A long neck principle for Riemannian spin manifolds with positive scalar curvature

We develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a ``long neck principle'' for a compact Riemannian spin $n$-manifold with boundary $X$, stating that if $\textrm{scal}(X)\geq n(n-1)$ and there is a nonzero degree map into the sphere $f\colon X\to S^n$ which is area decreasing, then the distance between the support of $\textrm{d} f$ and the boundary of $X$ is at most $\pi/n$. This answers, in the spin setting, a question recently asked by Gromov. As a second application, we consider a Riemannian manifold $X$ obtained by removing $k$ pairwise disjoint embedded $n$-balls from a closed spin $n$-manifold $Y$. We show that if $\textrm{scal}(X)>\sigma>0$ and $Y$ satisfies a certain condition expressed in terms of higher index theory, then the radius of a geodesic collar neighborhood of $\partial X$ is at most $\pi \sqrt{(n-1)/(n\sigma)}$. Finally, we consider the case of a Riemannian $n$-manifold $V$ diffeomorphic to $N\times [-1,1]$, with $N$ a closed spin manifold with nonvanishing Rosenberg index. In this case, we show that if $\textrm{scal}(V)\geq\sigma>0$, then the distance between the boundary components of $V$ is at most $2\pi \sqrt{(n-1)/(n\sigma)}$. This last constant is sharp by an argument due to Gromov.


Introduction and main results
The study of manifolds with positive scalar curvature has been a central topic in differential geometry in recent decades. On closed spin manifolds, the most powerful known obstruction to the existence of such metrics is based on the index theory for the spin Dirac operator. Indeed, the Lichnerowicz formula [Lic63] implies that, on a closed spin manifold Y with positive scalar curvature, the spin Dirac operator is invertible and hence its index must vanish.
When X is a compact Riemannian manifold with boundary of dimension at least three, it is well known by classical results of Kazdan and Warner [KW75a, KW75b,KW75c] that X always carries a metric of positive scalar curvature. In order to use topological information to study metrics of positive scalar curvature on X, we need extra geometric conditions. When X is equipped with a Riemannian metric with a product structure near the boundary, it is well known [APS75a, APS75b,APS76] that the Dirac operator with global boundary conditions is elliptic. This fact has been extensively used in the past decades to study metrics of positive scalar curvature in the spin setting.
The purpose of this paper is to systematically extend the spin Dirac operator technique to the case when the metric does not necessarily have a product structure near the boundary and the topological information is encoded by bundles supported away from the boundary. As an application, we prove some metric inequalities with scalar curvature on spin manifolds with boundary, following the point of view recently proposed by Gromov. 1.1. Some questions by Gromov on manifolds with boundary. Let (X, g) be a compact oriented n-dimensional Riemannian manifold with boundary and let f : (X, g) → (S n , g 0 ) be a smooth area decreasing map, where g 0 denotes the standard round metric on the sphere. The "length of the neck" of (X, f ) is defined as the distance between the support of the differential of f and the boundary of X.
The long neck problem [Gro19,page 87] consists in the following question. Question 1.1 (Long Neck Problem). What kind of a lower bound on scal g and a lower bound on the "length of the neck" of (X, f ) would make deg(f ) = 0?
Remark 1.2. In this case, the topological obstruction is the existence of an area decreasing map f : (X, g) → (S n , g 0 ) of nonzero degree. The extra geometric information is given by the "length of the neck" of (X, f ) and the lower bound of scal g . The main motivation of this paper is to prove this inequality in the case when X is spin.
We will now review two conjectures recently proposed by Gromov, which are related to the long neck problem. Let Y be a closed n-dimensional manifold. Let X be the n-dimensional manifold with boundary obtained by removing a small ndimensional ball from Y . Observe that X is a manifold with boundary ∂X ∼ = S n−1 . Let g be a Riemannian metric on X. For R > 0 small enough, denote by B R (∂X) the geodesic collar neighborhood of ∂X of width R. Gromov proposed the following conjecture [Gro18, Conjecture D', 11.12].
Conjecture 1.4. Let Y be a closed n-dimensional manifold such that Y minus a point admits no complete metric of positive scalar curvature. Let X be the manifold with boundary obtained by removing a small n-dimensional ball from Y . Let g be a Riemannian metric on X whose scalar curvature is bounded from below by a constant σ > 0. Then there exists a constant c > 0 such that if there exists a geodesic collar neighborhood B R (∂X) of width R, then Let us now consider a second situation related to the long neck principle. Let N be a closed manifold. A band over N is a manifold V diffeomorphic to N × [−1, 1]. If g is a Riemannian metric on V , we say that (V, g) is a Riemannian band over N and define the width of V by setting Remark 1.6. In general, one can ask whether, under the same hypotheses of Conjecture 1.5, there exists a constant c n , depending only on the dimension n of the manifold N , such that the inequality holds. Gromov proved [Gro18, Optimality of 2π/n, page 653] that the constant c n = 2π n − 1 n is optimal.
1.2. Codimension zero obstructions. Let (X, g) be a compact n-dimensional Riemannian spin manifold with boundary whose scalar curvature is bounded from below by a constant σ > 0. The first main result of this paper consists in a "long neck principle" in this setting. Our method is based on the analysis of the incomplete Riemannian manifold X o = X \ ∂X. The topological information is encoded by a pair of bundles with metric connections E and F over X o which have isomorphic typical fibers and are trivializable outside a compact submanifold with boundary L ⊂ X o . Our topological invariant is given by the index of a twisted spin Dirac operator D E,F LD on the double L D of L, constructed using the pair (E, F ). In order to relate this invariant to the geometry of the manifold X, we make use of extra data. We use the distance function from the deleted boundary ∂X of X o to construct a rescaling function ρ in such a way that the Dirac operator of X o , rescaled by the function ρ, is essentially self-adjoint. We also make use of a potential, i.e. a smooth function φ : X o → [0, ∞) which vanishes on L and is locally constant in a neighborhood of the deleted boundary ∂X. Using these extra data, we construct a Fredholm operator P E,F ρ,φ on X o whose index coincides with the index of D E,F LD . A vanishing theorem for the operator P E,F ρ,φ allows us to give conditions on scal g and dist(K, ∂V ) in such a way that the index of D E,F L D must vanish. Our method can be regarded as an extension to a certain class of incomplete manifolds of the technique of Gromov and Lawson [GL80,GL83].
