The Hijazi Inequalities on Complete Riemannian Spin^c Manifolds

Abstract

We extend the Hijazi type inequality, involving the energy-momentum tensor, to the eigenvalues of the Dirac operator on complete Riemannian Spin^c 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.

1. Introduction

On a compact Riemannian Spin^c manifold (𝑀^𝑛,𝑔) of dimension 𝑛⩾2, any eigenvalue 𝜆 of the Dirac operator satisfies the Friedrich type inequality [1, 2]𝜆²⩾𝑛4(𝑛−1)inf_𝑀𝑆−𝑐_𝑛|Ω|, (1.1) where 𝑆 denotes the scalar curvature of 𝑀, 𝑐_𝑛=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∇_𝑋𝜆𝜓=−¹𝑛𝑐𝑋⋅𝜓,Ω⋅𝜓=𝑖^𝑛2|Ω|𝜓. (1.2) 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 curvature𝜆²⩾inf_𝑀14𝑐𝑆−^𝑛4|Ω|+ℓ^𝜓2, (1.3) where ℓ^𝜓 is the field of symmetric endomorphisms associated with the field of quadratic forms denoted by 𝑇^𝜓, called the Energy-Momentum tensor. It is defined on the complement set of zeroes of the eigenspinor 𝜓, for any vector field 𝑋 by 𝑇^𝜓(𝑋)=Re𝑋⋅∇_𝑋𝜓𝜓,𝜓². (1.4) Equality holds in (1.3) if and only, for all 𝑋∈Γ(TM), we have∇_𝑋𝜓=−ℓ^𝜓(𝑋)𝑐⋅𝜓,Ω⋅𝜓=𝑖^𝑛2|Ω|𝜓, (1.5) where 𝜓 is an eigenspinor associated with the first eigenvalue 𝜆₁. By definition, the trace tr(ℓ^𝜓) of ℓ^𝜓, 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, |ℓ^𝜓|²⩾(tr(ℓ^𝜓))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 satisfies𝜆²⩾𝑛4()𝜇𝑛−1¹, (1.6) where 𝜇₁ is the first eigenvalue of the perturbed Yamabe operator defined by 𝐿^Ω=𝐿−𝑐_𝑛|Ω|_𝑔=4((𝑛−1)/(𝑛−2))Δ+𝑆−𝑐_𝑛|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 satisfies𝜆²⩾14𝜇₁+inf_𝑀ℓ^𝜓2. (1.7) Equality in (1.7) holds if and only, for all 𝑋∈Γ(TM), we have∇_𝑋𝜑=−ℓ^𝜑(𝑋)⋅𝑐𝜑,Ω⋅𝜓=𝑖^𝑛2|Ω|_𝑔𝜓, (1.8) where 𝜑=𝑒^{−((𝑛−1)/2)𝑢}𝜓, the spinor field 𝜓 is the image of 𝜓 under the isometry between the spinor bundles of (𝑀^𝑛,𝑔) and (𝑀^𝑛,𝑔=𝑒^2𝑢𝑔), 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 (𝑀^𝑛,𝑔) be a complete Riemannian Spin^c 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 Spin^c manifolds of finite volume and equality also holds if and only if the eigenspinor associated with the first eigenvalue 𝜆₁ is a Killing Spin^c 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 (𝑀^𝑛,𝑔) be a complete Riemannian Spin^c 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 Spin^c 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 Spin^c 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 Spin^c manifolds the Hijazi type (1.6) obtained by Grosse in [9] on spin manifolds.

