TY - JOUR
AU - Nakad, Roger
AB - Abstract We study boundary value problems for the Dirac operator on Riemannian $$\mathrm {Spin^c}$$ manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by Bär and Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound of Hijazi–Montiel–Zhang, involving the mean curvature of the boundary, for the spectrum of the Dirac operator on the noncompact boundary of a $$\mathrm {Spin^c}$$ manifold, and the limiting case is studied. 1. Introduction In the last years, the spectrum of the Dirac operator on hypersurfaces of $${\mathop {\rm Spin}}$$ manifolds has been intensively studied. Indeed, many extrinsic upper bounds have been obtained (see [1, 2, 4, 6, 7, 10] and references therein) and more recently in [16–20, 30], extrinsic lower bounds for the hypersurface Dirac operator are established. From these spectral estimates and their limiting cases, many topological and geometric informations on the hypersurface are derived. In [16], Hijazi, Montiel and Zhang investigated the spectral properties of the Dirac operator on a compact manifold with boundary for the Atiyah–Patodi–Singer type boundary condition (or shortly APS-boundary condition) corresponding to the spectral resolution of the classical Dirac operator of the boundary hypersurface. They proved that, on the compact boundary $$\Sigma = \partial M$$ of a compact Riemannian $${\mathop {\rm Spin}}$$ manifold $$(M^{n+1}, g)$$ of nonnegative scalar curvature $${\rm scal} ^M$$, the first nonnegative eigenvalue of the Dirac operator on the boundary satisfies   \[\lambda _1 \geq \frac n2 \inf _\Sigma H,\] (1) where the mean curvature of the boundary $$H$$ is calculated with respect to the inner normal and assumed to be nonnegative. Equality holds in (1) if and only if $$H$$ is constant and every eigenspinor associated with the eigenvalue $$\lambda _1$$ is the restriction to $$\Sigma$$ of a parallel spinor field on $$M$$ (and hence $$M$$ is Ricci-flat). As application of the limiting case, they gave an elementary $${\mathop {\rm Spin}}$$ proof of the famous Alexandrov theorem: The only closed embedded hypersurface in $$\mathbb {{R}} ^{n+1}$$of constant mean curvature is the sphere of dimension $$n$$. Furthermore, Inequality (1) does not only give an extrinsic lower bound on the first nonnegative eigenvalue, but can also be seen as an obstruction to positive scalar curvature of the interior given only in terms of a neighbourhood of the boundary. More precisely, let a neighbourhood of the boundary $$\Sigma$$ be equipped with a metric of nonnegative scalar curvature and such that the boundary has nonnegative mean curvature. If the lowest positive eigenvalue of the Dirac operator on the boundary is smaller than $$({n}/{2})\inf _{\Sigma } H$$, then the metric cannot be extended to all of $$M$$ such that the scalar curvature remains nonnegative. In this paper, we extend the lower bound (1) to noncompact boundaries of Riemannian $$\mathrm {Spin^c}$$ manifolds under suitable geometric assumptions, see Theorem  1.2. When shifting from the compact case to the noncompact case, many obstacles occur. Moreover, when shifting from the classical $${\mathop {\rm Spin}}$$ geometry to $$\mathrm {Spin^c}$$ geometry, the situation is more general since the spectrum of the Dirac operator will not only depend on the geometry of the manifold but also on the connection of the auxiliary line bundle associated with the fixed $$\mathrm {Spin^c}$$ structure. When we consider a Riemannian $${\mathop {\rm Spin}}$$ or $$\mathrm {Spin^c}$$ manifold with noncompact boundary, the main technical difference to the compact case is that we cannot restrict all our computations to smooth spinors. For compact manifolds, this is possible by using the spectral decomposition of $$L^2$$ by an eigenbasis. For complete manifolds, eigenspinors do not have to exist or even if they do, in general, they do not form an orthonormal basis of $$L^2$$ since continuous spectrum can occur. Additionally, the proof of Inequality (1) in the closed case uses the existence of a solution of a boundary value problem defined under the $$APS$$-boundary condition. While for noncompact boundaries the idea of $$APS$$-boundary conditions can be carried over to noncompact boundaries by using the spectral theorem, it is not clear to us whether they actually define an actual boundary condition, see Example  4.16. To circumvent all these problems, a large part of the paper is devoted to give a generalization of the theory of boundary value problems for noncompact boundaries, see Section  4. We stick to the part of the theory that gives existence of solutions of such boundary value problem, cf. Remark  4.15. For complete manifolds with closed boundary, the theory of boundary value problems is given by Bär and Ballmann [9]. They did not only restrict to the classical Dirac operator but they generalized the traditional theory of elliptic boundary value problems to Dirac-type operators. Additionally, they proved a decomposition theorem for the essential spectrum, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem. In Section  4, we will classify boundary conditions for a Riemannian $$\mathrm {Spin^c}$$ manifold $$(M^{n+1}, g)$$ with noncompact boundary $$\Sigma := \partial M$$ and of bounded geometry, see Definition  2.2. Indeed, we prove in Section  4 that the trace map or the restriction map $$R: \varphi \mapsto \varphi |_\Sigma ,$$ where $$\varphi$$ is a compactly supported smooth spinor on $$M$$ can be extended to a bounded operator   \[R: {\mathop {\rm dom}} \, D_{\mathrm {max}} \longrightarrow H_{-{1}/{2}} (\Sigma , \mathbb {S}_M|_\Sigma ).\] Here, $${\mathop {\rm dom}} \, D_{\text {max}}$$ is the maximal domain of the Dirac operator on $$M$$, $$\mathbb {S}_M|_\Sigma$$ is the restriction of the $$\mathrm {Spin^c}$$ bundle $$\mathbb {S}_M$$ to $$\Sigma$$ and for $$H_{-{1}/{2}} (\Sigma , \mathbb {S}_M|_\Sigma ),$$ see Definition  3.5. The map $$R$$ is not surjective. But in Theorem 3.13, we show that there is an extension map $$\tilde {\mathcal {E}}$$, a right inverse to the restriction map $$R: \Gamma _c^\infty (M, \mathbb {S}_M) \to \Gamma _c^\infty (M, \mathbb {S}_M)$$, such that $$\tilde {\mathcal {E}}R$$ is a bounded linear operator from $${\mathop {\rm dom}} \, D_{\text {max}}$$ to itself. The definition of $$\tilde {\mathcal {E}}$$ uses the extension map for closed boundaries introduced by Bär and Ballmann [9] as local building blocks. This will allow one to equip $$R({\mathop {\rm dom}} \ D_{\mathrm {max}})$$ with a norm $$\Vert .\Vert _{\check {R}}$$ that turns it into a Hilbert space. With these ingredients, we can then classify the closed extensions of the Dirac operator $$D_{cc}$$ acting on smooth compactly supported spinors on $$M$$: For every closed extension of the Dirac operator acting on smooth compactly supported spinors on $$M$$ the set $$B:=R({\mathop {\rm dom}} \, D)\subset H_{-{1}/{2}} (\Sigma , \mathbb {S}_M|_\Sigma )$$ is closed in $$(R({\mathop {\rm dom}} \, D_{\mathrm {max}}), \Vert .\Vert _{\check {R}})$$. Conversely, every closed linear subset $$B\subset (R({\mathop {\rm dom}} \, D_{\mathrm {max}}),\Vert .\Vert _{\check {R}})$$ gives the domain $${\mathop {\rm dom}} \, D_B$$ of a closed extension. Such subsets $$B$$ are called a boundary conditions. Then, we generalize the existence result for boundary value problems to our noncompact setting. For this, we need the notion of $$B$$-coercivity at infinity, see Definition  4.17. This notion generalizes the notion of coercivity at infinity for closed boundaries as used in [9], where this assumption is also needed when characterizing the Fredholmness of the Dirac operator. The $$B$$-coercivity at infinity condition will, in general, depend on the boundary condition $$B$$ and under some additional assumptions, it coincides with the coercivity at infinity condition used in [9]. Theorem 1.1 Let$$M$$be a Riemannian$$\mathrm {Spin^c}$$manifold with boundary$$N$$. Let$$(M,N)$$and the auxiliary line bundle$$L$$over$$M$$be of bounded geometry, cp. Definitions  2.2 and 2.3. Let$$B\subset R({\mathop {\rm dom}} \, D_{\rm max})$$be a boundary condition, and let the Dirac operator  \[D_B\colon \mathrm {dom} D_B\subset L^2(M, \mathbb {S}_M) \longrightarrow L^2(M, \mathbb {S}_M)\]be$$B$$-coercive at infinity. Let$$P_B$$be a projection from$$R({\mathop {\rm dom}} \, D_{\mathrm {max}})$$to$$B$$. Then, for all$$\psi \in L^2(M, \mathbb {S}_M)$$and$$\tilde {\rho }\in {\mathop {\rm dom}} \, D_{\mathrm {max}},$$where$$\psi -D\tilde {\rho }\in (\ker (D_{B})^* )^\perp$$the boundary value problem  \[\left\{ \begin {array}{ll}D\varphi =\psi & {\text {on}} \ M,\\ (\operatorname {Id} - P_{B})R\varphi = (\operatorname {Id} - P_{B})R\tilde {\rho } & {\text {on}} \ \Sigma \end {array}\right .\]has a unique solution$$\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}},$$up to elements of the kernel$$\ker D_B$$. Note that projection just means a linear operator that restricted to $$B$$ acts as identity operator. Theorem  1.1 will be one of the main ingredients to generalize Inequality (1) to our noncompact setting. As boundary condition $$B$$ we will not take the APS-boundary condition as in the closed case but another one: $$B_\pm$$, cf. Section  5. For closed boundaries, the $$B_\pm$$ boundary condition was introduced in [18] to prove a conformal version of (1). Using Theorem  1.1 for the boundary condition $$B_\pm$$ and the $$\mathrm {Spin^c}$$ Reilly inequality on possibly open boundary domains, we obtain the following theorem. Theorem 1.2 Let$$(M^{n+1},g)$$be a complete Riemannian$$\mathrm {Spin^c}$$manifold with boundary$$\Sigma$$and$$L$$be the auxiliary line bundle associated to the$$\mathrm {Spin^c}$$-structure. Assume that$$(M,\Sigma )$$and$$L$$are of bounded geometry. Moreover, we assume that$$\Sigma$$has nonnegative mean curvature$$H$$with respect to its inner unit normal field of$$\Sigma ,$$the Dirac operator$$D$$is$$(B_+ )$$- or$$(B_-)$$-coercive at infinity and that$${\rm scal} ^M +2{i} \Omega \cdot$$is a nonnegative operator where$${i} \Omega$$denotes the curvature$$2$$-form of$$L$$. Then, the infimum$$\lambda _1$$of the nonnegative part of the spectrum of the Dirac operator on$$\Sigma$$satisfies  \[\lambda _1\geq \frac {n}{2} \inf _{\Sigma } H.\]If$$\lambda _1 \geq 0$$is an eigenvalue, then equality holds if and only if$$H$$is constant and any eigenspinor corresponding to$$\lambda _1$$is the restriction of a parallel$$\mathrm {Spin^c}$$spinor$$\varphi$$on$$M$$. The paper is structured as follows: In Section  2, we give all the preliminaries as, for example, the $$\mathrm {Spin^c}$$ Dirac operator and the assumption on the bounded geometry. In Section  3, we review the trace and extension theorem for Sobolev spaces on manifolds of bounded geometry and appropriate noncompact boundary, the spectral decomposition of the Dirac operator on the boundary and analyse an extension map for the maximal domain of the Dirac operator. The theory of boundary values will be generalized to our noncompact setting in Section  4. The special boundary condition $$B_\pm$$ needed to proof the desired inequality is examined in Section  5. In Section  6, we study the coercivity condition for the Dirac operator. Then, we review the spinorial Reilly inequality in order to ready to proof the inequality in Section  8. 2. Notations and preliminaries In this section, we briefly review some basic facts about $$\mathrm {Spin^c}$$ geometry. Then, we give the necessary preliminaries on Sobolev spaces on manifolds with boundary, the Trace Theorem and its implications, some basics of spectral theory, and we recall the closed range theorem. The $$\mathrm {Spin^c}$$Dirac operator. Let $$(M^{n+1}, g)$$ be an $$(n+1)$$-dimensional Riemannian $$\mathrm {Spin^c}$$ manifold with boundary. On such a manifold, we have a Hermitian complex vector bundle $$\mathbb {S}_M$$ endowed with a natural scalar product $$\langle .,.\rangle$$ and with a connection $$\nabla$$ that parallelizes the metric. Moreover, the bundle $$\mathbb {S}_M$$, called the $$\mathrm {Spin^c}$$ bundle, is endowed with a Clifford multiplication denoted by ‘ $$\cdot$$ ’, $$\cdot : TM \rightarrow \mathrm {End}_\mathbb {{C}} (\mathbb {S}_M)$$, such that at every point $$x\in M$$, ‘ $$\cdot$$ ’ defines an irreducible representation of the corresponding Clifford algebra. Hence, the complex rank of $$\mathbb {S}_M$$ is $$2^{[({n+1})/{2}]}$$. Given a $$\mathrm {Spin^c}$$ structure on $$(M^{n+1}, g)$$, one can prove that the determinant line bundle $$\mathrm {det}\ \mathbb {S}_M$$ has a root of index $$2^{[({n+1})/{2}]-1}$$, see [13, Section 2.5]. We denote by $$L$$ this root line bundle over $$M$$ and call it the auxiliary line bundle associated with the $$\mathrm {Spin^c}$$ structure. Locally, a $${\mathop {\rm Spin}}$$ structure always exists. We denote by $$\mathbb {S}_M^{'}$$ the (possibly globally non-existent) spinor bundle. Moreover, the square root of the auxiliary line bundle $$L$$ always exists locally. But, $$\mathbb {S}_M = \mathbb {S}_M^{'} \otimes L^{1/2}$$, see [13, Appendix D; 23]. This essentially means that, while the spinor bundle and $$L^{1/2}$$ may not exist globally, their tensor product (the $$\mathrm {Spin^c}$$ bundle) is defined globally. Thus, the connection $$\nabla$$ on $$\mathbb {S}_M$$ is the twisted connection of the one on the spinor bundle (coming from the Levi-Civita connection) and a fixed connection on $$L$$. We denote by $$\Gamma _c^\infty (M, \mathbb {S}_M)$$ the set of all compactly supported smooth spinors on $$M$$. This allows boundary values if $$\partial M \neq \emptyset$$. The set of smooth spinors that are compactly supported in the interior of $$M$$ is denoted by $$\Gamma _{cc}^\infty (M, \mathbb {S}_M)$$. For abbreviation, we set $$L^2=L^2(M)=L^2(M, \mathbb {S}_M)$$ and $$L^2(\Sigma )=L^2(\Sigma , \mathbb {S}_M|_\Sigma )$$ and analogously for other function spaces. Moreover, $$(.,.)$$ shall always denote the $$L^2$$-scalar product on $$M$$ and $$(.,.)_\Sigma$$ the one on $$\Sigma$$. With these ingredients, we may define the Dirac operator $$D$$ acting on the space of smooth sections of $$\mathbb {S}_M$$, denoted by $$\Gamma ^\infty (M, \mathbb {S}_M),$$ by the composition of the metric connection and the Clifford multiplication. In local coordinates, this reads as   \[D =\sum _{j=1}^{n+1} e_j \cdot \nabla _{e_j}{, }\] where $${\{ }e_j{\} }_{j=1,\ldots , n+1}$$ is an orthonormal basis of $$TM$$. It is a first-order elliptic operator satisfying for all smooth spinors $$\varphi ,\psi$$ on $$M$$ at least one of them being compactly supported   \[(D\psi , \varphi )-(\psi , D\varphi )=-\int _{{\partial M}} \langle \nu \cdot \psi |_{\partial M}, \varphi |_{\partial M}\rangle \,ds,\] (2) where $$(.,.)