Theorem A. Let (X, g) be a compact n-dimensional Riemannian spin manifold with boundary. Let f : X → S n be a smooth area decreasing map. If n is odd, we make the further assumption that f is constant in a neighborhood of ∂X. Suppose that the scalar curvature of g is bounded from below by a constant σ > 0. Moreover, suppose that (1.6) scal g ≥ n(n − 1) on supp(df )
Remark 1.7. Theorem A answers Question 1.1 when X is spin and even-dimensional.
Remark 1.8. Condition (1.7) implies that f is constant in a neighborhood of each connected component of ∂X so that the degree of f is well defined. The extra assumption when n is odd is needed, at least with the argument used in this paper, to reduce the odd-dimensional case to the even-dimensional case. We believe it is possible to drop this extra assumption.
Remark 1.9. It is an interesting question whether, in dimension at most eight, it is possible to drop the spin assumption from Theorem A by using the minimal hypersurface technique of Schoen and Yau [SY79]. In fact, it is not clear whether this method can be used to approach the long neck problem, due to the difficulties, pointed out in [CS19]. arising when the minimal hypersurface technique is used to treat maps that are area contracting.
We now consider a higher version of the long neck principle. Let Y be a closed n-dimensional spin manifold with fundamental group Γ. There is a canonical flat bundle L Y over Y , called the Mishchenko bundle of Y , whose typical fiber is C * Γ, the maximal real group C * -algebra of Γ. The Rosenberg index [Ros83, Ros86a,Ros86b] of Y is the class α(Y ) ∈ KO n (C * Γ), obtained as the index of the spin Dirac operator twisted with the bundle L Y . Here, KO n (C * Γ) is the real K-theory of C * Γ. The class α(Y ) is the most general known obstruction to the existence of metrics of positive scalar curvature on Y . Denote by / D Y,C * Γ the spin Dirac operator twisted with the bundle C * Γ, the trivial bundle on Y with typical fiber C * Γ. We assume that We use Condition (1.8) to establish a "higher neck principle". Let D 1 , . . . , D N be pairwise disjoint disks embedded in Y . Consider the compact manifold with boundary If g is a Riemannian metric on X, the normal focal radius of ∂X, denoted by rad ⊙ g (∂X), is defined as follows. For R > 0 small enough, denote by B R (S n−1 j ) the geodesic collar neighborhood of S n−1 j of width R. Define rad ⊙ g (∂X) as the supremum of the numbers R > 0 such that there exist pairwise disjoint geodesic collar neighborhoods B R (S n−1 1 ), . . . , B R (S n−1 N ). Theorem B. Let Y , Γ and X be as above. Suppose the Rosenberg index α(Y ) does not coincide with the index of / D Y,C * Γ . Moreover, suppose g is a Riemannian metric on X whose scalar curvature is bounded from below by a constant σ > 0. Then (1.9) rad ⊙ g (∂X) ≤ π n − 1 nσ . 1.3. Codimension one obstructions. Let us now consider an n-dimensional Riemannian band (V, g) over a closed spin manifold N . Let ∂ ± V and width(V ) denote the same objects as in Subsection 1.1. In this case, our obstruction is the Rosenberg index of the (n − 1)-dimensional spin manifold N . In analogy with the case of codimension zero obstructions, we consider the incomplete manifold V o = V \ ∂V and fix a rescaling function ρ and a potential ψ. We also assume that ψ is compatible with the band V . This means that there exist constants λ − < 0 < λ + such that ψ = λ − in a neighborhood of the deleted negative boundary component ∂ − V and ψ = λ + in a neighborhood of the deleted positive boundary component ∂ + V . We use these extra data to construct a Fredholm operator B ρ,ψ on V o whose index coincides with α(N ). From a vanishing theorem for the index of the operator B ρ,ψ , we deduce the following result. Theorem D. Let N be a closed (n−1)-dimensional spin manifold with fundamental group Γ. Suppose the Rosenberg index α(N ) ∈ KO n (C * Γ) does not vanish. Let V be a Riemannian band over N whose scalar curvature is bounded from below by a constant σ > 0. Then The paper is organized as follows. In Section 2, we prove a K-theoretic additivity formula for the index in the setting of manifolds complete for a differential operator. In Section 3, we study rescaled Dirac operators and prove a Lichnerowicz-type inequality in this situation. In Section 4, we construct the operator P E,F ρ,φ and prove a formula to compute its index. In Section 5, we prove a vanishing theorem for the operator P E,F ρ,φ and use it to prove Theorem A, Theorem B, and Theorem C. Finally, in Section 6 we construct the operator B ρ,ψ and use it to prove Theorem D.
Acknowledgment. I am very thankful to Thomas Schick for many enlightening discussions and suggestions.

A K-theoretic additivity formula for the index
This section is devoted to the analytical background of this paper. In Subsection 2.1, we recall some preliminary notions on differential operators acting on bundles of modules over C * -algebras and fix notation. In Subsection 2.2, we consider a differential operator P on a not necessarily complete Riemannian manifold M . In order to ensure that P has self-adjoint and regular closure, we make use of the notion of completeness of M for P , developed by Higson and Roe [HR00] and extended to the C * -algebra setting by Ebert [Ebe16]. When P 2 is uniformly positive at infinity, by results of Ebert [Ebe16] the closure of P is Fredholm and its index is well defined. In Subsection 2.3, we extend to this slightly more general class of operators a K-theoretic additivity formula due to Bunke [Bun95].
2.1. Differential operators linear over C * -algebras. Throughout this paper, A denotes a complex unital C * -algebra. We will also consider the case when A is endowed with a Real structure. We are mostly interested in the following two types of Real C * -algebras. The first one is the Real Clifford algebra Cl n,m : see [Sch93, Section 1.2] and [Ebe16, page 4] for details. The second one is the maximal group C * -algebra C * Γ associated to a countable discrete group Γ. This is the completion of the group algebra C[Γ] with respect to the maximal norm and is endowed with a canonical Real structure induced by complex conjugation: see [Ebe16, Section 1.1] and [HR00, Definition 3.7.4].