Theorem 1.3. Let (𝑀^𝑛,𝑔) be a complete Riemannian Spin^c manifold of dimension 𝑛≥5 with finite volume. Furthermore, assume that 𝑆−𝑐_𝑛|Ω| is bounded from below. If 𝜆 is in the essential spectrum of the Dirac operator 𝜎_ess(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 have𝜆⁺≥4𝜋Area𝑀²,,𝑔 (1.9) where 𝜆⁺=inf_𝜑∈𝐶^∞_𝑐(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 or Spin^c 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 Spin^c 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 Spin^c manifold carrying a parallel spinor [13, 14]. Moreover, in low dimensions, the existence, on a spin or Spin^c 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 Spin^c 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 Spin^c Manifolds
Let (𝑀^𝑛,𝑔) be a connected oriented Riemannian manifold of dimension 𝑛⩾2 without boundary. Furthermore, let 𝑆𝑂𝑀 be the 𝑆𝑂_𝑛-principal bundle over 𝑀 of positively oriented orthonormal frames. A Spin^c structure of 𝑀 is a Spin^𝑐_𝑛-principal bundle (Spin^c𝑀,𝜋,𝑀) and an 𝕊¹-principal bundle (𝕊¹𝑀,𝜋,𝑀) together with a double covering given by 𝜃∶Spin^c𝑀→𝑆𝑂𝑀×_𝑀𝕊¹𝑀 such that 𝜃(𝑢𝑎)=𝜃(𝑢)𝜉(𝑎), for every 𝑢∈Spin^c𝑀 and 𝑎∈Spin^𝑐_𝑛, where 𝜉 is the 2-fold covering of Spin^𝑐_𝑛 over 𝑆𝑂_𝑛×𝕊¹. Let Σ𝑀∶=Spin^c𝑀×_𝜌𝑛Σ_𝑛 be the associated spinor bundle, where Σ_𝑛=ℂ^2[𝑛/2] and 𝜌_𝑛∶Spin^𝑐_𝑛→End(Σ_𝑛) 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 Γ_𝑐(Σ𝑀). The spinor bundle Σ𝑀 is equipped with a natural Hermitian scalar product, denoted by ⟨⋅,⋅⟩, satisfying 𝑋⋅𝜓,𝜑=−𝜓,𝑋⋅𝜑forevery𝑋∈Γ(𝑇𝑀),𝜓,𝜑∈Γ(Σ𝑀), (2.1) where 𝑋⋅𝜓 denotes the Clifford multiplication of 𝑋 and 𝜓. With this Hermitian scalar product we define an 𝐿²-scalar product 𝜓,𝜑=_𝑀𝜓,𝜑v_𝑔, (2.2) 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 𝜔^𝑀 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 𝐹_𝐴=𝑑𝐴, that is, 𝐹_𝐴=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] 𝑐𝑖Ω⋅𝜓,𝜓⩾−^𝑛2|Ω|_𝑔𝜓², (2.3) where |Ω|_𝑔 is the norm of Ω given by |Ω|²_𝑔=∑_𝑖<𝑗(Ω_𝑖𝑗)². Moreover, equality holds in (2.3) if and only if 𝑐Ω⋅𝜓=𝑖^𝑛2|Ω|_𝑔𝜓. (2.4) The Dirac operator is a first-order elliptic operator locally given by 𝐷=^𝑛_𝑖=1𝑒_𝑖⋅∇_𝑒𝑖. (2.5) 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 𝐷²=∇^∗1∇+4𝑆Id_Γ(Σ𝑀)+𝑖2Ω⋅, (2.6) where ∇^∗ is the adjoint of ∇ and Ω⋅ is the extension of the Clifford multiplication to differential forms given by (𝑒^∗_𝑖∧𝑒^∗_𝑗)⋅𝜓=𝑒_𝑖⋅𝑒_𝑗⋅𝜓. For the Friedrich connection ∇^𝑓_𝑋𝜓=∇_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] 𝐷−𝑓²𝜓=Δ^𝑓𝑆𝜓+4+𝑛−1𝑛𝑓²𝑖𝜓+2Ω⋅𝜓−𝑛−1𝑛2𝑓𝐷𝜓+∇𝑓⋅𝜓, (2.7) 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 𝜎_𝑝(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 𝜎_𝑑(𝐷) and essential spectrum 𝜎_ess(𝐷).
A complex number 𝜆 lies in the essential spectrum of 𝐷 if there exists a sequence of smooth compactly supported spinors 𝜓_𝑖 which are orthonormal with respect to the 𝐿²-product and (𝐷−𝜆)𝜓_𝑖𝐿2⟶0. (2.8) The essential spectrum contains all eigenvalues of infinite multiplicity. In contrast, the discrete spectrum 𝜎_𝑑(𝐷)∶=𝜎_𝑝(𝐷)⧵𝜎_ess(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 𝜓_𝑖∈Γ_𝑐(Σ𝑀) such that 𝜓_𝑖 converges 𝐿²-weakly to 0 with ‖(𝐷−𝜆)𝜓_𝑖‖_𝐿²→0 and ‖(𝐷−𝜆)²𝜓_𝑖‖_𝐿²→0.

