{n+1}. As we stated in the Introduction, S.Y. Cheng and Yau [6] proved that let M be an n-dimensional oriented complete hypersurface with constant scalar curvature n(n − 1)r, if the sectional curvature of M is non-negative, then M is isometric to either a totally umbilical hypersurface or a Riemannian product S^{k}(c) × R^{n−k} 1 ≤ k ≤ n − 1, where S^{k}(c) denotes a sphere of constant curvaturte c. In the Section 3, we investigated oriented complete hypersurfaces with constant scalar curvature and with two distinct principal curvatures. And we know that there are many complete hypersurfaces of this kind which are not isometric to S^{n−1}(c) × R and S^{1}(c) × R^{n−1}. In this section, we shall characterize the Riemannian product S^{n−1}(c) × R and S^{1}(c) × R^{n−1}.

Theorem 4.1. Let M be an n-dimensional oriented complete hypersurface in R^{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 is isometric to the Riemannian product S^{n−1}(c) × R or S^{1}(c) × R^{n−1}.

Corollary 4.1. Let M be an n-dimensional (n > 3) oriented complete locally conformally flat hypersurface with constant scalar curvature n(n − 1)r in R^{n+1}. If S ≥ \frac{n(n−1)r}{n−2} , then M is isometric to one of the following hypersurfaces:

(1) R^{n},

(2) S^{1}(c) × R^{n−1},

(3) S^{n−1}(c) × R.

Proof. Since M is a locally conformally flat hypersurface in R^{n+1} and n > 3, from a result of Cartan-Schouten, we know that M is of at most two distinct principal curvatures and one of them is of at least n − 1 multiplicities.

(a). In the case r = 0.

From (3.2), we have λ = 0 on M or μ = −\frac{(n−2)λ}{2} on M.

If λ = 0 on M, from the Gauss equation (2.2) we have that the sectional curvature of M is non-negative. The result of S.Y. Cheng and Yau [6] yields that M is isometric to R^{n} or R^{n−1} × S^{1}(c).

If μ = −\frac{(n−2)λ}{2} on M, from the proof of Theorem 3.2 and the Theorem 4.1 we obtain that M is isometric to S^{1}(c) × R^{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 the Gauss equation, 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 is of two distinct principal curvatures and one of them is of n − 1 multiplicities. From the Theorem 4.1, we know that M is isometric to S^{n−1}(c) × R. That is, Corollary 4.1 is proved.

Proof of Theorem 4.1. If w is constant, then λ is constant because of w^{−n} = λ^{2} − r. Hence, the Theorem 4.1 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 (3.19), that is

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

we know that w(s) is a function defined in (−∞, +∞) because M is complete and the integral curve of the principal vector field e_{n} corresponding to the principal curvature μ is a geodesic from Lemma 3.1. If r ≤ 0, then from (4.1), 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}.

From the arguments in the Section 3, we know λ^{2} − r > 0 and we can assume λ > 0. Since

(4.2)  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 (4.1), 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 (4.1), we have

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

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

(4.4)  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 (4.1),

(4.5)  C > \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}, 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

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

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 4.1 is completed.

References

1. Alexandrov, A. D., Uniqueness theorems for surfaces in the large, V. Vestnik, Leningrad Univ. 13 (1958), 5-8.

2. Cartan, E., La deformation des hypersurfaces dans l’espace cinforme reel a n ≥ 5 dimensions, Bull. Soc. Math. France, 45 (1917), 57-121.

3. Cheng, Q. M., Hypersurfaces in a unit sphere S^{n+1}(1) with constant scalar curvature, Preprint 1999.

4. Cheng, Q. M. and Nakagawa, H., Totally umbilical hypersurfaces, Hiroshima Math. J. 20 (1990), 1-10.

5. Cheng, Q. M. and Wan, Q. R., Complete hypersurfaces of R^{4} with constant mean curvature, Monatshefte fur Math. 118 (1994), 171-204.

6. Cheng, S. Y. and Yau, S. T., Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), 195-204.

7. do Carmo, M. Dajczer, M. and Mercuri, F., Compact conformally flat hypersurfaces, Trans. Amer. Math. Soc. 288 (1985), 189-203.

8. Hopf, H., Differential Geometry in the large, vol. 1000, Lecture Notes in Math., Springer-Verlag, 1983.

9. Hsiang, W. Y., Generalized rotational hypersurfaces of constant mean curvature in the Euclidean spaces I, J. Diff. Geom. 17 (1982), 337-356.

10. Liebmann, H., Uder die Verbiegung der geschlossenen Flachen positier Krummung, Math. Ann. 53 (1900), 61-112.

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

12. Pinkall, U., Compact conformally flat hypersurfaces, in Conformal Geometry (Kulkarni, R.S. and Pinkall, U., eds), Aspects of Math., vol. E12, Max-Plamck-Ins. fur Math., 1988, pp. 217-236.

13. Reilly, R., Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J. 26 (1977), 459-472.

14. Ros, A., Compact hypersurfaces with constant scalar curvature and a congruence theorem, J. Diff. Geom. 27 (1988), 215-220.

15. Schouten, J. A., Uber die Konforme Abbildung n-dimensionaler Mannigfaltigkeiter mit quadratischer Maβ bestimmung auf eine Mannigfaltigkeit mit euklidischer Maβ bestimmung, Math. Z. 11 (1921), 58-88.

16. Suyama, Y., Explicit representation of compact conformally flat hypersurfaces, Tohoku Math. J. 50 (1998), 179-196.

17. Wente, H. C., Counterexample to a conjecture of H. Hopf, Pacific J. Math. 121 (1986), 193-243.

18. Yau, S. T. (ed), Seminar on Differential Geometry, vol. 102, Annals of Math. Studies, Princeton Univ. Press, 1982.

Qing-Ming Cheng Department of Mathematics, Faculty of Science, Josai University, Sakado, Saitama 350-0295, Japan e-mail: cheng@math.josai.ac.jp