$$ is the $$L^2$$-scalar product given by $$(\varphi , \psi )=\int _M \langle \varphi , \psi \rangle \,dv$$, $$\partial M$$ is the boundary of $$M$$, $$|_{ \partial M}$$ denotes the restriction to the boundary, $$\nu$$ the inner unit normal vector of the embedding $$\partial M \hookrightarrow M$$, and $$dv$$ (respectively, $$ds$$) is the Riemannian volume form of $$M$$ (respectively, of $$\partial M$$). Hence, if $$\partial M = \emptyset$$, then the Dirac operator is formally self-adjoint with respect to the $$L^2$$-scalar product. An important tool when examining the Dirac operator on $$\mathrm {Spin^c}$$ manifolds is the Schrö dinger–Lichnerowicz formula:   \[D^2 = \nabla ^* \nabla + \frac 14 {\rm scal} ^M\, \mathrm {Id}_{\Gamma (\mathbb {S}_M)}+ \frac {{i} }{2}\Omega \cdot ,\] (3) where $$\nabla ^* $$ is the adjoint of $$\nabla$$ with respect to the $$L^2$$-scalar product, $${i} \Omega$$ is the curvature of the auxiliary line bundle $$L$$ associated with a fixed connection ($$\Omega$$ is a real $$2$$-form on $$M$$) and $$\Omega \cdot$$ is the extension of the Clifford multiplication to differential forms. Example 2.1 (i) A $${\mathop {\rm Spin}}$$ structure can be seen as a $$\mathrm {Spin^c}$$ structure with trivial auxiliary line bundle $$L$$ and trivial connection (and so $${i} \Omega =0$$). (ii) Every almost complex manifold $$(M^{2m = n+1}, g, J)$$ of complex dimension $$m$$ has a canonical $$\mathrm {Spin^c}$$ structure. In fact, the complexified cotangent bundle $$T^*M\otimes \mathbb {C} = \Lambda ^{1,0} M \oplus \Lambda ^{0,1}M$$ decomposes into the $$\pm {i}$$-eigenbundles of the complex linear extension of the complex structure $$J$$. Thus, the spinor bundle of the canonical $$\mathrm {Spin^c}$$ structure is given by   \[\mathbb {S}_M = \Lambda ^{0,* } M =\bigoplus _{r=0}^m \Lambda ^{0,r}M,\] where $$\Lambda ^{0,r}M = \Lambda ^r(\Lambda ^{0,1}M)$$ is the bundle of $$r$$-forms of type $$(0, 1)$$. The auxiliary line bundle of this canonical $$\mathrm {Spin^c}$$ structure is given by $$L = (K_M)^{-1}= \Lambda ^m (\Lambda ^{0,1}M)$$, where $$K_M$$ is the canonical bundle of $$M$$ (see [13, 15, 21, 23]). Let $$\alpha$$ be the Kähler form defined by the complex structure $$J$$, that is, $$\alpha (X, Y)= g(X, JY)$$ for all vector fields $$X,Y\in \Gamma (TM).$$ The auxiliary line bundle $$L= (K_M)^{-1}$$ has a canonical holomorphic connection induced from the Levi-Civita connection whose curvature form is given by $${i} \Omega = {i} \rho$$, where $$\rho$$ is the Ricci $$2$$-form given by $$\rho (X, Y) = \mathrm {Ric} (X, JY)$$. Here, $$\mathrm {Ric}$$ denotes the Ricci tensor of $$M$$. For any other $$\mathrm {Spin^c}$$ structure on $$M^{2m}$$, the spinorial bundle can be written as [13, 15]   \[\mathbb {S}_M = \Lambda ^{0,* }M\otimes \mathcal L,\] where $$\mathcal L^2 = K_M\otimes L$$ and $$L$$ is the auxiliary bundle associated with this $$\mathrm {Spin^c}$$ structure. In this case, the $$2$$-form $$\alpha$$ can be considered as an endomorphism of $$\mathbb {S}_M$$ via Clifford multiplication and we have the well-known orthogonal splitting $$\mathbb {S}_M = \bigoplus _{r=0}^{m}\mathbb {S}_M^r,$$ where $$\mathbb {S}_M^r$$ denotes the eigensubbundle corresponding to the eigenvalue $${i} (m-2r)$$ of $$\alpha$$, with complex rank $$\binom {m}{k}$$. The bundle $$\mathbb {S}_M^r$$ corresponds to $$\Lambda ^{0, r}M\otimes \mathcal L$$. For the canonical $$\mathrm {Spin^c}$$ structure, the subbundle $$\mathbb {S}_M^0$$ is trivial. Hence and when $$M$$ is a Kähler manifold, this $$\mathrm {Spin^c}$$ structure admits parallel spinors (constant functions) lying in $$\mathbb {S}_M^0$$ (see [21]). Of course, we can define another $$\mathrm {Spin^c}$$ structure for which the spinor bundle is given by $$\Lambda ^{* , 0} M =\bigoplus _{r=0}^m \Lambda ^r (T_{1, 0}^*M)$$ and the auxiliary line bundle by $$K_M$$. This $$\mathrm {Spin^c}$$ structure is called the anti-canonical $$\mathrm {Spin^c}$$ structure.Any $$\mathrm {Spin^c}$$ structure on $$(M^{n+1}, g)$$ induces a $$\mathrm {Spin^c}$$ structure on its boundary $$\Sigma = \partial M$$ and we have   \[\left\{ \begin {array}{ll}{\mathbb {S}_M}{|_\Sigma } \simeq \mathbb {S}_{\Sigma } & {\text {if\ $n$ is even,}}\\ {\mathbb {S} ^+_M}{|_\Sigma } \simeq \mathbb {S}_{\Sigma } & {\text {if\ $n$ is odd.}} \end {array}\right .\] We recall that if $$n$$ is odd, the spinor bundle $$\mathbb {S}_M$$ splits into   \[\mathbb {S}_M = {\mathbb {S} ^+_M} \oplus {\mathbb {S} ^-_M},\] by the action of the complex volume element. Moreover, Clifford multiplication with a vector field $$X$$ tangent to $$\Sigma$$ is given by   \[X\bullet \varphi = (X\cdot \nu \cdot \psi ){|_\Sigma },\] where $$\psi \in \Gamma ^\infty (M, \mathbb {S}_M)$$ (or $$\psi \in \Gamma ^\infty (\mathbb {S} ^+_M)$$ if $$n$$ is odd), $$\varphi$$ is the restriction of $$\psi$$ to $$\Sigma$$, ‘ $$\bullet$$’ is the Clifford multiplication on $$M$$. When $$n$$ is odd we also obtain $${\mathbb {S} ^-_M} \simeq \mathbb {S}_{\Sigma }$$. In this case, the Clifford multiplication by a vector field $$X$$ tangent to $$\Sigma$$ is given by $$X\bullet \varphi = - (X\cdot \nu \cdot \psi ){|_\Sigma }$$ and hence we have $${\mathbb {S}_M}{|_\Sigma } \simeq \mathbb {S}_{\Sigma } \oplus \mathbb {S}_{\Sigma }$$. Moreover, the corresponding auxiliary line bundle $$L^\Sigma$$ on $$\Sigma$$ is the restriction to $$\Sigma$$ of the auxiliary line bundle $$L$$ and $${i} \Omega ^\Sigma = {{i} \Omega }{|_\Sigma }$$. We denote by $$\nabla ^\Sigma$$ the spinorial Levi-Civita connection on $$\mathbb {S}_{\Sigma }$$. For all smooth vector fields $$X\in \Gamma ^\infty (T\Sigma )$$ and for every smooth spinor field $$\psi \in \Gamma ^\infty (M, \mathbb {S}_M)$$, we consider $$\varphi = \psi {|_\Sigma }$$ and we have the following $$\mathrm {Spin^c}$$ Gauss formula [15, 22, 23]:   \[(\nabla _X\psi ){|_\Sigma } = \nabla ^{\Sigma }_X \varphi + \tfrac 12 II(X)\bullet \varphi ,\] where $$II$$ denotes the Weingarten map with respect to $$\nu$$. Moreover, let $$D$$ and $$D^\Sigma$$ be the Dirac operators on $$M$$ and $$\Sigma$$. After denoting any smooth spinor and its restriction to $$\Sigma$$ by the same symbol, we have on $$\Sigma$$ (see [15, 22, 23]) that   \[\tilde {D}^{\Sigma } \varphi = \frac {n}{2}H\varphi -\nu \cdot D\varphi -\nabla _{\nu }\varphi ,\] (4)  \[\tilde {D}^{\Sigma }(\nu \cdot \varphi ) = -\nu \cdot \tilde {D}^{\Sigma } \varphi ,\] (5) where $$H = (1/n) \,{\mathrm {tr}}(II)$$ denotes the mean curvature and $$\tilde {D}^{\Sigma } = D^\Sigma$$ if $$n$$ is even and $$\tilde {D}^{\Sigma }=D^\Sigma \oplus (-D^\Sigma )$$ if $$n$$ is odd. Note that $$\sigma (\tilde {D}^{\Sigma })={\{ }\pm \lambda \,|\, \lambda \in \sigma (D^\Sigma ){\} },$$ where $$\sigma (A)$$ denotes the spectrum of an operator $$A$$. Bounded geometry. In this paragraph, we recall the definition of manifolds of bounded geometry. Definition 2.2 [25, Definition 2.2] Let $$(M^{n+1},g)$$ be a complete Riemannian manifold with boundary $$\Sigma$$. We say that $$(M,\Sigma )$$ is of bounded geometry if the following is fulfilled. The curvature tensor of $$M$$ and all its covariant derivatives are bounded. The injectivity radius of $$\Sigma$$ is positive. There is a collar around $$\Sigma$$, that is: there is $$r_\partial >0$$ such that the geodesic collar   \[F: U_\Sigma =[0,r_\partial )\times \Sigma \longrightarrow M,\quad (t,x)\longmapsto \exp _x(t\nu )\] is a diffeomorphism onto its image where $$\nu$$ is the inner unit normal field on $$\Sigma$$. We equip $$U_\Sigma$$ with the induced metric and will identify $$U_\Sigma$$ with its image. There exists $$\varepsilon >0$$ such that the injectivity radius of each point $$x\in M\setminus U_\Sigma$$ is greater or equal than $$\varepsilon$$. The mean curvature of $$\Sigma$$ and all its covariant derivatives are bounded. Definition 2.3 (cp. [26, A.1.1] together with [12, Theorem B]) Let $$E$$ be a hermitian vector bundle over $$M,$$ where $$(M,\Sigma )$$ is of bounded geometry. Then, $$E$$ is said to be of bounded geometry if its curvature and all its covariant derivatives are bounded. Remark 2.4 Note that the above definition contains the usual definition of manifold of bounded geometry without boundary. Moreover, if $$(M,g)$$ is of bounded geometry, then $$(\Sigma , g|_\Sigma )$$ is also of bounded geometry [25, Corollary 2.24]. For the spinor bundle $$\mathbb {S}_M^{'}$$ associated with a $${\mathop {\rm Spin}}$$ structure, the bounded geometry follows automatically from the bounded geometry of $$M$$ (see [3, Section 3.1.3]). For a $$\mathrm {Spin^c}$$ manifold the situation is more general since the $$\mathrm {Spin^c}$$ bundle $$\mathbb {S}_M$$ does not only depend on the geometry of the underlying manifold, but also on the geometry of the auxiliary line bundle $$L$$. But, $$\mathbb {S}_M = \mathbb {S}_M^{'}\otimes L^{1/2}$$, where $$\mathbb {S}_M^{'}$$ is the locally defined spinor bundle, $$L^{{1}/{2}}$$ is locally defined too and $$\mathbb {S}_M$$ is globally defined. Thus, the assumption that $$L$$ is of bounded geometry assures that $$\mathbb {S}_M$$ is also of bounded geometry. Assumption for the rest of the paper:$$(M,\Sigma )$$ and $$L$$ are of bounded geometry. The Sobolev space $$H_1$$on manifolds with boundary. We define the $$H_1 = H_1 (M, \mathbb {S}_M)$$-norm on $$\Gamma _{c}^\infty (M, \mathbb {S}_M)$$ by   \[\Vert \varphi \Vert _{H_1(M, \mathbb {S}_M)}^2 = \Vert \varphi \Vert _{L^2(M, \mathbb {S}_M)}^2 + \Vert \nabla \varphi \Vert _{L^2(M, \mathbb {S}_M)}^2.\] Finally, we define $$H_1=H_1 (M, \mathbb {S}_M)$$ as the closure of $$\Gamma _c^\infty (M, \mathbb {S}_M)$$ with respect to the $$H_1$$-norm defined above. Using the Lichnerowicz formula (3), the Gauß theorem $$(\nabla ^* \nabla \varphi , \varphi )= \Vert \nabla \varphi \Vert ^2_{L^2}+ \int _{\Sigma } \langle \nabla _\nu \varphi ,\varphi \rangle \,ds$$, (2) and (4), we obtain another description of the $$H_1$$-norm: For all $$\varphi \in \Gamma _c^\infty (M, \mathbb {S}_M)$$, we have   \[\Vert \varphi \Vert _{H_1}^2 = \Vert \varphi \Vert _{L^2}^2 + \Vert D\varphi \Vert _{L^2}^2-\int _{M} \frac {{\rm scal} ^M}{4}|\varphi |^2 \,dv - \int _{M} \frac {i}{2} \langle \Omega \cdot \varphi , \varphi \rangle \,dv + \int _{{\Sigma }} \langle \varphi |_\Sigma , D^W (\varphi |_\Sigma ) \rangle \,ds,\] (6) where $$D^W = \tilde {D}^{\Sigma } - (n/2) H$$ is the so-called Dirac–Witten operator. Note that due to the local expression of $$D$$ and the Cauchy–Schwarz inequality, we always have   \[\Vert D\varphi \Vert _{L^2}^2 \leq \int _M \left (\sum _{i=1}^{n+1} |\nabla _{e_i} \varphi |\right )^2 dv\leq (n+1)\Vert \nabla \varphi \Vert _{L^2}^2,\] (7) for all $$\varphi \in H_1(M, \mathbb {S}_M)$$. Spectral theory. Most of the following can be found in [8]. In this paragraph, we shortly review the spectral theory of the Dirac operator $$D\colon H_1(N, \mathbb {S}_N)\subset L^2(N, \mathbb {S}_N) \to L^2(N, \mathbb {S}_N)$$ on a complete Riemannian $$\mathrm {Spin^c}$$ manifold $$N$$ without boundary. Note that we assume that $$N$$ is of bounded geometry, and hence the graph norm of $$D$$, $$\Vert .\Vert _D$$, and the $$H_1$$-norm are equivalent. Then, $$D$$ is self-adjoint and the spectrum is real. A real number $$\lambda$$ is an eigenvalue of $$D$$ if there exists a non-zero spinor $$\varphi \in H_1$$ with $$D\varphi = \lambda \varphi$$. Then, $$\varphi$$ is called an eigenspinor to the eigenvalue $$\lambda$$. Standard local elliptic regularity theory gives that an eigenspinor is always smooth. The set of all eigenvalues is denoted by $$\sigma _p(D^\Sigma ),$$ the point spectrum. If $$N$$ is closed, then the Dirac operator has a pure point spectrum. But on open manifolds, the spectrum might have a continuous part. In general, the spectrum, denoted by $$\sigma (D),$$ is composed of the point, the continuous and the residual spectrum. In case of a self-adjoint operator, as we have, there is no residual spectrum. Often another decomposition of the spectrum is used, the one into discrete spectrum $$\sigma _d(D)$$ and essential spectrum $$\sigma _{\rm ess}(D)$$. A real number $$\lambda$$ lies in the essential spectrum of $$D$$ if there exists a sequence of smooth compactly supported spinors $$\varphi _i$$ which $$\Vert \varphi _i\Vert _{L^2}=1$$, $$\varphi _i$$ converge weakly to zero and   \[\Vert (D - \lambda )\varphi _i \Vert _{ L^2} \longrightarrow 0.\] The essential spectrum contains amongst other elements all eigenvalues of infinite multiplicity. In contrast, the discrete spectrum $$\sigma _{d}(D):= \sigma _{p}(D) \smallsetminus \sigma _{\rm ess}(D)$$ consists of all eigenvalues of finite multiplicity. Closed Range Theorem. Next, we want to recall briefly (a part of) the Closed Range Theorem for later use. Theorem 2.5 [29, p.205] Let$$T: X\to Y$$be a closed linear operator between Banach spaces$$X,Y$$. Then, the range$${\mathop {\rm ran}} (T)$$of$$T$$is closed in$$Y$$if and only if$${\mathop {\rm ran}} (T)=\ker (T^* )^\perp ,$$where$$T^* $$is the adjoint operator of$$T$$and$$\ker (T^* )$$is the kernel of$$T^* $$.A linear operator $$T: X\to Y$$ between Banach spaces is called Fredholm if its kernel is finite-dimensional and its image has finite codimension. 3. Trace theorems and extensions We consider the restriction operator   \[\begin {array}{rl} R: \Gamma _c^\infty (M, \mathbb {S}_{M}) &\longrightarrow \Gamma _c^\infty (\Sigma , \mathbb {S}_{M}|_\Sigma ){,} \\ \varphi &\longmapsto \varphi |_{\Sigma }. \end {array}\] If it is clear from the context that $$R\varphi$$ is considered instead of $$\varphi$$, then we will sometimes abbreviate by using $$\varphi$$ only. The first part of this section will be devoted to see how the restriction operator $$R$$ extends to a bounded linear operator between the Sobolev spaces $$H_1(M, \mathbb {S}_M)$$ and $$H_{{1}/{2}} (\Sigma , \mathbb {S}_M|_\Sigma )$$. This theorem is known as Trace Theorem and is a very classical result for $$\mathbb {{R}} ^n_+ $$ and compact manifolds with boundary. After reviewing the Euclidean result and basic definitions, we will shortly review how this result extends to manifolds $$(M,\Sigma )$$ of bounded geometry. In particular, the restriction operator will have a bounded linear right inverse, that is called extension operator $$\mathcal {E}$$. For more details on the definition of bounded geometry on manifolds with boundary, see [25]. For the equivalence of all those different definitions of Sobolev-norms involved here and the corresponding theorems for submanifolds (not necessarily hypersurfaces), see [14]. For our purpose, Sobolev spaces will not be sufficient later on. The maximal domain of the Dirac operator is bigger than $$H_1(\mathbb {S}_M)$$. The rest of this section is devoted to define an extension operator $$\tilde {\mathcal {E}}$$ such that $$\tilde {\mathcal {E}}R: \Gamma ^\infty _{c}(M, \mathbb {S}_M)\to \Gamma ^\infty _{c}(M, \mathbb {S}_M)$$ extends to a bounded operator with respect to the graph norm of $$D$$. For the definition of $$\tilde {\mathcal {E}}$$ we will use the special extension map introduced by Bär and Ballmann [9] for closed boundaries. 