Spinor Bundles Associated with Conformally Related Metrics
The conformal class of 𝑔 is the set of metrics 𝑔=𝑒^2𝑢𝑔, for a real function 𝑢 on 𝑀. At a given point 𝑥 of 𝑀, we consider a 𝑔-orthonormal basis {𝑒₁,…,𝑒_𝑛} of 𝑇_𝑥𝑀. The corresponding 𝑔-orthonormal basis is denoted by {𝑒₁=𝑒^−𝑢𝑒₁,…,𝑒_𝑛=𝑒^−𝑢𝑒_𝑛}. This correspondence extends to the Spin^c 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 𝐷𝑒^{−((𝑛−1)/2)𝑢}𝜓=𝑒^{−((𝑛+1)/2)𝑢}𝐷𝜓. (2.9) The norms of any real 2-form Ω with respect to 𝑔 and 𝑔 are related by |Ω|_𝑔=𝑒^−2𝑢|Ω|_𝑔. (2.10) Hijazi [6] showed that on a spin manifold the energy-momentum tensor verifies ℓ^𝜑2_𝑔=𝑒^−2𝑢ℓ^𝜑2_𝑔=𝑒^−2𝑢ℓ^𝜓2_𝑔, (2.11) 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 𝑑𝜑≤∇𝜑. (2.12) 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|)|≤𝑘_𝑃|∇𝜑|, where 𝑘_𝑃 is a constant depending on the operator 𝑃 and 0<𝑘_𝑃<1. Without the assumption that 𝜑∈ker𝑃, we get away from the zero set of 𝜑𝑑𝜑≤𝑃𝜑+𝑘_𝑃∇𝜑. (2.13) A proof of (2.13) can be found in [8, 17, 18] or [9]. In [17], the constant 𝑘_𝑃 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 𝑘_𝐷=𝑘_𝐷−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 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 𝜂_𝑖∶𝑀→[0,1] which is zero on 𝑀⧵𝐵_2𝑖(𝑝) and equal 1 on 𝐵_𝑖(𝑝), where 𝐵_𝑙(𝑝) is the ball of center 𝑝 and radius 𝑙. In between, the function is chosen such that |∇𝜂_𝑖|≤4/𝑖 and 𝜂_𝑖∈𝐶^∞_𝑐(𝑀). While 𝜂_𝑖 cuts off 𝜓 at infinity, we