1. Introduction
On a compact Riemannian Spin
c manifold (
Mn,
g) of dimension
n⩾2, any eigenvalue
λ of the Dirac operator satisfies the Friedrich type inequality [
1,
2]
()
where
S denotes the scalar curvature of
M,
cn = 2[
n/2]
1/2 and
iΩ is the curvature form of the connection on the line bundle given by the Spin
c structure. Equality holds if and only if the eigenspinor
ψ associated with the first eigenvalue
λ1 is a Spin
c Killing spinor; that is, for every
X ∈ Γ(
TM), the eigenspinor
ψ satisfies
()
Here,
X ·
ψ denotes the Clifford multiplication and ∇ the spinorial Levi-Civita connection [
3,
4]. In [
5], it is shown that on a compact Riemannian Spin
c manifold any eigenvalue
λ of the Dirac operator to which is attached an eigenspinor
ψ satisfies the Hijazi type inequality [
6] involving the Energy-Momentum tensor and the scalar curvature
()
where
ℓψ is the field of symmetric endomorphisms associated with the field of quadratic forms denoted by
Tψ, called the Energy-Momentum tensor. It is defined on the complement set of zeroes of the eigenspinor
ψ, for any vector field
X by
()
Equality holds in (
1.3) if and only, for all
X ∈ Γ(TM), we have
()
where
ψ is an eigenspinor associated with the first eigenvalue
λ1. By definition, the trace tr (
ℓψ) of
ℓψ, where
ψ is an eigenspinor associated with an eigenvalue
λ, is equal to
λ. Hence, (
1.3) improves (
1.1) since by the Cauchy-Schwarz inequality, |
ℓψ|
2⩾(tr (
ℓψ))
2/
n =
λ2/
n. It is also shown that the sphere equipped with a special Spin
c structure satisfies the equality case in (
1.3) but equality in (
1.1) cannot occur.
In the same spirit as in [
7], Herzlich and Moroianu (see [
1]) generalized the Hijazi inequality [
7], involving the first eigenvalue of the Yamabe operator
L, to the case of compact Spin
c manifolds of dimension
n⩾3: any eigenvalue
λ of the Dirac operator satisfies
()
where
μ1 is the first eigenvalue of the perturbed Yamabe operator defined by
LΩ =
L −
cn | Ω|
g = 4((
n − 1)/(
n − 2))Δ +
S −
cn | Ω|
g. The limiting case of (
1.6) is equivalent to the limiting case in (
1.1). The Hijazi inequality [
6], involving the energy-momentum tensor and the first eigenvalue of the Yamabe operator, is then proved by the author in [
5] for compact Spin
c manifolds. In fact, any eigenvalue of the Dirac operator to which is attached an eigenspinor
ψ satisfies
()
Equality in (
1.7) holds if and only, for all
X ∈ Γ(TM), we have
()
where
, the spinor field
is the image of
ψ under the isometry between the spinor bundles of (
Mn,
g) and
, and
ψ is an eigenspinor associated with the first eigenvalue
λ1 of the Dirac operator. Again, (
1.7) improves (
1.6). In this paper we examine these lower bounds on open manifolds, and especially on complete Riemannian Spin
c manifolds. We prove the following.
Theorem 1.1. Let (Mn, g) be a complete Riemannian Spinc manifold of finite volume. Then, any eigenvalue λ of the Dirac operator to which is attached an eigenspinor ψ satisfies the Hijazi type (1.3). Equality holds if and only if the eigenspinor associated with the first eigenvalue λ1 satisfies (1.5).
The Friedrich type (1.1) is derived for complete Riemannian Spinc manifolds of finite volume and equality also holds if and only if the eigenspinor associated with the first eigenvalue λ1 is a Killing Spinc spinor. This was proved by Grosse in [8, 9] for complete spin manifolds of finite volume. Using the conformal covariance of the Dirac operator, we prove the following.
Theorem 1.2. Let (Mn, g) be a complete Riemannian Spinc manifold of finite volume and dimension n > 2. Any eigenvalue λ of the Dirac operator to which is attached an eigenspinor ψ satisfies the Hijazi type (1.7). Equality holds if and only if (1.8) holds.
