CURVATURES OF COMPLETE

HYPERSURFACES IN SPACE FORMS*

Qing-Ming Cheng

Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan (cheng@ms.saga-u.ac.jp)

Abstract. In this paper, we investigated that 3-dimensional complete minimal hypersurfaces with constant Gauss-Kronecker curvature in a space form M^{4}(c) (c ≤ 0). We proved that if the scalar curvature of a such hypersurface is bounded from below, then its Gauss-Kronecker curvature vanishes identically. Examples of complete minimal hypersurfaces in the Euclidean space E^{4} and the hyperbolic space H^{4}(c) with vanishing Gauss-Kronecker curvature are presented also, which are not totally geodesic. It is also proved that totally umbilical hypersurfaces are the only complete hypersurfaces with non-zero constant mean curvature and with zero quasi-Gauss-Kronecker curvature in a space form M^{4}(c) (c ≤ 0) if the scalar curvature is bounded from below. In particular, we classified complete hypersurfaces with constant mean curvature and with constant quasi-Gauss-Kronecker curvature in a space form M^{4}(c) (c ≤ 0) if the scalar curvature r satisfies r ≥ \frac{2}{3} c.

1. Introduction

Let M^{n} be an n-dimensional hypersurface in a space form M^{n+1}(c) of constant curvature c. Where M^{n+1}(c) denotes the (n + 1)-dimensional sphere S^{n+1}(c) of constant curvature c, the (n + 1)-dimensional Euclidean space E^{n+1} or the (n + 1)- dimensional hyperbolic space H^{n+1}(c) of constant curvature c according as c > 0, c = 0 or c < 0. It is well known that curvature properties are very important in the research of differential geometry. In particular, we know that n-dimensional closed hypersurfaces in E^{n+1} with non-vanishing Gauss-Kronecker curvature at every point of M^{n} are diffeomorphic to a sphere. When n = 3, Peng and Terng [11] and Cheng [4] proved that Cartan’s isoparametric minimal hypersurfaces with zero Gauss-Kronecker curvature are the only complete minimal hypersurfaces in S^{4}(1) with constant scalar curvature and three distinct principal curvatures at each point.

Key words and phrases. complete minimal hypersurfaces, isoparametric hypersurfaces, Gauss-Kronecker curvature, quasi-Gauss-Kronecker curvature and scalar curvature.

2000 Mathematics Subject Classification 53C42

On the other hand, de Almeida and Brito [1] considered closed minimal hypersurfaces in S^{4}(1) with constant Gauss-Kronecker curvature. They proved that the Clifford torus S^{2} × S^{1} and the boundary of the tube of a minimal 2-dimensional surface in S^{4}(1) with geodesic radius π/2 are the only closed minimal hypersurfaces in S^{4}(1) with constant Gauss-Kronecker curvature and non-vanishing second fundamental form. Recently, Lusala [8] studied complete minimal hypersurfaces with zero Gauss-Kronecker curvature in S^{4}(1). A local classification of minimal hypersurfaces with zero Gauss-Kronecker curvature in S^{4}(1) is given under additional conditions. It is also natural to investigate complete minimal hypersurfaces in E^{n+1} and H^{n+1}(c) with constant Gauss-Kronecker curvature. It is our purpose to study complete hypersurfaces with constant mean curvature in E^{4} and H^{4}(c). In section 3, we shall prove that if M^{3} is a 3-dimensional complete minimal hypersurface with constant Gauss-Kronecker curvature in M^{4}(c) (c ≤ 0), then the scalar curvature is not bounded from below or the Gauss-Kronecker curvature vanishes identically. Examples of complete minimal hypersurfaces with zero Gauss-Kronecker curvature in E^{4} and H^{4}(c) are given, which are not totally geodesic. Hence, we can not expect to prove that totally geodesic hypersurfaces are the only complete minimal hypersurfaces with zero Gauss-Kronecker curvature in M^{4}(c) (c ≤ 0). In section 4, we shall study complete hypersurfaces with non-zero constant mean curvature in M^{4}(c) (c ≤ 0). In [2], the author introduced a notation of quasi-Gauss-Kronecker curvature for hypersurfaces in space forms and studied compact hypersurfaces in S^{4}(1). In the section 4, we consider quasi-Gauss-Kronecker curvature of complete hypersurfaces in M^{4}(c) (c ≤ 0). It will be proved that totally umbilical hypersurfaces are the only complete hypersurfaces with non-zero constant mean curvature and with zero quasi-Gauss-Kronecker curvature in M^{4}(c) (c ≤ 0) if the scalar curvature is bounded from below. It is well known that the totally umbilical spheres, the totally geodesic hyperplane E^{3} and the cylinder E^{k} × S^{3−k} for, k = 1, 2, are complete hypersurfaces with constant mean curvature, constant quasi-Gauss-Kronecker curvature and nonnegative scalar curvature in E^{4}. It is natural to ask whether these hypersurfaces are the only complete hypersurfaces with constant mean curvature in E^{4}, which have constant quasi-Gauss-Kronecker curvature and non-negative scalar curvature. In the section 4, we shall answer this problem affirmatively. In fact, we shall prove a more general result, that is, we classified complete hypersurfaces with constant mean curvature and with constant quasi-Gauss-Kronecker curvature in M^{4}(c) (c ≤ 0) if the scalar curvature r satisfies r ≥ \frac{2}{3}c. We should remark the quasi-Gauss-Kronecker curvature is invarant under conformal transformation.

