PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 102(116) (2017), 175–193 DOI: https://doi.org/10.2298/PIM1716175S

WILLMORE SPACELIKE SUBMANIFOLDS IN AN INDEFINITE SPACE FORM N_{q}^{n+p} (c)

Shichang Shu and Junfeng Chen

Abstract. Let N_{q}^{n+p} (c) be an (n+p)-dimensional connected indefinite space form of index q (1 ⩽ q ⩽ p) and of constant curvature c. Denote by φ : M → N_{q}^{n+p} (c) the n-dimensional spacelike submanifold in N_{q}^{n+p} (c), φ : M → N_{q}^{n+p} (c) is called a Willmore spacelike submanifold in N_{q}^{n+p} (c) if it is a critical submanifold to the Willmore functional W (φ) = ∫_{M} ρ^{n} dv = ∫_{M} (S − nH^{2})^{\frac{n}{2}} dv, where S and H denote the norm square of the second fundamental form and the mean curvature of M and ρ^{2} = S − nH^{2}. If q = p, in [14 ], we proved some integral inequalities of Simons’ type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in a Lorentzian space form N_{p}^{n+p} (c). In this paper, we continue to study this topic and prove some integral inequalities of Simons’ type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in an indefinite space form N_{q}^{n+p} (c) (1 ⩽ q < p).

1. Introduction

Let N_{q}^{n+p} (c) be an (n+p)-dimensional connected indefinite space form of index q (1 ⩽ q ⩽ p) and of constant curvature c. If c > 0, c = 0 or c < 0, it is denoted by S_{q}^{n+p} (c), R_{q}^{n+p} or H_{q}^{n+p} (c). A submanifold M in N_{q}^{n+p} (c) is said to be spacelike if the induced metric on M from that of the ambient space is positive definite. Let φ: M → N_{q}^{n+p} (c) be an n-dimensional spacelike submanifold in N_{q}^{n+p} (c). If q = p and M is a complete maximal spacelike submanifold in N_{p}^{n+p} (c), from [6], we know that M is totally geodesic for c ⩾ 0, thus the class of all such submanifolds is very small. If 0 ⩽ q < p, from [1] and [4], we know that if M is a complete minimal submanifold in sphere S^{m}(c) m > n, which is embeded in S_{q}^{m+q} (c) as a totally geodesic spacelike submanifold such that m − n + q = p, then M is a complete maximal spacelike submanifold in S_{q}^{n+p} (c), thus, we see that the class of complete maximal spacelike submanifold in S_{q}^{n+p} (c) is very large. Therefore, if 0 ⩽ q < p, the topic of studying spacelike submanifold in S_{q}^{n+p} (c) is also interesting and

2010 Mathematics Subject Classification: Primary 53C42; Secondary 53C40. Key words and phrases: indefinite space form, Willmore spacelike submanifold, totally um-

bilical, Euler-Lagrange equation.

Communicated by Michael Kunzinger.

important. But as far as we know, the results of this topic are less well established. In [1], Alias and Romero studied compact maximal spacelike submanifold M in S_{q}^{n+p} (c) and proved that if the Ricci curvature of M satisfying Ric(M) ⩾ (n − 1)c, then M is totally geodesic. Cheng–Ishikawa [4] also studied compact maximal spacelike submanifold in S_{q}^{n+p} (c) and obtained some important results in terms of the pinching conditions on the scalar curvature, sectional curvature and Ricci curvature, respectively.

Denote by h_{ij}^{α} , S, H⃗ and H the second fundamental form, the norm square of the second fundamental form, the mean curvature vector and the mean curvature of M and denote by ρ^{2} the nonnegative function ρ^{2} = S − nH^{2}, we define the Willmore functional (see [2, 8, 11]):

W (φ) = ∫_{M} ρ^{n}dv = ∫_{M} (S − nH^{2})^{\frac{n}{2}} dv,

which vanishes if and only if M is a totally umbilical spacelike submanifold. It was shown in [9] that this functional is an invariant under the conformal transformations of a conformal space. The points of M are called the critical points of Willmore functional W (φ) if W′(φ) = 0. If the critical points of W (φ) are submanifolds in N_{q}^{n+p} (c), we call them Willmore spacelike submanifolds. Obviously, we notice that the totally umbilical spacelike submanifold is a Willmore spacelike submanifold, but, conversely, it is not true.

Since any minimal submanifold in a unit sphere S^{n+p} (c) is not necessarily Willmore submanifold, due to their backgrounds in mathematics, we know that Willmore submanifolds in a unit sphere have been extensively studied in recent years (see [8, 13]). In indefinite or Lorentzian geometry, we also see that any maximal spacelike submanifold in N_{q}^{n+p} (c) (1 ⩽ q ⩽ p) is not necessarily Willmore spacelike submanifold, thus the study of Willmore spacelike submanifold in N_{q}^{n+p} (c) (1 ⩽ q ⩽ p) is also interesting and important. In [14], if q = p, we proved some integral inequalities of Simons’ type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in a Lorentzian space form N_{p}^{n+p} (c). In this paper, we shall continue to study this topic and prove some integral inequalities of Simons’ type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in an indefinite space form N_{q}^{n+p} (c) (1 ⩽ q < p).

Denote by K and Q the functions which assign to each point of M the infimum of the sectional curvature and the Ricci curvature at the point, we obtain the following:

Theorem 1.1. Let φ: M → N_{q}^{n+p} (c) be an n(n ⩾ 2)-dimensional compact

Willmore spacelike submanifold in the indefinite space form N_{q}^{n+p} (c), c > 0 and 1 ⩽ q < p.

(1) If p − q = 1, then

∫_{M} ρ^{n}\{n(c − H^{2}) − ( 2 − \frac{1}{p} ) ρ^{2}\}dv ⩽ 0.

In particular, if

ρ^{2} ⩽ \frac{n}{2 − \frac{1}{p}} (c − H^{2}),

then M is totally umbilical or M lies in the totally geodesic spacelike submanifold S^{n+1} (c) of S_{q}^{n+q+1} (c) and is isometric to the Clifford torus S^{k}(\frac{1}{\sqrt{2}}c) × S^{k}(\frac{1}{\sqrt{2}}c);

(2) If p − q > 1, then

∫_{M} ρ^{n}\{n(c − H^{2}) − \frac{3}{2}ρ^{2}\}dv ⩽ 0.

In particular, if

ρ^{2} ⩽ \frac{2n}{3}(c − H^{2}),

then M is totally umbilical or M lies in the totally geodesic spacelike submanifold S^{4}(c) of S_{q}^{4+q} (c) and is isometric to the Veronese surface.

Theorem 1.2. Let φ: M → N_{q}^{n+p} (c) be an n(n ⩾ 2)-dimensional compact

Willmore spacelike submanifold in the indefinite space form N_{q}^{n+p} (c) (1 ⩽ q < p). Then the following integral inequality holds

∫_{M} ρ^{n}\{ K − \frac{n − 2}{\sqrt{n(n − 1)}} Hρ − \frac{1}{n}( 1 − \frac{1}{p − q} ) ρ^{2}\} dv ⩽ 0.

In particular, if

K ⩾ \frac{n − 2}{\sqrt{n(n − 1)}} Hρ + \frac{1}{n}( 1 − \frac{1}{p − q} ) ρ^{2},

then M is totally umbilical or M is a maximal spacelike submanifold in N_{q}^{n+p} (c) with parallel second fundamental form.

Theorem 1.3. Let φ: M → N_{q}^{n+p} (c) be an n(n ⩾ 2)-dimensional compact

Willmore spacelike submanifold in the indefinite space form N_{q}^{n+p} (c) (1 ⩽ q < p). Then the following integral inequality holds

∫_{M} ρ^{n}\{Q − (n − 2)c − nH^{2} − \frac{1}{n}( 3 − \frac{p + q}{(p − q)q} ) ρ^{2}\}dv ⩽ 0.

In particular, if

Q ⩾ (n − 2)c + nH^{2} + \frac{1}{n}( 3 − \frac{p + q}{(p − q)q} ) ρ^{2},

then M is totally umbilical or M is a maximal spacelike submanifold in N_{q}^{n+p} (c) with parallel second fundamental form.

