DOI: 10.1515/auom-2015-0040
An. Şt. Univ. Ovidius Constanţa
Vol. 23(2),2015, 241–257
Spectral characterization and Schrödinger
operator of space-like submanifolds
Shichang Shu and Tianmin Zhu
Abstract
In this paper, we would like to study space-like submanifolds in a de
Sitter spaces Spn+p (1). We define and discuss three Schrödinger operators LH , LR , LR/H and obtain some spectral characterizations of totally
umbilical space-like submanifolds in terms of the first eigenvalue of the
Schrödinger operators LH , LR and LR/H respectively.
1
Introduction
Let Mpn+p (c) be an (n + p)-dimensional connected semi-Riemannian manifold
of constant curvature c whose index is p. It is called an indefinite space form
of index p and simply a space form when p = 0. If c > 0, we call it a de Sitter
space of index p, and denote it by Spn+p (c). A submanifold in a de Sitter space
is said to be space-like if the induced metric on the submanifold is positive
definite. It was pointed out by Marsden and Tipler [11] and Stumbles [16] that
space-like hypersurfaces with constant mean curvature in arbitrary space-time
get interested in the relativity theory. Therefore, space-like hypersurfaces in a
de Sitter space have recently been investigated by many mathematicians from
both physics and mathematical points of view, see for instance [1, 4, 10, 12].
We know that hypersurfaces with constant mean curvature in a Riemannian manifold M n+1 (c) of constant sectional curvature c are critical points of
Key Words: Space-like submanifold, Schrödinger operator, mean curvature, scalar curvature, first eigenvalue.
2010 Mathematics Subject Classification: Primary: 53C42; Secondary: 53A10.
Received: November, 2012.
Revised: September, 2014.
Accepted: September, 2014.
241
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
242
the area functional under variations that keep constant a certain volume function. Barbosa, do Carmo and Eschenburg [3] studied the stability for hypersurfaces of constant mean curvature in Riemannian manifolds. In analogy with
the case of constant mean curvature, questions of stability can be considered
for hypersurfaces with constant scalar curvature. In [2], Alencar, do Carmo
and Colares extended the study of stability to hypersurfaces with constant
scalar curvature. As researched in C. Wu [17] for minimal submanifolds in a
unit sphere, A.A. Barros et al. [6] and Cheng [8] studied the first eigenvalues
of some Schrödinger operators of submanifolds with parallel mean curvature
vector or hypersurfaces with constant scalar curvature in a unit sphere and
obtained some spectral characterizations of so √
called Veronese surface, Clifford
torus or Riemannian product S m (r) × S n−m ( 1 − r2 ), 1 ≤ m ≤ n − 1.
In connection with the stability for hypersurfaces with constant mean curvature or constant scalar curvature in Riemannian manifolds, Barbosa-Oliker
[4] and Liu-Deng [10] studied the stability for space-like hypersurfaces with
constant mean curvature or constant scalar curvature in Lorentz manifolds.
From Barbosa-Oliker [4] and [5], we know that constant mean curvature spacelike hypersurfaces are solutions to a variational problem. In fact, they are critical points of the area functional for variations that leave constant a certain
volume function.
In the present paper, we would like to study space-like submanifolds in a de
Sitter spaces Spn+p (1). We will define and discuss three Schrödinger operators
LH , LR , LR/H and obtain some spectral characterizations of totally umbilical
space-like submanifolds in terms of the first eigenvalue of the Schrödinger
operators LH , LR and LR/H respectively.
Take an immersion ψ : M n → Spn+p (1) and choose a suitable pseudoorthonormal frame field {e1 , . . . , en , en+1 , . . . , en+p } adapted to the immersion
ψ and its associated coframe {ω1 , . . . , ωn , ωn+1 , . . . , ωn+p }. Now recall the
symmetric tracelessPtensor introduced by Cheng and Yau in [9], which may
α
be given by φ =
i,j,α φij ωi ⊗ ωj ⊗ ωα , where i, j = 1, . . . , n, α = n +
1
α
α
α
1, . . . , n + p, φij = hij − n trH α δij , H α = (hα
ij ) and hij are the coefficients
of the P
second fundamental form in the direction eα . It is easy to verify that
2
2
2
2
|φ|2 = i,j,α (φα
ij ) = |h| − nH , where |h| denotes the squared norm of the
second fundamental form of M n . We know that |φ|2 ≡ 0 if and only if M n is
totally umbilical.
