Complete submanifolds in Euclidean spaces with parallel mean curvature vector

Qing-Ming Cheng* and Kazuhiro Nonaka

Abstract

In this paper, we prove that n-dimensional complete and connected submanifolds with parallel mean curvature vector H in the (n+p)-dimensional Euclidean space E^{n+p} are the totally geodesic Euclidean space E^{n}, the totally umbilical sphere S^{n}(c) or the generalized cylinder S^{n−1}(c) × E^{1} if the second fundamental form h satisfies ⟨h⟩^{2} ≤ n^{2}|H|^{2}/(n − 1).

1 Introduction

The purpose of this paper is to study n-dimensional complete and connected submanifolds in an (n + p)-dimensional Euclidean space E^{n+p}. In 1900, Liebmann proved that compact surfaces with constant Gauss curvature in E^{3} are the standard spheres. In 1953, Hopf proved that compact surfaces with constant mean curvature and with genus zero in E^{3} are the standard spheres. Hopf’s result was extended to complete surfaces in E^{3} by Klotz-Osserman as following.

Klotz-Osserman’s Theorem (see [5]). Let M be a complete and connected surface with constant mean curvature in E^{3}. If the Gauss curvature of M is nonnegative, then M is the plane E^{2}, the sphere S^{2}(c) or the cylinder S^{1}(c) × E^{1}.

From the equation of Gauss, we know that the Gaussian curvature of a surface M in E^{3} is nonnegative if and only if ⟨h⟩^{2} ≤ n^{2} |H|^{2} /(n − 1), n = 2, where ⟨h⟩ is the length of the second fundamental form h of M and |H| is the mean curvature of M. It is natural to extend the result due to Klotz-Osserman to higher dimensions and higher codimensions. In this paper, we shall prove the following result.

Main Theorem 1. Let M be an n-dimensional complete and connected submanifold with parallel mean curvature vector H in E^{n+p}, n ≥ 3. If the second fundamental form h of M satisfies

2000 Mathematics Subject Classification 53C42.

Key words and phrases: submanifolds, parallel mean curvature vector, second fundamental form, maximum principle.

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

⟨h⟩^{2} ≤ \frac{n^{2} |H|^{2}}{n − 1} ,  (1.1)

then M is the totaly geodesic Euclidean space E^{n}, the totally umbilical sphere S^{n}(c) or the generalized cylinder S^{n−1}(c) × E^{1} in E^{n+1}.

Remark 1. In [9], Shen intended to prove the above theorem by making use of the result of Motomiya [7]. But, since the result of Motomiya is wrong (see the section 2), the proof of Shen about the above theorem is not valid. In the section 4, we shall give a proof of the Main Theorem 1.

2 The Maximum Principles

In this section, we shall mention the maximum principles which play on an important role in the study of differential geometry on Riemannian manifolds. First of all, we state the well known theorem which is called Hopf’s maximum principle as the following.

Hopf’s maximum principle :

Let M be a compact Riemannian manifold. If a C^{2}-function f on M satisfies Δf ≥ 0 or Δf ≤ 0, then f is a constant function, where Δ denotes the Laplacian on M.

As a generalization of Hopf’s maximum principle, Omori [8] and Yau [10] proved a very important theorem, which is called the generalized maximum principle.

The generalized maximum principle :

Let M be a complete and connected Riemannian manifold with Ricci curvature bounded from below. If a C^{2}-function f on M is bounded from above, then for all ε > 0, there exists a point x ∈ M such that

sup f − ε < f (x) ,  (2.1)
||grad f (x) || < ε ,  (2.2)
Δf (x) < ε .  (2.3)

In [7], Motomiya intended to improve the generalized maximum principle due to Omori and Yau as the following.

Motomiya’s wrong result :

Let M be a complete and connected Riemannian manifold with Ricci curvature bounded from below. If a C^{2}-function f is bounded from above and has no maximum on M, then for all ε > 0, there exists a point x ∈ M such that

sup f − ε < f (x) < sup f − ε/2 ,  (2.4)
||grad f (x) || < ε ,  (2.5)
Δf (x) < ε .  (2.6)

But this result is wrong. In fact, Cheng and Wu gave the following counter example in [2].

Counter example of Motomiya’s result: Let M = E^{2}, f (x, y) = − exp (cx) (c ≥ 2). Obviously, M is a complete and connected Riemannian manifold with Ricci curvature zero and f is a smooth function bounded from above by zero and f has no maximum on M. So, if for ε > 0, there exists a point q = (x, y) ∈ M such that (2.4), (2.5) and (2.6) hold , then from (2.5) and (2.6), we would have

||grad f(q) || = c exp(cx) < ε ,
Δf(q) = −c^{2} exp(cx) < ε.