2. Preliminaries

Let N_{q}^{n+p} (c) be an (n + p)-dimensional indefinite space form with index q(1 ⩽ q ⩽ p). Let M be an n-dimensional connected spacelike submanifold immersed in N_{q}^{n+p} (c). We choose a local field of semi-Riemannian orthonormal frames e_{1}, …, e_{n+p} in N_{q}^{n+p} (c) such that at each point of M, e_{1}, …, e_{n} span the tangent space of M and form an orthonormal frame there. We use the following convention on the range of indices:

1 ⩽ A, B, C, … ⩽ n + p, 1 ⩽ i, j, k, … ⩽ n, n + 1 ⩽ α, β, γ, … ⩽ n + p.

Let ω_{1}, …, ω_{n+p} be its dual frame field so that the semi-Riemannian metric of N_{q}^{n+p} (c) is given by ds̄^{2} = ∑ _{A} ε_{A}ω_{A}^{2}, where ε_{A} = 1 for 1 ⩽ A ⩽ n + p − q and ε_{A} = −1 for n + p − q + 1 ⩽ A ⩽ n + p. Then the structure equations of N_{q}^{n+p} (c) are given by

dω_{A} = − ∑_{B} ε_{B}ω_{AB} ∧ ω_{B},  ω_{AB} + ω_{BA} = 0,

dω_{AB} = − ∑_{C} ε_{C}ω_{AC} ∧ ω_{CB} − \frac{1}{2} ∑_{C,D} ε_{C}ε_{D}K_{ABCD} ω_{C} ∧ ω_{D},
K_{ABCD} = cε_{A}ε_{B}(δ_{AD} δ_{BC} − δ_{AC} δ_{BD} ).

If we restrict this form to M, then ω_{α} = 0, n + 1 ⩽ α ⩽ n + p and

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

The second fundamental form II, the mean curvature vector H⃗ of M are defined by

(2.1)  II = ∑_{α,i,j} ε_{α} h_{ij}^{α} ω_{i}ω_{j} e_{α} , H⃗ = ∑_{α} ε_{α} H^{α} e_{α} , H^{α} = \frac{1}{n} ∑_{k} h_{kk}^{α}.

The norm square of the second fundamental form and the mean curvature of M are defined by

S = |II|^{2} = ∑_{i,j,α} (ε_{α} h_{ij}^{α} )^{2} = ∑_{i,j,α} (h_{ij}^{α} )^{2}, H = |H⃗| = \frac{1}{n} \sqrt{∑_{α} ( ∑_{k} h_{kk}^{α}) ^{2}}.

The Gauss equations are

(2.2)  R_{ijkl} = c(δ_{il} δ_{jk} − δ_{ik} δ_{jl}) + ∑_{α} ε_{α} (h_{il}^{α} h_{jk}^{α} − h_{ik}^{α} h_{jl}^{α}),

(2.3)  R_{jk} = (n − 1)cδ_{jk} + ∑_{α} ε_{α} ( ∑_{i} h_{ii}^{α} h_{jk}^{α} − ∑_{i} h_{ik}^{α} h_{ji}^{α} ) .

Defining the first and the second covariant derivatives of h_{ij}^{α} , say h_{ijk}^{α} and h_{ijkl}^{α} by

∑_{k} h_{ijk}^{α} ω_{k} = dh_{ij}^{α} − ∑_{k} h_{ik}^{α} ω_{kj} − ∑_{k} h_{jk}^{α} ω_{ki} − ∑_{β} ε_{β}h_{ij}^{β} ω_{βα} ,

∑_{l} h_{ijkl}^{α} ω_{l} = dh_{ijk}^{α} − ∑_{m} h_{mjk}^{α} ω_{mi} − ∑_{m} h_{imk}^{α} ω_{mj}

− ∑_{m} h_{ijm}^{α} ω_{mk} − ∑_{β} ε_{β}h_{ijk}^{β} ω_{βα} ,

we have the Codazzi equations and the Ricci identities

h_{ijk}^{α} = h_{ikj}^{α} , (2.4)

h_{ijkl}^{α} − h_{ijlk}^{α} = − ∑_{m} h_{im}^{α} R_{mjkl} − ∑_{m} h_{jm}^{α} R_{mikl} − ∑_{β} ε_{β}h_{ij}^{β} R_{βαkl} . (2.5)

The Ricci equations are

(2.6)  R_{αβij} = − ∑_{m} (h_{im}^{α} h_{mj}^{β} − h_{jm}^{α} h_{mi}^{β} ).

The Laplacian of h_{ij}^{α} is defined by Δh_{ij}^{α} = ∑ _{k} h_{ijkk}^{α} . From (2.5), we obtain for any α, n + 1 ⩽ α ⩽ n + p,

Δh_{ij}^{α} = ∑_{k} h_{kkij}^{α} − ∑_{k,m} h_{km}^{α} R_{mijk} − ∑_{k,m} h_{im}^{α} R_{mkjk} − ∑_{k,β} ε_{β}h_{ik}^{β} R_{βαjk} .

For the fix index α(n + 1 ⩽ α ⩽ n + p), we introduce an operator □^{α} due to Cheng-Yau [3] by

(2.7)  □^{α} f = ∑_{i,j} (nH^{α} δ_{ij} − h_{ij}^{α} )f_{i,j} .

Since M is compact, the operator □^{α} is self-adjoint (see [3]) if and only if

(2.8)  ∫_{M} (□^{α} f)g dv = ∫_{M} f(□^{α} g) dv,

where f and g are any smooth functions on M. We need the following Lemma (see [12]):

Lemma 2.1. Let A, B be symmetric n × n matrices satisfying AB = BA and trA = trB = 0. Then

| trA^{2}B| ⩽ \frac{n − 2}{\sqrt{n(n − 1)}} (trA^{2})(trB^{2})^{1/2} ,

and the equality holds if and only if (n − 1) of the eigenvalues x_{i} of B and the corresponding eigenvalues y_{i} of A satisfy |x_{i}| = (trB^{2})^{1/2} /\sqrt{n(n − 1)} , x_{i}x_{j} ⩾ 0, y_{i} = (trA^{2})^{1/2} /\sqrt{n(n − 1)} .

By the same method as in the proof of [8, Lemma 4.2], we also have the following:

Lemma 2.2. Let φ: M → N_{q}^{n+p} (c) be an n-dimensional (n ⩾ 2) spacelike submanifold in N_{q}^{n+p} (c)(1 ⩽ q ⩽ p). Then we have

|∇h|^{2} ⩾ \frac{3n^{2}}{n + 2} |∇^{⊥} H⃗|^{2},

where |∇h|^{2} = ∑ _{i,j,k,α} (h_{ijk}^{α} )^{2}, |∇^{⊥} H⃗|^{2} = ∑ _{i,α} (H_{,i}^{α})^{2}.

3. Euler–Lagrange equation of Willmore spacelike submanifolds

From [9, Theorem 4.1], we know that the Euler–Lagrange equation of Willmore spacelike submanifolds in terms of invariants of conformal metric g is stated as following: a spacelike submanifold is a Willmore spacelike submanifold if and only if

(3.1) ∑_{i,j,k,l,β} g_{αβ} g^{ik} g^{jl} ( B_{ij,kl}^{β} + A_{ij} B_{kl}^{β} + ∑_{r,q,γ,ν} g_{γν} g^{rq}B_{ir}^{β} B_{qj}^{γ} B_{kl}^{ν}) = 0, ∀α,

where 1 ⩽ i, j, k, l, r, q ⩽ n, n + 1 ⩽ α, β, γ, ν ⩽ n + p, (g_{ij} ) = (I_{n}), (g_{αβ} ) = (I_{p−q} ) ⊕ (−I_{q}), (g_{ij} ) = (g^{ij} )^{−1} and (g_{αβ} ) = (g^{αβ} )^{−1} (see [9]). From [9, (3.23)], we have (1−n)C_{i}^{α} = ∑ _{k} B_{ik,k}^{α} . Thus, by a simply calculation, we may rewrite (3.1) as

(3.2)  (1 − n) ∑_{i} C_{i,i}^{α} + ∑_{i,j} A_{ij} B_{ij}^{α} + ∑_{β} ∑_{i,j,k} ε_{β}B_{ik}^{α} B_{kj}^{β} B_{ij}^{β} = 0, ∀α,

where ε_{β} = g_{ββ} and ε_{β} = 1 for n+1 ⩽ β ⩽ n+p−q and ε_{β} = −1 for n+p−q +1 ⩽ β ⩽ n + p.