3.1. Trace and extension for Sobolev spaces Trace Theorem for functions on $$\mathbb {R}_{+ }^{n+1}={\{ } (x_0,x_1,\ldots , x_n)\in \mathbb {R}^{n+1}\,|\,x_0\geq 0{\} }$$. We identify the boundary of $$\mathbb {{R}}_+ ^{n+1}$$ with $$\mathbb {{R}} ^n$$. First, we repeat the definition of the Sobolev spaces $$H_s(\mathbb {{R}} ^n, \mathbb {{C}} ^r)$$: Definition 3.1 [27, Definition 3.1] Let $$s\in \mathbb {R}$$. The $$H_s:=H_{s}^2$$-norm of a compactly supported function $$f:\mathbb {R}^n \mapsto \mathbb {C}^r$$ is defined as   \[\Vert f\Vert _{H_s(\mathbb {R}^n, \mathbb {C}^r)}^2:= \int _{\mathbb {R}^n} \left |\hat {f}(\xi )\right |^2 (1+ |\xi |)^s d \xi {,}\] where $$\hat {f}(x):=(2\pi )^{-{n}/{2}} \int _{\mathbb {R}^n} e^{-ix\cdot \xi } f(\xi )\, d\xi$$ denotes the Fourier transform of $$f$$. The space $$H_s(\mathbb {R}^n, \mathbb {C}^r)$$ is then defined as the completion of $$\Gamma _c^\infty (\mathbb {R}^n, \mathbb {C}^r)$$, the space of smooth compactly supported functions on $$\mathbb {R}^n$$ with values in $$\mathbb {C}^r$$, with respect to the $$H_s$$-norm. The spaces $$H_s(\mathbb {{R}} ^{n+1}_+ , \mathbb {C}^r)$$ are defined analogously. Theorem 3.2 [24, Theorems 7.34 and 7.36; 27, Theorem I.3.4; 28, p.138, Remark 1] Let$$s>\frac {1}{2}$$. The restriction map for complex-valued smooth functions$$R: \Gamma _c^\infty (\mathbb {{R}} ^{n+1}_+ )\to \Gamma _c^\infty (\mathbb {{R}} ^{n}),$$$$f\to f|_{\mathbb {{R}} ^n}$$extends to a bounded linear operator from$$H_s(\mathbb {{R}} ^{n+1}_+ )$$to$$H_{(s-{1}/{2})} (\mathbb {{R}} ^{n})$$. Moreover, there is an extension operator$$\mathcal {E}: H_{(s-{1}/{2})} (\mathbb {{R}} ^{n})\to H_s(\mathbb {{R}} ^{n+1}_+ )$$that is a bounded linear operator and a right inverse to$$R$$. The generalization of this theorem to vector-valued Sobolev spaces follows immediately by the following: Let $$f=(f_1, \ldots , f_r): \mathbb {R}^n\to \mathbb {C}^r$$. Then, the norms $$\Vert f\Vert _{H_{s}(\mathbb {R}^n, \mathbb {C}^r)}$$ and $$\sum _{i=1}^r \Vert f_i\Vert _{H_{s}(\mathbb {R}^n, \mathbb {C})}$$ are equivalent. Trace theorem on manifolds of bounded geometry. From now on, let $$M$$ be a Riemannian manifold possibly with boundary and of bounded geometry, as in Definition  2.2. Moreover, let $$E$$ be a hermitian vector bundle over $$M$$. We assume that $$E$$ is also of bounded geometry, see Definition  2.3. To obtain a trace theorem for sections in $$E$$ we need coordinates of the manifold that are adapted to the structure of the boundary. Those will be Fermi coordinates and there will be a adapted synchronous trivialization of $$E$$. This will allow that we can use the trace theorem on $$\mathbb {{R}} ^n$$ on the individual charts to obtain the trace theorem on $$(M,\Sigma )$$. In the following, we restrict to trace theorems for Sobolev spaces over $$L^2$$, for more general domains as Sobolev spaces over $$L^p$$ or Triebel–Lizorkin spaces see [14]. Before we define Sobolev spaces for sections of $$E$$, we introduce Fermi coordinates adapted to the boundary and a corresponding synchronous trivialization of the vector bundle. Definition 3.3 [14, Definition 4.3 and Lemma 4.4; 25, Definition 2.3] Let $$(M^n,\Sigma )$$ be of bounded geometry, see Definition  2.2 and the notions defined therein. Let $$r=\min {\{ } \frac {1}{2} r_{\Sigma }, \frac {1}{4}r_M, \frac {1}{2}r_\partial {\} },$$ where $$r_{\Sigma }$$ is the injectivity radius of $${\Sigma }$$ and $$r_M$$ the one of $$M$$. Let $$p^{\Sigma }_\alpha \in {\Sigma }$$ and $$p_\beta \in M$$ be points such that the metric balls $$B_{r}^{\Sigma }(p^{\Sigma }_\alpha )$$ in $${\Sigma }$$ (that is, with respect to the metric $$g|_{\Sigma }$$) give a uniformly locally finite cover of $${\Sigma };$$ the metric balls $$B_{r}(p_\beta )$$ in $$M$$ cover $$M\setminus U_r(\Sigma ),$$ where $$U_r(\Sigma ):=F([0,r)\times \Sigma )$$ and those balls are uniformly locally finite on all of $$M$$. Let $$(U_\gamma )_{\gamma }$$ be a locally finite covering of $$M$$ where each $$U_\gamma$$ is of the form $$B_{r}(p_\beta )$$ or $$U^{\Sigma }_{p_\alpha ^{\Sigma }}=F([0,2r) \times B^{\Sigma }_{2r} (p_\alpha ^{\Sigma }))$$. By construction, the covering $$(U_\gamma )_{\gamma }$$ is locally finite. Coordinates on $$U_\gamma$$ are chosen to be geodesic normal coordinates around $$p_\beta$$ in case $$U_\gamma =B_{r}(p_\beta )$$. Otherwise, coordinates are given by Fermi coordinates   \[\kappa _\alpha : U^{\Sigma }_{p_\alpha ^{\Sigma }}:=[0, 2r) \times B_{2r}(0)\subset \mathbb {R}^{n} \longrightarrow U^{\Sigma }_{p_{\alpha }^{\Sigma }},\quad (t,x)\longmapsto \exp _{\exp ^{\Sigma }_{p_{\alpha }^{\Sigma }}(x)} (t\nu ){,}\] where $$\nu$$ is the inner normal field of $$\Sigma$$ and $$\exp ^{\Sigma }$$ is the exponential map on $${\Sigma }$$ with respect to the induced metric. We call such coordinates $$(U_\gamma , \kappa _\gamma )_\gamma$$ Fermi coordinates for $$(M,\Sigma )$$. If $$U_\gamma =B_r(p_\gamma )$$, then $$E|_{U_\gamma }$$ is trivialized via parallel transport along radial geodesic and we identify $$E|_{U_\gamma }$$ with the trivial $$\mathbb {C}^r$$-bundle over $$U_\gamma$$. Otherwise, $$E|_{U_\gamma }$$ is trivialized via parallel transport along radial geodesic of the boundary and along the normal direction. The obtained trivialization is denoted by $$(\xi _\gamma )_\gamma$$. In case of manifolds without boundary, the Definition of $$\xi _\gamma$$ in  3.3 is the usual definition of synchronous trivialization as found in [3, Section 3.1.3]. Note that by construction the restriction of a synchronous trivialization of $$E$$ over a manifold $$M$$ to its boundary $$\Sigma$$ gives a synchronous trivialization of $$E|_{\Sigma }$$. Lemma 3.4 [14, Lemma 4.8] There is a partition of unity$$h_\gamma$$subordinated to the Fermi coordinates introduced above fulfilling: For all$$k\in \mathbb {{N}}$$there is$$c_k>0$$such that for all$$\gamma$$and all multi-indices$$\mathfrak {a}=(\mathfrak {a}_1,\ldots , \mathfrak {a}_n)$$with$$|\mathfrak {a}|\leq k$$  \[|D^\mathfrak {a} (h_\gamma \circ \kappa _\gamma )|\leq c_k.\]Here,$$D^\mathfrak {a}= {\partial ^{\mathfrak {a}_1}}/{(\partial x_1)^{\mathfrak {a}_1}}\cdots {\partial ^{\mathfrak {a}_n}}/{(\partial x_n)^{\mathfrak {a}_n}},$$where$$x_i$$are the coordinates. Now, we have all the ingredients to define Sobolev spaces on $$E$$ via local pullback to vector-valued functions over $$\mathbb {R}^n$$. Definition 3.5 [14, Definition 5.9] Let $$s\in \mathbb {R}$$. Let $$(U_\gamma )_{\gamma }$$ be a covering of $$M$$ together with a synchronous trivialization $$\xi _\gamma$$ of $$E$$ as defined above. Moreover, let the covering be locally finite, and let $$h_\gamma$$ be a partition of unity subordinated to $$U_\gamma$$ as in Lemma  3.4. Then,   \[\Vert \varphi \Vert _{H_s(M,E)}^2:=\sum _{\alpha } \Vert \xi _{\alpha * }(h_\alpha \varphi ) \Vert _{H_s(\mathbb {R}_+ ^n,\mathbb {C}^r)}^2.\] Note that up to equivalence the $$H_s$$-norm does not depend on the choices of $$(U_\gamma , h_\gamma , \xi _\gamma )$$, cp. [14, Theorem 4.9, 5.11 and Lemma 5.13]. Remark 3.6 For $$s\in \mathbb {N,}$$ the definition of $$H_s(M,E)$$ from above is equivalent to the usual definition given by   \[\Vert \varphi \Vert _{H_s(M,E)}:=\sum _{i=0}^s \Vert \underbrace {\nabla ^E\cdots \nabla ^E}_{i\ \mathrm {times}} \varphi \Vert _{L^2(M,E)},\] cp. [14, Theorem 5.7; 25]. For $$s\leq t,$$ we have $$\Vert \varphi \Vert _{H_s(M,E)} \leq \Vert \varphi \Vert _{H_t(M,E)}$$. That is seen for $$M=\mathbb {R}^n_+ $$ immediately using $$(1+ |\xi |)^s\leq (1+ |\xi |)^t$$. For general $$M$$, one just lifts this result by using a partition of unity and a synchronous trivialization. Let $$D^\Sigma : \Gamma _c^\infty (\Sigma , \mathbb {S}_{\Sigma }) \to \Gamma _c^\infty (\Sigma , \mathbb {S}_{\Sigma })$$ be the Dirac operator on $$\Sigma$$. For any $$s\in \mathbb {R}$$, there is a unique closed extension of $$D^\Sigma$$ from $$H_{s}(\Sigma , \mathbb {S}_\Sigma ) \to H_{s-1}(\Sigma , \mathbb {S}_\Sigma )$$. Theorem 3.7 Let$$M^{n}$$be a Riemannian manifold with boundary$$\Sigma$$. Assume that$$(M, \Sigma )$$is of bounded geometry and that$$E$$is a hermitian vector bundle over$$M$$that is also of bounded geometry. Then, for all$$s\in \mathbb {R}$$with$$s>\frac {1}{2},$$the operator$$R: \Gamma _c^\infty (M, E) \to \Gamma _c^\infty (\Sigma , E|_\Sigma )$$with$$\varphi \mapsto \varphi |_\Sigma$$extends to a bounded linear operator from$$H_{s}(M, E)$$to$$H_{s-{1}/{2}}(\Sigma , E|_\Sigma )$$. Moreover, there is a bounded right inverse$$\mathcal {E}: H_{s-{1}/{2}}(\Sigma , E_\Sigma )\to H_s(M,E)$$of the trace map$$R: H_s(M,E)\to H_{s-{1}/{2}}(\Sigma , E|_\Sigma )$$. In particular,$$\mathcal {E} (\Gamma _c^\infty (\Sigma , E|_\Sigma )) \subset \Gamma _c^\infty (M, E_M)$$. Proof This theorem is a special case of [14, Theorem 5.14]. We shortly sketch the basic idea: We choose a covering $$U_\gamma$$ together with a synchronous trivialization $$\xi _\gamma$$ of $$E$$ and a subordinated partition of unity $$h_\gamma$$ as in Definition  3.3 and Lemma  3.4. The restrictions $$U_\gamma \cap \Sigma$$ then cover $$\Sigma$$. Let $$\varphi \in H_s(M,E)$$. Then, for all $$\alpha ,$$ we have $$\xi _{\alpha * }(h_\alpha \varphi )\in H_s(\mathbb {R}_+ ^n, \mathbb {C}^r)$$. Thus, there exists a $$C>0$$ with $$\Vert R(\xi _{\gamma * }(h_\gamma \varphi ))\Vert _{H_{s-{1}/{2}}(\mathbb {R}^{n-1}, \mathbb {C}^r)}\leq C \Vert \xi _{\gamma * }(h_\gamma \varphi )\Vert _{H_{s}(\mathbb {R}_+ ^n, \mathbb {C}^r)}$$. With $$R(\xi _{\alpha * }(h_\alpha \varphi ))=\xi _{\alpha * }(h_\alpha R\varphi )$$ we get after summing up that $$\Vert R\varphi \Vert _{H_{s-{1}/{2}}(\Sigma , E|_\Sigma )}\leq C \Vert \varphi \Vert _{H_{s}(M,E)}$$ since $$\xi _\alpha$$ is still a synchronous trivialization for $$E|_\Sigma$$. The rest is proved analogously as the Trace Theorem using the original Euclidean version of the extension map $$\mathcal {E}: H_{s-{1}/{2}}(\mathbb {{R}} ^{n-1})\to H_s(\mathbb {{R}} ^n)$$. The last inclusion follows immediately from $$\mathcal {E} (\Gamma _c^\infty (\mathbb {{R}} ^{n-1})) \subset \Gamma _c^\infty (\mathbb {{R}} ^n)$$. The last theorem gives immediately the following corollary. Corollary 3.8 The map$$\mathcal {E}R: \Gamma _c^\infty (M, E)\to \Gamma _c^\infty (M,E)$$extends to a bounded linear map$$\mathcal {E}R: H_s(M, E)\to H_s(M,E)$$for all$$s>\frac {1}{2}$$. Lemma 3.9 The$$L^2$$-product$$(\varphi , \psi )= \int _\Sigma \langle \varphi ,\psi \rangle \,dv$$for$$\varphi , \psi \in \Gamma _c^\infty (\Sigma , E|_\Sigma )$$extends to a perfect pairing$$H_s(\Sigma ,E|_\Sigma ) \times H_{-s}(\Sigma ,E|_\Sigma )\to \mathbb {C}$$for all$$s\in \mathbb {R}$$. Proof This is also proved in the same way as above, by lifting the corresponding result from the Euclidean case [27, Section I.3]. The Trace Theorem now allows one to extend the allowed domain for the spinors in the Equalities (6) and (2). Lemma 3.10 For all$$\varphi ,\psi \in H_1(M, \mathbb {S}_M),$$Equalities (6) and (2) hold. Proof The proof is a more or less straightforward usage of the Trace Theorem  3.7 and the corresponding equalities on $$\Gamma _c^\infty (M,\mathbb {S}_M)$$. Indeed, let $$\varphi _i$$ be a sequence in $$\Gamma _c^ \infty (M, \mathbb {S}_M)$$ with $$\varphi _i\to \varphi$$ in $$H_1(M, \mathbb {S}_M)$$. The Trace Theorem  3.7 gives $$R\varphi _i\to R\varphi$$ in $$H_{{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma )$$ and, hence, $$\tilde {D}^{\Sigma } R\varphi _i\to \tilde {D}^{\Sigma }R\varphi$$ in $$H_{-{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma )$$, cf. Remark  3.6(iii). Clearly, $$\Vert \varphi _i-\varphi \Vert _{H_1}\to 0$$ and with (7), this implies $$\Vert \varphi _i-\varphi \Vert _{D}\to 0$$. Moreover, the bounded geometry of $$(M, \Sigma )$$ implies   \[\int _{M} {{\rm scal} ^M}|\varphi _i|^2 \,dv \longrightarrow \int _{M} {{\rm scal} ^M}|\varphi |^2 \,dv,\quad \int _{\Sigma } H|\varphi _i|^2 \,ds \longrightarrow \int _{\Sigma } H|\varphi |^2 \,ds,\] and   \[\left | \int _{M} \langle \Omega \cdot \varphi _i, \varphi _i \rangle \,dv - \int _{M} \langle \Omega \cdot \varphi , \varphi \rangle \,dv\right | \leq (\Vert \varphi _i-\varphi \Vert _{L^2}\Vert \varphi \Vert _{L^2}+ \Vert \varphi _i\Vert _{L^2}\Vert \varphi _i-\varphi \Vert _{L^2})\sup _M |\Omega |\longrightarrow 0.\] Note that due to the bounded geometry of $$L$$, $$\sup _M |\Omega |$$ is finite. It remains to consider the term $$\int _{\Sigma } \langle R\varphi , \tilde {D}^{\Sigma }R\varphi \rangle \,ds$$. First, we note that due to the pairing in Lemma 3.9, the Trace Theorem  3.7, and $$\tilde {D}^{\Sigma }: H_{{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma ) \to H_{-{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma )$$, this expression is finite for all $$\varphi \in H_{1}(M, \mathbb {S}_M)$$. Abbreviating $$R\varphi$$ by $$\varphi$$, we have   \[\begin {array}{rl} | (\tilde {D}^{\Sigma }\varphi _i,\varphi _i)_\Sigma - (\tilde {D}^{\Sigma }\varphi ,\varphi )_\Sigma | & \leq | (\tilde {D}^{\Sigma }\varphi _i, \varphi - \varphi _i)_\Sigma | + |(\tilde {D}^{\Sigma }\varphi -\tilde {D}^{\Sigma }\varphi _i,\varphi )_\Sigma |\\ & \leq \Vert \tilde {D}^{\Sigma }\varphi _i\Vert _{H_{-{1}/{2}}} \Vert \varphi - \varphi _i\Vert _{H_{{1}/{2}}} + \Vert \tilde {D}^{\Sigma }\varphi -\tilde {D}^{\Sigma }\varphi _i\Vert _{H_{-{1}/{2}}} \Vert \varphi \Vert _{H_{{1}/{2}}}, \end {array}\] which gives the convergence of the last term. This proves Equality (6) for all $$\varphi \in H_1(M, \mathbb {S}_M)$$. Now, let $$\varphi _i,\psi _j$$ be sequences in $$\Gamma _c^\infty (M, \mathbb {S}_M)$$ with $$\varphi _i\to \varphi$$ and $$\psi _j\to \psi$$ in $$H_1(M, \mathbb {S}_M)$$. Then,   \[\begin {array}{rl} |(D\psi _j,\varphi _i)-(D\psi ,\varphi )|&\leq |(D\psi _j,\varphi _i)-(D\psi _j,\varphi )|+ |(D\psi _j,\varphi )-(D\psi ,\varphi )|\\ &\leq \Vert D\psi _j\Vert _{L^2} \Vert \varphi _i-\varphi \Vert _{L^2}+ \Vert D(\psi _j-\psi )\Vert _{L^2} \Vert \varphi \Vert _{L^2}. \end {array}\] Using (7) and that $$\varphi _i$$ and $$\psi _j$$ are uniformly bounded in $$H_1$$, we get for a certain constant $$C>0$$ that   \[|(D\psi _j,\varphi _i)-(D\psi ,\varphi )|\leq C\Vert \varphi _i-\varphi \Vert _{L^2}+C\Vert \psi _j-\psi \Vert _{H_1}\longrightarrow 0.\] Analogously, one obtains $$(\psi _j, D\varphi _i)\to (\psi , D\varphi )$$. Moreover, using again the Trace Theorem  3.7, we obtain   \[\begin {array}{rl} \left | \int _{\Sigma } \langle \nu \cdot R\psi _j, R\varphi _i\rangle - \langle \nu \cdot R\psi _j, R\varphi \rangle \,ds\right | &\leq \Vert R\psi _j\Vert _{L^2(\Sigma )} \Vert R(\varphi _i-\varphi )\Vert _{L^2(\Sigma )}\\ &\leq C \Vert \psi _j\Vert _{H_1} \Vert \varphi _i-\varphi \Vert _{H_1}\longrightarrow 0. \end {array}\] In the same way, $$| \int _{\Sigma } \langle \nu \cdot R\psi _j, R\varphi \rangle - \langle \nu \cdot R\psi , R\varphi \rangle \,ds|\to 0$$. Hence,   \[\left | \int _{\Sigma } \langle \nu \cdot R\psi _j, R\varphi _i\rangle - \langle \nu \cdot R\psi , R\varphi \rangle \,ds\right |\longrightarrow 0.