Before announcing our main results, we introduce three Schrödinger operators:
n(n − 2)
1
H|φ|,
(1.1)
LH = −∆ + |φ|2 − p
p
n(n − 1)
LR = − +
1 − H2 2
1
|φ|4 +
|φ| ,
npH
H
(1.2)
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
LR/H = −L +
1
1 − H2 2
|φ|4 +
|φ| ,
npH
H
243
(1.3)
where ∆ is the Laplace-Beltrami operator, is the differential operator f =
P
n
n+1
i,j=1 (nHδij − hij )fij and L = + (k/2n)∆ is a differential operator
defined by the linear combination of ∆ and .
Now we state the spectral characterizations of totally umbilical space-like
submanifolds in terms of the first eigenvalue of the Schrödinger operators LH ,
LR and LR/H as follows:
Theorem 1.1. Let M n be an n-dimensional compact orientable spacelike submanifold in an (n+p)-dimensional de Sitter space Spn+p (1) with nonzero
H
the first eigenvalue of the
parallel mean curvature vector field. Denote by λL
1
2
H
Schrödinger operator LH . If λL
≥
−n(1
−
H
),
then
M n is totally umbilical.
1
Theorem 1.2. Let M n be an n-dimensional compact orientable space-like
submanifold in an (n+p)-dimensional de Sitter space Spn+p (1) with constant
scalar curvature n(n − 1)R, R < 1 and parallel normalized mean curvature
R
vector field. Denote by λL
the first eigenvalue of the Schrödinger operator
1
n−2
R
R
√
LR . Then λL
≤
max
|φ|3 , and λL
= √ n−2 max |φ|3 if and only
1
1
n(n−1)
n(n−1)
if M n is totally umbilical.
Theorem 1.3. Let M n be an n-dimensional compact orientable space-like
submanifold in an (n+p)-dimensional de Sitter space Spn+p (1) with parallel
normalized mean curvature vector field and let the scalar curvature n(n − 1)R
be proportional to the mean curvature H of M n , that is, there exists a conL
stant k such that n(n − 1)R = kH. Denote by λ1 R/H the first eigenvalue
L
of the Schrödinger operator LR/H . Then λ1 R/H ≤ √ n−2 max |φ|3 , and
n(n−1)
L
λ1 R/H
= √ n−2
3
n(n−1)
max |φ| if and only if M
n
is totally umbilical.
Corollary 1.4. Let M n be an n-dimensional compact space-like hypersurface in an (n + 1)-dimensional de Sitter space S1n+1 (1) and let the scalar curvature n(n−1)R be proportional to the mean curvature H of M n , that is, there
L
exists a constant k such that n(n−1)R = kH. Denote by λ1 R/H the first eigenLR/H
value of the Schrödinger operator LR/H . Then λ1
≤ √ n−2 max |φ|3 ,
n(n−1)
and
L
λ1 R/H
= √ n−2
n(n−1)
3
max |φ| if and only if M
n
is totally umbilical.
Remark 1.5. If p = 1, then Theorem 1.1 and Theorem 1.2 reduce to
Theorem 1 and Theorem 2 of [15], respectively. Therefore, we generalize the
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SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
previous results obtained by [15] to general submanifolds with higher codimension.
2
Preliminaries
Let Spn+p (1) be an (n + p)-dimensional de Sitter space with index p and constant curvature 1 . Let M n be an n-dimensional connected space-like submanifold immersed in Spn+p (1). We choose a local field of semi-Riemannian
orthonormal frames e1 , . . . , en+p in Spn+p (1) such that at each point of M n ,
e1 , . . . , en span the tangent space of M n 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 ω1 , . . . , ωn+p be its dual frame
so that the
metric
P semi-Riemannian
P 2fieldP
2
, where εi = 1 and
ωi −
ωα2 =
ε A ωA
of Spn+p (1) is given by ds2 =
α
i
A
εα = −1. Then the structure equations of Spn+p (1) are given by
X
dωA =
εB ωAB ∧ ωB , ωAB + ωBA = 0,
(2.1)
B
dωAB =
X
εC ωAC ∧ ωCB −
C
1X
εC εD KABCD ωC ∧ ωD ,
2
(2.2)
C,D
KABCD = εA εB (δAC δBD − δAD δBC ).
