# The Hijazi Inequalities on Complete RiemannianSpinc Manifolds

Authors: Roger Nakad
Journal: Advances in Mathematical Physics
 (2011)

---

## Abstract

We extend the Hijazi type inequality, involving the energy-momentum
tensor, to the eigenvalues of the Dirac operator on complete Riemannian Spinc manifolds
without boundary and of finite volume. Under some additional assumptions,
using the refined Kato inequality, we prove the Hijazi type inequality for elements of
the essential spectrum. The limiting cases are also studied.


---

## Body

## 1. Introduction

On a compact RiemannianSpinc manifold (Mn,g) of dimension n⩾2, any eigenvalue λ of the Dirac operator satisfies the Friedrich type inequality [1, 2](1.1)λ2⩾n4(n-1)inf⁡M(S-cn|Ω|),
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 Spinc structure. Equality holds if and only if the eigenspinor ψ associated with the first eigenvalue λ1 is a Spinc Killing spinor; that is, for every X∈Γ(TM), the eigenspinor ψ satisfies(1.2)∇Xψ=-λ1nX⋅ψ,Ω⋅ψ=icn2|Ω|ψ.
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 Spinc 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(1.3)λ2⩾inf⁡M(14S-cn4|Ω|+|lψ|2),
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 (1.4)Tψ(X)=Re〈X⋅∇Xψ,ψ|ψ|2〉.
Equality holds in (1.3) if and only, for all X∈Γ(TM), we have(1.5)∇Xψ=-lψ(X)⋅ψ,Ω⋅ψ=icn2|Ω|ψ,
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 Spinc 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 Spinc manifolds of dimension n⩾3: any eigenvalue λ of the Dirac operator satisfies(1.6)λ2⩾n4(n-1)μ1,
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 Spinc manifolds. In fact, any eigenvalue of the Dirac operator to which is attached an eigenspinor ψ satisfies(1.7)λ2⩾14μ1+inf⁡M|lψ|2.
Equality in (1.7) holds if and only, for all X∈Γ(TM), we have(1.8)∇¯Xφ¯=-lφ¯(X)⋅¯φ¯,Ω⋅ψ=icn2|Ω|gψ,
where φ¯=e-((n-1)/2)uψ¯, the spinor field ψ¯ is the image of ψ under the isometry between the spinor bundles of (Mn,g) and (Mn,g¯=e2ug), 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 Spinc 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(1.9)λ+≥4πArea(M2,g),
where λ+=infφ∈Cc∞(M)((D2φ,φ)/(φ,φ)) (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 orSpinc 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 concerningSpinc manifolds, the Dirac operator and its conformal covariance. Then, we recall the refined Kato inequality which is crucial for the proof.The Dirac Operator onSpinc 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 Spinnc-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 a∈Spinnc, where ξ is the 2-fold covering of Spinnc over SOn×𝕊1. Let ΣM:=SpincM×ρnΣn be the associated spinor bundle, where Σn=ℂ2[n/2] and ρn:Spinnc→End(Σn) 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
(2.1)〈X⋅ψ,φ〉=-〈ψ,X⋅φ〉foreveryX∈Γ(TM),ψ,φ∈Γ(ΣM),
where X·ψ denotes the Clifford multiplication of X and ψ. With this Hermitian scalar product we define an L2-scalar product
(2.2)(ψ,φ)=∫M〈ψ,φ〉vg,
for any spinors ψ and φ in Γc(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 ofA 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]
(2.3)〈iΩ⋅ψ,ψ〉⩾-cn2|Ω|g|ψ|2,
where |Ω|g is the norm of Ω given by |Ω|g2=∑i<j(Ωij)2. Moreover, equality holds in (2.3) if and only if
(2.4)Ω⋅ψ=icn2|Ω|gψ.
The Dirac operator is a first-order elliptic operator locally given by
(2.5)D=∑i=1nei⋅∇ei.
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
(2.6)D2=∇*∇+14SIdΓ(ΣM)+i2Ω⋅,
where ∇* is the adjoint of ∇ and Ω· is the extension of the Clifford multiplication to differential forms given by (ei*∧ej*)·ψ=ei·ej·ψ. For the Friedrich connection ∇Xfψ=∇Xψ+(f/n)X·ψ, where f is real valued function one gets a Schrödinger-Lichnerowicz type formula similar to the one obtained by Friedrich in [2]
(2.7)(D-f)2ψ=Δfψ+(S4+n-1nf2)ψ+i2Ω⋅ψ-n-1n(2fDψ+∇f⋅ψ),
where Δf is the spinorial Laplacian associated with the connection ∇f.
A complex numberλ 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 numberλ 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
(2.8)‖(D-λ)ψi‖L2⟶0.
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 Spinc 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∈Γc(ΣM) such that ψi converges L2-weakly to 0 with ∥(D-λ)ψi∥L2→0 and ∥(D-λ)2ψi∥L2→0.Spinor Bundles Associated with Conformally Related Metrics
The conformal class ofg is the set of metrics g¯=e2ug, 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 g¯-orthonormal basis is denoted by {e¯1=e-ue1,…,e¯n=e-uen}. 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 g¯. Then, for any spinor field ψ and φ, one has 〈ψ¯,φ¯〉=〈ψ,φ〉, where 〈·,·〉 denotes the natural Hermitian scalar products on Γ(ΣM), and on Γ(ΣM¯). The corresponding Dirac operators satisfy
(2.9)D¯(e-((n-1)/2)uψ¯)=e-((n+1)/2)uDψ¯.
The norms of any real 2-form Ω with respect to g and g¯ are related by
(2.10)|Ω|g¯=e-2u|Ω|g.
Hijazi [6] showed that on a spin manifold the energy-momentum tensor verifies
(2.11)|lφ¯|g¯2=e-2u|lφ|g2=e-2u|lψ|g2,
where φ=e-((n-1)/2)u)ψ. We extend the result to a Spinc 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
(2.12)|d(|φ|)|≤|∇φ|.
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∈kerP, we get away from the zero set of φ(2.13)|d|φ||≤|Pφ|+kP|∇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 kD=kD-λ=(n-1)/n.


