Scalar and mean curvature comparison via the Dirac operator

We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary. In the situations we consider, we thereby give refined answers to questions on metric inequalities recently proposed by Gromov. This includes optimal estimates for Riemannian bands and for the long neck problem. In the case of bands over manifolds of non-vanishing $\hat{\mathrm{A}}$-genus, we establish a rigidity result stating that any band attaining the predicted upper bound is isometric to a particular warped product over some spin manifold admitting a parallel spinor. Furthermore, we establish scalar- and mean curvature extremality results for certain log-concave warped products. The latter includes annuli in all simply-connected space forms. On a technical level, our proofs are based on new spectral estimates for the Dirac operator augmented by a Lipschitz potential together with local boundary conditions.

Manifolds of positive scalar curvature have been a central topic in differential geometry and topology in recent decades.On complete spin manifolds, a particularly powerful tool in the study of positive scalar curvature metrics has been the spinor Dirac operator which facilitates a fruitful exchange between geometry and topology.This technique exploits the tension between, on the one hand, the Schrödinger-Lichnerowicz Simone Cecchini and Rudolf Zeidler formula = D 2 D r r C 1  4 scal; which implies invertibility of the spinor Dirac operator = D when the scalar curvature is uniformly positive and, on the other hand, index theory in the sense of Atiyah and Singer, which in various situations yields differential-topological obstructions to invertibility.Until recently, and with the notable exceptions of sharp Dirac eigenvalue estimates (see Friedrich [15] and Hijazi, Montiel and Roldán [27]), sharp K-area estimates (see Goette and Semmelmann [16] and Llarull [31]), and approaches to the positive mass theorem based on an idea of Witten [41], the strongest applications of the Dirac operator technique in positive scalar curvature geometry have been of a fundamentally qualitative nature.Indeed, there is a substantial body of celebrated literature addressing existence questions of positive scalar curvature metrics on a given manifold, or more generally, studying the topology of the space of positive scalar curvature metrics via the Dirac method; see Botvinnik, Ebert and Randal-Williams [7], Gromov and Lawson [23;21], Lichnerowicz [30] and Stolz [39] for a selection.However, Gromov [18; 19] recently directed the focus towards more quantitative questions and proposed studying the geometry of scalar curvature via various metric inequalities which have similarities to classical Riemannian comparison geometry.This resulted in a number of conjectures, a few of which we now recall.
Conjecture 1.1 [19, page 103, long neck problem] Let .M; g/ be a compact connected n-dimensional Riemannian manifold with boundary whose scalar curvature is bounded below by n.n 1/.Suppose that ˆW M !S n is a smooth area-nonincreasing map which is locally constant near the boundary.If dist g .supp.dˆ/; @M / n ; then the mapping degree of ˆis zero.
Conjecture 1.2 [18, 11.12, Conjecture D 0 ] Let X be a closed manifold of dimension n and such that X n fp 0 g with p 0 2 X does not admit a complete metric of positive scalar curvature.Let M be the manifold with boundary obtained from X by removing an open ball around p 0 .Then for any Riemannian metric of scalar curvature n.n 1/ > 0 on M, the width of a geodesic collar neighborhood of @M is bounded above by =n.
Conjecture 1.3 [18, 11.12, Conjecture C] Let M be a closed connected manifold of dimension n 1 ¤ 4 such that M does not admit a metric of positive scalar curvature.Let g be a Riemannian metric on V D M OE 1; 1 of scalar curvature bounded below by n.n 1/ D scal S n .Then width.V; g/ Ä 2 n ; where width.V; g/ WD dist g .@V; @ C V / is the distance between the two boundary components of V with respect to g.
Note that we have strengthened the bounds in Conjectures 1.1 and 1.2 compared to the original sources.All these constants are optimal, as we shall discuss below.
Gromov's first definitive result on these questions [18] was a proof of Conjecture 1.3 for the torus and related manifolds via the geometric measure theory approach to positive scalar curvature going back to the minimal hypersurface method of Schoen and Yau [38].In fact, it initially appeared that the Dirac operator technique was not suitable to such quantitative questions, in particular because they involve manifolds with boundary.However, in recent articles of the authors [43; 10; 42], we have demonstrated that the Dirac operator method can in principle be used to approach Conjectures 1.1-1.3.A slightly different Dirac operator approach based on quantitative K-theory leading to similar (nonsharp) estimates for bands was subsequently given by Guo, Xie and Yu [24].
In the present article, we advance the spinor Dirac operator method further and put forward a novel point of view towards Conjectures 1.1-1.3,which brings the mean curvature of the boundary into the focus of attention.That is, under the assumption that the scalar curvature is bounded below by n.n 1/ and that suitable index-theoretic invariants do not vanish, we establish a precise quantitative relationship between the mean curvature of the boundary and the relevant distance quantity appearing in situations related to Conjectures 1.1-1.3.More precisely, we show that in each case there exist constants c n .l/> 0, depending on a distance parameter l > 0 and the dimension of the manifold, such that if the mean curvature is bounded below by c n .l/,then the relevant distance is at most l.The crucial property of these constants is that c n .l/! 1 as l approaches the conjectured distance bound.In other words, as the relevant distance tends to the threshold, the mean curvature tends to 1 somewhere at the boundary.The geometric intuition behind this behavior is that the metric must collapse as the critical threshold is approached.Moreover, our new point of view allows us to establish rigidity results for certain extremal cases of the predicted quantitative relationship between scalar curvature, mean curvature and distance.
On a technical level, one ingredient is to augment the spinor Dirac operator -similarly to our previous approaches -by a potential defined in terms of a distance function.This procedure modifies the classical Schrödinger-Lichnerowicz formula in a way that allows us to relate distance estimates to spectral properties of the modified operator.However, the crucial new ingredient is that we study a tailor-made boundary value problem associated to the augmented Dirac operator.This enables us to use spinorial techniques to not only quantitatively control the scalar curvature using a differential expression of the potential, but also bring the mean curvature of the boundary into play.The main principle behind our new approach is that we can compare certain spin manifolds to model spaces which are suitable warped products, provided that one can produce a nontrivial solution of a boundary value problem associated to the augmented Dirac operator on the given manifold.Moreover, up to a constant, the potential directly corresponds to the mean curvature of the cross sections in the model warped product space.
We develop this approach in a general setting that allows us to treat the results related to Conjectures 1.1-1.3 as well as further novel results in an essentially unified way.To this end, we introduce a new abstract geometric structure which we call a relative Dirac bundle.This is a Dirac bundle S !M in the sense of Gromov and Lawson [23, Section 1] (see also Lawson and Michelsohn [28]) together with a suitable bundle involution 2 C1 .M n K; End.S //, which is defined outside a compact subset K M ı of the Simone Cecchini and Rudolf Zeidler interior of the manifold; see Section 2 for details.The use of this structure is twofold.Firstly, together with a suitable function W M !R, it allows us to define the potential term necessary for the precise quantitative estimates.This leads to the Callias 1 operator where D is the Dirac operator associated to the Dirac bundle S .Secondly, the involution can be used to define natural chiral boundary conditions, which are crucial for the development of a suitable index theory for relative Dirac bundles (see Section 3) and allow spectral estimates for B (see Section 4) relating the mean curvature with the value of the function along the boundary.These boundary conditions are related to the treatment of the cobordism theorem via a boundary value problem as in [6, Section 21; 4, Section 6.3].Our chirality also allows for an auxiliary choice of sign for each boundary component, reminiscent of the boundary conditions considered by Freed [14].This additional choice will be relevant in the proofs of our results related to Conjecture 1.3.
A further notable observation is that our construction has a vague formal similarity to -bubbles or generalized soap bubbles, which have recently led to substantial advances via the geometric measure theory approach to scalar curvature; see Gromov [19,Section 5], Chodosh and Li [12], Gromov [20], Lesourd, Unger and Yau [29] and Zhu [46; 45].Indeed, the latter can be viewed as an augmentation of the minimal hypersurface method by suitable potentials.
In Sections 1.1-1.3 of the introduction, we present a simplified overview of our main results.In the main body of the article, these are derived by working with a suitable relative Dirac bundle and choosing a potential that is appropriate for the situation at hand.

Length of the neck
Here we present our main geometric results related to Conjectures 1.1 and 1.2.We improve the upper bound of =n to an estimate depending on the mean curvature of the boundary.
In our first result we estimate the length of the neck of a Riemannian manifold with boundary.Recall that for a smooth map of Riemannian manifolds ˆW M !N, the area contraction constant at p 2 M is defined to be the norm of the induced map We say the map is area nonincreasing if the area contraction constant is Ä 1 at every point.If M is compact and N is closed, both connected and oriented, where n D dim M D dim N 2, then a smooth map ˆW M !N that is locally constant near the boundary @M has a well-defined mapping degree deg.ˆ/ 2 Z. 2 Moreover, given a Riemannian manifold .M; g/, we denote the mean curvature of its boundary @M by H g (or simply H if the metric is implicit); see also (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) in Section 2 for our sign and normalization conventions.Theorem 1.4 (see Section 5) Let .M; g/ be a compact connected Riemannian spin manifold with nonempty boundary, n D dim M 2 even, and let ˆW M !S n be a smooth area-nonincreasing map.Assume that scal g n.n 1/.Moreover, suppose there exists l 2 .0;=n/ such that H g tan 1 2 nl and dist g supp.dˆ/;@M l; compare Figure 1.Then deg.ˆ/D 0.
The statement of Theorem 1.4 is sharp; we discuss this in Section 5, Proposition 5.2.In this context, a subtle point hidden in the statement of the theorem is that we also rule out the equality situation dist g supp.dˆ/;@M D l under these scalar and mean curvature bounds if deg.ˆ/ ¤ 0. This is in contrast to the situations of the other conjectures, where the equality situations can be realized; compare Remark 1.13 below.Addressing this detail requires a considerably more precise analysis than in the earlier approach from [10, Theorem A].
Corollary 1.5 Let .M; g/ be a compact connected Riemannian spin manifold with boundary, let n D dim M 2 be even, and let ˆW M !S n be a smooth area-nonincreasing map.Assume that scal g n.n 1/.
This corollary is a direct consequence of Theorem 1.4 because tan 1 2 nl ! 1 as l !=n.Thus this refines the original approach to Conjecture 1.1 from [10, Theorem A].Proposition 5.2 also shows that the constant =n is optimal here.For our second result, we introduce the notion of y A-area, which is a generalization of the notion of K-area introduced in [17, Section 4].For a Hermitian bundle E, denote by R E the curvature of the connection on E. Definition 1.6 Let .X; g/ be a closed even-dimensional oriented Riemannian manifold.The y A-area of .X; g/ is the supremum of the numbers kR E k 1 1 ; ranging over all Hermitian bundles E with metric connections such that R X y A.X / ^ch.E/ ¤ 0, where y A.X / denotes the y A-form of X and ch.E/ denotes the Chern character form of E.