Hence,

−\frac{ε}{c} < − exp(cx) = f(q) .  (2.7)

On the other hand, since sup f = 0, from (2.4), we have

−ε < f(q) < −\frac{ε}{2}.  (2.8)

Taking c ≥ 2, we find that (2.7) and (2.8) are contradictory.

The Motomiya’s wrong result was used by Shen in [9] in order to prove the Main Theorem 1 in the section 1. Hence, Shen’s proof is not valid. In the section 4, we will give a proof of the Main Theorem 1 by making use of the generalized maximum principle due to Omori and Yau.

3 Definitions and Fundamental Equations

Let M be an n-dimensional submanifold in an (n + p)-dimensional Riemannian manifold N of constant curvature k. We denote by g̃, g the Riemannian metric on N, M, respectively and by ˜∇, ∇ the Riemannian connection on N, M, with respect to g̃, g, respectively. Then the formula of Gauss (3.1) and the formula of Weingarten (3.2) hold.

˜∇_{X}Y = ∇_{X}Y + h (X, Y) ,  (3.1)
˜∇_{X}ξ = −A_{ξ}(X) + ∇_{X}^{⊥}ξ,  (3.2)

where X, Y are any tangent vector fields on M and ξ is any normal vector field on M. We call h the second fundamental form of M and A_{ξ} the second fundamental tensor of M with respect to ξ and ∇^{⊥} the normal connection of M. A normal vector field ξ of M is said to be parallel in the normal bundle, or simply, parallel if we have ∇^{⊥} ξ = 0 identically. From the formula of Gauss (3.1) and the formula of Weingarten (3.2), we have immediately,

g̃(h (X, Y) , ξ) = g (A_{ξ} (X) , Y) .  (3.3)

Let E_{1}, E_{2}, ⋯, E_{n} be orthonormal tangent vectors at a point x ∈ M and ξ_{1}, ξ_{2}, ⋯, ξ_{p} be orthonormal normal vectors at x ∈ M, then

H = \frac{1}{n} ∑_{α=1}^{p}(trace A_{α}) ξ_{α}  (3.4)

is a normal vector at x, which is independent of the choice of the othonormal basis E_{1}, ⋯, E_{n}, ξ_{1}, ⋯, ξ_{p}, where A_{α} = A_{ξ}_{α}, trace A_{α} = ∑_{i=1}^{n} g (A_{α} (E_{i}) , E_{i}). The vector field H is called mean curvature vector field of M. Obviously, if the mean curvature vector field H of M is parallel in the normal bundle, then the mean curvature |H| of M is constant.

We denote by X_{i} = ∂/∂x^{i} (i = 1, 2, ⋯, n) the natural frame of submanifold M and by K̃, K the curvature tensor of N, M, respectively. Let ξ_{1}, ξ_{2}, ⋯, ξ_{p} be orthonormal normal vector fields of M and h^{α} be the corresponding second fundamental form, that is,

h (X, Y) = h^{α} (X, Y) ξ_{α}.

Now, since the ambient space N is of constat curvature k, then we have

K̃(X̃, Ỹ)Z̃ = k{g̃(Z̃, Ỹ)X̃ − g̃(Z̃, X̃)Ỹ},  (3.5)

for any tangent vector fields X̃, Ỹ, Z̃ on N. Thus, from the formula of Gauss (3.1), the formula of Weingarten (3.2) and (3.3), we have the equation of Gauss as following.

\begin{aligned}K(X, Y, Z, W) & = k\{g(X, W)g(Y, Z) − g(X, Z)g(Y, W)\} \\ & + g̃(h (X, W) , h (Y, Z)) − g̃(h (X, Z) , h (Y, W)) , \end{aligned}  (3.6)

where K(X, Y, Z, W) = g (K (X, Y) Z, W) and X, Y, Z, W are any tangent vector fields on M. In local components,

K_{kjiℓ} = k (g_{kℓ}g_{ji} − g_{ki}g_{jℓ}) + ∑_{α=1}^{p}(h_{kℓ}^{α}h_{ji}^{α} − h_{ki}^{α}h_{jℓ}^{α}).  (3.7)

Moreover, from (3.5), we see that the normal components of K̃(X, Y)Z vanish, then we have ,

(∇_{X}h^{α})(Y, Z)ξ_{α} + h^{α}(Y, Z)∇_{X}^{⊥}ξ_{α} = (∇_{Y} h^{α})(X, Z)ξ_{α} + h^{α}(X, Z)∇_{Y}^{⊥}ξ_{α}.  (3.8)

We define the covariant derivative, denoted by \overline{∇}, to be

(\overline{∇}_{X}h)(Y, Z) = (∇_{X}h^{α})(Y, Z)ξ_{α} + h^{α}(Y, Z)∇_{X}^{⊥}ξ_{α}.  (3.9)

In general, for example, for a tensor fields, say T, of type (0.3) with values in the normal bundle T^{⊥}(M) of M, we define

(\overline{∇}_{W} T)(X, Y, Z) = (∇_{W} T^{α})(X, Y, Z)ξ_{α} + T^{α}(X, Y, Z)∇_{W}^{⊥} ξ_{α}.

