ARCHIVUM MATHEMATICUM (BRNO)
Tomus 47 (2011), 77–82
THE FIRST EIGENVALUE OF SPACELIKE SUBMANIFOLDS
IN INDEFINITE SPACE FORM Rpn+p
Yingbo Han and Shuxiang Feng
Abstract. In this paper, we prove that the first eigenvalue of a complete
spacelike submanifold in Rpn+p with the bounded Gauss map must be zero.
1. Introduction
n
Let M be a complete noncompact Riemannian manifold and Ω ⊂ M n be a
domain with compact closure and nonempty boundary ∂Ω. The Dirichlet eigenvalue
λ1 (Ω) of Ω is defined by
R |∇f |2 dM
2
λ1 (Ω) = inf ΩR
:
f
∈
L
(Ω)
{0}
,
1,0
f 2 dM
M
where dM is the volume element on M n and L21,0 (Ω) the completion of C0∞ with
respect to the norm
Z
Z
kϕk2Ω =
ϕ2 dM +
|∇ϕ|2 dM .
M
M
If Ω1 ⊂ Ω2 are bounded domains, then λ1 (Ω1 ) ≥ λ1 (Ω2 ) ≥ 0. Thus one may define
the first Dirichlet eigenvalue of M n as the following limit
λ1 (M ) = lim λ1 B(p, r) ≥ 0 ,
r→∞
where B(p, r) is the geodesic ball of M n with radius r centered at p. It is clear that
the definition of λ1 (M ) does not depend on the center point p. It is interesting to
ask that for what geometries a noncompact manifold M n has zero first eigenvalue.
Cheng and Yau [1] showed that λ1 (M ) = 0 if M n has polynomial volume growth.
In [5], B. Wu proved the following result.
Theorem A. Let M n be a complete spacelike hypersurface in R1n+1 whose Gauss
map is bounded, then λ1 (M ) = 0.
In this note, we discover that Wu’s result still holds for higher codimensional
complete spacelike submanifolds in Rpn+p . In fact, we prove
2010 Mathematics Subject Classification: primary 53C42; secondary 53B30.
Key words and phrases: spacelike submanifolds, the first eigenvalue.
The first author is supported by NSFC grant No. 10971029 and NSFC-TianYuan Fund. No.
11026062.
Received September 17, 2010. Editor O. Kowalski.
78
Y. HAN AND S. FENG
Theorem 3.1. Let M n be a complete spacelike submanifold in Rpn+p whose Gauss
map is bounded, then λ1 (M ) = 0.
2. The geometry of pseudo-Grassmannian
In this section we review some basic properties about the geometry of pseudo-Grassmannian. For details one referred to see [6, 3].
Let Rpn+p be the (n + p)-dimensional pseudo-Euclidean space with index p,
where, for simplicity, we assume that n ≥ p. The case n < p can be treated
similarly. We choose a pseudo-Euclidean frame field {e1 ,P
. . . , en+p } such
P that
the pseudo-Euclidean metric of Rpn+p is given by ds2 =
(ωi )2 − α ωα =
i
P
2
A εA (ωA ) , where {ω1 , . . . , ωn=p } is the dual frame field of {e1 , . . . , en+p }, εi = 1
and εα = −1. Here and in the following we shall use the following convention on
the ranges of indices:
1 ≤ i, j, · · · ≤ n;
n + 1 ≤ α, β, · · · ≤ n + p ;
1 ≤ A, B, · · · ≤ n + p .
The structure equations of Rpn+p are given by
X
deA = −
εA ωAB eB ,
B
dωA = −
X
ωAB ∧ ωB ,
ωAB + ωBA = 0 ,
B
dωAB = −
X
εC ωAC ∧ ωCB .
C
Let Gpn,p be the pseudo-Grassmannian of all spacelike n-subspace in Rpn+p , and
p
n+p
g
G
. They are
n,p be the pseudo-Grassmannian of all timelike p-subspace in R
p
specific Cartan-Hadamard manifolds, and the canonical Riemannian metric on
p
g
Gpn,p and G
n,p is
X
dsG = dsG
(ωαi )2 .
e=
iα
Rpn+p .
0
Let 0 be the origin of
Let SO (n + p, p) denote the identity component
of the Lorentzian group O(n + p, p). SO0 (n + p, p) can be viewed as the manifold
consisting of all pseudo-Euclidean frames (0; ei , eα ), and SO0 (n + p, p)/SO(n) ×
p
p
g
SO(p) can be viewed as Gp or G
can be represented by
n,p . Any element in G
n,p
n,p
p
g
a unit simple n-vector e1 ∧ · · · ∧ en , while any element in G
n,p can be represented
by a unit simple p-vector en+1 ∧ · · · ∧ en+p . They are unique up to an action of
SO(n) × SO(p). The Hodge star ∗ provides an one to one correspondence between
p
p
g
Gpn,p and G
n,p . The product h, i on Gn,p for e1 ∧ · · · ∧ en , v1 ∧ · · · ∧ vn is defined by
he1 ∧ · · · ∧ en , v1 ∧ · · · ∧ vn i = det hei , vj i .
p
g
The product on G
n,p can be defined similarly.
Now we fix a standard pseudo-Euclidean frame ei , eα for Rpn+p , and take g0 =
p
g
e ∧ · · · ∧ e ∈ Gp , ge = ∗g = e
∧ ··· ∧ e
∈G
n,p . Then we can span the
1
n
n,p
0
0
n+1
n+p
n+p
THE FIRST EIGENVALUE OF SPACELIKE SUBMANIFOLDS IN Rp
79
spacelike n-subspace g in a neighborhood of g0 by n spacelike vectors fi :
X
fi = e i +
ziα eα ,
α
where (ziα ) are the local coordinates of g. By an action of SO(n) × SO(p) we can
assume that