define another cutoff near the zeros of 𝜓. Let 𝜌_𝑎,𝜖 be the function 𝜌_𝑎,𝜖(𝑥)=0for𝜖𝑟<𝑎𝜖,1−𝛿ln𝑟for1𝑎𝜖≤𝑟≤𝜖,for𝜖<𝑟, (3.1) where 𝑟=𝑑(𝑥,Υ) is the distance from 𝑥 to Υ. The constant 0<𝑎<1 is chosen such that 𝜌_𝑎,𝜖(𝑎𝜖)=0, that is, 𝑎=𝑒^−1/𝛿. Then, 𝜌_𝑎,𝜖 is continuous, constant outside a compact set and Lipschitz. Hence, for 𝜑∈Γ(Σ𝑀) the spinor 𝜌_𝑎,𝜖𝜑 is an element in 𝐻^𝑟₁(Σ𝑀) for all 1≤𝑟≤∞. Now, consider Ψ∶=𝜂_𝑖𝜌_𝑎,𝜖𝜓∈𝐻^𝑟₁(Σ𝑀). These spinors are compactly supported on 𝑀⧵Υ. Furthermore, 𝑔=𝑒^2𝑢𝑔=ℎ^4/(𝑛−2)𝑔 with ℎ=|𝜓|^{(𝑛−2)/(𝑛−1)} is a metric on 𝑀⧵Υ. Setting Φ∶=𝑒^{−((𝑛−1)/2)𝑢}Ψ (𝜑=𝑒^{−((𝑛−1)/2)𝑢}𝜓), (2.3), (2.10), (2.11), and the Schrödinger-Lichnerowicz formula imply ∇^ℓΦΦ²_𝑔=𝐷Φ²_𝑔−14_𝑀−Υ𝑆Φ²v_𝑔−_𝑀−Υℓ^Φ2Φ²v_𝑔−_𝑀−Υ𝑖2Ω⋅Φ,Φv_𝑔⩽𝐷Φ²_𝑔−14_𝑀𝑆𝑒^2𝑢−𝑐_𝑛|Ω|_𝑔|Ψ|²𝑒^−𝑢v_𝑔−_𝑀ℓ^Ψ2|Ψ|²𝑒^−𝑢v_𝑔=𝐷Φ²_𝑔−14_𝑀ℎ⁻¹𝐿^Ωℎ|Ψ|²𝑒^−𝑢v_𝑔−_𝑀ℓ^Ψ2|Ψ|²𝑒^−𝑢v_𝑔, (3.2) where ∇^ℓ𝜑_𝑋𝜑 is the spinor field defined in [6] by ∇^ℓ𝜑_𝑋𝜑∶=∇_𝑋𝜑+ℓ^𝜑(𝑋)⋅𝜑, and where we used |Φ|²v_𝑔=𝑒^𝑢|Ψ|²v_𝑔 and 𝑆𝑒^2𝑢−𝑐_𝑛|Ω|_𝑔=ℎ⁻¹𝐿^Ωℎ (see [5]). Using 𝐷𝜑=𝜆𝑒^−𝑢𝜑 and ⟨∇(𝜂_𝑖𝜌_𝑎,𝜖)⋅𝜑,𝜑⟩∈𝐶^∞(𝑀,𝑖ℝ), we calculate 𝐷Φ²_𝑔=∇𝜂_𝑖𝜌_𝑎,𝜖⋅𝜑²_𝑔+𝜆²_𝑀𝜂²_𝑖𝜌²_𝑎,𝜖𝑒^{−(𝑛+2)𝑢}𝜑²v_𝑔. (3.3) Inserting (3.3) and ‖∇^ℓΦΦ‖²_𝑔⩾0 in the above inequality, we get ∇𝜂_𝑖𝜌_𝑎,𝜖⋅𝜑²_𝑔≥14_𝑀ℎ⁻¹𝐿^Ωℎ|Ψ|²𝑒^−𝑢v_𝑔+_𝑀ℓ^Ψ2|Ψ|²𝑒^−𝑢v_𝑔−𝜆²_𝑀𝜂²_𝑖𝜌²_𝑎,𝜖𝜑²𝑒^{−(𝑛+2)𝑢}v_𝑔. (3.4) Moreover, we have ‖∇(𝜂_𝑖𝜌_𝑎,𝜖)⋅𝜑‖²_𝑔=∫_𝑀|∇(𝜂_𝑖𝜌_𝑎,𝜖)⋅𝜓|²𝑒^−𝑢v_𝑔. Thus, with 𝑒^𝑢=|𝜓|^2/(𝑛−1) the above inequality reads _𝑀∇𝜂_𝑖𝜌_𝑎,𝜖²𝜓^{2((𝑛−2)/(𝑛−1))}v_𝑔≥14_𝑀𝜂_𝑖𝜌_𝑎,𝜖𝜓^{(𝑛−2)/(𝑛−1)}𝐿^Ω𝜂_𝑖𝜌_𝑎,𝜖𝜓^{(𝑛−2)/(𝑛−1)}v_𝑔−𝜆²_𝑀𝜂²_𝑖𝜌²_𝑎,𝜖𝜓^{2((𝑛−2)/(𝑛−1))}v_𝑔−𝑛−1𝑛−2_𝑀∇𝜂_𝑖𝜌_𝑎,𝜖²𝜓^{2((𝑛−2)/(𝑛−1))}v_𝑔+_𝑀ℓ^𝜓2𝜓^{2((𝑛−2)/(𝑛−1))}𝜂²_𝑖𝜌²_𝑎,𝜖v_𝑔. (3.5) Hence, we obtain 2𝑛−3𝑛−2_𝑀∇𝜂_𝑖𝜌_𝑎,𝜖²𝜓^{2((𝑛−2)/(𝑛−1))}v_𝑔≥𝜇₁4+inf_𝑀ℓ^𝜓2−𝜆²_𝑀𝜂²_𝑖𝜌²_𝑎,𝜖𝜓^{2((𝑛−2)/(𝑛−1))}v_𝑔, (3.6) where 𝜇₁ is the infimum of the spectrum of the perturbed conformal Laplacian. With |𝜂_𝑖∇𝜌_𝑎,𝜖+𝜌_𝑎,𝜖∇𝜂_𝑖|²≤2𝜂²_𝑖|∇𝜌_𝑎,𝜖|²+2𝜌²_𝑎,𝜖|∇𝜂_𝑖|², we have 𝑘_𝑀𝜂²_𝑖∇𝜌_𝑎,𝜖²+𝜌²_𝑎,𝜖∇𝜂_𝑖²𝜓^{2((𝑛−2)/(𝑛−1))}v_𝑔≥𝜇₁4+inf_𝑀ℓ^𝜓2−𝜆²𝜂_𝑖𝜌_𝑎,𝜖𝜓^{(𝑛−2)/(𝑛−1)2}, (3.7) where 𝑘=2((2𝑛−3)/(𝑛−2)). Next, we examine the limits when 𝑎 goes to zero. Recall that Υ∩𝐵_2𝑖(𝑝) is bounded, closed (𝑛−2)-𝐶^∞-rectifiable and has still locally finite (𝑛−2)-dimensional Hausdorff measure. For fixed 𝑖, we estimate _𝑀∇𝜌_𝑎,𝜖²𝜂²_𝑖𝜓^{2((𝑛−2)/(𝑛−1))}v_𝑔≤sup_{𝐵2𝑖(𝑝)}𝜓^{2((𝑛−2)/(𝑛−1))}_{𝐵2𝑖(𝑝)}∇𝜌_𝑎,𝜖²v_𝑔. (3.8) Furthermore, we set ℬ_𝜖,𝑝∶={𝑥∈𝐵_𝜖∣𝑑(𝑥,𝑝)=𝑑(𝑥,Υ)} with 𝐵_𝜖∶={𝑥∈𝑀∣𝑑(𝑥,Υ)≤𝜖}. For 𝜖 sufficiently small each ℬ_𝜖,𝑝 is star shaped. Moreover, there is an inclusion ℬ_𝜖,𝑝↪𝐵_𝜖(0)⊂ℝ² via the normal exponential map. Then, we can calculate _{𝐵𝜖∩𝐵2𝑖(𝑝)}∇𝜌_𝑎,𝜖²v_𝑔≤vol_𝑛−2Υ∩B_2𝑖𝑝sup_{𝑥∈Υ∩𝐵2𝑖(𝑝)ℬ𝜖,𝑥⧵ℬ𝑎𝜖,𝑥}∇𝜌_𝑎,𝜖²v_𝑔′≤𝑐vol_𝑛−2Υ∩𝐵_2𝑖𝑝_{𝐵𝜖(0)⧵𝐵𝑎𝜖(0)}∇𝜌_𝑎,𝜖²v_𝑔𝐸≤𝑐^′𝜖_𝑎𝜖𝛿²𝑟𝑑𝑟=−𝑐^′𝛿²ln𝑎=𝑐^′𝛿⟶0for𝑎⟶0, (3.9) where vol_𝑛−2 denotes the (𝑛−2)-dimensional volume and 𝑔′=𝑔_{|ℬ𝜖,𝑝}. The positive constants 𝑐 and 𝑐^′ arise from vol_𝑛−2(Υ∩𝐵_2𝑖(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 