COMPLETE MAXIMAL SPACELIKE SURFACES

IN AN ANTI-DE SITTER SPACE H_{2}^{4}(c)^{∗}

(RUNNING TITLE:

COMPLETE MAXIMAL SPACELIKE SURFACES)

Qing-Ming Cheng

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

Abstract. In this paper, we prove that if M^{2} is a complete maximal spacelike surface of an anti-de Sitter space H_{2}^{4}(c) with constant scalar curvature, then S = 0, S = \frac{−10c}{11} , S = \frac{−4c}{3} or S = −2c, where S is the squared norm of the second fundamental form of M^{2}. And

(1) S = 0 if and only if M^{2} is the totally geodesic surface H^{2}(c);

(2) S = \frac{−4c}{3} if and only if M^{2} is the hyperbolic Veronese surface;

(3) S = −2c if and only if M^{2} is the hyperbolic cylinder of the totally geodesic surface H_{1}^{3}(c) of H_{2}^{4}(c).

1. Introduction

Let M_{p}^{n+p}(c) be an n + p-dimensional connected semi-Riemannian manifold of index p and of constant curvature c, which is called as an indefinite space form of index p. The standard models of indefinite space forms are given as follows. In an (n + p)-dimensional real vector space R^{n+p} with the standard basis, the scalar product ⟨, ⟩ is given by

⟨x, y⟩ = ∑_{i=1}^{n} x_{i}y_{i} − ∑_{j=n+1}^{n+p} x_{j}y_{j},

where x = (x_{1}, x_{2}, ⋯, x_{n+p}) and y = (y_{1}, y_{2}, ⋯, y_{n+p}). Then (R^{n+p}, ⟨, ⟩) is an indefinite Euclidean space, which is denoted by R_{p}^{n+p}.

Key words and phrases. anti-de Sitter space, maximal spacelike surfaces, hyperbolic Veronese surfaces and constant scalar curvature.

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

1991 Mathematics Subject Classification 53C40, 53C42

Typeset by AMS-TEX

Let S_{p}^{n+p}(c) for c > 0 be the hypersurface in R_{p}^{n+p+1} given as

⟨x, x⟩ = \frac{1}{c} =: r_{0}^{2}.

Then we know that the S_{p}^{n+p}(c) inherits an indefinite Riemannian metric induced through R_{p}^{n+p+1} and has constant curvature c. This is called a de Sitter space of constant curvature c with index p.

On the other hand, let H_{p}^{n+p}(c) for c < 0 be the hypersurface in R_{p+1}^{n+p+1} given as

⟨x, x⟩ = \frac{1}{c} =: −r_{0}^{2}.

Then we also know that the H_{p}^{n+p}(c) inherits an indefinite Riemannian metric induced through R_{p+1}^{n+p+1} and has constant curvature c. This is called an anti-de Sitter space of constant curvature c with index p.

Let M^{n} be an n-dimensional Riemannian manifold immersed in M_{p}^{n+p}(c). A submanifold M^{n} of M_{p}^{n+p}(c) is said to be spacelike if the induced metric on M^{n} from that of the ambient space is positive definite.

E. Calabi [1] first studied the Bernstein problem for a maximal spacelike entire graph in the Minkowski space R_{1}^{n+1} and proved that it has to be hyperplane. when n ≤ 4. S.Y. Cheng and S.T. Yau [6] proved that the conclusion remains true for all n. As a generalization of the Bernstein type problem, T. Ishihara [8] proved that a complete spacelike maximal submanifold M^{n} of M_{p}^{n+p}(c) (c ≥ 0) is totally geodesic.

On the other hand, there exist many examples of complete maximal spacelike submanifolds in the anti-de Sitter space H_{p}^{n+p}(c), which are not totally geodesic. For examples, we consider the following examples.

Example 1. We consider the mapping defined by

u_{1} = \frac{1}{\sqrt{−3c}}yz, u_{2} = \frac{1}{\sqrt{−3c}}zx, u_{3} = \frac{1}{\sqrt{−3c}}xy,

u_{4} = \frac{1}{2\sqrt{−3c}}(x^{2} − y^{2}), u_{5} = \frac{1}{6\sqrt{−c}}(x^{2} + y^{2} + 2z^{2}),

where (x, y, z) is the natural coordinate system in R_{1}^{3} and (u_{1}, u_{2}, u_{3}, u_{4}, u_{5}) is the natural coordinate system R_{3}^{5}. This defines a complete maximal spacelike isometric immersion of H^{2}(\frac{c}{3}) into H_{2}^{4}(c), where H^{n_{i}}(c_{i}) is an n_{i}-dimensional hyperbolic space of constant curvature c_{i}, which is called the hyperbolic Veronese surface.

Example 2. Let n_{1}, ⋯, n_{p+1} be positive integers and n = n_{1} + ⋯ + n_{p+1}. Let x_{i} be a point of H^{n_{i}}(\frac{nc}{n_{i}}). Then x = (x_{1}, ⋯, x_{p+1}) is a vector in R_{p+1}^{n+p+1} with ⟨x, x⟩ = \frac{1}{c}. This also defines a complete maximal spacelike isometric immersion of H^{n_{1}}(\frac{nc}{n_{1}} ) × ⋯ × H^{n_{p+1}}(\frac{nc}{n_{p+1}} ) into H_{p}^{n+p}(c).

Hence this case of complete maximal spacelike submanifolds in the anti-de Sitter H_{p}^{n+p}(c) is very different from the ones in the indefinite Euclidean space R_{p}^{n+p} and the de Sitter space S_{p}^{n+p}(c). Hence, the investigation of complete maximal spacelike submanifolds in H_{p}^{n+p}(c) would be very interesting.

T. Ishihara [8] characterized the complete maximal spacelike submanifolds H^{n_{1}}(\frac{nc}{n_{1}} ) × ⋯ × H^{n_{p+1}}(\frac{nc}{n_{p+1}} ) of H_{p}^{n+p}(c), that is, he proved that let M^{n} be an ndimensional complete maximal spacelike submanifold in H_{p}^{n+p}(c), then S ≤ −npc and S = −npc if and only if M^{n} = H^{n_{1}}(\frac{nc}{n_{1}} ) × ⋯ × H^{n_{p+1}}(\frac{nc}{n_{p+1}} ), where S is the squared norm of the second fundamental form of M^{n}. When p = 1, the Bernstein type properties of complete maximal spacelike hypersurfaces in H_{1}^{n+1}(c) are also studied in [3] and [4].

In particular, if n = 2, we know that the well known examples of complete maximal spacelike surfaces in the anti-de Sitter space H_{2}^{4}(c) are the totally geodesic surface H^{2}(c) with S = 0 and the hyperbolic Veronese surface with S = \frac{−4c}{3}. Therefore, it is natural to ask whether there exist the other complete maximal spacelike surfaces with S = constant in H_{2}^{4}(c), which are different from the above ones. If there exist such surfaces, can we determine all of the value of S? In this paper we shall answer these problems.