\] This proves Equality (2) for all $$\varphi , \psi \in H_1(M, \mathbb {S}_M)$$. 3.2. Extension and the graph norm Spectral decomposition of the boundary. Let $$(M,\Sigma )$$ be of bounded geometry. Then, $$(\Sigma , g|_\Sigma )$$ is complete and, thus, the Dirac operator $$D^\Sigma$$ on $$\mathbb {S}_\Sigma$$, and thus also $$\tilde {D}^{\Sigma }$$ on $$\mathbb {S}_M|_\Sigma$$, is self-adjoint. Let $${\{ }E_I{\} }_{I\subset \mathbb {R}}$$ be the family of projector-valued measures belonging to the self-adjoint operator   \[\tilde {D}^{\Sigma }: H_1(\Sigma , \mathbb {S}_M|_\Sigma ) \subset L^2(\Sigma , \mathbb {S}_M|_\Sigma )\longrightarrow L^2(\Sigma , \mathbb {S}_M|_\Sigma ).\] We define for a connected (not necessarily bounded) interval $$I\in \mathbb {{R}}$$ the spectral projection   \[\pi _{I}: L^2(\Sigma , \mathbb {S}_M|_{\Sigma })\longrightarrow L^2(\Sigma , \mathbb {S}_M|_{\Sigma }),\quad \varphi \longmapsto E_{I}\varphi\] and the spaces   \[\Gamma ^{\rm APS}_{I}={\{ } \varphi \in L^2(\Sigma , \mathbb {S}_M|_{\Sigma })\,|\, \varphi =\pi _{I} \varphi {\} }.\] Next, we will show that for every $$s\in \mathbb {R}$$ the spectral projections extend to bounded linear maps from $$H_s(\Sigma , \mathbb {S}_M|_\Sigma )$$ to itself: First, we note that the spectral projections commute with $$\tilde {D}^{\Sigma }$$. Moreover, since $$(\Sigma , g|_\Sigma )$$ has bounded geometry, the norm $$\Vert \varphi \Vert _{L^2}^2+ \Vert D^k \varphi \Vert _{L^2}^2$$ and the $$H_k$$-norm are equivalent on $$\Gamma _c^\infty (\Sigma , \mathbb {S}_M|_\Sigma )$$ for $$k\in \mathbb {N}_0$$, cp. [3, Lemma 3.1.6]. Hence, $$\pi _I$$ restricts to a bounded linear map from $$H_k(\Sigma , \mathbb {S}_M|_\Sigma )$$ to itself for $$k\in \mathbb {N}_0$$. Let now $$k$$ be a negative integer, $$\varphi \in \Gamma _c^\infty (\Sigma , \mathbb {S}_M|_\Sigma )$$ and $$\psi \in H_{-k}(\Sigma , \mathbb {S}_M|_\Sigma )$$. Using that $$\pi _I$$ is symmetric with respect to $$L^2$$-product on $$(\Sigma , \mathbb {S}_M|_\Sigma )$$ and Lemma  3.9, we obtain   \[|(\pi _I \varphi , \psi )_\Sigma |=|(\varphi , \pi _I \psi )_\Sigma |\leq C\Vert \varphi \Vert _{H_{-k}(\Sigma )}\Vert \pi _I \psi \Vert _{H_k(\Sigma )}\leq C'\Vert \varphi \Vert _{H_{-k}(\Sigma )}\Vert \psi \Vert _{H_k(\Sigma )}.\] Thus, $$\pi _I$$ extends to a bounded linear map from $$H_k(\Sigma , \mathbb {S}_M|_\Sigma )$$ to itself for all nonnegative integers $$k$$. Then, by Riesz–Thorin Interpolation Theorem we obtain that $$\pi _I: H_s(\Sigma , \mathbb {S}_M|_\Sigma )\to H_s(\Sigma , \mathbb {S}_M|_\Sigma )$$ for all $$s\in \mathbb {R}$$. We abbreviate $$\pi _{>}=\pi _{(0,\infty )}$$ and $$\pi _{\leq }=\pi _{(-\infty , 0]}$$. As in [9, Section 5], we define, for $$\varphi \in \Gamma _c^\infty (\Sigma , \mathbb {S}_M|_\Sigma ),$$  \[\Vert \varphi \Vert _{\check {H}}^2= \Vert \pi _{\leq }\varphi \Vert _{H_{1/2}(\Sigma )}^2 + \Vert \pi _{>}\varphi \Vert _{H_{-{1}/{2}}(\Sigma )}^2\ {\text { and }} \Vert \varphi \Vert _{\hat {H}}^2= \Vert \pi _{\leq }\varphi \Vert _{H_{-{1}/{2}}(\Sigma )}^2 + \Vert \pi _{>}\varphi \Vert _{H_{{1}/{2}}(\Sigma )}^2\] and the spaces   \[\check {H}:=\overline {\Gamma _c^\infty (\Sigma , \mathbb {S}_M|_\Sigma )}^{\Vert .\Vert _{\check {H}}}\quad {\text {and}}\quad \hat {H}:=\overline {\Gamma _c^\infty (\Sigma , \mathbb {S}_M|_\Sigma )}^{\Vert .\Vert _{\hat {H}}}.\] (8) Local description of the graph norm on $$(M,\Sigma )$$. Let $$(M,g)$$ be a manifold with boundary $$\Sigma$$. Let $$(U_\gamma , \kappa _\gamma , \xi _\gamma , h_\gamma )_\gamma$$ be Fermi coordinates on $$(M,g)$$ together with a synchronous trivialization $$\xi _\gamma$$ and a partition of unity $$h_\gamma$$. Lemma 3.11 On$$\Gamma ^\infty _c(M, \mathbb {S}_M)$$the norms$$\Vert .\Vert _D$$and$$(\sum _\gamma \Vert h_\gamma .\Vert _D^2)^{1/2}$$are equivalent. Proof All the constants $$c_i$$ involved here are positive. Let $$\varphi \in \Gamma _c^\infty (M,\mathbb {S}_M)$$. Since the cover $$U_\gamma$$ is uniformly locally finite the norms $$\Vert .\Vert _{L^2}$$ and $$(\sum _\gamma \Vert h_\gamma . \Vert _{L^2}^2)^{{1}/{2}}$$ are equivalent. Thus,   \[\begin {array}{rl} \Vert D\varphi \Vert _{L^2}^2& \leq c_1 \sum _\gamma \Vert h_\gamma D\varphi \Vert _{L^2}^2=c_1 \sum _\gamma \Vert D(h_\gamma \varphi )-\nabla h_\gamma \cdot \varphi \Vert _{L^2}^2\\ &\leq c_2 \sum _\gamma (\Vert D(h_\gamma \varphi )\Vert _{L^2}^2 + \Vert \nabla h_\gamma \cdot \varphi \Vert _{L^2}^2)\leq c_3 \sum _\gamma (\Vert D(h_\gamma \varphi )\Vert _{L^2}^2 + \Vert \varphi |_{U_\gamma }\Vert _{L^2}^2)\\ &\leq c_3\sum _\gamma \Vert D(h_\gamma \varphi )\Vert _{L^2}^2 +c_4 \Vert \varphi \Vert _{L^2}^2{, } \end {array}\] where the end of the second line follows by Lemma  3.4, and the last inequality follows since the cover $$U_\gamma$$ is uniformly locally finite. Hence, $$\Vert \varphi \Vert _D^2\leq c_5 \sum _\gamma \Vert h_\gamma \varphi \Vert _D^2$$. Conversely, we obtain analogously   \[\sum _\gamma \Vert D(h_\gamma \varphi )\Vert _{L^2}^2= \sum _\gamma \Vert h_\gamma D \varphi + \nabla h_\gamma \cdot \varphi \Vert _{L^2}^2\leq c_6 \Vert \varphi \Vert _{D}^2.\] Lemma 3.12 Let$$(\Sigma , g|_\Sigma )$$be a manifold of bounded geometry. Then, there is an$$\varepsilon >0$$smaller than the injectivity radius of$$\Sigma$$and a constant$$c>0$$such that for all$$x\in \Sigma$$and$$\varphi \in \Gamma ^\infty _c(B_\varepsilon (x)\subset N, \mathbb {S}_N),$$we have$$\Vert D^N\varphi \Vert _{L^2}>c\Vert \varphi \Vert _{L^2}$$. Proof Let $$\exp ^{\Sigma }_x: B_\varepsilon (0)\subset \mathbb {{R}} ^n\to B_\varepsilon (x)\subset \Sigma$$ be the exponential map. Set $$\tilde {g}:=(\exp ^{\Sigma }_x)^*g|_{B_{\varepsilon }(x)}$$. We will compare the Dirac operator $$D^{\tilde {g}}$$ with $$D^E$$, [5, Proposition 3.2]:   \[D^{\tilde {g}}\varphi = D^E\varphi + \sum _{ij} (b_i^j-\delta _i^j)\partial _i \cdot \nabla _{\partial _j}\varphi + \frac {1}{4}\sum _{ijk} \tilde {\Gamma }_{ij}^k e_i\cdot e_j\cdot e_k\cdot \varphi {, }\] where $$\varphi$$ is a smooth spinor over $$B_{\varepsilon }(0)$$, $$\partial _i$$ and $$e_i$$ form an orthonormal basis with respect to the Euclidean metric and $$\tilde {g}$$, respectively. Moreover, $$e_i=b_i^j\partial _j$$, $$\nabla$$ is the Levi-Civita connection with respect to the Euclidean metric, and $$\tilde {\Gamma }_{ij}^k$$ are the Christoffel symbols with respect to the metric $$\tilde {g}$$. By [5, (11)–(13) and below], $$|b_i^j-\delta _i^j|\leq C r^2$$ and $$|\tilde {\Gamma }_{ij}^k|\leq Cr,$$ where $$r$$ is the Euclidean distance to the origin and $$C$$ can be chosen to depend only on the global curvature bounds of $$g$$. Moreover, note that there is a positive constant $$C$$ also depending only on the global curvature bounds of $$g$$ such that $$C^{-1}\leq f\leq C$$ where $${\rm dvol}_{\tilde {g}}= f{\rm dvol}_{g_E}$$. Thus, for $$\varepsilon$$ small enough we can estimate for all smooth spinors $$\varphi$$ compactly supported in $$B_\varepsilon (0)$$ that   \[\begin {array}{rl} \frac {\Vert D^{\tilde {g}} \varphi \Vert _{L^2(\tilde {g})}^2 }{\Vert \varphi \Vert _{L^2(\tilde {g})}^2} &\geq C_1 \frac {\Vert D^E \varphi \Vert _{L^2(g_E)}^2}{\Vert \varphi \Vert _{L^2(g_E)}^2} - C_2\varepsilon ^2 \frac {\Vert \nabla \varphi \Vert _{L^2(g_E)}^2}{\Vert \varphi \Vert _{L^2(g_E)}^2} - C_3\varepsilon \\ &\geq C_4 \frac {\Vert D^E \varphi \Vert _{L^2(g_E)}^2}{\Vert \varphi \Vert _{L^2(g_E)}^2} - C_5\varepsilon {, } \end {array}\] where the last step uses the equivalence of the graph norm and the $$H_1$$-norm. Let $$A$$ be such that $$\Vert D^E \psi \Vert _{L^2(g_E)}^2 \geq A \Vert \psi \Vert _{L^2(g_E)}^2$$ for smooth spinors compactly supported in $$B_{\varepsilon }(0)$$. Then, one can always choose $$\varepsilon$$ small enough that $$C_4A-C_5\varepsilon \geq 2^{-1}C_4A=:c$$ Thus, the same is true for $$D^g$$ on $$B_\varepsilon (x)\subset \Sigma$$. Let $$(\hat {M}, \hat {N}=\partial \hat {M})$$ be manifold of bounded geometry with closed boundary. Let $$\mathcal {E}_{\rm BB}$$ be an extension map as defined in [9, (43)]. Let $$D$$ and $$D^{\hat {N}}$$ be the Dirac operators on $$\hat {M}$$ and $$\hat {N}$$, respectively. By [9, Lemma 6.1, 6.2, (41) and below], we have, for all $$\varphi \in \Gamma _c^\infty (\hat {M}, \mathbb {S}_{\hat {M}}|_{\hat {N}}),$$  \[\Vert \mathcal {E}_{\rm BB}R\varphi \Vert _D \leq C \Vert \varphi \Vert _D.\] (9) Note that $$C$$ can be chosen to depend only on curvature bounds of $$(\hat {M}, \hat {N})$$ including mean curvature, the injectivity radii of $$\hat {M}$$ and $$\hat {N}$$, respectively, and the spectral gap of $$D^{\hat {N}}$$. We now come back to our pair $$(M, N)$$: Let $$\varepsilon , c>0$$ be constants such that Lemma  3.12 is fulfilled. Let $$(U_\gamma , \kappa _\gamma , \xi _\gamma , h_\gamma )$$ be Fermi coordinates together with a subordinated partition of unity such that there are $$x_\gamma \in \Sigma$$ with $$U_\gamma \cap \Sigma \subset B_{\varepsilon } (x_\gamma )$$. Let $$\hat {U}_\gamma$$ be a manifold with closed boundary $$\hat {U}'_\gamma :=\partial \hat {U}_\gamma$$ such that $$\tilde {U}_\gamma :=U_\gamma \cup (\bigcup _{\alpha ; U_\alpha \cap U_\gamma \neq \varnothing } U_\alpha )$$ can be isometrically embedded in $$\hat {U}_\gamma$$, $$\tilde {U}_\gamma \cap \Sigma \subset \hat {U}'_\gamma$$, such that the spectral gap of the Dirac operator on $$\hat {U}'_\gamma$$ is at least $$[-c, c]$$ and such that the curvature, mean curvature of the boundary and the injectivity radii are still uniformly bounded in $$\gamma$$. Define the map $$\tilde {\mathcal {E}}: \Gamma _c^\infty (\Sigma , \mathbb {S}_M|_\Sigma )\to \Gamma _c^\infty (M, \mathbb {S}_M)$$ by   \[\tilde {\mathcal {E}}\psi = \sum _{\gamma ,\alpha ;\ U_\gamma '\neq \varnothing , U_\gamma \cap U_\alpha \neq \varnothing } h_\alpha \mathcal {E}_{\rm BB} (h_\gamma |_{\Sigma } \psi ){, }\] where $$h_\gamma |_{\Sigma } \varphi$$ is understood to be a spinor on $$U_\gamma \cap N \subset \hat {U}'_\gamma$$ and then $$\mathcal {E}_{\rm BB} (h_\gamma |_{\Sigma } \psi )$$ is a spinor on $$\hat {U}_\gamma$$. The only reason why $$\sum _\alpha h_\alpha$$ appears in the definition is to assure that each summand can be seen to live on $$M$$ and that $$R\tilde {\mathcal {E}}=\operatorname {Id}$$. Note that just using $$h_\gamma$$ in front of $$\mathcal {E}_{\rm BB}$$ would be enough to first requirement but not the second. Proposition 3.13 Using the notations from above, there is a positive constant$$C$$such that, for all$$\varphi \in \Gamma _c^\infty (M, \mathbb {S}_M),$$  \[\Vert \tilde {\mathcal {E}}R\varphi \Vert _D\leq C\Vert \varphi \Vert _D.\] Proof We abbreviate $$h_\gamma ':=h_\gamma |_\Sigma$$. Using (in this order) the definition of $$\tilde {\mathcal E}$$, Lemma  3.11 the uniform local finiteness of the cover $$U_\gamma$$, (9), and again Lemma  3.11 we estimate   \[\begin {array}{rl} \Vert \tilde {\mathcal E}R\varphi \Vert _D^2 & \leq C_1 {\left \Vert \sum _{\gamma , U'_\gamma \neq \varnothing } \mathcal {E}_{\rm BB} R(h_\gamma \varphi )\right \Vert }_D^2\leq C_2\sum _{\gamma , U'_\gamma \neq \varnothing } \Vert \mathcal {E}_{\rm BB} R(h_\gamma \varphi )\Vert _D^2\\ & \leq C_3\sum _{\gamma , U'_\gamma \neq \varnothing } \Vert h_\gamma \varphi \Vert _{D}^2\leq C \Vert \varphi \Vert _D^2. \end {array}\] 4. Boundary value problems The general theory of boundary value problems for elliptic differential operators of order 1 on complete manifolds with closed boundary can be found in [9]. The aim of this section is to generalize a part of this theory to noncompact boundaries on manifolds of bounded geometry. We restrict to the part that gives existence of solutions of boundary value problems as in Theorem  1.2. The property needed to assure a solution to such a problem is the closedness of the range. For that, we introduce a type of coercivity condition which, in general, can depend on the boundary values (that is not the case for closed boundaries). Moreover, we restrict to the classical $$\mathrm {Spin^c}$$ Dirac operator. In the first part, we first give some generalities on domains of the Dirac operator and introduce a coercivity condition that implies closed range of the Dirac operator. Then, we extend the trace map $$R$$ to the whole maximal domain of the Dirac operator and give some examples and properties of boundary conditions. In particular, we will introduce two boundary conditions $$B_\pm$$ which will be used to prove Theorem  1.2 in Section  8. At the end, we give an existence result for boundary value problems in our context. General domains and closed range. Let $$D$$ be the Dirac operator acting on $$\Gamma _{cc}^\infty (M, \mathbb {S})$$ on a manifold $$M$$ with boundary $$\Sigma$$. If we want to emphasize that $$D$$ acts on the domain $$\Gamma _{cc}^\infty (M, \mathbb {S})$$, then we shortly write $$D_{cc}$$. We denote the graph norm of $$D$$ by   \[\Vert \varphi \Vert _D^2=\Vert \varphi \Vert _{L^2}^2+ \Vert D \varphi \Vert _{L^2}^2.\] By $$D_{\mathrm {max}}:=(D_{cc})^* $$ we denote the maximal extension of $$D$$. Here, $$A^* $$ denotes the adjoint operator of $$A$$ in the sense of functional analysis. Note that $$H_1(M, \mathbb {S}_M)\subset {\mathop {\rm dom}} \, D_{\mathrm {max}}$$ and   \[{\mathop {\rm dom}} \, D_{\mathrm {max}}={\{ }\varphi \in L^2(M, \mathbb {S}_M)\,|\, \exists \tilde {\varphi }\in L^2(M, \mathbb {S}_M) \forall \psi \in \Gamma _{cc}^\infty (M, \mathbb {S}_M): (\tilde {\varphi }, \psi )=(\varphi , D\psi ){\} },\] and together with $$\Vert . \Vert _D$$, the space $${\mathop {\rm dom}} \, D_{\mathrm {max}}$$ is a Hilbert space. Moreover, we denote by $$D_{\mathrm {min}}:= (D_{cc})^{* * } =\bar {D}_{cc}^{\Vert .\Vert _D}$$ the minimal extension of $$D$$. Here, $$\bar {A}^{\Vert .