Simone Cecchini and Rudolf Zeidler
The y A-area of .X; g/ depends on the metric g.However, since X is compact, the notion of having infinite y A-area is independent of g.A closed spin manifold of infinite K-area also has infinite y A-area.An important class of examples consists of even-dimensional compactly enlargeable manifolds, eg the 2m-dimensional torus T 2m .For the notion of enlargeability and more examples, we refer to [22; 23].More generally, if X has infinite K-area and Y has nonvanishing y A-genus, then X Y has infinite y A-area.This includes, for example, the Cartesian product T 2m Y , where Y is the K3-surface.Now suppose X is a closed n-dimensional enlargeable spin manifold.By [10, Theorem C], X n fp 0 g with p 0 2 X does not admit a complete metric of positive scalar curvature.Moreover, the double of M WD X nB n , with B n an open n-ball embedded in X , is enlargeable as well.Therefore, the next theorem in particular refines the upper bound of Conjecture 1.2 in the case of even-dimensional enlargeable spin manifolds, using information from the mean curvature of the boundary.
Theorem 1.7 (see Section 6) Let .M; g/ be a compact connected n-dimensional Riemannian spin manifold with boundary such that the double of M has infinite y A-area.Suppose that scal g > 0 and that there exist positive constants Ä and l, with 0 < l < =.
p Än/, such that Then the boundary @M admits no open geodesic collar neighborhood N Â M of width strictly greater than l such that scal g Ä n.n 1/ on N .
The estimate in this theorem is also optimal; see Remark 1.13 below.
Remark 1.8 Similarly as before, one deduces from this that the case l =.n p Ä/ is ruled out independently of mean curvature restrictions.This also follows from the techniques in [10, Theorem B], and see the discussion in [11].
In particular, Theorem 1.7 implies that a manifold with boundary whose double has infinite y A-area cannot carry any metric of positive scalar curvature and mean convex boundary.This also follows from recent results of Bär and Hanke [5].

Estimates of bands
Here we exhibit our main results related to Conjecture 1.3.Similarly as above, we are able to improve the upper bound of 2 =n to a bound depending on the mean curvature of the boundary.We will formulate our result for manifolds which are not only cylinders.We say that a band is a compact manifold V together with a decomposition @V D @ V t @ C V , where @ ˙V are unions of components.This notion goes back to Gromov [18], where such manifolds are called compact proper bands.A map V !V 0 is a band map if it takes @ ˙V to @ ˙V 0 .The width width.V; g/ of a Riemannian band .V; g/ is the distance between @ V and @ C V with respect to g.
We now focus on a class of bands to which our results apply and which is simple to describe.An overtorical band [18, Section 2] is a band V together with a smooth band map V !T n 1 OE 1; 1 of nonzero mapping degree, where T n 1 denotes the torus of dimension n 1.More generally, we can also consider y A-overtorical bands [42], which are defined similarly but replacing the usual mapping degree by the y A-degree in the sense of [22,Definition 2.6].
Theorem 1.9 (cf Corollaries 7.7 and 7.8) Let n be odd and .V; g/ be an n-dimensional y A-overtorical spin band.Suppose that scal g n.n 1/.If the mean curvature of the boundary satisfies either H g tan 1  4 nl for some 0 < l < 2 =n, or H g j @ V 0 and H g j @ C V tan 1 2 nl for some 0 < l < =n, then width.V; g/ Ä l.
Again, as the expression tan 1  4 nl tends to 1 as l ! 2 =n, we obtain the strict version of the original estimate desired by Conjecture 1.3.This also follows from [42,Corollary 1.5].If, in addition, we assume that one of the boundary components is mean convex, then we can even obtain a strict bound of =n.
Corollary 1.10 (cf Corollary 7.9) Let n be odd and .V; g/ be an n-dimensional y A-overtorical spin band.Suppose that scal g n.n 1/.Then we always have width.V; g/ < 2 =n.Moreover, if @ V is mean convex, then width.V; g/ < =n.
Remark 1.11The statement of Theorem 1.9 exhibited here in the introduction is a special case of the more general Theorem 7.6 comparing the mean curvature to arbitrary values of the form tan 1  2 nt ˙ , where =n < t < t C < =n, to get a corresponding width bound t C t .
Remark 1.12 Our methods apply to a more general class of bands of infinite vertical y A-area which, in particular, includes bands diffeomorphic to M OE 1; 1 with M being a closed spin manifold of infinite y A-area in the sense of Definition 1.6.See Section 7 for details.
Remark 1. 13 The band estimates given in Theorem 1.9 are sharp.This follows from the warped product metric ' 2 g T n 1 C dx ˝dx on T n 1 OE l; l for any 0 < l < =n, where g T n 1 is the flat torus metric and '.t/ D cos 1 2 nt 2=n , as indicated by Gromov in [18, page 653].By rescaling a given arbitrary metric g M on any manifold M allowed in Conjecture 1.3, the warped product metric ' 2 g M C dx ˝dx in fact shows optimality of this estimate on any band diffeomorphic to M OE 1; 1. Incidentally, this construction also shows optimality of the estimate in Theorem 1.7 by forgetting the band structure and simply considering M D T n 1 OE l; l with this warped product metric.

Extremality and rigidity results
In view of the band width estimates, it is natural to investigate the extremal case.The class of y A-overtorical bands discussed above includes the special case of y A-bands, that is, bands such that y A.@ V / ¤ 0 (and thus, by bordism invariance also y A.@ C V / ¤ 0).In this special case, we prove the following rigidity theorem stating that the extremal case can only be achieved by the warped product construction discussed in Remark 1.13 over a Ricci flat manifold.

Simone Cecchini and Rudolf Zeidler
Theorem 1.14 (cf Corollary 9.2) Let .V; g/ be an n-dimensional band which is a spin manifold and satisfies y A.@ V / ¤ 0. Suppose that scal g n.n 1/.Let 0 < d < =n and assume furthermore that one of the following conditions holds: either width.V; g/ 2d and we have H g j @V tan 1 2 nd , or width.V; g/ d and we have H g j @ V 0 and H g j @ C V tan 1 2 nd .Then .V; g/ is isometric to a warped product .M I; ' 2 g M C dx ˝dx/, where I D OE d; d or I D OE0; d , '.t/ D cos .nt=2/2=n and g M is some Riemannian metric on M which carries a nontrivial parallel spinor.
In particular, g M is Ricci-flat.
Again, we also have a more general version of this theorem involving arbitrary mean curvature bounds of the form tan 1  2 nt ˙ ; see Theorem 9.1.The study of extremality questions about scalar curvature has a long history initiated by Gromov's K-area inequalities [17].Llarull [31] proved sharp inequalities using the Dirac operator, which imply that the round metric on the sphere is scalar-curvature extremal, meaning it cannot be enlarged without a decrease in the scalar curvature at some point.Llarull's technique and results were subsequently refined and generalized by Goette and Semmelmann [16] and remain of central importance in contemporary research on scalar curvature; see for instance [18,Section 10;19,Section 4;44].Lott [32] recently extended the technique of Llarull and Goette and Semmelmann to even-dimensional manifolds with boundary using the Dirac operator with (local) boundary conditions.The following results combine the technique of Goette and Semmelmann with our machinery to obtain a new kind of extremality result for a large class of warped product manifolds.
Let M be a manifold with boundary @M.We say that a Riemannian metric g M on M is scalar-mean extremal if every metric g on M which satisfies g g M , scal g scal g M and H g H g M already satisfies H g M D H g and scal g M D scal g .Moreover, we say that g M is scalar-mean rigid if any such metric must satisfy g D g M .Theorem 1.15 (cf Corollary 10.3) Let n be odd and .M; g M / be an .n1/-dimensional Riemannian spin manifold of nonvanishing Euler-characteristic whose Riemannian curvature operator is nonnegative.Let ' W OEt ; t C ! .0;1/ be a smooth strictly logarithmically concave function and consider the warped product metric g V D ' 2 g M C dx ˝dx on V WD M OEt ; t C .Then any metric g on V which satisfies (i) g g V , (ii) scal g scal g V , and g M C dx ˝dx for some metric z g M on M such that scal z g M D scal g M .In particular, g V is scalar-mean extremal.
If , in addition, the metric g M satisfies Ric g M > 0, then g V is scalar-mean rigid.
Note that strictly logarithmically concave means that log.'/ 00 < 0. Geometrically, this means that the mean curvature of M fxg in the warped product (with respect to the normal field @ x ) is strictly increasing.Remark 1. 16 We have a more general version of this theorem involving distance-nonincreasing maps of nonzero degree to these spaces; see Theorem 10.2.
A particularly interesting special case of manifolds which can be written as such log-concave warped products are annuli in spaces of constant curvature.Indeed, as a direct consequence of Theorem 1.15, we obtain the following result.
Corollary 1.17 (cf Corollary 10.5) Let n 3 be odd and .M Ä ; g Ä / the n-dimensional simply connected space form of constant sectional curvature Ä 2 R. Let 0 < t < t C < t 1 , where t 1 D C1 if Ä Ä 0, and Consider the annulus around some basepoint p 0 2 M Ä .Then the metric g Ä is scalar-mean rigid on A t ;t C .
Similar statements for log-concave warped products (and, in particular, punctured space forms) have been suggested by Gromov in [19,Section 5.4] based on considerations with -bubbles.Moreover, different rigidity statements for hyperbolic space have been studied in the context of the positive mass theorem; see for instance Andersson and Dahl [1], Chruściel and Herzlich [13], Min-Oo [33] and Sakovich and Sormani [37].
Remark 1.18These results also appear quite similar to Lott's main result in [32] (apart from the fact that they apply in complementary parities of the dimension).They are, however, of an essentially different geometric nature.On the one hand, [32] proves area extremality rather than merely (length) extremality as in our results.Our technique here relies on the existence of a suitable Lipschitz function and does not appear to be readily applicable to the study of area-nonincreasing maps.On the other hand, we do not require that the curvature operator of the metric on V itself is nonnegative or that the second fundamental form of the boundary is nonnegative.Indeed, as the examples of hyperbolic annuli show, our results include manifolds of negative sectional curvature.Similarly, suitable spherical annuli are examples which have negative second fundamental form at the boundary.One should note that on a manifold without boundary, a metric of negative scalar curvature can never be scalar extremal -simply rescaling using a constant > 1 provides a counterexample.However, in our example of an annulus in hyperbolic space, the outer boundary component has positive mean curvature, which thwarts such a rescaling argument.This shows that the presence of the outer boundary component is crucial for our result to hold in the negative curvature example.More generally, in all the examples covered by Corollary 1.17 it is the case that at least one quantity is positive among the scalar curvature in the interior and the mean curvatures on the two boundary components.
Simone Cecchini and Rudolf Zeidler