Main Theorem. Let M^{2} be a complete maximal spacelike surface of an anti-de Sitter space H_{2}^{4}(c) with constant scalar curvature, then S = 0, S = \frac{−10c}{11}, S = \frac{−4c}{3} or S = −2c, where S is the squared norm of the second fundamental form of M^{2}. And

(1) S = 0 if and only if M^{2} is the totally geodesic surface H^{2}(c);

(2) S = \frac{−4c}{3} if and only if M^{2} is the hyperbolic Veronese surface;

(3) S = −2c if and only if M^{2} is the hyperbolic cylinder of the totally geodesic surface H_{1}^{3}(c) of H_{2}^{4}(c).

Remark 1. It is still open for the author whether there exist complete maximal spacelike surfaces of the anti-de Sitter space H_{2}^{4}(c) with S = \frac{−10c}{11}.

2. Preliminaries

Let M^{n} be an n-dimensional spacelike submanifold of an anti-de Sitter space H_{p}^{n+p}(c) of dimension n+p and with index p. We choose a local orthonormal frame field e_{1}, ⋯, e_{n}, e_{n+1}, ⋯, e_{n+p} in H_{p}^{n+p}(c), restricted to M^{n}, so that e_{1}, …, e_{n} are tangent to M^{n}. With respect to the above frame field of H_{p}^{n+p}(c), let ω_{1}, ⋯, ω_{n+p} denote the dual coframe field. Then

(2.1)  ω_{α} = 0 for any α = n + 1, ⋯, n + p.

It follows from Cartan’s Lemma that

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

The structure equations of M^{n} are given by

(2.3)  \{ \begin{matrix}dω_{i} + ∑_{j} ω_{ij} ∧ ω_{j} = 0, w_{ij} + ω_{ji} = 0, \\ dω_{ij} + ∑_{k} ω_{ik} ∧ ω_{kj} = Ω_{ij}, \\ Ω_{ij} = \frac{1}{2} ∑_{k,l} R_{ijkl}ω_{k} ∧ ω_{l}, \end{matrix}

(2.4)  R_{ijkl} = c(δ_{ik}δ_{jl} − δ_{il}δ_{jk}) − ∑_{β}(h_{ik}^{β}h_{jl}^{β} − h_{il}^{β}h_{jk}^{β}),

where Ω_{ij} (resp. R_{ijkl}) denotes the curvature form (resp. the components of the curvature tensor) of M^{n}.

We have also the structure equations of the normal bundle of M^{n}.

(2.5)  \{ \begin{matrix}dω_{αβ} + ∑_{γ} ω_{αγ} ∧ ω_{γβ} = Ω_{αβ}, \\ Ω_{αβ} = \frac{1}{2} ∑_{k,l} R_{αβkl}ω_{k} ∧ ω_{l}, \end{matrix}

(2.6)  R_{αβkl} = − ∑_{t}(h_{tk}^{α}h_{tl}^{β} − h_{tl}^{α}h_{tk}^{β}).

The second fundamental form h of M^{n} is given by

h = ∑_{i,j,α} h_{ij}^{α}w_{i}w_{j}e_{α}

We recall \frac{1}{n} ∑_{α}(∑_{i} h_{ii}^{α})e_{α} the mean curvature vector. If ∑_{i} h_{ii}^{α} = 0 for all α, then M^{n} is said to be maximal. The Codazzi equation and Ricci formulas for the second fundamental form and its covariant derivatives are given by

(2.7)  h_{ijk}^{α} = h_{ikj}^{α} = h_{jik}^{α},
(2.8)  h_{ijkl}^{α} − h_{ijlk}^{α} = ∑_{m} h_{im}^{α}R_{mjkl} + ∑_{m} h_{mj}^{α}R_{mikl} − ∑_{β} h_{ij}^{β}R_{αβkl},

(2.9)  h_{ijklm}^{α} − h_{ijkml}^{α} = ∑_{r} h_{rjk}^{α}R_{rilm} + ∑_{r} h_{irk}^{α}R_{rjlm}

+ ∑_{r} h_{ijr}^{α}R_{rklm} − ∑_{β} h_{ijk}^{β}R_{αβlm},

where h_{ijk}^{α}, h_{ijkl}^{α} and h_{ijklm}^{α} are the coefficients of the first, the second and the third covariant derivatives of the second fundamental form of M^{n}, respectively. If M^{n} is maximal, the scalar curvature is given by

(2.10)  R = n(n − 1)c + ∑_{i,j,α}(h_{ij}^{α})^{2}.

Hence the scalar curvature is constant if and only if S = ∑_{i,j,α}(h_{ij}^{α})^{2} is constant.

The following Generalized Maximum Principle due to Omori [9] and Yau [12] will be used in this paper.

Generalized Maximum Principle (cf. Omori [9] and Yau [12]). Let M^{n} be an n-dimensional complete Riemannian manifold whose Ricci curvature is bounded from below. Let F be a C^{2}-function bounded from above on M^{n}, then there exists a sequence {p_{m}} of points in M^{n} such that

lim_{m→∞} F (p_{m}) = sup F, lim_{m→∞} |∇F |(p_{m}) = 0, lim_{m→∞} sup ΔF (p_{m}) ≤ 0.

3. Proof of Main Theorem

In this section, we assume n = p = 2. We first compute some local formulas in order to prove Main Theorem. Let S_{3} := ∑_{ij}(h_{ij}^{3})^{2} and S_{4} := ∑_{ij}(h_{ij}^{4})^{2}. We know that S_{3}S_{4} is a function defined globally on M^{2}. For arbitrary fixed point p in M^{2} we can choose e_{1} and e_{2} such that

(3.1)  h_{ij}^{3} = λ_{i}δ_{ij}.

Since M is maximal we get λ_{1} = −λ_{2} =: λ. Let

S_{αβ} = ∑_{i,j} h_{ij}^{α}h_{ij}^{β}.

We know that the (2 × 2)-matrix (S_{αβ}) is symmetric. Hence we can assume that it is diagonal for a suitable choice of e_{3} and e_{4}. Thus setting μ := h_{11}^{4} = −h_{22}^{4} and μ_{1} = h_{12}^{4}, we have

(3.2)  ∑_{i,j} h_{ij}^{3}h_{ij}^{4} = 2λμ = 0.