2. Preliminaries

Let M^{n} be an n-dimensional hypersurface in a space form M^{n+1}(c). M^{n+1}(c) denotes the (n + 1)-dimensional sphere S^{n+1}(c) of constant curvature c, the (n + 1)- dimensional Euclidean space E^{n+1} or the (n + 1)-dimensional hyperbolic space H^{n+1}(c) of constant curvature c, according as c > 0, c = 0 or c < 0. We take a local orthonormal frame field {e_{1}, …, e_{n+1}} in M^{n+1}(c), restricted to M^{n}, so that e_{1}, …, e_{n} are tangent to M^{n}. Let ω_{1}, ⋯, ω_{n+1} denote the dual coframe fields in M^{n+1}(c). 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 M^{n+1}(c) 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} + Ω_{AB}, Ω_{AB} = −\frac{1}{2} ∑_{C,D=1}^{n+1} R_{ABCD}ω_{C} ∧ ω_{D},

R_{ABCD} = c(δ_{AC}δ_{BD} − δ_{AD}δ_{BC}),

where Ω_{AB} (resp. R_{ABCD}) denotes the curvature form (resp. the components of the curvature tensor) of M^{n+1}(c). Then, on M^{n},

ω_{n+1} = 0.

It follows from Cartan’s Lemma that

ω_{n+1i} = ∑_{j} h_{ij}ω_{j}, h_{ij} = h_{ji}.  (2.1)

The second fundamental form α and the mean curvature H of M^{n} are defined by

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

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

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

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

r = n(n − 1)c + n^{2}H^{2} − S  (2.3)

where r is the scalar curvature of M^{n} and S = ∑_{ij=1}^{n} h_{ij}^{2} is the square of the length of the second fundamental form of M^{n}.

The function K = det(h_{ij}) is called the Gauss-Kronecker curvature of M^{n}. Let μ_{ij} = h_{ij} − Hδ_{ij}, then K_{q} = det(μ_{ij}) = det(h_{ij} − Hδ_{ij}) is called quasi-Gauss-Kronecker curvature. It is obvious that when M^{n} is minimal, the quasi-Gauss-Kronecker curvature K_{q} is nothing but the Gauss-Kronecker curvature K. We can prove that the quasi-Gauss-Kronecker curvature is a conformal invariant. Note that the Gauss-Kronecker curvature is not a conformal invariant generally.

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

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

and

K = det(h_{ij}) = λ_{1}λ_{2} ⋯ λ_{n},  (2.5)
K_{q} = det(μ_{ij}) = (λ_{1} − H)(λ_{2} − H) ⋯ (λ_{n} − H) = μ_{1}μ_{2} ⋯ μ_{n},  (2.6)

where μ_{i} = λ_{i} − H.

By taking the exterior differentiation of (2.1) and defining h_{ijk} by

∑_{k=1}^{n} h_{ijk}ω_{k} = dh_{ij} − ∑_{k=1}^{n} h_{ik}ω_{kj} − ∑_{k=1}^{n} h_{jk}ω_{ki},  (2.7)

we obtain Codazzi equation by straightforward computations

h_{ijk} = h_{ikj} = h_{jik}.  (2.8)

We take the exterior differentiation of (2.7) and define h_{ijkl} by

∑_{l=1}^{n} h_{ijkl}ω_{l} = dh_{ijk} − ∑_{l=1}^{n} h_{ljk}ω_{li} − ∑_{l=1}^{n} h_{ilk}ω_{lj} − ∑_{l=1}^{n} h_{ijl}ω_{lk}.  (2.9)

Then Ricci formula for the second fundamental form is given by

h_{ijkl} − h_{ijlk} = ∑_{m=1}^{n} h_{mj}R_{mikl} + ∑_{m=1}^{n} h_{im}R_{mjkl}.  (2.10)

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

Δh_{ij} = ∑_{k=1}^{n} h_{ijkk}.  (2.11)

The following Generalized Maximum Principle due to Omori and Yau will be used.

Generalized Maximum Principle ( Omori [9] and Yau [12]). Let M^{n} be a complete Riemannian manifold whose sectional curvature is bounded from below and f ∈ C^{2}(M) a function bounded from above on M^{n}. Then for any ϵ > 0, there exists a point p ∈ M^{n} such that

f(p) ≥ sup f − ϵ, ∥gradf∥(p) < ϵ, ∇_{i}∇_{i}f(p) < ϵ, for i = 1, 2, ⋯, n.

3. Complete minimal hypersurfaces

In this section, we shall study complete minimal hypersurfaces with zero Gauss-Kronecker curvature in E^{4} and H^{4}(c). First, we prove the following:

Theorem 3.1. Let M^{3} be a 3-dimensional complete minimal hypersurface with constant Gauss-Kronecker curvature in M^{4}(c) (c ≤ 0). If the scalar curvature is bounded from below, then the Gauss-Kronecker curvature vanishes identically.

