SURGERY AND THE SPINORIAL τ-INVARIANT

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, when the metrics are normalized to unit volume. This invariant is a spinorial analogue of Schoen’s σ-constant, also known as the smooth Yamabe invariant.

We prove that if N is obtained from M by surgery of codimension at least 2 then τ(N) ≥ min{τ(M), Λ_{n}}, where Λ_{n} is a positive constant depending only on n = dim M. Various topological conclusions can be drawn, in particular that τ is a spin-bordism invariant below Λ_{n}. Also, below Λ_{n} the values of τ cannot accumulate from above when varied over all manifolds of dimension n.

Contents

1. Introduction 2 1.1. Spin manifolds and Dirac operators 2 1.2. The τ-invariant 2 1.3. The σ-constant 3 1.4. Geometric constants 3 1.5. Joining manifolds 4 1.6. Surgery and bordism 4 1.7. Variants of the results 6 2. Preliminaries 6 2.1. Notation for balls and neighbourhoods 6 2.2. Joining manifolds along submanifolds 7 2.3. Comparing spinors for different metrics 8 2.4. Regularity results 9 2.5. The associated variational problem 9 3. Preparations for proofs 10 3.1. Removal of singularities 10 3.2. Limit spaces and limit solutions 11 3.3. Dirac spectral bounds on products with spheres 13 3.4. Approximation by local product metrics 13 4. Proofs 17 4.1. Proof of Theorem 1.2 17 4.2. Proof of Theorem 1.1 28

Date: August 26, 2008. 2000 Mathematics Subject Classification. 53C27 (Primary) 55N22, 57R65 (Secondary). Key words and phrases. Dirac operator, eigenvalue, surgery. We thank Victor Nistor for helpful discussions about Sobolev inequalities on non-compact

manifolds. Also Mattias Dahl thanks the Institut Élie Cartan, Nancy, for its kind hospitality and support during visits when this work was initiated.

References 28

1. Introduction

1.1. Spin manifolds and Dirac operators. 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.

As explained in [19, 10, 14] one associates the spinor bundle Σ_{ρ}^{g}M to the spin structure, together with a Riemannian metric g on M and a complex irreducible representation ρ of the Clifford algebra over ℝ^{n}. The Dirac operator D_{ρ}^{g} is a selfadjoint 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 spectrum 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. In odd dimensions there are two choices ρ^{+} and ρ^{−}. Exchanging the representation results in reversing the spectrum, that is if λ is an eigenvalue of D_{ρ^{+}}^{g} then −λ is an eigenvalue of D_{ρ^{−}}^{g} with the same multiplicity, and vice versa. This has no effect if n ≡ 1 mod 4 since the real/quaternionic structure on Σ_{ρ}^{g}M anticommutes with the Dirac operator and the spectrum therefore is symmetric, see [10, Section 1.7]. However, in dimensions n ≡ 3 mod 4 the choice of ρ matters. In this case we choose the representation such that Clifford multiplication of e_{1} · e_{2} ⋯ 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.

1.2. The τ-invariant. We denote by λ_{1}^{+}(Dg̃) the first non-negative eigenvalue of Dg̃. For a metric g on M we define

λ_{min}^{+}(M, g) := inf λ_{1}^{+}(Dg̃)Vol(M, g̃)^{1/n},

where the infimum is taken over all metrics g̃ 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 unit sphere with its standard metric. We have

λ_{min}^{+}(S^{n}, σ^{n}) = \frac{n}{2}ω_{n}^{1/n},

where ω_{n} is the volume of (S^{n}, σ^{n}). Moreover it is shown in [2, 6] that

λ_{min}^{+}(M, g) ≤ λ_{min}^{+}(S^{n}, σ^{n})

for any compact Riemannian spin manifold (M, g). Together with Inequality (1) below we get

τ^{+}(S^{n}) = λ_{min}^{+}(S^{n}, σ^{n}) = \frac{n}{2}ω_{n}^{1/n},

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 [2] that if the kernel of D^{g} is trivial, that is if D^{g} is invertible, then λ_{min}^{+}(M, g) > 0. It follows that τ^{+}(M) > 0 if and only if there is 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].

For compact Riemannian spin manifolds (M_{1}, g_{1}) and (M_{2}, g_{2}) we denote by M_{1} ⨿ M_{2} the disjoint union of M_{1} and M_{2} with the natural metric g_{1} ⨿ g_{2}. It is not difficult to see that

λ_{min}^{+}(M_{1} ⨿ M_{2}, g_{1} ⨿ g_{2}) = min{λ_{min}^{+}(M_{1}, g_{1}), λ_{min}^{+}(M_{2}, g_{2})}.

This implies

τ^{+}(M_{1} ⨿ M_{2}) = min{τ^{+}(M_{1}), τ^{+}(M_{2})}.

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. If M has dimension ≢ 3 mod 4 this does not change the first positive eigenvalue of D since the spectrum is symmetric, so we then have λ_{min}^{+}(−M, g) = λ_{min}^{+}(M, g) and τ^{+}(−M) = τ^{+}(M). For manifolds M of dimension ≡ 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).

1.3. The σ-constant. The τ-invariant is a spinorial analogue of the σ-constant [17, 21] which is defined for a compact manifold M by

σ(M) := sup inf \frac{∫ Scalg̃ dvg̃}{Vol(M, g̃)^{\frac{n−2}{n}}} ,

where the infimum runs over all metrics g̃ in a conformal class and the supremum runs over all conformal classes. σ(M) is also known as the smooth Yamabe invariant of M. When σ(M) is positive it can be computed in a way analogous to τ^{+}(M) using the lowest eigenvalue of the conformal Laplacian L^{g} = 4\frac{n−1}{n−2}Δ^{g}+Scal^{g} instead of λ_{1}^{+}(D^{g}). Hijazi’s inequality [12, 13] gives a comparison of the two invariants,

τ^{±}(M)^{2} ≥ \frac{n}{4(n − 1)}σ(M).  (1)

For M = S^{n} equality is attained in (1). Upper bounds for τ^{±}(M) may help to determine the σ-constant.

Surgery formulas for the σ-constant analogous to those obtained in this paper have been proved in [5].

1.4. Geometric constants. We are going to prove a surgery formula for the invariant τ^{+}. This formula involves geometric constants Λ_{n,k} which we now define.

For a complete spin manifold (V, g) we set

λ̃_{min}^{+}(V, g) := inf λ ∈ [0, ∞],

where the infimum is taken over all λ ∈ (0, ∞) for which there is a non-zero spinor field φ ∈ L^{∞}(V) ∩ L^{2}(V) ∩ C_{loc}^{1}(V) such that ∥φ∥_{L^{\frac{2n}{n−1}} (V)} ≤ 1, and

D^{g}φ = λ|φ|^{\frac{2}{n−1}} φ.  (2)

If there are no such solutions of (2) on V then λ̃_{min}^{+}(V, g) = ∞.

For a positive integer k we let ξ^{k} denote the Euclidean metric on ℝ^{k}. For c ∈ ℝ we denote by η_{c}^{k+1} := e^{2ct}ξ^{k} + dt^{2} the hyperbolic metric of sectional curvature −c^{2} on ℝ^{k+1}. As above σ^{n−k−1} denotes the metric of sectional curvature 1 on S^{n−k−1}. We define the product metric

G_{c} := η_{c}^{k+1} + σ^{n−k−1}

on ℝ^{k+1} × S^{n−k−1}, and we define our geometric invariants as

Λ_{n,k} := inf_{c∈[−1,1]} λ̃_{min}^{+}(ℝ^{k+1} × S^{n−k−1}, G_{c}),

and

Λ_{n} := min_{0≤k≤n−2} Λ_{n,k}.

Note that the infimum could as well be taken over c ∈ [0, 1] since G_{c} and G_{−c} are isometric. It is easy to see that Λ_{n,0} = λ_{min}^{+}(S^{n}, 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 − 2 we have Λ_{n,k} > 0.

1.5. 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} and M_{2} be spin manifolds of dimension n and let N be obtained by joining M_{1} and M_{2} along a common submanifold as described in Section 2.2. The manifold N is spin and from the construction there is a natural choice of spin structure on N. The following results make it possible to compare τ^{+}(M_{1} ⨿ M_{2}) and τ^{+}(N).

Theorem 1.2. Let (M_{1}, g_{1}) and (M_{2}, g_{2}) be compact Riemannian spin manifolds of dimension n for which both D^{g_{1}} and D^{g_{2}} have trivial kernel. Let W be a compact spin manifold of dimension k embedded into M_{1} and M_{2} with trivializations of the corresponding normal bundles given. Assume that 0 ≤ k ≤ n − 2, and let N be obtained by joining M_{1} and M_{2} along W. Then there is a family of metrics g_{θ}, θ ∈ (0, θ_{0}) on N satisfying

min{λ_{min}^{+}(M_{1} ⨿ M_{2}, g_{1} ⨿ g_{2}), Λ_{n,k}} ≤ lim inf_{θ→0} λ_{min}^{+}(N, g_{θ}))
≤ lim sup_{θ→0} λ_{min}^{+}(N, g_{θ})
≤ λ_{min}^{+}(M_{1} ⨿ M_{2}, g_{1} ⨿ g_{2}).

Taking the supremum over all metrics on M_{1} ⨿ M_{2} the first inequality gives us the following corollary.

Corollary 1.3. In the situation of Theorem 1.2 we have

τ^{+}(N) ≥ min{τ^{+}(M_{1} ⨿ M_{2}), Λ_{n,k}} ≥ min{τ^{+}(M_{1}), τ^{+}(M_{2}), Λ_{n}}.

Note that these estimates on τ^{+} would be trivial without Theorem 1.1.

1.6. Surgery and bordism. Performing surgery on a spin manifold is a special case of joining manifolds, this is discussed in more detail in Section 2.2. From 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

\overline{τ}^{+}(M) := min\{τ^{+}(M), Λ_{n}\}.

We also define

\overline{τ}(M) := min\{τ^{+}(M), τ^{−}(M), Λ_{n}\}.