Now, the Hijazi type (1.6) can be derived for complete Riemannian Spinc manifolds of finite volume and dimension n > 2 and equality holds if and only if the eigenspinor associated with the first eigenvalue λ1 is a Killing Spinc spinor. This was also proved by Grosse in [8, 9] for complete spin manifolds of finite volume and dimension n > 2. On complete manifolds, the Dirac operator is essentially self-adjoint and, in general, its spectrum consists of eigenvalues and the essential spectrum. For elements of the essential spectrum, we also extend to Spinc manifolds the Hijazi type (1.6) obtained by Grosse in [9] on spin manifolds.
Theorem 1.3. Let (Mn, g) be a complete Riemannian Spinc manifold of dimension n ≥ 5 with finite volume. Furthermore, assume that S − cn | Ω| is bounded from below. If λ is in the essential spectrum of the Dirac operator σess(D), then λ satisfies the Hijazi type (1.6).
For the 2-dimensional case, Grosse proved in [
8] that for any Riemannian spin surface of finite area, homeomorphic to
ℝ2, we have
()
where
(in the compact case,
λ+ coincides with the first eigenvalue of the square of the Dirac operator). Recently, in [
10], Bär showed the same inequality for any connected 2-dimensional Riemannian manifold of genus 0, with finite area and equipped with a spin structure which is
bounding at infinity. A spin structure on
M is said to be
bounding at infinity if
M can be embedded into
𝕊2 in such a way that the spin structure extends to the unique spin structure of
𝕊2.
Studying the energy-momentum tensor on a compact Riemannian spin or Spinc manifolds has been done by many authors, since it is related to several geometric situations. Indeed, on compact spin manifolds, Bourguignon and Gauduchon [11] proved that the energy-momentum tensor appears naturally in the study of the variations of the spectrum of the Dirac operator. Hence, when deforming the Riemannian metric in the direction of this tensor, the eigenvalues of the Dirac operator are then critical. Using this, Kim and Friedrich [12] obtained the Einstein-Dirac equation as the Euler-Lagrange equation of a certain functional. The author extends these last results to compact Spinc manifolds [13]. Even it is not a computable geometric invariant, the energy-momentum tensor is, up to a constant, the second fundamental form of an isometric immersion into a Spinc manifold carrying a parallel spinor [13, 14]. Moreover, in low dimensions, the existence, on a spin or Spinc manifold M, of a spinor ψ satisfying (1.5) is, under some additional assumptions, equivalent to the existence of a local immersion of M into ℝ3, 𝕊3, ℂP2, 𝕊2 × ℝ or some others manifolds [14–16].
2. Preliminaries
In this section, we briefly introduce basic notions concerning Spinc manifolds, the Dirac operator and its conformal covariance. Then, we recall the refined Kato inequality which is crucial for the proof.
The Dirac Operator on Spinc Manifolds Let (Mn, g) be a connected oriented Riemannian manifold of dimension n⩾2 without boundary. Furthermore, let SOM be the SOn-principal bundle over M of positively oriented orthonormal frames. A Spinc structure of M is a -principal bundle (SpincM, π, M) and an 𝕊1-principal bundle (𝕊1M, π, M) together with a double covering given by θ : SpincM → SOM×M𝕊1M such that θ(ua) = θ(u)ξ(a), for every u ∈ SpincM and , where ξ is the 2-fold covering of over SOn × 𝕊1. Let be the associated spinor bundle, where and the complex spinor representation. A section of ΣM will be called a spinor and the set of all spinors will be denoted by Γ(ΣM) and those of compactly supported smooth spinors by Γc(ΣM). The spinor bundle ΣM is equipped with a natural Hermitian scalar product, denoted by 〈·, ·〉, satisfying
()
where
X ·
ψ denotes the Clifford multiplication of
X and
ψ. With this Hermitian scalar product we define an
L2-scalar product
()
for any spinors
ψ and
φ in Γ
c(
ΣM). Additionally, given a connection 1-form
A on
𝕊1M,
A :
T(
𝕊1M) →
iℝ and the connection 1-form
ωM on
SOM for the Levi-Civita connection ∇
M, we consider the associated connection on the principal bundle
SOM×
M𝕊1M, and hence a covariant derivative ∇ on Γ(
ΣM) [
3].