(2.3)
If we restrict these form to M n , then
n + 1 ≤ α ≤ n + p.
ωα = 0,
From Cartan’s Lemma we have
X
ωαi =
hα
ij ωj ,
α
hα
ij = hji .
(2.4)
(2.5)
j
The connection forms of M n are characterized by the structure equations
dωi =
n
X
ωij ∧ ωj , ωij + ωji = 0,
(2.6)
j=1
dωij =
X
k
ωik ∧ ωkj −
1X
Rijkl ωk ∧ ωl ,
2
k,l
(2.7)
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
Rijkl = (δik δjl − δil δjk ) −
X
α
α α
(hα
ik hjl − hil hjk ),
245
(2.8)
α
where Rijkl are the components of the curvature tensor of M n .
Denote by h the second fundamental form of M n . Then
X
h=
hα
ij ωi ⊗ ωj ⊗ eα .
(2.9)
i,j,α
Denote by ξ, H and |h|2 the mean curvature vector field, the mean curvature and the squared norm of the second fundamental form of M n . Then they
are defined by
s
X
1 XX α 2
1XX α
2
(
hii )eα , H = |ξ| =
(
hii ) , |h|2 =
(hα
ξ=
ij ) . (2.10)
n α i
n
α
i
i,j,α
Moreover, the normal curvature tensor Rαβkl , the Ricci curvature tensor
Rik and the scalar curvature n(n − 1)R are expressed as
X
β
β
α
Rαβkl =
(hα
(2.11)
km hml − hlm hmk ),
m
Rik = (n − 1)δik −
XX
X
α
α
(
hα
hα
ll )hik +
ij hjk ,
α
(2.12)
α,j
l
n(n − 1)R = n(n − 1) + |h|2 − n2 H 2 ,
(2.13)
where R is the normalized scalar curvature.
α
Define the first and the second covariant derivatives of hα
ij , say hijk and
α
hijkl by
X
α
hα
ijk ωk = dhij +
k
X
l
α
hα
ijkl ωl = dhijk +
X
hα
ik ωkj +
k
X
m
X
hα
jk ωki −
X
k
hα
mjk ωmi +
X
hα
imk ωmj +
m
hβij ωβα ,
(2.14)
β
X
hα
ijm ωmk −
X
m
hβijk ωβα .
β
(2.15)
We obtain the Codazzi equation by straightforward computations
α
hα
ijk = hikj .
(2.16)
It follows that the Ricci identities hold
X
X
X β
α
hα
hα
hα
hij Rαβkl .
ijkl − hijlk =
im Rmjkl +
jm Rmikl +
m
m
β
(2.17)
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
α
The Laplacian of hα
ij is defined by ∆hij =
P
k
hα
ijkk . From (2.17) we obtain
for any α, n + 1 ≤ α ≤ n + p,
X
X
X
X β
∆hα
hα
hα
hα
hik Rαβjk .
ij =
kkij +
km Rmijk +
im Rmkjk +
k
k,m
246
k,m
(2.18)
k,β
In the case of the mean curvature vector ξ has no zero, we know that ξ/H is
a normal vector field defined globally on M n . We define |µ|2 and |τ |2 by
X
X X
2
|µ|2 =
(hn+1
− Hδij )2 , |τ |2 =
(hα
(2.19)
ij ) ,
ij
α>n+1 i,j
i,j
respectively. Then |µ|2 and |τ |2 are functions defined on M n globally, which
do not depend on the choice of the orthonormal frame {e1 , . . . , en }. We have
|h|2 = nH 2 + |µ|2 + |τ |2 .
(2.20)
Since the normalized mean curvature vector field is parallel, we choose en+1 =
ξ/H, then
X
X
trH n+1 =
hn+1
= nH, trH α =
hα
(2.21)
ii = 0 (α ≥ n + 2).
ii
i
i
From (2.8), (2.11), (2.18) and (2.21), by a direct calculation we have (see [7])
X
X
1
2
2
2
∆|h|2 =
(hα
hn+1
ijk ) +
ij (nH)ij + nc(|h| − nH )
2
i,j
i,j,k,α
X
X
2
− nH
tr(Hα Hn+1 ) +
[tr(Hα Hβ )]2
α
+
X
(2.22)
α,β
N (Hα Hβ − Hβ Hα ),
α,β
t
where Hα denote the matrix (hα
ij ) for all α, N (A) = tr(AA ), for all matrix
A = (aij ).