Theorem 3.1. For α = 3, 4, we have

(3.3)  Δh_{ij}^{α} = (S + 2c)h_{ij}^{α} − 2 ∑_{l,t,β≠α} h_{lt}^{α}h_{tj}^{β}h_{il}^{β} + ∑_{l,t,β≠α} h_{tl}^{α}h_{tl}^{β}h_{ij}^{β},

(3.4)  \frac{1}{2}ΔS = ∑_{i,j,k,α}(h_{ijk}^{α})^{2} + (S + 2c)S + 2S_{3}S_{4},

(3.5)  \frac{1}{2}ΔS_{3} = ∑_{i,j,k}(h_{ijk}^{3})^{2} + (S + 2c)S_{3} + S_{3}S_{4},

where S = S_{3} + S_{4} = ∑_{i,j,α}(h_{ij}^{α})^{2} is the squared norm of the second fundamental form of M^{2} and S_{3} = ∑_{i,j}(h_{ij}^{3})^{2} and S_{4} = ∑_{i,j}(h_{ij}^{4})^{2}.

Proof. For any α,

Δh_{ij}^{α} = ∑_{l} h_{ijll}^{α} = ∑_{l} h_{lijl}^{α}

= ∑_{l} h_{lilj}^{α} + ∑_{l,t} h_{ti}^{α}R_{tljl} + ∑_{l,t} h_{lt}^{α}R_{tijl} − ∑_{l,β} h_{li}^{β}R_{αβjl}

= ∑_{l,t} h_{ti}^{α}[c(δ_{tj}δ_{ll} − δ_{tl}δ_{lj}) − ∑_{β}(h_{tj}^{β}h_{ll}^{β} − h_{tl}^{β}h_{lj}^{β})]

+ ∑_{l,t} h_{tl}^{α}[c(δ_{tj}δ_{il} − δ_{tl}δ_{ij}) − ∑_{β}(h_{tj}^{β}h_{il}^{β} − h_{tl}^{β}h_{ij}^{β})]

+ ∑_{t,l,β} h_{li}^{β}(h_{tj}^{α}h_{tl}^{β} − h_{tl}^{α}h_{tj}^{β})

= (2c + S)h_{ij}^{α} − 2 ∑_{l,t,β≠α} h_{lt}^{α}h_{tj}^{β}h_{il}^{β} + ∑_{l,t,β≠α} h_{tl}^{α}h_{tl}^{β}h_{ij}^{β},

\frac{1}{2}ΔS = ∑_{i,j,k,α}(h_{ijk}^{α})^{2} + ∑_{i,j,α} h_{ij}^{α}Δh_{ij}^{α}

= ∑_{i,j,k,α}(h_{ijk}^{α})^{2} + (2c + S)S + 2S_{3}S_{4},

\frac{1}{2}ΔS_{3} = ∑_{i,j,k}(h_{ijk}^{3})^{2} + ∑_{i,j} h_{ij}^{3}Δh_{ij}^{3}

= ∑_{i,j,k}(h_{ijk}^{3})^{2} + (2c + S)S_{3} + S_{3}S_{4}.

This finishes the Proof of Theorem 3.1. Theorem 3.2.

\frac{1}{2}Δ ∑_{i,j,k,α}(h_{ijk}^{α})^{2} (3.6)

= ∑_{i,j,k,l,α}(h_{ijkl}^{α})^{2} + (\frac{9}{2}S + 7c) ∑_{i,j,k,α}(h_{ijk}^{α})^{2}

+ 3|∇S|^{2} − 5 ∑_{α} S_{α} ∑_{i,j,k}(h_{ijk}^{α})^{2}.

Since M^{2} is maximal, for any α, we have

h_{11}^{α} + h_{22}^{α} = 0.

Hence

(3.7)  h_{11l}^{α} = −h_{22l}^{α}, h_{11lk}^{α} = −h_{22lk}^{α} for any l, k.

In the sequel, we will often use the formula (3.7). Proof of Theorem 3.2.

∑_{i,j,k,α} h_{ijk}^{α}Δh_{ijk}^{α} = ∑_{i,j,k,l,α} h_{ijk}^{α}h_{ijkll}^{α} (3.8)

= ∑_{i,j,k,l,α} h_{ijk}^{α}[h_{ijlkl}^{α} + ∇_{l}(∑_{t} h_{tj}^{α}R_{tikl}

+ ∑_{t} h_{ti}^{α}R_{tjkl} − ∑_{β} h_{ij}^{β}R_{αβkl})]

= ∑_{i,j,k,l,α} h_{ijk}^{α}[h_{ijllk}^{α} + ∑_{t} h_{tjl}^{α}R_{tikl}

+ ∑_{t} h_{til}^{α}R_{tjkl} + ∑_{t} h_{ijt}^{α}R_{tlkl} − ∑_{β} h_{ijl}^{β}R_{αβkl}]

+ ∑_{i,j,k,l,α} h_{ijk}^{α}[∑_{t} h_{tjl}^{α}R_{tikl} + ∑_{t} h_{til}^{α}R_{tjkl} − ∑_{β} h_{ijl}^{β}R_{αβkl}]

+ ∑_{i,j,k,l,α} h_{ijk}^{α}[∑_{t} h_{tj}^{α}∇_{l}R_{tikl} + ∑_{t} h_{ti}^{α}∇_{l}R_{tjkl} − ∑_{β} h_{ij}^{β}∇_{l}R_{αβkl}]

= ∑_{i,j,k,α} h_{ijk}^{α}{∇_{k}[(S + 2c)h_{ij}^{α}

− 2 ∑_{l,t,β≠α} h_{lt}^{α}h_{tj}^{β}h_{il}^{β} + ∑_{l,t,β≠α} h_{tl}^{α}h_{tl}^{β}h_{ij}^{β}]}

+ 2 ∑_{i,j,k,l,t,α} h_{ijk}^{α}h_{tjl}^{α}R_{tikl} + 2 ∑_{i,j,k,l,t,α} h_{ijk}^{α}h_{til}^{α}R_{tjkl}

+ ∑_{i,j,k,l,t,α} h_{ijk}^{α}h_{ijt}^{α}R_{tlkl} − 2 ∑_{i,j,k,l,α,β} h_{ijk}^{α}h_{ijl}^{β}R_{αβkl}

+ ∑_{i,j,k,l,α} h_{ijk}^{α}[∑_{t} h_{tj}^{α}∇_{l}R_{tikl} + ∑_{t} h_{ti}^{α}∇_{l}R_{tjkl} − ∑_{β} h_{ij}^{β}∇_{l}R_{αβkl}].