## 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 Theorem1.2.
Letψ∈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 ηi∈Cc∞(M). While ηi cuts off ψ at infinity, we define another cutoff near the zeros of ψ. Let ρa,ϵ be the function
(3.1)ρa,ϵ(x)={0forr<aϵ,1-δln⁡ϵrforaϵ≤r≤ϵ,1forϵ<r,
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 H1r(ΣM) for all 1≤r≤∞. Now, consider Ψ:=ηiρa,ϵψ∈H1r(ΣM). These spinors are compactly supported on M∖Υ. Furthermore, g¯=e2ug=h4/(n-2)g with h=|ψ|(n-2)/(n-1) is a metric on M∖Υ. Setting Φ¯:=e-((n-1)/2)uΨ¯ (φ=e-((n-1)/2)uψ), (2.3), (2.10), (2.11), and the Schrödinger-Lichnerowicz formula imply
(3.2)‖∇¯lΦ¯Φ¯‖g¯2=‖D¯Φ¯‖g¯2-14∫M-ΥS¯|Φ¯|2vg¯-∫M-Υ|lΦ¯|2|Φ¯|2vg¯-∫M-Υ〈i2Ω⋅¯Φ¯,Φ¯〉vg¯⩽‖D¯Φ¯‖g¯2-14∫M(S¯e2u-cn|Ω|g)|Ψ|2e-uvg-∫M|lΨ|2|Ψ|2e-uvg=‖D¯Φ¯‖g¯2-14∫M(h-1LΩh)|Ψ|2e-uvg-∫M|lΨ|2|Ψ|2e-uvg,
where ∇Xℓφφ is the spinor field defined in [6] by ∇Xℓφφ:=∇Xφ+ℓφ(X)·φ, and where we used |Φ¯|2vg¯=eu|Ψ|2vg and S¯e2u-cn|Ω|g=h-1LΩh (see [5]). Using D¯φ¯=λe-uφ¯ and 〈∇(ηiρa,ϵ)·¯φ¯,φ¯〉∈C∞(M,iℝ), we calculate
(3.3)‖D¯Φ¯‖g¯2=‖∇(ηiρa,ϵ)⋅¯φ¯‖g¯2+λ2∫Mηi2ρa,ϵ2e-(n+2)u|φ|2vg.
Inserting (3.3) and ∥∇¯ℓΦ¯Φ¯∥g¯2⩾0 in the above inequality, we get
(3.4)‖∇(ηiρa,ϵ)⋅¯φ¯‖g¯2≥14∫M(h-1LΩh)|Ψ|2e-uvg+∫M|lΨ|2|Ψ|2e-uvg-λ2∫Mηi2ρa,ϵ2|φ|2e-(n+2)uvg.
Moreover, we have ∥∇(ηiρa,ϵ)·¯φ¯∥g¯2=∫M|∇(ηiρa,ϵ)·ψ|2e-uvg. Thus, with eu=|ψ|2/(n-1) the above inequality reads
(3.5)∫M|∇(ηiρa,ϵ)|2|ψ|2((n-2)/(n-1))vg≥14∫Mηiρa,ϵ|ψ|(n-2)/(n-1)LΩ(ηiρa,ϵ|ψ|(n-2)/(n-1))vg-λ2∫Mηi2ρa,ϵ2|ψ|2((n-2)/(n-1))vg-n-1n-2∫M|∇(ηiρa,ϵ)|2|ψ|2((n-2)/(n-1))vg+∫M|lψ|2|ψ|2((n-2)/(n-1))ηi2ρa,ϵ2vg.
Hence, we obtain
(3.6)2n-3n-2∫M|∇(ηiρa,ϵ)|2|ψ|2((n-2)/(n-1))vg≥(μ14+inf⁡M|lψ|2-λ2)∫Mηi2ρa,ϵ2|ψ|2((n-2)/(n-1))vg,
where μ1 is the infimum of the spectrum of the perturbed conformal Laplacian. With |ηi∇ρa,ϵ+ρa,ϵ∇ηi|2≤2ηi2|∇ρa,ϵ|2+2ρa,ϵ2|∇ηi|2, we have
(3.7)k∫M(ηi2|∇ρa,ϵ|2+ρa,ϵ2|∇ηi|2)|ψ|2((n-2)/(n-1))vg≥(μ14+inf⁡M|lψ|2-λ2)‖ηiρa,ϵ|ψ|(n-2)/(n-1)‖2,
