COMPLETE SPACE-LIKE HYPERSURFACES IN LOCALLY SYMMETRIC LORENTZ SPACES

Jin Ok Baek, Qing-Ming Cheng and Young Jin Suh

Abstract. The purpose of this paper is to study complete space-like hypersurfaces with constant mean curvature in a locally symmetric Lorentz space satisfying some curvature conditions. We give an optimal estimate of the squared norm of the second fundamental form of such hypersurfaces. Furthermore, the totally umbilical hypersurfaces are characterized.

1. Introduction

Let M_{s}^{m} be an m-dimensional connected semi-Riemannian manifold of index s(≥0). It is called a semi-definite space of index s. In particular, M_{1}^{m} is called a Lorentz space. A hypersurface M of a Lorentz space is said to be space-like if the induced metric on M from that of the Lorentz space is positive definite. When the Lorentz space M_{1}^{m} is of constant curvature c, we call it Lorentz space form, denoted by M_{1}^{m}(c).

It is well known that a maximal space-like entire graph in Minkowski space R_{1}^{n+1} is a linear hyperplane (cf. Calabi [2] and Cheng-Yau [7]). As a generalization of the result above, Chouquet-Bruhat and et. [12] and Ishihara [14] proved that totally geodesic hypersurfaces are the only complete space-like maximal hypersurfaces in M_{1}^{n+1}(c), c ≥ 0.

On the other hand, it was pointed out by Marsden and Tipler[17] and Stumbles [23] that space-like hypersurfaces with constant mean curvature in arbitrary

Typeset by AMS-TEX 2 COMPLETE SPACE-LIKE HYPERSURFACES

2001 Mathematics Subject Classification. Primary 53C42; Secondary 53C15.

Key words and phrases: constant mean curvature, space-like hypersurface, second fundamental form and Locally symmetric Lorentz space

The First Author and the Third Author are Supported by Grant Proj. No. R14-2002-003-01000-0 from Korea Science & Engineering Foundation.

The Second Author is partially Supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture, Japan.

spacetime get interested in the relativity theory. Space-like hypersurfaces with constant mean curvature are convenient as initial hypersurfaces for the Cauchy problem in arbitrary spacetime and for studying the propagation of gravitational radiation. Hence, complete space-like hypersurfaces with constant mean curvature in a Lorentz space form M_{1}^{n+1}(c) are extensively investigated by many differential geometers in various view points; For example, Akutagawa [1] and Ramanathan [18] proved complete space-like hypersurfaces with constant mean curvature H in the de Sitter space S_{1}^{n+1}(c) must be totally umbilical if n^{2}H^{2} < 4(n − 1)c when n > 2, and H^{2} ≤ 1 when n = 2. For further development in this directions, see Cheng [3], Cheng [4], Cheng and Ishikawa [5], Cheng and Nakagawa [6], Ki, Kim and Nakagawa [15] Nishikawa [19], Montiel [18] and so on.

As standard models of complete space-like hypersurfaces with constant mean curvature in M_{1}^{n+1}(c), there are four classes of complete hypersurfaces H^{k}(c_{1}) × S^{n−k}(c_{2}), R^{k} × S^{n−k}(c_{2}), H^{k}(c_{1}) × R^{n−k} and H^{k}(c_{1}) × H^{n−k}(c_{2}), where k = 0, 1, ⋯, n, according as c > 0, = 0 or < 0. In particular, H^{1}(c_{1}) × M^{n−k}(c_{2}) is called a hyperbolic cylinder and H^{n−1}(c_{1}) × S^{1}(c_{2}) is called a spherical cylinder.

It is important and natural to study complete space-like hypersurfaces with constant mean curvature in the more general Lorentz spaces since they have important meaning in the relatively theory. First of all, we shall consider several examples of Lorentz spaces which are not Lorentz space form.

Example 1. We consider the semi-Riemannian product manifold

H_{1}^{k}(−\frac{c_{1}}{n})×M^{n+1−k}(c_{2}), c_{1} > 0.

Its sectional curvature is given by

K′(u_{1}, u_{b}) = −\frac{c_{1}}{n} , K′(u_{a}, u_{b}) = −\frac{c_{1}}{n} , K′(u_{a}, u_{r}) = 0, K′(u_{r}, u_{s}) = c_{2},

where a, b, ... = 2, ..k, r, s, .. = k + 1, ...., n + 1 − k, and u_{1} and u_{a}, u_{r} denotes time-like and space-like vector respectively.

Example 2. We consider the semi-Riemannian product manifold

R_{1}^{k}×S^{n+1−k}(1).

Its sectional curvature is given by

K′(u_{1}, u_{a}) = 0, K′(u_{a}, u_{b}) = 0, K′(u_{1}, u_{r}) = 0, K′(u_{r}, u_{s}) = 1,

where a, b, ... = 2, ..., k and r, s, .. = k + 1, ...., n + 1 − k. In particular, R_{1}^{1}×S^{n}(1) is so-called Einstein Static Universe. Of course, it is not Lorentzian space form.

Next, we shall show a general example of Lorentz space which is called Robertson-Walker spacetime.

Example 3. Let I denote an open interval of R_{1}^{1} and f > 0 a smooth function defined on the interval I. For a three-dimensional Riemannian manifold of constant curvature c, c = −1, 0, 1, we construct a Lorentz space M(c, f) as the warped product

M(c, f) = I×_{f}M^{3}(c),

which is called Robertson-Walker spacetime (See O’Neill [21] page 343–345). Thus, by a direct computation, we have that its sectional curvature is given by

K′(v, w) = (\frac{f′}{f})^{2} + \frac{c}{f^{2}}

for any space-like vector v and w and

K′(e_{0}, v) = \frac{f′′}{f}

for any time-like vector e_{0} and any space-like vector v.

In this paper, we shall consider n + 1-dimensional Lorentz spaces M′ of index 1. Let ∇′, K′ and R′ denote the semi-Riemannian connection, sectional curvature and the curvature tensor on M′, respectively. For constants c_{1}, c_{2} and c_{3}, we consider Lorentz spaces which satisfy the following:

(1) for any space-like vector u and any time-like vector v,

K′(u, v) = −\frac{c_{1}}{n} ,

(2) for any space-like vectors u and v,

K′(u, v) ≥ c_{2},

(3)

|∇′R′| ≤ \frac{c_{3}}{n} .