If n ≢ 3 mod 4 then \overline{τ}(M) = \overline{τ}^{+}(M). As noted before, all results for \overline{τ}^{+}(M) also hold for \overline{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 ≥ 2. Then

τ^{+}(N) ≥ min{τ^{+}(M), Λ_{n,k}} ≥ min{τ^{+}(M), Λ_{n}}.

Corollary 1.4 tells us that

\overline{τ}^{+}(N) ≥ \overline{τ}^{+}(M), \overline{τ}(N) ≥ \overline{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, VII Theorem 3] for dimension 3, and [18, VIII Proposition 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 \overline{τ}(N) ≥ \overline{τ}(M). In particular, if M is also connected we have \overline{τ}(N) = \overline{τ}(M).

Corollary 1.5 can also be shown in dimension 2 with similar arguments [7, Theorem 1.3].

The spin bordism group Ω_{n}^{spin} is the set of equivalence classes of spin bordant manifolds of dimension n with disjoint union as addition. Since every element in Ω_{n}^{spin} can be represented by a connected manifold we obtain a well-defined map \overline{τ} : Ω_{n}^{spin} → [0, Λ_{n}] which sends the equivalence class [M] of a connected spin manifold M to \overline{τ}(M).

Corollary 1.6. There is a positive constant ε_{n} such that

τ^{+}(M) ∈ {0} ∪ [ε_{n}, λ_{min}^{+}(S^{n}, σ^{n})].

for all spin manifolds M of dimension n.

Proof. The spin bordism group Ω_{n}^{spin} is finitely generated [22, page 336]. This implies that the kernel of the map α : Ω_{n}^{spin} → KO_{n} is also finitely generated. Let [N_{1}], …, [N_{r}] be generators of this kernel, we assume that the manifolds N_{i} are all connected. Since τ(M) = 0 if and only if α(M) ≠ 0 we obtain the corollary for

ε_{n} := min\{Λ_{n}, \overline{τ}(N_{1}), …, \overline{τ}(N_{r})\}.

□

The α-map is injective when n < 8, and then ε_{n} = Λ_{n}. 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 observations might be interesting.

First, if M is a spin manifold with τ^{+}(M) < Λ_{n}, then it follows from Corollary 1.5 that the σ-constant of any manifold N spin bordant to M satisfies

σ(N) ≤ \frac{4(n − 1)}{n}τ^{+}(M)^{2}.

For the next observation we define

S(t) := \{[M] ∈ Ω_{n}^{spin} | \overline{τ}(M) ≥ t\}, S^{+}(t) := \{[M] ∈ Ω_{n}^{spin} | \overline{τ}^{+}(M) ≥ t\},

and

T (t) := \{[M] ∈ Ω_{n}^{spin} | \overline{τ}(M) > t\}, T^{+}(t) := \{[M] ∈ Ω_{n}^{spin} | \overline{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_{n}^{spin} for t ∈ [0, Λ_{n}] and T (t) is a subgroup of Ω_{n}^{spin} for t ∈ [0, Λ_{n}). If n ≡ 3 mod 4, then S^{+}(t) and T^{+}(t) are submonoids.

Corollary 1.8. The values of \overline{τ} cannot accumulate from above.

Proof. Assume that t_{i} := \overline{τ}(M_{i}), i ∈ ℕ, is a decreasing sequence of values of \overline{τ} which converges to a limit t_{∞}. We want to show that t_{i} = t_{∞} for all but finitely many i.

We have S(t_{i}) ⊂ S(t_{i+1}), and hence ⋃_{i} S(t_{i}) = T (t_{∞}) is a subgroup of the finitely generated group Ω_{n}^{spin}. It is thus finitely generated itself and we choose a finite set of generators. There must then be an I ∈ ℕ such that S(t_{I}) contains this finite set, and thus S(t_{I}) = T (t_{∞}). Hence [M_{i}] ∈ S(t_{I}) for all i, which implies t_{i} ≥ t_{I}. We conclude that t_{i} = t_{I} = t_{∞} for i ≥ I. □

We do not know whether \overline{τ}^{+} can accumulate from above in dimensions n ≡ 3 mod 4.

1.7. Variants 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 the kernel of D^{g} is non-trivial for any metric g on M, and hence τ^{+}(M) = 0. For a connected spin manifold M the index theorem implies that the kernel of the Dirac operator has at least dimension

a(M) := \begin{cases}|Â(M)|, & if n ≡ 0 mod 4; \\ 1, & if n ≡ 1 mod 8 and α(M) ≠ 0; \\ 2, & if n ≡ 2 mod 8 and α(M) ≠ 0; \\ 0, & otherwise. \end{cases}

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 [2] 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}^{+}.

2. Preliminaries

2.1. 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 Riemannian manifold (M, g) and a subset S ⊂ M we let U^{g}(S, r) := ⋃_{x∈S} B^{g}(x, r) denote the r-neighbourhood of S. Again, if the Riemannian metric is clear from the context we abbreviate to U(S, r).

2.2. Joining manifolds along submanifolds. We are now going to describe how two manifolds are joined along a common submanifold with trivialized normal bundle. Strictly speaking this is a differential topological construction, but since we work with Riemannian manifolds we will make the construction adapted to the Riemannian metrics and use distance neighbourhoods defined by the metrics etc.

Let (M_{1}, g_{1}) and (M_{2}, g_{2}) be complete Riemannian manifolds of dimension n. Let W be a compact manifold of dimension k, where 0 ≤ k ≤ n. We assume that W is embedded in both M_{1} and M_{2} with trivializations of the normal bundle, we desribe these embeddings as follows.

Let w̄_{i} : W × ℝ^{n−k} → T M_{i}, i = 1, 2, be smooth embeddings. We assume that w̄_{i} restricted to W × {0} maps to the zero section of T M_{i} (which we identify with M_{i}) and thus gives an embedding W → M_{i}. The image of this embedding is denoted by W′_{i}. Further we assume that w̄_{i} restrict to linear isomorphisms {p} × ℝ^{n−k} → N_{w̄_{i}(p,0)}W′_{i} for all p ∈ W_{i}, where NW′_{i} denotes the normal bundle of W′_{i} defined using g_{i}.

We set w_{i} := exp^{g_{i}} ∘w̄_{i}. For i = 1, 2 this gives embeddings w_{i} : W ×B^{n−k}(R_{max}) → M_{i} for some R_{max} > 0. We have W′_{i} = w_{i}(W × {0}) and we define the disjoint union

(M, g) := (M_{1} ⨿ M_{2}, g_{1} ⨿ g_{2}),

and

W′ := W′_{1} ⨿ W′_{2}.

Let r_{i} be the function on M_{i} giving the distance to W′_{i}. Then r_{1} ∘ w_{1}(w, x) = r_{2}∘w_{2}(w, x) = |x| for w ∈ W, x ∈ B^{n−k}(R_{max}). Let r be the function on M defined by r(x) := r_{i}(x) for x ∈ M_{i}, i = 1, 2. For 0 < ε we set U_{i}(ε) := {x ∈ M_{i} : r_{i}(x) < ε} and U(ε) := U_{1}(ε) ∪ U_{2}(ε). For 0 < ε < θ we define

N_{ε} := (M_{1} \ U_{1}(ε)) ∪ (M_{2} \ U_{2}(ε))/∼,

and

U_{ε}^{N}(θ) := (U(θ) \ U(ε))/∼

where ∼ indicates that we identify x ∈ ∂U_{1}(ε) with w_{2} ∘ w_{1}^{−1}(x) ∈ ∂U_{2}(ε). Hence

N_{ε} = (M \ U(θ)) ∪ U_{ε}^{N}(θ).

We say that N_{ε} is obtained from M_{1}, M_{2} (and w̄_{1}, w̄_{2}) by a connected sum along W with parameter ε.

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_{2} → [0, ∞) also defines a continuous function r : N_{ε} → [ε, ∞) whose definition depends on ε. We will also keep the ε-subscript for U_{ε}^{N}(θ) as important estimates for spinors will be carried out on U_{ε}^{N}(θ). As the embeddings w_{1} and w_{2} 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_{2} \ w_{2}(W × B^{n−k})) coincides with the restriction of the given spin structure on M_{1} ⨿ M_{2}. If W is not connected, then this choice is not unique. The statements of our theorem hold for any such spin structure on N.

The surgery operation on a manifold is a special case of taking connected sum along a submanifold. Indeed, let M be a compact manifold of dimension n and let M_{1} = M, M_{2} = S^{n}, W = S^{k}. Let w_{1} : S^{k} × B^{n−k} → M be an embedding defining a surgery and let w_{2} : S^{k} × B^{n−k} → S^{n} be the standard embedding. Since S^{n} \ w_{2}(S^{k} × B^{n−k}) is diffeomorphic to B^{k+1} × S^{n−k−1} we have in this situation that N is obtained from M using surgery on w_{1}, see [18, Section VI.9].

2.3. 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, b_{g′}^{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 then the map b_{g′}^{g} lifts to a map β_{g′}^{g} of Spin(n)-principal bundles,

[figure: Spin(M, g); Spin(M, g′); SO(M, g); SO(M, g′); β_{g′}^{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

β_{g′}^{g} : Σ^{g}M = Spin(M, g) ×_{ρ} Σ_{n} → Spin(M, g′) ×_{ρ} Σ_{n} = Σ^{g′}M,
ψ = [s, φ] ↦ [β_{g′}^{g}s, φ] = β_{g′}^{g}ψ,

where (ρ, Σ_{n}) is the complex spinor representation, and where [s, φ] ∈ Spin(M, g)×_{ρ} Σ_{n} denotes the equivalence class of (s, φ) ∈ Spin(M, g) × Σ_{n} for the equivalence relation given by the action of Spin(n). The map β_{g′}^{g} of Hermitian vector bundles is fiberwise an isometry.

We define the Dirac operator ^{g}D^{g′} acting on sections of the spinor bundle for g by

gD^{g′} := (β_{g′}^{g})^{−1} ∘ D^{g′} ∘ β_{g′}^{g}.