From [9] or [10], we have the following relations of the connections of the conformal metric e^{2τ}du · du and induced metric du · du

(3.3)  ω_{i} = e^{τ}θ_{i}, ω_{ij} = θ_{ij} + τ_{i}θ_{j} − τ_{j} θ_{i}, ω_{αβ} = θ_{αβ} ,

where e^{2τ} = \frac{n}{n−1} (S −nH^{2}). We know that the relations of the conformal invariants and the induced invariants are

(3.4)  e^{2τ}C_{i} = H^{α} τ_{i} − H_{,i}^{α} − ∑_{j} h_{ij}^{α} τ^{j} ,

(3.5)  e^{2τ}A_{ij} = τ_{i}τ_{j} − τ_{i,j} − ∑_{α} H^{α} h_{ij}^{α} − \frac{1}{2}( ∑_{k} τ^{k}τ_{k} − H^{2} − c) I_{ij} ,
(3.6)  e^{τ}B_{ij}^{α} = h_{ij}^{α} − H^{α} I_{ij} ,

where τ_{i,j} is the Hessian of τ with respect to the first fundamental form I, τ^{i} = ∑ _{j} I^{ij} τ_{j}, (I^{ij} ) = (I_{ij} )^{−1} , H_{i}^{α} = e_{i}(H^{α}) and c = 0 for R_{q}^{n+p} (c), c > 0 for S_{q}^{n+p} (c) and c < 0 for H_{q}^{n+p} (c) (see [10])

From (3.3) and (3.4), by a similar calculation of Li [8], we have

∑_{j} e^{τ}C_{i,j}^{α} θ_{j} = ∑_{j} C_{i,j}^{α} ω_{j} = dC_{i}^{α} + ∑_{j} C_{j}^{α} ω_{ji} + ∑_{β} C_{i}^{β}ω_{βα}

= dC_{i}^{α} + ∑_{j} C_{j}^{α} θ_{ji} + ∑_{j} C_{j}^{α} (τ_{j} θ_{i} − τ_{i}θ_{j}) + ∑_{β} C_{i}^{β}θ_{βα} ,

therefore, we have

(3.7) e^{τ}C_{i,j}^{α} = e^{−2τ} ( − 2H^{α} τ_{i}τ_{j} + 2τ_{j} ∑_{k} h_{ik}^{α} τ_{k} + 2τ_{j} H_{,i}^{α} + H_{,j}^{α} τ_{i} + H^{α} τ_{i,j}

− ∑_{k} h_{ik,j}^{α} τ_{k} − ∑_{k} h_{ik}^{α} τ_{k,j} − H_{,ij}^{α} ) + ∑_{k} C_{k}^{α} τ_{k}δ_{ij} − τ_{i}C_{j}^{α} .

WILLMORE SPACELIKE SUBMANIFOLDS… 181

From (3.7), we see that

(3.8)  e^{3τ} ∑_{i} C_{i,i}^{α} = (n − 3)( H^{α} |∇τ|^{2} − ∑_{i,k} h_{ik}^{α} τ_{k}τ_{i})

− 2(n − 2) ∑_{i} H_{,i}^{α} τ_{i} + H^{α} Δτ − ∑_{i,k} h_{ik}^{α} τ_{k,i} − Δ^{⊥} H^{α} .

From (3.5) and (3.6), we have

(3.9) e^{3τ} ( ∑_{i,j} A_{ij} B_{ij}^{α} + ∑_{β} ∑_{i,j,k} ε_{β}B_{ik}^{α} B_{kj}^{β} B_{ij}^{β} )

= ∑_{i,j} [τ_{i}τ_{j} −τ_{i,j} −∑_{β} H^{β}h_{ij}^{β} − \frac{1}{2}( ∑_{k} τ^{k}τ_{k}−H^{2}−c) I_{ij} ](h_{ij}^{α} −H^{α} I_{ij} )

+ ∑_{β} ∑_{i,j,k} ε_{β}(h_{ik}^{α} − H^{α} I_{ik} )(h_{kj}^{β} − H^{β}I_{kj} )(h_{ij}^{β} − H^{β}I_{ij} )

= ∑_{i,j} h_{ij}^{α} (τ_{i}τ_{j} − τ_{i,j} ) + H^{α} [Δτ − |∇τ|^{2} + n ∑_{β} (1 + 2ε_{β})(H^{β})^{2}]

+ ∑_{β} ∑_{i,j,k} ε_{β}h_{ik}^{α} h_{kj}^{β} h_{ij}^{β} − ∑_{β,i,j} (1 + 2ε_{β})H^{β}h_{ij}^{β} h_{ij}^{α} − H^{α} ∑_{β,i,j} ε_{β}(h_{ij}^{β} )^{2}.

From (3.2), (3.8) and (3.9), we see that

(3.10)  (n− 2)^{2}bigg(∑_{i,j} h_{ij}^{α} τ_{i}τ_{j} − H^{α} |∇τ|^{2}) + 2(n − 1)(n − 2) ∑_{i} H_{,i}^{α} τ_{i}

+ (n − 2)( ∑_{i,j} h_{ij}^{α} τ_{i,j} − H^{α} Δτ) + (n − 1)Δ^{⊥} H^{α}

− H^{α} ∑_{β,i,j} ε_{β}(h_{ij}^{β} )^{2} − ∑_{β,i,j} (1 + 2ε_{β})H^{β}h_{ij}^{β} h_{ij}^{α}

+ nH^{α} ∑_{β} (1 + 2ε_{β})(H^{β})^{2} + ∑_{β} ∑_{i,j,k} ε_{β}h_{ik}^{α} h_{kj}^{β} h_{ij}^{β} = 0.

Putting ρ^{2} = S − nH^{2}, we have e^{2τ} = \frac{n}{n−1} (S − nH^{2}) = \frac{n}{n−1} ρ^{2}. Thus e^{τ} = \sqrt{\frac{n}{n−1}} ρ and τ = ln(\sqrt{\frac{n}{n−1}} ρ). From (3.10), we see that

(3.11)  \frac{ρ^{n−2}}{n − 1} \{ − H^{α} ∑_{β,i,j} ε_{β}(h_{ij}^{β} )^{2} − ∑_{β,i,j} (1 + 2ε_{β})H^{β}h_{ij}^{β} h_{ij}^{α}

+ ∑_{β} ∑_{i,j,k} ε_{β}h_{ik}^{α} h_{kj}^{β} h_{ij}^{β} + nH^{α} ∑_{β} (1 + 2ε_{β})(H^{β})^{2}}

+ ρ^{n−2} Δ^{⊥} H^{α} + \frac{n − 2}{n − 1} ρ^{n−2} ∑_{i,j} (lnρ)_{i,j} (h_{ij}^{α} − H^{α} δ_{ij} )

+ 2(n − 2)ρ^{n−2} ∑_{i} (lnρ)_{i}H_{,i}^{α}

+ \frac{(n − 2)^{2}}{n − 1} ρ^{n−2} ∑_{i,j} (lnρ)_{i}(lnρ)_{j} (h_{ij}^{α} − H^{α} δ_{ij} ) = 0.