\Vert _D}$$ denotes the closure of the set $$A$$ with respect to the graph norm. Any closed linear subset of $$\mathrm {dom}\, D_{\mathrm {max}}$$ between $$\mathrm {dom}\, D_{\mathrm {min}}$$ and $$\mathrm {dom}\, D_{\mathrm {max}}$$ gives the domain of a closed extension of $$D\colon \Gamma _{cc}^\infty (M, \mathbb {S}_M) \to \Gamma _{cc}^\infty (M, \mathbb {S}_M)$$. Before examining those domains, let us extend the trace map to $${\mathop {\rm dom}} \, D_{\rm max}$$: Extension of the trace map. The Trace Theorem  3.7 extends the trace map   \[\begin {array}{rl} R: \Gamma _c^\infty (M,\mathbb {S}_M) &\to \Gamma _c^\infty (\Sigma ,\mathbb {S}_M|_\Sigma ){,} \\ \varphi &\mapsto \varphi |_\Sigma \end {array}\] to a bounded map $$R:H_1(M,\mathbb {S}_M)\to H_{{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma )$$. Here, we will extend $$R$$ further to $${\mathop {\rm dom}} \, D_{\mathrm {max}}$$. This will generalize the corresponding result [9, Theorem 6.7(ii)] for closed boundaries to noncompact boundaries. Moreover, we give some auxiliary lemmata which are found in [9] for closed boundaries. Some of the proofs and the order of obtaining them will be a little bit different since we do not use (and cannot use, cf. Example  4.16(iv)) the projection to the negative spectrum. Note that in this part we could use an arbitrary extension map as given by Theorem  3.7 and are not restricted to the explicit one defined via the eigenvalue decomposition of $$\tilde {D}^{\Sigma }$$ on closed boundaries used in [9]. Lemma 4.1 The space$$\Gamma _c^\infty (M,\mathbb {S}_M)$$is dense in$${\mathop {\rm dom}} \, D_{\mathrm {max}}$$with respect to the graph norm. Proof For a closed boundary, this is done in [9, Theorem 6.7(i)]. We use a different proof here. Let $$\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}$$. Let $$K_i$$ be a compact exhaustion of $$M$$ that comes together with smooth cut-off functions $$\eta _i:M \to [0,1]$$ such that $$\eta _i=1$$ on $$K_i$$, $$\eta _i=0$$ on $$M\setminus K_{i+1}$$ and $$\max |d\eta _i|\leq {2}/{i}$$. Then, $$\varphi _i=\eta _i\varphi$$ are compactly supported sections in $${\mathop {\rm dom}} \, D_{\max }$$ fulfilling   \[\begin {array}{rl} \Vert \varphi _i-\varphi \Vert _D^2 &= \Vert \varphi _i-\varphi \Vert _{L^2}^2 + \Vert D\varphi _i-D\varphi \Vert _{L^2}^2\\ &\leq \Vert (1-\eta _i)\varphi \Vert _{L^2}^2 + \left (\Vert (1-\eta _i) D\varphi \Vert _{L^2} + \frac {2}{i} \Vert \varphi \Vert _{L^2} \right )^2\to 0. \end {array}\] Each $$\varphi _i$$ has now compact support in $$K_{i+1}$$. Thus, there is a sequence $$\varphi _{ij}\in \Gamma _c^\infty (K_{i+1}, \mathbb {S}_M)$$ with $$\varphi _{ij}\to \varphi _i$$ in the graph norm on $$K_{i+1}$$. Choose $$j=j(i)\geq i$$ such that $$\Vert \varphi _{ij}-\varphi _i\Vert _D\to 0$$ as $$i\to \infty$$. Then, $$\Vert \varphi _{ij}-\varphi \Vert _D \leq \Vert \varphi _{ij}-\varphi _i\Vert _D + \Vert \varphi _i- \varphi \Vert _D \to 0$$, too. Then   \[\begin {array}{rl} \Vert \eta _j\varphi _{ij}-\varphi _{ij}\Vert _D^2 &\leq \Vert (1-\eta _j)\varphi _{ij}\Vert _{L^2}^2 + (\Vert (1-\eta _j)D \varphi _{ij} \Vert _{L^2} + \Vert d\eta _j\cdot \varphi _{ij}\Vert _{L^2})^2\\ &\leq (\Vert \varphi _{ij}-\varphi _i\Vert _{L^2} + \Vert (1-\eta _j)\eta _i\varphi \Vert _{L^2})^2 + \left (\Vert D(\varphi _{ij}-\varphi _i)\Vert _{L^2}\right . \\ &\quad \left . + \,\Vert (1-\eta _j)(\eta _i D\varphi +d\eta _i \cdot \varphi )\Vert _{L^2} + \frac {2}{j} \Vert \varphi _{ij}-\varphi _i\Vert _{L^2} + \frac {2}{j} \Vert \varphi \Vert _{L^2}\right )^2 \longrightarrow 0 \end {array}\] for $$i\to \infty$$. Thus, we have a sequence $$\hat {\varphi }_i:=\eta _{j(i)}\varphi _{ij(i)}\in \Gamma _c^ \infty (M, \mathbb {S}_M)$$ such that $$\hat {\varphi }_i\to \varphi$$ in the graph norm as $$i\to \infty$$. Note that the proof of Lemma  4.1 only uses the completeness of $$M$$ and not the bounded geometry. Theorem 4.2 The trace map$$R: \Gamma _{c}^\infty (M, \mathbb {S}_M) \to \Gamma _{c}^\infty (\Sigma , \mathbb {S}_M|_\Sigma )$$can be extended to a bounded operator  \[R: {\mathop {\rm dom}} \, D_{\mathrm {max}} \longrightarrow H_{-{1}/{2}} (\Sigma , \mathbb {S}_M|_\Sigma ).\] Proof Let $$\varphi \in \Gamma _c^\infty (M, \mathbb {S}_M)$$ and $$\psi \in H_{{1}/{2}} (\Sigma , \mathbb {S}_M|_\Sigma )$$. Then, by Theorem  3.7, the spinor $$\mathcal {E}\psi \in H_1(M, \mathbb {S}_M)$$. Thus, we can use Lemma  3.10, (7) and Theorem  3.7 to obtain   \[\begin {array}{rl} |(\varphi |_\Sigma , \nu \cdot \psi )_\Sigma |&=|(D\varphi , \mathcal {E} (\nu \cdot \psi ))-(\varphi , D\mathcal {E}(\nu \cdot \psi ))|\\ &\leq \Vert D\varphi \Vert _{L^2}\Vert \mathcal {E}(\nu \cdot \psi )\Vert _{L^2}+ \Vert \varphi \Vert _{L^2}\Vert D \mathcal {E}(\nu \cdot \psi )\Vert _{L^2}\\ &\leq 2\Vert \varphi \Vert _{D}\Vert \mathcal {E}(\nu \cdot \psi )\Vert _{D} \leq C \Vert \varphi \Vert _{D}\Vert \mathcal {E}(\nu \cdot \psi )\Vert _{H_1} \leq C' \Vert \varphi \Vert _{D}\Vert \nu \cdot \psi \Vert _{H_{{1}/{2}}(\Sigma )}. \end {array}\] Together with Lemma  3.9, this implies   \[\Vert \varphi |_\Sigma \Vert _{H_{-{1}/{2}}(\Sigma )}\leq C' \Vert \varphi \Vert _{D}.\] Since $$\Gamma _c^\infty (M, \mathbb {S}_M)$$ is dense in $${\mathop {\rm dom}} \, D_{\mathrm {max}}$$ with respect to the graph norm, cf. Lemma  4.1, the claim follows. Remark 4.3 Note that $$R$$ is not surjective here. For closed boundaries the image was specified in [9, Theorems 1.7 and 6.7(ii)]. For noncompact boundaries the image will be further considered in Lemma  4.8 and below. Lemma 4.4 Equality (2) holds for all$$\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}$$and$$\psi \in H_1(M,\mathbb {S}_M)$$. Proof The proof is done as the one of Lemma  3.10 starting with $$\psi _j, \varphi _i\in \Gamma _c^\infty (M, \mathbb {S}_M),$$ where $$\psi _j\to \psi$$ in $$H_1$$ and $$\varphi _i\to \varphi$$ in the graph norm of $$D$$ and using the (extended) Trace Theorem  4.2. The only difference is seen in the estimate of the boundary integrals which now read, for example,   \[\left | \int _{\Sigma } \langle \nu \cdot R\psi _j, R\varphi _i - R\varphi \rangle \,ds\right |\leq \Vert R\psi _j\Vert _{H_{{1}/{2}}(\Sigma )} \Vert R(\varphi _i-\varphi )\Vert _{H_{-{1}/{2}}(\Sigma )} \leq C\Vert \psi _j\Vert _{H_1} \Vert \varphi _i-\varphi \Vert _{D}\longrightarrow 0{, }\] where the last inequality uses both versions of the Trace Theorems   3.7 and 4.2.The next lemma gives a full description of $${\mathop {\rm dom}} \, D_{\mathrm {min}}$$. Lemma 4.5 The$$H_1$$-norm and the graph norm$$\Vert . \Vert _D$$are equivalent on  \[{\{ }\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}\,|\,R\varphi =0{\} }.\]In particular ,   \[\begin {array}{rl} {\mathop {\rm dom}} \, D_{\mathrm {min}}&=\overline {\Gamma _{cc}^\infty (M, \mathbb {S}_M)}^{\Vert .\Vert _D}= \overline {\Gamma _{cc}^\infty (M, \mathbb {S}_M)}^{\Vert .\Vert _{H_1}}={\{ }\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}\,|\,R\varphi =0{\} }\\ &={\{ }\varphi \in H_1(M, \mathbb {S}_M)\,|\,R\varphi =0{\} }. \end {array}\] Proof First, we show the equivalence on $${\{ }\psi \in \Gamma _{c}^\infty (M, \mathbb {S}_M)\,|\,R\psi =0{\} }$$: Let $$\varphi$$ be in this set. Then, by (6) we have   \[\Vert \varphi \Vert _{H_1}^2=\Vert \varphi \Vert _{L^2}^2+ \Vert D\varphi \Vert _{L^2}^2 -\int _{M} \frac {{\rm scal} ^M}{4}|\varphi |^2 \,dv - \int _{M} \frac {{i} }{2} \langle \Omega \cdot \varphi , \varphi \rangle \,dv\leq C \Vert \varphi \Vert _D^2,\] where we used that $$M$$ and $$L$$ are of bounded geometry and, hence, $$|{\rm scal} ^M|$$ and $$|\Omega |$$ are uniformly bounded on all of $$M$$. The reverse inequality was seen in (7). From the definition of $${\mathop {\rm dom}} \, D_{\mathrm {min}}$$ and the equivalence of the norms from above, we already have $${\mathop {\rm dom}} \, D_{\mathrm {min}}=\overline {\Gamma _{cc}^\infty }^{\Vert .\Vert _D}=\overline {\Gamma _{cc}^\infty }^{\Vert .\Vert _{H_1}}$$. From the Trace Theorem  4.2, we obtain   \[\overline {\Gamma _{cc}^\infty }^{\Vert .\Vert _D}\subset {\{ }\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}\,|\,R\varphi =0{\} }.\] Next, we show that $$D\colon {\{ }\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}\,|\,R\varphi =0{\} } \to L^2(M, \mathbb {S}_M)$$ already equals $$D_{\mathrm {min}}$$. First, we note that by the Trace Theorem  4.2, $$D$$ is a closed extension of $$D_{cc}$$. Hence, it suffices to show that $$D^* =D_{\mathrm {max}}$$. By definition, we have   \[{\mathop {\rm dom}} \, D^* ={\{ } \vartheta \in L^2(M,\mathbb {S}_M)\,|\,\exists \chi \in L^2(M, \mathbb {S}_M)\, \forall \psi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}, R\psi =0: (\vartheta , D\psi )=(\chi ,\psi ){\} }.\] Let $$\vartheta \in {\mathop {\rm dom}} \, D_{\mathrm {max}}$$. By Lemma  4.1, there exists a sequence $$\vartheta _i\in \Gamma _c^\infty (M,\mathbb {S}_M)$$ with $$\vartheta _i\to \vartheta$$ in the graph norm. Hence, for all $$\psi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}$$ with $$R\psi =0,$$ we have $$(\vartheta , D\psi )=\lim _{i\to \infty }(\vartheta _i,D\psi )$$. Then, by Lemma  4.4 and $$R\psi =0$$, we obtain   \[(\vartheta , D\psi )=\lim _{i\to \infty }(D\vartheta _i,\psi )=(D\vartheta ,\psi ){, }\] which implies that $$\vartheta \in {\mathop {\rm dom}} \, D^* $$. Thus, $$D^* =D_{\mathrm {max}}$$ and $$D=D_{\mathrm {min}}$$. Together with   \[{\mathop {\rm dom}} \, D_{\mathrm {min}}=\overline {\Gamma _{cc}^\infty }^{\Vert .\Vert _{H_1}}\subset {\{ }\varphi \in H_1(M,\mathbb {S}_M)\,|\,R\varphi =0{\} }\subset {\{ }\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}\,|\,R\varphi =0{\} }={\mathop {\rm dom}} \, D_{\mathrm {min}},\] the rest of the lemma follows. Now, we can describe $$H_1$$ in terms of its image under the trace map. Lemma 4.6 We have$$H_1(M, \mathbb {S}_M)={\{ }\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}\,|\,R\varphi \in H_{{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma ){\} }$$. Proof The inclusion ‘ $$\subset$$’ is clear from the Trace Theorem  3.7. It remains to prove ‘ $$\supset$$’: Let $$\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}$$ with $$R\varphi \in H_{1/2} (\Sigma , \mathbb {S}_M|_\Sigma )$$. Then, Theorem  3.7 implies that $$\psi :=\mathcal {E}R\varphi \in H_1 (M, \mathbb {S}_M)$$. Thus, $$\varphi -\psi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}$$ and $$R(\varphi -\psi )=0$$. But due to Lemma 4.5, $$\varphi -\psi \in H_1(M, \mathbb {S}_M)$$ and, hence, $$\varphi \in H_1(M, \mathbb {S}_M)$$. In Proposition  3.13, we have shown that there is a linear map $$\tilde {\mathcal E}$$ such that $$\tilde {\mathcal {E}}R: \Gamma _c^\infty (M,\mathbb {S}_M)\to \Gamma _c^\infty (M,\mathbb {S}_M)$$ fulfills for all $$\varphi \in \Gamma _c^\infty (M, \mathbb {S}_M)$$  \[\Vert \tilde {\mathcal {E}} R\varphi \Vert _D^2\leq C\Vert \varphi \Vert _D^2.\] (10) Thus, $$\tilde {\mathcal {E}}R$$ extends uniquely to a bounded linear map   \[\tilde {\mathcal {E}}R: {\mathop {\rm dom}} \, D_{\rm max} \longrightarrow {\mathop {\rm dom}} \, D_{\rm max} .\] (11) Note that $$\tilde {\mathcal {E}}|_{H_{1/2}}$$ is an extension map in the sense of Theorem  3.7 as can be seen in the following: Let $$\psi \in H_{1/2} (\Sigma , \mathbb {S}_M|_\Sigma )$$. By Lemma  4.6, there is a $$\varphi \in H_1(M, \mathbb {S}_M)$$ with $$R\varphi =\psi$$. Thus, by Lemma  4.5 $$\tilde {\mathcal {E}}\psi -\varphi \in {\mathop {\rm dom}} \, D_{\rm min}\subset H_1(M, \mathbb {S}_M)$$. In particular, $$\tilde {\mathcal {E}}|_{H_{{1}/{2}}}: H_{1/2} (\Sigma , \mathbb {S}_M|_\Sigma )\to H_1(M, \mathbb {S}_M)$$. From now, we choose any extension map $$\mathcal {E}$$ fulfilling (10). Obviously, all those maps lead to equivalent norms $$\Vert \mathcal {E}R.\Vert _D$$. Conjecture 4.7 Every extension map in the sense of Theorem  3.7 fulfills (10) with an appropriate constant $$C$$. On $$R({\mathop {\rm dom}} \, D_{\mathrm {max}})$$, we set   \[\Vert \psi \Vert _{\check {R}}:= \Vert \mathcal {E}R\varphi \Vert _D{, }\] where $$R\varphi =\psi$$. By Theorem  3.13 and (11), this is well defined. Lemma 4.8 The space$$\check {R}:=(R({\mathop {\rm dom}} \, D_{\mathrm {max}}), \Vert . \Vert _{\check {R}})$$is a Hilbert space with$$\check {R}= \overline {\Gamma _c^\infty (\Sigma , \mathbb {S}_M|_\Sigma )}^{\Vert . \Vert _{\check {R}}}$$. Proof From the definition of $$\Vert . \Vert _{\check {R}}$$, the linearity of the maps $$\mathcal {E}$$ and $$R$$, and the fact that $$({\mathop {\rm dom}} \, D_{\mathrm {max}}, \Vert . \Vert _D)$$ is a Hilbert space, we get immediately that $$\Vert .\Vert _{\check {R}}$$ is a norm on $$R({\mathop {\rm dom}} \, D_{\mathrm {max}})$$. Moreover, $$\Vert .\Vert _{\check {R}}$$ comes from a scalar product $$(\varphi ,\psi )_{\check {R}}:=(\mathcal {E}\varphi ,\mathcal {E}\psi )_D\colon =(\mathcal {E}\varphi ,\mathcal {E}\psi ) + (D\mathcal {E}\varphi ,D\mathcal {E}\psi )$$. To show that $$\check {R}$$ is a Hilbert space it remains to show completeness: For that, we consider a Cauchy sequence $$\psi _i$$ in $$\check {R}$$. Then, there is a sequence $$\varphi _i\in {\mathop {\rm dom}} \, D_{\mathrm {max}}$$ with $$R\varphi _i=\psi _i$$. With the definition of the $$\check {R}$$-norm, we get that $$\mathcal {E}R\varphi _i$$ is a Cauchy sequence in $$({\mathop {\rm dom}} \, D_{\mathrm {max}}, \Vert .\Vert _D)$$ and, hence, there is a $$\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}$$ with $$\mathcal {E}R\varphi _i\to \varphi$$ with respect to the graph norm. By Theorem  3.13, we obtain   \[\Vert \mathcal {E}R(\varphi _i-\varphi )\Vert _D=\Vert \mathcal {E}R(\mathcal {E}R\varphi _i-\varphi )\Vert _D\leq C\Vert \mathcal {E}R\varphi _i -\varphi \Vert _D\longrightarrow 0.\] Thus, $$\mathcal {E}R\varphi =\varphi$$ and $$\Vert \psi _i-R\varphi \Vert _{\check {R}}= \Vert \mathcal {E} (R\varphi _i-R\varphi )\Vert _D\to 0$$. Hence, $$\psi _i\to \psi$$ in the $$\check {R}$$-norm. Clearly, $$\overline {\Gamma _c^\infty (\Sigma , \mathbb {S}_M|_\Sigma )}^{\Vert . \Vert _{\check {R}}}\subset R({\mathop {\rm dom}} \, D_{\rm max})$$. Let now $$\psi \in R({\mathop {\rm dom}} \, D_{\rm max})$$. Then, there is a $$\varphi \in {\mathop {\rm dom}} \, D_{\rm max}$$ with $$R\varphi =\psi$$. By Lemma  4.1, there is a sequence $$\varphi _i\in \Gamma _c^\infty (M, \mathbb {S}_M)$$ with $$\Vert \varphi _i-\varphi \Vert _D\to 0$$ as $$i\to \infty$$. Thus, by Theorem 3.13 the sequence $$\psi _i:= R\varphi _i\in \Gamma _c^\infty (\Sigma , \mathbb {S}_M|_\Sigma )$$ converges to $$\psi$$ in the $$\check {R}$$-norm. Remark 4.9 (i) The proof of Proposition  3.13 and [9, Lemma 6.1] implies   \[\Vert \tilde {\mathcal E} R\varphi \Vert _D^2\leq C' \sum _{\gamma , \hat {U}'_\gamma \neq \varnothing } \Vert R(h_\gamma \varphi )\Vert _{\check {H}(\hat {U}'_\gamma )}^2 =: C' \Vert R\varphi \Vert ^2_{\check {H}_\gamma }.\] On the other hand, by [9, Lemma 6.2, (41) and below] $$\Vert R(h_\gamma \varphi )\Vert _{\check {H}(\hat {U}'_\gamma )}^2\leq C \Vert h_\gamma \varphi \Vert _D^2$$, where $$C$$ again only depends on the curvature bounds of $$(M,\Sigma )$$ and the spectral gap $$c$$ on $$\hat {U}'_\gamma$$. Thus, together with Lemma  3.11 the norms $$\Vert .\Vert _{\check {R}}$$ and $$\Vert .\Vert _{\check {H}_\gamma }$$ are equivalent. (ii) Using (i) and [9, Lemma 6.3], we see   \[\Vert \tilde {\mathcal E} (\nu \cdot R\varphi )\Vert _D^2\leq C' \sum _{\gamma , U'_\gamma \neq \varnothing } \Vert \nu \cdot R(h_\gamma \varphi )\Vert _{\check {H}(\hat {U}'_\gamma )}^2 = C' \sum _{\gamma , U'_\gamma \neq \varnothing } \Vert R(h_\gamma \varphi )\Vert _{\hat {H}(\hat {U}'_\gamma )}^2=: \Vert R\varphi \Vert _{\hat {H}_\gamma }^2.\] Together with [9, Lemma 6.1], we obtain for all $$\varphi \in \Gamma _c^\infty (M,\mathbb {S}_M)$$  \[\Vert \tilde {\mathcal E} (\nu \cdot R\varphi )\Vert _D^2\leq C \Vert \varphi \Vert _D^2\] and, thus, $$\Vert \psi \Vert _{\hat {R}}:= \Vert \mathcal {E}(\nu \cdot R\varphi )\Vert _D$$ also gives rise to a norm on $$R({\mathop {\rm dom}} \, D_{\mathrm {max}})$$. Moreover, the analogous statement of Lemma  4.8 holds for $$\hat {R}:=(R({\mathop {\rm dom}} \, D_{\mathrm {max}}), \Vert . \Vert _{\hat {R}})$$, and we have $$\Vert \psi \Vert _{\check {R}}=\Vert \nu \cdot \psi \Vert _{\hat {R}}$$. In particular, we get as in (i) that the norms $$\Vert \tilde {\mathcal E} (\nu \cdot .)