𝜌²_𝑎,𝜖𝑓↗𝑓, and thus by the monotone convergence theorem, we obtain when 𝑎→0, _𝐾𝜌²_𝑎,𝜖𝑓v_𝑔⟶_𝐾𝑓v_𝑔. (3.10) When applied to the functions 𝜌²_𝑎,𝜖|∇𝜂_𝑖|²|𝜓|^{2((𝑛−2)/(𝑛−1))}, with 𝐾=𝐵_2𝑖(𝑝), we get _{𝐵2𝑖(𝑝)}𝜌²_𝑎,𝜖∇𝜂_𝑖²𝜓^{2((𝑛−2)/(𝑛−1))}v_𝑔⟶_{𝐵2𝑖(𝑝)}∇𝜂_𝑖²𝜓^{2((𝑛−2)/(𝑛−1))}v_𝑔, (3.11) as 𝑎→0 and thus, 𝑘_𝑀∇𝜂_𝑖²𝜓^{2((𝑛−2)/(𝑛−1))}v_𝑔≥𝜇₁4+inf_𝑀ℓ^𝜓2−𝜆²_𝑀𝜂²_𝑖𝜓^{2((𝑛−2)/(𝑛−1))}v_𝑔. (3.12) 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))}v_𝑔 is bounded. With |∇𝜂_𝑖|≤4/𝑖, we get the result. Equality is attained if and only if ‖∇^ℓΦΦ‖²_𝑔→0 for 𝑖→∞, 𝑎→0 and Ω⋅𝜓=𝑖(𝑐_𝑛/2)|Ω|_𝑔𝜓. But we have 0⟵∇^ℓΦΦ²_𝑔=𝜂_𝑖𝜌_𝑎,𝜖∇^ℓΦ𝜑+∇𝜂_𝑖𝜌_𝑎,𝜖⋅𝜑_𝑔≥𝜂_𝑖𝜌_𝑎,𝜖∇^ℓ𝜑𝜑_𝑔−∇𝜂_𝑖𝜌_𝑎,𝜖⋅𝜑_𝑔. (3.13) Since ‖∇(𝜂_𝑖𝜌_𝑎,𝜖)⋅𝜑‖_𝑔→0, we conclude that ∇^ℓ𝜑𝜑 has to vanish on 𝑀⧵Υ.

Remark 3.1. By the Cauchy-Schwarz inequality, we have ℓ^𝜓2⩾trℓ^𝜓2𝑛=𝜆²𝑛, (3.14) where tr denotes the trace of ℓ^𝜓. 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+|ℓ^𝜓|² and equality in (3.14) is satisfied. Hence, it is easy to check that 𝑇^𝜓𝑒_𝑖,𝑒_𝑗=0for𝑖≠𝑗,𝑇^𝜓𝑒_𝑖,𝑒_𝑖𝜆=±𝑛. (3.15) Finally, ℓ^𝜓(𝑋)=±(𝜆/𝑛)𝑋 and ℓ^𝜑(𝑋)=𝑒^−𝑢ℓ^𝜓(𝑋)=±(𝜆/𝑛)𝑒^−𝑢𝑋. By (1.8) we get that 𝜑 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 𝜉 defined by 𝑖𝑔𝜉,𝑋=𝑋⋅𝜑,𝜑_𝑔forevery𝑋∈Γ(𝑇𝑀). (3.16) 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] Ω𝜉,⋅=𝜉²𝛼(⋅)=−4𝜆₁𝑑𝑒^−𝑢(⋅),Ω⋅𝜑=−𝑖𝐹𝜑−𝑖𝜉²𝛼⋅𝜑. (3.17) But equality in (1.1) is achieved so Ω⋅𝜑=𝑖(𝑐_𝑛/2)|Ω|_𝑔𝜑, 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 