In [9, Theorem 20] 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

gD^{g′}ψ = D^{g}ψ + A_{g′}^{g}(∇^{g}ψ) + B_{g′}^{g}(ψ),  (3)

where A_{g′}^{g} ∈ hom(T^{∗}M ⊗ Σ^{g}M, Σ^{g}M) satisfies

|A_{g′}^{g}| ≤ C|g − g′|_{g},  (4)

and B_{g′}^{g} ∈ hom(Σ^{g}M, Σ^{g}M) satisfies

|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^{2}g for a positive smooth function F the formula simplifies considerably, and one obtains

gD^{g′}(F^{− \frac{n−1}{2}} ψ) = F^{− \frac{n+1}{2}} D^{g}ψ,  (6)

see for instance [15, 8].

2.4. Regularity results. By standard elliptic theory we have the following lemma (see for example [3, Chapter 3] where the corresponding results of [11] are adapted to the Dirac operator).

Lemma 2.1. Let (V, g) be a Riemannian spin manifold and Ω ⊂ V an open set with compact closure in V. Let also r ∈ (1, ∞). Then there is a constant C so that

∫_{Ω} |∇^{g}φ|^{r} dv^{g} ≤ C (∫_{Ω} |D^{g}φ|^{r} dv^{g} + ∫_{Ω} |φ|^{r} dv^{g})  (7)

for all φ ∈ Γ(Σ^{g}Ω) which are of class C^{1} and compactly supported in Ω.

For a compact Riemannian manifold with invertible Dirac operator we have the following special case.

Lemma 2.2. Let (V, g) be a compact Riemannian spin manifold such that D^{g} is invertible. Then there exists a constant C such that

∫_{V} |∇^{g}φ|^{\frac{2n}{n+1}} dv^{g} ≤ C ∫_{V} |D^{g}φ|^{\frac{2n}{n+1}} dv^{g}  (8)

for all φ ∈ Γ(Σ^{g}V) of class C^{1}.

2.5. The associated variational problem. Let (M, g) be a compact spin manifold of dimension n with ker D^{g} = {0}. We define the functional J^{g} acting on smooth spinor fields ψ ∈ Γ(Σ^{g}M) by

J^{g}(ψ) := \frac{(∫_{M} |Dψ|^{\frac{2n}{n+1}} dv^{g})^{\frac{n+1}{n}}}{∫_{M}⟨Dψ, ψ⟩ dv^{g}} ,

whenever the denominator is non-zero. Using techniques from [20] it was proved in [2] that

λ_{min}^{+}(M, g) = inf_{ψ} J^{g}(ψ),  (9)

where the infimum is taken over the set of smooth spinor fields satisfying

∫_{M}⟨Dψ, ψ⟩ dv^{g} > 0.

If g and g̃ = F^{2}g are conformal metrics on M and if J^{g} and J g̃ are the associated functionals, then by Relation (6) one computes that

J g̃(F^{− \frac{n−1}{2}} ψ) = J^{g}(ψ)  (10)

for smooth ψ ∈ Γ(Σ^{g}M)).

The following result gives a universal upper bound on λ_{min}^{+}(M, g).

Proposition 2.3. Let (M, g) be a compact spin manifolds of dimension n ≥ 2. Then

λ_{min}^{+}(M, g) ≤ λ_{min}^{+}(S^{n}, σ^{n}) = \frac{n}{2} ω_{n}^{1/n},  (11)

where ω_{n} is the volume of (S^{n}, σ^{n}).

Proposition 2.3 was proven for n ≥ 3 in [2] using geometric methods. In the case n = 2 the article [2] 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^{g}(ψ_{ε}) ≤ λ_{min}^{+}(S^{n}, 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 1.2.

Theorem 2.4 ([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^{2,α}(ΣM)∩ C^{∞}(ΣM \ φ^{−1}(0)) where α ∈ (0, 1) ∩ (0, 2/(n − 1)] such that ∥φ∥_{L^{\frac{2n}{n−1}} (M)} = 1 and

Dφ = λ_{min}^{+}(M, g)|φ|^{\frac{2}{n−1}} φ.

Furthermore the infimum in the definition of λ_{min}^{+}(M, g) is attained by the generalized conformal metric g̃ = |φ|^{4/(n−1)}g, see [1] for details.

3. Preparations for proofs

3.1. Removal of singularities. The following theorem gives a condition for when singularities of solutions to Dirac equations can be removed.

Theorem 3.1. Let (V, g) be a (not necessarily complete) Riemannian spin manifold and let S be a compact submanifold of V of codimension m ≥ 2. Assume that φ ∈ L^{p}(Σ(V \ S)), p ≥ m/(m − 1), satisfies the equation

Dφ = ρ

weakly on V \S where ρ ∈ L^{1}(Σ(V \S)) = L^{1}(ΣV). Then this equation holds weakly on V. 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

∫_{V} ⟨φ, Dψ⟩ dv = ∫_{V} ⟨ρ, ψ⟩ dv.  (12)

Recall that for ε > 0 we denote the set of points in V of distance less than ε to S by U(S, ε). We choose a smooth cut-off function χ_{ε} : V → [0, 1] with support in U(S, 2ε), χ_{ε} = 1 on U(S, ε), and |gradχ_{ε}| ≤ 2/ε. We then have

∫_{V} ⟨φ, Dψ⟩ dv − ∫_{V} ⟨ρ, ψ⟩ dv = ∫_{V} ⟨φ, D((1 − χ_{ε})ψ + χ_{ε}ψ)⟩ dv − ∫_{V} ⟨ρ, ψ⟩ dv
= ∫_{V} ⟨Dφ, (1 − χ_{ε})ψ⟩ dv + ∫_{V} ⟨φ, χ_{ε}Dψ⟩ dv
+ ∫_{V} ⟨φ, gradχ_{ε} · ψ⟩ dv − ∫_{V} ⟨ρ, ψ⟩ dv
= − ∫_{V} ⟨ρ, χ_{ε}ψ⟩ dv + ∫_{V} ⟨φ, χ_{ε}Dψ⟩ dv
+ ∫_{V} ⟨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 follows that

|∫_{V} ⟨φ, Dψ⟩ dv − ∫_{V} ⟨ρ, ψ⟩ dv| ≤ ( sup_{U(S,2ε)} |ψ|) ∫_{U(S,2ε)} |ρ| dv

+ ( sup_{U(S,2ε)} |Dψ| + \frac{2}{ε} sup_{U(S,2ε)} |ψ|) ∫_{U(S,2ε)} |φ| dv

≤ o(1) + \frac{C}{ε} ∥φ∥_{L^{p}(U(S,2ε))}Vol(U(S, 2ε))^{1/q}

≤ o(1) + C∥φ∥_{L^{p}(U(S,2ε))}ε^{(m/q)−1},

where o(1) denotes a term tending to 0 as ε → 0. Since p ≥ m/(m−1) is equivalent to m/q ≥ 1 we see that (12) holds. □

Applying Theorem 3.1 to the non-linear Dirac equation in Theorem 2.4 we get the following corollary.

Corollary 3.2. Let V and S be as in Theorem 3.1. Then any L^{p}-solution, p = 2n/(n − 1), of

Dφ = λ|φ|^{p−2}φ  (13)

on V \ S is also a weak L^{p}-solution of (13) on V.

3.2. Limit spaces and limit solutions. In the proofs of the main theorems we will construct limit solutions of a Dirac equation on certain limit spaces. For this we need the following two lemmas. In the statement of these results, in order to simplify the notations, we write α → 0 instead of α_{i} → 0 as i → ∞ when (α_{i})_{i∈ℕ} is a sequence of positive numbers converging to 0. In the same way, the subsequences of (α_{i}) will also be denoted by (α).

Lemma 3.3. Let V be an n-dimensional manifold. Let (p_{α}) be a sequence of points in V 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^{2}(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^{γ_{α}}(p_{α}, b_{α}^{−1}r)

with Θ_{α}(0) = p_{α} such that the metric Θ_{α}^{∗}(b_{α}^{2}γ_{α}) tends to the Euclidean metric ξ^{n} in C^{1}(B^{n}(r)).

Proof. Denote by exp_{p_{α}}^{γ_{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 V and U_{α} is a neighbourhood of the origin in ℝ^{n}. We set

Θ_{α} : B^{n}(r) ∋ x ↦ exp_{p_{α}}^{γ_{α}}(b_{α}^{−1}x) ∈ B^{γ_{α}}(p_{α}, b_{α}^{−1}r).

It is easily checked that Θ_{α} is the desired diffeomorphism. □

Lemma 3.4. Let V 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 ⊂ V as α → 0. Assume that (U_{α}) is an increasing sequence of subdomains of V such that ∪_{α}U_{α} = V. Let ψ_{α} ∈ Γ(Σ^{g_{α}}U_{α}) be a sequence of spinors of class C^{1} such that ∥ψ_{α}∥_{L^{∞}(U_{α})} ≤ C where C does not depend on α, and

D^{g_{α}}ψ_{α} = λ_{α}|ψ_{α}|^{\frac{2}{n−1}} ψ_{α}  (14)

where the λ_{α} are positive numbers which tend to λ̄ ≥ 0. Then there exists a spinor ψ ∈ Γ(Σ^{g}V) of class C^{1} such that

D^{g}ψ = λ̄|ψ|^{\frac{2}{n−1}} ψ  (15)

on V and a subsequence of (β_{g}^{g_{α}}ψ_{α}) tends to ψ in C^{0}(K) for any compact set K ⊂ V. In particular

∥ψ∥_{L^{∞}(K)} = lim_{α→0} ∥ψ_{α}∥_{L^{∞}(K)},  (16)

and

∫_{K} |ψ|^{r} dv^{g} = lim_{α→0} ∫_{K} |ψ_{α}|^{r} dv^{g_{α}}  (17)

for any compact set K and any r ≥ 1.