We need the following Lemma
Lemma 2.1 ([13]). Let A, B be symmetric n × n matrices satisfying
AB = BA and trA = trB = 0. Then
|trA2 B| ≤ p
n−2
n(n − 1)
(trA2 )(trB 2 )1/2 .
(2.23)
and the equality holds if and only if (n − 1) of the eigenvalues xp
i of B and
the corresponding eigenvalues yi of A satisfy |xi | = (trB 2 )1/2 / n(n − 1),
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
247
p
xi xj ≥ 0, yi = (trA2 )1/2 / n(n − 1).
We consider a differential operator acting on function f defined by
X
f =
(nHδij − hn+1
(2.24)
ij )fij ,
i,j=1
where df =
P
i
fi ωi ,
P
fij ωj = dfi +
i,j
P
fj ωji . From Cheng and Yau [9], we
j
know that the operator is self-adjoint if M is compact.
We can prove that is a elliptic operator. In fact, we choose a local
orthonormal frame field {e1 , . . . , en } such that hn+1
= λi δij . Since R < 1,
ij
from (2.13), we have |h|2 < n2 H 2 . If there is one i such that nH − λi ≤ 0,
then n2 H 2 ≤ λ2i ≤ |h|2 , this is a contradiction. Thus, we have nH − λi > 0
for any i and the operator is elliptic.
From Proposition 3.6 in Section 3, we know that the differential operator
L = +(k/2n)∆ is elliptic. We also know that the Laplace-Beltrami operator
L
LR
H
∆ is always elliptic. Let λL
and λ1 R/H be the first eigenvalues of the
1 , λ1
Schrödinger operators LH , LR and LR/H respectively. Since ∆, and L are
elliptic operators, from (1.1), (1.2) and (1.3), we know that LH , LR and LR/H
LR
H
are elliptic operators. We can use the min-max characterization of λL
1 , λ1
LR/H
and λ1
, as
R
f LH (f )dv
LH
∞
n
MRn
; f ∈ C (M ), f 6≡ 0 ,
(2.25)
λ1 = min
f 2 dv
Mn
R
f LR (f )dv
∞
n
MRn
R
λL
;
f
∈
C
(M
),
f
≡
6
0
,
(2.26)
=
min
1
f 2 dv
Mn
R
f LR/H (f )dv
LR/H
∞
n
M nR
λ1
; f ∈ C (M ), f 6≡ 0 .
(2.27)
= min
f 2 dv
Mn
3
Proof of Theorems
We firstly prove the following:
Proposition 3.1. Let M n be an n-dimensional space-like submanifold in
an (n + p)-dimensional de Sitter space Spn+p (1). Suppose that the normalized
248
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
mean curvature vector field is parallel. Then
1
(nH) ≥ − n(n − 1)∆R + (|∇h|2 − n2 |∇H|2 )
2
1
n(n − 2)
H|φ| + |φ|2 }.
+ |φ|2 {n − nH 2 − p
p
n(n − 1)
(3.1)
Proof. By a simple calculation and from (2.13), we obtain
X
(nH) =
(nHδij − hn+1
ij )(nH)ij
(3.2)
i,j
X
1
= ∆(n2 H 2 ) − n2 |∇H|2 −
hn+1
ij (nH)ij
2
i,j
X
1
1
hn+1
= − n(n − 1)∆R + ∆|h|2 − n2 |∇H|2 −
ij (nH)ij .
2
2
i,j
1
α
α
Set φα
ij = hij − n trH δij and consider the symmetric tensor φ =
P
i,j,α
φα
ij ωi ωj eα .
We easy know that φ is traceless and
N (Φα ) = N (Hα ) −
X
1
(trHα )2 , |φ|2 =
N (Φα ) = |h|2 − nH 2 ,
n
α
(3.3)
where Φα denotes the matrix (φα
ij ).
Since the normalized mean curvature vector field is parallel, choosing
en+1 = ξ/H, from (2.21), we infer that
α
φn+1
= hn+1
− Hδij , φα
ij = hij , (α ≥ n + 2),
ij
ij
N (Φn+1 ) = N (Hn+1 ) − nH 2 , N (Φα ) = N (Hα ), (α ≥ n + 2),
3
3
(3.4)
3
tr(Hn+1 ) = tr(Φn+1 ) + 3HN (Φn+1 ) + nH .