For Hilbert A-modules H and H ′ , we denote by L A (H, H ′ ) the space of adjointable operators from H to H ′ and by K A (H, H ′ ) the subspace of the compact ones. We also use the notation L A (H) := L A (H, H) and K A (H) := K A (H, H). For the properties of Hilbert A-modules and adjointable operators, we refer to [Lan95] and [WO93,Section 15].
Let (M, g) be a Riemannian manifold. Let W be a bundle of finitely generated projective Hilbert A-modules with inner product on M and let P : Γ(M ; W ) → Γ(M ; W ) be a formally self-adjoint differential operator of order one. If W is Z 2graded, we require that the operator P is odd with respect to the grading. If A has a Real structure, we require that W is a bundle of finitely generated projective Real Hilbert A-modules and the operator P is real, i.e. P κ(w) = κ(P w) for all w ∈ Γ(M ; W ), where κ is the involution defining the Real structure. For more details, we refer to [Ebe16, Sections 1.1 and 1.2]. We are mostly interested in the two types of operators described in the following examples. For the background material on Dirac operators twisted with bundles of Hilbert A-modules, we refer to [Sch05, Section 6.3]. We finally recall a particular instance of this construction, which is relevant for the geometric applications of this paper. Let M be a closed n-dimensional spin manifold with fundamental group Γ. Let L Γ be the Mishchenko bundle over M . The bundle L Γ has typical fiber C * Γ and is equipped with a canonical flat connection. The class index / D M,L Γ ∈ KO n (C * Γ) is called the Rosenberg index of M and is denoted by α(M ). For more details, see [Ros07] and [Sto02].
Remark 2.3. To be precise, index / D M,E is a class in KO n (A R ), where A R is the real C * -algebra consisting of the fixed points of the involution of A. With a slight abuse of notation, we denote a Real C * -algebra and its fixed point algebra by the same symbol.
Remark 2.4. The fixed point algebra of C * Γ with respect to the canonical involution is the maximal real C * -algebra of Γ, which in this paper will be denoted by the same symbol.
2.2. Manifolds which are complete for a differential operator. Let (M, g) be a Riemannian manifold. Let W → M be a bundle of finitely generated projective Hilbert A-modules with inner product and let P : Γ(M ; W ) → Γ(M ; W ) be a formally self-adjoint differential operator of order one. We regard P as a symmetric unbounded operator on L 2 (M ; W ) with initial domain Γ c (M ; W ). We will now give a condition so that its closureP : dom(P) → L 2 (M ; W ) is self-adjoint and regular. For the background material on unbounded operators on Hilbert A-modules and the notion of regularity, see [Lan95].
Definition 2.5. A coercive function is a proper smooth function h : M → R which is bounded from below. Definition 2.6. We say that the pair (M, P) is complete, or that M is complete for P, if there exists a coercive function h : M → R such that the commutator [P, h] is bounded.
Remark 2.7. The notion of completeness of a manifold for an operator depends only on the principal symbol of the operator. This means that if (M, P) is complete and Φ : W → W is a fiberwise self-adjoint bundle map, then (M, P +Φ) is also complete.
The next theorem, due to Ebert, gives the wanted sufficient condition. It is a generalization to operators linear over C * -algebras of a result of Higson and Roe [HR00, Proposition 10.2.10].
Assume (M, P) is complete and denote the self-adjoint and regular closure of P by the same symbol. Assume also there is a Z 2 -grading W = W + ⊕ W − and the operator P is odd with respect to this grading, i.e. it is of the form are formally adjoint to one another. Finally, assume P is elliptic.
To simplify the notation, in the remaining part of this section we set H := L 2 (M ; W ). We say that the operator P 2 is uniformly positive at infinity if there exist a compact subset K ⊂ M and a constant c > 0 such that In this case, by [Ebe16, Theorem 2.41] the operator P P 2 +1 −1/2 ∈ L A (H) odd is Fredholm. We denote its index by index (P).
In the next lemma, we collect some properties of the operator P that will be needed in the proof of the additivity formula.
Lemma 2.9. The operator P 2 +1 + t 2 is invertible for every t ≥ 0. Moreover, P 2 +1+t 2 −1 is a positive element of L A (H) and there is the absolutely convergent integral representation Finally, we have the estimates Proof. The first part of the lemma and Inequality (2.4) follow from [Ebe16, Proposition 1.21]. Inequalities (2.5) and (2.6) follow from Part (2) of [Ebe16, Theorem 1.19].

Cut-and-paste invariance.
For i = 1, 2, let M i be a Riemannian manifold, i be a Z 2 -graded bundle of finitely generated projective Hilbert A-modules with inner product and let P i be an odd formally self-adjoint elliptic differential operator of order one. We assume that (M i , P i ) is complete and that P 2 i is uniformly positive at infinity so that its index is well defined.
We make the following assumption.
Assumption 2.10. The operators coincide near the separating hypersurfaces. This means that there exist tubular neighborhoods U(N 1 ) and U(N 2 ) respectively of N 1 and N 2 and an isometry Γ : U(N 1 ) → U(N 2 ) such that Γ| N1 : N 1 → N 2 is a diffeomorphism and Γ is covered by a bundle isometry This assumption allows us to do the following cut-and-paste construction. Cut the manifolds M i and the bundles W i along N i . Use the map Γ to interchange the boundary components and construct the Riemannian manifolds Moreover, using the map Γ to glue the bundles, we obtain Z 2 -graded bundles and odd formally self-adjoint elliptic differential operators of order one P 3 and P 4 . Observe that the pairs (M 3 , P 3 ) and (M 4 , P 4 ) are complete and the operators P 2 3 and P 2 4 are uniformly positive at infinity. Therefore, the indices of P 3 and P 4 are well defined. The next theorem is a slight generalization of [Bun95, Theorem 1.2].

Proof. Use the notation
In order to prove the thesis, we need to show that index (F ) = 0.
Pick cutoff functions χ Ui and χ Vi such that where z ∈ L A (H) is the Z 2 -grading. As explained in [CB18, Subsection 3.1] and in the proof of [Bun95, Theorem 1.14], in order to show that index (F ) = 0, it suffices to show that X F + F X ∈ K A (H). To this end, it is enough to verify the compactness of operators of the form Using Assumption 2.10, the operators χ P 3 − P 1 χ and (χ P 1 − P 1 χ) ρ define the same element in L A (H 3 , H 1 ), that we denote by [P, χ]. Using the integral representation (2.3) and the computations in [Bun95, page 13], we obtain . Using Inequalities (2.4), (2.6) and (2.5) and [Ebe16, Theorem 2.33 and Remark 2.35], we deduce that the operator Q 3,1 (t) is compact and absolutely integrable. By (2.7), a * F 3 − F 1 a * ∈ K A (H 3 , H 1 ), which concludes the proof.