It can be easily checked that

(3.12)  ρ^{n−2} Δ^{⊥} H^{α} + \frac{n − 2}{n − 1} ρ^{n−2} ∑_{i,j} (lnρ)_{i,j} (h_{ij}^{α} − H^{α} δ_{ij} )

+ 2(n − 2)ρ^{n−2} ∑_{i} (lnρ)_{i}H_{,i}^{α}

+ \frac{(n − 2)^{2}}{n − 1} ρ^{n−2} ∑_{i,j} (lnρ)_{i}(lnρ)_{j} (h_{ij}^{α} − H^{α} δ_{ij} )

= −\frac{1}{n − 1} ∑_{i,j} (ρ^{n−2} )_{i,j} (nH^{α} δ_{ij} − h_{ij}^{α} ) + ρ^{n−2} Δ^{⊥} H^{α}

+ 2 ∑_{i} (ρ^{n−2} )_{i}H_{,i}^{α} + H^{α} Δ(ρ^{n−2} ).

From (3.11) and (3.12), we may obtain the Euler–Lagrange equation of Willmore spacelike submanifolds in N_{q}^{n+p} (c) in terms of the induced invariants:

Theorem 3.1. Let φ: M → N_{q}^{n+p} (c) be an n-dimensional spacelike submanifold in N_{q}^{n+p} (c). Then M is an n-dimensional Willmore spacelike submanifold if and only if for n + 1 ⩽ α, β ⩽ n + p,

(3.13)  ρ^{n−2} { − H^{α} ∑_{β,i,j} ε_{β}(h_{ij}^{β} )^{2} − ∑_{β,i,j} (1 + 2ε_{β})H^{β}h_{ij}^{β} h_{ij}^{α}

+ ∑_{β} ∑_{i,j,k} ε_{β}h_{ik}^{α} h_{kj}^{β} h_{ij}^{β} + nH^{α} ∑_{β} (1 + 2ε_{β})(H^{β})^{2}}

+ (n − 1)ρ^{n−2} Δ^{⊥} H^{α} + 2(n − 1) ∑_{i} (ρ^{n−2} )_{i}H_{,i}^{α}

+ (n − 1)H^{α} Δ(ρ^{n−2} ) − □^{α} (ρ^{n−2} ) = 0.

where Δ(ρ^{n−2} ) = ∑ _{i}(ρ^{n−2} )_{i,i} , □^{α} (ρ^{n−2} ) = ∑ _{i,j} (ρ^{n−2} )_{i,j} (nH^{α} δ_{ij} − h_{ij}^{α} ) and (ρ^{n−2} )_{i,j} is the Hessian of ρ^{n−2} with respect to the induced metric.

Remark 3.1. In the proof of (3.13), since we denote e^{2τ} = \frac{n}{n−1} (S − nH^{2}) =

\frac{n}{n−1} ρ^{2}, it follows that ρ^{2} ≠ 0, that is, (3.13) holds only for ρ^{2} ≠ 0. But, if ρ^{2} = 0, we should notice that (3.13) also holds. Thus, in the following discussion, we agree that the Euler–Lagrange equation of Willmore spacelike submanifolds (3.13) holds for all ρ^{2}. But, if n = 3 and n = 5, we need to assume that M has no umbilical points to guarantee (ρ^{n−2} )_{i,j} to be continuous on M.

Proposition 3.1. Every maximal spacelike surface φ: M → N_{q}^{2+p} (c) in N_{q}^{2+p} (c) is a Willmore spacelike surface.

In fact, if n = 2, since H = 0, from (2.1), we see that H^{α} = 0 and ∑ _{k} h_{kk}^{α} = 0. On the other hand, since R_{ij} = \frac{R}{2} δ_{ij} , from Gauss equation (2.3), we have ∑ _{β,j} ε_{β}h_{jk}^{β} h_{ij}^{β} = cδ_{ik} + ∑ _{β,j} ε_{β}h_{jj}^{β} h_{ik}^{β} − R_{ik}, thus

∑_{β} ∑_{i,j,k} ε_{β}h_{ik}^{α} h_{kj}^{β} h_{ij}^{β} = ∑_{i,k} h_{ik}^{α} ( cδ_{ik} + ∑_{β,j} ε_{β}h_{jj}^{β} h_{ik}^{β} − R_{ik} )

= ( c − \frac{R}{2}) ∑_{i} h_{ii}^{α} + ∑_{β,i,k} ε_{β}h_{ik}^{α} h_{ik}^{β} ( ∑_{j} h_{jj}^{β} ) = 0,

it follows that (3.13) holds and Proposition 3.1 is concluded.

Example 3.1. If 0 ⩽ q < p = q + 1, since we know that the Clifford torus S^{k}( \sqrt{\frac{k}{n}} c) × S^{n−k} ( \sqrt{\frac{n−k}{n}} c) is a complete minimal hypersurface in sphere S^{n+1} (c)

which is embeded in S_{q}^{n+1+q} (c) as a totally geodesic spacelike submanifold such

that 1 + q = p, then S^{k}( \sqrt{\frac{k}{n}} c) × S^{n−k} ( \sqrt{\frac{n−k}{n}} c) is a complete maximal spacelike

submanifold in S_{q}^{n+q+1} (c), where 1 ⩽ k ⩽ n−1. Since S^{k}( \sqrt{\frac{k}{n}} c) ×S^{n−k} ( \sqrt{\frac{n−k}{n}} c)

lies in the totally geodesic spacelike submanifold S^{n+1} (c) of S_{q}^{n+q+1} (c), we know that h_{ij}^{α} = 0 for α = n + 2, …, n + q + 1. Thus, if and only if n = 2k then

∑_{β} ∑_{i,j,k} ε_{β}h_{ik}^{α} h_{kj}^{β} h_{ij}^{β} = ∑_{i,j,k} h_{ik}^{n+1} h_{kj}^{n+1} h_{ij}^{n+1} = ∑_{i} λ_{i}^{3}

= k( \sqrt{\frac{n − k}{k} c}) ^{3} + (n − k)( − \sqrt{\frac{k}{n − k} c}) ^{3} = 0,

where h_{ij}^{n+1} = λ_{i}δ_{ij} , \sqrt{\frac{n−k}{k} c} and −\sqrt{\frac{k}{n−k} c} are the two distinct principal curvatures

of S^{k}( \sqrt{\frac{k}{n}} c) × S^{n−k} ( \sqrt{\frac{n−k}{n}} c) ⊂ S^{n+1} (c) with multiplicities k and n − k, respec-

tively. We also see that ρ^{2} = S −nH^{2} = ∑ _{i} λ_{i}^{2} = nc is constant. Thus, (3.13) holds if and only if n = 2k, that is the Clifford torus S^{k}( \frac{1}{\sqrt{2}}c) × S^{k}( \frac{1}{\sqrt{2}}c), 1 ⩽ k ⩽ n − 1, is a maximal Willmore spacelike submanifold in S_{q}^{n+q+1} (c).

Example 3.2. From [5] and [1], we know that the Veronese surface is a minimal surface in S^{4}(c) which is embeded in S_{q}^{4+q} (c) as a totally geodesic spacelike submanifold such that 2 + q = p, then the Veronese surface is a maximal spacelike surface in S_{q}^{2+p} (c), where p = 2 + q. From Proposition 3.1, we know that it is a Willmore spacelike surface in S_{q}^{4+q} (c).

4. Basic integral equalities

Define tensors

h̃_{ij}^{α} = h_{ij}^{α} − H^{α} δ_{ij} , (4.1)

σ̃_{αβ} = ∑_{i,j} h̃_{ij}^{α} h̃_{ij}^{β} , σ_{αβ} = ∑_{i,j} h_{ij}^{α} h_{ij}^{β} . (4.2)

Then the (p × p)-matrix (σ̃_{αβ} ) is symmetric and can be assumed to be diagonized for a suitable choice of e_{n+1} , …, e_{n+p} . We set

(4.3)  σ̃_{αβ} = σ̃_{α} δ_{αβ} .

By a direct calculation, we have

(4.4) ∑_{k} h̃_{kk}^{α} = 0, σ̃_{αβ} = σ_{αβ} − nH^{α} H^{β}, ρ^{2} = ∑_{α} σ̃_{α} = S − nH^{2},
(4.5)  −H^{α} ∑_{β,i,j} ε_{β}(h_{ij}^{β} )^{2} − ∑_{β,i,j} (1 + 2ε_{β})H^{β}h_{ij}^{β} h_{ij}^{α}

+ ∑_{β} ∑_{i,j,k} ε_{β}h_{ik}^{α} h_{kj}^{β} h_{ij}^{β} + nH^{α} ∑_{β} (1 + 2ε_{β})(H^{β})^{2}

= ∑_{β} ∑_{i,j,k} ε_{β} h̃_{ik}^{α} h̃_{kj}^{β} h̃_{ij}^{β} − ∑_{i,j,β} H^{β} h̃_{ij}^{β} h̃_{ij}^{α} .

