Eigenvalues of the Dirac Operator on Manifolds with Boundary

Commun. ath. Phys. 1, 55 65 001 Communications in athematical Physics Springer-Verlag 001 Eigenvalues of the Dirac Operator on anifolds with Boundary Oussama Hijazi 1, Sebastián ontiel, Xiao Zhang 3 1 Institut Élie Cartan, Université Henri Poincaré, Nancy I, B.P. 39, 54506 Vandœuvre-Lès-Nancy Cedex, France. E-mail: hijazi@iecn.u-nancy.fr Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain. E-mail: smontiel@goliat.ugr.es 3 Institute of athematics, Academy of athematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, P.R. China. E-mail: xzhang@math08.math.ac.cn Received: August 000 / Accepted: 15 arch 001 Abstract: Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. For the local boundary conditions, limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary. 1. Introduction It is well known that the spectrum of the Dirac operator on closed spin manifolds detects subtle information on the geometry and the topology of such manifolds see for example 6,8]. In 31, 33, 7, 30], basic properties of the hypersurface Dirac operator are established. This hypersurface Dirac operator appears as the boundary term in the integral Schrödinger Lichnerowicz formula.3 for compact spin manifolds with compact boundary. In fact, the hypersurface Dirac operator is, up to a zero order operator, the intrinsic Dirac operator of the boundary. In this paper, we examine the classical local boundary conditions and certain Atiyah Patodi Singer boundary conditions for the Dirac operator. Here, the spectral resolution of the intrinsic Dirac operator of the boundary is used to define the APS boundary conditions. We first prove self-adjointness and ellipticity of such conditions. Then, systematic use of the modified Levi Civita connections, introduced in 10, 4, 33,11,7,30], is made see also 8,15] for the Dirac operators on submanifolds. Under appropriate curvature assumptions, these modified connections combined with formula.3, yield the corresponding estimates for compact spin manifolds with boundary. The limiting cases are then studied. Such estimates are obtained in Sects. 3 and 4. In Sect. 3 we consider both the local and the above mentioned APS boundary conditions. We first introduce the modified connection 3.1 which allows to establish a Friedrich s type inequality, in case the mean curvature of the boundary is nonnegative. Under the local boundary conditions,

56 O. Hijazi, S. ontiel, X. Zhang the limiting case is then characterized by the existence of a Killing spinor on the compact manifold with minimal boundary see 3.5. Then the energy-momentum tensor is used to define the modified connection 3.7, from which one can deduce inequality 3.9. Finally, in Sect. 4, under the local boundary conditions, the conformal aspect is examined. For example, generalizations of the conformal lower bounds in, 4] are obtained see Remark 9. It might be useful to mention that local and global boundary conditions are introduced in 5] to get optimal extrinsic lower bounds for the first nonnegative eigenvalue of the intrinsic Dirac operator of the boundary. oreover, in 6], the conformal aspect of this setup is examined where a conformal extrinsic lower bound is given.. The Elliptic Boundary Conditions Let be an n-dimensional Riemannian spin manifold with boundary endowed with its induced Riemannian and spin structures. Denote by S the spinor bundle of. Let resp. be the Levi Civita connection of resp. and denote by the same symbol their corresponding lift to the spinor bundle S. Consider the Dirac operator D of defined by on S. It is known 9] that there exists a positive definite Hermitian metric on S which satisfies, for any covector field X ƔT, and any spinor fields ϕ,ψ ƔS, the relation X ϕ, X ψ = X ϕ, ψ,.1 where denotes Clifford multiplication. The connection is compatible with the metric,. Fix a point p and an orthonormal basis {e α } of T p with e 0 the outward normal to and e i tangent to such that for 1 i, j n, i e j p = 0 e j p = 0. Let {e α } be the dual coframe. Then, for 1 i, j n, i e j p = h ij e 0, i e 0 p = h ij e j, where h ij = i e 0,e j are the components of the second fundamental form at p, and we have i = i 1 h ij e 0 e j.. Let H = h ii be the unnormalized mean curvature of. In the above notation, the standard sphere Sr n = Bn1 r has positive mean curvature H = n r. By.1, e0 e j ϕ,ψ = ϕ, e j e 0 ψ. Therefore. implies dϕ, ψ e i = i ϕ,ψ ϕ, i ψ 1 = i ϕ,ψ ϕ, i ψ 1. Hence the connection is also compatible with the metric,. Denote by D the Dirac operator of. In the above orthonormal coframe {e i } of, D = e i i. Thus D is self-adjoint with respect to the metric,. The relation. implies that i e 0 ϕ = e 0 i ϕ.