Proof. Since n = 3 and M^{3} is minimal, we know that the Gauss-Kronecker curvature K satisfies 3K = f_{3} = ∑_{i=1}^{3} λ_{i}^{3}. If the Gauss-Kronecker curvature K is non-zero constant, then we have, from (2.2), (2.10) and (2.11),

0 = ΔK = \frac{1}{3}Δf_{3} = −(S − 3c)f_{3} + 2 ∑_{i=1}^{3} λ_{i}h_{ijk}^{2}.  (3.1)

Since the scalar curvature is bounded from below, we infer that from Gauss equation (2.3), the squared norm S of the second fundamental form is bounded from above. From (2.2), we know that the sectional curvature is bounded from below. Applying the Generalized Maximum Principle due to Omori and Yau in the section 2 to S, we see that there exists a sequence {p_{m}} of points in M^{3} such that

lim_{m→∞} S(p_{m}) = sup S, lim_{m→∞} ∥grad S∥(p_{m}) = 0.  (3.2)

Since S is bounded, we know that {λ_{i}(p_{m})} is a bounded sequence for any i, (i = 1, 2, 3). Hence, we can assume that lim_{m→∞} λ_{i}(p_{m}) = λ̃_{i} by taking a subsequence if necessary. Because M^{3} is minimal and has constant Gauss-Kronecker curvature, we have, from 3K = f_{3} = ∑_{i=1}^{3} λ_{i}^{3},

∑_{i=1}^{3} h_{iik} = 0, for any k,  (3.3)

∑_{i=1}^{3} λ_{i}^{2}h_{iik} = 0, for any k.  (3.4)

From (3.2), we have

∑_{i=1}^{3} λ_{i}(p_{m})h_{iik}(p_{m}) = a_{k}(p_{m}), for any k  (3.5)

such that lim_{m→∞} a_{k}(p_{m}) = 0 for k = 1, 2, 3.

If these λ̃_{i}’s are distinct, then there exists a positive number N such that when m > N, these λ_{i}(p_{m})’s are distinct also. We solve the following linear system of equations with 3 unknown h_{11k}(p_{m}), h_{22k}(p_{m}), h_{33k}(p_{m}), for k = 1, 2, 3,

\{ \begin{matrix}∑_{i=1}^{3} h_{iik}(p_{m}) = 0, \\ ∑_{i=1}^{3} λ_{i}^{2}(p_{m})h_{iik}(p_{m}) = 0, \\ ∑_{i=1}^{3} λ_{i}(p_{m})h_{iik}(p_{m}) = a_{k}(p_{m}). \end{matrix}  (3.6)

Thus, we obtain

h_{11k}(p_{m}) = −\frac{\{λ_{2}(p_{m}) + λ_{3}(p_{m})\}a_{k}(p_{m})}{(λ_{3}(p_{m}) − λ_{1}(p_{m}))(λ_{2}(p_{m}) − λ_{1}(p_{m}))},

h_{22k}(p_{m}) = −\frac{\{λ_{1}(p_{m}) + λ_{3}(p_{m})\}a_{k}(p_{m})}{(λ_{1}(p_{m}) − λ_{2}(p_{m}))(λ_{3}(p_{m}) − λ_{2}(p_{m}))},

h_{33k}(p_{m}) = −\frac{\{λ_{1}(p_{m}) + λ_{2}(p_{m})\}a_{k}(p_{m})}{(λ_{1}(p_{m}) − λ_{3}(p_{m}))(λ_{2}(p_{m}) − λ_{3}(p_{m}))}.

Hence, we infer lim_{m→∞} h_{iik}(p_{m}) = 0, for i, k = 1, 2, 3. Therefore, we have

lim_{m→∞} ∑_{i,j,k=1}^{3} λ_{i}(p_{m})h_{ijk}^{2}(p_{m})  (3.7)

= \frac{1}{3} lim_{m→∞} ∑_{i,j,k=1}^{3}(λ_{i}(p_{m}) + λ_{j}(p_{m}) + λ_{k}(p_{m}))h_{ijk}^{2}(p_{m})

= lim_{m→∞}{ ∑_{i≠j=1}^{3}(2λ_{i}(p_{m}) + λ_{j}(p_{m}))h_{iij}^{2}(p_{m})

+ ∑_{i=1}^{3} λ_{i}(p_{m})h_{iii}^{2}(p_{m})} = 0.

Hence, from (3.1), we conclude that (sup S − 3c)f_{3} = 0. This is impossible from c ≤ 0.

Next, we consider the case that two of these λ̃_{i}’s are equal. We can assume that λ̃_{1} = λ̃_{2} = λ̃. Thus, λ̃_{3} = −2λ̃. And there exists a large number N such that when m > N, λ_{3}(p_{m}) ≠ λ_{1}(p_{m}), λ_{3}(p_{m}) ≠ λ_{2}(p_{m}). With a similar proof as in the first case we have

h_{33k}(p_{m}) = −\frac{\{λ_{1}(p_{m}) + λ_{2}(p_{m})\}a_{k}(p_{m})}{(λ_{1}(p_{m}) − λ_{3}(p_{m}))(λ_{2}(p_{m}) − λ_{3}(p_{m}))}.