From (2.22), (3.3) and (3.4), we have
X
X
1
2
2
2
∆|h|2 ≥
(hα
hn+1
ijk ) +
ij (nH)ij + n(c − H )|φ|
2
i,j
i,j,k,α
X
X
2
− nH
tr(Φα Φn+1 ) +
[tr(Φα Φβ )]2 .
α
(3.5)
α,β
Since we choose en+1 = ξ/H, we have P
ωαn+1 = 0 for P
all α. Consequently
n+1
n+1
Rαn+1jk = 0, from (2.11), we have
hα
=
hα
ij hik
ik hij , that is,
i
i
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
249
Hα Hn+1 = Hn+1 Hα . Thus Φα Φn+1 = Φn+1 Φα . Since matrices Φα and
Φn+1 are traceless, by Lemma 2.1, we have
X
tr(Φ2α Φn+1 ) ≤ p
α
n−2
|µ||φ|2 ≤ p
|φ|3 ,
n(n − 1)
n(n − 1)
n−2
(3.6)
where the following
|µ|2 ≤ |h|2 − nH 2 = |φ|2 ,
(3.7)
is used. By the Cauchy–Schwarz inequality, we have
X
[tr(Φα Φβ )]2 ≥
X
[N (Φα )]2 ≥
α
α,β
1 4
|φ| ,
p
(3.8)
From (3.5), (3.6) and (3.8), we have
X
X
1
2
∆|h|2 ≥
(hα
hn+1
ijk ) +
ij (nH)ij
2
i,j
(3.9)
i,j,k,α
n(n − 2)
1
+ |φ|2 {n − nH 2 − p
H|φ| + |φ|2 }.
p
n(n − 1)
From (3.2) and (3.9), we see that (3.1) is true.
We state a Proposition which can be proved by making use of the similar
method due to C. Wu [17] or A.A. Barros et al. [6] for Riemannian manifold.
Proposition 3.2. Let M n be an n-dimensional space-like submanifold in
an (n+p)-dimensional de Sitter space Spn+p (1). Then there holds the following
|∇|φ|2 |2 ≤
4n|φ|2
|∇φ|2 .
n+2
(3.10)
From the author [14], we also have the following:
Proposition 3.3. Let M n be an n-dimensional space-like submanifold in
an (n + p)-dimensional de Sitter space Spn+p (1). Suppose that the normalized
scalar curvature R is constant and R ≤ 1. Then
|∇h|2 ≥ n2 |∇H|2 .
(3.11)
Remark 3.4. If |∇h|2 = n2 |∇H|2 and R < 1, from (2.13) and the proof
of Proposition 3.3 (see [14]), we easily see that
0 ≤ n3 (n − 1)(1 − R)|∇H|2 ≤ S(|∇h|2 − n2 |∇H|2 ).
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
250
Thus, we have ∇H = 0 and H is constant.
Proof of theorem 1.1 From the definition of (2.10), we know that
H ≥p
0. For every ε > 0, from (2.25), we can introduce a smooth function
H
fε = ε + |φ|2 as the test function to estimate λL
1 . We easily have
1
1
∆fε = p
∆|φ|2 −
|∇|φ|2 |2 .
2 )3/2
2
4(ε
+
|φ|
2 ε + |φ|
(3.12)
Since we know that submanifolds with nonzero parallel mean curvature vector
field also have parallel normalized mean curvature vector field and H is nonzero
constant, from (3.9), we easily see that
n(n − 2)
1
1
∆|φ|2 ≥ |φ|2 {n − nH 2 − p
H|φ| + |φ|2 }.
2
p
n(n − 1)
By Proposition 3.2, (3.12) and (3.13), we have
1
1
fε ∆fε = ∆|φ|2 −
|∇|φ|2 |2
2
4(ε + |φ|2 )
n(n − 2)H
1
|φ| + n(1 − H 2 )}
≥|φ|2 { |φ|2 − p
p
n(n − 1)
1
+ |∇φ|2 −
|∇|φ|2 |2
4(ε + |φ|2 )
n(n − 2)H
1
≥|φ|2 { |φ|2 − p
|φ| + n(1 − H 2 )}
p
n(n − 1)
+ {1 −
n|φ|2
}|∇φ|2 .
