Universität Regensburg Mathematik

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1 Universität Regensburg Mathematik Surgery and the spinorial τ-invariant Bernd Ammann, Mattias Dahl and Emmanuel Humbert Preprint Nr. 7/007

2 SURGERY AND THE SPINORIAL τ-inariant BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT Abstract. We associate to a compact spin manifold M a real-valued invariant τ(m) by taking the supremum over all conformal classes of the infimum inside each conformal class of the first positive Dirac eigenvalue, normalized to volume 1. This invariant is a spinorial analogue of Schoen s σ-constant, also known as the smooth Yamabe number. We prove that if N is obtained from M by surgery of codimension at least, then τ(n) min{τ(m), Λ n} with Λ n > 0. arious topological conclusions can be drawn, in particular that τ is a spin-bordism invariant below Λ n. Below Λ n, the values of τ cannot accumulate from above when varied over all manifolds of a fixed dimension. Contents 1. Introduction 1.1. Spin manifolds, Dirac operators and choice of Clifford representation 1.. The τ-invariant 1.3. The σ-constant Geometric constants Joining manifolds Surgery and bordism ariants of the results 6. Preliminaries 6.1. Notation for balls and neighbourhoods 6.. Joining manifolds along submanifolds 7.3. Comparing spinors for different metrics 8.4. Regularity results 9.5. The associated variational problem 9 3. Preparations for proofs Removal of singularities Limit spaces and limit solutions Dirac spectral bounds on products with spheres Approximation by local product metrics Proofs Proof of Theorem Proof of Theorem Date: November 7, Mathematics Subject Classification. 53C7 (Primary) 55N, 57R65 (Secondary). Key words and phrases. Dirac operator, eigenvalue, surgery. We thank ictor Nistor, Penn State University, for some helpful discussions about Sobolev inequalities on non-compact manifolds with ends that are asymptotic to products of hyperbolic spaces and spheres. Also Mattias Dahl thanks the Institut Élie Cartan, Nancy, for its kind hospitality and support during visits when this work was initiated. 1

3 BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT References 8 1. Introduction 1.1. Spin manifolds, Dirac operators and choice of Clifford representation. Let M be a compact n-dimensional spin manifold without boundary. We will always consider spin manifolds as equipped with an orientation and a spin structure. The existence of these structures is equivalent to the vanishing of the first and second Stiefel-Whitney class. In some of the literature, the word spin only means that such structures exist. However, we use the word spin in the sense that M actually comes with a choice of orientation and spin structure. As explained in [19, 10, 14] one associates to the spin structure, to a Riemannian metric g on M and to a complex irreducible representation ρ of the Clifford algebra over R n a complex vector bundle, the spinor bundle Σ g ρ (M). The Dirac operator Dg ρ is a self-adjoint elliptic first order differential operator acting on smooth sections of the spinor bundle Σ g (M). It has a spectrum consisting only of real eigenvalues of finite multiplicity. The Dirac operator depends on the choice of spin structure, on the metric g and a priori on the representation ρ. In even dimensions n, the representation ρ is unique (up to equivalence). In odd dimensions there are two choices ρ{+} and ρ{ }. Exchanging the representation results in reversing the spectrum, i.e. if λ is an eigenvalue of D g ρ{+} then λ is an eigenvalue of Dg ρ{ } with the same multiplicity, and vice versa. This has no effect if n 1 mod 4 since the real/quaternionic structure on Σ g (M) anti-commutes with the Dirac operator and the spectrum therefore is symmetric, see [10, Section 1.7]. However, in dimension n 3 mod 4 the choice of ρ matters. In this case we choose the representation such that Clifford multiplication of e 1 e e n acts as the identity, where e 1,..., e n denotes the standard basis of R n. We thus can and will suppress ρ in the notation. 1.. The τ-invariant. We denote by λ + 1 (D g ) the first non-negative eigenvalue of D g. For a metric g on M we define λ + min (M, g) := inf λ+ 1 (D g )ol(m, g) 1 n where the infimum is taken over all metrics g conformal to g. Further we define τ + (M) := sup λ + min (M, g) where the supremum is taken over all metrics g on M. This yields an invariant of the spin manifold M, observe that we do not require M to be connected. We begin by noting some simple properties of the invariant τ +. Let (S n, σ n ) denote the sphere with its standard metric. We have λ + min (Sn, σ n ) = n ω1/n n where ω n is the volume of (S n, σ n ). Moreover it is shown in [, 6] that λ + min (M, g) λ+ min (Sn, σ n ) for any compact Riemannian spin manifold. Together with Inequality (1) below we get τ + (S n ) = λ + min (Sn, σ n ) = n ω1/n n

4 SURGERY AND THE SPINORIAL τ-inariant 3 so for all compact spin manifolds M we have τ + (M) τ + (S n ). If the kernel of D g is non-trivial, then obviously λ + min (M, g) = 0. Conversely, it was shown in [] that if D g is invertible, then λ + min (M, g) > 0. It follows that τ + (M) > 0 is equivalent to the existence of a metric g on M for which the Dirac operator D g is invertible. It is a further fact that τ + (M) = 0 precisely when α(m) 0, where α(m) is the alpha-invariant which equals the index of the Dirac operator for any metric on M, see [4]. If (M 1, g 1 ) and (M, g ) are compact Riemannian spin manifolds we denote by M 1 M the disjoint union of M 1 and M equipped with the natural metric g 1 g. It is not difficult to see that This implies λ + min (M 1 M, g 1 g ) = min{λ + min (M 1, g 1 ), λ + min (M, g )}. τ + (M 1 M ) = min{τ + (M 1 ), τ + (M )}. We denote by M the manifold M equipped with the opposite orientation. The Dirac operator changes sign when the orientation of the manifold is reversed. In dimensions 3 mod 4 this does not change the first positive eigenvalue of D since the spectrum is symmetric, so in those dimensions we have λ + min ( M, g) = λ + min (M, g) and τ + ( M) = τ + (M). For dimensions 3 mod 4 we define λ min (M, g) and τ (M) similar to λ + min (M, g) and τ + (M) by replacing λ + 1 by the absolute value of the first non-positive eigenvalue. We then have λ + min ( M, g) = λ min (M, g) and τ + ( M) = τ (M) The σ-constant. The τ-invariant is a spinorial analogue of the σ-constant [17, 1] which is defined for a compact manifold M by Scal g dv g σ(m) := sup inf ol(m, g) n n where the infimum runs over all metrics g in a conformal class and the supremum runs over all conformal classes (σ(m) is also known as the Yamabe invariant of M.) When σ(m) is positive it can be computed in a way analogous to τ + (M) using the smallest eigenvalue of the conformal Laplacian L g = 4 n g + Scal g instead of λ + 1 (Dg ). Hijazi s inequality [1, 13] gives a comparison of the two invariants, τ ± (M) n σ(m). (1) 4(n 1) For M = S n equality is attained in this inequality. Upper bounds for τ ± (M) may help to determine the σ-constant. We are currently working out an analogous surgery formula for the σ-invariant under surgeries of codimension at least 3. If the dimension of the surgery is smaller than [n/] 1, then the techniques of the present article can be carried over to the σ-invariant, but for higher-dimensional surgeries other techniques will be used. See [5] for an announcement of these results.