From (4.1), (4.4) and (4.5), Euler–Lagrange equation (3.13) can be rewritten as

Proposition 4.1. Let φ: M → N_{q}^{n+p} (c) be an n-dimensional spacelike submanifold in N_{q}^{n+p} (c). Then M is a Willmore spacelike submanifold if and only if for n + 1 ⩽ α ⩽ n + p

(4.6)  □^{α} (ρ^{n−2} ) = (n − 1)ρ^{n−2} Δ^{⊥} H^{α} + 2(n − 1) ∑_{i} (ρ^{n−2} )_{i}H_{,i}^{α}

+ (n − 1)H^{α} Δ(ρ^{n−2} )

+ ρ^{n−2} ( ∑_{β} ∑_{i,j,k} ε_{β} h̃_{ik}^{α} h̃_{kj}^{β} h̃_{ij}^{β} − ∑_{i,j,β} H^{β} h̃_{ij}^{β} h̃_{ij}^{α} ) .

Setting f = nH^{α} in (2.7), we have

(4.7)  □^{α} (nH^{α}) = ∑_{i,j} (nH^{α} δ_{ij} − h_{ij}^{α} )(nH^{α})_{i,j}

= ∑_{i} (nH^{α})(nH^{α})_{i,i} − ∑_{i,j} h_{ij}^{α} (nH^{α})_{i,j} .

We also have

(4.8)  \frac{1}{2}Δ(nH)^{2} = \frac{1}{2}Δ ∑_{α} (nH^{α})^{2} = \frac{1}{2} ∑_{α} Δ(nH^{α})^{2}

= \frac{1}{2} ∑_{α,i} [(nH^{α})^{2}]_{i,i} = ∑_{α,i} [(nH^{α})_{,i}]^{2} + ∑_{α,i} (nH^{α})(nH^{α})_{i,i}

= n^{2}|∇^{⊥} H⃗|^{2} + ∑_{α,i} (nH^{α})(nH^{α})_{i,i} .

Therefore, from (4.7), (4.8), we get

(4.9) ∑_{α} □^{α} (nH^{α}) = \frac{1}{2}Δ(nH)^{2} − n^{2}|∇^{⊥} H⃗|^{2} − ∑_{i,j,α} h_{ij}^{α} (nH^{α})_{i,j}

= \frac{1}{2}Δ[n(n − 1)H^{2} − ρ^{2} + S] − n^{2}|∇^{⊥} H⃗|^{2} − ∑_{i,j,α} h_{ij}^{α} (nH^{α})_{i,j}

= \frac{1}{2}ΔS+ \frac{1}{2}n(n−1)ΔH^{2}− \frac{1}{2}Δρ^{2}−n^{2}|∇^{⊥} H⃗|^{2}−∑_{i,j,α} h_{ij}^{α} (nH^{α})_{i,j} .

From (2.4) and (2.5), we have

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

= |∇h|^{2} + ∑_{i,j,α} h_{ij}^{α} (nH^{α})_{i,j} − ∑_{α} ∑_{i,j,k,l} h_{ij}^{α} h_{kl}^{α}R_{lijk}

− ∑_{α} ∑_{i,j,k,l} h_{ij}^{α} h_{li}^{α}R_{lkjk} − ∑_{α,β} ∑_{i,j,k} ε_{β}h_{ij}^{α} h_{ki}^{β}R_{βαjk} .

Putting (4.10) into (4.9), we have

(4.11) ∑_{α} □^{α} (nH^{α}) = |∇h|^{2} − n^{2}|∇^{⊥} H⃗|^{2} + \frac{1}{2}n(n − 1)ΔH^{2} − \frac{1}{2}Δρ^{2}

−∑_{α} ∑_{i,j,k,l} h_{ij}^{α} (h_{kl}^{α}R_{lijk} + h_{li}^{α}R_{lkjk} )−∑_{α,β} ∑_{i,j,k} ε_{β}h_{ij}^{α} h_{ki}^{β}R_{βαjk} .

Multiplying (4.11) by ρ^{n−2} and taking integration, using (2.8), we have

(4.12)  ∑_{α} ∫_{M} (nH^{α})□^{α} (ρ^{n−2} )dv = ∫_{M} ρ^{n−2} (|∇h|^{2} − n^{2}|∇^{⊥} H⃗|^{2})dv

+ \frac{1}{2}n(n − 1) ∫_{M} ρ^{n−2} ΔH^{2}dv − \frac{1}{2} ∫_{M} ρ^{n−2} Δρ^{2}dv

− ∫_{M} ρ^{n−2} ∑_{α} ∑_{i,j,k,l} h_{ij}^{α} (h_{kl}^{α}R_{lijk} + h_{li}^{α}R_{lkjk} )dv

− ∫_{M} ρ^{n−2} ∑_{α,β} ∑_{i,j,k} ε_{β}h_{ij}^{α} h_{ki}^{β}R_{βαjk} dv.

Substituting Willmore equation (4.6) into (4.12) and making use of the following:

∫_{M} ρ^{n−2} ∑_{α} H^{α} △^{⊥} H^{α} dv = \frac{1}{2} ∫_{M} ρ^{n−2} ∑_{α} Δ^{⊥} (H^{α})^{2}dv − ∫_{M} ρ^{n−2} ∑_{i,α} (H_{,i}^{α})^{2}dv

= \frac{1}{2} ∫_{M} ρ^{n−2} ΔH^{2}dv − ∫_{M} ρ^{n−2} |∇H⃗|^{2}dv,

∫_{M} H^{2}Δ(ρ^{n−2} )dv = ∫_{M} ∑_{α} (H^{α})^{2} ∑_{i} (ρ^{n−2} )_{i,i} dv

= ∑_{α,i} ∫_{M} (H^{α})^{2}(ρ^{n−2} )_{i,i} dv = − ∑_{α,i} ∫_{M} (ρ^{n−2} )_{i}((H^{α})^{2})_{,i}dv

186 SHU AND CHEN

= −2 ∫_{M} ∑_{α} H^{α} ∑_{i} (ρ^{n−2} )_{i}H_{,i}^{α} dv,

−\frac{1}{2} ∫_{M} ρ^{n−2} Δρ^{2}dv = −\frac{1}{2} ∑_{i} ∫_{M} ρ^{n−2} (ρ^{2})_{i,i} dv

= \frac{1}{2} ∑_{i} ∫_{M} (ρ^{2})_{i}(ρ^{n−2} )_{i}dv = (n − 2) ∫_{M} ρ^{n−2} |∇ρ|^{2}dv,

we have, by a direct calculation, the following:

Proposition 4.2. Let φ: M → N_{q}^{n+p} (c) be an n-dimensional spacelike submanifold in N_{q}^{n+p} (c). Then

(4.13)  ∫_{M} ρ^{n−2} (|∇h|^{2} − n|∇^{⊥} H⃗|^{2})dv + (n − 2) ∫_{M} ρ^{n−2} |∇ρ|^{2}dv

− ∫_{M} ρ^{n−2} ∑_{α,β} nH^{α} ( ∑_{i,j,k} ε_{β} h̃_{ik}^{α} h̃_{kj}^{β} h̃_{ij}^{β} − H^{β} σ̃_{αβ} ) dv

− ∫_{M} ρ^{n−2} ∑_{α} ∑_{i,j,k,l} h_{ij}^{α} (h_{kl}^{α}R_{lijk} + h_{li}^{α}R_{lkjk} )dv

− ∫_{M} ρ^{n−2} ∑_{α,β} ∑_{i,j,k} ε_{β}h_{ij}^{α} h_{ki}^{β}R_{βαjk} dv = 0.