(3.9)  ∑_{i,j,k,α} h_{ijk}^{α}{∇_{k}[(S + 2c)h_{ij}^{α} − 2 ∑_{l,t,β≠α} h_{lt}^{α}h_{tj}^{β}h_{il}^{β} + ∑_{l,t,β≠α} h_{tl}^{α}h_{tl}^{β}h_{ij}^{β}]}

= (S + 2c) ∑_{i,j,k,α}(h_{ijk}^{α})^{2} + ∑_{i,j,k,α} h_{ij}^{α}h_{ijk}^{α}∇_{k}S

− 2 ∑_{i,j,k,t,l,α,β≠α} h_{ijk}^{α}h_{tlk}^{α}h_{tj}^{β}h_{il}^{β} − 2 ∑_{i,j,k,t,l,α,β≠α} h_{ijk}^{α}h_{tl}^{α}h_{tjk}^{β}h_{il}^{β}
− 2 ∑_{i,j,k,t,l,α,β≠α} h_{ijk}^{α}h_{tl}^{α}h_{tj}^{β}h_{ilk}^{β} + ∑_{i,j,k,t,l,α,β≠α} h_{ijk}^{α}h_{tlk}^{α}h_{tl}^{β}h_{ij}^{β}
+ ∑_{i,j,k,t,l,α,β≠α} h_{ijk}^{α}h_{tl}^{α}h_{tlk}^{β}h_{ij}^{β} + ∑_{i,j,k,t,l,α,β≠α} h_{ijk}^{α}h_{ijk}^{β}h_{tl}^{α}h_{tl}^{β}

= (S + 2c) ∑_{i,j,k,α}(h_{ijk}^{α})^{2} + \frac{1}{2}|∇S|^{2} − 2 ∑_{i,j,k,l,t} h_{ijk}^{3}h_{tlk}^{3}h_{tj}^{4}h_{il}^{4}

− 4 ∑_{i,j,k,t} h_{ijk}^{4}h_{tjk}^{3}h_{ti}^{4}h_{ii}^{3} − 4 ∑_{i,j,k,t} h_{ijk}^{4}h_{tik}^{3}h_{tj}^{4}h_{jj}^{3} + ∑_{i,j,k,l,t} h_{ijk}^{3}h_{tlk}^{3}h_{tl}^{4}h_{ij}^{4}

+ \frac{S_{3}}{2} ∑_{i,j,k}(h_{ijk}^{4})^{2} + 4λ ∑_{i,j,k} h_{11k}^{4}h_{ij}^{4}h_{ijk}^{3} (by (3.2))

= (S + 2c) ∑_{i,j,k,α}(h_{ijk}^{α})^{2} + \frac{1}{2}|∇S|^{2} + \frac{S_{3}}{2} ∑_{i,j,k}(h_{ijk}^{4})^{2}

− 2 ∑_{i,j,k,l,t} h_{ijk}^{3}h_{tlk}^{3}h_{tj}^{4}h_{il}^{4} − 8 ∑_{i,j,k,t} h_{ijk}^{4}h_{tjk}^{3}h_{ti}^{4}h_{ii}^{3}

+ ∑_{k}(∑_{i,j} h_{ijk}^{3}h_{ij}^{4})^{2} + 4λ ∑_{i,j,k} h_{11k}^{4}h_{ij}^{4}h_{ijk}^{3}.

∑_{i,j,k,l,t,α} h_{ijk}^{α}h_{ijt}^{α}R_{tlkl} (3.10)

= ∑_{i,j,k,l,t,α} h_{ijk}^{α}h_{ijt}^{α}[c(δ_{tk}δ_{ll} − δ_{tl}δ_{lk}) − ∑_{β}(h_{tk}^{β}h_{ll}^{β} − h_{tl}^{β}h_{lk}^{β})]

= (\frac{S}{2} + c) ∑_{i,j,k,α}(h_{ijk}^{α})^{2}.

− 2 ∑_{i,j,k,l,α,β} h_{ijk}^{α}h_{ijl}^{β}R_{αβkl} (3.11)

= 2 ∑_{i,j,k,l,α,β} h_{ijk}^{α}h_{ijl}^{β} ∑_{t}(h_{tk}^{α}h_{tl}^{β} − h_{tl}^{α}h_{tk}^{β})

= −4 ∑_{i,j,k,l} h_{ijk}^{3}h_{ijl}^{4}h_{lk}^{4}(h_{ll}^{3} − h_{kk}^{3}).

(3.12)  2 ∑_{i,j,k,l,t,α} h_{ijk}^{α}h_{tjl}^{α}R_{tikl} + 2 ∑_{i,j,k,l,t,α} h_{ijk}^{α}h_{til}^{α}R_{tjkl}

= 2 ∑_{i,j,k,l,t,α} h_{ijk}^{α}h_{tjl}^{α}[c(δ_{tk}δ_{il} − δ_{tl}δ_{ik}) − ∑_{β}(h_{tk}^{β}h_{il}^{β} − h_{tl}^{β}h_{ik}^{β})]

+ 2 ∑_{i,j,k,l,t,α} h_{ijk}^{α}h_{itl}^{α}[c(δ_{tk}δ_{jl} − δ_{tl}δ_{jk}) − ∑_{β}(h_{tk}^{β}h_{jl}^{β} − h_{tl}^{β}h_{jk}^{β})]

= 4c ∑_{i,j,k,α}(h_{ijk}^{α})^{2} − 4 ∑_{i,j,k,l,t,α,β} h_{ijk}^{α}h_{tjl}^{α}h_{tk}^{β}h_{il}^{β}

+ 4 ∑_{i,j,k,l,t,α,β} h_{ijk}^{α}h_{tjl}^{α}h_{tl}^{β}h_{ik}^{β}

= 4c ∑_{i,j,k,α}(h_{ijk}^{α})^{2} + 2S_{3} ∑_{i,j,k,α}(h_{ijk}^{α})^{2}

− 4 ∑_{i,j,k,t,l} h_{ijk}^{3}h_{tjl}^{3}(h_{tk}^{4}h_{il}^{4} − h_{tl}^{4}h_{ik}^{4})

− 4 ∑_{i,j,k,t,l} h_{ijk}^{4}h_{tjl}^{4}(h_{tk}^{4}h_{il}^{4} − h_{tl}^{4}h_{ik}^{4}).

∑_{i,j,k,l,α} h_{ijk}^{α}[∑_{t} h_{tj}^{α}∇_{l}R_{tikl} + ∑_{t} h_{ti}^{α}∇_{l}R_{tjkl} (3.13)

− ∑_{β} h_{ij}^{β}∇_{l}R_{αβkl}]

= − ∑_{i,j,k,t,l,α,β} h_{ijk}^{α}h_{tj}^{α}∇_{l}(h_{tk}^{β}h_{il}^{β} − h_{tl}^{β}h_{ik}^{β})

− ∑_{i,j,k,t,l,α,β} h_{ijk}^{α}h_{ti}^{α}∇_{l}(h_{tk}^{β}h_{jl}^{β} − h_{tl}^{β}h_{jk}^{β})

+ ∑_{i,j,k,t,l,α,β} h_{ijk}^{α}h_{ij}^{β}∇_{l}(h_{tk}^{α}h_{tl}^{β} − h_{tl}^{α}h_{tk}^{β})

