Eigenvalue estimates for Dirac operator with the generalized APS boundary condition

0.14 MB·English

by  Daguang Chen

EIGENVALUE ESTIMATES FOR DIRAC OPERATOR WITH THE
GENERALIZED APS BOUNDARY CONDITION
7
0
0 DAGUANG CHEN
2

n Abstract. Undertwoboundaryconditions: thegeneralizedAtiyah-Patodi-Singerbound-
a
aryconditionandthemodifiedgeneralizedAtiyah-Patodi-Singerboundarycondition,we
J
get the lower bounds for the eigenvalues of the fundamental Dirac operator on compact
9
spin manifolds with nonempty boundary.
2

]
G
D
1. introduction
.
h
t LetM beacompactRiemannianspinmanifoldwithoutboundary. In1963,Lichnerowicz
a
m [21] proved that any eigenvalue of the Dirac operator satisfies
[ 1
λ2 > infR,
1 4 M
v
3 where R is the positive scalar curvature of M. The first sharp estimate for the smallest
4 eigenvalue λ of the Dirac operator D was obtained by Friedrich [8] in 1980. The idea of
8
the proof is based on using a suitably modified Riemannian spin connection. He proved
1
0 the inequality
7 n
0 λ2 ≥ infR,
/ 4(n−1) M
h
t on closed manifolds (Mn,g) with the positive scalar curvature R. Equality gives an
a
m Einstein metric. In 1986, combining the technique of the modified spin connection with
a conformal change of the metric, Hijazi [11] showed, for n ≥ 3
:
v
n
Xi λ2 ≥ µ ,
1
4(n−1)
r
a
where µ is the first eigenvalue of the conformal Laplacian given by L = 4(n−1)△ + R.
1 n−2
When the equality holds, the manifold becomes an Einstein manifold.
Let M be a compact Riemannian spin manifold with nonempty boundary. In this
case, the boundary conditions for spinors become crucial to make the Dirac operator
elliptic, and, in general, two types of the boundary conditions are considered, i.e. the
local boundary condition and the Atiyah-Patodi-Singer (APS) boundary condition (see,
e.g.[6]). In 2001, Hijazi, Montiel and Zhang [16] proved the generalized version of -
Friedrich-type inequalities under both the local boundary condition and the Atiyah--
Patodi-Singer boundary condition. In 2002, Hijazi, Montiel and Rold´an [15] investigated
the Friedrich-type inequality for eigenvalues of the fundamental Dirac operator under
fourellipticboundaryconditionsandundersomecurvatureassumptions(thenon-negative
Date: 21 Feb-2006.
Key words and phrases. eigenvalue, Dirac operator, boundary condition, ellipticity.
1
mean curvature). In particular, they introduced a new globalboundary condition, namely
the modified Atiyah-Patodi-Singer (mAPS) boundary condition.
Inthe present paper, we study the spectrum of the fundamental Diracoperator on com-
pact Riemannian spin manifolds with nonempty boundary, under the two new boundary
conditions: the generalized Atiyah-Patodi-Singer (gAPS) boundary condition and the
modified generalized Atiyah-Patodi-Singer (mgAPS) boundary condition. Following the
terminology of [6] those are obtained composing a so-called spectral projection with the
identity in the first case, and with the zero order differential operator Id +γ(e ) in the
0
second case (where γ(e ) is the Clifford product with the unit normal e on the boundary
0 0
∂M).
2. Preliminaries
Let (M,g) be an (n+1)-dimentional Riemannian spin manifold with nonempty bound-
ary ∂M. We denote by ∇ the Levi-Civit`a connection on the tangent bundle TM. For the
given structure SpinM (and so a corresponding orientation) on manifold M, we denote
by S the associated spinor bundle, which is a complex vector bundle of rank 2[n+21]. Then
let
γ : Cl(M) −→ EndC(S)
be the Clifford multiplication, which is a fibre preserving algebra morphism. It is well
known [22] that there exists a natural Hermitian metric (,) on the ’spinor bundle S which
satisfies the relation
(γ(X)ϕ,γ(X)ψ) = |X|2(ϕ,ψ), (2.1)
for any vector field X ∈ Γ(TM), and for any spinor fields ϕ,ψ ∈ Γ(S). We denote also by
∇thespinorialLevi-Civit`aconnectionactingonthespinorbundleS. Thentheconnection
∇ is compatible with the Hermitian metric (,) and Clifford multiplication ”γ”. Recalling
the fundamental Dirac operator D is the first order elliptic differential operator acting on
the spinor bundle S, which is locally given by
n
D = γ(e )∇ , (2.2)
i i
i=0
X
where {e ,e ,··· ,e } is a local orthonormal frame of TM.