\Vert _D$$ and $$\Vert .\Vert _{\hat {H}_\gamma }$$ are equivalent. Remark 4.10 Note that by Theorems   4.2 and  4.6   \[H_{{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma )\subset (R({\mathop {\rm dom}} \, D_{\mathrm {max}}), \Vert .\Vert _{\check {R}\, ({\text {resp.\ }} \hat {R})}) \subset H_{-{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma ).\] Moreover, the perfect pairing of $$\hat {H}_\gamma$$ and $$\check {H}_\gamma$$, induced by the pairing of $$H_{{1}/{2}}$$ and $$H_{-{1}/{2}}$$, gives immediately the following lemma. Lemma 4.11 The$$L^2$$-product on$$\Gamma _c^\infty (\Sigma , \mathbb {S}_M|_\Sigma )$$extends uniquely to a perfect pairing$$\check {R}\times \hat {R} \to \mathbb {{C}}$$.So for now, we have seen that the $$\check {R}$$-norm is equivalent to the norm $$\Vert .\Vert _{\check {H}_\gamma }$$, cp. Remark  4.9(i) where the second norm comes with an appropriate trivialization of the manifold near the boundary, see before Proposition 3.13. But we also think that as in the closed case there should be a ‘ more intrinsic’ equivalent norm: Conjecture 4.12 The $$\check {R}$$-norm on $$R({\mathop {\rm dom}} \, D_{\rm max})$$ is equivalent to the $$\check {H}$$-norm as defined in (8). Moreover, $$\check {H}=R({\mathop {\rm dom}} \, D_{\rm max})$$ as vector spaces. Boundary conditions. In this part, we show that each closed extension of $$D_{cc}$$ can be realized by a closed linear subset of $$\check {R}$$, and we give some examples. Lemma 4.13 Let$$D$$be a closed extension of$$D_{cc}$$with$$B:=R({\mathop {\rm dom}} \, D)\subset H_{-{1}/{2}} (\Sigma , \mathbb {S}_M|_\Sigma )$$. Then, its domain$${\mathop {\rm dom}} \, D$$equals$${\mathop {\rm dom}} \, D_B\colon ={\{ } \varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}\,|\,R\varphi \in B{\} },$$and$$B$$is a closed linear subset of$$\check { R}$$. Conversely, for every closed linear subset$$B\subset \check { R}$$the operator$$D_B\colon {\mathop {\rm dom}} \, D_B \to L^2(M,\mathbb {S}_M)$$is a closed extension of$$D_{cc}$$.Owing to this lemma, a closed subspace $$B$$ of $$\check { R}$$ is called boundary condition. Proof Let $$D$$ be a closed extension of $$D_{cc}$$ with domain $${\mathop {\rm dom}} \, D$$ and $$B:= R({\mathop {\rm dom}} \, D)$$. Clearly, $${\mathop {\rm dom}} \, D\subset {\mathop {\rm dom}} \, D_B$$. We have to show that also the converse is true: Let $$\varphi \in {\mathop {\rm dom}} \, D_B$$. Then, there exists $$\psi \in {\mathop {\rm dom}} \, D$$ with $$R \varphi = R\psi$$. By Lemma  4.5, $$\varphi -\psi \in {\mathop {\rm dom}} \, D_{\mathrm {min}}\subset {\mathop {\rm dom}} \, D$$ and, hence, $$\varphi \in {\mathop {\rm dom}} \, D$$. This implies that $${\mathop {\rm dom}} \, D={\mathop {\rm dom}} \, D_B$$. Moreover, from (11) and the definition of the $$\check {R}$$-norm the maps $$R: {\mathop {\rm dom}} \, D_{\mathrm {max}} \to \check { R}$$ and $$\mathcal {E}: \check { R}\to {\mathop {\rm dom}} \, D_{\mathrm {max}}$$ are continuous. Hence, if $${\mathop {\rm dom}} \, D$$ is closed in $${\mathop {\rm dom}} \, D_{\mathrm {max}}$$, then the set $$B=\mathcal {E}^{-1}({\mathop {\rm dom}} \, D)$$ is closed in $$R({\mathop {\rm dom}} \, D_{\mathrm {max}})$$. Conversely, if $$B$$ is closed in $$\check { R}$$, then $${\mathop {\rm dom}} \, D=R^{-1}(B)$$ is closed in $${\mathop {\rm dom}} \, D_{\mathrm {max}}$$. Lemma 4.14 Let$$B$$be a boundary condition such that$$B\subset H_{{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma )$$. Then, the$$H_1$$-norm and the graph norm$$\Vert .\Vert _D$$are equivalent on$${\mathop {\rm dom}} \, D_B$$. Proof Since $$B$$ is a boundary condition, $${\mathop {\rm dom}} \, D_B$$ is closed in $$({\mathop {\rm dom}} \, D_{\mathrm {max}}, \Vert . \Vert _D)$$. Moreover, by $$B\subset H_{{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma )$$, Lemma  4.6 and (7), $${\mathop {\rm dom}} \, D_B$$ is closed in $$(H_1(M, \mathbb {S}_M), \Vert .\Vert _{H_1})$$. Thus, $$({\mathop {\rm dom}} \, D_B, \Vert .\Vert _D)$$ and $$({\mathop {\rm dom}} \, D_B, \Vert .\Vert _{H_1})$$ are both Hilbert spaces. By (7), the identity map $$\operatorname {Id} \colon ({\mathop {\rm dom}} \, D_B, \Vert .\Vert _{H_1})\to ({\mathop {\rm dom}} \, D_B, \Vert .\Vert _D)$$ is a bijective bounded linear map. From the bounded inverse theorem, we know that also the inverse is bounded. Hence, the $$H_1$$- and the graph norm are equivalent on $${\mathop {\rm dom}} \, D_B$$. Remark 4.15 The definition of $${\mathop {\rm dom}} \, D_B$$ in [9, Section 7] uses $$H_1^D\colon = \overline {\Gamma _c^\infty (M,\mathbb {S}_M)}^{\Vert .\Vert _{H_1^D}}$$ instead of $$H_1$$ where the $$H_1^D$$-norm is given by   \[\Vert \varphi \Vert _{H_1^D}^2=\Vert \chi \varphi \Vert _{H_1}^2 + \Vert \varphi \Vert _{L^2}^2 + \Vert D\varphi \Vert _{L^2}^2.\] Here, $$\chi$$ denotes an appropriate cut-off function such that $$\chi \varphi$$ only lives on a small collar of the boundary. Since we work with the classical Dirac operator on $$\mathrm {Spin^c}$$ manifolds and assume $$(M,\Sigma )$$ and $$L$$ being of bounded geometry, the $$H_1$$- and the $$H_1^D$$-norm coincide. Bär and Ballmann consider a more general situation where it suffices that $$M$$ is only complete but not necessarily of bounded geometry. Then, the $$H_1^D$$-norm is needed. We could also switch to this more general setup when dropping the condition (i) and (iii) in Definition  2.2 while still assuming that $$(\Sigma , g|_\Sigma )$$ is of bounded geometry and that the curvature tensor and its derivatives are bounded on $$U_\Sigma$$. For that situation, we would also obtain Theorem  1.2. But in order to simplify notation we stick to the bounded geometry of $$(M,\Sigma )$$. Example 4.16 Minimal and maximal extension.$$B={0}$$ gives the minimal extension $$D_{B=0}=D_{\mathrm {min}}$$, cf. Lemma  4.5. The maximal extension is obtained with $$B=R({\mathop {\rm dom}} \, D_{\mathrm {max}})$$. $$D_{B=H_{{1}/{2}}}\colon H_1(M,\mathbb {S}_M)\to L^2(M,\mathbb {S}_M)$$ is an extension of $$D_{cc}$$ but not closed (if the boundary is non-empty): Since $$\Gamma _c^\infty (M,\mathbb {S}_M)\subset H_1$$ and $$\Gamma _c^\infty (M,\mathbb {S}_M)$$ dense in $${\mathop {\rm dom}} \, D_{\mathrm {max}}$$, the closure of $${D}_{B=H_{{1}/{2}}}$$ is $$D_{\mathrm {max}}$$. [18, Section 6] Let $$P_\pm : L^2 (\Sigma , \mathbb {S}_M|_\Sigma ) \to L^2 (\Sigma , \mathbb {S}_M|_\Sigma ), \ \varphi \mapsto \frac {1}{2}(\varphi \pm {i} \nu \cdot \varphi )$$ and   \[D_\pm \colon {\mathop {\rm dom}} \, D_\pm :={\{ }\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}\,|\,P_\pm R\varphi =0{\} }\to L^2(M,\mathbb {S}_M).\] In Section  5, we will show that $$D_\pm$$ is a closed extension and that $$D_\pm =D_{B_\pm }$$, where   \[B_\pm ={\{ }\varphi \in H_{1/2}(\Sigma ,\mathbb {S}_M|_\Sigma )\,|\, P_\pm \varphi =0{\} }.\] Each $$\varphi$$ decomposes uniquely into $$\varphi =P_+ \varphi +P_-\varphi$$, and if $$\varphi \in H_{1/2}(\Sigma , \mathbb {S}_M|_\Sigma )$$, then $$P_\pm \varphi \in H_{1/2}(\Sigma , \mathbb {S}_M|_\Sigma )$$, too. This assures that the boundary condition $$B_\pm$$ is honestly larger than the trivial boundary condition $$B={\{ }0{\} }$$. More properties of this boundary condition can be found in Section  5. APS-boundary conditions. An obvious way to generalize the APS-boundary conditions for a closed boundary to our situation is given by the following: Let $$(M,\Sigma )$$ be of bounded geometry. We use the notations introduced in Section  3.7. We set $$B^{\mathrm {APS}}_{\geq a}=R({\mathop {\rm dom}} \, D_{\mathrm {max}})\cap \Gamma _{[a,\infty )}^{\mathrm {APS}}$$ and $$B^{\mathrm {APS}}_{<a}=R({\mathop {\rm dom}} \, D_{\mathrm {max}})\cap \Gamma _{ (-\infty , a]}^{\rm APS}$$, respectively. In the same ways, let $$B^{\mathrm {APS}}_{\leq a}$$ and $$B^{\mathrm {APS}}_{>a}$$ be defined. If a neighbourhood of $$a$$ is in the spectrum of $$D^\Sigma$$, $$B^{\mathrm {APS}}_{<a}$$ and $$B^{\mathrm {APS}}_{>a}$$ will not be closed. We conjecture that for $$(M,\Sigma )$$ of bounded geometry the sets $$B^{\mathrm {APS}}_{\geq a}$$ and $$B^{\mathrm {APS}}_{\leq a}$$ define boundary conditions. But actually we do not know. Boundary value problems. In this part, we want to prove Theorem  1.1. For that, we need to define first the notion coercivity at infinity: Definition 4.17 A closed linear operator $$D\colon {\mathop {\rm dom}} \, D\subset L^2(M,\mathbb {S}_M)\to L^2(M,\mathbb {S}_M)$$ is said to be $$({\mathop {\rm dom}} \, D)$$-coercive at infinity if there is a $$c>0$$ such that   \[\forall \varphi \in {\mathop {\rm dom}} \, D\cap \left (\ker D\right )^\perp :\, \Vert D\varphi \Vert _{L^2} \geq c\Vert \varphi \Vert _{L^2}{,}\] where $$\!\!\phantom {,}^\perp$$ denotes the orthogonal complement in $$L^2$$. Note that in case that $$D$$ is the Dirac operator on a complete manifold without boundary, coercivity at infinity follows immediately if $$0$$ is not the essential spectrum. Conversely, if the Dirac operator is coercive at infinity, then either $$0$$ is not in the essential spectrum or the kernel is infinite-dimensional. For manifolds with boundary, $$D$$ is, in general, no longer self-adjoint. Thus, the spectrum is, in general, complex and this translation to the essential spectrum is not possible. In Section  6, we will compare this coercivity condition with the originally one used in [9, Defintion 8.2] for closed boundaries. But first, we will see how this condition forces the range of the operator to be closed which is crucial in order to apply the Closed Range Theorem  2.5 and show existence of preimages for linear operator as we will need in Theorem  1.1. Lemma 4.18 If the closed linear operator$$D\colon {\mathop {\rm dom}} \, D\subset L^2(M,\mathbb {S}_M)\to L^2(M,\mathbb {S}_M)$$is$$({\mathop {\rm dom}} \, D)$$-coercive at infinity, then the range is closed. Proof Let $$\varphi _i$$ be a sequence in $${\mathop {\rm dom}} \, D$$ with $$D\varphi _i\to \psi$$ in $$L^2$$. We have to show that $$\psi$$ is in the image of $$D$$. Without loss of generality, we can assume that $$\varphi _i \perp \ker D$$. Then, $$({\mathop {\rm dom}} \, D)$$-coercivity at infinity gives that $$\varphi _i$$ is bounded in $$L^2$$ and, thus, also in the graph norm of $$D$$. Thus, $$\varphi _i\to \varphi$$ weakly in $$\Vert . \Vert _D$$. Let $$\eta \in {\mathop {\rm dom}} \, D^* $$. Then, $$(D\varphi ,\eta )=\lim _{i\to \infty } (D\varphi _i, \eta )=\lim _{i\to \infty } (\varphi _i, D^* \eta )= (\varphi , D^* \eta )$$. Thus, $$\varphi \in {\mathop {\rm dom}} \, D$$ and closedness of $${\mathop {\rm dom}} \, D$$ then implies that $$D\varphi =\psi$$. We are now ready to prove the following theorem. Theorem (Theorem  1.1) Let$$B$$be a boundary condition, and let the Dirac operator  \[D_B\colon {\mathop {\rm dom}} \, D_B\subset L^2(M, \mathbb {S}_M) \longrightarrow L^2(M, \mathbb {S}_M)\]be$$B$$-coercive at infinity. Let$$P_{B}\colon R({\mathop {\rm dom}} \, D_{\mathrm {max}})\to B$$be a projection. Then, for all$$\psi \in L^2(M, \mathbb {S}_M)$$and$$\tilde {\rho }\in {\mathop {\rm dom}} \, D_{\mathrm {max}}$$where$$\psi -D\tilde {\rho }\in (\ker \, (D_{B})^* )^\perp ,$$the boundary value problem  \[\left\{ \begin {array}{ll}D\varphi =\psi & {\text {on\ }} M,\\ (\operatorname {Id} - P_{B})R\varphi = (\operatorname {Id} - P_{B}) R\tilde {\rho } & {\text {on\ }} \Sigma \end {array}\right .\]has a solution$$\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}$$that is unique up to elements of the kernel$$\ker \, D_B$$. Projection only means here that $$P_B$$ is linear and $$P_B|_B=\operatorname {Id}$$. Proof Since $$D$$ is $$B$$-coercive at infinity, its range is closed by Lemma  4.18. Thus, due to the Closed Range Theorem  2.5, the spinor $$\psi -D\tilde {\rho }\in \mathrm {ran}\, D_B$$. Hence, there exists $$\hat {\varphi }\in {\mathop {\rm dom}} \, D_B$$ with $$D\hat {\varphi }= \psi -D\tilde {\rho }$$. Setting $$\varphi =\hat {\varphi }+ \tilde {\rho }$$, we obtain $$\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}$$, $$D\varphi =\psi$$ and $$(\operatorname {Id} - P_{B})R\varphi = (\operatorname {Id} - P_{B})R\hat {\varphi }+ (\operatorname {Id} - P_{B})R\tilde {\rho }= (\operatorname {Id} - P_{B})R\tilde {\rho }$$. Corollary 4.19 Let$$B$$be a boundary condition such that$$B\subset H_{{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma )$$. We assume that the Dirac operator$$D\colon {\mathop {\rm dom}} \, D_B\subset L^2(M, \mathbb {S}_M) \to L^2(M, \mathbb {S}_M)$$is$$B$$-coercive at infinity. Let$$P_B\colon H_{{1}/{2}}(\Sigma ,\mathbb {S}_M|_\Sigma )\to B$$be a projection. Moreover, assume that$$\psi \in L^2(M, \mathbb {S}_M)$$and$$\rho \in H_{{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma )$$satisfy  \[(\psi ,\chi )+ (\nu \cdot \rho , R\chi )_\Sigma =0\] (12)for all$$\chi \in \ker \, (D_B)^* $$. Then, the boundary value problem  \[\left\{ \begin {array}{ll}D\varphi =\psi & {\text {on }} \ M,\\ (\operatorname {Id} - P_{B})R\varphi = (\operatorname {Id} - P_{B}){\rho }& {\text {on }} \ \Sigma \end {array}\right .\]has a solution$$\varphi \in H_1(M,\mathbb {S}_M)$$that is unique up to elements of the kernel$$\ker \, D_B$$. Proof By Lemma  4.6, $$B\subset H_{{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma )$$ implies $${\mathop {\rm dom}} \, D_B\subset H_1(M,\mathbb {S}_M)$$. We set $$\tilde {\rho }=\mathcal {E}\rho$$. By the Trace Theorem  3.7, $$\tilde {\rho }\in H_1(M,\mathbb {S}_M)$$. Moreover, by Lemma  4.4 the integrability condition (12) implies that $$\psi -D\tilde {\rho }\in (\ker \, (D_{B})^* )^\perp$$. Hence, together with the Closed Range Theorem there is $$\hat {\varphi }\in {\mathop {\rm dom}} \, D_B\subset H_1(M,\mathbb {S}_M)$$ with $$D\hat {\varphi }= \psi -D\tilde {\rho }$$. Thus, as in the proof of Theorem  1.1 $$\varphi =\hat {\varphi }+ \tilde {\rho }$$ gives a solution which is now in $$H_1(M,\mathbb {S}_M)$$. Remark 4.20 To give a full generalization of the theory given in [9] it would be interesting to examine the following questions. Consider general boundary conditions, in particular we would like to identify the image of the extended trace map in Theorem  4.2. Give a generalization of the definition for elliptic boundary conditions for noncompact boundaries (of bounded geometry) and study them. Consider, more generally, complete Dirac-type operators as in [9]. 