Higher index theory and future directions
In the present article we are only working with classical Dirac-type operators on finite-dimensional Hermitian vector bundles rather than using higher index theory, which would involve bundles with coefficients in infinite-dimensional C -algebras.This is in contrast to our previous work [43; 10; 42].The main rationale behind this change of perspective is that it allows a quicker and more accessible exposition of the novel geometric arguments we want to exhibit through this article.At the same time, this does not sacrifice too much of the possible generality of the statements, because many examples which are usually approached via higher index theory can already be dealt with via more classical notions like enlargeability or infinite K-area.It does, however, restrict the parity of the dimension in some of the results.
Note that our central structure of a relative Dirac bundle can be straightforwardly generalized to C -algebra coefficients.In this way it would be possible to reformulate all arguments from [43; 10; 42] in terms of this concept.This will be partly explained in [11].Since this notion is fairly general, we also expect that it can be applied in many other geometric contexts we have not considered thus far.However, extending the results from this article to higher index theory requires some new analytic work because the present state of the art in the literature on boundary value problems for Dirac-type operators, where we mostly rely on Bär and Ballmann [4; 3], does not cover the case of infinite-dimensional bundles.Since this aspect is orthogonal to the geometric arguments presented here, we will address it separately in future work.
Acknowledgements We thank Misha Gromov for useful comments on an earlier version of this manuscript and pointing us to the examples showing optimality of Theorem 1.4.We also would like to thank Bernd Ammann for a brief helpful correspondence and the referee for valuable suggestions improving the exposition.

Relative Dirac bundles
In this section, we set up the differential geometric preliminaries underlying the rest of this article.In particular, we introduce our new concept of a relative Dirac bundle.
We begin by fixing some notation.Let .M; g/ be a Riemannian manifold and let E !M be a Hermitian vector bundle.The space of smooth sections will be denoted by C 1 .M; E/ and the subspace of compactly supported smooth sections by C 1 c .M; E/.We denote fiberwise inner products by h ; i and fiberwise norms by j j.Next we recall the notion of a classic Dirac bundle in the sense of Gromov and Lawson.Definition 2.1 [23, Section 1], [28] A (Z=2-graded) Dirac bundle over M is a Hermitian vector bundle S !M with a metric connection r W C 1 .M; S / !C 1 .M; T M ˝S/ (endowed with a parallel and orthogonal Z=2-grading S D S C ˚S ) and a parallel bundle map cW T M !End.S/, called Clifford multiplication, such that c.!/ is antiselfadjoint (and odd), and c.!/ 2 D j!j 2 for all ! 2 T M.
For simplicity, we sometimes denote Clifford multiplication by !u WD c.!/u.Let the associated Dirac operator be D D P n iD1 c.e i /r e i W C 1 .M; S/ !C 1 .M; S/, where e 1 ; : : : ; e n is a local frame and e 1 ; : : : ; e n the dual coframe.If we have a Z=2-grading where D ˙W C 1 .M; S ˙/ !C 1 .M; S / are formally adjoint to one another.
We now turn to relative Dirac bundles, an augmentation of a Dirac bundle.
Definition 2.2 Let K M ı be compact subset in the interior.A relative Dirac bundle with support K is a Z=2-graded Dirac bundle S !M together with an odd, selfadjoint, parallel bundle involution and such that admits a smooth extension to a bundle map on an open neighborhood of M n K.
The final technical requirement in particular ensures the existence of a unique continuous extension of to the topological boundary of K. We do not consider the further extension to a neighborhood of M n K part of the data, we just require its existence.
It follows directly from the definition that the Dirac operator anticommutes with the involution where it is defined.
One main use of this structure is that it will allow us to associate local boundary conditions to any choice of a sign for each connected component of @M.Definition 2.3 (boundary chirality for relative Dirac bundles) For a relative Dirac bundle S !M and a locally constant function s W @M !f˙1g, we say that the endomorphism is the boundary chirality on S associated to the choice of signs s.Here, 2 C 1 .@M;TM j @M / is the inward-pointing unit normal field.
Note that is a selfadjoint even involution, which anticommutes with c. [ / but commutes with c.!/ for all ! 2 T .@M/.This defines local boundary conditions for sections u 2 C 1 .M; S/ by requiring that (2-2) .uj @M / D uj @M : In Section 3, we will observe that these boundary conditions are elliptic.The next lemma will be used to show that they are selfadjoint.

Simone Cecchini and Rudolf Zeidler
Lemma 2.4 Let S !M be a relative Dirac bundle and s W @M !f˙1g a choice of signs.Then L. / WD fu 2 S j @M j .u/D ug is a Lagrangian subbundle of Sj @M with respect to the fiberwise symplectic form .u; v/ 7 !hu; c. [ /vi.In other words, c. [ /L. / D L. / ? .
Proof Note that L. / is the image of the orthogonal bundle projection 1 2 .1C/.Since L. / ? is the image of the complementary projection 1  2 .1 / and anticommutes with c. [ /, we obtain c. [ /L. / D L. / ? .
In the following, we recall several standard formulas on Dirac bundles for later use.
Green's formula [40, Proposition 9.1] For the Dirac operator, and, for the connection, (2-4) where r r D P n iD1 r 2 e i ;e i is the connection laplacian and R D X i<j c.e i / c.e j / R r .ei ; e j / is a bundle endomorphism linearly depending on the curvature tensor R r of r.
Penrose operator and Friedrich inequality [8, Section 5.2] We have that for all u 2 C 1 .M; S /, where P W C 1 .M; S/ !C 1 .M; T M ˝S/ is the Penrose operator defined as In particular, we have the Friedrich inequality where equality holds if and only if Pu D 0, that is, u satisfies the twistor equation Boundary Dirac operator [4, Appendix 1], [27,Introduction] Let be the interior unit normal field.We turn S @ D S j @M into a Dirac bundle via the Clifford multiplication and connection, r @ D r C 1 2 c @ .r [/ for 2 T@M: Denote the corresponding Dirac operator by (2)(3)(4)(5)(6)(7)(8)(9) AW C 1 .@M;S @ / !C 1 .@M;S @ /; where A D This is the canonical boundary operator associated to the Dirac bundle S !M. Note that A is even with respect to the grading on S @ restricted from the grading on S. It satisfies and, if S !M is endowed with the structure of a relative Dirac bundle, then where is defined as in (2-1) with respect to any choice of signs.This implies that, if u 2 C 1 .M; S/ satisfies the boundary condition (2-2), then where H is the mean curvature of @M with respect to .To avoid any confusion about signs and normalization, let us be explicit about our convention for the mean curvature: Here r is the shape operator, II denotes the second fundamental form and we use a local orthonormal frame e 1 ; : : : ; e n 1 on @M.
Finally, in the remainder of this section, we discuss two concrete geometric examples of relative Dirac bundles, which are relevant for our main results.In analogy with Gromov and Lawson [23], we make the following assumption: (2-15) There exists a compact set K M ı and a parallel unitary bundle isomorphism tW Ej M nK !F j M nK which extends to a smooth bundle map on a neighborhood of M n K.
In this case, we say that .E; F / is a GL pair with support K. Note that condition (2-15) implies that S is a relative Dirac bundle with involution where y ˝stands for the graded tensor product of operators; compare [ where ˝Rr E˚F e i ;e j / is an even endomorphism of the bundle S which depends linearly on the curvature of the connection on E ˚F ; compare [28,Theorem 8.17].
Example 2.6 Let .V; g/ be an odd-dimensional Riemannian spin band and let = S V !V be the associated complex spinor bundle, endowed with the connection induced by the Levi-Civita connection.Let E !M be a Hermitian bundle equipped with a metric connection.Then S WD .= where c = S is the Clifford multiplication on = S V .Moreover, S turns into a relative Dirac bundle with involution globally defined on V (that is, the support is empty).The Dirac operator on S is given by is the spinor Dirac operator on .V; g/ twisted with the bundle E. As in the previous example, the curvature term from (2-5) is of the form where ˝Rr E e i ;e j /: 3 Index theory for relative Dirac bundles In this section, we introduce the Callias operators associated to relative Dirac bundles, and review the necessary analysis to develop an index theory for them.
We again start by briefly fixing the notation we are going to use.Let E !M be a Hermitian vector bundle over a smooth manifold M with compact boundary @M.The L 2 -inner product of two sections The corresponding L 2 -norm will be denoted by kuk WD .u;u/ 1=2 .The space of square-integrable sections, denoted by L 2 .M; E/, can be identified with the completion of C 1 c .M; E/ with respect to this norm.We denote the space of locally square-integrable sections by L 2 loc .M; E/.Similarly, the latter comes endowed with the family of seminorms k k K ranging over compact subsets K Â M, where We will also use Sobolev spaces to a limited extent; see for instance [40,Chapter 4] for a detailed introduction.The Sobolev space H 1 loc .M; E/ consists of all sections u 2 L 2 loc .M; E/ such that ru, a priori defined as a distributional section over the interior of M, is also represented by a locally square Simone Cecchini and Rudolf Zeidler integrable section.The space H 1 loc .M; E/ is topologized using the family of seminorms k k H 1 K defined by , where K ranges over all compact subsets of M. Since the boundary is assumed to be compact, the restriction C 1 c .M; E/ !C 1 .@M;Ej @M /, u 7 !uj @M , extends to a continuous linear operator W H 1 loc .M; E/ !H 1=2 .@M;Ej @M / by the trace theorem; see eg [40, Chapter 4, Proposition 4.5].Here H 1=2 .@M;Ej @M / denotes a fractional Sobolev space for instance in the sense of [40,Chapter 4,Section 3].An equivalent (and in our setup more relevant) description in the presence of a suitable Dirac operator on the boundary is given in [4,Section 4.1].
Next we turn to the main player, the Callias operator associated to a relative Dirac bundle and a suitable potential function.To this end, fix a complete Riemannian manifold M with compact boundary @M and S !M a relative Dirac bundle with support K.This means that K M ı is compact and the involution is defined on M n K; see Definition 2.2.
Let be an admissible potential.Then extends by zero to a continuous bundle map on all of M. The Callias operator associated to these data is the differential operator Since is Lipschitz, the commutator OED; extends to a bounded operator on L 2 .M; S /; compare for example [3,Lemma 3.1].In fact, by Rademacher's theorem [35], the differential d exists almost everywhere and is an L 1 -section of the cotangent bundle.In this view, the commutator is given by the formula OED; D c.d / as an element of L 1 .M; End.S //.Using that the involution anticommutes with the Dirac operator, a direct computation then yields To be very precise, this expression implicitly uses the requirement in Definition 2.2 that extends to a smooth section on a neighborhood of M n K.However, as d D 0 on the interior of K, we only need the values of the continuous extension of to M n K to specify c.d / , and this is only relevant if the topological boundary of K has positive Lebesgue measure.
Observe that the distance function x from any fixed subset of M is 1-Lipschitz, so that jdxj Ä 1 almost everywhere.In our applications, we will consider potentials of the form D Y ı x, where Y is a smooth function on R and x is a distance function from a geometrically relevant compact region of M.
For a choice of signs s W @M !f˙1g, we let B ;s denote the operator B on the domain C 1 ;s .M; S/ WD fu 2 C 1 c .M; S/ j .uj@M / D uj @M g; where WD s c. [ / as in (2-1).
By definition, S D S C ˚S is Z=2-graded and the differential operator B can be written as where B ˙ are differential operators C 1 .M; S ˙/ !L 2 .M; S /.Since is even with respect to the grading, B ;s also decomposes similarly into operators .M; S /: Note that -even ignoring regularity -B ;s does not preserve its domain.
In the following, we will discuss the necessary analysis and index theory for these operators.We will mostly rely on the general framework of elliptic boundary value problems for Dirac-type operators due to Bär and Ballmann [4; 3].However, as we will need to allow potentials which are a priori only Lipschitz for some of our applications, we will take a slight detour and first apply the results of Bär and Ballmann [4] only to the Dirac operator D and Callias operators B with smooth potentials, before deducing the desired statements for the general case.
We Proof This is a direct consequence of the fact that the difference B D D is in L 1 .M; End.S// and (fiberwise) selfadjoint, and thus extends to a selfadjoint bounded operator on L 2 .M; S/.