The curvature of A is an imaginary valued 2-form denoted by FA = dA, that is, FA = iΩ, where Ω is a real valued 2-form on 𝕊1M. We know that Ω can be viewed as a real-valued 2-form on M [3]. In this case, iΩ is the curvature form of the associated line bundle L. It is the complex line bundle associated with the 𝕊1-principal bundle via the standard representation of the unit circle. For any spinor ψ and any real 2-form Ω, we have [1]
()
where |Ω|
g is the norm of Ω given by
. Moreover, equality holds in (
2.3) if and only if
()
The Dirac operator is a first-order elliptic operator locally given by
()
It is an elliptic and formally self-adjoint operator with respect to the
L2-scalar product; that is, for all spinors
ψ,
φ, at least one of which is compactly supported on
M, we have (
Dψ,
φ) = (
ψ,
Dφ). An important tool when examining the Dirac operator is the Schrödinger-Lichnerowicz formula
()
where ∇
* is the adjoint of ∇ and Ω· is the extension of the Clifford multiplication to differential forms given by
. For the Friedrich connection
, where
f is real valued function one gets a Schrödinger-Lichnerowicz type formula similar to the one obtained by Friedrich in [
2]
()
where Δ
f is the spinorial Laplacian associated with the connection ∇
f.
A complex number λ is an eigenvalue of D if there exists a nonzero eigenspinor ψ ∈ Γ(ΣM)∩L2(ΣM) with Dψ = λψ. The set of all eigenvalues is denoted by σp(D), the point spectrum. We know that if M is closed, the Dirac operator has a pure point spectrum but on open manifolds, the spectrum might have a continuous part. In general, the spectrum of the Dirac operator σ(D) is composed of the point, the continuous and the residual spectrum. For complete manifolds, the residual spectrum is empty and σ(D) ⊂ ℝ. Thus, for complete manifolds, the spectrum can be divided into point and continuous spectrum. But often another decomposition of the spectrum is used: the one into discrete spectrum σd(D) and essential spectrum σess(D).
A complex number λ lies in the essential spectrum of D if there exists a sequence of smooth compactly supported spinors ψi which are orthonormal with respect to the L2-product and
()
The essential spectrum contains all eigenvalues of infinite multiplicity. In contrast, the discrete spectrum
σd(
D): =
σp(
D)∖
σess(
D) consists of all eigenvalues of finite multiplicity. The proof of the next property can be found in [
8]: on a Spin
c complete Riemannian manifold, 0 is in the essential spectrum of
D −
λ if and only if 0 is in the essential spectrum of (
D −
λ)
2, and in this case, there is a normalized sequence
ψi ∈ Γ
c(
ΣM) such that
ψi converges
L2-weakly to 0 with
and
.
Spinor Bundles Associated with Conformally Related Metrics The conformal class of g is the set of metrics , for a real function u on M. At a given point x of M, we consider a g-orthonormal basis {e1, …, en} of TxM. The corresponding -orthonormal basis is denoted by . This correspondence extends to the Spinc level to give an isometry between the associated spinor bundles. We put a “–” above every object which is naturally associated with the metric . Then, for any spinor field ψ and φ, one has , where 〈·, ·〉 denotes the natural Hermitian scalar products on Γ(ΣM), and on . The corresponding Dirac operators satisfy
()
The norms of any real 2-form Ω with respect to
g and
are related by
()
Hijazi [
6] showed that on a spin manifold the energy-momentum tensor verifies
()
where
φ =
e−((n−1)/2)u)ψ. We extend the result to a Spin
c manifold and get the same relation.
Refined Kato Inequalities On a Riemannian manifold (M, g), the Kato inequality states that away from the zeros of any section φ of a Riemannian or Hermitian vector bundle E endowed with a metric connection ∇, we have
()
This could be seen as follows 2 |
φ| |
d(|
φ|)| = |
d(|
φ | )
2 | = 2 | 〈∇
φ,
φ〉 | ≤ 2 |
φ∥∇
φ|. In [
17], refined Kato inequalities were obtained for sections in the kernel of first order elliptic differential operators
P. They are of the form |
d(|
φ|)| ≤
kP | ∇
φ|, where
kP is a constant depending on the operator
P and 0 <
kP < 1. Without the assumption that
φ ∈
ker P, we get away from the zero set of
φ
()
A proof of (
2.13) can be found in [
8,
17,
18] or [
9]. In [
17], the constant
kP is determined in terms of the conformal weights of the differential operator
P. For the Dirac operator
D and for
D −
λ, where
λ ∈
ℝ, we have
.