Hence, we infer lim_{m→∞} h_{33k}(p_{m}) = 0. From (3.1), we have

\frac{1}{2}(sup S − 3c)f_{3} = lim_{m→∞} ∑_{i,j,k=1}^{3} λ_{i}(p_{m})h_{ijk}^{2}(p_{m})

= \frac{1}{3} lim_{m→∞} ∑_{i,j,k=1}^{3}(λ_{i}(p_{m}) + λ_{j}(p_{m}) + λ_{k}(p_{m}))h_{ijk}^{2}(p_{m})

= lim_{m→∞}{ ∑_{i≠j=1}^{3}(2λ_{i}(p_{m}) + λ_{j}(p_{m}))h_{iij}^{2}(p_{m}) + ∑_{i=1}^{3} λ_{i}(p_{m})h_{iii}^{2}(p_{m})}

= lim_{m→∞}{(2λ_{1}(p_{m}) + λ_{2}(p_{m}))h_{112}^{2}(p_{m}) + (2λ_{2}(p_{m}) + λ_{1}(p_{m}))h_{221}^{2}(p_{m})

+ λ_{1}(p_{m})h_{111}^{2}(p_{m}) + λ_{2}(p_{m})h_{222}^{2}(p_{m})}.

Since lim_{m→∞} λ_{i}(p_{m}) = λ̃ ≠ 0 (for i = 1, 2), we have that {h_{iik}^{2}(p_{m})}’s are bounded. By putting lim_{m→∞} h_{iik}(p_{m}) = h̃_{iik} by taking a subsequence of {p_{m}} if necessary, we infer

\frac{1}{2}(sup S − 3c)f_{3} = 4λ̃(h̃_{111}^{2} + h̃_{222}^{2})

because of lim_{m→∞}{h_{11k}(p_{m})+h_{22k}(p_{m})} = 0 for k = 1, 2, 3. Since f_{3} = 3K = −6λ̃^{3}, we have f_{3}λ̃ < 0. Hence, we obtain

0 > \frac{1}{2}(sup S − 3c)f_{3}λ̃ = 4λ̃^{2}(h̃_{111}^{2} + h̃_{222}^{2}) ≥ 0.

This is impossible. Hence, we obtain that K = 0. Thus, we finished the proof of Theorem 3.1.

Next, we shall give examples of complete minimal hypersurfaces in H^{4}(−1) and E^{4} with zero Gauss-Kronecker curvature, which are not totally geodesic.

We consider Minkowski space R_{1}^{5} as the real vector space R^{5} endowed with the Lorentzian metric ⟨, ⟩ given by

⟨x, y⟩ = ∑_{i=1}^{4} x_{i}y_{i} − x_{5}y_{5}  (3.8)

for x, yϵR^{5}. Then, H^{4}(−1) can be defined as the following hyperquadric of R_{1}^{5}

H^{4}(−1) = {xϵR_{1}^{5}; |x|^{2} = −1}.

In this way, the hyperbolic space H^{4}(−1) inherits from ⟨, ⟩ a metric which makes it an Riemannian manifold of constant sectional curvature −1 (refer to B. O′Neill [10]).

Moreover, if x ∈ H^{4}(−1), we can put T_{x}H^{4}(−1) = {v ∈ R_{1}^{5}; ⟨v, x⟩ = 0}.

Example 3.1. Let C_{1}, C_{2}, C_{3}, C_{4}, C_{5} be constant orthogonal vectors in R_{1}^{5} such that \frac{1}{2} = ⟨C_{1}, C_{1}⟩ = ⟨C_{2}, C_{2}⟩ = ⟨C_{3}, C_{3}⟩ = ⟨C_{4}, C_{4}⟩ and ⟨C_{5}, C_{5}⟩ = −1. We define the immersion

ϕ : R_{+1}^{3} ⟶ H^{4}(−1) ⊂ R_{1}^{5},

by

ϕ(u, v, z) = \frac{1}{\sqrt{z^{2} − 1}}\{cos(\sqrt{2}u)C_{1} + sin(\sqrt{2}u)C_{2} + cos(\sqrt{2}v)C_{3} + sin(\sqrt{2}v)C_{4} + zC_{5}\},

where R_{+1}^{3} = {(u, v, z) ∈ R^{3}; z > 1}. Thus, we can prove that ϕ is a minimal immersion from R_{+1}^{3} into the hyperbolic space H^{4}(−1) with zero Gauss-Kronecker curvature. The principal curvatures of the immersion are given by λ_{1}(u, v, z) = −\sqrt{z^{2} − 1}, λ_{2}(u, v, z) = \sqrt{z^{2} − 1} and λ_{3}(u, v, z) = 0. In fact, since ϕ(u, v, z) = \frac{1}{\sqrt{z^{2}−1}}\{cos(\sqrt{2}u)C_{1} + sin(\sqrt{2}u)C_{2} + cos(\sqrt{2}v)C_{3} + sin(\sqrt{2}v)C_{4} + zC_{5}\}, we define e_{1}, e_{2}, e_{3}, e_{4}, e_{5} as following, for (u, v, z) ∈ R_{+}^{3},