In general, for a matrix A = (a_{ij} ) we denote by N(A) the square of the norm of A, that is, N(A) = trace(A · A^{t}) = ∑ _{i,j} (a_{ij} )^{2}. Clearly, N(A) = N(T^{t}AT) for any orthogonal matrix T. From (2.6), we have

(4.14)  − ∑_{α,β} ∑_{i,j,k} ε_{β}h_{ij}^{α} h_{ki}^{β}R_{βαjk} = − ∑_{α,β} ∑_{i,j,k,l} ε_{β}h_{ij}^{α} h_{ki}^{β}(h_{kl}^{β}h_{lj}^{α} − h_{jl}^{β} h_{lk}^{α})
= − \frac{1}{2} ∑_{α,β,j,k} ε_{β}( ∑_{l} h_{kl}^{β}h_{lj}^{α} − ∑_{l} h_{kl}^{α}h_{lj}^{β} ) ^{2}
= − \frac{1}{2} ∑_{α,β,j,k} ε_{β}( ∑_{l} h̃_{kl}^{β} h̃_{lj}^{α} − ∑_{l} h̃_{kl}^{α} h̃_{lj}^{β} ) ^{2}
= − \frac{1}{2} ∑_{α,β} ε_{β}N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α}),

where Ã_{α} := (h̃_{ij}^{α} ) = (h_{ij}^{α} − H^{α} δ_{ij} ).

By use of (2.2), (2.6), (4.1), (4.2), (4.4) and (4.14), we conclude that

− ∑_{α} ∑_{i,j,k,l} h_{ij}^{α} (h_{kl}^{α}R_{lijk} + h_{li}^{α}R_{lkjk} ) = ncρ^{2} − ∑_{α,β} ε_{β}σ_{αβ}^{2} (4.15)

+ n ∑_{α,β} ∑_{i,j,k} ε_{β}H^{β}h_{kj}^{β} h_{ij}^{α} h_{ik}^{α} − ∑_{α,β,i,j,k} ε_{β}h_{ji}^{α} h_{ik}^{β} R_{βαjk}

WILLMORE SPACELIKE SUBMANIFOLDS… 187

= ncρ^{2} − ∑_{α,β} ε_{β} σ̃_{αβ}^{2} − 2n ∑_{α,β} ∑_{i,j} ε_{β}H^{α} H^{β} h̃_{ij}^{α} h̃_{ij}^{β}

− n^{2} ∑_{α} (H^{α})^{2} ∑_{β} ε_{β}(H^{β})^{2} + n ∑_{α,β} ∑_{i,j,k} ε_{β}H^{β} h̃_{kj}^{β} h̃_{ij}^{α} h̃_{ik}^{α}

+ nρ^{2} ∑_{β} ε_{β}(H^{β})^{2} + 2n ∑_{α,β} ∑_{i,j} ε_{β}H^{α} H^{β} h̃_{ij}^{α} h̃_{ij}^{β}

+ n^{2} ∑_{α} (H^{α})^{2} ∑_{β} ε_{β}(H^{β})^{2} − \frac{1}{2} ∑_{α,β} ε_{β}N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α})

= ncρ^{2} − ∑_{α,β} ε_{β} σ̃_{αβ}^{2} + nρ^{2} ∑_{β} ε_{β}(H^{β})^{2}

+ n ∑_{α,β} ∑_{i,j,k} ε_{β}H^{β} h̃_{kj}^{β} h̃_{ij}^{α} h̃_{ik}^{α} − \frac{1}{2} ∑_{α,β} ε_{β}N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α}).

Putting (4.14) and (4.15) into (4.13), we have the following:

Proposition 4.3. Let φ: M → N_{q}^{n+p} (c) be an n-dimensional spacelike submanifold in N_{q}^{n+p} (c). Then

(4.16) ∫_{M} ρ^{n−2} (|∇h|^{2} − n|∇^{⊥} H⃗|^{2})dv + (n − 2) ∫_{M} ρ^{n−2} |∇ρ|^{2}dv

+ n ∫_{M} ρ^{n−2} ( ∑_{α,β} H^{α} H^{β} σ̃_{αβ} + ρ^{2} ∑_{β} ε_{β}(H^{β})^{2}) dv + nc ∫_{M} ρ^{n}dv

− ∫_{M} ρ^{n−2} ( ∑_{α,β} ε_{β}N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α}) + ∑_{α,β} ε_{β} σ̃_{αβ}^{2} ) dv = 0.

5. Proofs of Theorems

Proof of Theorem 1.1. (1) If p − q = 1, from Lemma 2.2 and (4.16), we have

(5.1) 0 = ∫_{M} ρ^{n−2} (|∇h|^{2} − \frac{3n^{2}}{n + 2} |∇^{⊥} H⃗|^{2})dv + ∫_{M} ρ^{n−2} (\frac{3n^{2}}{n + 2} − n)|∇^{⊥} H⃗|^{2}dv

+ (n − 2) ∫_{M} ρ^{n−2} |∇ρ|^{2}dv

+ n ∫_{M} ρ^{n−2} { ∑_{α,β} H^{α} H^{β} σ̃_{αβ} + 2ρ^{2}(H^{n+1} )^{2} − H^{2}ρ^{2}} dv + nc ∫_{M} ρ^{n}dv

+ ∫_{M} ρ^{n−2} { ∑_{α=n+2}^{n+p} ∑_{β=n+2}^{n+p} N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α}) − 2σ̃_{n+1 n+1}^{2}

+ ∑_{α=n+1}^{n+p} ∑_{β=n+1}^{n+p} σ̃_{αβ}^{2} } dv

⩾ −n ∫_{M} ρ^{n−2} H^{2}ρ^{2}dv + nc ∫_{M} ρ^{n}dv − 2 ∫_{M} ρ^{n−2} ρ^{4}dv + ∫_{M} ρ^{n−2} \frac{1}{p}ρ^{4}dv

= ∫_{M} ρ^{n}\{n(c − H^{2}) − ( 2 − \frac{1}{p}) ρ^{2}\}dv,

where the inequality N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α}) ⩾ 0 for any α, β, σ̃_{n+1n+1}^{2} ⩽ ρ^{4} and ∑ _{α,β} σ̃_{αβ}^{2} = ∑ _{α} σ̃_{α}^{2} ⩾ \frac{1}{p}( ∑ _{α} σ̃_{α} )^{2} = \frac{1}{p}ρ^{4} is used.

In particular, if ρ^{2} ⩽ \frac{n}{2− \frac{1}{p}} (c − H^{2}), from (5.1), we see that ρ^{2} = 0 and M is

totally umbilical or ρ^{2} = \frac{n}{2− \frac{1}{p}} (c − H^{2}). In the latter case, we have from (5.1) that

∫_{M} ρ^{n−2} ∑ _{α,β} H^{α} H^{β} σ̃_{αβ} dv = 0, that is

(5.2)  ∫_{M} ρ^{n−2} ∑_{α} (H^{α})^{2} σ̃_{α} dv = 0.

If ρ^{2} = 0, that is M is totally umbilical, otherwise, if ρ^{2} ≠ 0, it follows from (5.2) that ∑ _{α} (H^{α})^{2} σ̃_{α} = 0, thus (H^{α})^{2} σ̃_{α} = 0 for all α. Therefore, we see that σ̃_{α} = 0 for all α (contradicts to ρ^{2} ≠ 0), or H^{α} = 0 for all α. Thus, we have H = 0, that is, M is a compact maximal spacelike submanifold in S_{q}^{n+p} (c), by Cheng and Ishikawa [4, Theorem 1] and Example 3.1, we know that M lies in the totally geodesic spacelike submanifold S^{n+1} (c) of S_{q}^{n+q+1} (c) and is isometric to the Clifford torus S^{k}( \frac{1}{\sqrt{2}}c) × S^{k}( \frac{1}{\sqrt{2}}c).