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 𝑔=𝑔, that is, 𝑒^𝑢=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 𝜓_𝑖∈Γ_𝑐(Σ𝑀) such that ‖(𝐷−𝜆)²𝜓_𝑖‖→0 and ‖(𝐷−𝜆)𝜓_𝑖‖→0, while ‖𝜓_𝑖‖=1. We may assume that |𝜓_𝑖|∈𝐶^∞_𝑐(𝑀). That can always be achieved by a small perturbation. Now let, 1/2≤𝛽≤1. Then |𝜓_𝑖|^𝛽∈𝐻²₁(𝑀). First, we will show that the sequence ‖𝑑(|𝜓_𝑖|^𝛽)‖ is bounded: by the Cauchy-Schwarz inequality, we have _{|𝜓𝑖|≠0}𝜓_𝑖^2𝛽−2(𝐷−𝜆)²𝜓_𝑖,𝜓_𝑖v_𝑔≤𝜓_𝑖^2𝛽−1_{{|𝜓𝑖|≠0}}(𝐷−𝜆)²𝜓_𝑖≤𝜓_𝑖^2𝛽−1(𝐷−𝜆)²𝜓_𝑖=(𝐷−𝜆)²𝜓_𝑖. (3.18) Using (2.3) and the Schrödinger-Lichnerowicz type (2.7), we obtain (𝐷−𝜆)²𝜓_𝑖≥_{|𝜓𝑖|≠0}𝜓_𝑖^2𝛽−2∇^𝜆𝜓_𝑖²v_𝑔+2𝛽−1_{|𝜓𝑖|≠0}𝜓_𝑖^2𝛽−3𝑑𝜓_𝑖⋅𝜓_𝑖,∇^𝜆𝜓_𝑖v_𝑔+𝑆4−𝑐_𝑛4|Ω|−𝑛−1𝑛𝜆²𝜓_𝑖^2𝛽v_𝑔−2𝑛−1𝑛𝜆𝜓_𝑖^2𝛽−1_{{|𝜓𝑖|≠0}}(𝐷−𝜆)𝜓_𝑖. (3.19) The Cauchy-Schwarz inequality and the refined Kato (2.12) for the connection ∇^𝜆 imply _{|𝜓𝑖|≠0}𝜓_𝑖^2𝛽−2∇^𝜆𝜓_𝑖²v_𝑔+2𝛽−1_{|𝜓𝑖|≠0}𝜓_𝑖^2𝛽−3𝑑𝜓_𝑖⋅𝜓_𝑖,∇^𝜆𝜓_𝑖v_𝑔≥2𝛽−1_{|𝜓𝑖|≠0}𝜓_𝑖^2𝛽−2𝑑𝜓_𝑖²v_𝑔=12𝛽−1𝛽^2|𝜓_𝑖^|≠0𝑑𝜓_𝑖^𝛽2v_𝑔. (3.20) Hence, we have (𝐷−𝜆)²𝜓_𝑖≥12𝛽−1𝛽^2|𝜓_𝑖^|≠0𝑑𝜓_𝑖^𝛽2v_𝑔+𝑆4−𝑐_𝑛4|Ω|−𝑛−1𝑛𝜆²𝜓_𝑖^2𝛽v_𝑔−2𝑛−1𝑛𝜆(𝐷−𝜆)𝜓_𝑖. (3.21) Since 𝑆−𝑐_𝑛|Ω| is bounded from below, ∫(𝑆−c_𝑛|Ω|)|𝜓_𝑖|^2𝛽v_𝑔≥inf(𝑆−𝑐_𝑛|Ω|)‖𝜓_𝑖‖^2𝛽_2𝛽≥min{inf(𝑆−𝑐_𝑛|Ω|),0} is also bounded. Thus, with ‖(𝐷−𝜆)𝜓_𝑖‖→0, we see that ‖𝑑|𝜓_𝑖|^𝛽‖ is also bounded. Next, we fix 𝛼=(𝑛−2)/(𝑛−1) and obtain 𝜇₁4−𝑛−1𝑛𝜆²≤𝜇₁4−𝑛−1𝑛𝜆²𝜓_𝑖^𝛼2≤14𝜓_𝑖^𝛼𝐿^Ω𝜓_𝑖^𝛼v_𝑔−𝑛−1𝑛𝜆²𝜓_𝑖^𝛼2=𝜓_𝑖^{2((𝑛−2)/(𝑛−1))−2}𝑛𝑑𝜓𝑛−1^𝑖2+12𝑑^∗𝑑𝜓_𝑖²+𝑆4−𝑐_𝑛4|Ω|−𝑛−1𝑛𝜆²𝜓_𝑖²v_𝑔, (3.22) where