Remark 2. We call the connection \overline{∇} on M the connection of van der Waerden Bortolotti.

Then We have the equation of Codazzi as following,

(\overline{∇}_{X}h)(Y, Z) = (\overline{∇}_{Y} h)(X, Z),  (3.10)

in local component,

\overline{∇}_{k}h_{ji}^{α} = \overline{∇}_{j}h_{ki}^{α}.  (3.11)

We denote by K^{N} the curvature tensor of the normal connection ∇^{⊥} on M, that is,

K^{N}(X, Y)ξ = ∇_{X}^{⊥}∇_{Y}^{⊥}ξ − ∇_{Y}^{⊥}∇_{X}^{⊥}ξ − ∇_{[X,Y]}^{⊥}ξ,

then, from the formula of Gauss (3.1), the formula of Weingarten (3.2) and (3.5), we have the equation of Ricci as following,

K^{N}(X, Y, ξ, η) = g ([A_{ξ}, A_{η}] (X), Y) ,  (3.12)

where

K^{N}(X, Y, ξ, η) = g̃(K^{N}(X, Y)ξ, η )
[A_{ξ}, A_{η}] = A_{ξ}A_{η} − A_{η}A_{ξ} .

If we set K_{jiβ}^{α}ξ_{α} = K^{N}(X_{j}, X_{i})ξ_{β} , then we shall give the local expression of the equation of Ricci as following,

K_{jiβ}^{α} = h_{jt}^{α}h_{i}^{t}_{β} − h_{it}^{α}h_{j}^{t}_{β} ,  (3.13)

where h_{j}^{k}_{α} = g^{ki}h_{ji}^{α}. In this paper, we shall use the Einstein convention, that is, repeated indices, with one upper index and one lower index, denote summation over its range.

We may define the second covariant derivative \overline{∇} \overline{∇}h of the second fundamental form h by

\begin{aligned}(\overline{∇}_{W} \overline{∇}_{X}h)(Y, Z) & = ∇_{W}^{⊥} (\overline{∇}_{X}h(Y, Z)) − (\overline{∇}_{X}h)(∇_{W} Y, Z) \\ & − (\overline{∇}_{X}h)(Y, ∇_{W} Z) − (\overline{∇}_{∇_{W} X}h)(Y, Z) , \end{aligned}  (3.14)

for any tangent vector fields X, Y, Z, W on M. By a direct computation, we find

\begin{matrix}(\overline{∇}_{W} \overline{∇}_{X}h)(Y, Z) − (\overline{∇}_{X} \overline{∇}_{W} h)(Y, Z) \\ = h(K(X, W)Y, Z) + h(Y, K(X, W)Z) − K^{N}(X, W)h(Y, Z) , \end{matrix}  (3.15)

in local component,

\overline{∇}_{ℓ}\overline{∇}_{k}h_{ji}^{α} − \overline{∇}_{k}\overline{∇}_{ℓ}h_{ji}^{α} = K_{kℓj}^{t}h_{ti}^{α} + K_{kℓi}^{t}h_{tj}^{α} − K_{kℓβ}^{α}h_{ji}^{β} .  (3.16)

If the mean curvature |H| of M is nowhere zero, then we may choose orthonormal normal vector fields ξ_{1}, ξ_{2}, ⋯, ξ_{p} such that H = |H|ξ_{1}. Obviously, if we choose ξ_{1}, ξ_{2}, ⋯, ξ_{p} as above, then we have trace A_{1} = n|H|, trace A_{α} = 0 (α = 2, 3, ⋯, p). In the following, for any tangent vector fields X, Y on M, we set

U(X, Y) = {h^{1}(X, Y) − |H| g(X, Y)}ξ_{1} ,  (3.17)
T (X, Y) = ∑_{α=2}^{p} h^{α}(X, Y)ξ_{α} ,  (3.18)

in local component,

U_{ji} = h_{ji}^{1} − |H| g_{ji} ,  (3.19)
T_{ji}^{α} = h_{ji}^{α} (α = 2, ⋯, p) .  (3.20)

From (3.19) and (3.20), we have immediately the following,

|U|^{2} = trace A_{1}^{2} − n |H|^{2} ,  (3.21)
|T |^{2} = ∑_{α=2}^{p} trace A_{α}^{2} ,  (3.22)
⟨h⟩^{2} = |U|^{2} + |T |^{2} + n|H|^{2} .  (3.23)

From these formulas, we know that |U|^{2}, |T |^{2} is independent of the choice of the orthonormal basis E_{1}, ⋯, E_{n}. Then, functions |U|^{2}, |T |^{2} are defined globally on M if H is nowhere zero. Moreover, From (3.16) and (3.22), we have the following lemma.