A rescaled Dirac operator
In this section, we present a general method to construct a complete pair on a Riemannian spin manifold. Our method is based on rescaling the possibly twisted spin Dirac operator. Moreover, we prove an estimate from below for the square of the rescaled twisted Dirac operator. Finally, in order to obtain a slight improvement of this estimate, we extend to operators linear over C * -algebras an inequality due to Friedrich [Fri80, Thm.A] on closed manifolds and generalized by Bär [Bär09, Theorem 3.1] to open manifolds. This improvement will be used in Sections 5 and 6 to obtain the factor (n − 1)/n in Theorems A, B, and D. Even if we mostly focus on the spin case, all the results of this section hold with the obvious modifications for any operator of Dirac type.
3.1. Admissible rescaling functions. Let (M, g) be a Riemannian manifold. Let V → M be a bundle of finitely generated projective Hilbert A-modules with inner product and let Z : Γ(M ; V ) → Γ(M ; V ) be a formally self-adjoint elliptic differential operator of order one such that Observe that Z ρ is a formally self-adjoint differential operator of order one and Therefore, Z ρ is elliptic. Remark 3.2. The property for a smooth function ρ of being an admissible rescaling function depends only on its behavior at infinity. Moreover, suppose ρ 1 , ρ 2 : M → (0, 1] are smooth functions such that ρ 1 is admissible and ρ 2 = bρ 1 outside of a compact set for some constant b > 0. Then ρ 2 is admissible as well. Proof. Since ρ is admissible, choose a coercive function h such that ρ 2 |dh| is in L ∞ (M ). By (3.1) and (3.3), we deduce Remark 3.4. When (M, g) is a complete Riemannian manifold, the function ρ = 1 is admissible and Proposition 3.3 implies the classical fact that a Dirac operator on (M, g) is essentially self-adjoint.
We now describe a method for constructing admissible rescaling functions on open Riemannian manifolds. In Sections 5 and 6, we will use this method together with the geometry at infinity of the manifolds to construct complete pairs.
) is a coercive function. By (3.5) and since γ α (t) = t α for t near 0, there exists a compact subset K ⊂ M such that Since τ 2α−1 ∈ L ∞ (M ) for 2α ≥ 1, the previous inequality and Remark 3.2 imply the thesis. (2) A is a Real C * -algebra, S M is the Cl n,0 -linear spinor bundle / S M , E, ∇ E is a bundle of finitely generated projective Real Hilbert A-modules with inner product and metric connection and Z is the twisted Cl n,0 -linear Dirac operator / D M,E described in Example 2.2. When there is no danger of confusion, we will denote the bundle S M simply by S. The operator Z is related to the scalar curvature of g through the classical Lichnerowicz formula

A Friedrich inequality for operators linear over
where ∇ * ∇ is the connection Laplacian of S ⊗E and R E : S ⊗E → S ⊗E is a bundle map depending linearly on the components of the curvature tensor F ∇ E of ∇ E .
In particular, if F ∇ E = 0 in a region Ω ⊂ M , then R E = 0 on Ω. See [LM89, §II.8] for more details. The next theorem provides a slight improvement of the estimate from below of Z 2 directly following from (3.6).
Theorem 3.6. Let (M, g), E, ∇ E and Z be as above. Suppose the scalar curvature of g is bounded from below by a constant σ. Set Then the inequality In order to prove Theorem 3.6, we first establish the following abstract inequality for Hilbert C * -modules.
where the last inequality is obtained by applying Inequality (3.9) to the terms Proof of Theorem 3.6. Let u ∈ Γ c (M ; S ⊗ E). Recall that the operator Z has the local expression Clifford multiplication by e i , and ∇ is the connection on S ⊗ E induced by the connections on S and E. At a point x ∈ M , using Lemma 3.7 we obtain By integrating the previous inequality, we get Since scal g ≥ σ, using the last inequality together with the Lichnerowicz formula (3.6), we deduce from which Inequality (3.8) follows.
3.3. A Lichnerowicz-type inequality for the rescaled operator. Let (M, g), S, E and Z be as in Subsection 3.2. For a smooth function ρ, let Z ρ be the rescaled operator defined by (3.2). In the next proposition, we state a Lichnerowicz-type inequality for the rescaled operator.  .7). Then the inequality The proof of this proposition is based on the following lemma.
Lemma 3.9. Let ξ : M → R be a smooth function. Then the inequality holds for every ω > 0 and every v ∈ Γ c (M ; S ⊗ E).
Proof of Proposition 3.8. It follows from Theorem 3.6 and Lemma 3.9, with ξ = ρ and v = ρu.

Generalized Gromov-Lawson operators
In this section, we study the geometric situation when M is a Riemannian spin manifold and (E, F ) is a pair of bundles with isomorphic typical fibers and whose supports are contained in the interior of a compact submanifold with boundary L ⊂ M . In Subsection 4.1, we define the class rel-ind(M ; E, F ) as the index of a suitable elliptic differential operator D E,F L D over the double L D of L. In Subsection 4.2, we use an admissible rescaling function ρ and a potential φ to define a Fredholm operator P When there is no danger of confusion, we will use the notation S and S ± instead of S M and S ± M . Let F, ∇ F be a second bundle of finitely generated projective Hilbert Amodules with inner product and metric connection over M . We make the following assumption.
Assumption 4.1. The bundles have isomorphic typical fibers and are trivializable at infinity. This means that there exist a finitely generated projective Hilbert Amodule V and a compact subset K ⊂ M such that where V → M denotes the trivial bundle with fiber V and d V denotes the trivial connection on V. In this case, we say that K is an essential support for (E, F ) and that M \ K is a neighborhood of infinity.
In this setting, we define a relative index following Gromov and Lawson [GL83]. Let L ⊂ M be a smooth compact submanifold with boundary, whose interior contains an essential support of (E, F ). Deform the metric and the spinor bundle in such a way that they have a product structure in a tubular neighborhood of ∂L.   Proof. Observe first that it suffices to prove the thesis when one of the submanifolds is contained in the interior of the other. To see this, consider a compact submanifold with boundary L ′′ ⊂ M whose interior contains both L and L ′ .