5. On the boundary condition $$B_\pm$$ In this section, we briefly recall and give some basic facts on $$P_\pm$$. Some of them can be found in [18, Section 6]. Moreover, we prove the claims of Example  4.16(iii). Lemma 5.1 Let$$P_\pm \colon L^2(\Sigma , \mathbb {S}_M|_\Sigma )\to L^2(\Sigma , \mathbb {S}_M|_\Sigma )$$be the map$$\varphi \mapsto \frac {1}{2}(\varphi \pm {i} \nu \cdot \varphi )$$and consider$$B_\pm := {\{ } \varphi \in H_{{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma ) \,|\,P_\pm \varphi =0{\} }$$. Then, the following hold: $$P_\pm$$are self-adjoint projections, orthogonal to each other and$$\nu P_\pm =P_\pm \nu = \mp {i} P_\pm ;$$ for all$$s\in \mathbb {R},$$$$P_\pm (\varphi )= \frac {1}{2}(\varphi \pm i\nu \cdot \varphi )$$gives an operator from$$H_s(\Sigma , \mathbb {S}_M|_\Sigma )$$to itself such that for all$$\varphi \in H_s(\Sigma , \mathbb {S}_M|_\Sigma )$$and$$\psi \in H_{-s}(\Sigma , \mathbb {S}_M|_\Sigma )$$we have$$(P_+ \varphi ,P_-\psi )_\Sigma =0$$and$$(P_\pm \varphi , \psi )_\Sigma = (\varphi , P_\pm \psi )_\Sigma ;$$ $$\tilde {D}^{\Sigma }P_{\pm }=P_{\mp } \tilde {D}^{\Sigma };$$ $$D_\pm$$$$($$see Example  4.16(iii) for the definition$$)$$is a closed extension of$$D_{cc};$$ $$D_\pm = D_{B_\pm };$$ $$(D_{B_\pm })^* =D_{B_\mp };$$ let each connected component of$$M$$have a non-empty boundary. Then,$$\ker D_{B_\pm } ={\{ }0{\} }$$. Proof Assertions (i) and (ii) follow directly by simple calculations, and (iii) follows directly from (5). For (iv), we have by definition of $$D_\pm$$ (see Example  4.16(iii)) that $$D_\pm =D_{\tilde {B}_\pm }$$, where $$\tilde {B}_\pm ={\{ }\varphi \in R({\mathop {\rm dom}} \, D_{\mathrm {max}}) \,|\,P_\pm \varphi =0{\} }$$. To show the closedness of $$D_\pm$$ we want to apply Lemma  4.13. For that, we have to show that $$\tilde {B}_\pm$$ is closed in $$\check {R}$$: Let $$\varphi _i\in \tilde {B}_\pm$$ with $$\varphi _i\to \varphi$$ in $$\check {R}$$. Then, we get, together with Remark  4.9(ii), that   \[\begin {array}{rl} \Vert P_\pm \varphi \Vert _{\check {R}}&= \Vert P_\pm (\varphi -\varphi _i)\Vert _{\check {R}}= \Vert \mathcal {E} P_\pm (\varphi -\varphi _i)\Vert _{D}\leq \tfrac {1}{2} (\Vert \mathcal {E} (\varphi -\varphi _i)\Vert _{D} + \Vert \mathcal {E} \nu \cdot (\varphi -\varphi _i)\Vert _{D})\\ &\leq C\Vert \mathcal {E} (\varphi -\varphi _i)\Vert _{D} = \Vert \varphi -\varphi _i\Vert _{\check {R}}\longrightarrow 0. \end {array}\] Hence, $$P_\pm \varphi =0$$ and $$\varphi \in \tilde {B}_\pm$$. For (v), we have clearly that $${\mathop {\rm dom}} \, D_{B_\pm }\subset {\mathop {\rm dom}} \, D_\pm$$. It remains to show that any $$\varphi \in {\mathop {\rm dom}} \, D_\pm$$ is already in $$H_1(M, \mathbb {S}_M)$$. By Lemma  4.1, there is a sequence $$\varphi _i\in \Gamma _c^\infty (M,\mathbb {S}_M)$$ with $$\varphi _i\to \varphi$$ in the graph norm. Consider $$\mathcal {E}P_\pm R\varphi _i$$. By the linearity of $$\mathcal {E}$$, (11) and Remark  4.9(ii), we obtain   \[\begin {array}{rl} \Vert \mathcal {E}P_\pm R \varphi _i\Vert _D&= \Vert \mathcal {E}P_\pm R(\varphi _i-\varphi )\Vert _D\\ &\leq \tfrac {1}{2}(\Vert \mathcal {E} R(\varphi _i-\varphi )\Vert _D + \Vert \mathcal {E}(\nu \cdot R(\varphi _i-\varphi )\Vert _D)) \leq C \Vert \varphi _i-\varphi \Vert _D\longrightarrow 0. \end {array}\] Hence, $$\psi _i:=\varphi _i - \mathcal {E}P_\pm R \varphi _i \to \varphi$$ in the graph norm. Since $$\psi _i \in {\mathop {\rm dom}} \, D_{B_\pm }$$, this implies that $${\mathop {\rm dom}} \, D_{B_\pm }$$ is dense in $${\mathop {\rm dom}} \, D_\pm$$. Moreover, note that with (iii) and (i) we have   \[\int _\Sigma \langle R\psi _i, \tilde {D}^{\Sigma }R\psi _i\rangle \,ds= \int _\Sigma \langle P_\mp R\psi _i, \tilde {D}^{\Sigma }P_\mp R\psi _i\rangle \, ds=\int _\Sigma \langle P_\mp R\psi _i, P_\pm \tilde {D}^{\Sigma } R\psi _i\rangle \,ds=0.\] Hence, together with the Lichnerowicz formula in Lemma  3.10, the bounded geometry, (i) and Lemma  3.10, we obtain   \[\begin {array}{rl} \Vert \psi _i-\psi _j\Vert _{H_1}^2=&\Vert \psi _i-\psi _j\Vert _D^2-\frac {1}{4}\int _M\langle ({\rm scal} ^M+2{i} \Omega \cdot )(\psi _i-\psi _j),(\psi _i-\psi _j)\rangle \,dv\\ &-\frac {n}{2}\int _\Sigma H|R(\psi _i-\psi _j)|^2 \,ds\\ \leq & C\Vert \psi _i-\psi _j\Vert _D^2 \mp {i} \frac {n}{2}\int _\Sigma \langle \nu \cdot R(\psi _i-\psi _j), H R(\psi _i-\psi _j)\rangle \\ \leq & C\Vert \psi _i-\psi _j\Vert _D^2. \end {array}\] Thus, $$\psi _i$$ is even a Cauchy sequence in $$H_1$$ which implies that $$\varphi$$ is already in $$H_1(M,\mathbb {S}_M)$$. Note that this implies in particular that $$B_\pm =\tilde {B}_\pm$$. For (vi), the domain of the adjoint is defined by   \[{\mathop {\rm dom}} \, (D_{+ })^* ={\{ }\vartheta \in L^2(M,\mathbb {S}_M)\,|\,\exists \chi \in L^2(M,\mathbb {S}_M)\, \forall \psi \in {\mathop {\rm dom}} \, D_{+ }: (\chi , \psi )=(\eta , D\psi ) {\} }.\] Since, $$\Gamma _{cc}^\infty (M,\mathbb {S}_M) \subset {\mathop {\rm dom}} \, D_{+ }$$, we get $${\mathop {\rm dom}} \, (D_{+ })^* \subset {\mathop {\rm dom}} \, D_{\mathrm {max}}$$. Thus,   \[{\mathop {\rm dom}} \, (D_{+ })^* ={\{ }\vartheta \in {\mathop {\rm dom}} \, D_{\mathrm {max}}\,|\,\forall \psi \in {\mathop {\rm dom}} \, D_{+ }: (D\vartheta , \psi )=(\vartheta , D\psi ) {\} }.\] Due to Lemma  4.4, the definition of $${\mathop {\rm dom}} \, D_{+ }$$ and (v), we obtain   \[{\mathop {\rm dom}} \, (D_{+ })^* = \left\{ \vartheta \in {\mathop {\rm dom}} \, D_{\mathrm {max}}\left | \forall \psi \in H_1(M,\mathbb {S}_M): \int _\Sigma \langle \nu \cdot R\vartheta , P_-R\psi \rangle \,ds=0 \right . \right\} .\] By (i) and (ii), we have   \[-\int _\Sigma \langle R\vartheta , \nu \cdot P_-R\psi \rangle \,ds={i} \int _\Sigma \langle R\vartheta , P_-R\psi \rangle \,ds={i} \int _\Sigma \langle P_-R\vartheta , R\psi \rangle \,ds\] and $$P_-R\vartheta \in H_{-1/2}(\Sigma , \mathbb {S}_M|_\Sigma )$$. Hence, together with Lemmas  3.9 and 4.6,   \[\begin {array}{rl} {\mathop {\rm dom}} \, (D_{+ })^* &=\left\{ \vartheta \in {\mathop {\rm dom}} \, D_{\mathrm {max}}\left | \forall \hat {\psi }\in H_{{1}/{2}}(\Sigma ,\mathbb {S}_M|_\Sigma ): \int _\Sigma \langle P_-R\vartheta , \hat {\psi }\rangle \,ds=0 \right .\right\} \\ &={\{ }\vartheta \in {\mathop {\rm dom}} \, D_{\mathrm {max}}\,|\,P_-R\vartheta =0 {\} }={\mathop {\rm dom}} \ D_-. \end {array}\] The assertion (vii) is proved as in the closed case [18, Proof of Corollary 6]: Let $$\varphi \in \ker D_{\pm }$$, that is, $$\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {max}}$$, $$D\varphi =0$$ on $$M$$, and $$P_\pm R\varphi =0$$ on $$\Sigma$$. Using this, (2), Lemma  4.4 and (i), we compute   \[\begin {array}{rl} 0&=\int _M \langle \varphi , {i} D\varphi \rangle \,dv - \int _M \langle D\varphi , {i} \varphi \rangle \,dv =\int _\Sigma \langle \nu \cdot R\varphi , {i} R\varphi \rangle \,ds \\ &=\int _\Sigma \langle \nu \cdot P_\mp R\varphi , {i} P_\mp R\varphi \rangle \,ds=\pm \int _\Sigma |R \varphi |^2 \,ds. \end {array}\] Hence, $$R\varphi =0$$ and $$\varphi \in {\mathop {\rm dom}} \, D_{\mathrm {min}}$$, cf. Lemma  4.5. But due to the strong unique continuation property of the Dirac operator [11, Section 1.2], $$D_{\mathrm {min}}\varphi =0$$ implies $$\varphi =0$$. 6. Examples and the coercivity condition In Definition  4.17, we defined when an operator $$D_B$$ is $$(\mathrm {dom}\, D_B)$$-coercive at infinity. When working with $$B$$, we will also use the short version, $$B$$-coercive at infinity. In this passage, we will compare this notion with the one of coercivity at infinity given in [9, Definition 8.2] as cited below and give some examples. Definition 6.1 [9, Definition 8.2] $$D\colon {\mathop {\rm dom}} \, D_{\mathrm {max}} \subset L^2(M,\mathbb {S}_M)\to L^2(M, \mathbb {S}_M)$$ is coercive at infinity if there is a compact subset $$K\subset M$$ and a constant $$c>0$$ such that   \[\Vert D\varphi \Vert _{L^2} \geq c\Vert \varphi \Vert _{L^2},\] for all $$\varphi \in \Gamma _c^\infty (M\setminus K, \mathbb {S}_M)$$. By [9, Lemma 8.4], $$D$$ is coercive at infinity for a closed boundary $$\Sigma$$ if and only if there is a compact subset $$K\subset M$$ and a constant $$c>0$$ such that for all $$\varphi \in \Gamma _{cc}^\infty (M\setminus K, \mathbb {S}_M)$$ we have $$\Vert D\varphi \Vert _{L^2} \geq c\Vert \varphi \Vert _{L^2}$$. For noncompact boundaries, just the ‘ only if’-direction survives since in contrast to closed boundaries there is no compact $$K$$ such that $$\Gamma _c^\infty (M\setminus K,\mathbb {S}_M)\subset \Gamma _{cc}^\infty (M,\mathbb {S}_M)$$. Before we compare those different coercivity conditions we give some examples: Example 6.2 By the unique continuation property, the kernel of $$D_{\mathrm {min}}$$ is trivial. Thus, together with Lemma  4.5, we have that $$D$$ is $$(B=0)$$ -coercive at infinity if and only if there is a constant $$c>0$$ such that for all $$\varphi \in \Gamma _{cc}^\infty (M,\mathbb {S}_M)$$  \[\Vert D\varphi \Vert _{L^2} \geq c\Vert \varphi \Vert _{L^2}.\] For closed boundaries, this implies coercivity at infinity by [9, Lemma 8.4] which was cited above. We will see that for closed boundaries also the converse is true, cf. Corollary  6.7. By Lemma  5.1, $$\ker D_{B_\pm }={\{ }0{\} }$$. Thus, $$D$$ is $$B_\pm$$-coercive at infinity if and only if there is a constant $$c>0$$ such that   \[\Vert D\psi \Vert _{L^2}\geq c\Vert \psi \Vert _{L^2}\] for all $$\psi \in H_1(M,\mathbb {S}_M)$$ with $$P_\pm R\psi =0$$. In particular, this implies $$(B=0)$$-coercivity at infinity. More generally, if $$B_1\subset B_2$$ and $$\ker D_{B_1}=\ker D_{B_2}$$, then $$B_2$$-coercivity at infinity implies $$B_1$$-coercivity at infinity. Lemma 6.3 Let$$D$$be coercive at infinity, and let$$B$$be a boundary condition. Assume that$${\mathop {\rm dom}} \, D_B\cap (\ker D_B)^\perp \subset H_1(M,\mathbb {S}_M)$$and that the$$H_1$$-norm and the graph norm are equivalent on$${\mathop {\rm dom}} \, D_B\cap (\ker D_B)^\perp$$. Then,$$D$$is$$B$$-coercive at infinity. Proof Since $$D$$ is coercive at infinity, there is a compact subset $$K\subset M$$ and a constant $$c>0$$ such that $$\Vert D\varphi \Vert _{L^2}\geq c\Vert \varphi \Vert _{L^2}$$ for all $$\varphi \in \Gamma _c^\infty (M\setminus K, \mathbb {S}_M)$$. Assume that $$D$$ is not $$B$$-coercive at infinity. Then, there is a sequence $$\varphi _i\in {\mathop {\rm dom}} \, D_B\cap (\ker D_B)^\perp$$ with $$\Vert \varphi _i\Vert _{L^2}=1$$ and $$\Vert D\varphi _i\Vert _{L^2}\to 0$$. By equivalence of the norms, $$\varphi _i$$ is also bounded in $$H_1$$. This implies $$\varphi _i \to \varphi$$ weakly in $$H_1$$ and, thus, locally strongly in $$L^2$$. Moreover, $$D\varphi =0$$. Together with $$\varphi _i\perp \ker D_B$$, this implies $$\varphi =0$$. Thus, for each compact subset $$K'\subset M$$ we have $$\int _{K'} |\varphi _i|^2 \,dv\to 0$$ as $$i\to \infty$$. Let $$\eta \colon M \to [0,1]$$ be a cut-off function and $$K'$$ be a compact subset such that $$K\subset K' \subset M$$ and $$\eta =0$$ on $$K$$, $$\eta =1$$ on $$M\setminus K'$$ and $$|d \eta |\leq a$$ for a constant $$a>0$$ big enough. Then, $${\mathop {\rm supp}} \, (\eta \varphi _i)\subset M\setminus K$$, $$\Vert D(\eta \varphi _i)\Vert _{L^2}\leq a\Vert \varphi _i\Vert _{L^2(K')}+ \Vert D\varphi _i\Vert _{L^2}\to 0$$ and   \[1\geq \Vert \eta \varphi _i\Vert _{L^2}\geq \Vert \varphi _i \Vert _{L^2}- \Vert (1-\eta )\varphi _i \Vert _{L^2}\geq 1-\Vert \varphi _i\Vert _{L^2(K')}\longrightarrow 1.\] By Lemma  4.1, we can choose a sequence $$(\varphi _{ij})_j\subset \Gamma _c^\infty (M,\mathbb {S}_M)$$ with $$\varphi _{ij}\to \varphi _i$$ in the graph norm as $$j\to \infty$$. Then, $$\eta \varphi _{ij}\to \eta \varphi _i$$ in the graph norm and $${\mathop {\rm supp}} \, (\eta \varphi _{ij})\in M\setminus K$$. Thus, we can find $$j=j(i)$$ such that $$\Vert D(\eta \varphi _{ij(i)})\Vert _{L^2}\to 0$$ and $$\Vert \eta \varphi _{ij(i)}\Vert _{L^2}\to 1$$ as $$i\to \infty$$. But this contradicts the assumption that $$D$$ is coercive at infinity. From the last lemma and Lemma  4.14, we obtain immediately the following corollary. Corollary 6.4 If$$D$$is coercive at infinity and$$B\subset H_{{1}/{2}}(\Sigma , \mathbb {S}_M|_\Sigma ),$$then$$D$$is$$B$$-coercive at infinity. Next, we give some (very restrictive) conditions that are sufficient to prove that $$B$$-coercivity at infinity implies coercivity at infinity. Those additional assumptions are needed to make sure that the $$\varphi _i$$ appearing in Definition  6.1 are in $${\mathop {\rm dom}} \, D_B$$. Lemma 6.5 Let$$B$$be a boundary condition with$$B\subset H_{1/2}(\Sigma , \mathbb {S}_M|_\Sigma )$$. Assume that there exists a compact subset$$K'\subset M$$with$$\Gamma _c^\infty (M\setminus K', \mathbb {S}_M)\subset {\mathop {\rm dom}} \, D_B$$. If$$D\colon \mathrm {dom}\, D_B\subset L^2(\Sigma , \mathbb {S}_M|_\Sigma ) \to L^2(\Sigma , \mathbb {S}_M|_\Sigma )$$has a finite-dimensional kernel and$$D$$is$$B$$-coercive at infinity, then$$D$$is coercive at infinity. Proof Assume that $$D$$ is not coercive at infinity. Then, for all compact subsets $$K\subset M$$ there exists a sequence $$\varphi _i\in \Gamma _c^\infty (M\setminus K, \mathbb {S})$$ with $$\Vert \varphi _i\Vert _{L^2}=1$$ and $$\Vert D\varphi _i\Vert _{L^2}\to 0$$. We choose $$K$$ such that $$K'\subset K$$. Then, all those $$\varphi _i\in {\mathop {\rm dom}} \, D_B$$. Thus, $$\varphi _i\to \varphi \in {\mathop {\rm dom}} \, D_B$$ weakly in the graph norm of $$D$$, $$\varphi \in \ker \, D_B$$ and $$\varphi =0$$ on $$K$$. We decompose $$\varphi _i=\varphi _i^k + \varphi _i^\perp ,$$ where $$\varphi _i^k \in \ker \, D_B$$ and $$\varphi _i^\perp \in (\ker D_B)^\perp$$. Then $$\Vert D\varphi _i^\perp \Vert _{L^2}\to 0$$. Moreover, we assume that the kernel is finite-dimensional, that is, $$\varphi _i^k=\sum _{j=1}^l a_{ij}\psi _j$$, where the $$\psi _j$$'s form an orthonormal basis of $$\mathrm {ker}\, D_B$$. Thus, $$\Vert \varphi _i^k\Vert _{L^2}^2=\sum _{j=1}^l|a_{ij}|^2$$. Assume now that $$\Vert \varphi _i^\perp \Vert _{L^ 2}\to 0$$. Then $$\varphi _i^\perp \to 0$$ in the graph norm. But $$\Vert \varphi _i\Vert _{L^2}=1$$. This implies that there is at least one $$j\in {\{ } 1, \ldots , l{\} }$$ with $$|a_{ij}|$$ is bounded away from zero for almost all $$i$$, that is, $$\varphi$$ cannot be zero everywhere. Since $$\varphi$$ is zero on $$K$$, this is a contradiction to the unique continuation principle. Thus, the assumption was wrong and there exists $$c>0$$ with $$\Vert \varphi _i^\perp \Vert _{L^2}>c$$ and $$D$$ is not $$B$$-coercive at infinity. Note that the assumption on the existence of $$K'$$ is very restrictive. If the boundary is closed, then it is automatically satisfied and we get the corollary below. If the boundary is noncompact, for a general $${\mathop {\rm dom}} \, D$$, for example, for the minimal domain of $$D$$, then it is not true. But there are also examples for manifolds with noncompact boundary and closed extension of $$D_{cc}$$ where the assumptions of the last lemma are satisfied: Example 6.6 Let $$(\Sigma , h)$$ be a complete Riemannian $${\mathop {\rm Spin}}$$ manifold. Let $$M_\infty =\Sigma \times \mathbb {{R}}$$ and $$M=\Sigma \times [0,\infty )$$ be equipped with product metric $$h+dt^2$$. Both manifolds are of bounded geometry. Since $$M_\infty$$ is complete with no boundary, the Dirac operator on $$M_\infty$$ is essentially self-adjoint. Assume