Simone Cecchini and Rudolf Zeidler
From now on, we will drop the notational distinction between B ;s and its closure x B ;s , and simply view B ;s as being defined on H Lemma 3.5 Let M be a complete Riemannian manifold with compact boundary @M and S !M be a relative Dirac bundle.Let 1 ; 2 W M !R be two admissible potentials and s W @M !f˙1g a choice of signs.Suppose that 1 and 2 agree outside a compact subset of M. Then ind.B 1 ;s / D ind.B 2 ;s /: The next theorem provides the main tool for computing ind.B ;s /.Theorem 3.6 Let M be a complete Riemannian manifold with @M D ∅ and S !M be a relative Dirac bundle of support K Â M. Let W M !R be an admissible potential.Let N M n K be a compact hypersurface with trivial normal bundle.Then we let M 0 be the manifold obtained from cutting M open along N so that @M 0 D N 0 t N 1 , where N 0 and N 1 are two disjoint copies of N. Pulling back all data via the quotient map M 0 !M induces a relative Dirac bundle S 0 !M 0 and an admissible potential 0 W M 0 !R, respectively.Let s W @M 0 !f˙1g be any choice of signs such that sj where B D D C denotes the Callias operator on M, and B 0 0 the corresponding operator on M 0 .
Proof Using Lemmas 3.3 and 3.5, it is again enough to prove the claim in the case when is smooth.We will prove this case as a consequence of the general splitting theorem due to Bär and Ballmann; see [4, Theorem 6.5; 3, Theorem 8.17].
First we note that, by the proof of Theorem 3.4 and the remarks preceding it, the boundary condition B C Â H 1=2 .@M 0; S 0C / defined by the chirality D s c. [ / restricted to S C is elliptic and its adjoint is the corresponding boundary condition B Â H 1=2 .@M 0; S 0 /.Thus we can also apply the theory of Bär and Ballmann [4] separately to B ˙ and B 0˙ 0 ;s .In this light, the theorem is a direct consequence of the general splitting theorem; see [4, Theorem 6.5; 3, Theorem 8.17].To see this, observe that 1 with respect to the decomposition H 1=2 .@M 0; S 0C / D H 1=2 .N; S C / ˚H1=2 .N; S C / coming from @M D N 0 t N 1 , where By construction, the interior normal field of @M 0 along N 0 is equal to the exterior normal field along N 1 and hence j N 0 D j N 1 .Thus .B C 0 / ?D B C 1 viewed as an L 2 -orthogonal complement in H 1=2 .N; S C /. Consequently, the hypotheses of [4, Theorem 6.5] are satisfied and we obtain ind.B C 0 ;s / D ind.B C /.This concludes the proof because, by definition, ind.B 0 0 ;s / D ind.B 0C 0 ;s / and ind.B / D ind.B C /.
Lemma 3.7 Let M be a complete Riemannian manifold with compact boundary @M and S !M be a relative Dirac bundle.Let W M !R be an admissible potential and s W @M !f˙1g a choice of signs such that (i) there exists C > 0 such that 2 jd j C on all of M, and Then B ;s is invertible.In particular, ind.B ;s / D 0.
Proof By Green's formula, for any u 2 C 1 ;s .M; S/, we have By continuity, the final estimate holds for all u 2 H 1 ;s .M; S/.Therefore, B ;s is invertible; compare [4, Corollary 5.9].Lemma 3.8 Let M be a compact Riemannian manifold with boundary @M and S !M be a relative Dirac bundle with empty support (that is, the involution is defined on all of M ).Let s W @M !f˙1g be a choice of signs.Then any Lipschitz function W M !R is an admissible potential and ind.B ;s / does not depend on the choice of the potential .Furthermore, if the sign s 2 f˙1g is constant on all of @M, then ind.B ;s / D 0 for any potential .
Proof Since M is compact and is defined on of all M, the condition of being an admissible potential is vacuous.Moreover, by Lemma 3.5, any two potentials yield the same index.Finally, if the choice of signs s is constant, then Lemma 3.7 implies vanishing of the index for the constant potential D s.Since the index does not depend on the potential, it must vanish for any choice of potential.
Let us now specialize to the case of Example 2.5.Let M be a compact even-dimensional Riemannian manifold with boundary.Let S be the Dirac bundle associated to a GL pair .E; F / over M. For an admissible potential , consider the Callias operator B .With the choice of sign s D 1, let us consider the index of B ;1 .In order to compute ind.B ;1 /, we make use of the following construction.Form the double dM WD M [ @M M of M, where M denotes the manifold M with opposite orientation.Observe that dM is a closed manifold equipped with a spin structure induced by the spin structure of M. Using condition (2-15), let V .E; F / !dM be a bundle on dM which outside a small collar neighborhood coincides with E over M and with F over M defined using the bundle isomorphism implicit in a GL pair.The relative index of .E; F / is the index of the spin Dirac operator on dM twisted with the bundle V .E; F /, that is, The computation of ind.B ;1 / is given by the next proposition.where the latter expression is described in the paragraph preceding this corollary.
Proof Let E 0 D V .E; F /.Moreover, extend the bundle F to a bundle with metric connection F 0 on dM such that F 0 j M D F .Consider the Z=2-graded Dirac bundle W 0 WD = S dM y ˝.E 0 ˚.F 0 / op / with associated Dirac operator D 0 .Observe that .Cut dM open along @M as in Theorem 3.6.Pulling back all data, we obtain the operators B E;F 0;1 on M and B F;F 0;1 on M .By Lemma 3.8, ind B F;F 0;1 D 0. By Lemma 3.5, ind.B E;F ;1 / D ind.B E;F 0;1 /.Therefore, the thesis follows using identity (3-5) and Theorem 3.6.
We now deal with Example 2.6 in a similar fashion.
Corollary 3.10 Consider the setup of Example 2.6 and choose the signs so that sj @ ˙V D ˙1.Then for any choice of potential , we have where = D @ V;Ej @ V denotes the corresponding twisted spinor Dirac operator on @ V , y A.@ V / is the y A-form of @ V and ch.E/ the Chern character form associated to E.
Proof First of all, the index does not depend on by Lemma 3.8 since V is compact.We can thus choose a function suitable for our purposes.Furthermore, let be the infinite band obtained from attaching infinite cylinders along the boundary parts @ ˙V .We extend the Riemannian metric on V to a complete metric on V 1 .Then the same construction as in Example 2.6 yields a relative Dirac bundle on V 1 extending the data on V .Now choose a smooth function However, Lemma 3.7 implies that ind.B V ˙;˙1;˙1 / D 0, and so ind.B V 1 ; 1 / D ind.B V; ;s /.Finally, it follows from [2, Corollary 1.9] that ind.B V 1 ; 1 / D ind.= D @ V;Ej @ V /; for a more general context compare also the partitioned manifold index theorem from [43, Appendix A].The last equality follows from the Atiyah-Singer index theorem [28,Theorem 13.10].

Spectral estimates
Our goal here is to establish spectral estimates for the Callias operator B from equation (3-1) associated to a relative Dirac bundle S !M and an admissible potential W M !R. The parts only concerning the Dirac operator are similar to estimates of imaginary eigenvalues in the context of the "MIT bag boundary conditions" due to Raulot [36].