Lemma 1. If the nonzero mean curvature vector H of submanifold M is parallel in the normal bundle, then we have

\begin{aligned}\frac{1}{2}Δ|T |^{2} & = ∑_{α=2}^{P}(K_{j}^{t}h_{it}^{α}h^{ji}_{α} − K_{kjit}h^{kt}_{α}h^{ji}_{α}) \\ & + ∑_{α,β=2}^{p} K_{jiβ}^{α}h_{t}^{j}_{β}h^{it}_{α} + |\overline{∇}T |^{2} , \end{aligned}  (3.24)

where K_{ji} = K_{tji}^{t} denotes the component of Ricci tensor of M, K_{j}^{t} = g^{ti}K_{ji}, h^{ji}_{α} = g^{jk}g^{iℓ}h_{kℓ}^{α}, Δ denotes the Laplacian on M.

Proof. Let ξ_{1}, ξ_{2}, ⋯, ξ_{p} be orthonormal normal vector fields on M such that H = |H|ξ_{1} . Then, from (3.22), we have

\frac{1}{2}Δ|T |^{2} = ∑_{α=2}^{p} g^{kℓ}(∇_{k}∇_{ℓ}h_{ji}^{α})h^{ji}_{α} + ∑_{α=2}^{p} g^{kℓ}(∇_{k}h_{ji}^{α})(∇_{ℓ}h^{ji}_{α}) ,  (3.25)

by virtue of trace A_{α}^{2} = h_{i}^{j}_{α}h_{j}^{i}_{α} .

If we set ∇_{X_{k}}^{⊥}ξ_{α} = ℓ_{kα}^{β}ξ_{β}, then we find immediately ℓ_{kα}^{β} = −ℓ_{kβ}^{α} by g̃(ξ_{α}, ξ_{β}) = δ_{αβ}. Since H = |H|ξ_{1} is parallel in the normal bundle, we have ℓ_{kα}^{1} = −ℓ_{k1}^{α} = 0. Hence, from the definition of \overline{∇}, we obtain

\begin{aligned}\frac{1}{2}Δ|T |^{2} & = ∑_{α=2}^{p} g^{kℓ}(\overline{∇}_{k}\overline{∇}_{ℓ}h_{ji}^{α})h^{ji}_{α} + ∑_{α=2}^{p} g^{kℓ}(\overline{∇}_{k}h_{ji}^{α})(\overline{∇}_{ℓ}h^{ji}_{α}) \\ & − ∑_{α,β=2}^{p} g^{kℓ}∇_{ℓ}(h_{ji}^{β}ℓ_{kβ}^{α})h^{ji}_{α} − ∑_{α,β=2}^{p} g^{kℓ}(∇_{ℓ}h^{ji}_{α})h_{ji}^{β}ℓ_{kβ}^{α} \\ & − ∑_{α,β,γ=2}^{p} g^{kℓ}h_{ji}^{β}h^{ji}_{α}ℓ_{kβ}^{γ}ℓ_{ℓγ}^{α} − ∑_{α,β,γ=2}^{p} g^{kℓ}h_{ji}^{β}h^{ji}_{γ}ℓ_{kβ}^{α}ℓ_{ℓγ}^{α} \\ & − ∑_{α,β=2}^{p} g^{kℓ}(∇_{k}h_{ji}^{β})ℓ_{ℓβ}^{α}h^{ji}_{α} − ∑_{α,β=2}^{p} g^{kℓ}(∇_{k}h_{ji}^{α})h^{ji}_{β}ℓ_{ℓβ}^{α} . \end{aligned}  (3.26)

Moreover, from ℓ_{kα}^{β} = −ℓ_{kβ}^{α} , we find

∑_{α,β=2}^{p} g^{kℓ}(∇_{k}h_{ji}^{β}ℓ_{kβ}^{α})h^{ji}_{α} + ∑_{α,β=2}^{p} g^{kℓ}(∇_{ℓ}h^{ji}_{α})h_{ji}^{β}ℓ_{kβ}^{α} = 0 ,

∑_{α,β,γ=2}^{p} g^{kℓ}h_{ji}^{β}h^{ji}_{α}ℓ_{kβ}^{γ}ℓ_{ℓγ}^{α} + ∑_{α,β,γ=2}^{p} g^{kℓ}h_{ji}^{β}h^{ji}_{γ}ℓ_{kβ}^{α}ℓ_{ℓγ}^{α} = 0 ,

∑_{α,β=2}^{p} g^{kℓ}(∇_{k}h_{ji}^{β})ℓ_{ℓβ}^{α}h^{ji}_{α} + ∑_{α,β=2}^{p} g^{kℓ}(∇_{k}h_{ji}^{α})h^{ji}_{β}ℓ_{ℓβ}^{α} = 0 .