When M′ satisfies conditions (1) and (2), we shall call that M′ satisfies condition (*). When M′ satisfies conditions (1), (2) and (3), we shall call that M′ satisfies condition (**).

Remark 1. It can be easily seen that if the Lorentz space M′ is locally symmetric, then the condition (3) holds.

Remark 2. The Lorentz space form M_{1}^{n+1}(c) satisfies the conditions (*) and (**), where −\frac{c_{1}}{n} = c_{2} = c.

Remark 3. These examples 1 and 2 satisfy the conditions (*) and (**) and the example 3 also satisfies the conditions (*) and (**) if we choose an appropriated function f (See O’Neill [21] page 343–345).

In [9] and [24], authors investigated complete space-like hypersurfaces M in a Lorentz space satisfying condition (**). They estimated the squared norm of the second fundamental form of M under some conditions. In this paper, we shall prove the following:

Theorem 1. Let M is a complete space-like hypersurface with constant mean curvature H in an n + 1-dimensional locally symmetric Lorentz space M′ satisfying the condition (*).

(1) If n^{2}H^{2} < 4(n − 1)c, where c = 2c_{2} + \frac{c_{1}}{n}, then c > 0, S ≡ nH^{2} and M is totally umbilical, where S denotes the squared norm of the second fundamental form of M.

(2) If n^{2}H^{2} = 4(n − 1)c, then c ≥ 0 and either S ≡ nH^{2} and M is totally umbilical, or sup S = nc.

(3) If n^{2}H^{2} > 4(n − 1)c and c < 0, then either S ≡ nH^{2} and M is totally

umbilical, or

nH^{2} < sup S ≤ S_{max},

J.O. BAEK, Q.-M. CHENG AND Y.J. SUH 5

where S_{max} = \frac{n}{2(n−1)}[n^{2}H^{2} − 2(n − 1)c + (n − 2)|H|\{n^{2}H^{2} − 4(n − 1)c\}^{\frac{1}{2}} ].

(4) If n^{2}H^{2} > 4(n − 1)c and c ≥ 0, then either S ≡ nH^{2} and M is totally umbilical, or

S_{max} ≥ sup S \{ \begin{matrix}> nH^{2}, & if H^{2} ≥ c, \\ ≥ S_{min}, & if H^{2} < c. \end{matrix}

where S_{min} = \frac{n}{2(n−1)}[n^{2}H^{2} − 2(n − 1)c − (n − 2)|H|\{n^{2}H^{2} − 4(n − 1)c\}^{\frac{1}{2}} ]. (5)

S ≡ \frac{n}{2(n − 1)}[n^{2}H^{2} − 2(n − 1)c + (n − 2)|H|\{n^{2}H^{2} − 4(n − 1)c\}^{\frac{1}{2}} ]

if and only if M is an isoparametric hypersurface with two distinct principal curvatures one of which is simple.

Theorem 2. Let M be an n (n > 2) dimensional complete space-like hypersurface with constant mean curvature H in an n + 1-dimensional locally symmetric Lorentz space M′ satisfying the condition (*). If the sectional curvature of M is not less than −(c_{2} + \frac{c_{1}}{n}), then, c ≥ 0, where c = 2c_{2} + \frac{c_{1}}{n}. Furthermore, if H^{2} ≥ c and

S < \frac{n}{2(n − 1)}[n^{2}H^{2} − 2(n − 1)c + (n − 2)|H|\{n^{2}H^{2} − 4(n − 1)c\}^{\frac{1}{2}} ]

hold, then M is totally umbilical.

Remark 4. We should notice that when M′ is a Lorentz space form M_{1}^{n+1}(c), a part of the similar results in Theorems 1 and 2 was obtained in Cheng and Nakagawa [6] and Ki, Kim and Nakagawa [15].

Remark 5. Euclidean space R^{n} which is defined by x_{1} = x_{n+2} + t is a totally umbilical space-like hypersurface of S_{1}^{n+1}(c) in R_{1}^{n+2}, where {x_{1}, ..., x_{n+2}} is the natural coordinate system in R_{1}^{n+2}. The mean curvature H satisfies H^{2} = c.

Remark 6. We consider a family of space-like hypersurfaces H^{k}(c_{1})×S^{n−k}(c_{2}) of S_{1}^{n+1}(c) which is defined by

\begin{matrix}H^{k}(c_{1})×S^{n−k}(c_{2}) = \{ & (x, y)∈S_{1}^{n+1}(c)⊂R_{1}^{n+2} = R_{1}^{k+1}×R^{n−k+1} : \\ |x|^{2} = −\frac{1}{c_{1}} , |y|^{2} = \frac{1}{c_{2}} \} \end{matrix}

6 COMPLETE SPACE-LIKE HYPERSURFACES

where c_{1} < 0, c_{2} > 0 and \frac{1}{c_{1}} + \frac{1}{c_{2}} = \frac{1}{c}. When k > 1, it is not of non-negative curvature. The number of distinct principal curvatures of such a hypersurface is exactly two. A principal curvature is equal to (c − c_{1})^{\frac{1}{2}} with the multiplicity k and the other is equal to (c − c_{2})^{\frac{1}{2}} with multiplicity n − k. We can prove that

S < \frac{1}{2(n − 1)}[n\{n^{2}H^{2} − 2(n − 1)c\} + (n − 2)n|H|\{n^{2}H^{2} − 4(n − 1)c\}^{\frac{1}{2}} ],

and H^{2} ≥ c if we choose apropriated c_{1} and c_{2}. Hence, the assumption on sectional curvature in Theorem 2 is essential.

Remark 7. Hyperbolic cylinder H^{1}(c_{1})×S^{n−1}(c_{2}) has non-negative curvature and satisfies H^{2} ≥ c if we choose apropriated c_{1} and c_{2}. The squared norm S of the second fundamental form α satisfies S = \frac{1}{2(n−1)}[n\{n^{2}H^{2} − 2(n − 1)c\} + (n − 2)n|H|\{n^{2}H^{2}−4(n−1)c\}^{\frac{1}{2}} ]. Hence, the estimate of S in Theorem 2 is best possible.