Simone Cecchini and Rudolf Zeidler
We start with a lemma computing the L 2 -norm of the Dirac operator applied to a section in terms of the Bochner-Lichnerowicz-Weitzenböck curvature term and a boundary term.In the following, we make extensive use of the notation and formulas introduced in Section 2. Lemma 4.1 Let S !M be a relative Dirac bundle and s W @M !f˙1g a choice of signs.Then for every u 2 C 1 c .M; S/, the following identity holds: Proof Using the Green and Weitzenböck formulas, we obtain The identity now follows from equations (2-13) and (2)(3)(4)(5)(6).
We now combine this with another application of Green's formula to get a corresponding computation for the Callias operator.(4-1) Using Green's formula (2-3) on the second term implies Combining this with Lemma 4.1 yields the desired identity (4-1).
Finally, this leads to the main result of this section.
Theorem 4.3 Let S !M be a relative Dirac bundle over a compact manifold M, let W M !R be an admissible potential and s W @M !f˙1g be a choice of signs.Then for every u 2 H 1 ;s .M; S/, the following estimate holds: Moreover, equality throughout both estimates in (4-2) holds if and only if Proof We first assume that u 2 C 1 ;s .M; S/.Then c. [ / uj @M D s uj @M D suj @M and huj @M ; Auj @M i D 0I see equations (2-1) and (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12).Thus (4-1) simplifies to the equality (4-4) Now we observe that both sides of the identity (4-4) are continuous in u with respect to the topology of H  and in particular, r r =jr j u D n u: Simone Cecchini and Rudolf Zeidler In the final remark of this section, we prepare another technical observation in the context of the extremality case of Theorem 4.3, which we will use on multiple occasions.Here we assume that the potential has a special form relevant for our applications.
Remark 4.5 Suppose that there is a 1-Lipschitz function x W M !I for some compact interval I Â R and let ' W I !.0;1/ be a smooth strictly logarithmically concave function such that x/ is an admissible potential for the relative Dirac bundle S !M we consider -the geometric meaning of this relationship will become apparent later; compare Section 8 below.Now suppose that we are in the same situation as in Remark 4.4, that is, we have an element u 2 ker.B ;s / Â H We also note that since d D f 0 .x/dx and f 0 .x/> 0, the second part of (4-3) implies that fc.dx/u D ug contains the set of points in M n K where jdxj D 1.But this is not a priori satisfied everywhere, just under the hypotheses of this remark, and so we will verify the condition c.dx/u D u directly in the specific applications when needed.

A long neck principle with mean curvature
In this section, we establish our long neck principle for Riemannian spin manifolds with boundary, relating the length of the neck to the scalar curvature in the interior and the mean curvature of the boundary.
Based on the technical preparations from the previous sections, we now directly enter into the proof of Theorem 1.4, starting with the following lemma.
Lemma 5.1 Suppose that M is a compact spin manifold with boundary and ˆW M !S n is a smooth map that is locally constant near @M and of nonzero degree, where n D dim M 2 is even.Set l D dist g .supp.dˆ/; @M / > 0. Then there exists a GL pair .E; F / in the sense of Example 2.5 such that 4 n.n 1/ at each point p 2 M, where a.p/ is the area-contraction constant of ât p, (ii) .E; F / has support K WD fp 2 M j dist g .p;@M / lg Ã supp.dˆ/,(iii) indrel.M I E; F / ¤ 0.
Proof We fix a basepoint 2 S n .Since ˆis locally constant on M n K and @M has only finitely many components, there exist finitely many distinct points q 1 ; : : : ; q k 2 S n such that ˆ.M n K/ D fq 1 ; : : : ; q k g.
. By continuity, ˆ. i / D fq i g and hence the closures i , i D 1; : : : ; k, form a family of pairwise disjoint closed subsets of M. Thus there exist open neighborhoods V i i that are still pairwise disjoint together with smooth functions i W M !OE0; 1 such that i D 1 on i and i D 0 on M n V i .Next we choose geodesic curves i W OE0; 1 !S n such that i .0/D is the basepoint and i .1/D q i .From this we obtain a smooth map (see Figure 2) ˆon each i and, since ‰ by definition locally factors through smooth curves, the induced map Llarull's argument [31] shows the existence of a Hermitian vector bundle E 0 !S n such that for any smooth map ‚W X !S n with X any spin manifold, we have , where a ‚ is the areacontraction function of ‚.Moreover, if X is closed and ‚ has nonzero degree, then ind.= D X ;‚ E 0 / ¤ 0. We now define Hermitian bundles E D ˆ E 0 and F D ‰ E 0 on M. Then F is a flat bundle because ‰ induces the zero map on 2-vectors.Thus Llarull's estimate for E D ˆ E 0 shows that (i) holds.
To see that (ii) holds, we choose pairwise disjoint open balls U 1 ; : : : ; U k S n such that q i 2 U i together with a unitary trivialization Note that M n K Â ˆ 1 .fq 1 ; : : : ; q k g/ \ ‰ 1 .fq 1 ; : : : ; q k g/ Â N by construction and thus N is an open neighborhood of M n K.The bundle isomorphism is not parallel on all of N , but as ˆD ‰ is locally constant on M n K, it follows that tj M nK is parallel.Thus the condition (2-15) from Example 2.5 is satisfied, showing that .E; F / is a GL pair with support K. Proof of Theorem 1.4 Suppose, by contradiction, that dist g .supp.dˆ/; @M / l and deg.ˆ/ ¤ 0. We then pick a GL pair .E; F / satisfying the conditions (i)-(iii) from Lemma 5.1.Moreover, restating the hypotheses of the theorem, we have the curvature bounds (iv) scal g n.n 1/ on M, We then consider the relative Dirac bundle S !M constructed out of the GL pair .E; F / as in Example 2.5.Recall from (2-18) that the relevant curvature term here takes the form To construct a suitable admissible potential, we first define the 1-Lipschitz function x W M !OE0; l; x.p/ WD min.distg .K; p/; l/: Since dist g .K; @M / l, we have that xj @M D l.Let '; f W OE0; =n/ !R given by '.t/ D cos 1 2 nt Since x is 1-Lipschitz, it follows that (5-1) 2 jd j D f .x/ 2 f 0 .x/jdxjf .x/ 2 f 0 .x/D 1 4 n 2 almost everywhere on M.Moreover, by (v), (5-2) 2 n H g : Then is an admissible potential for the relative Dirac bundle S .Corollary 3.9 together with (iii) implies that the corresponding Callias operator subject to the sign s D 1 satisfies ind.B ;1 / D indrel.M I E; F / ¤ 0: In particular, there exists an element 0 ¤ u 2 ker.B ;1 /.To analyze u further, we let Then, by definition, U 0 Â U are open subsets in the interior of M and U Â K. Since ˆhas nonzero degree, the set U must be nonempty.Furthermore, as the map ˆis locally constant near the boundary and M is connected, the intermediate value theorem implies that U 0 is also nonempty.Then using our main spectral estimates (4-2) from Theorem 4.3, we obtain j .u/j 2 vol @M and continuing the estimate, using that D 0 on U and R E˚F D 0 on M n U , leads to 0 by (iv) and (5-1) where in the last step we used (iv) and (i) together with the fact that the area-contraction constant is at most 1 2 on U 0 by definition.We conclude that we are in the equality situation of Theorem 4.3.Furthermore, since the last two integrands are separately nonnegative, we also deduce u D 0 on U 0 and f 0 .x/hu;.1 c.dx/ /ui D 0 on M nU .Since f 0 .x/> 0 and jdxj Ä 1, the latter implies that c.dx/u D u almost everywhere on M n U .Hence it follows from (4-8) in Remark 4.5 above that the modified section w D ' 1=2 u has a constant norm.But since u vanishes on the nonempty open subset U 0 and ' > 0, this implies that u vanishes almost everywhere, a contradiction.

Simone Cecchini and Rudolf Zeidler
The precise analysis of the equality situation in the proof above is necessary to rule out the case dist g .dˆ;@M / D l.If we only wanted to establish the nonstrict estimate dist g .dˆ;@M / Ä l for deg.ˆ/ ¤ 0, then this could be proved in a simpler way along the same lines as in [10, Proof of Theorem A] by directly showing that the operator B ;1 must be invertible if dist g .dˆ;@M / > l.
We now show that Theorem 1.4 is in fact sharp.The proof of the following proposition is an almost verbatim adaption of a construction due to Gromov and Lawson [23,Proposition 6.7]; see also [28, Chapter IV, Proposition 6.10].
In particular, Theorem 1.4 is sharp.
Proof We again work with the example from Remark 1.13; that is, consider V WD T n 1 OE l; l endowed with the metric g V D ' 2 g T n 1 C dx ˝dx, where g T n 1 is the flat torus metric and '.t/ D cos 1 2 nt 2=n .
Then scal g V D n.n 1/ and H g V D tan 1 2 nl .Let S 1 be the circle of radius 1 and 2 S 1 a basepoint.We choose a smooth map W OE l; l !S 1 of degree one such that takes OE l; l n ."; "/ to the basepoint 2 S 1 .For any ı > 0 we can find a finite covering z T n 1 !T n 1 together with a ı-Lipschitz map hW z T n 1 !S n 1 of nonzero degree.Then we set M WD z T n 1 OE l; l endowed with the lifted metric g WD ' 2 g z T n 1 Cdx ˝dx.This still satisfies scal g D n.n 1/ and H g D tan 1 2 nl .Let ‚W S n 1 S 1 !S n be a smooth map of degree 1 which factors through the smash product S n 1 ^S1 .If ı is sufficiently small, then it follows that the composition is .ıC /-area-contracting for some constant C > 0 which only depends on " and the Lipschitz constants of and ‚; this can be seen using [23,Proposition 6.3].By having chosen ı sufficiently small, we can thus arrange that f is area-nonincreasing.Moreover, it follows that deg.ˆ/D deg.h/ ¤ 0. Finally, the support of dˆis contained in z T n 1 OE "; " by construction; hence dist g .supp.dˆ/; @M / l ", as claimed.
Note that unlike the extremal examples for the band and collar width estimates discussed in Remark 1.13, the bounds from Theorem 1.4 are only approximately realized.Indeed, it is not possible to do better because -unlike in our other results -Theorem 1.4 actually rules out the equality situation.