Dirac Operator on anifolds with Boundary 57 Hence D e 0 ϕ = e 0 D ϕ. Consider the integral form of the Schrödinger Lichnerowicz formula for a compact manifold with compact boundary e ϕ, 0 D ϕ 1 H ϕ = ϕ R 4 ϕ Dϕ..3 It is well-known that there are basically two types of elliptic boundary conditions for the Dirac operator: The local boundary condition and the global Atiyah Patodi Singer APS boundary condition. Such boundary conditions are used in the positive mass theorem for black holes, Penrose conjecture in general relativity and the index theory in topology 13, 14, 0, 1, 34]. The APS boundary condition exists on any spin manifold with boundary 4] see also 16 19], while the local boundary condition requires certain additional structures on manifolds such as the existence of a Lorentzian structure or a chirality operator, etc 1,13,1]. Now we shall show that the local boundary condition exists on certain spin manifolds with a boundary chirality operator. An operator Ɣ defined on C, S is said to be a boundary chirality operator if it satisfies the following conditions: Ɣ = Id,.4 e i Ɣ = 0,.5 e 0 Ɣ = Ɣ e 0,.6 e i Ɣ = Ɣ e i,.7 Ɣ ϕ,ɣ ψ = ϕ, ψ..8 If is a spacelike hypersurface of a spacetime manifold with timelike covector T, then we can let Ɣ = T e 0, where e 0 is the normal covector on. Recall that see 1] for example, an operator F defined on C, S is called a chirality operator on if for all X ƔT, and any spinor fields ϕ,ψ ƔS, one has F = Id, X F = 0, X F = F X, F ϕ,f ψ = ϕ, ψ. Note that such an operator exists if the spin manifold is even dimensional. It is easy to see that if has a chirality operator F, then Ɣ = F e 0 is a boundary chirality operator. In this paper, we consider the following boundary conditions: The local boundary condition. As the eigenvalues of the chirality operator Ɣ are ±1, the corresponding eigenspaces provide local boundary conditions. Ɣ loc = {ϕ C, S, Ɣ ϕ = ϕ }, } Ɣ loc = {ϕ C, S, Ɣ ϕ = ϕ

58 O. Hijazi, S. ontiel, X. Zhang TheAPS type boundary condition. The operator e 0 D is self-adjoint with respect to the induced metric, on. Therefore it has a discrete real spectrum. Let ϕ k k N be the spectral resolution of e 0 D, i.e., e 0 D ϕ k = λ k ϕ k, and consider the corresponding L -orthogonal subspaces Ɣ± APS spanned by the positive and negative eigenspaces of e 0 D, i.e., { Ɣ APS = ϕ C, S, ϕ = c k ϕ k }, λ k >0 { Ɣ APS = ϕ C, S, ϕ = c k ϕ k }. λ k <0 We will consider the APS type boundary conditions corresponding to the projections onto these subspaces. Recall that the original Atiyah Patodi Singer APS boundary condition refers to the spectral resolution of e 0 D H/ instead of e 0 D see for example 1]. Note that if ϕ,ψ Ɣ loc ±, then e 0 ϕ,ψ = e 0 Ɣ ϕ,ɣ ψ = Ɣ e 0 ϕ,ɣ ψ = e 0 ϕ,ψ, hence e 0 ϕ,ψ = 0. On the other hand, if ϕ,ψ Ɣ± APS, then e0 ϕ Ɣ APS, therefore e 0 ϕ,ψ = 0. These facts imply that the Dirac operator D is self-adjoint under either the local boundary conditions or the APS boundary negative and positive conditions. oreover, it has real eigenvalues. Now we define the H k Sobolev norm by ϕ H k = α ϕ L ϕ H k 1, where α is a multi-index. α =k Proposition 1. Under the local boundary condition ϕ Ɣ± loc or the APS boundary condition ϕ Ɣ APS, the Dirac operator D satisfies elliptic estimates: For any k 1, δ>0, there exists C k,δ such that ϕ H k 1 δ Dϕ L C k,δ ϕ H k 1..9 Proof. Note that for any ϕ Ɣ loc or ϕ Ɣloc, D Ɣ ϕ = Ɣ D ϕ, thus ϕ, e 0 D ϕ = Ɣ ϕ,e 0 D Ɣ ϕ = Ɣ ϕ,e 0 Ɣ D ϕ = ϕ, e 0 D ϕ.