Acknowledgement. Authors would like to express their gratitude to the referee for his/her valuable suggestions.

2. Preliminaries

First of all, we review basic formulas on space-like hypersurfaces in a Lorentz space. Let (M′, g′) be an (n + 1)-dimensional Lorentz space, i.e., an indefinite Riemannian manifold of index 1. Throughout this paper, manifolds are always assumed to be connected and geometric objects are assumed to be of class C^{∞}. For any point x in M′ we choose a local field of orthonormal frames {e_{A}} = {e_{0}, e_{1}, ⋯, e_{n}} on a neighborhood of x. Here and in the sequel, the following convention on the range of indices will be used throughout this paper, unless otherwise stated:

\begin{aligned}A, B, ⋯ & = 0, 1, ⋯, n, \\ i, j, ⋯ & = 1, ..., n. \end{aligned}

Let {ω_{A}} = {ω_{0}, ω_{1}, ⋯, ω_{n}} denote the dual frame fields of {e_{A}} on M′. Then metric tensor g′ of M′ satisfies g′(e_{A}, e_{B}) = ϵ_{A}δ_{AB}, where ϵ_{0} = −1 and ϵ_{j} = 1.

The canonical forms ω_{A} and the connection forms ω_{AB} of M′ satisfy the structure equations:

(2.1)  dω_{A} + ∑ ϵ_{B}ω_{AB}∧ω_{B} = 0, ω_{AB} + ω_{BA} = 0,
(2.2)  dω_{AB} + ∑ ϵ_{C}ω_{AC}∧ω_{CB} = Ω′_{AB},

(2.3)  Ω′_{AB} = −\frac{1}{2} ∑ ϵ_{C}ϵ_{D}R′_{ABCD}ω_{C}∧ω_{D},

where Ω′ = Ω′_{AB} (resp. R′_{ABCD}) denotes the Riemannian curvature form (resp. the components of the Riemannian curvature tensor R′) of M′. The components R′_{CD} of the Ricci tensor and the scalar curvature r′ are given by

(2.4)  R′_{CD} = ∑_{B} ϵ_{B}R′_{BCDB},

(2.5)  r′ = ∑_{A} ϵ_{A}R′_{AA},

respectively. The components R′_{ABCD;E} of the covariant derivative of the Riemannian curvature tensor R′ are defined by

\begin{matrix}∑_{E} ϵ_{E}R′_{ABCD;E}ω_{E} = dR′_{ABCD} \\ − ∑_{E} ϵ_{E}(R′_{EBCD}ω_{EA} + R′_{AECD}ω_{EB} \\ + R′_{ABED}ω_{EC} + R′_{ABCE}ω_{ED}). \end{matrix} (2.6)

A plane section P′ of the tangent space T_{x}M′ of M′ at any point x is said to be nondegenerate, provided that g_{x}|P′ is non-degenerate. It is easily seen that P′ is nondegenerate if and only if it has a basis {u, v} such that g(u, u)g(v, v) − g(u, v)^{2}≠0. The sectional curvature of the non-degenerate plane section P′ spanned by u and v is denoted by K′(P′) = K′(u, v). The semi-definite Riemannian manifold M′ is said to be of constant curvature if its sectional curvature K′(P′) is constant for all P′ and for all points of M′. M′ is called a semi-definite space form if it is of constant curvature. An m-dimensional semi-definite space form of constant

8 COMPLETE SPACE-LIKE HYPERSURFACES

curvature c and of index s is denoted by M_{s}^{m}(c). The standard models of semidefinite space forms are the following three kinds: the semi-definite Euclidean space R_{s}^{m}, the semi-definite spherical space S_{s}^{m}(c) or the semi-definite hyperbolic space H_{s}^{m}(c), according as c = 0, c > 0 or c < 0. The Riemannian curvature tensor R_{ABCD} of the semi-definite space form M_{s}^{m}(c) is given by

(2.7)  R_{ABCD} = cϵ_{A}ϵ_{B}(δ_{AD}δ_{BC} − δ_{AC}δ_{BD}).

Now, let (M′, g′) be an (n + 1)-dimensional Lorentz space and let M be an ndimensional space-like hypersurface of M′. We choose a local field of orthonormal frames {e_{A}} = {e_{0}, e_{1} ⋯, e_{n}} in such a way that restricted to M, e_{1}, ..., e_{n} are tangent to M and the other is normal to M. Namely, e_{1}, ⋯, e_{n} are space-like vectors and the other e_{0} is a time-like vector. Let {ω_{A}} be its dual frame field. Then the indefinite Riemannian metric tensor g′ of M′ is given by g′ = ∑_{A}ϵ_{A}ω_{A}⊗ω_{A}. The connection forms on M′ are denoted by ω_{AB}, where ϵ_{0} = −1 and ϵ_{j} = 1.

Restricting these forms to the space-like hypersurface M in M′, we have

(2.8)  ω_{0} = 0

and the induced metric g of M is given by g = ∑ ω_{j}⊗ω_{j}. From (2.1), (2.8) and the Cartan lemma, we have

(2.9)  ω_{0i} = ∑_{j} h_{ij}ω_{j}, h_{ij} = h_{ji}.

The quadratic form α = − ∑_{i,j} h_{ij}ω_{i}⊗ω_{j}⊗e_{0} with values in the normal bundle and H = \frac{1}{n} ∑_{j=1}^{n} h_{jj} are called second fundamental form and mean curvature of the hypersurface M, respectively. When principal curvatures of M are constant, M is called isoparametric. The connection forms {ω_{ij}} of M are characterized by the structure equation of M:

(2.10)  dω_{i} + ∑_{j} ω_{ij}∧ω_{j} = 0, ω_{ij} + ω_{ji} = 0,

\begin{matrix}d & ω_{ij} + ∑_{k} ω_{ik}∧ω_{kj} = Ω_{ij}, \\ Ω_{ij} = −\frac{1}{2} ∑_{k,l} R_{ijkl}ω_{k}∧ω_{l}. \end{matrix} (2.11)

From (2.3) and (2.11), we have Gauss equation

(2.12)  R_{ijkl} = R′_{ijkl} − (h_{il}h_{jk} − h_{ik}h_{jl}).