e_{1} =\sqrt{2}\{− sin(\sqrt{2}u)C_{1} + cos(\sqrt{2}u)C_{2}\},
e_{2} =\sqrt{2}\{− sin(\sqrt{2}v)C_{3} + cos(\sqrt{2}v)C_{4}\},
e_{3} =\frac{−z}{\sqrt{z^{2} − 1}}\{cos(\sqrt{2}u)C_{1} + sin(\sqrt{2}u)C_{2}
+ cos(\sqrt{2}v)C_{3} + sin(\sqrt{2}v)C_{4}\} − \frac{1}{\sqrt{z^{2} − 1}}C_{5},
e_{4} = − \{cos(\sqrt{2}u)C_{1} + sin(\sqrt{2}u)C_{2}\}
+\{cos(\sqrt{2}v)C_{3} + sin(\sqrt{2}v)C_{4}\},
e_{5} =\frac{1}{\sqrt{z^{2} − 1}}\{cos(\sqrt{2}u)C_{1} + sin(\sqrt{2}u)C_{2}
+ cos(\sqrt{2}v)C_{3} + sin(\sqrt{2}v)C_{4} + zC_{5}\}.

Then e_{1}, e_{2}, e_{3}, e_{4}, e_{5} is an orthonormal basis of R_{1}^{5} at point ϕ(u, v, z) such that e_{1}, e_{2}, e_{3}, e_{4} is an orthonormal basis of T_{ϕ(u,v,z)}H^{4}(−1) and e_{1}, e_{2}, e_{3} are tangent to the hypersurface. By a direct calculation, we infer

dϕ = \frac{du}{\sqrt{z^{2} − 1}}e_{1} + \frac{dv}{\sqrt{z^{2} − 1}}e_{2} + \frac{dz}{z^{2} − 1}e_{3}.

Hence, ω_{1} = \frac{du}{\sqrt{z^{2}−1}}, ω_{2} = \frac{dv}{\sqrt{z^{2}−1}} and ω_{3} = \frac{dz}{z^{2}−1} become a dual coframe of e_{1}, e_{2}, e_{3}. The following can also be proved by a simple computation:

de_{i} = ∑_{j=1}^{5} ω_{ij}ϵ_{j}e_{j}, for i = 1, 2, 3, 4, 5,

where ϵ_{1} = ϵ_{2} = ϵ_{3} = ϵ_{4} = 1, ϵ_{5} = −1 and

(ω_{ij}) = ( \begin{matrix}0 & 0 & \frac{zdu}{\sqrt{z^{2}−1}} & du & −\frac{du}{\sqrt{z^{2}−1}} \\ 0 & 0 & \frac{zdv}{\sqrt{z^{2}−1}} & −dv & −\frac{dv}{\sqrt{z^{2}−1}} \\ −\frac{zdu}{\sqrt{z^{2}−1}} & −\frac{zdv}{\sqrt{z^{2}−1}} & 0 & 0 & −\frac{dz}{\sqrt{z^{2}−1}} \\ −du & dv & 0 & 0 & 0 \\ \frac{du}{\sqrt{z^{2}−1}} & \frac{dv}{\sqrt{z^{2}−1}} & \frac{dz}{\sqrt{z^{2}−1}} & 0 & 0 \end{matrix} ) .

Hence, we obtain that components h_{ij} of the second fundamental form of the hypersurface are given by

(h_{ij}) = ( \begin{matrix}−\sqrt{z^{2} − 1} & 0 & 0 \\ 0 & \sqrt{z^{2} − 1} & 0 \\ 0 & 0 & 0 \end{matrix} ) .

Thus, we infer that the principal curvatures of this hypersurface are given by λ_{1} = −\sqrt{z^{2} − 1}, λ_{2} = \sqrt{z^{2} − 1} and λ_{3} = 0. Therefore, the hypersurface is minimal and the Gauss-Kronecker curvature is zero identically. This is not difficult to prove that the above hypersurface is complete.

By using similar method, we can prove that the following is a complete minimal hypersurface in E^{4} with zero Gauss-Kronecker curvature.

Example 3.2. Let C_{1}, C_{2}, C_{3}, C_{4} be constant orthogonal vectors in E^{4} such that \frac{1}{2} = ⟨C_{1}, C_{1}⟩ = ⟨C_{2}, C_{2}⟩ = ⟨C_{3}, C_{3}⟩ = ⟨C_{4}, C_{4}⟩. We define the immersion

ψ : R_{+0}^{3} ⟶ E^{4},

by

ψ(u, v, z) = \frac{1}{z}\{cos(\sqrt{2}u)C_{1} + sin(\sqrt{2}u)C_{2} + cos(\sqrt{2}v)C_{3} + sin(\sqrt{2}v)C_{4}\},

where R_{+0}^{3} = {(u, v, z) ∈ R^{3}; z > 0}. Thus, we can prove that ψ is a minimal immersion from R_{+0}^{3} into the Euclidean space E^{4} with zero Gauss-Kronecker curvature. The principal curvatures of the immersion are given by λ_{1}(u, v, z) = −z, λ_{2}(u, v, z) = z and λ_{3}(u, v, z) = 0.