we used the definition of 𝜇₁ as the infimum of the spectrum of 𝐿^Ω and |𝜓_𝑖|^𝛼𝑑^∗𝑑(|𝜓_𝑖|^𝛼)=(𝛼/2)|𝜓_𝑖|^2𝛼−2𝑑^∗𝑑(|𝜓_𝑖|²)−𝛼(𝛼−2)|𝜓_𝑖|^2𝛼−2|𝑑(|𝜓_𝑖|)|². Next, using the following: 12𝑑^∗𝑑𝜓_𝑖,𝜓_𝑖⩽𝐷²𝜓_𝑖,𝜓_𝑖−14𝑆−𝑐_𝑛|Ω|𝜓_𝑖²−∇𝜓_𝑖²,∇^𝜆𝜓_𝑖²=∇𝜓_𝑖²𝜆−2𝑛Re(𝐷−𝜆)𝜓_𝑖,𝜓_𝑖−𝜆²𝑛𝜓_𝑖², (3.23) we have 𝜇₁4−𝑛−1𝑛𝜆²≤𝜓_𝑖^{2((𝑛−2)/(𝑛−1))−2}𝑛𝑑𝜓𝑛−1^𝑖2−∇^𝜆𝜓_𝑖²v_𝑔+𝜓_𝑖^{2((𝑛−2)/(𝑛−1))−2}(𝐷−𝜆)²𝜓_𝑖,𝜓_𝑖v_𝑔+121−𝑛𝜆𝜓_𝑖^{2((𝑛−2)/(𝑛−1))−2}Re(𝐷−𝜆)𝜓_𝑖,𝜓_𝑖v_𝑔. (3.24) The limit of the last two summands vanish since 𝜓_𝑖^{2((𝑛−2)/(𝑛−1))−2}(𝐷−𝜆)²𝜓_𝑖,𝜓_𝑖v_𝑔≤(𝐷−𝜆)²𝜓_𝑖𝜓_𝑖^{(𝑛−3)/(𝑛−1)}𝜓⟶0,_{𝑖2((𝑛−2)/(𝑛−1))−2}Re(𝐷−𝜆)𝜓_𝑖,𝜓_𝑖v_𝑔≤(𝐷−𝜆)𝜓_𝑖𝜓_𝑖^{(𝑛−3)/(𝑛−1)}⟶0. (3.25) For the other summand, we use the Kato type (2.13) 𝑑𝜓≤(𝐷−𝜆)𝜓+𝑘∇^𝜆𝜓, (3.26) which holds outside the zero set of 𝜓, and where 𝑘=(𝑛−1)/𝑛. Thus, for 𝑛≥5, we can estimate 𝜓_𝑖^2𝛼−2𝑛𝑑𝜓𝑛−1^𝑖2−∇^𝜆𝜓_𝑖²v_𝑔=𝜓_𝑖^2𝛼−2𝑘⁻¹𝑑𝜓_𝑖−∇^𝜆𝜓_𝑖𝑘⁻¹𝑑𝜓_𝑖+∇^𝜆𝜓_𝑖v_𝑔≤𝑘⁻¹_{{|𝑑(|𝜓𝑖|)|≥𝑘|∇𝜆𝜓𝑖|}}𝜓_𝑖^2𝛼−2(𝐷−𝜆)𝜓_𝑖𝑘⁻¹𝑑𝜓_𝑖+∇^𝜆𝜓_𝑖v_𝑔≤2𝑘⁻²_{{|𝑑(|𝜓𝑖|)|≥𝑘|∇𝜆𝜓𝑖|}}𝜓_𝑖^2𝛼−2(𝐷−𝜆)𝜓_𝑖𝑑𝜓_𝑖v_𝑔≤2𝑘⁻²𝑛−1)𝜓𝑛−3(𝐷−𝜆_𝑖𝑑𝜓_𝑖^{(𝑛−3)/(𝑛−1)}. (3.27) For 𝑛≥5, we have 1≥(𝑛−3)/(𝑛−1)≥1/2 and, thus, ‖𝑑|𝜓_𝑖|^{(𝑛−3)/(𝑛−1)}‖ is bounded. Together with ‖(𝐷−𝜆)𝜓_𝑖‖→0, we obtain the following: for all 𝜖>0, there is an 𝑖₀ such that for all 𝑖≥𝑖₀, we have 𝜓_𝑖^{2((𝑛−2)/(𝑛−1))−2}𝑛𝑑𝜓𝑛−1^𝑖2−∇^𝜆𝜓_𝑖²v_𝑔≤𝜖. (3.28) Hence, we have 𝜇₁/4≤((𝑛−1)/𝑛)𝜆².

Acknowledgment

The author would like to thank Oussama Hijazi for his support and encouragements.