Proof. Let K be a compact subset of V and let Ω be an open set in V with compact closure such that K ⊂ Ω. Let χ ∈ C^{∞}(V) with 0 ≤ χ ≤ 1 be compactly supported in Ω and satisfy χ = 1 on a neighbourhood Ω̃ of K. Set φ_{α} = (β_{g_{α}}^{g})^{−1}ψ_{α}. Using Equations (14) and (3) we get

D^{g}(χφ_{α}) = grad^{g}χ · φ_{α} + χλ_{α}|φ_{α}|^{\frac{2}{n−1}} φ_{α} − χA_{g_{α}}^{g}(∇^{g}φ_{α}) − χB_{g_{α}}^{g}(φ_{α}). (18)

Using the fact that |a + b + c|^{r} ≤ 3^{r}(|a|^{r} + |b|^{r} + |c|^{r}) for a, b, c ∈ ℝ, r ≥ 1, we see that

\begin{aligned}|D^{g}(χφ_{α})|^{r} & ≤ 3^{r}(|grad^{g}χ · φ_{α} + χλ_{α}|φ_{α}|^{\frac{2}{n−1}} φ_{α}|^{r} \\ & + |χA_{g_{α}}^{g}(∇^{g}φ_{α})|^{r} + |χB_{g_{α}}^{g}(φ_{α})|^{r}) \end{aligned}

for r ≥ 1. Since ∥φ_{α}∥_{L^{∞}(V)} = ∥ψ_{α}∥_{L^{∞}(V)} ≤ C we have

|grad^{g}χ · φ_{α} + χλ_{α}|φ_{α}|^{\frac{2}{n−1}} φ_{α}|^{r} ≤ C.

By Relations (4) and (5), and since lim_{α→0} ∥g_{α} − g∥_{C^{1}(Ω)} = 0, we get

\begin{aligned}|χA_{g_{α}}^{g}(∇^{g}φ_{α})|^{r} + |χB_{g_{α}}^{g}(φ_{α})|^{r} & ≤ o(1) (|∇^{g}(χφ_{α})|^{r} + |grad^{g}χ · φ_{α}|^{r} + |χφ_{α}|^{r}) \\ & ≤ o(1) (|∇^{g}(χφ_{α})|^{r} + C) , \end{aligned}

where o(1) tends to 0 with α. It follows that

|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

∫_{Ω} |∇^{g}(χφ_{α})|^{r} dv^{g} ≤ C + o(1) ∫_{Ω} |∇^{g}(χφ_{α})|^{r} dv^{g}.

In particular (χφ_{α}) is bounded in H_{0}^{1,r}(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^{g_{α}}Ω) of class C^{0,a}. We take the inner product of (18) with a smooth spinor φ̃ which is compactly supported in Ω̃ and integrate over Ω. Since χ = 1 on the support of φ̃ the result is

\begin{aligned}∫_{Ω}⟨φ_{α}, D^{g}φ̃⟩ dv^{g} & = ∫_{Ω} λ_{α}|φ_{α}|^{\frac{2}{n−1}} ⟨φ_{α}, φ̃⟩ dv^{g} \\ & − ∫_{Ω}⟨A_{g_{α}}^{g}(∇^{g}φ_{α}), φ̃⟩ dv^{g} − ∫_{Ω}⟨B_{g_{α}}^{g}(φ_{α}), φ̃⟩ dv^{g}. \end{aligned}

Taking the limit α → 0 and again using (4) and (5) we get

∫_{Ω}⟨ψ_{K}, D^{g}φ̃⟩ dv^{g} = ∫_{Ω} λ̄|ψ_{K}|^{\frac{2}{n−1}} ⟨ψ_{K}, φ̃⟩ dv^{g}.

Hence, ψ_{K} satisfies Equation (15) 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} = V. 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}|_{K_{m−1}} = ψ_{m−1}. We define ψ on V by ψ := ψ_{m} on K_{m}. By taking a diagonal subsequence of we get that (φ_{α}) tends to ψ in C^{0} on any compact set K ⊂ V.

The relations (16) and (17) follow immediately since β_{g_{α}}^{g} is an isometry, since φ_{α} = (β_{g_{α}}^{g})^{−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 3.4. □

3.3. 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 B^{2} to the boundary. The proof is a simple application of the formula for the squared Dirac operator on a product manifold together with the lower bound of its spectrum on the standard sphere.

Lemma 3.5. Let (V, g) be a complete Riemannian spin manifold. Then any L^{2}- spinor ψ on (V × S^{m}, g + σ^{m}) satisfies

∫_{V ×S^{m}} |Dψ|^{2} dv^{g+σ^{m}} ≥ \frac{m^{2}}{4} ∫_{V ×S^{m}} |ψ|^{2} dv^{g+σ^{m}}.

3.4. Approximation by local product metrics. In this paragraph we will see how to change the metrics g_{i} to product form g_{i} = h_{i} + dr_{i}^{2} + r_{i}^{2}σ^{n−k−1} in a neighbourhood of W′_{i} in M_{i} without changing λ_{min}^{+}(M_{i}, g_{i}) much.

Lemma 3.6. Let (V, g) be a compact Riemannian manifold of dimension n and let S be a closed submanifold of dimension k, where 0 ≤ k ≤ n − 2. Assume that a trivialization of the normal bundle of S is given and assume that D^{g} is invertible. Then there exists a sequence (ε_{i})_{i∈ℕ} of positive real numbers converging to 0 and a sequence (g_{ε_{i}}) of metrics on V such that

lim_{i→∞} λ_{min}^{+}(V, g_{ε_{i}}) = λ_{min}^{+}(V, g)

and

g_{ε_{i}} = h + dr^{2} + r^{2}σ^{n−k−1}

on U^{g}(S, ε_{i}). Here h is the restriction of the metric g to S and r(x) = d^{g}(S, x).

Proof. Using the trivialization of the normal bundle we identify a neighbourhood of S with S × B^{n−k}(R_{max}) as described in Section 2.2. In this neighbourhood we define the metric \overline{g} := h + dr^{2} + r^{2}σ^{n−k−1}. Recall that U^{g}(S, ε) denotes the set of points x ∈ V such that r(x) < ε and let χ_{ε} ∈ C^{∞}(M), 0 ≤ χ ≤ 1, be a cut-off function such that χ = 1 on U^{g}(S, ε), χ = 0 on M \ U^{g}(S, 2ε), and |dχ_{ε}| ≤ 2/ε. We define

g_{ε} := χ_{ε}\overline{g} + (1 − χ_{ε})g.

Then g_{ε} has product form on U^{g}(S, ε). For convenience we introduce the notation λ_{ε} := λ_{min}^{+}(V, g_{ε}) and λ := λ_{min}^{+}(V, g). Let (ε_{i})_{ı∈ℕ} be a sequence of positive numbers tending to 0 such that the limit lim_{i→∞} λ_{ε_{i}} exists. In the following, we write ε → 0 instead of ε_{i} → 0 as i → ∞. In the same way, (ε) will denote the successive subsequences of (ε_{i}) we will need. With this notations, let λ̄ := lim_{ε→0} λ_{ε} which exists after possibly taking a subsequence.

We begin by proving that

λ̄ ≤ λ,  (19)

which is the simpler part of the proof. Let J := J^{g} and J_{ε} := J^{g_{ε}} be the functionals associated to g and g_{ε}, and let δ > 0 be a small number. We set χ′_{ε} := 1 − χ_{2ε}, so that χ′_{ε} = 1 on V \ U^{g}(S, 4ε), χ′_{ε} = 0 on U^{g}(S, 2ε), and |dχ′_{ε}| ≤ 1/ε. We see that g = g_{ε} on the support of η′_{ε}. Let ψ be a smooth spinor such that J(ψ) ≤ λ + δ. We then have

∫_{V} ⟨D^{g}(χ′_{ε}ψ), χ′_{ε}ψ⟩ dv^{g} = ∫_{V} χ_{ε}^{′2}⟨D^{g}ψ, ψ⟩ dv^{g} + ∫_{V} ⟨grad^{g}χ′_{ε} · ψ, χ′_{ε}ψ⟩ dv^{g}.

Since the last term here is purely imaginary we obtain

lim_{ε→0} ∫_{V} ⟨D^{g}(χ′_{ε}ψ), χ′_{ε}ψ⟩ dv^{g} = lim_{ε→0} Re ∫_{V} χ_{ε}^{′2}⟨D^{g}ψ, ψ⟩ dv^{g} = ∫_{V} ⟨D^{g}ψ, ψ⟩ dv^{g}. (20)

We compute

\begin{aligned}∫_{V} |D^{g}(χ′_{ε}ψ)|^{\frac{2n}{n+1}} dv^{g} & = ∫_{V \U^{g}(S,4ε)} |D^{g}ψ|^{\frac{2n}{n+1}} dv^{g} \\ & + ∫_{U^{g}(S,4ε)\U^{g}(S,2ε)} |grad^{g}χ′_{ε} · ψ + χ′_{ε}D^{g}ψ|^{\frac{2n}{n+1}} dv^{g}. \end{aligned} (21)

Using the fact that |a + b|^{\frac{2n}{n+1}} ≤ 2^{\frac{2n}{n+1}} (|a|^{\frac{2n}{n+1}} + |b|^{\frac{2n}{n+1}} ) for a, b ∈ ℝ we have

|grad^{g}χ′_{ε} · ψ + χ′_{ε}D^{g}ψ|^{\frac{2n}{n+1}} ≤ 2^{\frac{2n}{n+1}} (|grad^{g}χ′_{ε}|^{\frac{2n}{n+1}} |ψ|^{\frac{2n}{n+1}} + |χ′_{ε}|^{\frac{2n}{n+1}} |D^{g}ψ|^{\frac{2n}{n+1}} )
≤ 2^{\frac{2n}{n+1}} (C_{1}ε^{− \frac{2n}{n+1}} + C_{2}) ,

where C_{1} and C_{2} are bounds on |ψ| and |Dψ|. Since Vol(U^{g}(S, 4ε) \ U^{g}(S, 2ε)) ≤ Cε^{n−k} ≤ Cε^{2} it follows that

lim_{ε→0} ∫_{U^{g}(S,4ε)\U^{g}(S,2ε)} |grad^{g}χ′_{ε} · ψ + χ′_{ε}D^{g}ψ|^{\frac{2n}{n+1}} dv^{g} = 0.

It is clear that lim_{ε→0} ∫_{V \U^{g}(S,4ε)} |D^{g}ψ|^{\frac{2n}{n+1}} dv^{g} = ∫_{V} |D^{g}ψ|^{\frac{2n}{n+1}} dv^{g} so Equation (21) tells us that

lim_{ε→0} ∫_{V} |D^{g}(χ′_{ε}ψ)|^{\frac{2n}{n+1}} dv^{g} = ∫_{V} |D^{g}ψ|^{\frac{2n}{n+1}} dv^{g}.