µ1


..


.
(ziα ) = 
.

µp 
0
From [3] we know that the normal geodesic g(t) between g0 and g has local
coordinates


tanh(λ1 t)


..


.
(ziα ) = 
,

tanh(λp t)
0
Pp
for real numbers λ1 . . . λp such that i=1 λ2i = 1. This means that g(t) is spanned by
f1 (t) = e1 + tanh(λ1 t)en+1 , . . . , fp (t) = ep + tanh(λp t)en+p , fp+1 = ep+1 , . . . , fn =
en . Consequently, g(t) can also be represented by a unit simple n-vector as following:
g(t) = cosh(λ1 t)e1 + sinh(λ1 t)en+1 ∧ · · · ∧ cosh(λp t e1
+ sinh(λp t)en+p ) ∧ ep+1 ∧ · · · ∧ en .
Set λα = λα−n , then it is clear that
cosh(λ1 t)e1 + sinh(λ1 t)en+1 , . . . , cosh(λp t)e1 + sinh(λp t)en+p , ep+1 , . . . , en ,
sinh(λn+1 t)e1 + cosh(λn+1 t)en+1 , . . . , sinh(λn+p t)ep + cosh(λn+p t)en+p
is again a pseudo-Euclidean frame for Rpn+p , so we have
g = ∗g(t) = sinh(λn+1 t)e1 + cosh(λn+1 t)en+1 ∧ · · · ∧ sinh(λn+p t)ep
g(t)
g
p
+ cosh(λn+p t)en+p ∈ G
n,p .
Thus we have
hg0 , gi = (−1)p h∗g0 , ∗gi = (−1)p hge0 , gei =
Y
cosh(λα t) .
α
In this note, we also need the following lemma,
Q
P
Lemma 2.1 ([4]). Let µ1 ≥ 1, . . . , µp ≥ 1 and α µα = C. Then α cosh2 (λα ) ≤
C 2 + p − 1, and the equality holds if and only if µi0 = C for some 1 ≤ i0 ≤ p and
µi = 1 for any i 6= i0 .
80
Y. HAN AND S. FENG
3. Main results for space-like submanifolds
In this note, we get the following result:
Theorem 3.1. Let M n be a complete space-like submanifold in Rpn+p whose Gauss
map is bounded, then we have λ1 (M ) = 0.
Proof. We choose a local frames e1 . . . , en+p in Rpn+p such that restricted to M n ,
e1 , . . . , en are tangent to M n , en+1 , . . . , en+p are normal to M n , the Gauss map
p
g
is defined by en+1 ∧ · · · ∧ en+p : M n → G
n,p . Let us fix p-vector and n-vector
p
g
an+1 ∧ · · · ∧ an+p ∈ G
,
a
∧
·
·
·
∧
a
∈
Gpn,p , where haα , aβ i = −δαβ and
n,p
1
n
hai , aj i = δij . We defined the projection Π : M n → Ran by
n+p
X
Π(x) = x +
(1)
hx, aα iaα ,
α=n+1
where h, i is the standard indefinite inner product on Rpn+p and Ran the totally
geodesic Euclidean n-space determined by a = an+1 ∧ · · · ∧ an+p which is defined
by
Ran = {x ∈ Rpn+p : hx, an+1 i = · · · = hx, an+p i = 0} .
(2)
It is clear from (1) that
dΠ(X) = X +
(3)
n+p
X
hX, aα iaα
α=n+1
for any tangent vector field on M n and consequently,
|dΠ(X)|2 = |X|2 +
(4)
n+p
X
hX, aα i2 .
α=n+1
From the equation (4), we know that the map Π : M n → Ran increases the distance.
If a map, from a complete Riemannian manifold M1 into another Riemannian
manifold M2 of same dimension, increases the distance, then it is a covering map
and M2 is complete (in [2, VIII, Lemma 8.1]). Hence Π is a covering map, but Ran
being simply connected this means that Π is in face a diffeomorphism between
M n and Ran , and thus M n is noncompact. Now assume that the Gauss map
p
g
en+1 ∧ · · · ∧ en+p : M n → G
n,p is bounded, then there exists ρ > 0 such that
1 ≤ (−1)p hen+1 ∧ · · · ∧ en+p , an+1 ∧ · · · ∧ an+p i ≤ ρ .
(5)
From Section 2 we know that by an action of SO(n) × SO(p) we can assume that
en+1 = sinh(λn+1 t)a1 + cosh(λn+1 t)an+1 , . . . , en+p