Using this observation, we will prove the theorem under the assumption that L is contained in the interior of L ′ . Consider the Riemannian spin manifolds (L D , g 1 ) and (L ′ D , g 2 ), where g 1 and g 2 are induced by g as explained above. Consider the operators D which concludes the proof. where L ⊂ M is a submanifold with boundary whose interior contains an essential support of (E, F ). This class will be used as a localized obstruction for the metric g to have positive scalar curvature under some extra geometric conditions. To this end, we will need information on the endomorphisms R E and R F that appear in the Lichnerowicz formula (3.6). We conculde this subsection presenting two examples where we can determine whether the class rel-ind(M ; E, F ) vanishes and we have control on the lower bound of the endomorphisms R E and R F . These two examples will be used in the geometric applications of Section 5 and Section 6.
and that the operators Q and R are odd with respect to these gradings. Fix an admissible rescaling function ρ for M and consider the rescaled operators Q ρ and R ρ defined by (3.2). Recall from Subsection 3.1 that Q ρ and R ρ are first order formally self-adjoint elliptic differential operators. Finally, observe that the operators Q ρ and R ρ are odd with respect to the grading (4.6), i.e. they are of the form formally adjoint respectively to Q + ρ , R + ρ . In order to construct a Fredholm operator out of the operators Q ρ and R ρ , we make use of a potential.
Definition 4.7. We say that a smooth function φ : M → [0, ∞) is a compatible potential if φ = 0 in a neighborhood of an essential support of (E, F ) and φ is constant and nonzero in a neighborhood of infinity.
Fix a compatible potential φ. By Assumption 4.1, φ defines bundle maps Define the operator P + ρ,φ : Γ(W + ) → Γ(W − ) through the formula Denote by P − ρ,φ its formal adjoint and consider the graded bundle W := W + ⊕ W − . The generalized Gromov-Lawson operator associated to our data is the operator P E,F ρ,φ : Γ(W ) → Γ(W ) defined as By construction, P E,F ρ,φ is an odd formally self-adjoint elliptic differential operator of order one. When there is no danger of confusion, we will denote P E,F ρ,φ simply by P ρ,φ . Theorem 4.8. For every admissible rescaling function ρ and every compatible potential φ, the pair (M, P ρ,φ ) is complete and the operator P 2 ρ,φ is uniformly positive at infinity.
The proof of this theorem is based on the following two lemmas. Lemma 4.9. Let U , V be Hilbert A-modules and let T : U → V be an adjointable operator such that T * T = b 2 id U , for some constant b > 0. Then for every η ∈ U and θ ∈ V we have where (· | ·) U and (· | ·) V are the A-valued inner products respectively of U and V .
Proof of Theorem 4.8. The completeness of the pair (M, P ρ,φ ) follows from Proposition 3.3 and Remark 2.7. Moreover, since ρ ≤ 1, from Lemma 4.10 we deduce Since φ is a compatible potential, the previous inequality implies that P 2 ρ,φ is uniformly positive at infinity.
From Theorem 4.8 and the results of Subsection 2.2, the class index (P ρ,φ ) is well defined, for every admissible rescaling function ρ and every compatible potential φ. In the case (I) from Subsection 4.1, index (P ρ,φ ) ∈ Z. In the case (II), index (P ρ,φ ) ∈ KO n (A).  In order to prove this theorem, we first establish some stability properties of the index of P ρ,φ .
We will now establish an additivity formula for generalized Gromov-Lawson operators, from which we will deduce Theorem 4.11. Let us consider the following situation. Let (E, F ) and (G, H) be two pairs over (M, g) satisfying Assumption 4.1. Let K ⊂ M be an essential support of both (E, F ) and (G, H) and let L ⊂ M be a compact submanifold with boundary whose interior contains K. Let φ be a compatible potential vanishing in a neighborhood of L and let ρ be an admissible rescaling function such that ρ = 1 in a neighborhood of L. Denote respectively by P E,F ρ,φ and P G,H ρ,φ the generalized Gromov-Lawson operators associated to these data. Lemma 4.13. In the above situation, we have Proof. Let S L D and D L D be the Z 2 -graded spinor bundle and odd Dirac operator associated to the spin manifold L D . Consider the Z 2 -graded bunlde where T − is the formal adjoint of T + . Observe that P G,E H,F is an odd formally self-adjoint elliptic differential operator of order one. By construction, Using We now relate the operator P G,E H,F to the operator P E,F ρ,φ on M . Consider the partitions M = L ∪ ∂L (M \ L) and L D = L ∪ ∂L L − . Modify the Riemannian metrics and the Clifford structures on M and L D in a tubular neighborhood of ∂L in such a way that Assumption 2.10 is satisfied. Using the cut-and-paste construction described in Subsection 2.3, we obtain the operator P E,F ρ,φ on M and the operator P G,E H,F on L D . From Identities (4.14), (4.15) and Theorem 2.11, we deduce which implies Identity (4.13).
Proof of Theorem 4.11. Let ρ be an admissible rescaling function, let φ be a compatible potential, and let L ⊂ M be a compact submanifold with boundary whose interior contains an essential support of (E, F ). Using Part (a) and Part (b) of Lemma 4.12, we assume that, in a neighborhood of L, we have φ = 0 and ρ = 1. Let V, d V be as in Assumption 4.1. By Part (c) of Lemma 4.12 and Identity (4.2), the indices of P which concludes the proof.

The long neck problem
This section is devoted to proving a long neck principle in the spin setting. Suppose (X, g) is a compact Riemannian spin manifold with boundary and let (E, F ) be a pair of bundles with isomorphic typical fibers and whose supports are contained in the interior of X. We will give conditions on the lower bound of scal g and the distance between the supports of E, F and ∂X so that rel-ind(X o ; E, F ) must vanish, where X o is the interior of X. To this end, we will use the distance function from ∂X to construct a generalized Gromov-Lawson operator P E,F ρ,φ on X o . In Subsection 5.1, we prove a vanishing theorem for the operator P E,F ρ,φ , from which we deduce an abstract long neck principle. As applications, we prove Theorem A and Theorem B respectively in Subsections 5.2 and 5.3. Finally, in Subsection 5.4 we use a generalized Gromov-Lawson operator on a complete manifold to prove Theorem C.