0 1 n
Since the normal bundle to the boundary hypersurface is trivial, the Riemannian man-
ifold ∂M is also spin and so we have a corresponding spinor bundle S∂M, a spinorial
Levi-Civit`a connection ∇∂M and an intrinsic Dirac operator D∂M = γ∂M(e )∇∂M. Then
i i
a simple calculation yields the spinorial Gauss formula
1
∇ ϕ = ∇∂Mϕ+ h γ(e )γ(e )ϕ,
i i 2 ij j 0
for 1 ≤ i,j ≤ n,ϕ ∈ Γ(S| ), e is a unit normal vector field compatible with the induced
∂M 0
orientation, h is the second fundamental form of the boundary hypersurface.
ij
It is easy to check (see [4][14][15][17]) that the restriction of the spinor bundle S of M
to its boundary is related to the intrinsic Hermitian spinor bundle S∂M by
S∂M if n is even
S := S| ∼=
∂M S∂M ⊕S∂M if n is odd.
(
2
For any spinor field ψ ∈ Γ(S) on the boundary ∂M, define on the restricted spinor bundle
S, the Clifford multiplication γS and the spinorial connection ∇S by
γS(e )ψ := γ(e )γ(e )ψ (2.3)
i i 0
1 1
∇Sψ := ∇ ψ − h γS(e )ψ = ∇ ψ − h γ(e )γ(e )ψ. (2.4)
i i 2 ij j i 2 ij j 0
One can easily find that ∇S is compatible with the Clifford multiplication γS, the induced
Hermitian inner product (,) from M and together with the following additional identity
∇S(γ(e )ψ) = γ(e )∇Sψ. (2.5)
i 0 0 i
As a consequence, the boundary Dirac operator D associated with the connection ∇S and
the Clifford multiplication γS, is locally given by
H
Dψ = γS(e )∇Sψ = γ(e )γ(e )∇ ψ + ψ, (2.6)
i ei i 0 ei 2
where H = h is the mean curvature of ∂M. In fact, γ(e )(D − 1H) = γ(e )∇ is the
ii 0 2 i i
hypersurface Diracoperatordefined by Witten[24]to provethe positive energyconjecture
P
in general relativity.
Now by (2.5), we have the supersymmetry property
Dγ(e ) = −γ(e )D. (2.7)
0 0
Hence, when ∂M is compact, the spectrum of D is symmetric with respect to zero and
coincides with the spectrum of D∂M for n even and with (SpecD∂M)∪(-SpecD∂M) for n
odd.
From [14] or [15], one gets the integral form of the Schr¨odinger-Lichnerowicz formula
for a compact Riemannian spin manifold with compact boundary
1 R
(ϕ,Dϕ)− H|ϕ|2 = |∇ϕ|2 + |ϕ|2 −|Dϕ|2. (2.8)
2 4
Z∂M Z∂M ZM
By (2.7), D is self-adjoint with respect to the induced Hermitian metric (,) on S.
Therefore, D has a discrete spectrum contained in R numbered like
··· ≤ λ ≤ ···λ < 0 ≤ λ ≤ λ ≤ ··· ≤ λ ≤ ···
−j −1 0 1 j
and one can find an orthonormal basis {ϕj}j∈Z of L2(∂M;S) consisting of eigenfunctions
of D (i.e.Dϕj = λjϕj,j ∈ Z) (see e.g. Lemma 1.6.3 in [9]). Such a system {λj;ϕj}j∈Z
is called a spectral decomposition of L2(∂M;S) generated by D, or, in short, a spectral
resolution of D. According to the definition 14.1 in [5], we have
Definition 2.1. For self-adjoint Dirac operator D and for any real b we shall denote by
P : L2(∂M;S) −→ L2(∂M;S)
≥b
the spectralprojection, that is, the orthogonalprojection ofL2(∂M;S) onto the subspace
spanned by { ϕj|λj ≥ b}, where {λj;ϕj}j∈Z is a spectral resolution of D.
Definition 2.2. For ϕ ∈ Γ(S) and b ≤ 0, the projective boundary conditions
P ϕ = 0,
≥b
Pmϕ = P (Id+γ(e ))ϕ = 0
≥b ≥b 0
3
are called the generalized Atiyah-Patodi-Singer (gAPS) boundary condition and the mod-
ified generalized Atiyah-Patodi-Singer (mgAPS) boundary condition respectively.