Together with Equation (20) this proves that lim_{ε→0} J(χ′_{ε}ψ) = J(ψ) ≤ λ+δ. Since g_{ε} = g on the support of χ′_{ε}ψ, we have J_{ε}(χ′_{ε}ψ) = J(χ′_{ε}ψ). Relation (19) now follows since λ_{ε} ≤ J_{ε}(χ′_{ε}ψ) and δ is arbitrary.

The second and harder part of the proof is to show that

λ̄ ≥ λ.  (22)

From Proposition 2.3 we know that λ_{ε} ≤ λ_{min}^{+}(S^{n}, σ^{n}), λ̄ ≤ λ_{min}^{+}(S^{n}, σ^{n}), and λ ≤ λ_{min}^{+}(S^{n}, σ^{n}). Inequality (22) is obvious if λ̄ = λ_{min}^{+}(S^{n}, σ^{n}). Hence we will assume λ_{ε} < λ_{min}^{+}(S^{n}, σ^{n}) for a sequence ε → 0. As the Dirac operator is invertible we know that (8) holds. By Theorem 2.4 there exists for all ε spinor fields ψ_{ε} ∈ Γ(Σ^{g_{ε}}v) of class C^{1} such that

D^{g_{ε}}ψ_{ε} = λ_{ε}|ψ_{ε}|^{\frac{2}{n−1}} ψ_{ε},  (23)

and

∫_{V} |ψ_{ε}|^{\frac{2n}{n−1}} dv^{g_{ε}} = 1.  (24)

Define φ_{ε} = (β_{g_{ε}}^{g})^{−1}ψ_{ε}. Since g_{ε} → g it is easily seen that the sequence (φ_{ε}) is bounded in L^{\frac{2n}{n−1}} (V, g). By (3) and (23) we have

D^{g}φ_{ε} = λ_{ε}|φ_{ε}|^{\frac{2}{n−1}} φ_{ε} − A_{g_{ε}}^{g}(∇^{g}φ_{ε}) − B_{g_{ε}}^{g}(φ_{ε}),  (25)

together with |a + b + c|^{\frac{2n}{n+1}} ≤ 3^{\frac{2n}{n+1}} (|a|^{\frac{2n}{n+1}} + |b|^{\frac{2n}{n+1}} + |c|^{\frac{2n}{n+1}} ) for a, b, c ∈ ℝ this implies

|D^{g}φ_{ε}|^{\frac{2n}{n+1}} ≤ C (λε^{\frac{2n}{n+1}} |φ_{ε}|^{\frac{2n}{n−1}} + |A_{g_{ε}}^{g}(∇^{g}φ_{ε})|^{\frac{2n}{n+1}} + |B_{g_{ε}}^{g}(φ_{ε})|^{\frac{2n}{n+1}} ) .  (26)

We also have

|A_{g_{ε}}^{g}(∇^{g}φ_{ε})| ≤ ∥g − g_{ε}∥_{C^{0}(V)}|∇^{g}φ_{ε}| ≤ Cε|∇^{g}φ_{ε}|,  (27)

and

|B_{g_{ε}}^{g}(φ_{ε})| ≤ ∥g − g_{ε}∥_{C^{1}(V)}|φ_{ε}| ≤ C|φ_{ε}|.  (28)

Indeed, since g and g_{ε} coincide on S, there exists a constant C so that ∥g − g_{ε}∥_{B^{g}(V,ε)} ≤ Cε. Together with the fact that |dχ_{ε}| ≤ 2/ε and using the definition of g_{ε}, this immediately implies that ∥g − g_{ε}∥_{C^{1}(V)} ≤ C. Using Relation (8) and integrating (26) we find that

\begin{aligned}∫_{V} |∇^{g}φ_{ε}|^{\frac{2n}{n+1}} dv^{g} & ≤ C(λε^{\frac{2n}{n+1}} ∫_{V} |φ_{ε}|^{\frac{2n}{n−1}} dv^{g} + ε^{\frac{2n}{n+1}} ∫_{V} |∇^{g}φ_{ε}|^{\frac{2n}{n+1}} dv^{g} \\ & + ∫_{V} |B_{g_{ε}}^{g}(φ_{ε})|^{\frac{2n}{n+1}} dv^{g}). \end{aligned}

As g and g_{ε} coincide on V \ B^{g}(S, 2ε) we conclude that B_{g_{ε}}^{g}(φ_{ε}) = 0 on this set. Together with (28) we have

∫_{V} |B_{g_{ε}}^{g}(φ_{ε})|^{\frac{2n}{n+1}} dv^{g} ≤ C ∫_{B^{g}(S,2ε)} |φ_{ε}|^{\frac{2n}{n+1}} dv^{g}

≤ C Vol(B^{g}(S, 2ε))^{\frac{2}{n+1}} (∫_{B^{g}(S,2ε)} |φ_{ε}|^{\frac{2n}{n−1}} dv^{g})^{\frac{n−1}{n+1}}
= o(1),

where o(1) tends to 0 with ε. Hence

∫_{V} |∇^{g}φ_{ε}|^{\frac{2n}{n+1}} dv^{g} ≤ C (λε^{\frac{2n}{n+1}} ∫_{V} |φ_{ε}|^{\frac{2n}{n−1}} dv^{g} + ε ∫_{V} |∇^{g}φ_{ε}|^{\frac{2n}{n+1}} dv^{g} + o(1)) . (29)

This implies in particular that (φ_{ε}) is bounded in H1^{\frac{2n}{n+1}} (V) and hence after passing

to a subsequence (φ_{ε}) converges weakly to a limit φ in H1^{\frac{2n}{n+1}} (V).

The next step is to prove that λ̄ = lim_{ε→0} λ_{ε} is not zero. To get a contradiction let us assume that λ̄ = 0. We then obtain from (29) that

∫_{V} |∇^{g}φ|^{\frac{2n}{n+1}} dv^{g} ≤ lim_{ε→0} ∫_{V} |∇^{g}φ_{ε}|^{\frac{2n}{n+1}} dv^{g} = 0.

So φ is parallel and since D^{g} is invertible we conclude φ = 0, in other words (φ_{ε}) converges weakly to zero in H1^{\frac{2n}{n+1}} (V). As this space embeds compactly into L^{\frac{2n}{n+1}} (V) we have

lim_{ε→0} ∥φ_{ε}∥_{L^{\frac{2n}{n+1}} (V)} = ∥φ∥_{L^{\frac{2n}{n+1}} (V)} = 0,

and hence (φ_{ε}) converges strongly to zero in H1^{\frac{2n}{n+1}} (V). As this space embeds continuously into L^{\frac{2n}{n−1}} (V) we conclude that the sequence converges strongly to zero in L^{\frac{2n}{n−1}} (V). This is impossible since by Relation (24) we easily get that

lim_{ε→0} ∥φ_{ε}∥_{L^{\frac{2n}{n−1}} (V)} = 1.

From this contradiction we conclude

λ̄ > 0.  (30)

From (25) we have

\begin{aligned}∥D^{g}φ_{ε}∥_{L^{\frac{2n}{n+1}} (V)} & ≤ λ_{ε}∥φ_{ε}∥L^{\frac{2n}{n−1}} (V)^{\frac{n−1}{n+1}} + ∥A_{g_{ε}}^{g}(∇^{g}φ_{ε})∥_{L^{\frac{2n}{n+1}} (V)} \\ & + ∥B_{g_{ε}}^{g}(φ_{ε})∥_{L^{\frac{2n}{n+1}} (V)}. \end{aligned}

We already proved above that

lim_{ε→0} ∥B_{g_{ε}}^{g}(φ_{ε})∥_{L^{\frac{2n}{n+1}} (V)} = 0.

Using Relation (27) we get similarily

lim_{ε→0} ∥A_{g_{ε}}^{g}(∇^{g}φ_{ε})∥_{L^{\frac{2n}{n+1}} (V)} = 0.

Moreover since dv^{g_{ε}} = (1 + o(1)) dv^{g} it follows from (24) that

λ_{ε}∥φ_{ε}∥L^{\frac{2n}{n−1}} (V)^{\frac{n+1}{n−1}} = λ_{ε}(1 + o(1)).

We conclude

∥D^{g}φ_{ε}∥_{L^{\frac{2n}{n+1}} (V)} ≤ λ_{ε} + o(1).  (31)

Starting from Equation (25) we can prove in a similar way that

∫_{V} ⟨D^{g}φ_{ε}, φ_{ε}⟩ dv^{g} ≥ λ_{ε} + o(1).  (32)

From (30), (31), and (32) it follows that λ ≤ lim_{ε→0} J(φ_{ε}) = λ̄. This ends the demonstration of (22), which together with (19) proves Lemma 3.6. □

− ln r = | ln ϵ| − |t|

f(r)

− ln θ − ln δ_{2}

ln A_{θ}

Figure 1. The function − ln r ↦ f(r)

4. Proofs

4.1. Proof of Theorem 1.2. This section is devoted to the proof of Theorem 1.2. Our goal is to construct a family of metrics (g_{θ}) with 0 < θ < θ_{0} which satisfies the conclusion of Theorem (1.2).

From Lemma 3.6 applied with V = M = M_{1} ⨿ M_{2} and S = W′ = w_{1}(W × {0}) ⨿ w_{2}(W × {0}) we may assume that

g = h + dr^{2} + r^{2}σ^{n−k−1}  (33)

in a neighbourhood U(R_{max}) of W′ where 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 \ W′ → ℝ^{+} such that

F (x) = \begin{cases}1, & if x ∈ M_{i} \ U_{i}(R_{1}); \\ r_{i}^{−1} & if x ∈ U_{i}(R_{0}) \ W′. \end{cases}

We further choose θ ∈ (0, R_{0}), later we will let θ → 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}(θ) and δ_{2} = δ_{2}(θ) with θ > δ_{2} > δ_{1} > 0 and a real number A_{θ} ∈ (θ^{−1}, δ_{2}^{−1}) such that

f(x) = \begin{cases}− ln r & if x ∈ U(R_{max}) \ U(θ); \\ ln A_{θ} & if x ∈ U(δ_{2}), \end{cases}

and such that

|r \frac{df}{dr} | = |\frac{df}{d(ln r)}| ≤ 1,

and

∥r \frac{d}{dr} (r \frac{df}{dr} )∥_{L^{∞}} = ∥\frac{d^{2}f}{d^{2}(ln r)}∥_{L^{∞}} → 0

as θ → 0. It follows that lim_{θ→0} A_{θ} = ∞.