A vanishing theorem on compact manifolds with boundary.
We consider the following setup. Let (X, g) be a compact n-dimensional Riemannian spin manifold with boundary ∂X. By removing the boundary, we obtain the open manifold X o := X \ ∂X. The metric g induces an incomplete metric on X o , that we denote by the same symbol. Let (E, ∇ E ) and (F, ∇ F ) be bundles of finitely generated projective Hilbert A-modules with inner product and metric connection over (X o , g). Suppose Assumption 4.1 is satisfied.
Definition 5.1. For an essential support K of (E, F ), a K-bounding function is a smooth function ν : We say that ν is a bounding function if it is a K-bounding function for some essential support K of (E, F ).
The next theorem states an abstract "long neck principle" for compact Riemannian spin manifolds with boundary.
Theorem 5.2. Suppose the scalar curvature of g is bounded from below by a constant σ > 0. Let K be an essential support of (E, F ) and let ν be a K-bounding function. Suppose that Then rel-ind(X o ; E, F ) = 0.
In order to prove this theorem, we will make use of the index theory developed in Section 4. For an admissible rescaling function ρ and a compatible potential φ on X o , let P ρ,φ : Γ c (X o ; W ) → Γ c (X o ; W ) be the associated generalized Gromov-Lawson operator. For the definition of the bundle W and the operator P ρ,φ and their properties, see Subsection 4.2. We start with proving an estimate for the operator P 2 ρ,φ in this setting. As in Section 3, we use the notationn = n/(n − 1) andσ = σ/4. Lemma 5.3. Let ρ be an admissible rescaling function, let φ be a compatible potential and let ν a bounding function. Suppose the scalar curvature of g is bounded from below by a constant σ > 0. Then, for every w ∈ Γ c (M ; W ) and every ω > 0, we have P 2 ρ,φ w, w ≥ Φ ω,ν ρ,φ w, w , where Φ ω,ν ρ,φ : M → R is the smooth function defined by the formula Proof. It follows from Lemma 4.10 and Proposition 3.8.
Corollary 5.4. Suppose the scalar curvature of g is bounded from below by a constant σ > 0. Moreover, suppose there exist an admissible rescaling function ρ, a compatible potential φ, a bounding function ν, and positive constants ω and c such that Then the class rel-ind(X o ; E, F ) vanishes.
Proof. From Lemma 5.3, Condition (5.5) implies that the operator P 2 ρ,φ is invertible. Now the thesis follows from Theorem 4.11.
Lemma 5.5. Suppose Λ > π/ √n σ. Then there exist positive constants ω, η and a smooth function Y : for t varying in a neighborhood of [η, Λ]. Moreover, η can be chosen arbitrary small.
Lemma 5.6. Suppose Λ > π/ √n σ. Then there exist a constant ω > 0 and smooth there exists a constant c > 0 such that the functions Z and Y satisfy the differential inequalitȳ for all t > 0.
Let U 0 be an open neighborhood of (0, ηΛ] such that Y = Y 0 on U 0 . Let U 1 be an open neighborhood of [ηΛ, ∞) such that Z = ηΛ on U 1 . We will prove Property (e) by analyzing separately these two open sets. Let us begin with U 0 . On this set, Y = Y 0 . Moreover, by Properties (ii) and (iii) we have Therefore, using Property (iv) we obtain (5.12)nσ for every t ∈ U 0 . Let us now analyze U 1 . On this set, Z = ηΛ. Therefore, using Property (3) of Lemma 5.5, we deduce that there exists a constant c > 0 such that for every t ∈ U 1 . Since U 0 ∪ U 1 = (0, ∞), (5.12) and (5.13) imply that Y and Z satisfy Property (e). This concludes the proof.
Proof of Theorem 5.2. By corollary 5.4, it suffices to show that, when Conditions (5.2) and (5.3) are satisfied, there exist an admissible rescaling function ρ, a compatible potential φ and positive constants ω and c such that where Φ ω,ν ρ,φ is the function defined by (5.4). Using Condition (5.3), pick a constant Λ satisfying Observe that these conditions imply that τ δ is positive and goes to 0 at infinity. Choose a constant Λ 1 such that Λ < Λ 1 < dist(K, ∂X). Consider the open sets Observe that X o = Ω 0 ∪ Ω 1 and that, by taking δ small enough, K ⊂ Ω 0 . Define functions ρ : X o → (0, 1] and φ : X o → [0, ∞) by setting By Proposition 3.5, Remark 3.2 and Property (a) of Lemma 5.6, ρ is an admissible rescaling function. Moreover, by Property (b) of Lemma 5.6, we have (i) ρ = ρ 0 on Ω 0 , for a constant ρ 0 > 0. By Properties (c) and (d) of Lemma 5.6, φ is a compatible potential and satisfies (ii) φ = 0 on Ω 0 . Since K ⊂ Ω 0 , using (i), (ii) and Condition (5.2) we deduce for x ∈ Ω 0 . Therefore, Inequality (5.14) holds on Ω 0 . In order to complete the proof, we will show that, for δ small enough, Inequality (5.14) holds on Ω 1 as well. Using (5.16), we have Since ν is a K-bounding function, ν = 0 on Ω 1 . By taking δ small enough and using Property (e) of Lemma 5.6, we deduce that there exists a constant c 1 > 0 such that for every x ∈ Ω 1 . This concludes the proof.

Proof of Theorem
for some constant θ 1 ∈ (θ, 1). Observe that K is an essential support of (E, F ). Let ν : X o → [0, θn(n − 1)/4] be a smooth function such that ν = θn(n − 1)/4 on supp(df ) and ν = 0 on X o \ K. Then ν is a K-bounding function and The last inequality, (1.7) and Theorem 5.2 imply that rel-ind(X o ; E, F ) = 0. The thesis now follows from (5.20). Suppose now that n is odd and the map f is constant in a neighborhood of ∂X. As in the proof of [LM89, Proposition 6.10], we fix a 1-contracting map µ : S n ×S 1 → S n+1 of degree one, which is constant on { * } × S n ∪ S 1 × { * ′ }. Here, * and * ′ are distinguished points respectively in S n and S 1 with f (∂X) ⊂ { * }. For R > 0, let S 1 R be the circle of radius R and consider the manifold X 1 := X × S 1 R equipped with the product metric, denoted by g 1 . Let f 1 : X 1 → S n+1 be the map given by the composition µ • (f × 1/R). Then f 1 is area decreasing for R large enough. Moreover, scal g1 = scal g ≥ σ and dist(supp(df 1 ), ∂X 1 ) = dist(supp(df ), ∂X). Now the thesis follows from the even-dimensional case.