5 4 BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT 1.4. Geometric constants. We are going to prove a surgery formula for the invariant τ +. This formula involves some geometric constants Λ n,k which we now define. For a complete spin manifold (, g) we set λ + min (, g) := inf λ [0, ] where the infimum is taken over all λ (0, ) for which there is a non-zero spinor field ϕ L L Cloc 1 such that ϕ L n 1 and D g ϕ = λ ϕ ϕ. () If there are no such solutions of () on then λ + min (, g) =. For k a positive integer we let ξ k be the Euclidean metric on R k. For c R we let ηc k+1 = e ct ξ k + dt be the hyperbolic metric of sectional curvature c on R k+1. As above σ n k 1 denotes the metric of curvature 1 on S n k 1. We define our geometric invariants as Λ n,k := inf c [ 1,1] λ + min (Rk+1 S n k 1, ηc k+1 + σ n k 1 ), Λ n := min 0 k n Λ n,k. Note that the infimum could as well be taken over c [0, 1]. It is easy to show that Λ n,0 = λ + min (Sn, σ n ). For k > 0 we are not able to compute these constants, but at least we can show that they are positive. Theorem 1.1. For 0 k n we have Λ n,k > Joining manifolds. We are going to study the behaviour of τ + when two compact Riemannian spin manifolds are joined along a common submanifold. Let (M 1, g 1 ) and (M, g ) be compact spin manifolds of dimension n and let N be obtained by joining M 1 and M along a submanifold of dimension k as described in Section.. The manifold N is spin and from the construction follows a natural choice of spin structure on N. The following results make it possible to compare τ + (M 1 M ) and τ + (N). Theorem 1.. Assume that k = dim W satisfies 0 k n. Let w i : W B n k M n i be embeddings and let N be obtained by joining M 1 and M along W. Assume further that both D g1 and D g have trivial kernel. Then there is a sequence of metrics g, 0, such that min{λ + min (M 1 M, g 1 g ), Λ n,k } lim 0 λ + min (N, g ) λ + min (M 1 M, g 1 g ). Taking the supremum over all metrics on M 1 M the first inequality gives us the following corollary. Corollary 1.3. We have τ + (N) min{τ + (M 1 M ), Λ n,k } min{τ + (M 1 ), τ + (M ), Λ n }. Note that these estimates on τ + would be trivial without Theorem Surgery and bordism. Performing surgery on a spin manifold is a special case of joining manifolds, this is discussed in more detail in Section. below. As a special case of Corollary 1.3 we get an inequality relating the τ-invariant before and after surgery. For a compact spin manifold M of dimension n we define τ + (M) := min{τ + (M), Λ n }.

6 SURGERY AND THE SPINORIAL τ-inariant 5 It will also be convenient to introduce τ(m) := min{τ + (M), τ (M), Λ n }. As already explained, in the case n 3 mod 4, one has τ(m) = τ + (M). As before, all results for τ + (M) also hold for τ (M) := min{τ (M), Λ n }. Corollary 1.4. Assume that M is a spin manifold of dimension n and that N is obtained from M by a surgery of codimension n k, then τ + (N) min{τ + (M), Λ n,k } min{τ + (M), Λ n }. The Corollary implies, in particular, τ + (N) τ + (M), τ(n) τ(m). Two compact spin manifolds M and N are spin bordant if there is a spin diffeomorphism from their disjoint union to the boundary of a spin manifold of one dimension higher, and this diffeomorphism respects the orientation of N and reverses that of M. This happens if and only if N can be obtained from M by a sequence of surgeries. To apply Corollary 1.4 we need to know when this sequence of surgeries can be chosen to include only surgeries of codimension at least two. The theory of handle decompositions of bordisms tells us that this can be done when N is connected, see [16, II Theorem 3] for dimension 3 and [18, III Theorem 3.1] for higher dimensions. Corollary 1.5. Let M and N be spin bordant manifolds of dimension at least 3 and assume that N is connected. Then τ(n) τ(m). In particular, if M is also connected we have τ(n) = τ(m). The corollary can also be shown in dimension with similar arguments [7, Theorem 1.3]. The spin bordism group Ω spin n is the set of equivalence classes of spin bordant manifolds of dimension n with disjoint union as addition. Since every element in Ω spin n can be represented by a connected manifold we obtain a well-defined map τ : Ω spin n [0, Λ n ] which sends the equivalence class [M] of a connected spin manifold M to τ(m). Corollary 1.6. There is a positive constant ε n such that τ + (M) {0} [ε n, λ + min (Sn, σ n )]. for all spin manifolds M of dimension n. Proof. The spin bordism group Ω spin n is finitely generated [, page 336]. This implies that the kernel of the map α : Ω spin n KO n is also finitely generated. Let [N 1 ],..., [N r ] be generators of this kernel, we assume that all N i are connected. Since τ(m) = 0 if and only if α(m) 0 we obtain the corollary for ε n := min{λ n, τ(n 1 ),..., τ(n r )}. The α-map is injective when n < 8, and then ε n = Λ n. Unfortunately, we do not know whether there are n N with ε n < Λ n. In other words, we do not know if there are n-dimensional manifolds M with 0 < τ + (M) < Λ n. If such manifolds exist, the following conclusions might be interesting.

7 6 BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT At first, if M is a spin manifold with τ + (M) < Λ n, then it follows from the Corollary 1.5 that the σ-invariant of any manifold N spin bordant to M satisfies and 4(n 1) σ(n) τ + (M). n For the next application we define S(t) := {[M] Ω spin n τ(m) t} S + (t) := {[M] Ω spin n τ + (M) t} T (t) := {[M] Ω spin n τ(m) > t} T + (t) := {[M] Ω spin τ + (M) > t}. Obviously S(t) = S + (t) and T (t) = T + (t) in dimensions n 3 mod 4. Corollary 1.7. S(t) is a subgroup of Ω spin n for t [0, Λ n ] and T (t) is a subgroup of Ω spin n for t [0, Λ n ). If n 3 mod 4, then S + (t) and T + (t) are subsemigroups. Corollary 1.8. The values of τ cannot accumulate from above. Proof. Assume that τ(m i ) = t i, i N is a sequence of values of τ. We want to show that the infimum inf t i is attained by a t i. Assume that the infimum is not attained. After passing to a subsequence we can assume that the sequence t i is decreasing. Then S(t i ) S(t i+1 ), hence S(t i ) = T (t ) is a subgroup of the finitely generated group Ω spin n. It is thus finitely generated itself. Hence there exists r N such that S(t r ) contains the finite set of generators, and thus S(t r ) = T (t ). Hence [M i ] S(t r ) for all i, which implies t i t r, i.e. we obtain the contradiction t r = inf t i. We do not know whether the same statement holds for τ + in dimensions n 3 mod ariants of the results. We already remarked earlier that if the alpha-genus α(m) of a spin manifold M does not vanish, then the index theorem tells us that for any metric g on M, the kernel of D g is non-trivial, and hence τ + (M) = 0. More exactly the index theorem implies for connected spin manifold M, that the kernel of the Dirac operator has at least dimension Â(M), if n 0 mod 4; 1, if n 1 mod 8 and α(m) 0; a(m) :=, if n mod 8 and α(m) 0; 0, otherwise. Let us modify the definition of τ + and use the k-th non-negative eigenvalue of the Dirac operator instead of the first one. The quantity thus obtained, denoted by τ + k (M) is zero if k a(m). It follows from [] and [4] that τ + a(m)+1 (M) > 0. We expect that our methods generalize to this situation and yield similar surgery formulas for τ + k.. Preliminaries.1. Notation for balls and neighbourhoods. We write B n (r) for the open standard ball of radius r around 0 in R n, and set B n := B n (1). For a Riemannian manifold (M, g) we let B g (p, r) denote the open ball of radius r around p M. If the Riemannian metric is clear from the context we will write B(p, r). For a n