(n + 2)(ε + |φ|2 )
Therefore, we have
1
n(n − 2)
H|φ|}fε2
fε LH fε = − fε ∆fε + { |φ|2 − p
p
n(n − 1)
1
n(n − 2)H
≤ − |φ|2 { |φ|2 − p
|φ| + n(1 − H 2 )}
p
n(n − 1)
n|φ|2
}|∇φ|2
(n + 2)(ε + |φ|2 )
1
n(n − 2)
+ { |φ|2 − p
H|φ|}(ε + |φ|2 ).
p
n(n − 1)
− {1 −
(3.13)
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
251
Using (2.25) with fε as a test function, we have
Z
Z
Z
LH
2
2
H
λL
(ε+|φ|
)dv
=
λ
f
dv
≤
fε LH (fε )dv
(3.14)
ε
1
1
Mn
Mn
Mn
Z
1
n(n − 2)H
≤−
|φ|2 { |φ|2 − p
|φ| + n(1 − H 2 )}dv
p
n(n − 1)
Mn
Z
n|φ|2
−
{1 −
}|∇φ|2 dv
2)
(n
+
2)(ε
+
|φ|
n
ZM
n(n − 2)
1
H|φ|}(ε + |φ|2 )dv.
+
{ |φ|2 − p
n(n − 1)
Mn p
Letting ε → ∞ in (3.14), we get
Z
Z
2
2
H
λL
|φ|
dv
≤
−n(1
−
H
)
1
Mn
Mn
|φ|2 dv −
Z
Mn
2
|∇φ|2 dv.
n+2
(3.15)
H
Since λL
≥ −n(1 − H 2 ), from (3.15), we have |∇φ|2 = 0. Proposition
1
3.2 implies that ∇|φ|2 = 0, that is, |φ|2 is constant. Therefore, we know
H
that p1 |φ|2 − √n(n−2) H|φ| is constant. From (1.1), we obtain that λL
=
1
n(n−1)
1
2
p |φ|
n(n−2)
−√
n(n−1)
H|φ|. So we have
−n(1 − H 2 ) ≤
that is
1 2
n(n − 2)
|φ| − p
H|φ|,
p
n(n − 1)
n(n − 2)
1 2
|φ| − p
H|φ| + n(1 − H 2 ) ≥ 0.
p
n(n − 1)
Therefore, we know that the equalities in (3.13), (3.8), (3.7) and (2.23) hold
and
1
n(n − 2)
|φ|2 { |φ|2 − p
H|φ| + n(1 − H 2 )} = 0.
p
n(n − 1)
This implies that |φ|2 = 0, that is, M n is totally umbilical, or
1 2
n(n − 2)
|φ| − p
H|φ| + n(1 − H 2 ) = 0.
p
n(n − 1)
In the second case, from the equalities in (3.8) and (3.7), we have |µ|2 = |φ|2
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
and p
P
252
[tr(Φα Φβ )]2 = |φ|4 . Thus, we have
α,β
|φ|4 =p{(trΦ2n+1 )2 +
X
[tr(Φα Φn+1 )]2
α>n+1
X
+
X
[tr(Φn+1 Φβ )]2 +
β>n+1
[tr(Φα Φβ )]2 }.
α,β>n+1
From (3.4), we have
(p − 1)|φ|4 + p
X
[tr(Φα Φn+1 )]2
α>n+1
+p
X
[tr(Φn+1 Φβ )]2 + p
β>n+1
X
[tr(Φα Φβ )]2 = 0.
α,β>n+1
Since p ≥ 1 and all parts of the above equality are nonnegative, we have
(p − 1)|φ|4 = 0. Thus, |φ|2 = 0 and M n is totally umbilical, or p = 1. If p = 1,
we know that M n is an n-dimensional compact orientable space-like hypersurface in an (n+1)-dimensional de Sitter space S1n+1 (1) with constant mean
curvature H. From the equalities in (2.23), we infer that M n has at most two
distinct constant principal curvatures. We conclude that M n is totally umbilical from the compactness of M n . This completes the proof of Theorem 1.1.