3. Proof of the Hijazi Type Inequalities
First, we follow the main idea of the proof of the original Hijazi inequality in the compact case [6, 7], and its proof on spin noncompact case obtained by Grosse [9]. We choose the conformal factor with the help of an eigenspinor and we use cutoff functions near its zero set and near infinity to obtain compactly supported test functions.
Proof of Theorem 1.2. Let ψ ∈ C∞(M, S)∩L2(M, S) be a normalized eigenspinor; that is, Dψ = λψ and ∥ψ∥ = 1. Its zero set Υ is closed and lies in a closed countable union of smooth (n − 2)-dimensional submanifolds which has locally finite (n − 2)-dimensional Hausdorff measure [19]. We can assume without loss of generality that Υ is itself a countable union of (n − 2)-submanifolds described above. Fix a point p ∈ M. Since M is complete, there exists a cutoff function ηi : M → [0,1] which is zero on M∖B2i(p) and equal 1 on Bi(p), where Bl(p) is the ball of center p and radius l. In between, the function is chosen such that |∇ηi| ≤ 4/i and . While ηi cuts off ψ at infinity, we define another cutoff near the zeros of ψ. Let ρa,ϵ be the function
()
where
r =
d(
x,
Υ) is the distance from
x to
Υ. The constant 0 <
a < 1 is chosen such that
ρa,ϵ(
aϵ) = 0, that is,
a =
e−1/δ. Then,
ρa,ϵ is continuous, constant outside a compact set and Lipschitz. Hence, for
φ ∈ Γ(
ΣM) the spinor
ρa,ϵφ is an element in
for all 1 ≤
r ≤
∞. Now, consider
. These spinors are compactly supported on
M∖
Υ. Furthermore,
with
h = |
ψ|
(n − 2)/(n−1) is a metric on
M∖
Υ. Setting
(
φ =
e−((n − 1)/2)uψ), (
2.3), (
2.10), (
2.11), and the Schrödinger-Lichnerowicz formula imply
()
where
is the spinor field defined in [
6] by
, and where we used
and
(see [
5]). Using
and
, we calculate
()
Inserting (
3.3) and
in the above inequality, we get
()
Moreover, we have
. Thus, with
eu = |
ψ|
2/(n−1) the above inequality reads
()
Hence, we obtain
()
where
μ1 is the infimum of the spectrum of the perturbed conformal Laplacian. With
, we have
()
where
k = 2((2
n − 3)/(
n − 2)). Next, we examine the limits when
a goes to zero. Recall that
is bounded, closed (
n − 2)-
C∞-rectifiable and has still locally finite (
n − 2)-dimensional Hausdorff measure. For fixed
i, we estimate
()
Furthermore, we set
ℬϵ,p : = {
x ∈
Bϵ∣
d(
x,
p) =
d(
x,
Υ)} with
Bϵ : = {
x ∈
M∣
d(
x,
Υ) ≤
ϵ}. For
ϵ sufficiently small each
ℬϵ,p is star shaped. Moreover, there is an inclusion
ℬϵ,p↪
Bϵ(0) ⊂
ℝ2 via the normal exponential map. Then, we can calculate
()
where vol
n−2 denotes the (
n − 2)-dimensional volume and
. The positive constants
c and
c′ arise from vol
n−2(
Υ∩
B2i(
p)) and the comparison of v
g′ with the volume element of the Euclidean metric. Furthermore, for any compact set
K ⊂
M and any positive function
f it holds
, and thus by the monotone convergence theorem, we obtain when
a → 0,
()
When applied to the functions
, with
K =
B2i(
p), we get
()
as
a → 0 and thus,
()
Next, we have to study the limit when
i →
∞: Since
M has finite volume and ∥
ψ∥ = 1, the Hölder inequality ensures that
is bounded. With |∇
ηi| ≤ 4/
i, we get the result. Equality is attained if and only if
for
i →
∞,
a → 0 and Ω ·
ψ =
i(
cn/2) | Ω|
gψ. But we have
()
Since
, we conclude that
has to vanish on
M∖
Υ.