Relative K-and y A-area
In this section, we aim to prove Theorem 1.7 from the introduction.We begin with a version of K-area for manifolds with boundary.The classical case has been introduced by Gromov in [17, Section 4].A slightly less general variant of the following was recently studied by Bär and Hanke [5].
Definition 6.1 Let .M; g/ be a compact orientable Riemannian manifold with boundary.An admissible pair of bundles over M is a pair of Hermitian bundles E; F ! M endowed with metric connections such that there exists a unitary parallel bundle isomorphism Ej @M Š !F j @M along @M.We say that an admissible pair .E; F / has a nontrivial Chern number, if there exists a polynomial p.c 0 ; c 1 ; : : : / in the Chern forms such that (6-1) Z M p.c 0 .E/; c 1 .E/; : : : / p.c 0 .F /; c 1 .F /; : : : / ¤ 0: The relative K-area of .M; @M / is the supremum of the numbers where .E; F / ranges over all admissible pairs of bundles which have a nontrivial Chern number.Remark 6.2 Since the bundles with all structures are isomorphic along the boundary, the Chern-Weil forms associated to E and F agree if pulled back to the boundary, and so (6-1) is a well-defined cohomological expression.Remark 6.3 As in the classical case, the property of having infinite relative K-area does not depend on the Riemannian metric.
We now relate the notion of infinite relative K-area for a manifold with boundary M with the notion of infinite K-area for its double dM.For a pair of Hermitian bundles E, F ! M which are suitably identified in a neighborhood of @M, as in Section 3 we denote by V .E; F / the Hermitian bundle on dM obtained by gluing E and F over a neighborhood of @M.Proposition 6.4 Let M be a compact manifold with boundary.The following conditions are equivalent: (a) The double dM has infinite K-area in the classical sense.
Proof We fix a smooth Riemannian metric on dM and endow M with the restricted metric for the purposes of this argument.Then the statement would become immediate, if -in the definition of K-areawe only considered bundles which restricted to a fixed tubular neighborhood @M OE 1; 1 Š U dM are of product structure, that is, they are of the form pr 1 E @ for some bundle E @ ! @M together with the pullback connection.Indeed, a bundle E ! dM of this form is just a pair of two bundles on M which are identified and of product structure on the corresponding collar neighborhood U \ M of @M.

Simone Cecchini and Rudolf Zeidler
Conversely, suppose E; F ! M have product structure and are identified on the collar neighborhood U \ M. Then the Chern numbers of V .E; F / agree with the corresponding relative Chern numbers (6-1).
The general case can be reduced to this observation by using a smooth map W dM !M such that is smoothly homotopic to the identity map on dM through maps which are the identity on @M and preserve each half of dM, and such that j U agrees with the projection onto @M.In fact, can be constructed on a slightly larger tubular neighborhood by manipulating the radial coordinate and setting it to be the identity on the rest of dM.Then, for any bundle E ! dM, the pullback E is of the special form described in the previous paragraph.Since is homotopic to the identity, E and E have the same Chern numbers (and the same applies to the relative Chern number (6-1) when passing from a pair of bundles .E; F / to .j M E; j M F /).Moreover, is a smooth map on a compact manifold, it admits some fixed Lipschitz constant L > 0, and hence passing from E to E only changes the norm of the curvature by at most a factor of L 2 .Hence, for the purposes of detecting infinite K-area, it makes no difference to restrict to the class of (pairs of) bundles described in the previous paragraph, and the proposition follows.
The notion of K-area is quite appealing because it is a purely bundle-theoretic property of the manifold and does not rely on spin structures or index theory.However, to apply it to positive scalar curvature geometry via spin geometry, the nonvanishing property (6-1) has to be translated (by potentially changing the bundle) into a property which can be used in the Atiyah-Singer index theorem for the Dirac operator.This is possible by the classical algebraic argument using Adams operations given in 17, Section 5 3  8 ; see also Bär and Hanke [5,Lemma 7] for a situation closer to the present context.In the following, we introduce an explicit notion to capture the resulting property.Definition 6.5 Let .M; g/ be a compact Riemannian manifold with boundary.The relative y A-area of .M; @M / is the supremum of the numbers kR E˚F k 1 1 ; where .E; F / ranges over all admissible pairs of bundles such that Remark 6.6 The notion of y A-area for closed manifolds was discussed in Definition 1.6.An analogous argument as in Proposition 6.4 shows that .M; @M / has infinite relative y A-area if and only if the double dM has infinite y A-area.
In the next proposition, we clarify the relationship between the notions of infinite relative K-area and infinite relative y A-area.
Proposition 6.7 Let M be a manifold with boundary of infinite relative K-area and N a closed manifold such that y A.N / ¤ 0. Then M N has infinite relative y A-area.In particular, infinite relative K-area implies infinite relative y A-area.
Proof Let " > 0. We start with an admissible pair of bundles .E; F / on M such that kR E˚F k 1 < " and such that (6-1) is satisfied.Arguing as in the proof of Proposition 6.4, we may assume without loss of generality that on a collar neighborhood U of the boundary the bundles E; F are of product structure and that there exists a parallel unitary bundle isomorphism ˆW Ej U ! F j U .Consider the bundle V .E; F / on dM.It follows that Z dM p.c 0 .V .E; F //; c 1 .V .E; F //; : : : / ¤ 0: Now using the fact that y A.dM / D 0 (as the double is nullbordant), the argument given in 17, Section 5 We can also relate this property to enlargeability.For an n-dimensional Riemannian manifold N with boundary, we say that .N; @N / is compactly area-enlargeable if for any " > 0, there exists a finite covering x N !N and an "-area-contracting map x M !S n of nonzero degree which is locally constant outside a compact subset of the interior.Proposition 6.8 If N is an even-dimensional compact manifold with .N; @N / compactly area-enlargeable and there exists a smooth map ˆW .M; @M / !.N; @N / of nonzero y A-degree, then .M; @M / has infinite relative y A-area.
Proof By passing to doubles and Proposition 6.4 and Remark 6.6, it suffices to consider the case that M and N are closed manifolds.Since N is compactly area-enlargeable, for any " > 0, there exists a Hermitian bundle E 0 !N such that kR E 0 k 1 < "=L 2 , where L is a Lipschitz constant for ˆ, and such that ch.E 0 / as a cohomology class is concentrated and nontrivial in the degrees 0 and dim N ; see for instance [25, page 313].Let E D ˆ E 0 and F to be the trivial bundle of the same rank.Then it follows that kR E˚F k 1 < " and Z In the following lemma, we spell out a technical consequence of infinite y A-area that will be used in our applications.
Simone Cecchini and Rudolf Zeidler Lemma 6.9 Let .M; g/ be an even-dimensional compact spin manifold of infinite relative y A-area and U Š @M .1; 0 Â M be an open collar neighborhood.Then for any sufficiently small " 1 ; " 2 > 0, there exists a pair of Hermitian vector bundles .E; F / on M such that: (i) There exists a parallel unitary bundle isomorphism tW Ej U " 1 !F j U " 1 , where U " 1 corresponds to @M . 1 C " 1 ; 0 in the tubular neighborhood.
(iii) ind.B ;1 / ¤ 0, where B ;1 is the operator considered in Corollary 3.9 associated to the relative Dirac bundle constructed from .E; F / as in Example 2.5.
Proof Let .E; F / be an admissible pair of bundles given by infinite y A-area satisfying (6-2) and (ii).Again arguing as in the proof of Proposition 6.4, we may assume that E and F are of product structure on the collar neighborhood and that also (i) is satisfied.We again consider the bundle V .E; F / on dM.Using Corollary 3.9 and the Atiyah-Singer index theorem, we deduce where the final nonvanishing statement is due to .Thus (iii) is also satisfied, completing the proof.
After all these technical preparations, we are now ready to give a proof of Theorem 1.7 from the introduction.
Proof of Theorem 1.7 For all sufficiently small d > 0, denote by N d the open geodesic collar neighborhood of @M of width d.Suppose, by contradiction, that N l 0 exists for some l < l 0 < =.We now use the function Y r to construct an admissible potential.Let x W M !OE0; ƒ be the distance function from K ƒ .Consider the Lipschitz function WD Y r ı x (which is actually smooth in the complement of @K ƒ ).Then j K ƒ D 0 and j @M D Y r .ƒ/.By (i) and , is an admissible potential satisfying  j @M > where ‚ ;n is the L 1 -function defined by By (ii) and since is constant on K ƒ , in the interior K ı ƒ we have ‚ By (iii), scal g Än.n 1/ on N l 0 and since x is 1-Lipschitz, in M n K ı ƒ we have ‚ Therefore, there exists a constant c > 0 such that kB uk ckuk for all u 2 H 1 ;1 .M; S/.It follows that ind.B ;1 / D 0, contradicting (iv).

Estimates of bands
In this section, we prove our statements related to Conjecture 1.3, that is, estimates of Riemannian bands under lower bounds on scalar and mean curvature.We start with reviewing Gromov's notion of a Riemannian band [18] and other relevant concepts to formulate our results more conveniently.Definition 7.1 (i) A band is a compact manifold V together with a decomposition @V D @ V t @ C V , where @ ˙V are unions of components.
(ii) A map ˆW V !V 0 between bands is called a band map if ˆ.@ ˙V / Â @ ˙V 0 .
(iii) The width width.V; g/ of a Riemannian band .V; g/ is the distance between @ V to @ C V with respect to g.
(iv) A width function for a Riemannian band .V; g/ is a 1-Lipschitz function x W V !OEt ; t C for some real numbers t < t C such that @ ˙V Â x Note that this x is already smooth in a neighborhood of @V .Moreover, if t C t < w, we can find " > 0 such that .1 C "/ 1 w D t C t .We can then approximate x by a smooth function z x W V !OE0; w which agrees with x near @V and is .1 C "/-Lipschitz.Then .1 C "/ 1 z x W V !OE0; t C t is a smooth width function.Translating to the interval OEt ; t C yields the desired result.
The following notion is an adaption of the ideas from Section 6 to the situation of bands.