Proof of theorem 1.2 Since n(n − 1)R is constant and R < 1, by
Proposition 3.1 and Proposition 3.3, we have
1
n(n − 2)
(nH) ≥ |φ|2 {n − nH 2 − p
H|φ| + |φ|2 }.
p
n(n − 1)
(3.16)
From the assertion in Section 2, we know that the operator is elliptic. Since
we assume that the normalized mean curvature vector en+1 = ξ/H is parallel,
we have H > 0. Thus, from (2.26), we introduce a smooth function f = H as
R
the test function to estimate λL
1 . By (1.2) and (3.16), we have
1
|φ|4 + (1 − H 2 )|φ|2
np
1
n−2
≤−
|φ|4 − (1 − H 2 )|φ|2 + p
H|φ|3
np
n(n − 1)
1
n−2
+
|φ|4 + (1 − H 2 )|φ|2 = p
H|φ|3 .
np
n(n − 1)
LR (H) = − (H) +
(3.17)
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
From (2.26) and (3.17), we have
Z
Z
2
R
λL
H
dv
≤
HLR (H)dv
1
n
Mn
ZM
n−2
p
≤
H 2 |φ|3 dv
n
n(n − 1)
M
Z
n−2
3
≤p
max |φ|
H 2 dv.
n(n − 1)
Mn
Thus, we have
R
√ n−2
If λL
1 =
n(n−1)
253
(3.18)
n−2
R
λL
max |φ|3 .
1 ≤ p
n(n − 1)
max |φ|3 , then the equalities in (3.18), (3.17), (3.16), (3.11),
(3.8), (3.7) and (2.23) hold. Since the operator is self-adjoint and M n is
compact, from the equality of (3.16), we obtain that
Z
1
n(n − 2)H
|φ|2 { |φ|2 − p
|φ| + n(1 − H 2 )}dv = 0.
p
n(n − 1)
Mn
This implies that |φ|2 = 0 and M n is totally umbilical, or
1 2 n(n − 2)H
|φ| − p
|φ| + n(1 − H 2 ) = 0.
p
n(n − 1)
(3.19)
In the second case, since R < 1 and the equality holds in (3.11), from the
Remark 3.4, we know that H is constant. From the equalities of (3.8), (3.7)
and (2.23), by the same assertion in the proof of Theorem 1.1, we know that
M n is an n-dimensional compact orientable space-like hypersurface in de Sitter
space S1n+1 (1) with at most two distinct constant principal curvatures. We
conclude that M n is totally umbilical from the compactness of M n . If M n is
1
R
totally umbilical, that is |φ| = 0, from (1.2), we know that λL
|φ|4 +
= npH
1
1−H 2
2
H |φ|
= 0. This completes the proof of Theorem 1.2.
We may also prove the following:
Proposition 3.5. Let M n be an n-dimensional space-like submanifold
in an (n + p)-dimensional de Sitter space Spn+p (1). If the scalar curvature
n(n − 1)R is proportional to the mean curvature H of M n , that is, there exists
a constant k such that n(n − 1)R = kH, then we have
|∇h|2 ≥ n2 |∇H|2 .
(3.20)
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
254
In particular, if |∇h|2 = n2 |∇H|2 , then H is constant.
Proof. For a fixed α, we choose a orthonormal frame P
field {ei } at each
α
2
2
point on M n so that hα
=
λ
δ
.
Then
we
have
|h|
=
(hα
ij
i ij
ij ) 6= 0. In
i,j,α
P
2
n
α
fact, if |h|2 = (λα
i ) = 0 at a point of M , then λi = 0 for all i and α at
i,α
this point. This implies H = 0 and R = 0 at this point. From (2.13), we have
n(n − 1) = 0. This is impossible. From (2.13) and n(n − 1)R = kH, we have
X
α
k∇i H = −2n2 H∇i H + 2
hα
kj hkji ,
j,k,α
(
X X
X
X
k
α 2
2
2
2
2
+ n2 H)2 |∇H|2 =
(
hα
(hα
(hα
kj hkji ) ≤
ij )
ijk ) = |h| |∇h| .
2
i
i,j,α
j,k,α
i,j,k,α
Thus, we have
k
1
+ n2 H)2 − n2 |h|2 ]|∇H|2 2
2
|h|
(k)2
1
=[
+ n3 (n − 1)]|∇H|2 2 ≥ 0.
4
|h|
|∇h|2 − n2 |∇H|2 ≥[(
If |∇h|2 = n2 |∇H|2 , we easily see that ∇H = 0 and H is constant. The proof
of Proposition 3.5 is completed.