From examples 3.1 and 3.2, we know that there exist complete minimal hypersurfaces with zero Gauss-Kronecker curvature which are not totally geodesic. Since their scalar curvatures are not bounded from below, we do not know whether the condition that scalar curvature is bounded from below yields that M^{3} is totally geodesic under the assumptions of theorem 3.1.

4. Complete hypersurfaces with non-zero constant mean curvature

In this section, we shall study complete hypersurfaces in M^{4}(c) (c ≤ 0) with non-zero constant mean curvature.

Theorem 4.1. If M^{3} is a 3-dimensional complete hypersurface with non-zero constant mean curvature and with zero quasi-Gauss-Kronecker curvature in M^{4}(c) (c ≤ 0), then M is totally umbilical if the scalar curvature is bounded from below.

Proof. Since K_{q} = μ_{1}μ_{2}μ_{3} and μ_{1} + μ_{2} + μ_{3} = 0, we have

3K_{q} = μ_{1}^{3} + μ_{2}^{3} + μ_{3}^{3} = B_{3}.  (4.1)

By a simple calculation, we have

B = ∑_{i=1}^{3} μ_{i}^{2} = S − 3H^{2}, B_{3} = 6H^{3} − 3HS + f_{3},  (4.2)

where f_{k} = ∑_{i=1}^{3} λ_{i}^{k}. From (2.2), (2.10) and (2.11), we can obtain

\frac{1}{2}ΔS = ∑_{i,j,k=1}^{3} h_{ijk}^{2} − S(S − 3c) − 9cH^{2} + 3Hf_{3},  (4.3)

\frac{1}{3}Δf_{3} = −(S − 3c)f_{3} + 3Hf_{4} − 3cHS + 2 ∑_{i,j,k=1}^{3} λ_{i}h_{ijk}^{2}.  (4.4)

From μ_{i} = λ_{i} − H and f_{4} = \frac{S^{2}}{2} − 9H^{2}S + \frac{27}{2}H^{4} + 4Hf_{3}, we have

\frac{1}{3}Δf_{3} = −(S − 3c − 12H^{2})f_{3}  (4.5)

+ 3H(−9H^{2}S + \frac{27}{2}H^{4} + \frac{S^{2}}{2} − cS)

+ 2H ∑_{i,j,k=1}^{3} h_{ijk}^{2} + 2 ∑_{i,j,k=1}^{3} μ_{i}h_{ijk}^{2}.

Since B_{3} = 3K_{q} = 0 and H is constant, we have, from (4.2)

Δf_{3} = 3HΔS.  (4.6)

Hence, from (4.5) and (4.6),

HΔS = −(S − 3c − 12H^{2})f_{3}

+ 3H(−9H^{2}S + \frac{27}{2}H^{4} + \frac{S^{2}}{2} − cS)

+ 2H ∑_{i,j,k=1}^{3} h_{ijk}^{2} + 2 ∑_{i,j,k=1}^{3} μ_{i}h_{ijk}^{2}.

Hence, we have, from (4.3),

− ∑_{i,j,k=1}^{3} μ_{i}h_{ijk}^{2} = \frac{1}{4}H(S − 3H^{2})^{2}.  (4.7)

Since the scalar curvature is bounded from below, we know that S is bounded. Applying the Generalized Maximum Principle due to Omori and Yau in the second section to S, we see that there exists a sequence {p_{m}} of points in M^{3} such that

lim_{m→∞} S(p_{m}) = sup S, lim_{m→∞} ∥grad S∥(p_{m}) = 0.  (4.8)

Since S is bounded, we know that {λ_{i}(p_{m})} is a bounded sequence for any i, (i = 1, 2, 3). Using a subsequence, if necessary, we can assume that lim_{m→∞} λ_{i}(p_{m}) = λ̃_{i}. Hence lim_{m→∞} μ_{i}(p_{m}) = μ̃_{i}. Since B_{3} = 3μ̃_{1}μ̃_{2}μ̃_{3} = 0, we can assume μ̃_{2} = 0. If μ̃_{1} = μ̃_{3} = 0, then sup S − 3H^{2} = sup B = 0. Hence, S ≥ 3H^{2} = sup S, that is, S = 3H^{2} is constant and M^{3} is totally umbilical.

Next, we shall prove that μ̃_{1} = −μ̃_{3} ≠ 0 can not occur. In fact, if μ̃_{1} = −μ̃_{3} ≠ 0 holds, then μ̃_{1}, μ̃_{2}, μ̃_{3} are distinct. Because M^{3} has constant mean curvature and constant quasi-Gauss-Kronecker curvature, we have

∑_{i=1}^{3} h_{iik} = 0, for any k,  (4.9)
∇_{k}f_{3} = 3H∇_{k}S, for any k.
∑_{i=1}^{3} λ_{i}^{2}h_{iik} = 3H ∑_{i=1}^{3} λ_{i}h_{iik}, for any k.  (4.10)

Hence, we obtain

\{ \begin{matrix}lim_{m→∞} ∑_{i=1}^{3} h_{iik}(p_{m}) = 0, & for any k, \\ lim_{m→∞} ∑_{i=1}^{3} λ_{i}^{2}(p_{m})h_{iik}(p_{m}) = 0, & for any k. \\ lim_{m→∞} ∑_{i=1}^{3} λ_{i}(p_{m})h_{iik}(p_{m}) = 0, & for any k. \end{matrix}  (4.11)

By making use of the similar assertion, we can obtain lim_{m→∞} h_{iik}(p_{m}) = 0, for i, k = 1, 2, 3. Hence

lim_{m→∞} ∑_{i,j,k=1}^{3} μ_{i}h_{ijk}^{2}(p_{m}) = 0.