where k=2((2n-3)/(n-2)). Next, we examine the limits when a goes to zero. Recall that Υ∩B2i(p)¯ is bounded, closed (n-2)-C∞-rectifiable and has still locally finite (n-2)-dimensional Hausdorff measure. For fixed i, we estimate
(3.8)∫M|∇ρa,ϵ|2ηi2|ψ|2((n-2)/(n-1))vg≤sup⁡B2i(p)|ψ|2((n-2)/(n-1))∫B2i(p)|∇ρa,ϵ|2vg.
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
(3.9)∫Bϵ∩B2i(p)|∇ρa,ϵ|2vg≤voln-2(Υ∩B2i(p))sup⁡x∈Υ∩B2i(p)∫Bϵ,x∖Baϵ,x|∇ρa,ϵ|2vg′≤cvoln-2(Υ∩B2i(p))∫Bϵ(0)∖Baϵ(0)|∇ρa,ϵ|2vgE≤c′∫aϵϵδ2rdr=-c′δ2ln⁡a=c′δ⟶0fora⟶0,
where voln-2 denotes the (n-2)-dimensional volume and g′=g|ℬϵ,p. The positive constants c and c′ arise from voln-2(Υ∩B2i(p)) and the comparison of vg′ with the volume element of the Euclidean metric. Furthermore, for any compact set K⊂M and any positive function f it holds ρa,ϵ2f↗f, and thus by the monotone convergence theorem, we obtain when a→0,
(3.10)∫Kρa,ϵ2fvg⟶∫Kfvg.
When applied to the functions ρa,ϵ2|∇ηi|2|ψ|2((n-2)/(n-1)), with K=B2i(p), we get
(3.11)∫B2i(p)ρa,ϵ2|∇ηi|2|ψ|2((n-2)/(n-1))vg⟶∫B2i(p)|∇ηi|2|ψ|2((n-2)/(n-1))vg,
as a→0 and thus,
(3.12)k∫M|∇ηi|2|ψ|2((n-2)/(n-1))vg≥(μ14+inf⁡M|lψ|2-λ2)∫Mηi2|ψ|2((n-2)/(n-1))vg.
Next, we have to study the limit when i→∞: Since M has finite volume and ∥ψ∥=1, the Hölder inequality ensures that ∫M|ψ|2((n-2)/(n-1))vg is bounded. With |∇ηi|≤4/i, we get the result. Equality is attained if and only if ∥∇¯ℓΦ¯Φ¯∥g¯2→0 for i→∞, a→0 and Ω·ψ=i(cn/2)|Ω|gψ. But we have
(3.13)0⟵‖∇¯lΦ¯Φ¯‖g¯2=‖ηiρa,ϵ∇¯lΦ¯φ¯+∇(ηiρa,ϵ)⋅¯φ¯‖g¯≥‖ηiρa,ϵ∇¯lφ¯φ¯‖g¯-‖∇(ηiρa,ϵ)⋅¯φ¯‖g¯.
Since ∥∇(ηiρa,ϵ)·¯φ¯∥g¯→0, we conclude that ∇¯ℓφ¯φ¯ has to vanish on M∖Υ.Remark 3.1.
By the Cauchy-Schwarz inequality, we have(3.14)|lψ|2⩾(tr⁡(lψ))2n=λ2n,
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 Spinc 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
(3.15)Tψ(ei,ej)=0fori≠j,Tψ(ei,ei)=±λn.
