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Extrinsic eigenvalue estimates for Dirac operator

In this note, we prove lower and upper bounds for Dirac operators of submanifolds in certain ambient manifolds in terms of conformal and extrinsic quantities.


Introduction
The eigenvalues of Dirac operators on spin manifolds are extensively studied. In 1980, Friedrich [9] first derived the lower bound of the first eigenvalues of a Dirac operator D in terms of the scalar curvature S M and dimension m of the underling manifold M m : Since then, various kinds of estimates in terms of intrinsic geometric quantities have been proved (see e.g. [11,13] and the references therein). A well known result of Hijazi [16] states that  [2] proved that where g M is the genus of M.
On the other hand, the submanifold theory for Dirac operators was introduced by Bär in [3]. Let M m ι ֒→M m+n be a closed oriented connected spin submanifold isometrically immersed in a Riemannian spin manifoldM m+n with fixed spin structures. Milnor's Lemma claims that there is a unique spin structure [23] on the normal bundle N of M inM. Denoted by∇, ∇ and ∇ ⊥ the Levi-Civita connections onM, M, N respectively. Denoted by ∇ ΣM , ∇ ΣM and ∇ ΣN the Levi-Civita connections on ΣM, ΣM and ΣN respectively. For every X, Y ∈ T M, definē Denoted byγ, γ, γ ⊥ the Clifford multiplications on ΣM, ΣM and ΣN respectively. Denoted byD, D, D ⊥ the Dirac operators on ΣM, ΣM and ΣN respectively. Let A µ be the shape operator of M inM with respect to the normal vector field µ, B be the second fundamental form of M inM and H be the normalized mean curvature vector of M inM. If M is a hypersurface ofM, we denote A be the shape operator of M inM with respect to the unit outward normal vector field. Finally, denote R(ι) be the normalized trace of the ambient sectional curvature on the tangent space, i.e., where {e i } is a local orthonormal frame of T M.
Bär [3] derived upper eigenvalue estimates for Dirac operators of closed hypersurfaces in real space forms. According to [3], we know that By D ΣN we mean the Dirac operator on M twisted with the bundle ΣN.
A spinor ψ onM is called a Killing spinor with Killing constant α ∈ C if Bär [3] proved that ifM admits a nontrivial Killing spinor with constant α ∈ R, then the first eigenvalue λ 1 D ΣN of D ΣN (in the sense all other eigenvalue λ of D ΣN satisfying |λ| ≥ |λ 1 | ) satisfies the following estimate If α ∈ √ −1R, he obtained the following estimate Especially, ifM is the Euclidean space R m+n , then IfM is the unit sphere S m+n (1), then Finally, ifM is the hyperbolic space H m+n (−1), then WhenM is the hyperbolic space H m+1 , the result has been improved by Ginoux (cf. [12]). It was proved that For hypersurface M m inM m+1 , given a spinor ψ onM m+1 with no zero on the hypersurface M, Ginoux, Habib and Raulot introduced in [14] a differential operator L ψ acting on smooth functions on M by S − 2Ric(ν, ν) ,S ,Ric and ν are the scalar curvature, the Ricci curvature ofM and the unit outward norm vector field of M inM respectively. It was proved in [14] that ifM admits a nontrivial twistor-spinor ψ with no zero on M, then λ 2 1 (D) ≤ λ 1 (L ψ ). Notice that if ψ is a Killing spinor, then which is independent of ψ. HereÅ is the traceless part of A.
For lower bounds estimates of submanifold Dirac operators, Hijazi and Zhang in [19,20] proved that for D H ϕ = λ H ϕ, ϕ ∈ Γ(ΣM)| M , it holds: where a is some real function on M, M ϕ = {x ∈ M|ϕ(x) 0}, and Under some extra conditions on the extrinsic curvature, they also obtained some lower bound in terms of the Yamabe constant, the curvature and volume of M (see [20] for details).
In this paper, we will prove lower and upper bound estimates for submanifold Dirac operators in terms of conformal and extrinsic quantities. Firstly, we have the following lower bound estimate: Here is the first eigenvalue of the operator L defined by

Moreover, the equality implies that the Ricci curvature of M satisfies
is invariant under the conformal change of the metricḡ. The equality implies that g M = 0 or g M = 1 andÅ = 0, i.e., M is a 2-sphere or a totally umbilici 2-torus.
• If m > 2, the operator is conformally invariant in the following sense. Ifḡ ′ = u 4/(m−2)ḡ is a metric conformal tō g, and L ′ is similarly defined with respect to the metricḡ ′ , then • If m = n = 2, then the first nonzero eigenvalue λ of D ΣN satisfies For a Dirac operator D, let λ i be the eigenvalues. We recall the conformal eigenvalue σ i (D) of D (cf. [1]) given by Here [g] stands for the conformal class of g. Similarly, for a second positive self adjoint elliptic operator L, we have the conformal eigenvalue We say that ψ is a twistor spinor onM if By definition, we know that each Killing spinor is a twistor spinor. For the upper bound of the Dirac operator D ΣN , we will prove the following • If m ≥ 3,

Preliminaries
We first compare the Dirac operator onM with the one on M. We will use notations in [3]. We also refer the reader to [6,15,17,18,19,20] and the references therein. Basic facts concerning Clifford algebras and spinor representations can be found in classical books [4,23].