Thus, we infer

\frac{1}{2}Δ|T |^{2} = ∑_{α=2}^{p} g^{kℓ}(\overline{∇}_{k}\overline{∇}_{ℓ}h_{ji}^{α})h^{ji}_{α} + ∑_{α=2}^{p} g^{kℓ}(\overline{∇}_{k}h_{ji}^{α})(\overline{∇}_{ℓ}h^{ji}_{α}) .  (3.27)

By using the equation of Codazzi (3.11) and (3.16), we know that formula (3.24) holds. This finished the proof of Lemma 1.

4 Proof of Main Theorem 1

Proof of Main Theorem 1 : Since the mean curvature vector H of M is parallel in the normal bundle, |H| is constant. We now consider the case |H| = 0 and |H| ≠ 0 separately.

Case(1) : |H| = 0 on M

From the assumption of the Main Theorem 1, we have

⟨h⟩^{2} ≤ \frac{n^{2}|H|^{2}}{n − 1} = 0 ,

that is, M is an n-dimensional totally geodesic submanifold E^{n} in E^{n+p}. Therefore, the Main Theorem 1 holds.

Case(2) : |H| ≠ 0 on M

Let ξ_{1}, ξ_{2}, ⋯, ξ_{p} be an orthonormal normal vector fields on M such that H = |H|ξ_{1}. Since the mean curvature vector H of M is parallel in the normal bundle, from the equation of Ricci (3.12), it is easy to prove that,

A_{1}A_{α} = A_{α}A_{1}

for each α. Then, substituting the equation of Gauss (3.7) and the equation of Ricci (3.13) into (3.24), we find

\frac{1}{2}Δ|T |^{2} = ∑_{α=2}^{p}(h_{k}^{k}_{1}h_{j}^{t}_{1}h_{it}^{α}h^{ji}_{α} − h_{k}^{t}_{1}h_{j}^{k}_{1}h_{it}^{α}h^{ji}_{α}
− h_{kt}^{1}h_{ji}^{1}h^{kt}_{α}h^{ji}_{α} + h_{ki}^{1}h_{jt}^{1}h^{kt}_{α}h^{ji}_{α})

+ ∑_{α,β=2}^{p}{−h_{k}^{t}_{β}h_{j}^{k}_{β}h_{it}^{α}h^{ji}_{α} − (h_{kt}^{β}h_{ji}^{β} − h_{ki}^{β}h_{jt}^{β})h^{kt}_{α}h^{ji}_{α}

+ (h_{jt}^{α}h_{i}^{t}_{β} − h_{it}^{α}h_{j}^{t}_{β})h_{k}^{j}_{β}h^{ik}_{α}\} + |\overline{∇}T |^{2}

= ∑_{α=2}^{p}{trace A_{1}trace A_{1}A_{α}^{2} − (trace A_{1}A_{α})^{2}}

− ∑_{α,β=2}^{p}{(h_{k}^{j}_{α}h_{i}^{k}_{β} − h_{k}^{j}_{β}h_{i}^{k}_{α})(h_{t}^{i}_{α}h_{j}^{t}_{β} − h_{t}^{i}_{β}h_{j}^{t}_{α})

+ (h_{j}^{i}_{α}h_{i}^{j}_{β}h_{k}^{t}_{α}h_{t}^{k}_{β})\} + |\overline{∇}T |^{2}

= ∑_{α=2}^{p}{trace A_{1}trace A_{1}A_{α}^{2} − (trace A_{1}A_{α})^{2}}

− ∑_{α,β=2}^{p}{trace [(A_{α}A_{β} − A_{β}A_{α})^{t}(A_{α}A_{β} − A_{β}A_{α})]

+ (trace A_{α}A_{β})^{2}\} + |\overline{∇}T |^{2} .

By setting N(A_{α}) = trace A_{α}^{t}A_{α} and S_{αβ} = trace A_{α}A_{β}, we have

\begin{aligned}\frac{1}{2}Δ|T |^{2} & = ∑_{α=2}^{p}\{trace A_{1}trace A_{1}A_{α}^{2} − (trace A_{1}A_{α})\} \\ & − ∑_{α,β=2}^{p}\{N(A_{α}A_{β} − A_{β}A_{α}) + S_{αβ}^{2}\} + |\overline{∇}T |^{2} . \end{aligned}  (4.1)