Components R_{ij} of Ricci tensor and scalar curvature r of M are given by

(2.13)  R_{ij} = ∑_{k=1}^{n} R′_{kijk} − nHh_{ij} + ∑_{k=1}^{n} h_{ik}h_{kj},

(2.14)  r = ∑_{j,k=1}^{n} R′_{kjjk} − n^{2}H^{2} + S,

where S = ∑_{i,j=1}^{n} h_{ij}^{2} denotes the squared norm of the second fundamental form of M.

By taking exterior differentiation of (2.9) and defining h_{ijk} by

(2.15)  ∑_{k} h_{ijk}ω_{k} = dh_{ij} − ∑_{k}(h_{kj}ω_{ki} + h_{ik}ω_{kj}),

we have Codazzi equation

(2.16)  h_{ijk} − h_{ikj} = R′_{0ijk}.

Similarly, defining h_{ijkl} by

(2.17)  ∑_{l} h_{ijkl}ω_{l} = dh_{ijk} − ∑_{l}(h_{ljk}ω_{li} + h_{ilk}ω_{lj} + h_{ijl}ω_{lk})

and differentiating (2.15) exteriorly, we have

∑k{dh_{ijk}∧ω_{k} + h_{ijk}(−∑_{l}ω_{kl}∧ω_{l})}
= − ∑_{k}[dh_{kj}∧ω_{ki} + h_{jk}{−∑_{l}ω_{kl}∧ω_{li}

− \frac{1}{2}∑_{l,m}R_{kilm}ω_{l}∧ω_{m}\} + dh_{ik}∧ω_{kj}

+ h_{ik}\{−∑_{l}ω_{kl}∧ω_{lj} − \frac{1}{2}∑_{l,m}R_{kjlm}ω_{l}∧ω_{m}\}].

Hence, we obtain Ricci formula for the second fundamental form of M:

(2.18)  h_{ijkl} − h_{ijlk} = −∑_{r}(h_{ir}R_{rjkl} + h_{jr}R_{rikl}).

10 COMPLETE SPACE-LIKE HYPERSURFACES

Now let us denote, by R′_{ABCD;E}, covariant derivative of R′_{ABCD}. Then, restricting on M, R′_{0ijk;l} is given by

(2.19)  R′_{0ijk;l} = R′_{0ijkl} + R′_{0i0k}h_{jl} + R′_{0ij0}h_{kl} + ∑_{m}R′_{mijk}h_{ml},

where R′_{0ijkl} denote the covariant derivative of R′_{0ijk} as a tensor on M so that

\begin{matrix}∑_{l}R′_{0ijkl}ω_{l} = & dR′_{0ijk} − ∑_{l}R′_{0ljk}ω_{li} − ∑_{l}R′_{0ilk}ω_{lj} \\ − ∑_{l}R′_{0ijl}ω_{lk}. \end{matrix}

Next, we compute the Laplacian △h_{ij} defined by

(2.20)  △h_{ij} = ∑_{k}h_{ijkk}.

From (2.16) and (2.18) it follows that

\begin{matrix}△h_{ij} = & ∑_{k}h_{ikjk} + ∑_{k}R′_{0ijkk} \\ = & ∑_{k}h_{kijk} + ∑_{k}R′_{0ijkk} \\ = & ∑_{k}\{h_{kikj} − ∑_{l}(h_{kl}R_{lijk} + h_{il}R_{lkjk}) + R′_{0ijkk}\}. \end{matrix}

From h_{kikj} = h_{kkij} + R′_{0kikj}, we obtain

\begin{matrix}△h_{ij} = & ∑_{k}h_{kkij} + ∑_{k}(R′_{0ijkk} + R′_{0kikj}) \\ − & ∑_{k,l}(h_{kl}R_{lijk} + h_{il}R_{lkjk}). \end{matrix}

By (2.12), (2.19) and the above equation, we obtain

(2.21)  △h_{ij} = ∑_{k}h_{kkij} + ∑_{k}(R′_{0ijk;k} + R′_{0kik;j})
− ∑_{k}(h_{jk}R′_{0i0k} + h_{kk}R′_{0ij0} + ∑_{l}h_{kl}R′_{kijl})
− ∑_{k}(h_{ij}R′_{0k0k} + h_{kj}R′_{0ki0} + ∑_{l}h_{jl}R′_{lkik})
− ∑_{k,l}(R′_{lkjk} − h_{lk}h_{jk} + h_{kk}h_{jl})h_{il}
− ∑_{k,l}(R′_{lijk} − h_{kl}h_{ij} + h_{kj}h_{il})h_{kl}
= ∑_{k}h_{kkij} + ∑_{k}(R′_{0ijk;k} + R′_{0kik;j})
− ∑_{k}(h_{kk}R′_{0ij0} + h_{ij}R′_{0k0k})
− ∑_{k,l}(2h_{kl}R′_{lijk} + h_{jl}R′_{lkik} + h_{il}R′_{lkjk})
+ nH∑_{l}h_{il}h_{lj} + Sh_{ij}.

J.O. BAEK, Q.-M. CHENG AND Y.J. SUH 11

The following Generalized Maximum Principle of Omori [20] and Yau [25] will play an important role in the proof of our Theorems (cf. [8] and [24]).

Generalized Maximum Principle. Let M be a complete Riemannian manifold whose Ricci curvature is bounded from below on M. Let F be a C^{2}-function bounded from above on M, then, for any ϵ > 0, there exists a point p ∈ M such that

(2.22)  |∇F (p)| < ϵ, △F (p) < ϵ and sup F − ϵ < F (p).

§3. Locally formulas

In this section, we assume that M′ is an (n+1)-dimensional Lorentz space satisfying condition (**) and M is a space-like hypersurface with constant mean curvature in M′. First of all, we calculate the Laplacian of the squared norm S of the second fundamental form α of M.