8 SURGERY AND THE SPINORIAL τ-inariant 7 Riemannian manifold (M, g) with subset S we let B g (S, r) := x S Bg (p, r), the r-neighbourhood of S. Again, if the Riemannian metric is clear from the context we abbreviate to B(S, r)... Joining manifolds along submanifolds. Let (M 1, g 1 ) and (M, g ) be compact Riemannian spin manifolds of dimension n. Let W be a compact k-dimensional spin manifold where 0 k n, and assume that w i : W B n k M i, i = 1, are embeddings such that w 1 is orientation reserving and w is orientation preversing and they both preserve spin structures. Let N be obtained by joining M 1 and M along W. The manifold N is naturally equipped with a spin structure. We define W := w 1 (W {0}) w (W {0}). When we restrict g 1 to w 1 (W {0}) and pull it back to W, we obtain a metric on W, called h 1. Similarly one obtains the metric h on W by restricting and pullingback g via w. Let r i denote the distance function in (M i, g i ) to w i (W {0}) and let { r 1 (x), if x M 1 ; r(x) := r (x), if x M. For 0 < ε we define U i (ε) := {x M i : r i (x) < ε} and U(ε) := U 1 (ε) U (ε). For 0 < ε < we define and N ε := (M 1 \ U 1 (ε)) (M \ U (ε))/, U N ε () := (U() \ U(ε))/ where indicates that we identify x U 1 (ε) with w w1 1 (x) U (ε). Hence N ε = (M 1 M \ U()) Uε N(). We say that N ε is obtained from M 1 M by a connected sum along W with parameter ε. The operation of doing surgery on a manifold is a special case of this construction. Indeed, let M be a compact spin manifold and set M 1 = M, M = S n, W = S k, w 1 : S k B n k M an embedding defining a surgery of dimension k, w : S k B n k S n the standard embedding and let N ε be obtained from M by a surgery along W. Since S n \ w (S k {0}) is diffeomorphic to B k+1 S n k 1 we have that N ε is obtained from M by a surgery of dimension k, see [18, Section I.9]. The diffeomorphism type of N ε is independent of ε, hence we will usually write N = N ε. However, in some situations where dropping the index ε might cause ambiguites we will write N ε. For example the function r : M 1 M [0, ) also defines a continuous function r : N ε [ε, ) whose definition depends on ε. We will also keep the ε-subscript for Uε N () as important estimates for spinors will be carried out on Uε N (), and these estimates are not invariant if one applies a diffeomorphism of M to Uε N() without applying it to the spinor. As the embeddings w 1 and w preserve the spin structure, the manifold N carries a spin structure such that its restriction to (M 1 \ w 1 (W B n k )) (M \ w (W B n k )) coincides with the restriction of the given spin structure on M 1 M. If W is not connected, then this choice is not unique. The statements of our theorem hold for any such spin structure on N.

9 8 BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT.3. Comparing spinors for different metrics. Let M be a spin manifold of dimension n and let g, g be Riemannian metrics on M. The goal of this paragraph is to identify the spinor bundles of (M, g) and (M, g ) following Bourguignon and Gauduchon [9]. There exists a unique endomorphism b g g of T M which is positive, symmetric with respect to g, and satisfies g(x, Y ) = g (b g g X, bg g Y ) for all X, Y T M. This endomorphism maps g-orthonormal frames at a point to g -orthonormal frames at the same point and we get a map b g g : SO(M, g) SO(M, g ) of SO(n)-principal bundles. If we assume that Spin(M, g) and Spin(M, g ) are equivalent spin structures on M the map b g g lifts to a map βg g of Spin(n)-principal bundles, β g g Spin(M, g) Spin(M, g ) SO(M, g) SO(M, g ) b g g. From this we get a map between the spinor bundles Σ g M and Σ g M denoted by the same symbol and defined by Σ g M = Spin(M, g) σ Σ n Spin(M, g ) σ Σ n = Σ g M ψ = [s, ϕ] [β g g s, ϕ] = βg g ψ where (σ, Σ n ) is the complex spinor representation, and where [s, ϕ] denotes an element of Spin(M, g) σ Σ n. Note that the map β g g is fiberwise an isometry. We define the Dirac operator D g acting on sections of the spinor bundle for g by g D g := (β g g ) 1 D g β g g In [9, Theorem 0] the operator g D g is computed in terms of D g and some extra terms which are small if g and g are close. Formulated in a way convenient for us the relationship is where A g g hom(t M Σ g M, Σ g M) satisfies and B g g hom(σg M, Σ g M) satisfies g D g ψ = D g ψ + A g g ( g ψ) + B g g (ψ) (3) A g g C g g g (4) B g g C( g g g + g (g g ) g ) (5) for some constant C. In the special case that g and g are conformal with g = F g for a positive smooth function F the formula simplifies considerably, and one obtains see for instance [15, 8]. g D g (F ψ) = F n+1 D g ψ (6)

10 SURGERY AND THE SPINORIAL τ-inariant 9.4. Regularity results. By standard elliptic theory we have the following lemma (see for example [3] where the corresponding results of [11] are adjusted to the Dirac operator). Lemma.1. Let (, g) be a Riemannian spin manifold and Ω an open set with compact closure in. Let also r (1, ). Then there is a constant C so that ( ) g ϕ r dv g C D g ϕ r dv g + ϕ r dv g (7) Ω Ω Ω for all ϕ Γ(Σ g Ω) which are of class C 1 and compactly supported in Ω. In case of a compact Riemannian manifold with invertible Dirac operator we have the following special case. Lemma.. Let (, g) be a compact Riemannian spin manifold with invertible Dirac operator. Then there exists a constant C such that g ϕ n n+1 dv g C D g ϕ n n+1 dv g. (8) for all ϕ Γ(Σ g ) of class C 1 ( )..5. The associated variational problem. Let (M, g) be a compact spin manifold of dimension n with ker D g = {0}. For ψ Γ(ΣM) we define ( ) n+1 n n Dψ n+1 M dv g J(ψ) := M Dψ, ψ. dvg whenever the denominator is non-zero. Using techniques from [0] it was proved in [] that λ + min (M, g) = inf J(ψ) (9) ψ where the infimum is taken over the set of smooth spinor fields satisfying Dψ, ψ dv g > 0. M If g and g = F g are conformal metrics on M and if J and J are the associated functionals, then by Relation (6) one computes that for all smooth ψ Γ(Σ g M)) J(F ψ) = J(ψ). (10) The following result gives a universal upper bound on λ + min (M, g). Proposition.3. Let (M, g) be a compact spin manifolds of dimension n. Then λ + min (M, g) λ+ min (Sn, σ n ) = n ω1/n n (11) where ω n is the volume of (S n, σ n ). The proposition was proven for n 3 in [] using geometric methods. In the case n = the article [] only provides a proof if ker D = {0}. Another method that yields the proposition in full generality is to construct for any p M and ε > 0 a suitable test spinor field ψ ε supported in B g (p, ε) satisfying J(ψ ε ) λ + min (Sn, σ n ) + o(ε), see [6] for details. If inequality (11) holds strictly then one can show that the infimum in equation (9) is attained by a spinor field ϕ. The following theorem will be a central ingredient in the proof of Theorem 1..