Thus, sup S = 3H^{2} from (4.7), that is, μ̃_{1} = μ̃_{3} = 0. This is impossible. We completed the proof of Theorem 4.1.

It is well known that the totally umbilical spheres, the totally geodesic hyperplane E^{3} and the cylinder E^{k}×S^{3−k} for, k = 1, 2, are complete hypersurfaces with constant mean curvature, constant quasi-Gauss-Kronecker curvature and non-negative scalar curvature in E^{4}. It is natural to ask whether these hypersurfaces are the only complete hypersurfaces with constant mean curvature in E^{4}, which have constant quasi-Gauss-Kronecker curvature and non-negative scalar curvature. Next we shall answer this problem affirmatively. First, we shall prove a more general result.

Theorem 4.2. If M^{3} is a 3-dimensional complete hypersurface with constant mean curvature and with constant quasi-Gauss-Kronecker curvature in M^{4}(c) (c ≤ 0),

then M^{3} is isoparametric if the scalar curvature r satisfies r ≥ \frac{2}{3}c.

In particular, we have

Corollary 4.1. If M^{3} is a 3-dimensional complete hypersurface with constant mean curvature and with constant quasi-Gauss-Kronecker curvature in E^{4}, then M^{3} is isometric to a sphere, the totally geodesic hyperplane E^{3} or a cylinder E^{k} × S^{3−k} for, k = 1, 2, if the scalar curvature is nonnegative.

In order to prove the Theorem 4.2, we shall use the following

Lemma 4.1 (cf. [3]). Let μ_{1}, μ_{2}, μ_{3} be real numbers with μ_{1} + μ_{2} + μ_{3} = 0. Then,

−\frac{B^{\frac{3}{2}}}{\sqrt{6}} ≦ B_{3} ≦ \frac{B^{\frac{3}{2}}}{\sqrt{6}}

and the equality holds if and only if at least two of μ_{i} are equal, where B = ∑_{i=1}^{3} μ_{i}^{2} and B_{3} = ∑_{i=1}^{3} μ_{i}^{3}.

Lemma 4.2. Let μ_{1}, μ_{2}, μ_{3} be real numbers with μ_{1} +μ_{2} +μ_{3} = 0 and μ_{1}μ_{2}μ_{3} = K_{q}. Then,

∑_{i=1}^{3} μ_{i}^{2} ≥ \{54K_{q}^{2}\}^{\frac{1}{3}} .

Proof. From elementary algebra, we know that

2Π_{i<j}(μ_{i} − μ_{j})^{2} = B^{3} − 54K_{q}^{2}.

Thus, we know that Lemma 4.2 is true.

Proof of Theorem 4.2. If the mean curvature H is zero, then from Gauss equation (2.3), we have S = 0 and c = 0, that is, M^{3} is a totally geodesic hyperplane E^{3} in E^{4}.

Next, we assume H > 0. If the quasi-Gauss-Kronecker curvature K_{q} is zero, from Theorem 4.1, we know that M^{3} is totally umbilical. Hence, we assume K_{q} ≠ 0. Since B_{3} = 3K_{q} is constant, by making use of the similar proof of Theorem 4.1, we have

− ∑_{i,j,k=1}^{3} μ_{i}h_{ijk}^{2} = −\frac{1}{2}B_{3}(S − 3c − 6H^{2}) + \frac{1}{4}H(S − 3H^{2})^{2}.  (4.13)

Since the scalar curvature is bounded from below, we know that S is bounded. Applying the Generalized Maximum Principle due to Omori and Yau in the section 2 to S, we see that there exists a sequence {p_{m}} of points in M^{3} such that

lim_{m→∞} S(p_{m}) = sup S,  (4.14)
lim_{m→∞} ∥grad S∥(p_{m}) = 0,  (4.15)
lim_{m→∞} sup ΔS(p_{m}) ≤ 0.  (4.16)

Since S is bounded, we know that {λ_{i}(p_{m})} is a bounded sequence for any i, (i = 1, 2, 3). Hence, we can assume, without loss of generality, that lim_{m→∞} λ_{i}(p_{m}) = λ̃_{i}. Hence lim_{m→∞} μ_{i}(p_{m}) = μ̃_{i}.

(1). If μ̃_{1}, μ̃_{2}, μ̃_{3} are distinct, by making use of the similar assertion as in the proof of Theorem 4.1, we can obtain

lim_{m→∞} ∑_{i,j,k=1}^{3} μ_{i}h_{ijk}^{2}(p_{m}) = 0.  (4.17)

Hence, we have, from (4.13),

0 = −B_{3}(sup S − 3c − 6H^{2}) + \frac{1}{2}H(sup S − 3H^{2})^{2}.  (4.18)

Where B_{3} = 3μ̃_{1}, μ̃_{2}, μ̃_{3} = 3K_{q} is constant. Since H > 0 and K_{q} is a non-zero constant, we consider the following two subcases.