Remark 3.1. By the Cauchy-Schwarz inequality, we have
()
where tr denotes the trace of
ℓψ. Hence, the Hijazi type (
1.6) can be derived. Equality is achieved if and only if the eigenspinor associated with the first eigenvalue
λ1 is a Spin
c Killing spinor. In fact, if equality holds then
λ2 = (
n/4(
n − 1))
μ1 = (1/4)
μ1+|
ℓψ|
2 and equality in (
3.14) is satisfied. Hence, it is easy to check that
()
Finally,
ℓψ(
X) = ±(
λ/
n)
X and
. By (
1.8) we get that
is a generalized Killing Spin
c spinor and hence a Killing Spin
c spinor for
n⩾4 [
1, Theorem 1.1]. The function
e−u is then constant and
ψ is a Killing Spin
c spinor. For
n = 3, we follow the same proof as in [
1]. First we suppose that
λ1 ≠ 0, because if
λ1 = 0, the result is trivial. We consider the Killing vector
defined by
()
In [
1], it is shown that
,
and
, where * is the Hodge operator defined on differential forms. Since
, the 2-form Ω can be written
, where
α is a real 1-form and
F a function. We have [
1]
()
But equality in (
1.1) is achieved so
, which implies that
is collinear to
and hence
is collinear to
. Moreover,
so
. It is easy to check that
which gives
. Because of
and
is collinear to
, we have
and finally
α = 0. Using (
3.17), we obtain
d(
e−u) = 0, that is,
e−u is constant, hence
is a Killing Spin
c spinor and finally
ψ is also a Spin
c Killing spinor.
Proof of Theorem 1.1. The proof of Theorem 1.1 is similar to Theorem 1.2. It suffices to take , that is, eu = 1. The Friedrich type (1.1) is obtained from the Hijazi type (1.6).
Next, we want to prove Theorem 1.3 using the refined Kato inequality.
Proof of Theorem 1.3. We may assume vol(M, g) = 1. If λ is in the essential spectrum of D, then 0 is in the essential spectrum of D − λ and of (D − λ) 2. Thus, there is a sequence ψi ∈ Γc(ΣM) such that ∥(D − λ) 2ψi∥→0 and ∥(D − λ)ψi∥→0, while ∥ψi∥ = 1. We may assume that . That can always be achieved by a small perturbation. Now let, 1/2 ≤ β ≤ 1. Then . First, we will show that the sequence ∥d(|ψi|β)∥ is bounded: by the Cauchy-Schwarz inequality, we have
()
Using (
2.3) and the Schrödinger-Lichnerowicz type (
2.7), we obtain
()
The Cauchy-Schwarz inequality and the refined Kato (
2.12) for the connection ∇
λ imply
()
Hence, we have
()
Since
S −
cn | Ω| is bounded from below,
is also bounded. Thus, with ∥(
D −
λ)
ψi∥→0, we see that ∥
d |
ψi|
β∥ is also bounded. Next, we fix
α = (
n − 2)/(
n − 1) and obtain
()
where we used the definition of
μ1 as the infimum of the spectrum of
LΩ and |
ψi|
αd*d(|
ψi|
α) = (
α/2) |
ψi|
2α−2d*d(|
ψi|
2) −
α(
α − 2) |
ψi|
2α−2 |
d(|
ψi|)|
2. Next, using the following:
()
we have
()
The limit of the last two summands vanish since
()
For the other summand, we use the Kato type (
2.13)
()
which holds outside the zero set of
ψ, and where
. Thus, for
n ≥ 5, we can estimate
()
For
n ≥ 5, we have 1 ≥ (
n − 3)/(
n − 1) ≥ 1/2 and, thus, ∥
d |
ψi|
(n−3)/(n−1)∥ is bounded. Together with ∥(
D −
λ)
ψi∥→0, we obtain the following: for all
ϵ > 0, there is an
i0 such that for all
i ≥
i0, we have
()
Hence, we have
μ1/4 ≤ ((
n − 1)/
n)
λ2.
Acknowledgment
The author would like to thank Oussama Hijazi for his support and encouragements.