11 10 BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT Theorem.4 ([1, 3]). Let (M, g) be a compact spin manifold of dimension n for which inequality (11) holds strictly. Then there exists a spinor field ϕ C,α (ΣM) C (ΣM \ ϕ 1 (0)) where α (0, 1) (0, /(n 1)] such that ϕ n = 1 and L Dϕ = λ + min (M, g) ϕ ϕ. Furthermore the infimum in the definition of λ + min (M, g) is attained by the generalized conformal metric g = ϕ 4/() g, see [1] for details. 3. Preparations for proofs 3.1. Removal of singularities. The following theorem gives conditions for when singularities of solutions to Dirac equations can be removed. Theorem 3.1. Let (, g) be a (not necessarily complete) Riemannian spin manifold and let S be a compact submanifold of of codimension m. Assume that ϕ L p (Σ( \ S)), p m/(m 1), satisfies the equation Dϕ = ρ weakly on \S where ρ L 1 (Σ( \S)) = L 1 (Σ ). Then this equation holds weakly on. In particular the singular support of the distribution Dϕ is empty. Proof. Let ψ be a smooth compactly supported spinor. We have to show that ϕ, Dψ dv = ρ, ψ dv. Recall that for ε > 0 we denote the set of points in of distance less than ε to S by B(S, ε). We choose a smooth cut-off function χ ε : [0, 1] with support in B(S, ε), χ ε = 1 on B(S, ε), and gradχ ε /ε. We then have ϕ, Dψ dv ρ, ψ dv = ϕ, D((1 χ ε )ψ + χ ε ψ) dv ρ, ψ dv = Dϕ, (1 χ ε )ψ dv + ϕ, χ ε Dψ dv + ϕ, gradχ ε ψ dv ρ, ψ dv = ρ, χ ε ψ dv + ϕ, χ ε Dψ dv + ϕ, gradχ ε ψ dv, where Dϕ = ρ is used in the last equality. Let q be related to p via 1/q + 1/p = 1. It then follows that ( ) ϕ, Dψ dv ρ, ψ dv sup ψ ρ dv B(S,ε) B(S,ε) ( ) + sup ψ + sup Dψ ϕ dv ε B(S,ε) B(S,ε) B(S,ε) o(1) + C ε ϕ L p ( ε(s))ol(b(s, ε)) 1/q o(1) + o(ε (m/q) 1 ),

12 SURGERY AND THE SPINORIAL τ-inariant 11 where o(1) denotes a term tending to 0 as ε 0. Since m/q 1 is equivalent to p m/(m 1) the statement follows. Applying this result to the non-linear equation in Theorem.4 we get the following corollary. Corollary 3.. Let and S be as above. Then any L p -solution, p = n/(n 1), of Dϕ = λ ϕ p ϕ (1) on \ S is also a weak L p -solution of (1) on. 3.. Limit spaces and limit solutions. In the proofs of the main theorems we will construct limit solutions of a partial differential equation on certain limit spaces. For this we need the following two lemmas. Lemma 3.3. Let be an n-dimensional manifold. Let (p α ) be a sequence of points in which converges to a point p as α 0. Let (γ α ) be a sequence of metrics defined on a neighbourhood O of p which converges to a metric γ 0 in the C (O)-topology. Finally, let (b α ) be a sequence of positive real numbers such that lim α 0 b α = +. Then for r > 0 there exists for α small enough a diffeomorphism Θ α : B n (r) B γα (p α, b 1 α r) with Θ α (0) = p α such that the metric Θ α (b α γ α) tends to the Euclidean metric ξ n in C 1 (B n (r)). Proof. Denote by exp α : U α O α the exponential map at the point p α defined with respect to the metric γ α. Here O α is a neighbourhood of p α in and U α is a neighbourhood of the origin in R n. We set Θ α : B n (r) x exp α (b 1 α x) B γα (p α, b 1 α r) It is easily checked that Θ α is the desired diffeomorphism. Lemma 3.4. Let an n-dimensional spin manifold. Let (g α ) be a sequence of metrics which converges to a metric g in C 1 on all compact sets K as α 0. Assume that (U α ) is an increasing sequence of subdomains of such that α U α =. Let ψ α Γ(Σ gα U α ) be a sequence of spinors of class C 1 (U α ) such that ψ α L (U α) C where C does not depend on α, and D gα ψ α = λ α ψ α ψα (13) where the λ α are positive numbers which tend to λ 0. Then there exists a spinor ψ Γ(Σ g ) of class C 1 ( ) such that D g ψ = λ ψ ψ (14) on and a subsequence of (βg gα ψ α ) tends to ψ in C 0 (K) for any compact set K. In particular ψ L (K) = lim ψ α L α 0 (K), (15) and ψ r dv g = lim ψ α K α 0 K r dv gα (16) for any compact set K and any r 1.

13 1 BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT Proof. Let K be a compact subset of and let Ω be an open set in with compact closure such that K Ω. Let χ C ( ) with 0 χ 1 be compactly supported in Ω and satisfy χ = 1 on a neighbourhood Ω of K. We set ϕ α = (β g g α ) 1 ψ α. Using Equations (13) and (3) we get D g (χϕ α ) = grad g χ ϕ α + χλ α ϕ α ϕα χa g g α ( g ϕ α ) χbg g α (ϕ α ). (17) and hence using the fact that a + b + c r 3 r ( a r + b r + c r ) for a, b, c R we get for r > 1 that D g (χϕ α ) r 3 r( grad g χ ϕ α + χλ α ϕ α Since ϕ α L ( ) = ψ α L ( ) C we have ϕα r + χa g g α ( g ϕ α ) r + χb g g α (ϕ α ) r). grad g χ ϕ α + χλ α ϕ α ϕα r C. By Relations (4) and (5) and since lim α 0 g α g C 1 (Ω) = 0 we get χa g g α ( g ϕ α ) r + χb g g α (ϕ α ) r o(1) ( g (χϕ α ) r + grad g χ ϕ α r + χϕ α r ) where o(1) tends to 0 with α. It follows that o(1) ( g (χϕ α ) r + C) D g (χϕ α ) r C + o(1) g (χϕ α ) r. Setting ϕ = χϕ α in Inequality (7) and again using that ϕ α L (Ω) is uniformly bounded we get that g (χϕ α ) r dv g C + o(1) g (χϕ α ) r dv g. Ω In particular (χϕ α ) is bounded in H 1,r 0 (Ω). Let a (0, 1). By the Sobolev Embedding Theorem this implies that a subsequence of (χϕ α ) converges in C 0,a (Ω) to ψ K Γ(Σ gα Ω) of class C 0,a. We take the inner product of (17) with a smooth spinor ϕ which is compactly supported in Ω and integrate over Ω. Since χ = 1 on the support of ϕ the result is ϕ α, D g ϕ dv g = λ α ϕ α ϕα, ϕ dv g Ω Ω A g g α ( g ϕ α ), ϕ dv g Bg g α (ϕ α ), ϕ dv g. Passing to the limit in α and again using (4) and (5) we get ψ K, D g ϕ dv g = λ ψ K ψk, ϕ dv g. Ω Ω Ω Hence, ψ K satisfies Equation (14) weakly on K. By standard regularity theorems we conclude that ψ K C 1 (K). Now we choose an increasing sequence of compact sets K m such that m K m =. Using the above arguments and taking successive subsequences it follows that (ϕ α ) converge to spinor fields ψ m on K m with ψ m Km 1 = ψ m 1. We define ψ on by ψ := ψ m on K m. By taking a diagonal subsequence of (ϕ α ) we get that (ϕ α ) tends to ψ in C 0 on any compact set K. Ω Ω