By virtue of

A_{1}A_{α} = A_{α}A_{1}

for each α, then, for a fixed α, A_{1} and A_{α} can be simultaneously diagonalized. We denote by ρ_{i}, ρ_{i}^{α} (i = 1, 2, ⋯, n) the principal curvatures of the second fundamental tensors A_{1}, A_{α}, respectively. Then we obtain

\begin{matrix}trace A_{1}trace A_{1}A_{α}^{2} − (trace A_{1}A_{α})^{2} \\ = ∑_{i,j=1}^{n} ρ_{i}ρ_{j}(ρ_{j}^{α})^{2} − ∑_{i,j=1}^{n} ρ_{i}ρ_{j}ρ_{i}^{α}ρ_{j}^{α} \\ = \frac{1}{2} ∑_{i,j=1}^{n} ρ_{i}ρ_{j}(ρ_{j}^{α})^{2} + \frac{1}{2} ∑_{i,j=1}^{n} ρ_{j}ρ_{i}(ρ_{i}^{α})^{2} − ∑_{i.j=1}^{n} ρ_{i}ρ_{j}ρ_{i}^{α}ρ_{j}^{α} \\ = \frac{1}{2} ∑_{i,j=1}^{n} ρ_{i}ρ_{j}(ρ_{i}^{α} − ρ_{j}^{α})^{2} . \end{matrix}  (4.2)

Condition (1.1) and (3.23) imply that

(∑_{i=1}^{n} ρ_{i})^{2} ≥ (n − 1) ∑_{i=1}^{n}(ρ_{i})^{2} + (n − 1)|T |^{2} .  (4.3)

The following lemma is an algebraic result.

Lemma 2 (Chen [1, p.55]). Let a_{1}, a_{2}, ⋯, a_{n}, b be n+1 (n > 1) real numbers satisfying the following inequality,

(∑_{i=1}^{n} a_{i})^{2} ≥ (n − 1) ∑_{i=1}^{n} a_{i}^{2} + b (resp >).

Then we have

2a_{i}a_{j} ≥ \frac{b}{n − 1} (resp >)

for any distinct i and j.

By applying the Lemma 2 to (4.3), we find

ρ_{i}ρ_{j} ≥ \frac{|T |^{2}}{2}  (4.4)

for any distinct i and j. From this inequality and (4.2), we obtain, for any α,

trace A_{1}trace A_{1}A_{α}^{2} − (trace A_{1}A_{α})^{2}

≥ \frac{1}{4}|T |^{2} ∑_{i,j=1}^{n}(ρ_{i}^{α} − ρ_{j}^{α})^{2}

= \frac{1}{2}|T |^{2}\{∑_{i,j=1}^{n}(ρ_{i}^{α})^{2} − ∑_{i,j=1}^{n} ρ_{i}^{α}ρ_{j}^{α}\}

= \frac{n}{2}|T |^{2} ∑_{i=1}^{n}(ρ_{i}^{α})^{2} − \frac{1}{2}|T |^{2} (∑_{i=1}^{n} ρ_{i}^{α})^{2}

= \frac{n}{2}|T |^{2}trace A_{α}^{2} − \frac{1}{2}|T |^{2}(trace A_{α})^{2}.

From trace A_{α} = 0 for α ≥ 2, we have

∑_{α=2}^{p}(trace A_{1}trace A_{1}A_{α}^{2} − (trace A_{1}A_{α})^{2}) ≥ \frac{n}{2}|T |^{4}  (4.5)

by (3.22).

On the other hand, from the following result, we have

∑_{α,β=2}^{p}\{N(A_{α}A_{β} − A_{β}A_{α}) + S_{αβ}^{2}\} ≤ \frac{3}{2}|T |^{4} .  (4.6)

Lemma 3 (Li, A. M. and Li, J. M. [6]). Let A_{1}, A_{2}, ⋯, A_{p} be symmetric ( n × n)-matrices. Then we have

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

and the equality holds if and only if one of the following conditions holds :

(i) A_{1} = A_{2} = ⋯ = A_{p} = 0,

(ii) only two of A_{1}, A_{2}, ⋯, A_{p} are different from zero. Moreover, assuming A_{1} ≠ 0, A_{2} ≠ 0, then S_{1} = S_{2} and there exists an orthogonal ( n × n)-matrix T such that

tT A_{1} T = \sqrt{\frac{S_{1}}{2}} (\begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 0 & −1 & 0 & ⋯ & 0 \\ 0 & 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & ⋯ & 0 \end{matrix}) , ^{t}T A_{2} T = \sqrt{\frac{S_{1}}{2}} (\begin{matrix}0 & 1 & 0 & ⋯ & 0 \\ 1 & 0 & 0 & ⋯ & 0 \\ 0 & 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & ⋯ & 0 \end{matrix}) ,

where S_{α} = N(A_{α}).