\frac{1}{2}△S = \frac{1}{2}△(∑_{i,j} h_{ij}^{2}) (3.1)

= ∑_{i,j,k}(h_{ijk}h_{ij})_{k}

= ∑_{i,j,k} h_{ijk}^{2} + ∑_{i,j,k} h_{ijkk}h_{ij}.

By (2.21), we have

(3.2)  \frac{1}{2}△S = ∑_{i,j,k} h_{ijk}^{2} + ∑_{i,j} [ ∑_{k}(R′_{0ij0;j} + R′_{0ijk;k})

− ∑_{k}(h_{kk}R′_{0ij0} + h_{ij}R′_{0k0k})

− ∑_{k,l}(2h_{kl}R′_{lijk} + h_{lj}R′_{lkik} + h_{li}R′_{kjkl})

+ nHh_{ij}^{2} + Sh_{ij}]h_{ij}.

Thus, we have

\begin{aligned}\frac{1}{2}△S = & ∑_{i,j,k} h_{ijk}^{2} + ∑_{i,j,k} h_{ij}(R′_{0kik;j} + R′_{0ijk;k}) \\ & − (∑_{i,j} nHh_{ij}R′_{0ji0} + S ∑_{k} R′_{0k0k}) \\ & − ∑_{i,j,k,l} 2(h_{ij}h_{kl}R′_{lijk} + h_{li}h_{ij}R′_{lkjk}) \\ & − nHh_{3} + S^{2}, \end{aligned} (3.3)

12 COMPLETE SPACE-LIKE HYPERSURFACES

where h_{3} = ∑_{j=1}^{n} λ_{j}^{3} and λ_{j}’s are principal curvatures of M.

Next, we will choose {e_{1}, ⋯, e_{n}} such that

(3.4)  h_{ij} = λ_{i}δ_{ij}.

By definition, we see

S = ∑_{i}λ_{i}^{2}.

Putting μ_{j} = λ_{j} − H, we have

(3.5)  ∑_{j} μ_{j} = 0, ∑_{j} μ_{j}^{2} = S − nH^{2}.

Therefore, for any j,

(3.6)  (λ_{j} − H)^{2} ≤ \frac{n − 1}{n}(S − nH^{2}).

Since

∑_{i,j,k}(R′_{0kik;j} + R′_{0ijk;k})h_{ij}

= ∑_{j,k} λ_{j}(R′_{0kjk;j} + R′_{0jjk;k})

≥ − ∑_{k} [|H| + \sqrt{\frac{n − 1}{n}(S − nH^{2})}](\sqrt{∑_{j} R′_{0kjk;j}^{2}} + \sqrt{∑_{j} R′_{0jjk;k}^{2}})

≥ −2\sqrt{n}[|H| + \sqrt{\frac{n − 1}{n}(S − nH^{2})}]|∇R′|,

we obtain

(3.7)  ∑_{i,j,k}(R′_{0kik;j} + R′_{0ijk;k})h_{ij} ≥ −\frac{2}{\sqrt{n}}[|H| + \sqrt{\frac{n − 1}{n}(S − nH^{2})}]c_{3}.

By (3.4) and condition (*), we have

\begin{matrix}−(∑_{i,j} & nHh_{ij}R′_{0ij0} + S∑_{k}R′_{0k0k}) = −∑_{k}nHλ_{k}(R′_{0kk0} − S∑_{k}R′_{0kk0}) \\ = & ∑_{k}(S − nHλ_{k})R′_{0kk0} = −∑_{k}(S − nHλ_{k})\frac{c_{1}}{n} . \end{matrix}

Hence it follows that

(3.8)  −(∑_{i,j} nHh_{ij}R′_{0ij0} + S∑_{k}R′_{0k0k}) = −c_{1}(S − nH^{2}).

Since

\begin{matrix}− ∑_{i,j,k,l} & (h_{ij}h_{kl}(R′_{lijk} + h_{li}h_{ij}R′_{lkjk}) \\ = & − ∑_{j,k}(λ_{j}λ_{k}R′_{kjjk} − λ_{k}^{2}R′_{kjjk}) \\ = & − ∑_{j,k}(λ_{j}λ_{k} − λ_{k}^{2})R′_{kjjk} = \frac{1}{2} ∑_{j,k}(λ_{j} − λ_{k})^{2}R′_{kjjk} \\ ≥ & \frac{c_{2}}{2} ∑_{j,k}(λ_{j} − λ_{k})^{2}, \end{matrix}

we obtain

(3.9) − ∑_{i,j,k,l}(h_{ij}h_{kl}(R′_{lijk} − h_{li}h_{ij}R′_{lkjk}) ≥ \frac{c_{2}}{2} ∑_{j,k}(λ_{j} − λ_{k})^{2} = c_{2}(nS − n^{2}H^{2}).

Thus, substituting (3.7), (3.8) and (3.9) into (3.3), we can prove the following

Lemma 3.1. Let M′ be an (n + 1)-dimensional Lorentz space satisfying the condition (**). If M is a space-like hypersurface with constant mean curvature H in M′, then we have

(3.10)  \frac{1}{2}△S≥ − \frac{2}{\sqrt{n}}[|H| + \sqrt{\frac{n − 1}{n}(S − nH^{2})}]c_{3}
+ (2nc_{2} + c_{1})(S − nH^{2}) + (S^{2} − nHh_{3}).

In particular, if M′ is locally symmetric, we have

(3.11)  \frac{1}{2}△S≥(2nc_{2} + c_{1})(S − nH^{2}) + (S^{2} − nHh_{3}).

14 COMPLETE SPACE-LIKE HYPERSURFACES

§4. Proofs of Theorems

This section presents proofs of our theorems.

Proof of Theorem 1. Since M′ is an (n + 1)-dimensional locally symmetric Lorentz space satisfying the condition (*), that is, for constants c_{1} and c_{2}, we have

(4.1)  K′(u, v) = −c_{1}/n

for any space-like vector u and any time-like vector v and

(4.2)  K′(u_{1}, u_{2}) ≥ c_{2}

for any space-like vectors u_{1} and u_{2}. Hence, we have, from (3.11),

(4.3)  \frac{1}{2}△S≥(2nc_{2} + c_{1})(S − nH^{2}) + (S^{2} − nHh_{3}).