Proof of Theorem B.
Let g be a Riemannian metric on X whose scalar curvature is bounded from below by a constant σ > 0. Consider the incomplete Riemannian manifold X o . Let (E, ∇ E ) and (F, ∇ F ) be the flat bundles on X o with typical fiber C * Γ constructed in Example 4.6. Recall that, since Y satisfies Condition (1.8), the class rel-ind(X o ; E, F ) ∈ KO n (C * Γ) does not vanish. Moreover, for 0 < R < rad ⊙ g (∂X), the geodesic collar neighborhoods B R (S n−1 1 ), . . . , B R (S n−1 N ) are pairwise disjoint and the closure of the set is an essential support of (E, F ) that we denote by K R . Since dist(K R , ∂X) = R and rel-ind(X o ; E, F ) = 0, by Theorem 5.2 we deduce from which Inequality (1.9) follows. In order to obtain a contradiction, we will construct a generalized Gromov-Lawson operator on M .
Since the metric g is complete, the function ρ = 1 is admissible. Let K be an essential support of (E, F ) and let φ : M → [0, ∞) be a smooth function such that φ = 0 in a neighborhood of K and φ = 1 in a neighborhood of infinity. Observe that, for λ > 0, λφ is a compatible potential. Denote by P λ the generalized Gromov-Lawson operator P 1,λφ . Let Φ λ : M → R be the smooth function defined by the formula Since the connections ∇ E and ∇ F are flat, from Lemma 4.10 and the Lichnerowicz formula (3.6) we deduce In order to prove the thesis, we will show that the function Φ λ is uniformly positive for λ small enough. In fact, in this case (5.24) and Theorem 4.11 imply rel-ind(M ; E, F ) = 0, contradicting (5.22).

Estimates on band widths
This last section is devoted to the proof of Theorem D. In Subsection 6.1, using rescaling functions and potentials in a similar fashion as in Subsection 4, we extend the theory of Callias-type operators to Riemannian manifolds which are not necessarily complete. More precisely, we develop a rescaled version of the real Callias-type operators used by Zeidler in [Zei19]. In Subsection 6.2, we focus on compact Riemannian spin bands and prove a Callias-type index theorem, stating that the index of a Callias-type operator on a compact Riemannian spin band coincides with the index of an elliptic differential operator on a separating hypersurface. Finally, in Subsection 6.3 we prove a vanishing theorem yielding Theorem D. Fix an admissible rescaling function ρ and a Callias potential ψ. Let Z ρ be the operator Z rescaled with the function ρ defined by Formula (3.2). The generalized Callias-type operator associated to these data is the first order elliptic differential operator B ρ,ψ : Γ(M ; / S M⊗ E⊗ Cl 0,1 ) → Γ(M ; / S M⊗ E⊗ Cl 0,1 ) defined as (6.1) B ρ,ψ := Z ρ⊗ 1 + ψ⊗ǫ, where ǫ denotes left-multiplication by the Clifford generator of Cl 0,1 .
Remark 6.2. When the metric g is complete, ρ = 1 is admissible. If the Callias potential ψ is a proper map with bounded gradient, B 1,ψ coincides with the operator used in [Zei19].
We now show that the operator B ρ,ψ has a well-defined index.
Theorem 6.3. For every admissible rescaling function ρ and every Callias potential ψ, the pair (M, B ρ,ψ ) is complete and the operator B 2 ρ,ψ is uniformly positive at infinity.
Proof. The completeness of the pair (M, B ρ,ψ ) follows from Proposition 3.3 and Remark 2.7. Moreover, we have (6.2) B 2 ρ,ψ = Z 2 ρ⊗ 1 + ρ 2 c(dψ)⊗ǫ + ψ 2 . Since ρ ≤ 1, from the previous identity we deduce Since ψ is a Callias potential, the last inequality implies that B 2 ρ,ψ is uniformly positive at infinity. From Theorem 6.3 and the results of Subsection 2.2, the class index (B ρ,ψ ) in KO n (A) is well defined, for every admissible rescaling function ρ and every Callias potential ψ.
Remark 6.4. When (M, g) is complete, ρ = 1 and the Callias potential ψ is a proper function with uniformly bounded gradient, the index of B 1,ψ coincides with the class index PM / D M,E , ψ used in [Zei19].
We conclude this subsection with some stability properties of the index of B ρ,φ .
Proposition 6.5. Suppose ρ is an admissible rescaling function and ψ is a Callias potential. Then Proof. For Parts (a) and (b), it suffices to consider the linear homotopies ρ t := tρ ′ + (1 − t)ρ and ψ t = tψ + (1 − t)ψ, with 0 ≤ t ≤ 1, and argue as in the proof of Lemma 4.12. Let us prove Part (c). Since M is compact, the constant function ψ ′ = 1 is a Callias potential. By Identity (6.2), the operator B 2 ρ,1 is uniformly positive and the index of B ρ,1 vanishes. Using Part (b) and the compactness of M again, we deduce that the index of B ρ,ψ vanishes as well.
6.2. Callias-type operators on Riemannian spin bands. We start with recalling the notion of band due to Gromov [Gro18, Section 2]. A band is a manifold V with two distinguished subsets ∂ ± V of the boundary ∂V . It is called proper if each ∂ ± V is a union of connected components of the boundary and ∂V = ∂ − V ⊔ ∂ + V . If V is a Riemannian manifold, we define the width of V as width(V ) := dist(∂ − V, ∂ + V ).
In order to define a generalized Callias-type operator in this setting, we proceed as in Subsection 5.1 and consider the open manifold V o := V \ ∂V . The metric g induces an incomplete metric on V o , that we denote by the same symbol. Given a collar neighborhood U + ⊂ V of ∂ + V , we say that U o + := U + \ ∂ + V ⊂ V o is a neighborhood of the positive boundary at infinity of V o . In a similar way, define the notion of a neighborhood of the negative boundary at infinity of V o . Definition 6.6. A Callias potential ψ on (V o , g) is called band compatible if there exist constants λ − and λ + , with λ − < 0 < λ + , such that the image of ψ is contained in [λ − , λ + ], ψ = λ − in a neighborhood of the negative boundary at infinity of V o and ψ = λ + in a neighborhood of the positive boundary at infinity of V o .