14 SURGERY AND THE SPINORIAL τ-inariant 13 The relations (15) and (16) follow immediately since βg g α is an isometry, since ϕ α = (βg g α ) 1 ψ α and since (g α ) (resp. (ϕ α )) tends to g (resp. ψ) in C 0 on K. This ends the proof of Lemma Dirac spectral bounds on products with spheres. In the following lemma we assume (in the case m = 1) that S 1 carries the spin structure which is obtained by restricting the unique spin structure on the -ball to its boundary. The proof is a simple application of the formula for the squared Dirac operator on a product manifold and the lower bound of its spectrum on the standard sphere. Lemma 3.5. Let (, g) be a complete Riemannian spin manifold. Then any L - spinor ψ on ( S m, g + σ m ) satisfies Dψ dv g+σ m m ψ dv g+σ m. S 4 m S m 3.4. Approximation by local product metrics. The goal of this paragraph is to prove that we can assume that in a neighbourhood of w i (W {0}) in M i we have g i = h i + dr i + r i σn k 1. We are going to prove Lemma 3.6. Let (, g) be a compact Riemannian manifold of dimension n and let S be a closed submanifold of dimension k where 0 k n with a trivialization of its normal bundle. Assume that D g is invertible. Then there exists a sequence (g α ) of metrics on such that and lim α 0 λ+ min (, g α) = λ + min (, g) g α = h + dr + r σ n k 1 in a neighbourhood B g (, α) of S. Here h is the restriction of the metric g to S and r(x) = d g (S, x). Proof. Define the metric G on a neighbourhood of S as G := h+dr +r σ n k 1. Let B g (, α) be the set of points x such that r(x) < α and let χ α C (M), 0 χ 1, be a cut-off function such that χ = 1 on B g (, α), χ = 0 on M \ B g (, α), and dχ α /α. We define g α := χ α G + (1 χ α )g. For convenience we introduce the notation λ α := λ + min (, g α) and λ := λ + min (, g). After possibly taking a subsequence we assume that lim α 0 λ α exists and we denote the limit by λ. We begin by proving that λ λ (18) which is the simpler part of the proof. Let J and J α be the functionals associated to g and g α and let δ > 0 be a small number. We set χ α := 1 χ α so that χ α = 1 on \ B g (, 4α), χ α = 0 on Bg (, α), and dχ α 1/α. We see that g = g α on the support of η α. Let ψ be a smooth spinor such that J(ψ) λ + δ. We then have D g (χ α ψ), χ α ψ dvg = χ α Dg ψ, ψ dv g + grad g χ α ψ, χ α ψ dvg.

15 14 BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT Since the last term here is purely imaginary we obtain lim D g (χ α ψ), α 0 χ α ψ dvg = lim Re χ α Dg ψ, ψ dv g = α 0 We compute D g (χ αψ) n n+1 dv g = \B g (,4α) + D g ψ n n+1 dv g B g (,4α)\B g (,α) D g ψ, ψ dv g. (19) grad g χ α ψ + χ α Dg ψ n n+1 dv g. Using the fact that a + b n n+1 n n+1 ( a n n+1 + b n n+1 ) for a, b R we have grad g χ α ψ + χ α Dg ψ n n+1 ( lim α 0 n n+1 n n+1 B g (,4α)\B g (,α) grad g χ α n n+1 ψ n n+1 + χ α n n+1 D g ψ n n+1 ( C 1 α n n+1 + C ) where C 1 and C are bounds on ψ and Dψ. Since ol(b g (, 4α) \ B g (, α)) Cα n k Cα it follows that grad g χ α ψ + χ αd g ψ n n+1 dv g = 0. It is clear that lim α 0 \B g (,4α) Dg ψ n n+1 dv g = Dg ψ n n+1 dv g so Equation (0) tells us that lim D g (χ n αψ) n+1 dv g = D g ψ n n+1 dv g. α 0 Together with Equation (19) this proves that lim α 0 J(χ αψ) = J(ψ) λ+δ. Since g α = g on the support of χ αψ, we have J α (χ αψ) = J(χ αψ). Relation (18) now follows since λ α J α (χ αψ) and δ is arbitrary. The second and harder part of the proof is to show that ) (0) λ λ. (1) Recall that due to Proposition.3 we know λ α λ + min (Sn, σ n ), λ λ + min (Sn, σ n ), and λ λ + min (Sn, σ n ). Inequality (1) is obvious for λ = λ + min (Sn, σ n ). Hence we will assume λ α < λ + min (Sn, σ n ) for a sequence α 0. As the Dirac operator is invertible we know that (8) holds. By Theorem.4 there exists for all α spinor fields ψ α Γ(Σ gα v) of class C 1 such that and D gα ψ α = λ α ψ α ψα () ψ α n dv g α = 1. (3) We set ϕ α = (βg g α ) 1 ψ α. Since g α g it is easily seen that the sequence (ϕ α ) is bounded in L n (, g). By (3) and () we have D g ϕ α = λ α ϕ α ϕα A g g α ( g ϕ α ) Bg g α (ϕ α ), (4)

16 SURGERY AND THE SPINORIAL τ-inariant 15 together with a + b + c n n+1 n n 3 n+1 ( a n+1 n + b n+1 n + c n+1 ) for a, b, c R this implies D g ϕ α n n+1 C ( We also have and λ n n+1 α ) ϕ α n + A g gα ( g ϕ α ) n n+1 + B g gα (ϕ α ) n n+1. (5) A g g α ( g ϕ α ) g g α C 0 ( ) g ϕ α Cα g ϕ α (6) B g g α (ϕ α ) g g α C1 ( ) ϕ α C ϕ α. (7) Indeed, since g and g α coincide on S, there exists a constant C so that g g α B g (,α) Cα. Together with the fact that dχ α /α and using the definition of g α, this immediately implies that g g α C 1 ( ) C. Using Relation (8) and integrating (5) we find that g ϕ α n n+1 dv g ( ) C λ n n+1 α ϕ α n dv g + α n n+1 g ϕ α n n+1 dv g + Bg g α (ϕ α ) n n+1 dv g. As g and g α coincide on \ B g (, α) we conclude that Bg g α (ϕ α ) = 0 on this set. Together with (7) we have B g g α (ϕ α ) n n+1 dv g C B g (,α) ϕ α n n+1 dv g C ol(b g (, α)) n+1 = o(1) where o(1) tends to 0 with α. Hence ( g ϕ α n n+1 dv g C λ n n+1 α ( ϕ α n dv g + α ϕ α n dv g B g (,α) ) n+1 ) g ϕ α n n+1 dv g + o(1) This implies in particular that (ϕ α ) is bounded in H n n+1 1 ( ) and hence after passing to a subsequence (ϕ α ) converges weakly to a limit ϕ in H n n+1 1 ( ). The next step is to prove that λ = lim α 0 λ α is not zero. To get a contradiction let us assume that λ = 0. We then obtain from (8) that g ϕ n n+1 dv g lim α 0 g ϕ α n n+1 dv g = 0. So ϕ is parallel and since D g is invertible we conclude ϕ = 0, in other words (ϕ α ) converges weakly to zero in H n n+1 1 ( ). As this space embeds compactly into L n n+1 ( ) we have lim ϕ α n = ϕ α 0 L n+1 ( ) L n+1 n = 0 ( ) and hence (ϕ α ) converges strongly to zero in H n n+1 1 ( ). As this space embeds continuously into L n ( ) we conclude that the sequence converges strongly to zero in L n ( ). This is impossible since by Relation (3) we easily get that lim α 0 ϕ α L n ( ) = 1. (8)