By virtue of (4.5), (4.6), we have

\frac{1}{2}Δ|T |^{2} ≥ \frac{n − 3}{2}|T |^{4} .  (4.7)

Condition (1.1) implies that |T | is bounded from above by n^{2}|H|^{2}/(n − 1) and by the Lemma 2, we can prove that the sectional curvatures of M are nonnegative. Hence, we can apply the generalized maximum principle due to Omori and Yau to function |T |^{2}. Then, there exists a sequence {x_{k}} ⊂ M such that

lim_{k→∞} |T |^{2}(x_{k}) = sup |T |^{2} ,  (4.8)
lim sup_{k→∞} Δ|T |^{2}(x_{k}) ≤ 0 .  (4.9)

From (4.7), (4.8) and (4.9), we see that

0 ≥ lim sup_{k→∞} Δ|T |^{2}(x_{k}) ≥ (n − 3)( lim_{k→∞} |T |^{2}(x_{k}))^{2} = (n − 3)(sup |T |^{2})^{2} ≥ 0 ,

that is,

(n − 3) sup |T |^{2} = 0 .  (4.10)

Hence, if n ≥ 4, then we have |T |^{2} = 0 on M.

Now we consider n = 3. From (4.7), we see that Δ|T |^{2} ≥ 0. From this inequality and (4.9), we have

lim sup_{k→∞} Δ|T |^{2}(x_{k}) = 0 .  (4.11)

Hence, the above inequalities become equalities when we take the limitation. From (1.1), we know that {h_{ji}^{α}(x_{k})} is a bounded sequence for any i, j and α. Therefore, we can assume that lim_{k→∞} h_{ji}^{α}(x_{k}) exists for any i ,j and α, if necessary, we can take subsequence of {x_{k}}. Then we set

Ā_{α} = lim_{k→∞} A_{α}(x_{k}) = ( lim_{k→∞} h_{j}^{i}_{α}(x_{k})) .

From (4.5) and (4.6), we have

\begin{matrix}∑_{α=2}^{p}(traceĀ_{1}traceĀ_{1}Ā_{α}^{2} − (traceĀ_{1}Ā_{α})^{2}) \\ = \frac{3}{2} sup |T |^{2} ∑_{α=2}^{p} trace Ā_{α}^{2} , \end{matrix}  (4.12)

∑_{α,β=2}^{p}(N(Ā_{α}Ā_{β} − Ā_{β}Ā_{α}) + (trace Ā_{α}Ā_{β})^{2}) = \frac{3}{2} (∑_{α=2}^{p} N(Ā_{α}))^{2} .  (4.13)

Applying the Lemma 3 to (4.13), we see that (i) Ā_{2} = Ā_{3} = ⋯ = Ā_{p} = 0 or

(ii) Ā_{2} ≠ 0, Ā_{3} ≠ 0, Ā_{4} = ⋯ = Ā_{p} = 0 and there exists an orthogonal matrix T such that

tT Ā_{2}T = \sqrt{\frac{\overline{S}_{2}}{2}} (\begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 0 \end{matrix}) ,  ^{t}T Ā_{3}T = \sqrt{\frac{\overline{S}_{2}}{2}} (\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) ,  (4.14)

where \overline{S}_{2} = trace Ā_{2}^{2}.

If (i) holds, we have sup |T |^{2} = ∑_{α=2}^{p} trace Ā_{α}^{2} = 0, that is, |T |^{2} = 0 on M.

Now we assume that (ii) holds. For each α, by virtue of

trace A_{1}trace A_{1}A_{α}^{2} − (trace A_{1}A_{α})^{2} ≥ \frac{3}{2}|T |^{2}trace A_{α}^{2} ,

(4.12) implies that

∑_{i,j=1}^{n} \overline{ρ}_{i}\overline{ρ}_{j}(\overline{ρ}_{i} ^{α} − \overline{ρ}_{j} ^{α})^{2} = \frac{sup |T |^{2}}{2} ∑_{i,j=1}^{n}(\overline{ρ}_{i} ^{α} − \overline{ρ}_{j} ^{α})^{2} ,  (4.15)

where \overline{ρ}_{i} = lim_{k→∞} ρ_{i}(x_{k}) , \overline{ρ}_{i} ^{α} = lim_{k→∞} ρ_{i} ^{α}(x_{k}). Moreover, from (4.14), we have \overline{ρ}_{i} ^{2} ≠ \overline{ρ}_{j} ^{2} for any distinct i and j. Then, from (4.4) and (4.15), we find

\overline{ρ}_{i}\overline{ρ}_{j} = \frac{1}{2} sup |T |^{2}  (4.16)

for distinct i and j. It can be easily verified that

∑_{i≠j} ρ_{i}ρ_{j} = 6|H|^{2} − |U|^{2}  (4.17)

by (3.21). From (4.16) and (4.17) we have immediately

lim_{k→∞} |U|^{2}(x_{k}) + 3 sup |T |^{2} = 6|H|^{2} .  (4.18)

On the other hand, by the Lemma 2, (4.16) implies that

(∑_{i=1}^{n} \overline{ρ}_{i})^{2} = 2 ∑_{i=1}^{n}(\overline{ρ}_{i})^{2} + 2 sup |T |^{2} ,

that is,

lim_{k→∞} ⟨h⟩^{2} (x_{k}) = \frac{9}{2}|H|^{2} .  (4.19)