= −2 ∑_{i,j,k,t,l,α,β} h_{ijk}^{α}h_{tj}^{α}(h_{tkl}^{β}h_{il}^{β} − h_{tl}^{β}h_{ikl}^{β})

+ ∑_{i,j,k,t,l,α,β} h_{ijk}^{α}h_{ij}^{β}(h_{tkl}^{α}h_{tl}^{β} − h_{tl}^{α}h_{tkl}^{β})

= S_{3} ∑_{i,j,k}(h_{ijk}^{3})^{2} + \frac{S_{3}}{2} ∑_{i,j,k}(h_{ijk}^{4})^{2}

+ ∑_{k}(∑_{i,j} h_{ij}^{4}h_{ijk}^{3})^{2} − 4λ ∑_{i,j,k} h_{11k}^{4}h_{ij}^{4}h_{ijk}^{3}

− 2 ∑_{i,j,k,t,l} h_{ijk}^{4}h_{tkl}^{4}h_{il}^{4}h_{tj}^{4} + 2 ∑_{i,j,k,t,l} h_{ijk}^{4}h_{ikl}^{4}h_{tl}^{4}h_{tj}^{4}

− 4 ∑_{i,j,k,t,l} h_{ijk}^{3}h_{ikl}^{4}h_{jl}^{4}(h_{ii}^{3} − h_{jj}^{3}).

Hence, (3.8)∼(3.13) yield

∑_{i,j,k,α} h_{ijk}^{α}Δh_{ijk}^{α} (3.14)

= (\frac{3}{2}S + 7c) ∑_{i,j,k,α}(h_{ijk}^{α})^{2} + \frac{1}{2}|∇S|^{2} + 3S_{3} ∑_{i,j,k,α}(h_{ijk}^{α})^{2}

− 4 ∑_{i,j,k,l} h_{ijk}^{3}h_{ijl}^{4}h_{lk}^{4}(h_{ll}^{3} − h_{kk}^{3})

− 4 ∑_{i,j,k,l} h_{ijk}^{3}h_{ikl}^{4}h_{jl}^{4}(h_{ii}^{3} − h_{jj}^{3})

− 2 ∑_{i,j,k,l,t} h_{ijk}^{3}h_{tlk}^{3}h_{tj}^{4}h_{il}^{4} − 8 ∑_{i,j,k,t} h_{ijk}^{4}h_{tjk}^{3}h_{ti}^{4}h_{ii}^{3}

− 4 ∑_{i,j,k,t,l} h_{ijk}^{3}h_{tjl}^{3}(h_{tk}^{4}h_{il}^{4} − h_{tl}^{4}h_{ik}^{4})

− 4 ∑_{i,j,k,t,l} h_{ijk}^{4}h_{tjl}^{4}(h_{tk}^{4}h_{il}^{4} − h_{tl}^{4}h_{ik}^{4})

− 2 ∑_{i,j,k,t,l} h_{ijk}^{4}h_{tkl}^{4}h_{il}^{4}h_{tj}^{4} + 2 ∑_{i,j,k,t,l} h_{ijk}^{4}h_{ikl}^{4}h_{tl}^{4}h_{tj}^{4}

+ 2 ∑_{k}(∑_{i,j} h_{ijk}^{3}h_{ij}^{4})^{2}.

8 ∑_{i,j,k,t} h_{ijk}^{4}h_{tjk}^{3}h_{ti}^{4}h_{ii}^{3} (3.15)

= 8λ ∑_{j,k,t}(h_{1jk}^{4}h_{tjk}^{3}h_{t1}^{4} − h_{2jk}^{4}h_{tjk}^{3}h_{t2}^{4})

= 8λ ∑_{j,k}(h_{1jk}^{3}h_{1jk}^{4}μ + h_{2jk}^{3}h_{1jk}^{4}μ_{1}

− h_{2jk}^{4}h_{1jk}^{3}μ_{1} + h_{2jk}^{4}h_{2jk}^{3}μ) = 8λμ_{1}(h_{1jk}^{4}h_{2jk}^{3} − h_{2jk}^{4}h_{1jk}^{3}) (by (3.2)) = 32λμ_{1}(h_{222}^{4}h_{111}^{3} − h_{111}^{4}h_{222}^{3}).

4 ∑_{i,j,k,l} h_{ijk}^{3}h_{ijl}^{4}h_{lk}^{4}(h_{ll}^{3} − h_{kk}^{3}) (3.16)

= 4 ∑_{i,j} h_{ij1}^{3}h_{ij2}^{4}h_{12}^{4}(h_{22}^{3} − h_{11}^{3})

+ 4 ∑_{i,j} h_{ij2}^{3}h_{ij1}^{4}h_{12}^{4}(h_{11}^{3} − h_{22}^{3})

= −8λμ_{1} ∑_{ij}(h_{1ij}^{3}h_{2ij}^{4} − h_{1ij}^{4}h_{2ij}^{3}) (by (3.2))

= 32λμ_{1}(h_{222}^{4}h_{111}^{3} − h_{111}^{4}h_{222}^{3}).

4 ∑_{i,j,k,l} h_{ijk}^{3}h_{ikl}^{4}h_{lj}^{4}(h_{ii}^{3} − h_{jj}^{3}) (3.17)

= 4 ∑_{i,j,} h_{12k}^{3}h_{1kl}^{4}h_{l2}^{4}(h_{11}^{3} − h_{22}^{3})

+ 4 ∑_{i,j,} h_{21k}^{3}h_{2kl}^{4}h_{1l}^{4}(h_{22}^{3} − h_{11}^{3})

= 8λ ∑_{k,l}(h_{12k}^{3}h_{1kl}^{4}h_{l2}^{4} − h_{12k}^{3}h_{2kl}^{4}h_{1l}^{4})

= 16λμ_{1}(h_{222}^{4}h_{111}^{3} − h_{111}^{4}h_{222}^{3}) (by (3.2)).