Definition 7.3 A band V is said to have infinite vertical y
A-area if for every " > 0, there exists a Hermitian vector bundle E !V such that kR E k 1 < " and such that we have (7-1) Z @ V y A.@ V / ^ch.Ej @ V / ¤ 0:

Example 7.4 A simple example of an infinite vertical y
A-area is an y A-band; that is, a band such that y A.@ V / ¤ 0. Another is a band of the form V D M OE 1; 1, where M has infinite y A-area.
Example 7.5 If N is a closed manifold that is compactly area-enlargeable and there exists a band map V !N OE0; 1 of nonzero y A-degree, then V has infinite vertical y A-area; compare Proposition 6.8.In particular, this includes the classes of overtorical bands introduced by Gromov in [18], and the generalization of y A-overtorical bands studied in [42].
Theorem 7.6 Let .V; g/ be a spin band of infinite vertical y A-area.Suppose that scal g n.n 1/ and let =n < t < t C < =n such that the mean curvature of @V satisfies Now for 0 < Ä 1, we set This is possible because the tangent function is increasing and t 0 d < t < t C < t 0 Cd by our choice of d .We now choose a Hermitian bundle E !V satisfying (7-1) and such that the corresponding Weitzenböck curvature endomorphism satisfies R E ı for some ı < 1 4 n 2 .1 2 0 /.Now we consider the Callias operator B ;s associated to the relative Dirac bundle from Example 2.6 (using the twisting bundle E) and with potential WD f 0 ı x and subject to the boundary conditions coming from the choice of signs s.@ ˙V / D ˙1.Now (7-1) and Corollary 3.10 imply that ind.B ;s / ¤ 0. On the other hand, we have 4 n 2 : Thus (4-2) together with scal g n.n 1/ implies that each u in the domain of B ;s satisfies Since the terms 1  2 n H g j @ ˙V ˙f 0 .˙d/at the boundary are nonnegative by (7-2) and (7-3), this implies that Z where C D 1 4 n 2 .1 2 0 / ı > 0 by the choice of ı > 0. This shows that the operator B ;s is invertible, a contradiction to ind.B ;s / ¤ 0. Corollary 7.7 Let .V; g/ be a spin band of infinite vertical y A-area.Suppose that scal g n.n 1/ and that the mean curvature of @V satisfies Proof This is a consequence of Theorem 7.6 by setting t ˙WD ˙1 2 l.
Corollary 7.8 Let .V; g/ be a spin band of infinite vertical y A-area.Suppose that scal g n.n 1/ and that the mean curvature of @V satisfies H g j @ V 0 and H g j @ C V tan 1 2 nl for some 0 < l < n : Proof This is also a consequence of Theorem 7.6 by setting t WD 0 and t C WD l.Since z g t 0 D g t 0 D g M this implies that g x D z g x D '.x/ 2 g M , as desired.
Then interior elliptic regularity for the operator D 2 1 4 n 2 implies that u is smooth in the interior of V .Since, by Lemma 8.4, u is nowhere vanishing and we have the equality c.dx/u D u, this implies that the covector field dx must also be smooth in the interior of V .Since we already know that x is smooth near the boundary, this just means that the function x is smooth everywhere and so we can indeed apply Theorem 8.3.
Corollary 9.2 Let .V; g/ be an n-dimensional band which is a spin manifold and satisfies y A.@ V / ¤ 0. Suppose that scal g n.n 1/.Let 0 < d < =n and assume furthermore that one of the following conditions holds: either width.V; g/ 2d and we have H g j @V tan 1 2 nd , or width.V; g/ d and we have H g j @ V 0 and H g j @ C V tan 1 2 nd .
Then .V; g/ is isometric to a warped product .M I; ' 10 Scalar-mean extremality and rigidity of warped products In this section, we prove our general extremality and rigidity results for logarithmically concave warped products.
As a preparation for the proof of the main theorem, we discuss a particularly relevant example of a twisting bundle E !V to be used in Theorem 8.3, namely the fiberwise spinor bundle on a warped product.
Remark 10.1 (twisting with the fiberwise spinor bundle) Consider a warped product band .V WD M OEt ; t C ; g WD ' 2 g M C dx ˝dx/; and let = S !V be the spinor bundle with respect to the metric g.Then we let E 0 WD = S be the same bundle endowed with the same bundle metric but with the "fiberwise spinor connection"

:
For each t 2 OEt ; t C , the twisted Dirac operator = D N ft g;E 0 j N ft g is the Euler characteristic operator of N and thus has nontrivial index because n 1 is even.Since the degree of ˆis nonzero, it follows that the index of = D @ V;Ej @ V is also nonzero.Hence, Corollary 3.10 shows that the Callias operator B ;s considered in Theorem 8.3 has nontrivial index and hence nontrivial kernel.Thus Theorem 8.3 applies and we obtain scal g D Än.n 1/ and .V; g/ is isometric to a warped product .M OEt ; t C ; ' 2 g M C dx ˝dx/; where M D x 1 .t0 / for some arbitrary but fixed t 0 2 OEt ; t C and some Riemannian metric g M on M. Using the warped product structure, we obtain Än.n 1/ D scal g D scal g M '.x/ 2 2.n 1/h 0 .x/n.n 1/h.x/ 2 : Together with (10-3), this completely determines the scalar curvature of g M , and we obtain scal g M D scal g N ı ˆjM .This proves the first part of the theorem.
To prove the second part, we first observe that due to x D y ı ˆ, under the isometry V Š M OEt ; t C , the map ˆis of the form .p; x/ 7 !.ˆx.p/; x/, where ˆt WD ˆjM ft g W M !N .Since kTˆk Ä 1 and Tˆ.0; @ x / D .@ˆx=@x;@ x /, it follows that @ˆx=@x D 0; that is, ˆt D ˆt0 for all t.Moreover, ˆt0 W M !N is 1-Lipschitz with respect to the metric '.t 0 / 2 g M and '.t 0 / 2 g N .Thus the same holds with respect to the metrics g M and g N .If we assume that Ric g N > 0, then the rigidity argument of Goette and Semmelmann [16, Section 1.2] (and note that our ˆt0 is length-nonincreasing), implies that ˆt0 is an isometry.Together with the warped product structure, all of this implies that ˆitself is an isometry.

1 :
The long neck problem.

Example 2 . 5
Let M be a compact even-dimensional Riemannian spin manifold with boundary and let = S M D = S C M ˚= S M be the associated Z=2-graded complex spinor bundle endowed with the Levi-Civita connection.Let E; F ! M be a pair of Hermitian bundles equipped with metric connections.Note that the bundle S WD = S M y ˝.E ˚F op / Simone Cecchini and Rudolf Zeidler is a Z=2-graded Dirac bundle, where the grading S D S C ˚S is given by S C WD .= S C M ˝E/ ˚. = S M ˝F / and S WD .= S C M ˝F / ˚. = S M ˝E/: C 1 c .M; = S ˝E/ and = D C F W C 1 .M; = S C ˝F / !C 1 .M; = S ˝F /, and = D E and = D F are formally adjoint to = D C E and = D C F , respectively.The Dirac operator on S is given by (2-16) D D Â 0 D D C 0 Ã W C 1 .M; S / !C 1 .M; S /; where D ˙W C 1 .M; S ˙/ !C 1 .M; S / are the operators defined as (2-17)

Corollary 3 . 9
Consider the setup of Example 2.5.Then for any choice of potential , we have ind.B ;1 / D indrel.M I E; F /;

(3- 5 )
ind D 0 D indrel.M I E; F / because the index of the Dirac operator on = S dM ˝F 0 vanishes.Observe also that D 0 D B E 0 ;F 0 0

Proposition 4 . 2
Let S !M be a relative Dirac bundle and W M !R be an admissible potential.Then for every u 2 C 1 c .M; S/, the following identity holds.

Remark 4 . 4
Suppose that u lies in the kernel of B and at the same time satisfies (4-3).Then Du D u and so (4-5) r u D n c. [ / u for all 2 TM: On each point where d ¤ 0, this implies r u D n c. [ / c Â d jd j Ã u;

Figure 2 :
Figure 2: Construction of the map ‰.

Finally
, let ‚W dM D M [ @M M !S n be the smooth map defined by ‚j M D ˆand ‚j M D ‰. Then deg.‚/D deg.ˆ/ ¤ 0 because ‰ has zero degree.By definition, we have ‚ E 0 D V .E; F / and thus indrel.M I E; F / D ind.= D dM;V .E;F / / D ind.= D dM;‚ E 0 / ¤ 0 by Llarull's argument.This shows that (iii) holds and concludes the proof of the lemma.
/ ^ch.F / ¤ 0:Here yA.M / denotes the y A-form of M and ch denotes the Chern character form.

ind.B ; 1 /
D indrel.M I E; F / D ind.= D dM;V .E;F / / D Z dM y A.dM / ^ch.V .E; F // ¤ 0; 2 g M C dx ˝dx/, where either I D OE d; d or I D OE0; d, and we have '.t/D cos 1 2 nt 2=n , while g M is some Riemannian metric on M which carries a nontrivial parallel spinor.In particular, g M is Ricci-flat.Proof This follows immediately from Theorem 9.1 by setting t ˙D ˙d in the first case, and t D 0 and t C D d in the second.