Proposition 3.6. Let M n be an n-dimensional space-like submanifold in
a de Sitter space Spn+p (1). If n(n − 1)R = kH and H > 0, then the differential
operator L = + (k/2n)∆ is elliptic.
Proof. For a fixed α, we choose a local orthonormal frame fieldP
{e1 , . . . , en }
α
at each point on M n so that hα
=
λ
δ
.
From
H
>
0,
nH
=
hn+1
and
ij
i ij
ii
i
P α
hii = 0 for n + 2 ≤ α ≤ n + p on M n , we have, for any i
i
(nH − λn+1
+ k/2n) =
i
X
λn+1
− λn+1
j
i
j
X
2
2 2
+ (1/2)[ (λα
j ) − n H + n(n − 1)]/(nH)
j,α
≥
X
λn+1
j
− λn+1
i
j
X
X
+ (1/2)[ (λjn+1 )2 − (
λn+1
)2 + n(n − 1)]/(nH)
j
j
j
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
255
X
X
=[(
λn+1
)2 − λn+1
(
λn+1
)
j
i
j
j
j
− (1/2)
X
λn+1
λn+1
j
l
+ (1/2)n(n − 1)](nH)−1
l6=j
X
X
=[ (λn+1
)2 + (1/2)
λn+1
λn+1
j
j
l
j
−
l6=j
X
λn+1
(
i
λn+1
)
j
+ (1/2)n(n − 1)](nH)−1
j
X
X
λn+1
=[ (λn+1
)2 + (1/2)
λn+1
+ (1/2)n(n − 1)](nH)−1
j
j
l
j6=i
l6=j
l,j6=i
X
X
=(1/2)[ (λn+1
)2 + (
λn+1
)2 + n(n − 1)](nH)−1 > 0.
j
j
j6=i
j6=i
Thus, L is an elliptic operator. The proof of Lemma 3.6 is completed.
Proof of theorem 1.3 Since n(n − 1)R = kH, by Proposition 3.1 and
Proposition 3.5, we have
1
n(n − 2)
H|φ| + |φ|2 }.
L(nH) ≥ |φ|2 {n − nH 2 − p
p
n(n − 1)
(3.21)
From Proposition 3.6, we know that the operator L is elliptic. Since we assume
that the normalized mean curvature vector en+1 = ξ/H is parallel, we have
H > 0. Thus, from (2.27), we introduce a smooth function f = H as the test
L
function to estimate λ1 R/H . By (1.3) and (3.21), we have
1
|φ|4 + (1 − H 2 )|φ|2
np
1
n−2
≤−
|φ|4 − (1 − H 2 )|φ|2 + p
H|φ|3
np
n(n − 1)
n−2
1
|φ|4 + (1 − H 2 )|φ|2 = p
+
H|φ|3 .
np
n(n − 1)
LR/H (H) = − L(H) +
(3.22)
Thus
L
λ1 R/H
Z
H 2 dv ≤
Mn
Z
HLR/H (H)dv
ZM
n
n−2
H 2 |φ|3 dv
n(n − 1)
Z
n−2
3
≤p
max |φ|
H 2 dv.
n(n − 1)
Mn
≤
Mn
p
(3.23)
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
L
We have λ1 R/H ≤ √ n−2
n(n−1)
L
max |φ|3 . If λ1 R/H = √ n−2
n(n−1)
256
max |φ|3 , then the
equalities in (3.23), (3.22), (3.21), (3.20), (3.8), (3.7) and (2.23) hold. By the
same method of the proof of Theorem 1.2, we know that if and only if M n is
totally umbilical. This completes the proof of Theorem 1.3.
Acknowledgment. Project supported by NSF of Shaanxi Educational
Department (2010JK532, 11JK0479) and NSF of Weinan Normal University
(2012YKS026).
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Shichang SHU,
School of Mathematics and Information Science,
Xianyang Normal University,
Xianyang, 712000, Shaanxi, P.R. China.
Email: shushichang@126.com
Tianmin ZHU,
College of Mathematics and Information Science,
Weinan Normal University,
Weinan, 714000, Shaanxi, P.R. China.
Email: wnzhutm@163.com
SPECTRAL CHARACTERIZATION AND SCHRÖDINGER OPERATOR
258