4 ∑_{i,j,k,t,l,α} h_{ijk}^{α}h_{tjl}^{α}(h_{tk}^{4}h_{il}^{4} − h_{tl}^{4}h_{ik}^{4}) (3.18)

= 4 ∑_{i,j,k,t,α} h_{ijk}^{α}h_{tj1}^{α}(h_{tk}^{4}h_{i1}^{4} − h_{t1}^{4}h_{ik}^{4})

+ 4 ∑_{i,j,k,t,α} h_{ijk}^{α}h_{tj2}^{α}(h_{tk}^{4}h_{i2}^{4} − h_{t2}^{4}h_{ik}^{4})

= 4 ∑_{i,j,k,α} h_{ijk}^{α}h_{1j1}^{α}(h_{1k}^{4}h_{i1}^{4} − h_{11}^{4}h_{ik}^{4})

+ 4 ∑_{i,j,k,t,α} h_{ijk}^{α}h_{1j2}^{α}(h_{1k}^{4}h_{i2}^{4} − h_{12}^{4}h_{ik}^{4})

+ 4 ∑_{i,j,k,α} h_{ijk}^{α}h_{2j1}^{α}(h_{2k}^{4}h_{i1}^{4} − h_{21}^{4}h_{ik}^{4})

+ 4 ∑_{i,j,k,α} h_{ijk}^{α}h_{2j2}^{α}(h_{2k}^{4}h_{i2}^{4} − h_{22}^{4}h_{ik}^{4})

= −8 ∑_{i,j,k,α} h_{ijk}^{α}h_{11j}^{α}h_{ik}^{4}h_{11}^{4}

+ 4 ∑_{i,j,k,α} h_{ijk}^{α}h_{1j1}^{α}(h_{1k}^{4}h_{i1}^{4} − h_{2k}^{4}h_{2i}^{4})

− 8 ∑_{i,j,k,α} h_{ijk}^{α}h_{12j}^{α}h_{ik}^{4}h_{12}^{4} + 8 ∑_{i,j,k,α} h_{ijk}^{α}h_{12j}^{α}h_{1i}^{4}h_{2k}^{4}

= −2S_{4} ∑_{i,j,k,α}(h_{ijk}^{α})^{2}.

For any α,

2 ∑_{i,j,k,l,t} h_{ijk}^{α}h_{tlk}^{α}h_{tj}^{4}h_{il}^{4} (3.19)

= 2 ∑_{i,j,k,l} h_{ijk}^{α}h_{1lk}^{α}h_{1j}^{4}h_{il}^{4} + 2 ∑_{i,j,k,l,t} h_{ijk}^{α}h_{2lk}^{α}h_{2j}^{4}h_{il}^{4}

= 2μ ∑_{i,k,l}(h_{1ik}^{α}h_{1lk}^{α} − h_{2ik}^{α}h_{2lk}^{α})h_{il}^{4} + 4μ_{1} ∑_{i,k,l} h_{1ik}^{α}h_{2lk}^{α}h_{il}^{4}

= 0.

2 ∑(∑_{i,j} h_{ijk}^{3}h_{ij}^{4})^{2} (3.20)

= 2 ∑_{k}(∑_{ij} h_{ij}^{4}h_{ijk}^{3})^{2} = S_{4} ∑_{i,j,k}(h_{ijk}^{3})^{2}.

2 ∑_{i,j,k,t,l} h_{ijk}^{4}h_{ikl}^{4}h_{tl}^{4}h_{tj}^{4} (3.21)

= 2 ∑_{i,j,k,t} h_{ijk}^{4}h_{ik1}^{4}h_{t1}^{4}h_{tj}^{4} + 2 ∑_{i,j,k,t} h_{ijk}^{4}h_{ik2}^{4}h_{t2}^{4}h_{tj}^{4}

= S_{4} ∑_{i,j,k}(h_{ijk}^{4})^{2}.

According to (3.14) ∼ (3.21), we get

∑_{i,j,k,α} h_{ijk}^{α}Δh_{ijk}^{α} (3.22)

= (\frac{9}{2}S + 7c) ∑_{i,j,k,α}(h_{ijk}^{α})^{2} + \frac{1}{2}|∇S|^{2}

− 80λμ_{1}(h_{111}^{3}h_{222}^{4} − h_{111}^{4}h_{222}^{3}).

Since, for any α,

∑_{i,j,k}(h_{ijk}^{α})^{2} = 4{(h_{111}^{α})^{2} + (h_{222}^{α})^{2}}

and

|∇S|^{2} = ∑_{l}(∇_{l}S)^{2}

= ∑_{l}(2 ∑_{i,j,α} h_{ij}^{α}h_{ijl}^{α})^{2}

= 16 ∑_{l}(λh_{11l}^{3} + μh_{11l}^{4} + μ_{1}h_{12l}^{4})^{2}

= 2 ∑_{i,j,k,α} S_{α}(h_{ijk}^{α})^{2} − 32λμ_{1}(h_{111}^{3}h_{222}^{4} − h_{111}^{4}h_{222}^{3}),

we obtain

(3.23)  −8λμ_{1}(h_{111}^{3}h_{222}^{4} − h_{111}^{4}h_{222}^{3}) = \frac{1}{4}|∇S|^{2} − \frac{1}{2} ∑_{i,j,k,α} S_{α}(h_{ijk}^{α})^{2}.

From

\frac{1}{2}Δ ∑_{i,j,k,α}(h_{ijk}^{α})^{2} = ∑_{i,j,k,α} h_{ijk}^{α}Δh_{ijk}^{α} + ∑_{i,j,k,l,α}(h_{ijkl}^{α})^{2},

(3.22) and (3.23) yields the Theorem 3.2. Lemma 1.

∑_{i,j,k,l,α}(h_{ijkl}^{α})^{2} = ∑_{i≠j,α}(h_{iijj}^{α} − h_{jjii}^{α})^{2} + ∑_{i≠j,α}(h_{iijj}^{α} + h_{jjii}^{α})^{2}

+ ∑_{i≠j,α}(h_{iiij}^{α} − h_{iiji}^{α})^{2} + ∑_{i≠j,α}(h_{iiij}^{α} + h_{iiji}^{α})^{2}.

Proof.

∑_{i,j,k,l,α}(h_{ijkl}^{α})^{2} = ∑_{i,α}(h_{iiii}^{α})^{2} + 3 ∑_{i≠j,α}(h_{iijj}^{α})^{2} + ∑_{i≠j,α}(h_{iiij}^{α})^{2} + 3 ∑_{i≠j,α}(h_{iiji}^{α})^{2}.

Since

h_{iiii}^{α} = −h_{jjii}^{α} for any α and j ≠ i

and

h_{iiij}^{α} = −h_{jjij}^{α} for any α and j ≠ i,

we know that Lemma 1 holds. Lemma 2.

h_{1122}^{3} − h_{2211}^{3} = λ(2c + S + S_{4}),
h_{1112}^{3} − h_{1121}^{3} = 0,
h_{1122}^{4} − h_{2211}^{4} = (2c + S)μ_{,}
h_{1112}^{4} − h_{1121}^{4} = −(2c + S + S_{3})μ_{1}.

Proof. From the Ricci formula (2.8), we have

h_{iijj}^{α} − h_{jjii}^{α} = h_{ijij}^{α} − h_{ijji}^{α}

= ∑_{t} h_{tj}^{α}R_{tiij} + ∑_{t} h_{it}^{α}R_{tjij} − ∑_{t} h_{ij}^{β}R_{αβij}

= (h_{ii}^{α} − h_{jj}^{α})c − \frac{S}{2} (h_{jj}^{α} − h_{ii}^{α})

− ∑_{t,β≠α} h_{tj}^{α}h_{ti}^{β}h_{ij}^{β} + ∑_{t,β≠α} h_{it}^{α}h_{tj}^{β}h_{ij}^{β}.