that the Dirac operator on $$M_\infty$$ is invertible. Let $$K'\subset M_\infty$$ be a compact subset that intersects $$\Sigma \times {\{ }0{\} }$$ in a subset of non-zero measure. Define $$\mathcal {L}$$ to be the linear span of $$\Gamma _c^\infty (M\setminus K', \mathbb {S}_M)\cup \Gamma _{cc}^\infty (M,\mathbb {S}_M)$$ and $${\mathop {\rm dom}} \, D_B\colon = \bar {\mathcal {L}}^{\Vert . \Vert _D}$$. Then, $$B=\overline {\Gamma _c^\infty (\Sigma \setminus K', \mathbb {S}_M|_\Sigma )}^{\Vert . \Vert _{\check {R}}}$$. Note that by construction $${\mathop {\rm dom}} \, D_B$$ is the domain of a closed extension of $$D_{cc}$$. But it is honestly smaller than $${\mathop {\rm dom}} \, D_{\rm max}$$ since all $$\varphi \in B$$ have to vanish on $$\Sigma \cap K'$$. In particular, by the strong unique continuation property of $$D$$ (see [11, Section 1.2]) $$D_B\colon {\mathop {\rm dom}} \, D_B \to L^2(M,\mathbb {S}_M)$$ has trivial kernel. It remains to show that $$D_B$$ is $$B$$-coercive at infinity, that is, there is $$c>0$$ such that for all $$\varphi \in \mathcal {L}$$ we have $$\Vert D\varphi \Vert _{L^2}\geq c\Vert \varphi \Vert _{L^2}$$. We will show this by contradiction, that is, we assume that there is a sequence $$\varphi _i\in \mathcal {L}$$ with $$\Vert \varphi _i\Vert _{L^2}=1$$ and $$\Vert D\varphi _i\Vert _{L^2}\to 0$$. We will construct a sequence of spinors on $$M_\infty$$. Let $$\tilde {\varphi }_i$$ be obtained from $$\varphi _i$$ by reflection along $$\Sigma$$. Clearly, $$\tilde {\varphi }_i\in L^2(M_\infty , \mathbb {S}_{M_\infty })$$. Moreover, note that $$\tilde {\varphi }_i$$ is everywhere continuous. Let $$\nu$$ be the inward normal vector field of $$M$$. For $$\psi \in \Gamma _c^\infty (M_\infty , \mathbb {S}_{M_\infty }),$$ we can estimate using (2)   \[\begin {array}{rl} |(\tilde {\varphi }_i, D\psi )_{L^2(M_\infty )}| & = \left |\int _{\Sigma \times (0,\infty )} \langle \tilde {\varphi }_i, D\psi \rangle + \int _{\Sigma \times (-\infty ,0)} \langle \tilde {\varphi }_i, D\psi \rangle \right |\\ & =\left |\int _{\Sigma \times (0,\infty )} \langle D\tilde {\varphi }_i, \psi \rangle + \int _{\Sigma } \langle \nu \cdot \tilde {\varphi }_i|_\Sigma , \psi |_\Sigma \rangle + \int _{\Sigma \times (-\infty ,0)} \langle D\tilde {\varphi }_i, \psi \rangle \right .\\ &\quad + \left . \int _{\Sigma } \langle -\nu \cdot \tilde {\varphi }_i|_\Sigma , \psi |_\Sigma \rangle \right |\\ &\leq 2\Vert D\varphi _i\Vert _{L^2(M)} \Vert \psi \Vert _{L^2(M_\infty )}\longrightarrow 0. \end {array}\] In particular, this means that $$\tilde {\varphi }_i\in H_1(M_\infty , \mathbb {S}_{M_\infty })$$ and that $$\Vert D\tilde {\varphi }_i\Vert _{L^2(M_\infty )}\to 0$$ while $$\Vert \tilde {\varphi }_i\Vert _{L^2(M_\infty )}=2$$. This gives a contradiction to the invertibility of the Dirac operator on $$M_\infty$$. Corollary 6.7 Let the boundary$$\Sigma$$be closed. If$$B$$is an elliptic boundary condition as defined in [9, Definition 7.5], then$$B$$-coercivity at infinity implies coercivity at infinity. In particular,$$D$$is$$(B=0)$$-coercive at infinity if and only if it is coercive at infinity. Proof If the boundary is closed and $$B$$ is elliptic, then $$D_B$$ has a finite kernel [9, Theorem 8.5]. The rest of the assumption in Lemma  6.5 is trivially fulfilled which gives the first claim. The rest follows with Corollary  6.4. For closed boundaries and spin manifolds, assuming uniformly positive scalar curvature at infinity is a sufficient condition to have that $$D$$ is coercive at infinity, see [9, Example 8.3]. For noncompact boundaries, we obtain the following lemma. Lemma 6.8 If$$\frac 12 {\rm scal} ^M + {i} \Omega \cdot$$is a positive operator, then the Dirac operator$$D$$is$$(B=0)$$-coercive at infinity. If$$\frac 12 {\rm scal} ^M + {i} \Omega \cdot$$is a positive operator and$$H\geq 0,$$then the Dirac operator$$D$$is$$B_\pm$$-coercive at infinity. Proof Let $$c>0$$ such that $$\frac 12 {\rm scal} ^M + {i} \Omega \cdot \geq 2c$$. The Lichnerowicz formula (6) and Lemma  3.10 give   \[\begin {array}{rl} \Vert D\varphi \Vert _{L^2}^2&=\Vert \nabla \varphi \Vert _{L^2}^2+ \int _{M} \frac {{\rm scal} ^M}{4}|\varphi |^2 \,dv + \int _{M} \frac {{i} }{2}<\Omega \cdot \varphi , \varphi >\,dv - \int _{\Sigma } \langle R\varphi ,\tilde {D}^{\Sigma }(R\varphi )\rangle \,ds\\ & \quad + \frac {n}{2}\int _{\Sigma } H|R\varphi |^2\,ds\geq c\Vert \varphi \Vert _{L^2}^2 -\int _{\Sigma } \langle R\varphi , \tilde {D}^{\Sigma }(R\varphi )\rangle \,ds + \frac {n}{2}\int _{\Sigma } H|R\varphi |^2\,ds, \end {array}\] for all $$\varphi \in H_1(M, \mathbb {S}_M)$$. Then, (i) follows directly with Lemma  4.5. For (ii), let now $$H\geq 0$$ and $$R\varphi \in B_\pm$$. Then, together with Lemma  5.1, it implies   \[\begin {array}{rl} \Vert D\varphi \Vert _{L^2}^2&\geq c\Vert \varphi \Vert _{L^2}^2 -\int _{\Sigma } \langle R\varphi , \tilde {D}^{\Sigma }(R\varphi )\rangle \,ds=c\Vert \varphi \Vert _{L^2}^2 -\int _{\Sigma } \langle P_\mp R\varphi , \tilde {D}^{\Sigma }(P_\mp R\varphi )\rangle \\ &= c\Vert \varphi \Vert _{L^2}^2 -\int _{\Sigma } \langle P_\mp R\varphi , P_\pm \tilde {D}^{\Sigma }(R\varphi )\rangle = c\Vert \varphi \Vert _{L^2}^2. \end {array}\] 7. $$\mathrm {Spin^c}$$ Reilly inequality on possibly open boundary domains In this section, we shortly review the spinorial Reilly inequality. This inequality together with those boundary value problems discussed in Section  4 will be the main ingredient in the proof of Theorem  1.2. Theorem 7.1 ($$\mathrm {Spin^c}$$ Reilly inequality) For all$$\psi \in H_1(M,\mathbb {S}_M),$$we have  \[\int _{\Sigma }\left (\langle \tilde {D}^{\Sigma }\psi ,\psi \rangle -\frac {n}{2} H|\psi |^2\right )ds\geq \int _{M}\left (\frac {1}{4} {\rm scal} ^M \vert \psi \vert ^2 + \frac 12\langle {i} \Omega \cdot \psi , \psi \rangle - \frac {n}{n+1}|D\psi |^2\right )dv,\] (13)where$$dv$$$$($$respectively,$$ds)$$is the Riemannian volume form of$$M$$$$($$respectively,$$\Sigma )$$. Moreover, equality occurs if and only if the spinor field$$\psi$$is a twistor–spinor , that is, if and only if$$P\psi =0,$$where$$P$$is the twistor operator acting on$$\mathbb {S}_M$$and is locally given by$$P_X\psi =\nabla _X\psi + ({1}/({n+1}))X\cdot D\psi$$for all$$X \in \Gamma (TM)$$. Proof The inequality is proved for $$\psi \in \Gamma _c^\infty (M, \mathbb {S}_M)$$ analogously as in the compact $${\mathop {\rm Spin}}$$ case [16, (17)]. For the convenience of the reader, we will shortly recall it here. Then, for all $$\psi \in H_1(M,\mathbb {S}_M),$$ the claim follows using the Trace Theorem  3.7 in the same way as in Lemma  3.10: We define $$1$$-forms $$\alpha$$ and $$\beta$$ on $$M$$ by $$\alpha (X) = \langle X\cdot D\psi , \psi \rangle$$ and $$\beta (X) = \langle \nabla _X\psi , \psi \rangle$$ for all $$X \in \Gamma ^\infty (TM)$$. Then, $$\alpha$$ and $$\beta$$ satisfy   \[\delta \alpha = \langle D^2\psi , \psi \rangle - \vert D \psi \vert ^2, \quad \delta \beta = - \langle \nabla ^* \nabla \psi , \psi \rangle + \vert \nabla \psi \vert ^2.\] Applying the divergence theorem with (3) and (4), we obtain   \[\int _{\Sigma }\left (\langle \tilde {D}^{\Sigma } \psi , \psi \rangle -\frac n2 H\vert \psi \vert ^2 \right ) ds = \int _{M} \left (\vert \nabla \psi \vert ^2 - \vert D\psi \vert ^2 + \frac 14 {\rm scal} ^M\ \vert \psi \vert ^2 + \frac {{i} }{2} \langle \Omega \cdot \psi , \psi \rangle \right ) dv.\] (14) On the other hand, for any spinor field $$\psi ,$$ we have   \[\vert \nabla \psi \vert ^2 = \vert P\psi \vert ^2 + \frac {1}{n+1} \vert D\psi \vert ^2.\] (15) Combining the identities (15), and (14) and $$\vert P\psi \vert ^2 \geq 0$$, the result follows. Equality holds if and only if $$\vert P\psi \vert ^2 = 0$$, that is, the spinor $$\psi$$ is a twistor–spinor. 8. A lower bound for the first nonnegative eigenvalue of the Dirac operator on the boundary In this section, we prove Theorem  1.2. For that we will not follow the original proof given in [16] due to our problems concerning the $$\mathrm {APS}$$-boundary conditions as remarked at the end of Example  4.16(iv). But we will use $$B_\pm$$ as given in Example  4.16(iii). Proof of Theorem  1.2 Since $$\Sigma$$ is of bounded geometry, $$\tilde {D}^{\Sigma }: H_1(\Sigma , \mathbb {S}_M|_\Sigma )\to L^2(\Sigma , \mathbb {S}_M|_\Sigma )$$ is self-adjoint and, hence, $$\lambda _1$$ is an eigenvalue or in the essential spectrum of $$\tilde {D}^{\Sigma }$$. In both cases, there is a sequence $$\varphi _i\in H_1(\Sigma , \mathbb {S}_M|_\Sigma )$$ with $$\Vert \varphi _i\Vert _{L^2(\Sigma )}=1$$ and $$\Vert (\tilde {D}^{\Sigma } -\lambda _1)\varphi _i\Vert _{L^2(\Sigma )}\to 0$$. Then, $$\varphi _i\to \varphi$$ weakly in $$L^2(\Sigma , \mathbb {S}_M|_\Sigma )$$. (In case that $$\varphi \neq 0$$, then $$\varphi$$ is an eigenspinor of $$\tilde {D}^{\Sigma }$$ to the eigenvalue $$\lambda _1$$ otherwise $$\lambda _1$$ is in the essential spectrum of $$\tilde {D}^{\Sigma }$$). We assumed that $$D$$ is $$B_-$$-coercive at infinity (everything which follows is also true when assuming $$B_+ $$-coercivity at infinity when switching the signs). Then, by Lemma  4.18, the range of $$D_{B_-}$$ is closed. Moreover, from Lemma  5.1 we have $$\ker \, (D_{B_-})^* = \ker \, D_{B_+ }={\{ }0{\} }$$. Thus, due to Corollary  4.19 for each $$i$$ there exists a unique $$\Psi _i\in H_1(M,\mathbb {S}_M)$$ with $$D\Psi _i=0$$ and $$P_+R\Psi _i=P_+ \varphi _i.$$ Using Theorem  7.1 and $${\rm scal} ^M+2{i} \Omega \cdot \geq 0$$, we obtain   \[0\leq \int _{\Sigma } \left (\langle \tilde {D}^{\Sigma }R\Psi _i,R\Psi _i\rangle -\frac {n}{2}H|R\Psi _i|^2\right ) ds.\] Moreover,   \[\begin {array}{rl} & (\tilde {D}^{\Sigma } (P_+R\Psi _i +P_-R\Psi _i), P_+R\Psi _i +P_-R\Psi _i)_\Sigma \\ & \quad = (\tilde {D}^{\Sigma } P_+R\Psi _i, P_-R\Psi _i)_\Sigma + (\tilde {D}^{\Sigma } P_- R\Psi _i, P_+R\Psi _i)_\Sigma \\ &\quad = (\tilde {D}^{\Sigma } P_+R\Psi _i, P_-R\Psi _i)_\Sigma + (P_- R\Psi _i, \tilde {D}^{\Sigma } RP_+ \Psi _i)_\Sigma , \end {array}\] where we used Lemma  5.1 and that $$\tilde {D}^{\Sigma }$$ is self-adjoint on $$H_1(\Sigma , \mathbb {S}_M|_\Sigma )$$. Hence, summarizing we get that   \[\begin {array}{rl} \frac {n}{2}\int _{\Sigma } H|R\Psi _i|^2\,ds &\leq 2\Re \int _{\Sigma } \langle \tilde {D}^{\Sigma } P_+R\Psi _i,P_-R\Psi _i\rangle \,ds=2\Re \int _{\Sigma } \langle P_-\tilde {D} \varphi _i,P_-R\Psi _i\rangle \,ds\\ &\leq 2\Re \int _{\Sigma } \langle P_-(\tilde {D}^{\Sigma }-\lambda _1) \varphi _i,P_-R\Psi _i\rangle \,ds+2\lambda _1 \Re \int _{\Sigma } \langle P_- \varphi _i,P_-R\Psi _i\rangle \,ds. \end {array}\] Using $$2\Re \int _{\Sigma } \langle P_- \varphi _i,P_-R\Psi _i\rangle ds\leq \Vert P_- \varphi _i \Vert _{L^2(\Sigma )}^2 + \Vert P_-R\Psi _i\Vert _{L^2(\Sigma )}^2$$ and $$\lambda _1\geq 0$$, we obtain   \[\frac {n}{2}\inf _\Sigma H \Vert R\Psi _i\Vert _{L^2(\Sigma )}^2 \leq 2\Vert (\tilde {D}^{\Sigma }-\lambda _1) \varphi _i\Vert _{L^2} \Vert R\Psi _i\Vert _{L^2}+ \lambda _1 (\Vert P_- \varphi _i \Vert _{L^2(\Sigma )}^2 + \Vert P_-R\Psi _i\Vert _{L^2(\Sigma )}^2).\] Moreover, $$(\tilde {D}^{\Sigma } P_\pm \varphi _i, P_\mp \varphi _i)=(P_\mp (\tilde {D}^{\Sigma } -\lambda _1) \varphi _i, P_\mp \varphi _i) + \lambda _1 \Vert P_\mp \varphi _i\Vert _{L^2}^2$$. Since $$\tilde {D}^{\Sigma }$$ is self-adjoint, $$\Re (\tilde {D}^{\Sigma } P_+ \varphi _i, P_-\varphi _i)= \Re (\tilde {D}^{\Sigma } P_-\varphi _i, P_+ \varphi _i)$$. Thus, together with   \[|(P_\mp (\tilde {D}^{\Sigma } -\lambda _1) \varphi _i, P_\mp \varphi _i)|\leq \Vert (\tilde {D}^{\Sigma } -\lambda _1) \varphi _i\Vert _{L^2}\Vert \varphi _i\Vert _{L^2}\longrightarrow 0\] as $$i\to \infty$$, this implies that $$\lim _{i\to \infty } \Vert P_-\varphi _i\Vert _ {L^2}= \lim _{i\to \infty } \Vert P_+ \varphi _i\Vert _ {L^2} =\frac {1}{2}$$ for $$\lambda _1\neq 0$$. Hence, for certain $$\varepsilon _i$$ with $$\varepsilon _i\to 0$$ as $$i\to \infty$$  \[\begin {array}{rl} \frac {n}{2}\inf _\Sigma H \Vert R\Psi _i\Vert _{L^2(\Sigma )}^2&\leq 2\Vert (\tilde {D}^{\Sigma }-\lambda _1) \varphi _i\Vert _{L^2} \Vert R\Psi _i\Vert _{L^2}+ \lambda _1 (\Vert P_+ \varphi _i \Vert _{L^2(\Sigma )}^2 + \varepsilon _i + \Vert P_-R\Psi _i\Vert _{L^2(\Sigma )}^2)\\ &\leq 2\Vert (\tilde {D}^{\Sigma }-\lambda _1) \varphi _i\Vert _{L^2} \Vert R\Psi _i\Vert _{L^2}+ \lambda _1 (\Vert P_+R\Psi _i \Vert _{L^2(\Sigma )}^2 + \varepsilon _i + \Vert P_-R\Psi _i\Vert _{L^2(\Sigma )}^2)\\ &\leq 2\Vert (\tilde {D}^{\Sigma }-\lambda _1) \varphi _i\Vert _{L^2} \Vert R\Psi _i\Vert _{L^2}+ \lambda _1 (\Vert R \Psi _i \Vert _{L^2(\Sigma )}^2 + \varepsilon _i). \end {array}\] Hence,   \[\frac {n}{2}\inf _\Sigma H \leq 2\Vert (\tilde {D}^{\Sigma }-\lambda _1) \varphi _i\Vert _{L^2} \Vert R\Psi _i\Vert _{L^2}^{-1}+ \lambda _1 (1+ \varepsilon _i \Vert R\Psi _i\Vert _{L^2}^{-2}).\] With $$\Vert R\Psi _i\Vert _{L^2}\geq \Vert P_+R\Psi _i\Vert _{L^2}=\Vert P_+ \varphi _i\Vert _{L^2} \to \frac {1}{2}$$, we finally get for $$i\to \infty$$  \[\frac {n}{2}\inf _\Sigma H \leq \lambda _1.\] Next, we collect all conditions that have to be fulfilled to obtain the equality $$\frac {n}{2}\inf _\Sigma H = \lambda _1$$: from the spinorial Reilly Inequality (13), $$\int _M |P\Psi _i|^2\,dv\to 0$$ which implies together with $$D\Psi _i=0$$ that $$\int _M |\nabla \Psi _i|^2\,dv\to 0$$; $$\int _M {\rm scal} ^M|\Psi _i|^2+2i\langle \Omega \cdot \Psi _i, \Psi _i\rangle \,dv\to 0;$$ $$\Vert \varphi _i - R\Psi _i\Vert _{L^2(\Sigma )}\to 0;$$ $$\int _{\Sigma } (H-\inf _\Sigma H) |R\Psi _i|^2 \,ds \to 0$$. In case that $$\lambda _1$$ is an eigenvalue of $$\tilde {D}^{\Sigma }$$ with eigenspinor $$\varphi$$, one can choose $$\varphi _i=\varphi$$ for all $$i$$. Then, $$\Psi _i=:\Psi$$ for all $$i$$ and those equality conditions reduce to $$\varphi =R\Psi$$, $$\Psi$$ is a parallel spinor on $$M$$, $$H$$ is constant and $$\int _M {\rm scal} ^M|\Psi |^2+2i\langle \Omega \cdot \Psi , \Psi \rangle \,dv= 0$$. Acknowledgements The work was initiated during the Junior Trimester Program in Differential Geometry at the Hausdorff Research Institute for Mathematics (HIM) in Bonn where the first author participated. The authors gratefully acknowledge the support of the HIM. The second author thanks the Institute of Mathematics of the University of Leipzig for its support and hospitality. The first author is very much indebted to Bernd Ammann, Ulrich Bunke and Cornelia Schneider for many enlightening discussions. We also want to thank the referee for very helpful comments. References 1 Agricola I. 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TI - Boundary value problems for noncompact boundaries of Spinc manifolds and spectral estimates
JF - Proceedings of the London Mathematical Society
DO - 10.1112/plms/pdu026
DA - 2014-06-27
UR - https://www.deepdyve.com/lp/oxford-university-press/boundary-value-problems-for-noncompact-boundaries-of-spinc-manifolds-JUvG0aaar0
SP - 946
EP - 974
VL - 109
IS - 4
DP - DeepDyve
ER -