Dirac Operator on anifolds with Boundary 59 Therefore ϕ, e 0 D ϕ = 0. If ϕ Ɣ APS, then e ϕ, 0 D ϕ = c k λ k 0. λ k <0 By the Ehrling Gagliardo Nirenberg inequality 1], for each ε > 0, there exists a constant C ε > 0 such that thus.3 implies ϕ L ε ϕ H 1 C ε ϕ L, ϕ H 1 1 δ Dϕ L C δ ϕ L..10 Then a standard argument gives.9. The following corollary is a direct consequence of the Sobolev embedding theorem ϕ C k C ϕ H k n. Corollary. Any eigenspinor of the Dirac operator which satisfies either the local boundary condition ϕ Ɣ± loc or the negative APS boundary condition ϕ ƔAP S is smooth. 3. Lower Bounds for the Eigenvalues In this section, we adapt the arguments used in 7] to the case of spin compact manifolds with boundary. In particular, we get generalizations of basic inequalities on the eigenvalues of the Dirac operator D under the local boundary conditions Ɣ± loc or the negative APS boundary condition Ɣ APS. For this, we use the integral identity.3 together with an appropriate modification of the Levi Civita connection. Let Dϕ = λϕ, where λ is a real constant or a real function. For any real functions a and u, we define a,u i = i a i u a n j ue i e j λ n ei. 3.1 Then a,u ϕ = ϕ λ n ϕ a 1 1 du ϕ n a i u R i ϕ, ϕ λ n R iϕ, e i ϕ 1 1 n = ϕ λ n ϕ a du ϕ a i u i ϕ. Define the functions R a,u by R a,u = R 4a u 4 a u 4 1 1 a du, 3. n

60 O. Hijazi, S. ontiel, X. Zhang where is the positive scalar Laplacian. Then we have a,u ϕ = ϕ λ Ra,u n ϕ R 4 4 a due 0 ϕ. Therefore.3 yields a,u ϕ = 1 1 n λ R a,u 4 ϕ, e 0 D ϕ ] ϕ ϕ a due 0 H ] ϕ. 3.3 Now we generalize Lemma.3 in 11] to the case where a is a real function. Lemma 3. Suppose there exist a spinor field ϕ ƔS, a real number λ and a real functions a and u on such that for all i, 1 i n, i ϕ = λ n ei ϕ a i uϕ a n j ue i e j ϕ. 3.4 Then ϕ is a real Killing spinor, i.e., either a = 0 or du = 0. In particular, the manifold is Einstein. Proof. First, observe that 3.4 implies Dϕ = λϕ. By the Ricci identity see 11], we have 1 R ij e i e j ϕ = e i D i ϕ D ϕ = e i e j j λ n ei a i u a n kue i e k = λ n ei e i e j δ ij j ϕ λ ϕ du da ϕ a uϕ aλdu ϕ 1 n ei e i e j δ ij e k j a k uϕ a j k uϕ a k u j ϕ 1 n = λ a n u a u ϕ n n n 4aλ du da ϕ du ϕ. n n This implies either a = 0ordu = 0. By 3.3 and Lemma 3, we obtain ϕ λ ϕ