Hence

h_{1122}^{3} − h_{2211}^{3} = λ(2c + S + S_{4}),
h_{1122}^{4} − h_{2211}^{4} = (2c + S)μ.

By the same proof, we can obtain

h_{1112}^{3} − h_{1121}^{3} = 2λμμ_{1} = 0 (by (3.2)),
h_{1112}^{4} − h_{1121}^{4} = −(2c + S + S_{3})μ_{1}.

Thus we complete the proof of Lemma 2.

Next, we shall prove the Main Theorem. Since the scalar curvature is constant if and only if S is constant, in the sequel, we assume that S is constant.

Proof of Main Theorem. If S = 0, then M is totally geodesic because S is constant. Next we assume S ≠ 0. Since S is constant, we have

∇_{l}S = 0, for l = 1, 2,

namely,

(3.24)  \{ \begin{matrix}2λh_{11l}^{3} + 2μh_{11l}^{4} + 2μ_{1}h_{12l}^{4} = 0. \\ 2λh_{111}^{3} + 2μh_{111}^{4} − 2μ_{1}h_{222}^{4} = 0, \\ 2λh_{222}^{3} + 2μh_{222}^{4} + 2μ_{1}h_{111}^{4} = 0. \end{matrix}

Hence we obtain

(3.25)  S_{3} ∑_{i,j,k}(h_{ijk}^{3})^{2} = S_{4} ∑_{i,j,k}(h_{ijk}^{4})^{2},

from

(3.26)  ∑_{i,j,k}(h_{ijk}^{α})^{2} = 4[(h_{111}^{α})^{2} + (h_{222}^{α})^{2}].

We know that S_{3}S_{4} are function defined globally on M. Because S is constant, from Gauss equation, we infer that the sectional curvature is bounded from below and that function S_{3}S_{4} is bounded because 0 ≤ S_{3}S_{4} ≤ S^{2}. Since M is complete, from the Generalized Maximum Principle due to Omori and Yau, we know that there exists a sequence {p_{m}} ⊂ M^{2} of points such that

(3.27)  lim_{m→∞}(S_{3}S_{4})(p_{m}) = inf(S_{3}S_{4}),

(3.28)  lim_{m→∞} |∇(S_{3}S_{4})|(p_{m}) = 0,

(3.29)  lim_{m→∞} inf Δ(S_{3}S_{4})(p_{m}) ≥ 0.

From Theorem 3.1, we have

(3.30)  ∑_{i,j,k,α}(h_{ijk}^{α})^{2} = −S(S + 2c) − 2S_{3}S_{4}.

because S is constant. Hence ∑_{i,j,k,α}(h_{ijk}^{α})^{2} is bounded and

(3.31)  lim_{m→∞} ∑_{i,j,k,α}(h_{ijk}^{α})^{2}(p_{m}) = sup ∑_{i,j,k,α}(h_{ijk}^{α})^{2},

(3.32)  lim_{m→∞} |∇ ∑_{i,j,k,α}(h_{ijk}^{α})^{2}|(p_{m}) = −2 lim_{m→∞} |∇(S_{3}S_{4})|(p_{m}) = 0.

From

(3.33)  Δ ∑_{i,j,k,α}(h_{ijk}^{α})^{2} = −2S_{3}ΔS_{4} − 4∇S_{3} · ∇S_{4} − 2S_{4}ΔS_{3}

and Theorem 3.1 and Theorem 3.2, we obtain that ∑_{i,j,k,l,α}(h_{ijkl}^{α})^{2} is bounded. Thus, we can assume lim_{m→∞} S_{3}(p_{m}) = S̃_{3}, lim_{m→∞} S_{4}(p_{m}) = S̃_{4}, lim_{m→∞} λ(p_{m}) = λ̃, lim_{m→∞} μ(p_{m}) = μ̃, lim_{m→∞} μ_{1}(p_{m}) = μ̃_{1}, lim_{m→∞} h_{ijk}^{α}(p_{m}) = h̃_{ijk}^{α} and lim_{m→∞} h_{ijkl}^{α}(p_{m}) = h̃_{ijkl}^{α}, by taking a subsequence if necessary. Since

lim_{m→∞} |∇(S_{3}S_{4})|(p_{m}) = 0

and

∇_{l}(S_{3}S_{4}) = S_{3}∇_{l}S_{4} + S_{4}∇_{l}S_{3},

we have

lim_{m→∞}(S_{3}∇_{l}S_{4} + S_{4}∇_{l}S_{3})(p_{m}) = 0.

From S = S_{3} + S_{4}, we get

∇_{l}S_{3} = −∇_{l}S_{4}.

Therefore,

(3.34)  lim_{m→∞}(S_{3} − S_{4})(∇_{l}S_{4})(p_{m}) = lim_{m→∞}(S_{4} − S_{3})(∇_{l}S_{3})(p_{m}) = 0.

Hence,

(3.35)  S̃_{3} = S̃_{4} or lim_{m→∞}(∇_{l}S_{4})(p_{m}) = lim_{m→∞}(∇_{l}S_{3})(p_{m}) = 0.

(1). In the case where S̃_{3} = S̃_{4}. From Theorem 3.1, we have

0 = lim_{m→∞}\{ ∑_{i,j,k,α}(h_{ijk}^{α})^{2} + (S + 2c)S + 2S_{3}S_{4}\}(p_{m}) = sup ∑_{i,j,k,α}(h_{ijk}^{α})^{2} + (\frac{3}{2}S + 2c)S.

Hence,

sup ∑_{i,j,k,α}(h_{ijk}^{α})^{2} = −(\frac{3}{2}S + 2c)S.

From Theorem 3.1, we have

∑_{i,j,k,α}(h_{ijk}^{α})^{2} = −(S + 2c)S − 2S_{3}S_{4}

= −(\frac{3}{2}S + 2c)S + \frac{1}{2}(S_{3} − S_{4})^{2}

≥ −(\frac{3}{2}S + 2c)S.

Hence, we have

inf ∑_{i,j,k,α}(h_{ijk}^{α})^{2} ≥ −(\frac{3}{2}S + 2c)S = sup ∑_{i,j,k,α}(h_{ijk}^{α})^{2},

that is,

(3.36)  ∑_{i,j,k,α}(h_{ijk}^{α})^{2} ≡ −(\frac{3}{2}S + 2c)S

is constant. Therefore, S_{3} ≡ S_{4} on M^{2} and they are constant. Hence, on M^{2},

∇_{l}S_{3} = ∇_{l}S_{4} = 0.

From (3.25), we have ∑_{i,j,k,α}(h_{ijk}^{α})^{2} = 0 on M^{2}. According to (3.36), we have S = \frac{−4c}{3}.