Algebra preliminaries
Let E be an oriented Euclidean vector space. If dim E = m is even, then the the complex Clifford algebra of E, denoted by Cl(E), has precisely one irreducible module, the spinor module ΣE with dimension 2 m/2 . When restricted to the even subalgebra Cl 0 (E) the spinor module decomposes into even and odd half- . On Σ ± E it acts as ±1. Here {e i } stand for a positively oriented orthonormal frame of E and γ E : Cl(E) −→ End(E) stands for the Clifford multiplication.
If m is odd there are exactly two irreducible modules, Σ 0 E and Σ 1 E, again called spinor modules. In this case Let E and F be two oriented Euclidean vector spaces. Let dim E = m and dim F = n. We will construct the spinor module of E ⊕ F from those of E and F. Case 1. m and n are both even.
In this case Put Σ j ≔ ΣE ⊗ Σ j F for j = 0, 1. As similar to Case 1, we can define γ j : E ⊕ F −→ End Σ j with obvious modification.
Case 3. m is odd and n is even.
This case is symmetric to the second one. Put Σ j ≔ Σ j E ⊗ ΣF and define Set Recall that there exists a vector space isomorphism Φ :

Geometric preliminaries
With respect to the orthogonal splitting TM| M = T M ⊕ N, the Gauss formula says The following equations are well known, i.e., Gauss equations, Codazzi equations and Ricci equations (cf. [25]). For all X, Y, Z ∈ T M, µ ∈ N, Using a standard formula (cf. [23]), we have Here {ν α } is a local orthonormal frame of the normal bundle N. DefineD Recall the Bochner formula (cf., [22,23]), Recall the curvature decomposition ofR. DenotedP by the Schouten tensor which is defined bȳ the Weyl tensorW is given bȳ Therefore, for every orthonormal 4-frame {e A , e B , e C , e D }, we havē Proof. A standard computation (cf. [23]) gives a formula R(e i , e j )e k , e l γ(e i · e j · e k · e l ) + 1 8 R ⊥ (e i , e j )ν α , ν β γ(e i · e j · ν α · ν β ). (2. 2) The first term is According to the Codazzi equation, we compute the second term as follows, where we used the factW i jαβ =R i jαβ , ∀i j, α β. Thus, the second term is i jαβγ (e i · e j · ν α · ν β ).
2. If m = 2, n = 2, Here Therefore, In particular, ifM is flat and M is minimal (cf. [21]), then Proof. The first remark is obvious. For the first part of the second remark, we refer the reader to H. Iriyeh's paper [21]. For the second part, we have The third part follows from the fact

Conformal transformation
Consider a conformal changeḡ ′ = e 2uḡ ofM, then there is an isometric between ΣM and ΣM ′ , ψ → ψ ′ , with Moreover, when restricted to the boundary, we have Here σ ∈ Γ(ΣM), τ ∈ Γ(ΣN). The first equation follows from the fact that the metric restricted to boundary also satisfies g ′ = e 2u g. The second equation can be proved as follows. According to the Gauss formula, In particular, ω ⊥ αβ (X) =ω αβ (X). Here ω ⊥ andω are the connection 1-forms on the normal bundle N and the target manifoldM respectively. Since we have the transformation formula between connection 1-forms Hence, ω αβ (X). Now according to definition of the connection on N, we get (cf. [23]) Now we can prove the following Lemma 2.2. The Dirac operator on the twisted bundle ΣM⊗ΣN is conformal invariant, i.e., for every ψ ∈ Γ (ΣM ⊗ ΣN) Proof. Without loss generality, set ψ = σ ⊗ τ, then according to (2.5), we have Hence by using (2.6), we get

Lower bound estimate
In this section, we will give a conformal lower bound of the first eigenvalue of the Dirac operator on the twisted bundle ΣM ⊗ ΣN, i.e., we will give a proof of Theorem 1.1.
Proof of Theorem 1.1. For every smooth function f , we have the following weighted Bochner formula (cf. [7]) and P ΣN is the twistor operator defined by First, we estimate the curvature term R ΣN ψ . According to the proof of Lemma 2.1, (2.4) implies that In particular, Suppose ψ is an eigenspinor of D ΣN associated with λ, i.e.,
In this case, we choose f as a solution of the following PDE on M. Therefore, according to (3.4), we get Moreover, if n = 2, according to Remark 2.1 A direct computation implies that if D ΣN ψ = λψ, then D ΣN ψ ± = λψ ∓ . Since λ 0, we get ψ ± 0 since ψ is a nontrivial eigenspinor. Using a similar argument mentioned before, one can proved that Therefore, Here we used two formulae and M κ N = 2πχ(N).
In this case, (3.4) implies that for every positive function u, Choose u as an eigenfunction of the operator L, i.e., Moreover, we can choose u satisfying M u 2 = vol(M).

Upper bound estimate
In this section, we want to bound the first conformal eigenvalue of the Dirac operator D ΣN by extrinsic data providedM admits a twistor spinor ψ, i.e., P X ψ ≔∇ ΣM X ψ + 1 m + nγ (X)Dψ = 0, ∀X ∈ TM.
When restricted to the boundary, we first prove the following Lemma. Å α (X) · ν α ψ.