Hierarchy of Variables

R_{max} > R_{1} > R_{0} > θ > δ_{2} > δ_{1} > ε > 0

We choose in the order R_{max}, R_{1}, R_{0}, θ, δ_{2}, δ_{1}, A_{θ} We can assume for example that ε = e^{−A_{θ}}δ_{1}. This implies |t| = A_{θ} ⇔ r_{i} = δ_{1}.

Figure 2. Hierarchy of variables

After these choices we set ε := e^{−A_{θ}}δ_{1}. We assume that N is obtained from M by a connected sum along W with parameter ε, as explained in Section 2.2. In particular, recall that U_{ε}^{N}(s) = U(s)\U(ε)/∼ for all s ≥ ε. On the set U_{ε}^{N}(R_{max}) = U(R_{max}) \ U(ε)/∼ we define the variable t by

t := − ln r_{1} + ln ε ≤ 0

on U_{1}(R_{max}) \ U(ε) and

t := ln r_{2} − ln ε ≥ 0

on U_{2}(R_{max}) \ U(ε). This implies

r_{i} = e^{|t|+ln ε} = εe^{|t|}.

The choices imply that t : U_{ε}^{N}(R_{max}) → ℝ is a smooth function with t ≤ 0 on U_{ε}^{N}(R_{max}) ∩ M_{1}, t ≥ 0 on U_{ε}^{N}(R_{max}) ∩ M_{2}, and t = 0 is the common boundary ∂U_{1}(ε) identified in N with ∂U_{2}(ε). Then Equation (33) tells us that

r^{−2}g = ε^{−2}e^{−2|t|}h_{i} + dt^{2} + σ^{n−k−1}.

Expressed in the new variable t we have

F (x) = ε^{−1}e^{−|t|}

if x ∈ U_{ε}^{N}(R_{0}) \ U_{ε}^{N}(θ) or in other words if |t| + ln ε ≤ ln R_{0}, and

f(t) = \begin{cases}−|t| − ln ε & if |t| + ln ε ∈ (θ, R_{max}), \\ ln A_{θ} & if |t| + ln ε ≤ ln δ_{2}, \end{cases}

and |df/dt| ≤ 1, ∥d^{2}f/dt^{2}∥_{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

g_{θ}(x) := \begin{cases}F^{2}g_{i} & if x ∈ M_{i} \ U_{i}(θ); \\ e^{2f(t)}h_{i} + dt^{2} + σ^{n−k−1} & if x ∈ U_{i}(θ) \ U_{i}(δ_{1}); \\ A_{θ}^{2}χ(A_{θ}^{−1}t)h_{2} + A_{θ}^{2}(1 − χ(A_{θ}^{−1}t))h_{1} + dt^{2} + σ^{n−k−1} & if x ∈ U_{i}(δ_{1}) \ U_{i}(ε). \end{cases}

(Recall that the h_{i} are defined as the pullback via w_{i} of the metric g_{i} on M_{i}, composed with restriction to W = W × {0}.)

On U_{ε}^{N}(R_{0}) we write g_{θ} as

g_{θ} = α_{t}^{2}h̃_{t} + dt^{2} + σ^{n−k−1},

where the metric h̃_{t} is defined for t ∈ ℝ by

h̃_{t} := χ(A_{θ}^{−1}t)h_{2} + (1 − χ(A_{θ}^{−1}t))h_{1},  (34)

and where

α_{t} := e^{f(t)}.  (35)

The rest of the proof consists of showing that (g_{θ}) is the desired family of metrics. We first choose a sequence (θ_{i})_{i∈ℕ} converging to 0 so that lim_{i→∞} λ_{θ_{i}} exists. To avoid complicated notation we write θ → 0 for the sequence (θ_{i})_{i∈ℕ} converging to zero and we will pass successively to subsequences without changing notation. Similarly lim_{θ→0} h(θ) should be read as lim_{i→∞} h(θ_{i}). We set λ := λ_{min}^{+}(M_{1} ⨿ M_{2}, g), λ_{θ} := λ_{min}^{+}(N, g_{θ}), and λ̄ := lim_{θ→0} λ_{θ}. Let J := J^{g} and J_{θ} := J^{g_{θ}} be the functionals associated respectively to g and g_{θ}.

The easier part of the argument is to show that

λ̄ ≤ λ.  (36)

For this let α > 0 be a small number. We choose a smooth cut-off function χ_{α} : M_{1} ⨿ M_{2} → [0, 1] such that χ_{α} = 1 on M_{1} ⨿ M_{2} \ U(2α), |dχ_{α}| ≤ 2/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_{θ} = F^{2}g and hence by Formula (10) we have

λ_{θ} ≤ J_{θ} (χ_{α}β_{g_{θ}}^{g}(F^{− \frac{n−1}{2}} ψ)) = J(χ_{α}ψ)

for θ < 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 Relation (36) follows.

Now we turn to the more difficult part of the proof, that

λ̄ ≥ min{λ, Λ_{n,k}}.  (37)

By Proposition 2.3 we can assume that λ_{θ} < λ_{min}^{+}(S^{n}, σ^{n}) for all θ, otherwise Relation (37) is trivial. From Theorem 2.4 we know that there exists a spinor field ψ_{θ} ∈ Γ(Σ^{g_{α}}N) of class C^{2} such that

∫_{N} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}} = 1

and

D^{g_{θ}}ψ_{θ} = λ_{θ}|ψ_{θ}|^{\frac{2}{n−1}} ψ_{θ}.  (38)

We let x_{θ} in N be such that |ψ_{θ}(x_{θ})| = m_{θ} where m_{θ} := ∥ψ_{θ}∥_{L^{∞}(N)}.

The proof continues divided in cases.

Case I. The sequence (m_{θ}) is not bounded.

After taking a subsequence, we can assume that lim_{θ→0} m_{θ} = ∞. We consider two subcases.

Subcase I.1. There exists a > 0 such that x_{θ} ∈ N \ U^{N}(a) for an infinite number of θ.

We recall that N \ U^{N}(a) = N_{ε} \ U_{ε}^{N}(a) = M_{1} ⨿ M_{2} \ U(a). By taking a subsequence we can assume that there exists x̄ ∈ M_{1}⨿M_{2}\U(a) such that lim_{θ→0} x_{θ} = x̄.

We let g′_{θ} := mθ^{\frac{4}{n−1}} g_{θ}. In a neighbourhood U of x̄ the metric g_{θ} = F^{2}g does not depend on θ. We apply Lemma 3.3 with O = U, α = θ, p_{α} = x_{θ}, p = x̄, γ_{α} = g_{θ} = F^{2}g, and b_{α} = mθ^{\frac{2}{n−1}} . Let r > 0. For θ small enough Lemma 3.3 gives us diffeomorphisms

Θ_{θ} : B^{n}(r) → B^{g_{θ}}(x_{θ}, m_{θ}^{− \frac{2}{n−1}} r)

such that the sequence of metrics (Θ_{θ}^{∗}(g′_{θ})) tends to the Euclidean metric ξ^{n} in C^{1}(B^{n}(r)). We let ψ′_{θ} := m_{θ}^{−1}ψ_{θ}. By (6) we then have

D^{g′_{θ}}ψ′_{θ} = λ_{θ}|ψ′_{θ}|^{\frac{2}{n−1}} ψ′_{θ}

on B^{g_{θ}}(x_{θ}, m_{θ}^{− \frac{2}{n−1}} r) and

∫_{B^{g_{θ}} (x_{θ},m_{θ}^{− \frac{2}{n−1}} r)} |ψ′_{θ}|^{\frac{2n}{n−1}} dv^{g′_{θ}} = ∫_{B^{g_{θ}} (x_{θ},m_{θ}^{− \frac{2}{n−1}} r)} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}}
≤ ∫_{N} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}}
= 1.

Here we used the fact that dv^{g′_{θ}} = mθ^{\frac{2n}{n−1}} dv^{g_{θ}}. Since

Θ_{θ} : (B^{n}(r), Θ_{θ}^{∗}(g′_{θ})) → (B^{g_{θ}}(x_{θ}, m_{θ}^{− \frac{2}{n−1}} r), g′_{θ})

is an isometry we can consider ψ′_{θ} as a solution of

D^{Θ_{θ}^{∗}(g′_{θ})}ψ′_{θ} = λ_{θ}|ψ′_{θ}|^{\frac{2}{n−1}} ψ′_{θ}

on B^{n}(r) with ∫_{B^{n}(r)} |ψ′_{θ}|^{\frac{2n}{n−1}} dv^{Θ_{θ}^{∗}(g′_{θ})} ≤ 1. Since ∥ψ_{θ}∥_{L^{∞}(B^{n}(r))} = |ψ′_{θ}(0)| = 1 we can apply Lemma 3.4 with V = ℝ^{n}, α = θ, g_{α} = Θ_{θ}^{∗}(g′_{θ}), and ψ_{α} = ψ′_{θ} (we may 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^{ξ^{n}}ψ = λ̄|ψ|^{\frac{2}{n−1}} ψ.

Furthermore by (17) we have

∫_{B^{n}(r)} |ψ|^{\frac{2n}{n−1}} dv^{ξ^{n}} = lim_{θ→0} ∫_{B^{g_{θ}} (x_{θ},m_{θ}^{− \frac{2}{n−1}} r)} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}} ≤ 1

for any r > 0. We conclude that ∫_{ℝ^{n}} |ψ|^{\frac{2n}{n−1}} 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^{2}ξ^{n} for a positive function Φ. We define φ := Φ^{− \frac{n−1}{2}} β_{σ^{n}}^{ξ^{n}}ψ. By Equation (6) it follows that φ ∈ L^{\frac{2n}{n−1}} (S^{n}) is a solution of

D^{σ^{n}}φ = λ̄|φ|^{\frac{2}{n−1}} φ  (39)

on S^{n} \ {pt} of class C^{1}. By Corollary 3.2 we know that φ can be extended to a weak solution of (39) on all S^{n} and by standard regularity theorems it follows that φ ∈ C^{1}(S^{n}). Let J^{σ^{n}} be the functional associated to (S^{n}, σ^{n}). By Equation (39) we have

λ_{min}^{+}(S^{n}, σ^{n}) ≤ J^{σ^{n}}(φ) = λ̄

where the inequality comes from Proposition 2.3. We have proved Relation (37) in this subcase.