(a). HB_{3} > 0. (b). HB_{3} < 0.

If HB_{3} > 0 holds, then we have B_{3} > 0. From Lemma 4.1, we have

0 < B_{3} < \frac{(sup S − 3H^{2})^{\frac{3}{2}}}{\sqrt{6}} .

Thus, we have sup S − 3c − 6H^{2} > 0 from (4.18). Hence,

0 > −\frac{(sup S − 3c − 6H^{2})}{\sqrt{6}} + \frac{H}{2} (sup S − 3H^{2})^{\frac{1}{2}} .

Therefore, we have

\frac{(sup S − 3c − 6H^{2})^{2}}{6} > \frac{H^{2}}{4} (sup S − 3H^{2}),

that is,

sup S^{2} − (6c + \frac{27H^{2}}{2} ) sup S + 9c^{2} + \frac{81}{2}H^{4} + 36cH^{2} > 0.

From r ≥ \frac{2}{3}c and Gauss equation (2.3), we have 9H^{2} + 8c ≥ 0. Hence, we infer

sup S > 3c + \frac{27H^{2}}{4} + \frac{3}{4}\sqrt{9H^{4} + 8cH^{2}},  (4.19)

or

sup S < 3c + \frac{27H^{2}}{4} − \frac{3}{4}\sqrt{9H^{4} + 8cH^{2}}.  (4.20)

From sup S − 3c − 6H^{2} > 0, we have

sup S > 3c + \frac{27H^{2}}{4} + \frac{3}{4}\sqrt{9H^{4} + 8cH^{2}}.

From Gauss equation (2.3), we have

S ≤ 6c − \frac{2}{3}c + 9H^{2}.

Hence,

6c − \frac{2}{3}c + 9H^{2} > 3c + \frac{27H^{2}}{4} + \frac{3}{4}\sqrt{9H^{4} + 8cH^{2}},

that is

3c − \frac{2}{3}c + \frac{9}{4}H^{2} > \frac{3}{4}\sqrt{9H^{4} + 8cH^{2}} ≥ 0.  (4.21)

Hence,

\frac{49}{9}c^{2} + 6cH^{2} > 0.  (4.22)

From (4.21) and (4.22), we know that this is impossible because of c ≤ 0. If B_{3}H < 0 holds, then we have B_{3} < 0. From Lemma 4.1, we have

0 < −B_{3} < \frac{(sup S − 3H^{2})^{\frac{3}{2}}}{\sqrt{6}} .

Thus, we have sup S − 3c − 6H^{2} < 0 from (4.18). By taking the same assertion as above, we have

sup S < 3c + \frac{27H^{2}}{4} − \frac{3}{4}\sqrt{9H^{4} + 8cH^{2}}.

From (4.3) and (4.16), we have

0 ≥ 3c(sup S − 3H^{2}) − (sup S − 3H^{2})^{2} + 3H^{2}(sup S − 3H^{2}) + 3HB_{3}.

From Lemma 4.1, we have

sup S − 3H^{2} = 0

or

3c + 3H^{2} − (sup S − 3H^{2}) − \sqrt{\frac{3}{2}}H(sup S − 3H^{2})^{\frac{1}{2}} ≤ 0.

Since

sup S < 3c + \frac{27H^{2}}{4} − \frac{3}{4}\sqrt{9H^{4} + 8cH^{2}},

we infer sup S − 3H^{2} = 0, that is, M^{3} is totally umbilical. This is also impossible because μ̃_{1}, μ̃_{2}, μ̃_{3} are distinct.

(2). If μ̃_{1} = μ̃_{2} = μ̃_{3} holds, then sup S = 3H^{2} and M^{3} is totally umbilical. (3). If two out of μ̃_{1}, μ̃_{2}, μ̃_{3} are equal, say μ̃_{1} = μ̃_{2} = −\frac{1}{2} μ̃_{3}, then, we have

sup S − 3H^{2} = (54K_{q}^{2})^{\frac{1}{3}} .

On the other hand, under conditions ∑_{i=1}^{3} μ_{i} = 0 and μ_{1}μ_{2}μ_{3} = K_{q}, we minimize the function F = μ_{1}^{2} + μ_{2}^{2} + μ_{3}^{2}. We infer F ≥ (54K_{q}^{2})^{\frac{1}{3}} from Lemma 4.2. An easy computation shows that F = S − 3H^{2}. Hence, we have S − 3H^{2} is constant. In this case, we have that M^{3} is an isoparametric hypersurface with S = 3c + \frac{27H^{2}}{4} ± \frac{3}{4} \sqrt{9H^{4} + 8cH^{2}}. This completed the proof of Theorem 4.2.

Acknowledgements. I would like to express my gratitude to the referee for his/her valuable suggestions and for telling me a paper of Aldir Brasil Jr [ Aldir Brasil Jr, Hypersurfaces with constant mean curvature and constant quasi-Gauss-Kronecker curvature in S^{4}, An. Acad. Brasil. Cienc. vol 71 (1999), 599-605]. Since there is no this journal in our library, I shall try to get it.

This research was partly supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Sicence and Techonlogy, Japan.

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