Let E, ∇ E be a bundle of finitely generated projective Real Hilbert A-modules with inner product and metric connection. We now study the properties of the generalized Callias-type operator B ρ,ψ , where ρ is an admissible rescaling function and ψ is a band compatible Callias potential. Proof. Consider the linear homotopy ψ t = tψ 1 + (1 − t)ψ 2 , with 0 ≤ t ≤ 1. Since ψ 1 and ψ 2 are band compatible, ψ t is a band compatible Callias potential for all t ∈ [0, 1]. The thesis follows by arguing as in the proof of Lemma 4.12.
Theorem 6.8. Let ρ be an admissible rescaling function and let ψ : V o → [λ − , λ + ] be a band compatible Callias potential. If a ∈ (λ − , λ + ) is a regular value of ψ, then Proof. Let U ∼ = ∂V × [0, 1) be a collar neighborhood of ∂V such that ψ is constant on U \ ∂V . Using Part (a) of Proposition 6.5, assume there exists a constant ρ 0 ∈ (0, 1] such that ρ = ρ 0 on the complement of U ′ ∼ = ∂V × [0, 1/4). Observe that the manifold N ∼ = ∂V × {1/2} is a closed separating hypersurface of V o . Moreover, Here, Y is a compact manifold with ∂Y = N + ⊔ N − and W = W − ⊔ W + , where W − is a neighborhood of the negative boundary at infinity with ∂W − ∼ = N − and W + is a neighborhood of the positive boundary at infinity with ∂W + ∼ = N + . Let us assume (after deformation near N ) that our data respect the product structure of a tubular neighborhood of N where the function ψ is constant. Consider the half-cylinders Observe that the bundles E| N± extend to bundles E Z± with metric connections on Z ± . Consider the manifolds where W − − and W − + denote respectively the manifolds W − and W + with opposite orientations. Let g 2 be the Riemannian metric coinciding with g on W − ± and being a product on the half cylinders Z ± . Let E 2 be the bundle with metric connection coinciding with E on W − − ⊔ W − + and with E Z± respectively on Z ± . Let ρ + 2 : M + 2 → (0, 1] be the smooth function coinciding with ρ on W − + and with ρ 0 on Z + . Observe that ρ + 2 is admissible, since the metric g 2 is complete on the cylindrical end Z + . Finally, let µ + be a positive constant such that λ 2 + > µ + . Then there exist a smooth function ψ + 2 : (M 2 ) + → [λ + , ∞) such that (i) ψ + 2 = λ + on a neighborhood of W − + ; (ii) ψ + 2 (t, x) ≥ µ + t for all (t, x) ∈ [0, ∞) × N + ⊂ Z + ; (iii) dψ + 2 ≤ µ + . By Properties (i) and (ii), ψ + 2 is a Callias potential. Let B ρ + 2 ,ψ + 2 be the associated generalized Callias-type operator. Since ρ + 2 ≤ 1 and ψ + 2 2 ≥ (λ + ) 2 > µ + , Properties (i)-(iii) imply that the function ψ + 2 2 − dψ + 2 is uniformly positive on (M 2 ) + . By Inequality (6.3), we deduce that the index class of B ρ + 2 ,ψ + 2 vanishes. Finally, observe that ψ + 2 is proper with uniformly bounded gradient on the cylindrical end Z + .
In a similar way, construct an admissible rescaling function ρ − 2 and a Callias potential ψ − 2 : (M 2 ) − → (−∞, λ − ] such that the index of the associated generalized Callias-type operator B ρ − 2 ,ψ − 2 vanishes and the function ψ − 2 is proper with uniformly bounded gradient on the cylindrical end Z − . Finally, observe that ρ ± 2 induce an admissible rescaling function ρ 2 on M 2 and ψ ± 2 induce a Callias potential ψ 2 on M 2 . Since M 2 = M − 2 ⊔ M + 2 , the index of the associated operator B ρ2,ψ2 vanishes. Observe that the manifolds V o and M 2 satisfy Assumption 2.10. Using the cutand-paste construction described in Subsection 2.3, we obtain Riemannian manifolds M 3 := Z − ∪ N− Y ∪ N+ Z + and M 4 := Y ∪ N Y − and generalized Callias-type operators B ρ3,ψ3 and B ρ4,ψ4 respectively on M 3 and M 4 . Since M 4 is a closed manifold, the index of B ρ4,ψ4 vanishes by Part (c) of Proposition 6.5.
Let us now analyze the operator B ρ3,ψ3 . By construction, ψ 3 is a proper smooth function whose gradient is uniformly bounded. Moreover, since a ∈ (λ − , λ + ), which concludes the proof.
We finally specialize to the case when (V, g) is a Riemannian band over a closed spin manifold N , i.e. V is diffeomorphic to N × [−1, 1]. Let L V , ∇ LV be the Mishchenko bundle of V endowed with the canonical flat connection. For an admissible rescaling function ρ and a band compatible Callias potential ψ, denote by B LV ρ,ψ the generalized Callias-type operator associated to these data. Corollary 6.9. Let N be a closed (n − 1)-dimensional spin manifold with fundamental group Γ. Let (V, g) be a Riemannian spin band over N . For an admissible rescaling function ρ and a band compatible Callias potential ψ, we have (6.4) index B LV ρ,ψ = α(N ) ∈ KO n−1 (C * Γ).
Since the bundle L V is flat, from Proposition 3.8 and Identity (6.2) we deduce (6.7) B LV ρ,φ for every w ∈ Γ c (V o ; / S M⊗ L V⊗ Cl 0,1 ) and every ω > 0. By corollary 6.9 and Inequality (6.7), it suffices to show that there exist an admissible rescaling function ρ, a band compatible Callias potential ψ and positive constants ω, c such that To this end, we will use Lemma 5.6 in a similar way as in the proof of Theorem 5.2. Using Condition (6.5), choose a constant Λ satisfying (6.9) 2π √n σ < 2Λ < width(V ).