Subcase I.2. For all a > 0 it holds that x_{θ} ∉ M_{1} ⨿ M_{2} \ U(a) for θ sufficiently small.

This means that x_{θ} belongs to U^{N}(a) if θ is sufficiently small. This subset is diffeomorphic to W ×I ×S^{n−k−1} where I is an interval. Through this diffeomorphism x_{θ} can be written as

x_{θ} = (y_{θ}, t_{θ}, z_{θ})

where y_{θ} ∈ W, t_{θ} ∈ (− ln R_{0} + ln ε, − ln ε + ln R_{0}), and z_{θ} ∈ S^{n−k−1}. By taking a subsequence we can assume that y_{θ}, \frac{t_{θ}}{A_{θ}}, and z_{θ} converge respectively to y ∈ W, T ∈ [−∞, +∞], and z ∈ S^{n−k−1}. We apply Lemma 3.3 with V = W, α = θ, p_{α} = y_{θ}, p = y, γ_{α} = h̃_{t_{θ}}, γ_{0} = h̃_{T} (we define h̃_{−∞} := h_{1} and h̃_{+∞} := h_{2}), and

b_{α} = mθ^{\frac{2}{n−1}} α_{t_{θ}}. The lemma provides diffeomorphisms

Θ_{θ}^{y} : B^{k}(r) → Bh̃_{t_{θ}} (y_{θ}, m_{θ}^{− \frac{2}{n−1}} α_{t_{θ}}^{−1}r)

for r > 0 such that (Θ_{θ}^{y})^{∗}(mθ^{\frac{4}{n−1}} α_{t_{θ}}^{2} h̃_{t_{θ}}) tends to the Euclidean metric ξ^{k} on B^{k}(r) as θ → 0. Next we apply Lemma 3.3 with V = S^{n−k−1}, α = θ, p_{α} = z_{θ}, γ_{α} = γ_{0} = σ^{n−k−1}, and b_{α} = mθ^{\frac{2}{n−1}} . For r′ > 0 we get the existence of diffeomorphisms

Θ_{θ}^{z} : B^{n−k−1}(r′) → B^{σ^{n−k−1}}(z_{θ}, m_{θ}^{− \frac{2}{n−1}} r′)

such that (Θ_{θ}^{z})^{∗}(mθ^{\frac{4}{n−1}} σ^{n−k−1}) converges to ξ^{n−k−1} on B^{n−k−1}(r′) as θ → 0. For r, r′, r′′ > 0 we define

\begin{aligned}U_{θ}(r, r′, r′′) & := Bh̃_{t_{θ}} (y_{θ}, m_{θ}^{− \frac{2}{n−1}} α_{t_{θ}}^{−1}r) × [t_{θ} − m_{θ}^{− \frac{2}{n−1}} r′′, t_{θ} + m_{θ}^{− \frac{2}{n−1}} r′′] \\ & × B^{σ^{n−k−1}}(z_{θ}, m_{θ}^{− \frac{2}{n−1}} r′) \end{aligned}

and

\begin{matrix}Θ_{θ} : B^{k}(r) × [−r′′, r′′] × B^{n−k−1}(r′) & → U_{θ}(r, r′, r′′) \\ (y, s, z) & ↦ (Θ_{θ}^{y}(y), t(s), Θ_{θ}^{z}(z)) , \end{matrix}

where t(s) := t_{θ} + mθ^{\frac{2}{n−1}} s. By construction Θ_{θ} is a diffeomorphism. As is readily seen

Θ_{θ}^{∗}(mθ^{\frac{4}{n−1}} g_{θ}) = (Θ_{θ}^{y})^{∗}(mθ^{\frac{4}{n−1}} α_{t}^{2}h̃_{t}) + ds^{2} + (Θ_{θ}^{z})^{∗}(mθ^{\frac{4}{n−1}} σ^{n−k−1}).  (40)

By construction of α_{t} one can verify that

lim_{θ→0} ∥\frac{α_{t_{θ}}}{α_{t}} − 1∥_{C^{1}([t_{θ}−m_{θ}^{− \frac{2}{n−1}} r′′,t_{θ}+m_{θ}^{− \frac{2}{n−1}} r′′])} = 0

for all R > 0 since \frac{df}{dt} and \frac{d^{2}f}{dt^{2}} are uniformly bounded. Moreover it is clear that

lim_{θ→0} |h̃_{t} − h̃_{t_{θ}} |_{C^{1}(Bh̃_{t_{θ}} (y_{θ},m_{θ}^{− \frac{2}{n−1}} α_{t_{θ}}^{−1}R))} = 0

uniformly in t ∈ [t_{θ} − m_{θ}^{− \frac{2}{n−1}} r′′, t_{θ} + m_{θ}^{− \frac{2}{n−1}} r′′]. As a consequence

lim_{θ→0} |(Θ_{θ}^{y})^{∗} (mθ^{\frac{4}{n−1}} (α_{t}^{2}h̃_{t} − α_{t_{θ}}^{2} h̃_{t_{θ}}))|_{C^{1}(B^{k}(r))} = 0

uniformly in t. This implies that the sequence (Θ_{θ}^{y})^{∗}(mθ^{\frac{4}{n−1}} α_{t}^{2}h̃_{t}) tends to the Euclidean metric ξ^{k} in C^{1}(B^{k}(r)) uniformly in t as θ → 0. From (40) we know

that the sequence (Θ_{θ}^{z})^{∗}(mθ^{\frac{4}{n−1}} σ^{n−k−1}) tends to the Euclidean metric ξ^{n−k−1} on

B^{n−k−1}(r′) as θ → 0. Returning to (40) we obtain that the sequence Θ_{θ}^{∗}(mθ^{\frac{4}{n−1}} g_{θ}) tends to ξ^{n} = ξ^{k}+ds^{2}+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^{ξ^{n}}ψ = λ̄|ψ|^{\frac{2}{n−1}} ψ

with ∫_{B^{n}(r)} |ψ|^{\frac{2n}{n−1}} dx ≤ 1 for all r ∈ ℝ^{+}. Lemma 3.4 tells us that |ψ(0)| = 1 so ψ does not vanish identically. As in Subcase I.1 we conclude that

λ ≤ λ_{min}^{+}(S^{n}, σ^{n}) ≤ λ̄.

This ends the proof of Theorem 1.2 in Case I.

Case II. There exists a constant C_{1} such that m_{θ} ≤ C_{1} for all θ.

Again we consider two subcases.

Subcase II.1. Assume that

lim inf_{θ→0} ∫_{N\U^{N} (a)} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}} > 0  (41)

for some number a > 0.

Let K a compact subset such that K ⊂ M_{1} ⨿ M_{2} \ W′. Choose a small number b such that K ⊂ M_{1} ⨿ M_{2} \ U(2b) = N \ U^{N}(2b). Let χ ∈ C^{∞}(M_{1} ⨿ M_{2}), 0 ≤ χ ≤ 1, be a cut-off function equal to 1 on M_{1} ⨿ M_{2} \ U(2b) and equal to 0 on U(b). Set ψ′_{θ} := F^{\frac{n−1}{2}} (β_{g_{θ}}^{g})^{−1}ψ_{θ}. Since g_{θ} = F^{2}g on the support of χ we have

D^{g}ψ′_{θ} = λ_{θ}|ψ′_{θ}|^{\frac{2}{n−1}} ψ′_{θ}

on this set. For r > 0 we have

∫_{M_{1}⨿M_{2}} |D^{g}(χψ′_{θ})|^{r} dv^{g}
= ∫_{M_{1}⨿M_{2}} |grad^{g}χ · ψ′_{θ} + χλ_{θ}|ψ′_{θ}|^{\frac{2}{n−1}} ψ′_{θ}|^{r} dv^{g}
≤ 2^{r} (∫_{M_{1}⨿M_{2}} |grad^{g}χ|^{r}|ψ′_{θ}|^{r} dv^{g} + λ_{θ}^{r} ∫_{M_{1}⨿M_{2}} χ^{r}|ψ′_{θ}|^{\frac{(n+1)r}{n−1}} dv^{g})
≤ C.

since m_{θ} ≤ C_{1}. Together with Relation (8) we get that the sequence (χψ′_{θ}) is bounded in H_{1}^{r}(M_{1} ⨿ M_{2}) 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′_{θ}) converges to ψ_{0} in C^{0}(K) and which satisfies

D^{g}ψ_{0} = λ̄|ψ_{0}|^{\frac{2}{n−1}} ψ_{0}.  (42)

Furthermore the convergence in C^{0} implies that

∫_{K} |ψ_{0}|^{\frac{2n}{n−1}} dv^{g} ≤ lim inf_{θ→0} ∫_{K} |ψ′_{θ}|^{\frac{2n}{n−1}} dv^{g} = lim inf_{θ→0} ∫_{K} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}} ≤ 1.

Repeating the same for a sequence of compact sets which exhausts M_{1} ⨿ M_{2} \ W′ and taking a diagonal subsequence we can extend ψ_{0} to M_{1} ⨿ M_{2} \ W′. Since ψ_{0} ∈ L^{\frac{2n}{n−1}} (M_{1} ⨿ M_{2} \ W′) = L^{\frac{2n}{n−1}} (M_{1} ⨿ M_{2}) we can use Theorem 3.2 to extend ψ_{0} to a weak solution of Equation (42) on M_{1} ⨿ M_{2}. Note here that since D^{g} is invertible we have λ̄ > 0. By standard regularity theorems we conclude that ψ_{0} ∈ C^{1}(M_{1} ⨿ M_{2}). By (41) we have

∫_{M_{1}⨿M_{2}\U(a)} |ψ_{0}|^{\frac{2n}{n−1}} dv^{g} = lim_{θ→0} ∫_{M_{1}⨿M_{2}\U(a)} |ψ′_{θ}|^{\frac{2n}{n−1}} dv^{g}

= lim_{θ→0} ∫_{M_{1}⨿M_{2}\U(a)} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}}

> 0,

and we conclude that ψ_{0} does not vanish identically. Equation (42) then leads to

λ ≤ J(ψ_{0}) = λ̄ (∫_{M_{1}⨿M_{2}} |ψ_{0}|^{\frac{2n}{n−1}} dv^{g})^{\frac{n+1}{n} −1} ≤ λ̄,

which proves Theorem 1.2 in this case.