Remark 2.1. We shall adopt the notation P = Id−P ; P = P −P for b < 0
<b ≥b [b,−b] ≥b >−b
or P = P −P for b > 0. It is well known that the spectral projection P is a
[−b,b] ≥−b >b ≥b
pseudo-differential operator of order zero (see e.g. the proposition 14.2 in [5]).
Remark 2.2. If b = 0, the gAPS boundary condition is exactly the APS boundary condi-
tion (see Atiyah,Patodi & Singer[1][2][3]).
In case of a closed manifold, the fundamental Dirac operator is a formally self-adjoint
operator and so its spectrum is discrete and real. While in case of a manifold with
nonempty boundary, we have the following formula
(Dϕ,ψ)− (ϕ,Dψ) = − (γ(e )ϕ,ψ). (2.9)
0
ZM ZM Z∂M
for ϕ,ψ ∈ Γ(∂M), and where e is the inner unit field along the boundary. In the
0
following, we’ll show ellipticity and self-adjointness for the Dirac operator under both the
gAPS boundary condition and the mgAPS boundary condition.
On one hand, if P ϕ = 0,P ψ = 0, for ϕ,ψ ∈ Γ(S) and b ≤ 0, then P (γ(e )ϕ) =
≥b ≥b <b 0
γ(e )P . Therefore
0 >−b
(γ(e )ϕ,ψ) = 0. (2.10)
0
Z∂M
On the other hand, for ϕ,ψ ∈ Γ(S) and b ≤ 0, under the mgAPS boundary conditions,
i.e.
P (Id+γ(e ))ϕ = 0 and P (Id+γ(e ))ψ = 0,
≥b 0 ≥b 0
a simple calculation implies that (2.10) also holds.
These facts imply that the fundamental Dirac operator D is self-adjoint under the -
gAPS boundary condition and the gmAPS boundary condition and hence it has real and
discrete eigenvalues.
The ellipticity of both gAPS and mgAPS boundary conditions the fundamental Dirac
operator D follows straightforward from [6,Prop. 14.2], [16,Prop. 1] and [16,Thm. 5].
Lemma 2.1. Under the gAPS boundary condition P ϕ = 0, the inequality
≥b
(ϕ,Dϕ) ≤ b |ϕ|2
Z∂M Z∂M
holds for b ≤ 0.
Proof. Let{λj;ϕj}j∈Z beaspectralresolutionofthehypersurface DiracoperatorD. Then
any ϕ ∈ Γ(S) can be expressed as follows,
ϕ = c ϕ ,
j j j
where cj = ∂M(ϕ,ϕj). Then we have P
R (ϕ,Dϕ) = λ |c |2 ≤ b |c |2 = b |ϕ|2.
j j j
Z∂M λj<b λj<b Z∂M
P P (cid:3)
4
3. lower bounds for the eigenvalues
In this section, we adapt the arguments used in [18] and [16] to the case of compact
Riemannian spin manifolds with nonempty boundary. We use the integral identity (2.8)
together with an appropriate modification of the Levi-Civit`a connection to obtain the
eigenvalue estimates.
Theorem 3.1. Let Mn be a compact Riemannian spin manifold of dimension n ≥ 2, with
the nonempty boundary ∂M, and let λ be any eigenvalue of D under the gAPS boundary
condition P ϕ = 0, for ϕ ∈ Γ(S) and b ≤ 0. If there exist real functions a,u on M such
≥b
that
1
b+a du(e ) ≤ H
0
2
on ∂M, then
n
λ2 > sup infR , (3.1)
a,u
4(n−1) a,u M
where
1
R := R−4a∆u+4∇a·∇u−4(1− )a2|du|2 (3.2)
a,u
n
∆ is the positive scalar Laplacian, R is scalar curvature of M.
Proof. Foranyrealfunctionaandu, definethemodifiedspinorialconnection (see[16][18])
on Γ(S) by
a λ
∇a,u = ∇ +a∇ u+ ∇ uγ(e )γ(e )+ γ(e ). (3.3)
i i i n j i j n i
A simple calculation yields
1 R
|∇a,uϕ|2 = (1− )λ2 − a,u |ϕ|2
n 4
ZM ZM h i (3.4)
H
+ (ϕ,Dϕ)+ a du(e )− |ϕ|2.