(2) If p − q > 1, from Lemma 2.2 and (4.16), we have

0 = ∫_{M} ρ^{n−2} ( |∇h|^{2} − \frac{3n^{2}}{n + 2} |∇^{⊥} H⃗|^{2}) dv (5.3)

+ ∫_{M} ρ^{n−2} ( \frac{3n^{2}}{n + 2} − n) |∇^{⊥} H⃗|^{2}dv + (n − 2) ∫_{M} ρ^{n−2} |∇ρ|^{2}dv

+ n ∫_{M} ρ^{n−2} { ∑_{α,β} H^{α} H^{β} σ̃_{αβ} + 2ρ^{2} ∑_{β=n+1}^{n+p−q} (H^{β})^{2} − H^{2}ρ^{2}} dv + nc ∫_{M} ρ^{n}dv

+ ∫_{M} ρ^{n−2} { − ∑_{α,β} N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α}) − ∑_{α,β} σ̃_{αβ}^{2}

+ 2 ∑_{α} ∑_{β=n+p−q+1}^{n+p} N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α}) + 2 ∑_{α} ∑_{β=n+p−q+1}^{n+p} σ̃_{αβ}^{2} } dv

⩾ −n ∫_{M} ρ^{n−2} H^{2}ρ^{2}dv + nc ∫_{M} ρ^{n}dv + ∫_{M} ρ^{n−2} ( − \frac{3}{2}ρ^{4}) dv

= ∫_{M} ρ^{n}\{n(c − H^{2}) − \frac{3}{2}ρ^{2}\}dv,

where the inequality [7]

− ∑_{α,β} N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α}) − ∑_{α,β} σ̃_{αβ}^{2} ⩾ −\frac{3}{2}ρ^{4},

is used.

In particular, if ρ^{2} ⩽ \frac{2n}{3} (c − H^{2}), from (5.3), we see that ρ^{2} = 0 and M is totally umbilical or ρ^{2} = \frac{2n}{3} (c − H^{2}). In the latter case, from (5.3), we also see that (5.2) holds. If ρ^{2} = 0, that is M is totally umbilical, otherwise, if ρ^{2} ≠ 0, it follows from (5.2) that ∑ _{α} (H^{α})^{2} σ̃_{α} = 0. By the same argument as above, we see that H^{α} = 0 and H = 0, that is, M is a compact maximal spacelike submanifold in S_{q}^{n+p} (c), by Cheng and Ishikawa [4, Theorem 1] and Example 3.2, we know that M lies in the totally geodesic spacelike submanifold S^{4}(c) of S_{q}^{4+q} (c) and is isometric to the Veronese surface. This completes the proof of Theorem 1.1. □

Proof of Theorem 1.2. For a fixed α, n + 1 ⩽ α ⩽ n + p, we can take a local orthonormal frame field {e_{1}, …, e_{n}} such that h_{ij}^{α} = λ_{i}^{α} δ_{ij} , then h̃_{ij}^{α} = μ_{i}^{α} δ_{ij} with μ_{i}^{α} = λ_{i}^{α} − H^{α}, ∑ _{i} μ_{i}^{α} = 0. Thus

(5.4)  − ∑_{α,i,j,k,l} h_{ij}^{α} (h_{kl}^{α}R_{lijk} − h_{li}^{α}R_{lkjk} ) = \frac{1}{2} ∑_{α,i,k} (λ_{i}^{α} − λ_{k}^{α})^{2}R_{kiik}

= \frac{1}{2} ∑_{α,i,k} (μ_{i}^{α} − μ_{k}^{α})^{2}R_{kiik} ⩾ nKρ^{2},

where K denotes the infimum of the sectional curvature of M and the equality in (5.4) holds if and only if R_{kiik} = K for any i ≠ k.

Let ∑ _{i}(h̃_{ii}^{β})^{2} = τ_{β}. Then τ_{β} ⩽ ∑ _{i,j} (h̃_{ij}^{β} )^{2} = σ̃_{β} . Since ∑ _{i} h̃_{ii}^{β} = 0, ∑ _{i} μ_{i}^{α} = 0 and ∑ _{i}(μ_{i}^{α})^{2} = σ̃_{α} . We have from Lemma 2.1 that

(5.5)  − ∑_{α,β,i,j,k} H^{α} ε_{β} h̃_{ik}^{α} h̃_{kj}^{β} h̃_{ij}^{β} = − ∑_{α,i,j,k} ∑_{β=n+1}^{n+p−q} H^{α} h̃_{ik}^{α} h̃_{kj}^{β} h̃_{ij}^{β}

+ ∑_{α,i,j,k} ∑_{β=n+p−q+1}^{n+p} H^{α} h̃_{ik}^{α} h̃_{kj}^{β} h̃_{ij}^{β}

= − ∑_{α,i} ∑_{β=n+1}^{n+p−q} H^{α} h̃_{ii}^{α} (μ_{i}^{β})^{2} + ∑_{α,i} ∑_{β=n+p−q+1}^{n+p} H^{α} h̃_{ii}^{α} (μ_{i}^{β})^{2}

⩾ − \frac{n − 2}{\sqrt{n(n − 1)}} ∑_{α} ∑_{β=n+1}^{n+p−q} |H^{α} |σ̃_{β} \sqrt{τ_{α}}

− \frac{n − 2}{\sqrt{n(n − 1)}} ∑_{α} ∑_{β=n+p−q+1}^{n+p} |H^{α} |σ̃_{β} \sqrt{τ_{α}}

= −\frac{n − 2}{\sqrt{n(n − 1)}} ∑_{α} |H^{α} |\sqrt{τ_{α}}( ∑_{β=n+1}^{n+p−q} σ̃_{β} + ∑_{β=n+p−q+1}^{n+p} σ̃_{β} )

⩾ − \frac{n − 2}{\sqrt{n(n − 1)}} ( \sqrt{∑_{α} (H^{α})^{2} ∑_{α} τ̃_{α}} ) ρ^{2} ⩾ −\frac{n − 2}{\sqrt{n(n − 1)}} Hρ^{3}.

190 SHU AND CHEN

From Chen–Do Carmo–Kobayashi [5, Lemma 1], we see that

(5.6) −\frac{1}{2} ∑_{α,β} ε_{β}N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α})

= −\frac{1}{2} ∑_{α} ∑_{β=n+1}^{n+p−q} N(Ã_{α} Ã_{β} −Ã_{β} Ã_{α}) + \frac{1}{2} ∑_{α} ∑_{β=n+p−q+1}^{n+p} N(Ã_{α} Ã_{β} −Ã_{β} Ã_{α})

= −\frac{1}{2} ∑_{α=n+1}^{n+p−q} ∑_{β=n+1}^{n+p−q} N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α})

+ \frac{1}{2} ∑_{α=n+p−q+1}^{n+p}  ∑_{β=n+p−q+1}^{n+p} N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α})
⩾ − \frac{1}{2} ∑_{α=n+1}^{n+p−q} ∑_{β=n+1}^{n+p−q} N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α}) ⩾ − ∑_{α≠β} σ̃_{α} σ̃_{β}

= −( ∑_{α=n+1}^{n+p−q} σ̃_{α} )^{2} + ∑_{α=n+1}^{n+p−q} σ̃_{α}^{2} ⩾ −( ∑_{α=n+1}^{n+p−q} σ̃_{α} )^{2} + \frac{1}{p − q} ( ∑_{α=n+1}^{n+p−q} σ̃_{α} )^{2}

= −( 1 − \frac{1}{p − q} ) ( ∑_{α=n+1}^{n+p−q} σ̃_{α} )^{2} ⩾ −( 1 − \frac{1}{p − q} ) ρ^{4}.

Thus, from (4.13), (4.14), (5.4), (5.5), (5.6) and Lemma 2.2, we have

(5.7) 0 ⩾ ∫_{M} ρ^{n−2} (|∇h|^{2} − \frac{3n^{2}}{n + 2} |∇^{⊥} H⃗|^{2})dv + ∫_{M} ρ^{n−2} (\frac{3n^{2}}{n + 2} − n)|∇^{⊥} H⃗|^{2}dv

+ (n − 2) ∫_{M} ρ^{n−2} |∇ρ|^{2}dv − ∫_{M} ρ^{n−2} \frac{n(n − 2)}{\sqrt{n(n − 1)}} Hρ^{3}dv

+ ∫_{M} ρ^{n−2} ∑_{α,β} nH^{α} H^{β} σ̃_{αβ} + ∫_{M} ρ^{n−2} nKρ^{2}dv

− \frac{1}{2} ∑_{α,β} ε_{β}N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α})

⩾ ∫_{M} ρ^{n}\{ nK − \frac{n(n − 2)}{\sqrt{n(n − 1)}} Hρ − ( 1 − \frac{1}{p − q} ) ρ^{2}\} dv.