Finally, ℓψ(X)=±(λ/n)X and ℓφ¯(X)=e-uℓψ(X)=±(λ/n)e-uX. By (1.8) we get that φ¯ is a generalized Killing Spinc spinor and hence a Killing Spinc spinor for n⩾4 [1, Theorem 1.1]. The function e-u is then constant and ψ is a Killing Spinc 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 ξ¯ defined by
(3.16)ig¯(ξ¯,X)=〈X⋅¯φ¯,φ¯〉g¯foreveryX∈Γ(TM).
In [1], it is shown that dξ¯=2λ1e-u(*ξ¯),∇|ξ¯|2=0 and ξ¯·¯φ¯=i|ξ¯|2φ¯, where * is the Hodge operator defined on differential forms. Since *ξ¯(ξ¯,·)=0, the 2-form Ω can be written Ω=Fξ¯+ξ¯∧α, where α is a real 1-form and F a function. We have [1]
(3.17)Ω(ξ¯,⋅)=|ξ¯|2α(⋅)=-4λ1d(e-u)(⋅),Ω⋅¯φ¯=-iFφ¯-i|ξ¯|2α⋅¯φ¯.
But equality in (1.1) is achieved so Ω·¯φ¯=i(cn/2)|Ω|g¯φ¯, which implies that Ω·¯φ¯ is collinear to φ¯ and hence α·¯φ¯ is collinear to φ¯. Moreover, d(e-u)(ξ¯)=-(1/4λ1)Ω(ξ¯,ξ¯)=0 so α(ξ¯)=0. It is easy to check that 〈α·¯φ¯,φ¯〉g¯=0 which gives α·¯φ¯⊥φ¯. Because of α·¯φ¯⊥φ¯ and α·¯φ¯ is collinear to φ¯, we have α·¯φ¯=0 and finally α=0. Using (3.17), we obtain d(e-u)=0, that is, e-u is constant, hence φ¯ is a Killing Spinc spinor and finally ψ is also a Spinc Killing spinor.Proof of Theorem1.1.
The proof of Theorem1.1 is similar to Theorem 1.2. It suffices to take g¯=g, that is, eu=1. The Friedrich type (1.1) is obtained from the Hijazi type (1.6).Next, we want to prove Theorem1.3 using the refined Kato inequality.Proof of Theorem1.3.
We may assumevol(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 |ψi|∈Cc∞(M). That can always be achieved by a small perturbation. Now let, 1/2≤β≤1. Then |ψi|β∈H12(M). First, we will show that the sequence ∥d(|ψi|β)∥ is bounded: by the Cauchy-Schwarz inequality, we have
(3.18)|∫|ψi|≠0|ψi|2β-2〈(D-λ)2ψi,ψi〉vg|≤‖|ψi|2β-1‖{|ψi|≠0}‖(D-λ)2ψi‖≤‖ψi‖2β-1‖(D-λ)2ψi‖=‖(D-λ)2ψi‖.
Using (2.3) and the Schrödinger-Lichnerowicz type (2.7), we obtain