17 16 BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT From this contradiction we conclude λ > 0. (9) From (4) we have D g ϕ α n λ L α ϕ n+1 α n+1 ( ) We already proved above that Using Relation (6) we get similarily L n ( ) + B g g α (ϕ α ) L n n+1 ( ). lim α 0 Bg g α (ϕ α ) L n n+1 ( ) = 0. lim α 0 Ag g α ( g ϕ α ) L n n+1 ( ) = 0. + A g g α ( g ϕ α ) L n n+1 ( ) Moreover since dv gα = (1 + o(1)) dv g it follows from (3) that We obtain that λ α ϕ α n+1 = λ L n α (1 + o(1)). ( ) D g ϕ α L n n+1 ( ) λ α + o(1). (30) Starting from Equation (4) we can prove in a similar way that D g ϕ α, ϕ α dv g λ α + o(1). (31) From (9), (30), and (31) it follows that λ lim α 0 J(ϕ α ) = λ. This ends the proof of (1). Together with (18) this proves Lemma Proofs 4.1. Proof of Theorem 1.. This section is devoted to the proof of Theorem 1.. Our goal is to construct a sequence of metrics (g ) which satisfies the conclusion of Theorem (1.) when is small. Applying Lemma 3.6 with = M = M 1 M and S = W = w 1 (W {0}) w (W {0}) it follows that we may assume that g = h + dr + r σ n k 1 (3) in a neighbourhood U(R max ) of W, R max > 0. We fix numbers R 0, R 1 R with R max > R 1 > R 0 > 0 and we choose a function F : M \ S R + such that { 1, if x M i \ U i (R 1 ); F (x) = ri 1 if x U i (R 0 ) \ S. We further choose (0, R 0 ), later we will let 0. It is not difficult to see that there is a smooth function f : U(R max ) R (depending only on r), real numbers δ 1 = δ 1 () and δ = δ () with > δ > δ 1 > 0 and a real number A [ ln, ln δ 1 ) such that { ln r if x U(R max ) \ U(); f(x) = ln A if x U(δ ),

18 SURGERY AND THE SPINORIAL τ-inariant 17 f(r) ln A ln ln δ Figure 1. The function ln r f(r) ln r = ln ɛ t and such that and r d dr r df dr = df d(ln r) 1, ( r df dr ) L = d f d (ln r) 0 L as 0. It follows that lim 0 A =. After these choices we set ε := e A δ 1. We assume that N is obtained from M by a connected sum along W with parameter ε, as explained in section.. In particular, recall that Uε N (s) = U(s) \ U(ε)/ for all s ε. On the set Uε N(R max) = U(R max ) \ U(ε)/ we define the variable t by on U 1 (R max ) and on U (R max ). This implies t := ln r 1 + ln ε 0 t := ln r ln ε 0 r i = e t +ln ε = εe t. The choices imply that t : U N ε (R max ) R is a smooth function with t 0 on U N ε (R max ) M 1, t 0 on U N ε (R max ) M, and t = 0 is the common boundary U 1 (ε) identified in N with U (ε). Then equation (3) tells us that r g = ε e t h i + dt + σ n k 1. Expressed in the new variable t we have F (x) = ε 1 e t

19 18 BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT Hierarchy of ariables R max > R 1 > R 0 > > δ > δ 1 > ε > 0 We choose in the order R max, R 1, R 0,, δ, δ 1, A We can assume for example that ε = e A δ 1. This implies t = A r i = δ 1. Figure. Hierarchy of variables if x U(R 0 ) \ U N () or in other words if t + ln ε ln R 0, and { t ln ε if t + ln ε (, R max ), f(t) = ln A if t + ln ε ln δ, and df/dt 1, d f/dt L 0. After choosing a cut-off function χ : R [0, 1] such that χ = 0 on (, 1] and χ = 1 on [1, ), we define F g i if x M i \ U i (); g (x) := e f(t) h i + dt + σ n k 1 if x U i () \ U i (δ 1 ); A χ(a 1 t)h + A (1 χ(a 1 t))h 1 + dt + σ n k 1 if x U i (δ 1 ) \ U i (ε). (Recall that h i were define to be the pullback via w i of the metric g i on M i, composed with restriction to W = W {0}.) On U(R 0 ) we write g as g = α t h t + dt + σ n k 1 where the metric h t is defined for t R by h t := χ(a 1 t)h + (1 χ(a 1 t))h 1 (33) and where α t := e f(t). (34) The rest of the proof is devoted to show that a subsequence of (g ) is the desired sequence of metrics. In the following we keep the notation (g ) for any subsequence of (g ). We set λ := λ + min (M 1 M, g), λ := λ + min (N, g ), and λ := lim 0 λ (after passing to a subsequence we can assume that this limit exists). Let J and J be the functionals associated respectively to λ and λ. The easier part of the argument is to show that λ λ. (35) For this let α > 0 be a small number. We choose a smooth cut-off function χ α : M 1 M [0, 1] such that χ α = 1 on M 1 M \ U(α), dχ α α, and χ α = 0 on U(α). Let ψ be a smooth non-zero spinor such that J(ψ) λ + δ where δ is a small positive number. On the support of χ α the metrics g and g α are conformal since g = F g and hence by Formula (10) we have ( ) λ J χ α βg g (F ψ) = J(χ α ψ) for < α. Proceeding exactly as in the first part of the proof of Lemma 3.6 we show that lim α 0 J(χ α ψ) = J(ψ) λ + δ. From this follows Relation (35). Now we consider the more difficult part of the proof, that λ min{λ, Λ n,k }. (36)

20 SURGERY AND THE SPINORIAL τ-inariant 19 By Proposition.3 we can assume that for all, λ < λ + min (Sn, σ n ). Otherwise Relation (36) is trivial. From Theorem.4 we know that there exists a spinor field ψ Γ(Σ gα N) of class C such that ψ n dv g = 1 and N D g ψ = λ ψ ψ. (37) We let x in N be such that ψ (x ) = m where m := ψ L (N). The proof continues divided in cases. Case I. The sequence (m ) is not bounded. After taking a subsequence, we can assume that lim 0 m = +. We consider two subcases. Subcase I.1. There exists a > 0 such that x N \ U N (a) for an infinite number of. We recall that N \ U N (a) = N ε \ Uε N (a) = M 1 M \ U(a). By taking a subsequence we can assume that there exists x M 1 M \U(a) such that lim x = x. We let g := m 4 g. In a neighbourhood U of x the metric g = F g does not depend on. We apply Lemma 3.3 with O = U, α =, p α = x, p = x, γ α = g = F g, and b α = m. Let r > 0. For small enough Lemma 3.3 gives us a diffeomorphism Θ : B n (r) B g (x, m r) such that the sequence of metrics (Θ (g )) tends to the Euclidean metric ξn in C 1 (B n (r)). We let ψ ψ. By (6) we then have on B g (x, m r) and := m 1 B g (x,m r) Here we used the fact that dv g D g ψ = λ ψ ψ ψ n dv g = B g (x,m = 1. N r) ψ n dv g n = m dv g. Since Θ : (B n (r), Θ (g )) (Bg (x, m r), g ) is an isometry we can consider ψ as a solution of D Θ (g ) ψ = λ ψ ψ ψ n dv g on B n (r) with B n (r) ψ n dv Θ (g ) 1. Since ψ L (B n (r)) = ψ (0) = 1 we can apply Lemma 3.4 with = R n, α =, g α = Θ (g ), and ψ α = ψ (we can apply this lemma since each compact set of R n is contained in some ball B n (r)). This shows that there exists a spinor ψ of class C 1 on (R n, ξ n ) which satisfies D ξn ψ = λ ψ ψ.