In particular, if

K ⩾ \frac{n − 2}{\sqrt{n(n − 1)}} Hρ + \frac{1}{n}( 1 − \frac{1}{p − q} ) ρ^{2},

from (5.7), we see that ρ^{2} = 0 and M is totally umbilical or K = \frac{n−2}{\sqrt{n(n−1)}} Hρ + \frac{1}{n} (1 − \frac{1}{p−q} )ρ^{2}. In the latter case, from (5.7), we know that (5.2) holds. If ρ^{2} = 0, that is M is totally umbilical, otherwise, if ρ^{2} ≠ 0, it follows from (5.2) that ∑ _{α} (H^{α})^{2} σ̃_{α} = 0. By the same argument as in the proof of Theorem 1.1, we see

WILLMORE SPACELIKE SUBMANIFOLDS… 191

that H^{α} = 0 and H = 0. It also follows from (5.7) that |∇h|^{2} = \frac{3n^{2}}{n+2} |∇^{⊥} H⃗|^{2} = 0, that is, the second fundamental form of M is parallel. This completes the proof of Theorem 1.2. □

Proof of Theorem 1.3. From (2.3) and (4.1), we have

R_{kk} = (n − 1)c + (n − 2) ∑_{α} ε_{α} H^{α} h̃_{kk}^{α} + (n − 1) ∑_{α=n+1}^{n+p−q} (H^{α})^{2}

− (n − 1) ∑_{α=n+p−q+1}^{n+p} (H^{α})^{2} − ∑_{i,α=n+1}^{n+p−q} (h̃_{ik}^{α})^{2} + ∑_{i,α=n+p−q+1}^{n+p} (h̃_{ik}^{α})^{2}

⩽ (n − 1)c + (n − 2) ∑_{α} ε_{α} H^{α} h̃_{kk}^{α} + (n − 1)H^{2}

− ∑_{i,α=n+1}^{n+p−q} (h̃_{ik}^{α})^{2} + ∑_{i,α=n+p−q+1}^{n+p} (h̃_{ik}^{α})^{2}.

Thus

nQ ⩽ ∑_{k} R_{kk} = n(n − 1)c + n(n − 1)H^{2} − ∑_{i,k,α=n+1}^{n+p−q} (h̃_{ik}^{α})^{2} + ∑_{i,k,α=n+p−q+1}^{n+p} (h̃_{ik}^{α})^{2}.

From (4.2) and (4.3), we have

(5.8) − ∑_{α=n+1}^{n+p−q} σ̃_{α} + ∑_{α=n+p−q+1}^{n+p} σ̃_{α} ⩾ nQ − n(n − 1)c − n(n − 1)H^{2}.

From (5.8), we see that

(5.9) − ( ∑_{α=n+1}^{n+p−q} σ̃_{α} ) ^{2} + \frac{1}{q}( ∑_{α=n+p−q+1}^{n+p}  σ̃_{α} ) ^{2}
= −( ∑_{α=n+1}^{n+p−q} σ̃_{α} ) ^{2} + ( ∑_{α=n+p−q+1}^{n+p}  σ̃_{α} ) ^{2} + ( \frac{1}{q} − 1) ( ∑_{α=n+p−q+1}^{n+p}  σ̃_{α} ) ^{2}
⩾ ( − ∑_{α=n+1}^{n+p−q} σ̃_{α} + ∑_{α=n+p−q+1}^{n+p}  σ̃_{α} ) ( ∑_{α=n+1}^{n+p−q} σ̃_{α} + ∑_{α=n+p−q+1}^{n+p}  σ̃_{α} ) − ( 1 − \frac{1}{q}) ρ^{4}

⩾ (nQ − n(n − 1)c − n(n − 1)H^{2})ρ^{2} − ( 1 − \frac{1}{q}) ρ^{4}.

By Chen–Do Carmo–Kobayashi [5, Lemma 1], we also see that

(5.10)  − ∑_{α=n+1}^{n+p−q} ∑_{β=n+1}^{n+p−q} N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α}) ⩾ −2( 1 − \frac{1}{p − q} ) ρ^{4}.

From Lemma 2.2, (4.3), (4.16), (5.9) and (5.10), we have

0 = ∫_{M} ρ^{n−2} (|∇h|^{2} − \frac{3n^{2}}{n + 2} |∇^{⊥} H⃗|^{2})dv (5.11)

+ ∫_{M} ρ^{n−2} ( \frac{3n^{2}}{n + 2} − n) |∇^{⊥} H⃗|^{2}dv + (n − 2) ∫_{M} ρ^{n−2} |∇ρ|^{2}dv

+ n ∫_{M} ρ^{n−2} { ∑_{α,β} H^{α} H^{β} σ̃_{αβ} + 2ρ^{2} ∑_{β=n+1}^{n+p−q} (H^{β})^{2} − H^{2}ρ^{2}} dv + nc ∫_{M} ρ^{n}dv

+ ∫_{M} ρ^{n−2} { − ∑_{α=n+1}^{n+p−q} ∑_{β=n+1}^{n+p−q} N(Ã_{α} Ã_{β} − Ã_{β} Ã_{α}) − ∑_{α=n+1}^{n+p−q} σ̃_{α}^{2} + ∑_{α=n+p−q+1}^{n+p} σ̃_{α}^{2} } dv

⩾ − n ∫_{M} ρ^{n−2} H^{2}ρ^{2}dv + nc ∫_{M} ρ^{n}dv + ∫_{M} ρ^{n−2} \{ − 2( 1 − \frac{1}{p − q} ) ρ^{4}\}dv

+ ∫_{M} ρ^{n−2} \{ − ( ∑_{α=n+1}^{n+p−q} σ̃_{α} ) ^{2} + \frac{1}{q}( ∑_{α=n+p−q+1}^{n+p}  σ̃_{α} ) ^{2}\} dv

⩾ − n ∫_{M} ρ^{n−2} H^{2}ρ^{2}dv + nc ∫_{M} ρ^{n}dv + ∫_{M} ρ^{n−2} \{ − 2( 1 − \frac{1}{p − q} ) ρ^{4}\}dv

+ ∫_{M} ρ^{n−2} \{(nQ − n(n − 1)c − n(n − 1)H^{2})ρ^{2} − ( 1 − \frac{1}{q}) ρ^{4}\}dv

= ∫_{M} nρ^{n}\{Q − (n − 2)c − nH^{2} − \frac{1}{n}( 3 − \frac{p + q}{(p − q)q} ) ρ^{2}\}dv.

Q ⩾ (n − 2)c + nH^{2} + \frac{1}{n}( 3 − \frac{p + q}{(p − q)q} ) ρ^{2},

from (5.11), we see that ρ^{2} = 0 and M is totally umbilical or Q = (n − 2)c + nH^{2} + \frac{1}{n} (3− \frac{p+q}{(p−q)q} )ρ^{2}. In the latter case, from (5.11), we know that (5.2) holds. If ρ^{2} = 0, that is M is totally umbilical, otherwise, if ρ^{2} ≠ 0, it follows from (5.2) that ∑ _{α} (H^{α})^{2} σ̃_{α} = 0. By the same argument as in the proof of Theorem 1.1, we see that H^{α} = 0 and H = 0. It also follows from (5.11) that |∇h|^{2} = \frac{3n^{2}}{n+2} |∇^{⊥} H⃗|^{2} = 0, that is, the second fundamental form of M is parallel. This completes the proof of Theorem 1.3. □

Acknowledgment. Project supported by NSF of Shaanxi Province (SJ08A31) and NSF of Shaanxi Education Department (11JK0479)

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School of Mathematics and Information Science (Received 10 09 2015) Xianyang Normal University (Revised 12 07 2016) Xianyang P.R. China

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