0
2
Z∂M Z∂M h i
Considering Lemma 2.1 and b+a du(e ) ≤ 1H, we infer
0 2
n
λ2 ≥ sup infR . (3.5)
a,u
4(n−1) a,u M
If theequality in(3.5)holds, thenbytheLemma 3in[16], we have a = 0 oru = constant
and b ≤ 1H. Since ϕ is a non-degenerated killing spinor, |ϕ|2 is a nonzero constant. Let
2
{λj;ϕj}j∈Z be a spectral resolution of the hypersurface Dirac operator D. Under the
gAPS boundary condition P ϕ = 0, one gets ϕ = c ϕ , where c = (ϕ,ϕ ). Then
≥b j j j ∂M j
λj<b
we have P R
1
0 = (ϕ,Dϕ)− H|ϕ|2
2
Z∂M Z∂M
1
= λ |c |2 − c c H(ϕ ,ϕ )
j j j k j k
2
λPj<b λjX,λk<b Z∂M
≤ (λ −b)|c |2 < 0.
j j
λj<b
P (cid:3)
This is a contradiction. Thus the inequality (3.1) holds.
5
Remark 3.1. If b = 0, thegAPSboundaryconditionbecomes theAPSboundarycondition
and then the theorem is exactly the Theorem 3.1 in [16].
Now we make use of the energy-momentum tensor, introduced in [12] and used in
[17][18][19], to get lower bounds for the eigenvalues of D. For any spinor field ϕ ∈ Γ(S),
we define the associated energy-momentum 2-tensor Q on the complement of its zero set
ϕ
by
1
Q = ℜ γ(e )∇ ϕ+γ(e )∇ ϕ,ϕ/|ϕ|2 .
ϕ,ij i j j i
2
Obviously, Qϕ,ij is a symmetric te(cid:0)nsor. If ϕ is the eigenspinor o(cid:1)f the Dirac operator D,
the tensor Q is well-defined in the sense of distribution.
ϕ
Theorem 3.2. Let Mn be a compact Riemannian spin manifold of dimension n ≥ 2,
whose boundary ∂M is nonempty, and let λ be any eigenvalue of D under the gAPS
boundary condition P ϕ = 0, for b ≤ 0,ϕ ∈ Γ(S). If there exist real functions a,u on M
≥b
such that
b+adu(e ) ≤ H/2
0
on ∂M, where H is the mean curvature of ∂M, then
1
λ2 ≥ sup inf( R +|Q |2),
a,u ϕ
a,u M 4
where R is given in (3.2).
a,u
Proof. Foranyrealfunctionaandu, definethemodifiedspinorialconnection (see[16][18])
on Γ(S) by
a
∇Q,a,u = ∇ +a∇ u+ ∇ uγ(e )γ(e )+Q γ(e ). (3.6)
i i i n j i j ϕ,ij j
One can easily compute
R
|∇Q,a,uϕ|2 = λ2 −( a,u +|Q |2) |ϕ|2
ϕ
4
ZM ZM h i (3.7)
+ (ϕ,Dϕ)+ adu(e )−H/2 |ϕ|2.
0
Z∂M Z∂M h i
(cid:3)
Considering the boundary conditions, we immediately arrive at the asserted formula.
Remark 3.2. By Cauchy-Schwarz inequality and trQ = λ, we have
ϕ
1 1
|Q |2 = |Q |2 ≥ |Q |2 ≥ (trQ )2 = λ2.
ϕ ϕ,ij ϕ,ii ϕ
n n
i,j i
P P
Therefore, one gets the inequality (3.5).
Remark 3.3. If b = 0, the above theorem becomes Theorem 5 in [16]. Under the gAPS
boundary condition, taking a = 0 or u = constant in (3.6) and assuming H ≥ 0, then
one gets the following inequality [12]
R
λ2 ≥ inf +|Q |2 .
ϕ
M 4
In the following, we state the eigenvalu(cid:0)e estimates(cid:1) for the Dirac operator under the
mgAPS boundary condition.
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Theorem 3.3. Let Mn be a compact Riemannian spin manifold of dimension n ≥ 2,
whose boundary ∂M is nonempty, and let λ be any eigenvalue of D under the mgAPS
boundary condition Pm = P (Id+γ(e )) = 0, for b ≤ 0. If there exist real functions a,u
≥b ≥b 0
on M such that
1
a du(e ) ≤ H (3.8)
0
2
on ∂M, then
n
λ2 ≥ sup infR , (3.9)
a,u
4(n−1) a,u M
where R is given in (3.2). Moreover, the equality holds if and only if M carries a
a,u
nontrivial real killing spinor field with a killing constant −λ < b and the boundary ∂M
n n−1
is minimal.