Let B = ∑_{i} μ_{i}^{2} and B_{3} = ∑_{i} μ_{i}^{3}. We have

(4.4)  B = S − nH^{2}, B_{3} = h_{3} − 3HB − nH^{3}.

Hence, we have

(4.5)  \frac{1}{2}△B≥(2nc_{2} + c_{1})B + (B + nH^{2})^{2} − nH(B_{3} + 3HB + nH^{3}).

because H is constant.

Let a_{1}, ⋯, a_{n} be real numbers satisfying ∑_{i} a_{i} = 0 and ∑_{i} a_{i}^{2} = B, then we can prove

(4.6)  | ∑_{i} a_{i}^{3}| ≤ \frac{n − 2}{\sqrt{n(n − 1)}}B^{\frac{3}{2}}

and the equality holds if and only if at least n − 1 of the a_{i}’s are equal. Therefore, we infer

(4.7)  \frac{1}{2}△B≥B\{B − \frac{n − 2}{\sqrt{n(n − 1)}}n|H|B^{\frac{1}{2}} + nc − nH^{2}\},

where c = 2c_{2} + \frac{c_{1}}{n}. Since M′ satisfies the condition (*) and M has constant mean curvature, from Gauss equation (2.13), we know that the Ricci curvature of M is

J.O. BAEK, Q.-M. CHENG AND Y.J. SUH 15

bounded from below. Since we do not know whether S is bounded yet, we consider a function F defined by, for any positive constant a, F = −\frac{1}{\sqrt{B+a}}. We know that F is bounded because of B ≥ 0. According to the Generalized Maximum Principle of Omori [20] and Yau [25] in section 2, for any ϵ_{m} > 0, there exists a point p_{m} ∈ M such that

(4.8)  ΔF (p_{m}) < ϵ_{m}, |∇F |(p_{m}) < ϵ_{m}, sup F − ϵ_{m} < F (p_{m}).

Since

∇_{i}F = \frac{1}{2} \frac{∇_{i}B}{(B + a)^{\frac{3}{2}}} ,

ΔF = \frac{1}{2} \frac{ΔB}{(B + a)^{\frac{3}{2}}} − \frac{3}{4} \frac{|∇B|^{2}}{(B + a)^{\frac{5}{2}}} ,

we have

|∇F | = \frac{1}{2}|F |^{3}|∇B|

and

\frac{1}{2}|F |^{4}ΔB = |F |ΔF + 3|∇F |^{2}.

From (4.8), we infer

\frac{1}{2}|F (p_{m})|^{4}ΔB(p_{m}) < |F (p_{m})|ϵ_{m} + 3ϵ_{m}^{2}.

For any positive constant 0 < ϵ < 1, letting ϵ_{m} → 0, we know that there exists a positive integer m_{0} such that when m > m_{0}, |F (p_{m})|ϵ_{m} + 3ϵ_{m}^{2} < ϵ because F is a bounded function. According to (4.7) and the above inequalities, we infer

(1 − ϵ)B^{2}(p_{m}) − \frac{n − 2}{\sqrt{n(n − 1)}}n|H|B^{\frac{3}{2}} (p_{m}) + (nc − nH^{2} − 2aϵ)B(p_{m}) − a^{2}ϵ < 0.

Thus, we know that {B(p_{m})} is a bounded sequence. Since lim_{m→∞} F (p_{m}) = sup F = − inf \frac{1}{\sqrt{B+a}} = −\frac{1}{\sqrt{sup B+a}}, we have lim_{m→∞} B(p_{m}) = sup B from the definition of F. Hence, B is bounded. Since a and ϵ are any positive constants, we infer

(4.9)  sup B\{sup B − \frac{n − 2}{\sqrt{n(n − 1)}}n|H| sup B^{\frac{1}{2}} + nc − nH^{2}\} ≤ 0.

If n^{2}H^{2} < 4(n − 1)c holds, then we have c > 0 and

sup B − \frac{n − 2}{\sqrt{n(n − 1)}}n|H| sup B^{\frac{1}{2}} + nc − nH^{2} > 0.

Hence, we obtain sup B = 0, that is, B ≡ 0. Thus, we infer that S ≡ nH^{2} and M is totally umbilical.

If n^{2}H^{2} = 4(n − 1)c holds, then we have c ≥ 0 and

sup B − \frac{n − 2}{\sqrt{n(n − 1)}}n|H| sup B^{\frac{1}{2}} + nc − nH^{2} = (sup B^{\frac{1}{2}} − \frac{n − 2}{2\sqrt{n(n − 1)}}n|H|)^{2} ≥ 0.

Hence, from (4.9), we have sup B = 0, that is, B ≡ 0 if sup B^{\frac{1}{2}} ≠ \frac{n−2}{2\sqrt{n(n−1)}}n|H|. Thus, we have that either B ≡ 0, that is, S ≡ nH^{2} and M is totally umbilical, or sup B^{\frac{1}{2}} = \frac{n−2}{2\sqrt{n(n−1)}}n|H|, namely, sup S = nc. Thus, we complete the proof of (2) in Theorem 1.

If n^{2}H^{2} > 4(n − 1)c and c < 0 hold, we know

sup B − \frac{n − 2}{\sqrt{n(n − 1)}}n|H| sup B^{\frac{1}{2}} + nc − nH^{2}

=(sup B^{\frac{1}{2}} − Bmin^{\frac{1}{2}})(sup B^{\frac{1}{2}} − Bmax^{\frac{1}{2}}),

where Bmin^{\frac{1}{2}} = ^{\frac{(n−2)n|H|−n\sqrt{n^{2}H^{2}−4(n−1)c}}{2\sqrt{n(n−1)}}} and Bmax^{\frac{1}{2}} = ^{\frac{(n−2)n|H|+n\sqrt{n^{2}H^{2}−4(n−1)c}}{2\sqrt{n(n−1)}}}.