= sinh(λn+p t)a1 + cosh(λn+p t)an+p ,
where
P
α
λ2α = 1 and t ∈ R.
n+p
THE FIRST EIGENVALUE OF SPACELIKE SUBMANIFOLDS IN Rp
81
Write
aα = a> −
(6)
n+p
X
haα , eβ ieβ ,
β=n+1
n
where a>
α denote the component of aα which is tangent to M , and α = n +
1, . . . , n + p. Since haα , aβ i = −δαβ , we have
−1=
(7)
2
|a>
α|
n+p
X
−
2
2
haα , eβ i2 = |a>
α | − cosh (λα t) ,
β=n+1
where α = n + 1, . . . , n + p. It follows from Lemma 2.1 and Eq. (5), (7), we have
(8)
1+
n+p
X
2
|a>
α| =
α=n+1
n+p
X
cosh2 (λα t) − p + 1 ≤
Y
cosh2 (λα t) ≤ ρ2 .
α=n+1
From Eq.(4) and (8), we have
(9)
2
2
|dΠ(X)| = |X| +
n+p
X
2
hX, a>
αi
2
≤ |X| (1 +
α=n+1
n+p
X
2
2
2
|a>
α | ) ≤ ρ |X| .
α=n+1
n
for any tangent vector field on M . Let B(p, r) is the geodesic ball of M n with
e p, ρr)
e p, ρr), where B(e
radius r centered at p ∈ M n . We claim that Π(B(p, r)) ⊂ B(e
denotes the geodesic ball of Ran with radius ρr centered at pe = Π(p). In fact, for
any qe ∈ Π(B(p, r)) let q ∈ B(p, r) be the unique point such that Π(q) = qe, and
γ : [a, b] → M n is the minimal geodesic joining p and q, then from (9) we have
Z b
Z b
e p, qe) ≤ L(Π ◦ r) =
dΠ γ 0 (t) dt ≤ ρ
d(e
|γ 0 (t)| dt = ρL(γ) = ρd(p, q) ≤ ρr ,
a
a
Ran
where de and d denote the distance in
and M n , respectively. This prove our
claim.
Let dV denotes the n-dimensional volume element on Ran . Using (3) and (6) it
follows that
Π∗ (dV )(X1 , . . . , Xn ) = det dΠ(X1 ), . . . , dΠ(Xn ), an+1 , . . . , an+p
= det(X1 , . . . , Xn , an+1 , . . . , an+p )
= (−1)p hen+1 ∧ · · · ∧ en+p , an+1 ∧ · · · ∧ an+p i
det(X1 , . . . , Xn , en+1 , . . . , en+p )
= (−1)p hen+1 ∧ · · · ∧ en+p , an+1 ∧ · · · ∧ an+p i
dM (X1 , . . . , Xn )
for any tangent vector fields X1 , . . . , Xn of M n . In other words,
(10)
Π∗ (dV ) = (−1)p hen+1 ∧ · · · ∧ en+p , an+1 ∧ · · · ∧ an+p idM ≥ dM .
82
Y. HAN AND S. FENG
e p, ρr) and Π : M n → Ran is diffeomorphism, it follows from
Since Π(B(p, r)) ⊂ B(e
Eq. (10) that
Z
n n
e
ρ r ωn = Vol B(e
p, ρr) ≥ Vol Π(B(p, r)) =
dV
Π(B(p,r))
Z
Z
(11)
=
Π∗ dV ≥
dM = Vol B(p, r) ,
B(p,r)
B(p,r)
where ωn denotes the volume of unit ball in Euclidean n-space. (11) means that
the order of the volume growth of M n is not larger than n, thus by [1] we see that
λ1 (M ) = 0.
References
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[2] Kobayashi, S., Nomizu, K., Foundations of differential geometry, John Wiley & Sons, Inc.,
1969.
[3] Wong, Y. C., Euclidean n-space in pseudo-Euclidean spaces and differential geometry of
Cartan domain, Bull. Amer. Math. Soc. (N.S.) 75 (1969), 409–414.
[4] Wu, B. Y., On the volume and Gauss map image of spacelike submanifolds in de Sitter space
form, J. Geom. Phys. 53 (2005), 336–344.
[5] Wu, B. Y., On the first eigenvalue of spacelike hypersurfaces in Lorentzian space, Arch. Math.
(Brno) 42 (2006), 233–238.
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103 (2000), 191–202.
College of Mathematics and information Science,
Xinyang Normal University,
Xinyang 464000, Henan, P. R. China
E-mail: yingbhan@yahoo.com.cn fsxhyb@yahoo.com.cn