(3.19)‖(D-λ)2ψi‖≥∫|ψi|≠0|ψi|2β-2|∇λψi|2vg+2(β-1)∫|ψi|≠0|ψi|2β-3〈d|ψi|⋅ψi,∇λψi〉vg+∫(S4-cn4|Ω|-n-1nλ2)|ψi|2βvg-2n-1nλ‖|ψi|2β-1‖{|ψi|≠0}‖(D-λ)ψi‖.
The Cauchy-Schwarz inequality and the refined Kato (2.12) for the connection ∇λ imply
(3.20)∫|ψi|≠0|ψi|2β-2|∇λψi|2vg+2(β-1)∫|ψi|≠0|ψi|2β-3〈d|ψi|⋅ψi,∇λψi〉vg≥(2β-1)∫|ψi|≠0|ψi|2β-2|d(|ψi|)|2vg=(2β-1)1β2∫|ψi|≠0|d(|ψi|β)|2vg.
Hence, we have
(3.21)‖(D-λ)2ψi‖≥(2β-1)1β2∫|ψi|≠0|d(|ψi|β)|2vg+∫(S4-cn4|Ω|-n-1nλ2)|ψi|2βvg-2n-1nλ‖(D-λ)ψi‖.
Since S-cn|Ω| is bounded from below, ∫(S-cn|Ω|)|ψi|2βvg≥inf(S-cn|Ω|)∥ψi∥2β2β≥min{inf(S-cn|Ω|),0} 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
(3.22)μ14-n-1nλ2≤(μ14-n-1nλ2)‖|ψi|α‖2≤14∫|ψi|αLΩ|ψi|αvg-n-1nλ2‖|ψi|α‖2=∫|ψi|2((n-2)/(n-1))-2[nn-1|d(|ψi|)|2+12d*d(|ψi|2)+(S4-cn4|Ω|-n-1nλ2)|ψi|2]vg,
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:
(3.23)12d*d〈ψi,ψi〉⩽〈D2ψi,ψi〉-14(S-cn|Ω|)|ψi|2-|∇ψi|2,|∇λψi|2=|∇ψi|2-2λnRe〈(D-λ)ψi,ψi〉-λ2n|ψi|2,
we have
(3.24)μ14-n-1nλ2≤∫|ψi|2((n-2)/(n-1))-2(nn-1|d(|ψi|)|2-|∇λψi|2)vg+∫|ψi|2((n-2)/(n-1))-2〈(D-λ)2ψi,ψi〉vg+∫2(1-1n)λ|ψi|2((n-2)/(n-1))-2Re〈(D-λ)ψi,ψi〉vg.
The limit of the last two summands vanish since
(3.25)|∫|ψi|2((n-2)/(n-1))-2〈(D-λ)2ψi,ψi〉vg|≤‖(D-λ)2ψi‖‖|ψi|(n-3)/(n-1)‖⟶0,|∫|ψi|2((n-2)/(n-1))-2Re〈(D-λ)ψi,ψi〉vg|≤‖(D-λ)ψi‖‖|ψi|(n-3)/(n-1)‖⟶0.
For the other summand, we use the Kato type (2.13)
(3.26)|d(|ψ|)|≤|(D-λ)ψ|+k|∇λψ|,
which holds outside the zero set of ψ, and where k=(n-1)/n. Thus, for n≥5, we can estimate
(3.27)∫|ψi|2α-2(nn-1|d(|ψi|)|2-|∇λψi|2)vg=∫|ψi|2α-2(k-1|d(|ψi|)|-|∇λψi|)(k-1|d(|ψi|)|+|∇λψi|)vg≤k-1∫{|d(|ψi|)|≥k|∇λψi|}|ψi|2α-2|(D-λ)ψi|(k-1|d(|ψi|)|+|∇λψi|)vg≤2k-2∫{|d(|ψi|)|≥k|∇λψi|}|ψi|2α-2|(D-λ)ψi||d(|ψi|)|vg≤2k-2n-1n-3‖(D-λ)ψi‖‖d(|ψi|(n-3)/(n-1))‖.
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
(3.28)∫|ψi|2((n-2)/(n-1))-2(nn-1|d|ψi||2-|∇λψi|2)vg≤ϵ.
Hence, we have μ1/4≤((n-1)/n)λ2.



---

*Source: 471810-2011-03-20.xml*