Then, from (3.23) and (4.19), we have

lim_{k→∞} |U|^{2}(x_{k}) + sup |T |^{2} = \frac{3}{2}|H|^{2} .  (4.20)

Solving the equations (4.18) and (4.20), we find

lim_{k→∞} |U|^{2}(x_{k}) = −\frac{3}{4}|H|^{2} < 0 .  (4.21)

This is contradictory to that |U|^{2} is nonnegative. That is, (ii) does not hold. Thus, |T |^{2} = 0 for n ≥ 3 on M. Hence, we have trace A_{α}^{2} = 0 ( α = 2, ⋯, p). Namely, we obtain

A_{2} = A_{3} = ⋯ = A_{p} = 0 .

Hence, if we denote by N_{1} the normal subbundle spanned by ξ_{2}, ξ_{3}, ⋯, ξ_{p} of the normal bundle T^{⊥}(M) of M, then M is totally geodesic with respect to N_{1}. Since the mean curvature vector H = |H|ξ_{1} of M is parallel in the normal bundle, setting

∇_{X}^{⊥}ξ_{α} = ω_{α}^{β}(X)ξ_{β}

for any tangent vector X on M, we find

∇_{X}^{⊥}ξ_{α} = ∑_{β=2}^{p} ω_{α}^{β}(X)ξ_{β} ∈ N_{1},

for α = 2, 3, ⋯, p, that is, the normal subbundle N_{1} is invariant under parallel translation with respect to the normal connection ∇^{⊥} of M. From the theorem 1 in [11], we conclude that M lies in an (n + 1)-dimensional totally geodesic submanifold E^{n+1} of E^{n+p}. We denote by |H′| the mean curvature of M in E^{n+1}. Since E^{n+1} is totally geodesic in E^{n+p}, we have |H| = |H′|, that is, the mean curvature |H′| of M in E^{n+1} is constant. Thus, M is a complete and connected hypersurface with constant mean curvature |H| in E^{n+1} and its sectional curvatures are nonnegative. From the following proposition 1, we know that M is the hypersphere S^{n}(c) or a generalized cylinder S^{n−k}(c) × E^{k} , (1 ≤ k ≤ n − 1), in E^{n+1}. Since S^{n−k}(c) × E^{k} for 1 < k ≤ n − 1

satisfies S > \frac{n^{2}|H|^{2}}{n − 1} , we infer that Main Theorem 1 is true.

Proposition 1 (Cheng, S. Y. and Yau, S. T. [3]). Let M be a complete and connected hypersurface with nonnegative sectional curvatures in E^{n+1}. If the mean curvature |H| of M is constant, then M is the hyperplane E^{n}, the hypersphere S^{n}(c) or a generalized cylinder S^{n−k}(c) × E^{k} , (1 ≤ k ≤ n − 1).

Therefore, the Main Theorem 1 is proved completely.

References

[1] Chen, B. Y.: Geometry of submanifolds, Marcel Dekker, INC. New York, 1973

[2] Cheng, Q. M. and Wu, B. Q.: The generalized maximum principle and conformally flat spaces, Northeastern Math. J., 8(1992), 54-56.

[3] Cheng, S. Y. and Yau, S. T.: Differential equations on Riemannian submanifolds and their geometric applications, Comm. Pure Appl. Math., 28(1975), 333-354.

[4] Hasanis, Th.: Conformally flat spaces and a piching problem on Ricci tensor, Proc.Amer. Math. Soc., 86(1982), 312-215.

[5] Klotz, T. and Osserman, R.: On complete surfaces in E^{3} with constant mean curvature, Comm. Math. Helv., 41(1966-67), 313-318.

[6] Li, A. M. and Li, J. M.: An intrinsic rigidity theorem for minimal submanifolds, Arch. Math., 58(1992), 582-594.

[7] Motomiya, K.: On functions which satisfy some differential inequalities on Riemannian manifolds, Nagoya Math. J., 81(1981), 57-72.

[8] Omori, H.: Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan, 19(1967), 205-214.

[9] Shen, B. Y.: Complete submanifolds in E^{n+p} with parallel mean curvature, Chin. Ann. of Math. Ser. B, 6(1985), 345-350.

[10] Yau, S. T.: Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., 28(1975), 201-228.

[11] Yau, S. T.: Submanifolds with constant mean curvature I, II, Amer. J. Math., 96(1974), 346-366; 97(1975), 76-100.

Qing-Ming Cheng Department of Mathematics Faculty of Science and Engineering Saga University Saga 840-8502, Japan Cheng@ms.saga-u.ac.jp

Kazuhiro Nonaka Graduate School of Science Josai University Sakado, Saitama 350-0295, Japan