(2). In the case where S̃_{3} ≠ S̃_{4}. We have, for l = 1, 2,

lim_{m→∞} ∇_{l}S_{3}(p_{m}) = lim_{m→∞} ∇_{l}S_{4}(p_{m}) = 0.

From |∇S_{α}|^{2} = 4S_{α} ∑_{i,j,k}(h_{ijk}^{α})^{2}, we have

lim_{m→∞} S_{3} ∑_{i,j,k}(h_{ijk}^{3})^{2}(p_{m}) = lim_{m→∞} S_{4} ∑_{i,j,k}(h_{ijk}^{4})^{2}(p_{m}) = 0,

that is

(3.37)  S̃_{3} lim_{m→∞} ∑_{i,j,k}(h_{ijk}^{3})^{2}(p_{m}) = S̃_{4} lim_{m→∞} ∑_{i,j,k}(h_{ijk}^{4})^{2}(p_{m}) = 0.

If

lim_{m→∞} ∑_{i,j,k}(h_{ijk}^{3})^{2}(p_{m}) = lim_{m→∞} ∑_{i,j,k}(h_{ijk}^{4})^{2}(p_{m}) = 0,

we have

sup ∑_{i,j,k,α}(h_{ijk}^{α})^{2} = 0.

Hence, from Theorem 3.1, we have

0 = S(S + 2c) + 2S_{3}S_{4} = S(\frac{3}{2}S + 2c) − \frac{1}{2}(S_{3} − S_{4})^{2}.

Hence S > \frac{−4c}{3} and S_{3}S_{4} is constant. Hence S_{3} and S_{4} are constant because S = S_{3} + S_{4} and S_{3}S_{4} are constant. Since S > 0, we can assume S_{3} > 0. From the proof of Theorem 3.1, we have, for α = 3, 4,

0 = \frac{1}{2}ΔS_{α} = ∑_{i,j,k}(h_{ijk}^{α})^{2} + (S + 2c)S_{α} + S_{3}S_{4}.

Hence,

(S + 2c)S_{α} + S_{3}S_{4} = 0.

Therefore, S = −2c and M^{2} is the hyperbolic cylinder of the totally geodesic hypersurface H_{1}^{3}(c) from the Theorem due to Ishihara [8].

Next we can assume lim_{m→∞} ∑_{i,j,k}(h_{ijk}^{4})^{2}(p_{m}) ≠ 0 without loss of the generality. We have S̃_{4} = 0. Because S = S_{3} + S_{4} > 0 is constant, we have S̃_{3} ≠ 0. Hence,

lim_{m→∞} ∑_{i,j,k}(h_{ijk}^{3})^{2}(p_{m}) = 0.

We shall prove S = \frac{−10c}{11} in this case. Since S̃_{4} = 0, we have

lim_{m→∞} ∑_{i,j,k,α}(h_{ijk}^{α})^{2}(p_{m}) + (2c + S)S = 0.

Hence,

sup ∑_{i,j,k,α}(h_{ijk}^{α})^{2} = −(S + 2c)S.

From (3.31) and (3.32), we know that lim_{m→∞} |∇ ∑_{i,j,k,α}(h_{ijk}^{α})^{2}| = 0. Hence,

lim_{m→∞} ∑_{i,j,k} h_{ijk}^{4}h_{ijkl}^{4}(p_{m}) = 0 for l = 1, 2,

because of lim_{m→∞} ∑_{i,j,k}(h_{ijk}^{3})^{2}(p_{m}) = 0. Thus we conclude

h̃_{111}^{4}h̃_{2211}^{4} = −h̃_{222}^{4}h̃_{1121}^{4}, h̃_{111}^{4}h̃_{1112}^{4} = h̃_{222}^{4}h̃_{1122}^{4}.

According to Lemma 2 we have

h̃_{1122}^{4} = h̃_{2211}^{4}, h̃_{1112}^{4} = h̃_{1121}^{4}.

Hence

(h̃_{111}^{4})^{2}h̃_{2211}^{4} + (h̃_{222}^{4})^{2}h̃_{2211}^{4} = 0.

Since ∑_{i,j,k}(h̃_{ijk}^{4})^{2} ≠ 0, then h̃_{2211}^{4} = h̃_{1122}^{4} = h̃_{1121}^{4} = h̃_{1112}^{4} = 0. On the other hand, since S is constant we have

∑_{i,j,α} h_{ij}^{α}h_{ijlk}^{α} + ∑_{i,j,α} h_{ijk}^{α}h_{ijl}^{α} = 0 for any l, k.

Hence

2λ̃h̃_{1112}^{3} = 0,

2λ̃h̃_{1121}^{3} = 0,

2λ̃h̃_{1122}^{3} = −\frac{1}{2} ∑_{i,j,k,α}(h̃_{ijk}^{α})^{2},

2λ̃h̃_{2211}^{3} = \frac{1}{2} ∑_{i,j,k,α}(h̃_{ijk}^{α})^{2}.

We infer

(h̃_{1122}^{3} + h̃_{2211}^{3})^{2} + (h̃_{1112}^{3} + h̃_{1121}^{3})^{2} = 0.

Therefore, from Lemma 2 we have

(3.38)  ∑_{i,j,k,l,α}(h̃_{ijkl}^{α})^{2} = (S + 2c)^{2}S.

From Theorem 3.1, we have

\frac{1}{2}Δ ∑_{i,j,k,α}(h_{ijk}^{α})^{2}

= −Δ(S_{3}S_{4})

= −S_{3}ΔS_{4} − 2∇S_{3} · ∇S_{4} − S_{4}ΔS_{3}.

Hence

(3.39)  lim_{m→∞} \frac{1}{2}Δ ∑_{i,j,k,α}(h_{ijk}^{α})^{2}(p_{m}) = − lim_{m→∞}(S_{3}ΔS_{4})(p_{m}) = 2S(S + 2c)S

because of S̃_{4} = 0 and lim_{m→∞} ∑_{i,j,k}(h_{ijk}^{3})^{2}(p_{m}) = 0. From (3.37), (3.38), (3.39) and Theorem 3.2, we have

2(S + 2c)S^{2} = \frac{1}{2} lim_{m→∞} Δ ∑_{i,j,k,α}(h_{ijk}^{α})^{2}(p_{m}) = (S + 2c)^{2}S − (\frac{9}{2}S + 7c)(S + 2c)S.

Thus S = −\frac{10c}{11}. From the above proof, we know that S = \frac{−4c}{3} if and only if S_{3} = S_{4} is constant on M^{2} and ∑_{i,j,k,α}(h_{ijk}^{α})^{2} ≡ 0. By making use of the similar method to one which was used in [7] by Chern, do Carmo and Kobayashi, we can prove that M^{2} is isometric to the hyperbolic Veronese surface. We complete the proof of Main Theorem.

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