21 0 BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT Furthermore by (16) we have ψ n dv ξ n = lim 0 B n (r) B g (x,m r) ψ n dv g 1 for any r > 0. We conclude that ψ n R n dv ξ n 1. Since ψ(0) = 1 we also see that ψ is not identically zero. As (R n, ξ n ) and (S n \ {pt}, σ n ) are conformal we can write σ n = Φ ξ n for a positive function Φ. We define ϕ := Φ β ξn σnψ. By Relation (6) it follows that ϕ L n (S n ) is a solution of D σn ϕ = λ ϕ ϕ (38) on S n \ {pt} of class C 1. By Corollary 3. we know that ϕ can be extended to a weak solution of (38) on all S n and by standard regularity theorems it follows that ϕ C 1 (S n ). Let J σn be the functional associated to (S n, σ n ). By Equation (38) we have λ + min (Sn, σ n ) J σn (ϕ) = λ where the inequality comes from Proposition.3. We have proved Relation (36) in this subcase. Subcase I.. For all a > 0 it holds that x / M 1 M \ U(a) for sufficiently small. This means that x belongs to U N (a) if is sufficiently small. This subset is diffeomorphic to W I S n k 1 where I is an interval. Hence x can be written as x = (y, t, z ) where y W, t ( ln R 0 + ln ε, ln ε + ln R 0 ), and z S n k 1. By taking a t subsequence we can assume that y, A, and z converge respectively to y W, T [, + ], and z S n k 1. We apply Lemma 3.3 with = W, α =, p α = y, p = y, γ α = h t, γ 0 = h T (we define h := h 1 and h + := h ), and b α = m α t. The lemma provides diffeomorphisms Θ y : Bk (r) B h t (y, m α 1 t r) for r > 0 such that (Θ y ) (m 4 α ht t ) tends to the Euclidean metric ξ k on B k (r) as 0. Next we apply Lemma 3.3 with = S n k 1, α =, p α = z, γ α = γ 0 = σ n k 1, and b α = m. For r > 0 we get the existence of diffeomorphisms Θ z : B n k 1 (r ) B σn k 1 (z, m r ) such that (Θ z ) (m 4 σ n k 1 ) converges to ξ n k 1 on B n k 1 (r ) as 0. For r, r, r > 0 we define and U (r, r, r ) := B h t (y, m α 1 B σn k 1 (z, m r ) t r) [t m r, t + m r ] Θ : B k (r) [ r, r ] B n k 1 (r ) U (r, r, r ) (y, s, z) (Θ y (y), t(s), Θz (z)),

22 SURGERY AND THE SPINORIAL τ-inariant 1 where t(s) := t + m s. By construction Θ is a diffeomorphism. As is readily seen Θ (m 4 g ) = (Θ y ) (m 4 α h t t ) + ds + (Θ z ) (m 4 σ n k 1 ). (39) By construction of α t one can verify that lim α t 0 1 α t C 1 ([t m r,t +m r ]) for all R > 0 since df dt and d f dt lim h t h t C 1 t (B h (y,m 0 = 0 are uniformly bounded. Moreover it is clear that = 0 α 1 t R)) uniformly in t [t m r, t + m r ]. As a consequence ( ( lim (Θy ) m 4 α h ) ) t t α ht C t = 0 1 (B k (r)) 0 uniformly in t. This implies that the sequence (Θ y ) (m 4 α h t t ) tends to the Euclidean metric ξ k in C 1 (B k (r)) uniformly in t as 0. From (39) we know that the sequence (Θ z ) (m 4 σ n k 1 ) tends to the Euclidean metric ξ n k 1 on B n k 1 (r ) as 0. Returning to (39) we obtain that the sequence Θ (m 4 g ) tends to ξ n = ξ k +ds +ξ n k 1 on B k (r) [ r, r ] B n k 1 (r ). As in Subcase I.1 we apply Lemma 3.4 to get a spinor ψ of class C 1 on R n which satisfies D ξn ψ = λ ψ ψ with n ψ B n (r) dx 1 for all r R +. Lemma 3.4 tells us that ψ(0) = 1 so ψ does not vanish identically. Just as in Subcase I.1 we deduce that λ λ + min (Sn, σ n ) λ. This ends the proof of Theorem 1. in Case I. Case II. There exists a constant C 1 such that m C 1 for all. As in Case I we consider two subcases. Subcase II.1. Assume that for some number a > 0. lim inf 0 N\U N (a) ψ n dv g > 0 (40) Let K a compact subset such that K M 1 M \ W. We choose a small number b such that K M 1 M \ U(b) = N \ U N (b). Let χ C (M 1 M ), 0 χ 1, be a cut-off function equal to 1 on M 1 M \ U(b) and equal to 0 on U(b). We set ψ := F (βg g ) 1 ψ. Since g = F g on the support of χ we have D g ψ = λ ψ ψ

23 BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT on this set. For some r > 0 we have D g (χψ ) r dv g M 1 M = M 1 M r ( C. grad g χ ψ + χλ ψ ψ r dv g grad g χ r ψ r dv g + λ r M 1 M χ r ψ (n+1)r dv g ) since m C 1. Together with Relation (8) we get that the sequence (χψ ) is bounded in H1 r(m 1 M ) for all r > 0. Proceeding as in the proof of Lemma 3.4 we get a C 1 spinor ψ 0 defined on K such that a subsequence of (ψ ) converges to ψ 0 in C 0 (K) and which satisfies K D g ψ 0 = λ ψ 0 ψ0. (41) Furthermore the convergence in C 0 implies that ψ 0 n dv g lim inf ψ n dv g = lim inf 0 0 K K ψ n dv g 1. Repeating the same for a sequence of compact sets K m which exhausts M 1 M \ W and taking a diagonal subsequence we can extend ψ 0 to M 1 M \ W. Since ψ 0 L n (M1 M \ W ) = L n (M1 M ) we can use Theorem 3. to extend ψ 0 to a weak solution of Equation (41). Note here that since D g is invertible we have λ > 0. By standard regularity theorems we conclude that ψ 0 C 1 (M 1 M ). By assumption (40) we have M 1 M \U(a) ψ 0 n dv g = lim 0 = lim 0 > 0. M 1 M \U(a) M 1 M \U(a) ψ n dv g ψ n dv g We conclude that ψ 0 0. If J denotes the functional associated to g then Equation (41) leads to ( ) n+1 λ J(ψ 0 ) = λ ψ 0 n n 1 dv g λ, M 1 M which proves Theorem 1. in this case. Subcase II.. We have for all a > 0. lim inf 0 N\U N (a) ψ n dv g = 0 (4) This case is the most difficult one and we proceed in several steps. The assumption here is that we have a sequence ( i ) which tends to zero as i with the property that the integral above tends to zero for all a > 0. We will abuse notation and write lim 0 for what should be a limit as i or a limit of a subsequence of it.