Since Bmin^{\frac{1}{2}} = ^{\frac{(n−2)n|H|−n\sqrt{n^{2}H^{2}−4(n−1)c}}{2\sqrt{n(n−1)}}} < 0 holds when c < 0, from (4.9), we

infer either sup B = 0, in this case, M is totally umbilical, or 0 < sup B^{\frac{1}{2}} ≤ \frac{(n−2)n|H|+n\sqrt{n^{2}H^{2}−4(n−1)c}}{2\sqrt{n(n−1)}}. Hence, we know that the assertion (3) in Theorem 1 is true from S = B + nH^{2}.

If n^{2}H^{2} > 4(n − 1)c and c ≥ 0 hold, we also have

sup B − \frac{n − 2}{\sqrt{n(n − 1)}}n|H| sup B^{\frac{1}{2}} + nc − nH^{2}

=(sup B^{\frac{1}{2}} − Bmin^{\frac{1}{2}})(sup B^{\frac{1}{2}} − Bmax^{\frac{1}{2}}),

(a) when H^{2} ≥ c, Since Bmin^{\frac{1}{2}} = ^{\frac{(n−2)n|H|−n\sqrt{n^{2}H^{2}−4(n−1)c}}{2\sqrt{n(n−1)}}} ≤ 0 holds, from (4.9),

we infer either sup B = 0, in this case, M is totally umbilical, or 0 < sup B^{\frac{1}{2}} ≤ \frac{(n−2)n|H|+n\sqrt{n^{2}H^{2}−4(n−1)c}}{2\sqrt{n(n−1)}}.

(b) when H^{2} < c, we have Bmin^{\frac{1}{2}} = ^{\frac{(n−2)n|H|−n\sqrt{n^{2}H^{2}−4(n−1)c}}{2\sqrt{n(n−1)}}} > 0. Hence,

we have , from (4.9), sup B = 0, in this case, M is totally umbilical, or Bmin^{\frac{1}{2}} ≤ sup B^{\frac{1}{2}} ≤ Bmax^{\frac{1}{2}}. Thus, we infer that the assertion (4) in Theorem 1 is true because of S = B + nH^{2}.

If S ≡ \frac{n}{2(n−1)}[n^{2}H^{2} − 2(n − 1)c + (n − 2)|H|\{n^{2}H^{2} − 4(n − 1)c\}^{\frac{1}{2}} ] holds, we know that these inequalities in the proof of Lemma 3.1 and (4.6) are equalities and S > nH^{2}. Hence, we have n^{2}H^{2} ≥ 4(n − 1)c from (1) in Theorem 1. Thus, we can infer that n−1 of the principal curvatures λ_{i} are equal. Since the mean curvature H is constant and S is constant, we infer that principal curvatures are constant on M. Thus, M is an isoparametric hypersurface with two distinct principal curvatures one of which is simple. This completes the proof of Theorem 1.

Proof of Theorem 2. According to Gauss equation (2.12), we have

\begin{matrix}R_{jkkj} = & R′_{jkkj} − (h_{jj}h_{kk} − h_{jk}h_{kj}) \\ = & R′_{jkkj} − h_{jj}h_{kk} \\ = & R′_{jkkj} − λ_{j}λ_{k}. \end{matrix}

Hence, we obtain

(4.10)  R′_{jkkj} = R_{jkkj} + λ_{j}λ_{k}.

Therefore,

(4.11)  −2 ∑_{i,j,k,l}(h_{ij}h_{kl}(R′_{lijk} − h_{li}h_{ij}R′_{lkjk})
= − 2 ∑_{j,k}(λ_{j}λ_{k}R′_{kjjk} − λ_{k}^{2}R′_{kjjk})
= − 2 ∑_{j,k}(λ_{j}λ_{k} − λ_{k}^{2})R′_{kjjk}
= ∑_{j,k}(λ_{j} − λ_{k})^{2}R′_{kjjk}
=\frac{1}{2} ∑_{j,k}(λ_{j} − λ_{k})^{2}R′_{kjjk} + \frac{1}{2} ∑_{j,k}(λ_{j} − λ_{k})^{2}(R_{kjjk} + λ_{j}λ_{k})
≥\frac{c_{2}}{2} ∑_{j,k}(λ_{j} − λ_{k})^{2} + \frac{1}{2} ∑_{j,k}(λ_{j} − λ_{k})^{2}(R_{kjjk} + λ_{j}λ_{k}).

By making use of the same proof as in the proof of Lemma 3.1, we have

\frac{1}{2}△S (4.12)

≥(nc_{2} + c_{1})(S − nH^{2}) + (S^{2} − nHh_{3})

+ \frac{1}{2} ∑_{j,k}(λ_{j} − λ_{k})^{2}(R_{kjjk} + λ_{j}λ_{i}).

Since the sectional curvature of M is not less than −(c_{2} + \frac{c_{1}}{n}), we have

\frac{1}{2} ∑_{j,k}(λ_{j} − λ_{k})^{2}(R_{kjjk} + λ_{j}λ_{i}) (4.13)

≥ −\frac{1}{2}(c_{2} + \frac{c_{1}}{n}) ∑_{j,k}(λ_{j} − λ_{k})^{2} + \frac{1}{2} ∑_{j,k}(λ_{j} − λ_{k})^{2}λ_{j}λ_{k}

= −(c_{2} + \frac{c_{1}}{n})(nS − n^{2}H^{2}) + (nHh_{3} − S^{2}).

Thus, we infer, from (4.12) and (4.13),

(4.14) \frac{1}{2}△S≥(nc_{2}+c_{1})(S−nH^{2})+(S^{2}−nHh_{3})+ \frac{1}{2} ∑_{j,k}(λ_{j}−λ_{k})^{2}(R_{kjjk}+λ_{j}λ_{i}) ≥ 0.

From Theorem 1, we know that S is bounded. Applying the Generalized Maximum Principle to S, we have that there exists a point sequence {p_{m}} ⊂ M such that

(4.15)  lim sup_{m→∞} △S(p)≤0, lim_{m→∞} |∇S|(p_{m}) = 0, lim_{m→∞} S(p_{m}) = sup S.