Proof. Assuming
Pmϕ = P (ϕ+γ(e )ϕ) = 0 and Pmψ = P (ψ +γ(e )ψ) = 0,
≥b ≥b 0 ≥b ≥b 0
for ϕ,ψ ∈ Γ(S) and b ≤ 0, then we have
P ψ +γ(e )P ψ = 0 and P ψ +γ(e )P ψ = 0,
>−b 0 <b [b,−b] 0 [b,−b]
i.e.
Pmψ = P ψ +γ(e )P ψ = 0 and P ψ = 0 (3.10)
≥b >−b 0 <b [b,−b]
These imply
Pmϕ = P (ϕ+γ(e )ϕ) = 0,
≥b >−b 0
Pmψ = P (ψ +γ(e )ψ) = 0.
≥b >−b 0
Then one gets
γ(e )ψ −ψ = P (γ(e )ψ −ψ). (3.11)
0 >−b 0
The relation (2.7) implies that
1
(Dψ,ψ) = (D(ψ+γ(e )ψ),ψ −γ(e )ψ).
0 0
2
The combination Pmψ = Pm ψ = P (ψ +γ(e )ψ) = 0 with (3.11) yields
≥b >−b >−b 0
(D(ψ +γ(e )ψ),ψ −γ(e )ψ) = 0.
0 0
Z∂M
This implies
(Dψ,ψ) = 0. (3.12)
Z∂M
From (3.3), (3.4) and (3.12), one can infer
1 R H
|∇a,uϕ|2 = (1− )λ2 − a,u |ϕ|2 + a du(e )− |ϕ|2. (3.13)
0
n 4 2
ZM ZM h i Z∂M h i
Then the inequality (3.9) holds.
If the equality occurs, we deduce
1
∇a,uϕ = 0 and H = adu(e ).
0
2
7
By the Lemma 3 in [16], we get a = 0 or u = constant. Therefore
λ
H = 0 and ∇ ϕ = − γ(e )ϕ.
i i
n
From the supersymmetry property (2.7), we get
n−1
D(ϕ+γ(e )ϕ) = − λ(ϕ+γ(e )ϕ).
0 0
n
Since Pmϕ = 0, we deduce that −n−1λ < b.
≥b n
Conversely, if M is a compact Riemannian spin manifold with minimal boundary ∂M
and a nontrivial killing spinor ϕ with a real killing constant −λ/n < b , then we have
n−1
Dϕ = λϕ. Moreover, from the fact that ∇ ϕ = −λγ(e )ϕ, we infer that the restriction
e0 n 0
of ϕ to the boundary satisfies
n−1
Dϕ = − γ(e )ϕ.
0
n
Finally, we have
n−1
D(ϕ+γ(e )ϕ) = − λ(ϕ+γ(e )ϕ).
0 0
n
Since the spinor field ϕ + γ(e )ϕ is an eigenspinor of D with a eigenvalue −(n−1)λ < b,
0 n
one can infer Pmϕ = 0. (cid:3)
≥b
Remark 3.4. Under the mgAPS boundary condition, taking a = 0 or u = constant in
(3.8) and (3.13), one gets Friedrich’s inequality
n
λ2 ≥ infR.
4(n−1) M
In particular, if we take b = 0, the above theorem is exactly the Theorem 5 in [15].
Using the energy-momentum tensor, (3.7) and (3.12), we get
R
|∇Q,a,uϕ|2 = λ2 −( a,u +|Q |2) |ϕ|2+ adu(e )−H/2 |ϕ|2.
ϕ 0
4
ZM ZM h i Z∂M h i
Thus we obtain the following theorem:
Theorem 3.4. Let Mn be a compact Riemannian spin manifold of dimension n ≥ 2, with
nonempty boundary ∂M, and let λ be any eigenvalue of D under the mgAPS boundary
condition Pmϕ = P (ϕ+γ(e )ϕ) = 0, for b ≤ 0, ϕ ∈ Γ(S). If there exist real functions
≥b ≥b 0
a,u on M such that
1
a du(e ) ≤ H
0
2
on ∂M, then
1
λ2 ≥ sup inf R +|Q |2 , (3.14)
a,u ϕ
a,u M 4
where R is given in (3.2). (cid:0) (cid:1)
a,u
Acknowledgments The author would like to express his gratitude to Professor X.
Zhang for his suggestions and support. He would also like to thank the referee for many
valuable suggestions.
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E-mail address: [e-mail]
Current address: Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese
Academy of Sciences, Beijing 100080,P.R. China.
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