24 SURGERY AND THE SPINORIAL τ-inariant 3 For positive a and let γ (a) := N\U N (a) ψ dv g U N (a) ψ dv g The first step is to establish an estimate we will need later. Step 1. For a > 0 we have ( ) 1 C 0 γ (a) + ψ 4 L (U N (a)) (43) where C 0 > 0 is a constant which does not depend on a. Let χ C (N), 0 χ 1 be a cut-off function with χ = 1 on U N (a) and χ = 0 on N \ U N (a) = M 1 M \ U(a). Since the definitions of U N (a) and U(a) use the distance to W for the metric g we can and do assume that dχ g a. In the metric g this gives dχ g = F 1 dχ g = r dχ g a a = 4. From Proposition 3.5 and Equation (37) it follows that (n 1 k) N Dg (χψ ) dv g 4 N χψ dv g N = dχ g ψ dv g + λ N χ ψ (n+1) dv g N χψ dv g 16 U N (a)\u N (a) ψ dv g + λ ψ 4 N χψ dv g 16 U N (a)\u N (a) ψ dv g U N (a) ψ dv g 16γ (a) + λ ψ 4 L (U N (a)). L (U N (a)) + λ ψ 4 L (U N (a)) N χψ dv g Using that λ λ + min (Sn, σ n ) by Proposition.3 we obtain Relation (43) with C 0 := This ends the proof of Step 1. 4 (n 1 k) max { 16, λ + min (Sn ) }. Step. There exist a sequence of positive numbers (a ) which tends to 0 with and constants 0 < m < M such that for all. By (4) we have for all a m ψ L (U N (a )) M (44) lim 0 N\U N (a) ψ n dv g = 0

25 4 BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT for all a > 0. Since ol ( ) N \ U N (a), g does not depend on if < a it follows that ( ) n lim ψ n dv g ol(n \ U N (a), g ) 1 n = 0 0 N\U N (a) for any a. Hence we can take a sequence (a ) which tends sufficiently slowly to 0 so that ( ) n lim ψ n dv g ol(n \ U N (a ), g ) 1 n = 0. (45) 0 N\U N (a ) Using the Hölder inequality we get N\U γ (a ) = N (a ) ψ dv g U N (a ) ψ dv g ( N\U N (a ) ψ n dv g ) n ol(n \ U N (a ), g ) 1 n. ψ L (U N (a )) U N (a ) ψ n dv g The numerator of this expression tends to 0 by Relation (45). Further by (45) we have lim ψ n dv g = lim ψ n dv g ψ n dv g 0 U N (a ) 0 N N\U N (a ) = 1. Together with the fact that ψ L (U N (a )) m C 1 we obtain that lim γ (a ) = 0. 0 From Relation (43) applied with a = a we know that ψ L (U N (a )) is bounded from below. Moreover we have by the assumption of Case II that ψ L (U N (a )) m C 1. This finishes the proof of Step. Step 3. We have λ Λ n,k. Let x be a point in the closure of U N (a ) such that ψ (x ) = ψ L (U N (a )). As in Subcase I. we write x = (y, t, z ) where y W, t ( ln R 0 + ln ε, ln ε + ln R 0 ), and z S n k 1. By restricting to a subsequence we can t assume that y, A, and z converge respectively to y W, T [, + ], and z S n k 1. We apply Lemma 3.3 with = W, α =, p α = y, p = y, γ α = h t, γ 0 = h T, and b α = α t (recall that h t and α t were defined in (33) and (34)) and conclude that there is a diffeomorphism Θ y : Bk (r) B h t (y, α 1 t r) for r > 0 such that (Θ y ) (α t ht ) converges to the Euclidean metric ξ k on B k (r). For r, r > 0 we define U (r, r ) := B h t (y, α 1 t r) [t r, t + r ] S n k 1 and Θ : B k (r) [ r, r ] S n k 1 U (r, r ) (y, s, z) (Θ y (y), t(s), z),

26 SURGERY AND THE SPINORIAL τ-inariant 5 where t(s) := t + s. By construction Θ is a diffeomorphism. Since g = α t h t + dt + σ n k 1 we see that Θ (g ) = α t α t (Θ y ) (α t ht ) + ds + σ n k 1. (46) We will now find the limit of Θ (g ) in the C 1 topology. We define c := lim 0 f (t ). Lemma 4.1. The sequence of metrics Θ (g ) tends to η k+1 c + σ n k 1 = e cs ξ k + ds + σ n k 1 in C 1 on B k (r) [ r, r ] S n k 1 for fixed r, r > 0. Proof. Recall that α t = e f(t). The intermediate value theorem says that f(t) f(t ) ( d dt f)(t )(t t ) r max d ξ [t r,t +r ] dt f(ξ) for all t [t r, t + r ]. On the other hand we claimed f (t) 0 for t, hence f(t) f(t ) ( d dt f)(t )(t t ) 0 C0 ([t r,t +r ]) for 0 (and r fixed). Furthermore d ( f(t) f(t ) ( d ) dt dt f)(t d )(t t ) = dt f(t) d dt f(t ) t d = t dt f(s) ds r 0 max d ξ [t r,t +r ] as 0. Together with c = lim 0 f (t ) we have f(t) f(t ) c(t t ) 0 C 1 ([t r,t +r ]) Exponentiation of functions [t r, t + r ] R is a continuous map C 1 ([t r, t + r ]) C 1 ([t r, t + r ]), dt f(ξ) f exp f. Hence α t e c(t t ) α t = e f(t) f(t) e c(t t ) 0 C 1 ([t r,t +r ]) C 1 ([t r,t +r ]) We now write α h t t = α t ( h t h t ) + α t α α ht t t. Using the fact that lim h t h C = 0 t 0 1 (B ht (y,α 1 t R)) uniformly for t [t r, t r ] we get that the sequence α t (Θ y α ) (α t t ht ) tends to e cs ξ k in C 1 on B k (r)). Going back to Relation (46), this proves Lemma 4.1. We continue with the proof of Step 3. As in subcases I.1 and I. we apply Lemma 3.4 with (, g) = (R k+1 S n k 1, ηc k+1 + σ n k 1 ), α =, and g α = Θ (g ) (we can apply this lemma since any compact subset of R k+1 S n k 1 is contained

27 6 BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT in some B k (r) [ r, r ] S n k 1 ). We obtain a C 1 spinor ψ which is a solution of D ηk+1 c +σ n k 1 ψ = λ ψ ψ on (R k+1 S n k 1, ηc k+1 + σ n k 1 ). From (16) it follows that ψ n dv η k+1 c +σ n k 1 1. R k+1 S n k 1 From (15) it follows that ψ L (R k+1 S n k 1 ) and from (15) and (44) it follows that ψ is non-zero. We want to show that ψ L (R k+1 S n k 1 ). From (16) we get that ψ dv η k+1 c +σ n k 1 = lim ψ dv g B k (r) [ r,r ] S n k 1 0 U (r,r ) (47) lim ψ dv g 0 U N (a) for some fixed number a > 0 independent of r, r and. Let χ be defined as in Step 1. Using the Hölder inequality, Proposition 3.5, and Equation (37) we see that (n 1 k) 4 N Dg (χψ ) dv g N χψ dv g N = dχ g ψ dv g + λ N χ ψ (n+1) dv g N χψ dv g 16 U N (a)\u N (a) ψ dv g + λ ψ L (U N (a)) U N (a) ψ n dv g U N (a) ψ dv g. We have λ ψ L (U N (a)) ψ n dv g λ + U N (a) min (Sn, σ n ) C 1 and U N (a)\u N (a) ( ψ dv g ψ n U N (a)\u N (a) ) n dv g ol ( U N (a) \ U N (a), g ) 1 n. ol ( U N (a) \ U N (a), g ) 1 n Since g does not depend on on U N (a) \ U N (a) for < a, we get the existence of a constant C such that (n 1 k) C 4 U N (a) ψ. dv g Together with (47) we obtain that k+1 ψ dv η c +σ n k 1 C B k (r) [ r,r ] S n k 1