Dirac Operator on anifolds with Boundary 61 Theorem 4. Let n be a compact Riemannian spin manifold of dimension n, with boundary, and let λ be any eigenvalue of D under either the local boundary condition Ɣ± loc or the negative APS boundary condition ƔAPS. If there exist real functions a, u on such that H a due 0 on, where H is the mean curvature of, then λ n 4n 1 sup inf R a,u, 3.5 a,u where R a,u is given in 3.. In the limiting case with the local boundary conditions, the associated eigenspinor is a real Killing spinor and is minimal. Note that by 5], under the APS boundary conditions equality in 3.5 could not hold. Now we make use of the energy-momentum tensor see 4] to get lower bounds for the eigenvalues of D. For any spinor field ϕ, we define the associated energy momentum -tensor Q ϕ on the complement of its zero set by, Q ϕ,ij = 1 R e i j ϕ e j i ϕ,ϕ/ ϕ. 3.6 If ϕ is an eigenspinor of D, the tensor Q ϕ is well-defined in the sense of distribution. Let Q,a,u i = i a i u a n j ue i e j Q ϕ,ij e j. 3.7 It is easy to prove that see 7] Q,a,u ϕ = ϕ Q ϕ ϕ a 1 1 du ϕ a i u i ϕ. n Therefore Q,a,u ϕ = λ Ra,u 4 ϕ, e 0 D ϕ Q ϕ ] ϕ a due 0 H ] ϕ. 3.8 Thus we have Theorem 5. Let n be a compact Riemannian spin manifold of dimension n, with boundary, and let λ be any eigenvalue of D under either the local boundary condition Ɣ± loc or the negative APS boundary condition ƔAPS. If there exist real functions a, u on such that H a due 0 on, where H is the mean curvature of, then Ra,u λ sup inf Q ϕ. 3.9 a,u 4 In the limiting case, one has H = adue 0 on.

6 O. Hijazi, S. ontiel, X. Zhang Remark 6. Under either the local boundary condition Ɣ± loc or the negative APS boundary condition Ɣ APS, assume that H 0. Take a = 0oruconstant in 3.5 and 3.9, then one gets Friedrich s inequality 10] λ n 4n 1 inf R 3.10 and the following inequality 4] R λ inf 4 Q ϕ. 3.11 4. Conformal Lower Bounds As in the previous section and under the local boundary conditions Ɣ± loc, we show that the conformal arguments used in 7] combined with the integral formula.3 yield to generalizations of all known lower bounds for the eigenvalues of the Dirac operator. Let g be the metric of. For any real function u on, consider a conformal metric ḡ = e u g. Denote by D the Dirac operator with respect to this conformal metric. If Dϕ = λϕ, then D ψ = λe u ψ, where ψ = e n 1 u ϕ. Note that a,u e i = ei ae u i u a n e u j u e i e j λ n e u e i, u = i e u ei e u ei u = e u u du, Re u = R n 1 u n 1n du, also, on, D e n u ϕ = e n u D ϕ, H = e u H n 1 due 0. Define the function R a,u by n 1 R a,u = R 4 a u 4 a u n 1n 4 na 41 1n a du, 4.1 where is the positive scalar Laplacian. Then apply 3.3 to the conformal metric g,to get a,u ψ v g = ḡ e u 1 1 λ R a,u n 4 { ψ, e 0 D ψ g ] ϕ v g adue 0 H ] } ψ g v g,

Dirac Operator on anifolds with Boundary 63 hence a,u ψ v g = ḡ e u 1 1 λ R ] a,u ϕ v g n 4 e u {ϕ, e 0 D ϕ a n 1 due 0 H ] ϕ } v g. 4. Note that a,u ψ = 0 implies i ϕ = λ n ei ϕ a n i uϕ 1 a n j ue i e j ϕ n see 11], we thus have either a = n or du = 0 by Lemma 3. Thus we obtain: Theorem 7. Let n be a compact Riemannian spin manifold of dimension n, with boundary, and let λ be any eigenvalue of D under the local boundary condition Ɣ± loc. If there exist real functions a, u on such that H a n 1 due 0 on, where H is the mean curvature of, then λ n 4n 1 sup inf R a,u, 4.3 a,u where the function R a,u is given in 4.1. In the limiting case, the associated eigenspinor is a real Killing spinor and either H = due 0 or H = 0 on. Since Q ϕ,ī j = e u Q ϕ,ij under the conformal transformation g = e u g, we apply 3.8 to the conformal metric g,toget Q,a,u ψ v g = ḡ R e λ u a,u 4 e u {ϕ, e 0 D ϕ a n 1 due 0 H Q ϕ ] ϕ v g ] ϕ } v g. 4.4 Thus we have Theorem 8. Let n be a compact Riemannian spin manifold of dimension n, with boundary, and let λ be any eigenvalue of D under the local boundary condition Ɣ± loc. If there exist real functions a, u on such that H a n 1 due 0 on, where H is the mean curvature of, then R λ a,u sup inf Q ϕ. 4.5 a,u 4 In the limiting case one has H = a n 1 due 0 on.

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