Issue 3 (
pages 1005-1499) 2024 1005 Homological mirror symmetry for hypertoric varieties, I: Conic equivariant sheaves MICHAEL MCBREEN and BEN WEBSTER 1065 Moduli spaces of Ricci positive metrics in dimension five MCFEELY JACKSON GOODMAN 1099 Riemannian manifolds with entire Grauert tube are rationally elliptic XIAOYANG CHEN 1113 On certain quantifications of Gromov's nonsqueezing theorem KEVIN SACKEL, ANTOINE SONG, UMUT VAROLGUNES and JONATHAN J ZHU 1153 Zariski dense surface groups in SL.2k C 1; Z/ D DARREN LONG and MORWEN B THISTLETHWAITE 1167 Scalar and mean curvature comparison via the Dirac operator SIMONE CECCHINI and RUDOLF ZEIDLER 1213 Symplectic capacities, unperturbed curves and convex toric domains DUSA MCDUFF and KYLER SIEGEL 1287 Quadric bundles and hyperbolic equivalence ALEXANDER KUZNETSOV 1341 Categorical wall-crossing formula for Donaldson-Thomas theory on the resolved conifold YUKINOBU TODA 1409 Nonnegative Ricci curvature, metric cones and virtual abelianness JIAYIN PAN 1437 The homology of the Temperley-Lieb algebras RACHAEL BOYD and RICHARD HEPWORTH 26, Appendix A].This means ..u C ˚u / ˝.e ˚f // D .uC ˚ u / ˝.t f ˚te/ for sections u ˙2 C 1 .M; = S Ṁ /, e 2 C 1 .M; E/ and f 2 C 1 .M; F /.The Dirac operator on S is described as follows.Let = D E and = D F be the operators obtained by twisting the complex spin Dirac operator on M respectively with the bundles E and F .Observe that = D E and = D F are odd operators, that is, they are of the form [3,eover,[3, Lemma 7.3]implies that the domain of the closure is given by the Sobolev space It also follows from ellipticity (also of the boundary condition) that the inclusion H 1 ;s .M; S/ ,! H 1 loc .M; S / is continuous.In particular, the trace operator is continuous on H 1 ;s .M; S /.The next two lemmas allow us to also describe the domain of the closure of B ;s , temporarily denoted by x B ;s , and show that compact perturbations of the potential do not alter any relevant properties.
1 ;s .M; S/, unless specified otherwise.Lemma 3.3 Let 1 and 2 be admissible potentials coinciding outside a compact set L. Then B 2 B 1 defines a compact operator H 1 ;s .M; S/ !L 2 .M; S/.In particular, viewed as maps H 1 ;s .M; S/ !L 2 .M; S/, the operator B 1 ;s is Fredholm if and only if B 2 ;s is.Proof Observe that, since 1 and 2 are Lipschitz and coincide in M n L, both Á WD 2 1 and its differential dÁ are essentially bounded and supported in the compact set L. Together with the estimate kD.B 2 B 1 /uk D kDÁ uk Ä kÁ Duk C k c.dÁ/uk Ä kÁk 1 kDuk C kdÁk 1 kuk; this implies that B 2 B 1 D Á defines a bounded operator H 1 ;s .M; S / !H 1 ;s .M; S/.Moreover, since Á is supported in the compact subset L and using continuity of the inclusion H 1 ;s .M; S/ ,! H 1 loc .M; S/, this actually means that B 2 B 1 is a bounded operator H 1 ;s .M; S/ !H 1 .L; S/.Finally, since the inclusion H 1 .L; S / ,! L 2 .M; S/ is compact by the Rellich lemma, B 2 B 1 yields a compact operator H 1 ;s .M; S/ !L 2 .M; S/.The second claim follows from classical properties of Fredholm operators.and 2 jd j is uniformly positive outside a compact subset because is admissible, the operator B is coercive at infinity in the sense of [4, Definition 5.1].Thus B ;s is a Fredholm operator by As another immediate consequence of Lemma 3.3, we obtain the following statement.
[4,orem 3.4Let M be a complete Riemannian manifold with compact boundary @M and S !M be a relative Dirac bundle.Let W M !R be an admissible potential and s W @M !f˙1g a choice of signs.Then the operator B ;s is selfadjoint and Fredholm.Proof Since for any given admissible potential we can always find a smooth admissible potential which agrees with the original one outside a compact subset, Lemmas 3.2 and 3.3 imply that we can assume without loss of generality that is smooth.As B has the same principal symbol as D, the canonical boundary Dirac operator A from (2-9) is also an adapted operator for B in the sense of[4, Section 3.2].Thus the discussion in the paragraphs preceding Lemma 3.2 applies verbatim with B replaced by D.
The first estimate of (4-2) now follows directly because jPuj 2 0. The second estimate follows from c.d / jd j.Moreover, equality in the first estimate is equivalent to R M jPuj 2 D 0 and thus Pu D 0. Equality for the second estimate is equivalent to Z 1 ;s .M; S/.Thus (4-4) still holds for all u 2 H 1 ;s .M; S/.M hu; .c.d / C jd j/ui vol M D 0: Since the selfadjoint bundle endomorphism c.d / C jd j is fiberwise nonnegative, this is equivalent to .c.d / C jd j/u D 0, as claimed.
1 ;s .M; S/ that realizes equality in Theorem 4.3.Then the section w WD '.x/ 1=2 u still lies in H 1 ;s .M; S/ because '.x/ 1=2 is a Lipschitz function.Using the elementary computation d '.x/ 1=2 ./ D '.x/ 1=2 A priori, this only holds in the weak sense, but it still implies that jwj 2 is a constant function if M is connected and we have K [ fc.dx/u D ug D M.
n dx./; we deduce from (4-5) that the (weak) covariant derivative of w satisfies, almost everywhere, (4-6) r w D n dx./ C c. [ / w for every smooth vector field on M. In particular, rw D 0 on K.Moreover, on the set fc.dx/u D ug WD fp 2 M n K j c.d p x/u p D u p g we deduce that r w D n .dx. / C c. [ / /w D n .dx. / C c. [ / c.dx//w D n c. [ ^dx/w; where in the last step we use the Clifford relation c.dx/ c. [ / C c. [ / c.dx/ D 2 dx./ and the notation c. [ ^dx/ WD 1 2 .c. [ / c.dx/ c.dx/ c. [ //. [ ^dx/w almost everywhere on fc.dx/u D ug: Finally, since c. [ ^dx/ is an antiselfadjoint bundle endomorphism, (4-7) further implies that d.jwj 2 /. / D 2hw; r wi D 0 almost everywhere on fc.dx/u D ug.Together with (

3 8
shows that (after altering the initial choice of ", passing to a bundle associated to V .E; F /, and restricting the new bundle on dM to obtain a new pair of bundles on M ), we can achieve that Z But this just means that (6-2) is satisfied for .E; F / on M and already shows that M has infinite y A-area.Finally, pulling back the bundles from M to M N via the projection pr M W M N !M yields the desired result for M N because Z /^ch.V .prM E; pr M F // D p Än/ such that scal g Än.n 1/ in N l 0 .Fix ƒ 2 .l;l 0 /.Then K ƒ WD M n N ƒ is a compact manifold with boundary such that scal g Än.n 1/ in M n K ƒ .For r 2 .0;p Än=2/, consider the function Y Let Ä 0 WD inf p2K ƒ scal g .p/> 0. By Lemma 6.9, Corollary 3.9 and Example 2.5, there exists a GL pair .E; F / and associated relative Dirac bundle S !M such that r .t/D r tan.r t/, with t varying in OE0; =. p Än//.Observe that Y r .0/D 0 and (i) .E; F / and thus S have support K ƒ , (ii) 4kR E˚F k 1 < Ä 0 , 4 Än 2 r 2 , (iv) ind.B ;1 / D indrel.M I E; F / ¤ 0 for any admissible potential .
Let B ;1 be the associated Callias operator subject to the boundary condition coming from the sign s D 1 and u 2 dom.B ;1 / D H 1 ;1 .M; S /.From (6-4), we deduce Z M jB uj 2 vol M Z M ‚ ;n juj 2 vol M ; Simone Cecchini and Rudolf ZeidlerIn other words, a width function is a band map V !OEt ; t C which is 1-Lipschitz.A width function x W V !OEt ; t C always satisfies t C t Ä width.V; g/.We also have the following converse: Lemma 7.2 Let .V; g/ be a band.Then there exists a width function x W V !OEt ; t C which is smooth near the boundary of V and is such that t C t D width.V; g/.Moreover, for every t < t C satisfying t C t < width.V; g/, there exists a smooth width function x W V !OEt ; t C .
[4,one Cecchini and Rudolf Zeidler are in the equality situation of Theorem 4.3.Since f 0 > 0 and 1 C c.dx/ 0 almost everywhere, we furthermore deduce from vanishing of the fourth integrand that c.dx/ u D u almost everywhere and thus This already proves (8-2) for w WD '.x/ 1=2 u.The identityand the fact that juj 2 D c' for c D jwj 2 constant is now a consequence of Remark 4.5 because (8-5) holds almost everywhere.Moreover, since juj 2 D c' > 0, vanishing of the first integrand inimplies that scal g D n.n 1/Ä.It also follows from (8-2) that jdxj 2 jwj 2 D j c.dx/wj 2 D j wj 2 D jwj 2 , and thus jdxj D 1 almost everywhere.Proof of Theorem 8.3 We will refer to Lemma 8.4 and its proof in the following argument.First of all, we obtain that jdxj D 1. Since x is assumed to be smooth, this already implies that V is diffeomorphic to M OEt ; t C with g corresponding to g x C dx ˝dx for some family of Riemannian metrics .gx/x2OEt;tC on M. To prove the theorem, it remains to compute the Hessian of x.To this end, we let 0 ¤ u 2 ker.B ; / and let w WD '.x/ 1=2 u as in Lemma 8.4.Since D f .x/ is smooth, boundary elliptic regularity (see for example[4, Theorem 4.4]) implies that u and thus w are smooth sections.Using equations (8-2) and (8-3) and the fact that is parallel, we compute for any smooth vector field on V , c.r dx/w D r .c.dx/w/ " ƒ‚ … dx./dx/ for each vector field .Hnce the Hessian of x is given by(8-6) r 2 x D h.x/.g dx ˝dx/: Using standard formulas for Riemannian distance functions (in the sense of [34, Section 3.2.2]), the identity (8-6) implies for the Lie derivative of g x that L @ x g x D 2h.x/g x I see for instance [34, Proposition 3.2.11(1)].Let z g x D '.x/ 2 g M , where g M WD '.t 0 / 2 gj M .Then z g x satisfies the same differential equation because L @ x z g [dx. / dx/w D h.x/ c. [ dx. / dx/w: Since w vanishes nowhere, this implies r dx D h.x/. [x D 2' 0 .x/'.x/g M D 2 ' 0 .x/'.x/ z g x D 2h.x/z g x : [16]ne Cecchini and Rudolf ZeidlerProof The main idea of the proof is to apply the argument of Goette and Semmelmann[16]fiberwise in combination with Theorem 8.3.Since ˆis a 1-Lipschitz band map, the function x D y ı ˆW V !OEt ; t C is a width function.We need to verify that we are in an instance of Setup 8.1.To this end, we set where pr N W V 0 !N is the projection onto the first factor.It is now a consequence of the discussion in C f .x/ 2 f 0 .x/D0;wherefWD12nh.In particular, this choice of functions satisfies Setup 8.1.Now we consider the fiberwise spinor bundle E 0 !V 0 constructed as in Remark 10.1 and let E WD ˆ E 0 be the pullback bundle on V .Now the main estimate of Goette and Semmelmann [16, Section 1.1] together with the description of the Weitzenböck curvature endomorphism of E 0 from (10-1) shows that we precisely have the estimateR E r X dx/ c.dx/ for X 2 TV: 2 ;