Hence, we have

(4.16)  lim_{m→∞} ∑_{j,k}(λ_{j} − λ_{k})^{2}(R_{kjjk} + (c_{2} + \frac{c_{1}}{n}))(p_{m}) = 0

and

(4.17)  lim_{m→∞} ∑_{j,k}(λ_{j} − λ_{k})^{2}R′_{kjjk} = c_{2} lim_{m→∞} ∑_{j,k}(λ_{j} − λ_{k})^{2}.

Since the function S = ∑_{j}λ_{j}^{2} is bounded, {λ_{j}(p_{m})} is a bounded sequence for any j. Thus, we can assume

(4.18)  lim_{m→∞} λ_{j}(p_{m}) = λ_{j_{0}},

for any j, if necessary, we can take a subsequence of {λ_{j}(p_{m})}. Hence, from (4.10) and (4.17), we have

(4.19) lim_{m→∞} ∑_{j,k}(λ_{j} − λ_{k})^{2}R_{kjjk}(p_{m}) = lim_{m→∞} ∑_{j,k}(λ_{j} − λ_{k})^{2}(c_{2} − λ_{j}λ_{k})(p_{m})

Therefore, we infer, from (4.16) and (4.19),

(4.20)  lim_{m→∞} ∑_{j,k}(λ_{j} − λ_{k})^{2}(c − λ_{j}λ_{i}) = 0.

If λ_{j_{0}} ≠ λ_{k_{0}}, from (4.17), we have

lim_{m→∞}(c_{2} − λ_{j}λ_{k})(p_{m}) = lim_{m→∞}(R′_{kjjk} − λ_{j}λ_{k})(p_{m}) = lim_{m→∞} R_{kjjk}(p_{m}) ≥ −(c_{2} + \frac{c_{1}}{n}).

Hence, in this case, we obtain

(4.21)  lim_{m→∞}(c − λ_{j}λ_{i})(p_{m}) ≥ 0.

Thus, (4.20) and (4.21) yield, for any i and j,

(4.22)  lim_{m→∞}(−λ_{i}λ_{j}(p_{m}) + c)(λ_{i} − λ_{j})^{2}(p_{m}) = 0.

By (4.18) and (4.22) we get

(4.23)  (−λ_{i_{0}}λ_{j_{0}} + c)(λ_{i_{0}} − λ_{j_{0}})^{2} = 0

for any i_{0} and j_{0}. By the simple algebraic calculation, it is clear that at most two of {λ_{j_{0}}}’s are distinct.

If all of {λ_{j_{0}}}’s coincides with each other, then, from (4.21), we have c ≥ λ_{j_{0}}^{2} ≥ 0. If two of the {λ_{j_{0}}}’s are distinct, which are denoted by λ and μ (λ≠μ). By (4.23),

they satisfy

(4.24)  −λμ + c = 0.

Now let us denote by r and s the number of indices λ_{j}(p_{m})→λ and μ_{j}(p_{m})→μ respectively. Then we want to assert that r = 1 or s = 1.

In fact, if r ≥ 2 and s≥2 hold, it follows from r≥2 that there are distinct indices i and j such that λ_{i}(p_{m})→λ and λ_{j}(p_{m})→λ (m→∞) and hence we have

lim_{m→∞}(−λ_{i}λ_{j} + c) = −λ^{2} + c, for i≠j.

By (4.18) and (4.21), we obtain c≥λ^{2} ≥ 0. Similarly, we have c≥μ^{2} ≥ 0, which implies that

c^{2}≥λ^{2}μ^{2}.

On the other hand, since we see that c = λμ in (4.24), it implies λ^{2} = μ^{2} = c. Furthermore, it turns out to be λ = ±μ. Because they are distinct, it yields c = λμ = −λ^{2}. Hence we obtain λ = μ = 0. This is impossible because of λ ≠ μ. Thus, our assertion is true.

Without loss of generality, we assume r = 1. Since n > 2, we have s≥2 and by the above discussion we obtain 0 < μ^{2} < c. This finishes the proof of the first part of assertions in Theorem 2.

Next, we shall prove the second part of the assertions in Theorem 2. From the above assertion, together with (4.24) it follows that λ^{2} > c. It is clear that we may assume that the mean curvature H is positive. Then λ and μ are positive because of c > 0 and (4.24). By defining c_{3} and c_{4} by λ^{2} = c − c_{3} and μ^{2} = c − c_{4}, respectively, we have c_{3} < 0, 0 < c_{4} < c and

(4.25)  (c − c_{3})(c − c_{4}) = c^{2}, i.e., \frac{1}{c_{3}} + \frac{1}{c_{4}} = \frac{1}{c}.

Since the mean curvature H is constant, we obtain

\begin{aligned}nH & = λ + (n − 1)μ = (c − c_{3})^{\frac{1}{2}} + (n − 1)(c − c_{4})^{\frac{1}{2}} \\ λμ & = c. \end{aligned}

Hence, we have

λ^{2} − nHλ + (n − 1)c = 0.

Since H^{2} ≥ c holds, we conclude

(4.26) λ = \frac{c}{μ} = \frac{nH + \sqrt{n^{2}H^{2} − 4(n − 1)c}}{2}, and μ = \frac{nH − \sqrt{n^{2}H^{2} − 4(n − 1)c}}{2(n − 1)}.

Hence, we infer

\begin{matrix}S = & λ^{2} + (n − 1)μ^{2} \\ = & \frac{1}{2(n − 1)}[n\{n^{2}H^{2} − 2(n − 1)c\} + (n − 2)n|H|\{n^{2}H^{2} − 4(n − 1)c\}^{\frac{1}{2}} ]. \end{matrix} (4.27)

It is a contradiction. Therefore,

λ_{j_{0}} = λ

for any j. This implies

sup S = nH^{2}.

From S − nH^{2} = B ≥ 0, we have S ≡ nH^{2}. Hence, M is totally umbilical. □

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

e-mail:  cheng@ms.saga-u.ac.jp

Jin Ok Baek and Young Jin Suh Department of Mathematics, Kyungpook National University, Taegu, 702-701, Korea

e-mail:  yjsuh@bh.kyungpook.ac.kr