Subcase II.2. We have

lim inf_{θ→0} ∫_{N\U^{N} (a)} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}} = 0  (43)

for all a > 0.

This case is the most difficult one and we proceed in several steps. The assumption here is that we have a sequence (θ_{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_{θ→0} for what should be a limit as i → ∞ or a limit of a subsequence.

For positive a and θ let

γ_{θ}(a) := \frac{∫_{N\U^{N} (a)} |ψ_{θ}|^{2} dv^{g_{θ}}}{∫_{U^{N} (a)} |ψ_{θ}|^{2} dv^{g_{θ}}}

The first step is to establish an estimate for γ_{θ}(a).

Step 1. There is a constant C_{0} so that

1 ≤ C_{0} (γ_{θ}(a) + ∥ψ_{θ}∥L^{∞}(U^{N} (2a))^{\frac{4}{n−1}})  (44)

for all a > 0.

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}(2a) = M_{1} ⨿ M_{2} \ U(2a). 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} ≤ 2/a. For the metric g_{θ} this gives

|dχ|_{g_{θ}} = F^{−1}|dχ|_{g} = r|dχ|_{g} ≤ 2a\frac{2}{a} = 4.

From Lemma 3.5 and Equation (38) it follows that

\frac{(n − k − 1)^{2}}{4} ≤ \frac{∫_{N} |D^{g_{θ}}(χψ_{θ})|^{2} dv^{g_{θ}}}{∫_{N} |χψ_{θ}|^{2} dv^{g_{θ}}}

= \frac{∫_{N} |dχ|_{g_{θ}}^{2}|ψ_{θ}|^{2} dv^{g_{θ}} + λ_{θ}^{2} ∫_{N} χ^{2}|ψ_{θ}|^{\frac{2(n+1)}{n−1}} dv^{g_{θ}}}{∫_{N} |χψ_{θ}|^{2} dv^{g_{θ}}}

≤ \frac{16 ∫_{U^{N} (2a)\U^{N} (a)} |ψ_{θ}|^{2} dv^{g_{θ}} + λ_{θ}^{2}∥ψ_{θ}∥L^{∞}(U^{N} (2a))^{\frac{4}{n−1}} ∫_{N} |χψ_{θ}|^{2} dv^{g_{θ}}}{∫_{N} |χψ_{θ}|^{2} dv^{g_{θ}}}

≤ \frac{16 ∫_{U^{N} (2a)\U^{N} (a)} |ψ_{θ}|^{2} dv^{g_{θ}}}{∫_{U^{N} (a)} |ψ_{θ}|^{2} dv^{g_{θ}}} + λ_{θ}^{2}∥ψ_{θ}∥L^{∞}(U^{N} (2a))^{\frac{4}{n−1}}

≤ 16γ_{θ}(a) + λ_{θ}^{2}∥ψ_{θ}∥L^{∞}(U^{N} (2a))^{\frac{4}{n−1}}.

Using that λ_{θ} ≤ λ_{min}^{+}(S^{n}, σ^{n}) by Proposition 2.3 we obtain Relation (44) with

C_{0} := \frac{4}{(n − k − 1)^{2}} max \{16, λ_{min}^{+}(S^{n})^{2}\} .

This ends the proof of Step 1.

Step 2. There exist a sequence of positive numbers (a_{θ}) which tends to 0 with θ and constants 0 < m < M such that

m ≤ ∥ψ_{θ}∥_{L^{∞}(U^{N} (2a_{θ}))} ≤ M  (45)

for all θ.

By (43) we have

lim_{θ→0} ∫_{N\U^{N} (a)} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}} = 0

for all a > 0. Since Vol (N \ U^{N}(a), g_{θ}) does not depend on θ if θ < a it follows that

lim_{θ→0} (∫_{N\U^{N} (a)} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}})^{\frac{n−1}{n}} Vol(N \ U^{N}(a), g_{θ})^{\frac{1}{n}} = 0

for all a. Hence we can take a sequence (a_{θ}) which tends sufficiently slowly to 0 so that

lim_{θ→0} (∫_{N\U^{N} (a_{θ})} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}})^{\frac{n−1}{n}} Vol(N \ U^{N}(a_{θ}), g_{θ})^{\frac{1}{n}} = 0.  (46)

Using the Hölder inequality we get

\begin{aligned}γ_{θ}(a_{θ}) & = \frac{∫_{N\U^{N} (a_{θ})} |ψ_{θ}|^{2} dv^{g_{θ}}}{∫_{U^{N} (a_{θ})} |ψ_{θ}|^{2} dv^{g_{θ}}} \\ & ≤ \frac{(∫_{N\U^{N} (a_{θ})} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}})^{\frac{n−1}{n}} Vol(N \ U^{N}(a_{θ}), g_{θ})^{\frac{1}{n}}}{∥ψ_{θ}∥_{L^{∞}(U^{N} (a_{θ}))}^{− \frac{2}{n−1}} ∫_{U^{N} (a_{θ})} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}}} . \end{aligned}

The numerator of this expression tends to 0 by Relation (46). Further by (46) we have

\begin{aligned}lim_{θ→0} ∫_{U^{N} (a_{θ})} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}} & = lim_{θ→0} ∫_{N} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}} − ∫_{N\U^{N} (a_{θ})} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}} \\ & = 1. \end{aligned}

Together with the fact that ∥ψ_{θ}∥_{L^{∞}(U^{N} (a_{θ}))} ≤ m_{θ} ≤ C_{1} we obtain that

lim_{θ→0} γ_{θ}(a_{θ}) = 0.

From Relation (44) applied with a = a_{θ} we know that ∥ψ_{θ}∥_{L^{∞}(U^{N} (2a_{θ}))} is bounded from below. Moreover, by the assumption of Case II we have that ∥ψ_{θ}∥_{L^{∞}(U^{N} (2a_{θ}))} ≤ m_{θ} ≤ C_{1}. This finishes the proof of Step 2.

Step 3. We have

λ̄ ≥ Λ_{n,k}.

Let x_{θ} be a point in the closure of U^{N}(2a_{θ}) such that |ψ_{θ}(x_{θ})| = ∥ψ_{θ}∥_{L^{∞}(U^{N} (2a_{θ}))}. As in Subcase I.2 we write x_{θ} = (y_{θ}, t_{θ}, z_{θ}) where y_{θ} ∈ W, t_{θ} ∈ (− ln R_{0} + ln ε, − ln ε + ln R_{0}), and z_{θ} ∈ S^{n−k−1}. By restricting to a subsequence we can assume that y_{θ}, t_{θ}/A_{θ}, and z_{θ} converge respectively to y ∈ W, T ∈ [−∞, +∞], and z ∈ S^{n−k−1}. We apply Lemma 3.3 with V = W, α = θ, p_{α} = y_{θ}, p = y, γ_{α} = h̃_{t_{θ}}, γ_{0} = h̃_{T}, and b_{α} = α_{t_{θ}} (recall that h̃_{t} and α_{t} were defined in (34) and (35)) and conclude that there is a diffeomorphism

Θ_{θ}^{y} : B^{k}(r) → Bh̃_{t_{θ}} (y_{θ}, α_{t_{θ}}^{−1}r)

for r > 0 such that (Θ_{θ}^{y})^{∗}(α_{t_{θ}}^{2} h̃_{t_{θ}}) converges to the Euclidean metric ξ^{k} on B^{k}(r). For r, r′ > 0 we define

U_{θ}(r, r′) := Bh̃_{t_{θ}} (y_{θ}, α_{t_{θ}}^{−1}r) × [t_{θ} − r′, t_{θ} + r′] × S^{n−k−1}

and

\begin{matrix}Θ_{θ} : B^{k}(r) × [−r′, r′] × S^{n−k−1} & → U_{θ}(r, r′) \\ (y, s, z) & ↦ (Θ_{θ}^{y}(y), t(s), z) , \end{matrix}

where t(s) := t_{θ} + s. By construction Θ_{θ} is a diffeomorphism. Since g_{θ} = α_{t}^{2}h̃_{t} + dt^{2} + σ^{n−k−1} we see that

Θ_{θ}^{∗}(g_{θ}) = \frac{α_{t}^{2}}{α_{t_{θ}}^{2}} (Θ_{θ}^{y})^{∗}(α_{t_{θ}}^{2} h̃_{t}) + ds^{2} + σ^{n−k−1}.  (47)

We will now find the limit of Θ_{θ}^{∗}(g_{θ}) in the C^{1} topology. We define c := lim_{θ→0} f′(t_{θ}).

Lemma 4.1. The sequence of metrics Θ_{θ}^{∗}(g_{θ}) tends to

G_{c} = η_{c}^{k+1} + σ^{n−k−1} = e^{2cs}ξ^{k} + ds^{2} + 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 tells us that

|f(t) − f(t_{θ}) − f′(t_{θ})(t − t_{θ})| ≤ \frac{r^{′2}}{2} max_{ξ∈[t_{θ}−r′,t_{θ}+r′]} |f′′(ξ)|

for all t ∈ [t_{θ} − r′, t_{θ} + r′]. On the other hand we assume that f′′(t) → 0 as θ → 0, so

∥f(t) − f(t_{θ}) − f′(t_{θ})(t − t_{θ})∥_{C^{0}([t_{θ}−r′,t_{θ}+r′])} → 0

as θ → 0 (and r′ fixed). Furthermore

|\frac{d}{dt} (f(t) − f(t_{θ}) − f′(t_{θ})(t − t_{θ}))| = |f′(t) − f′(t_{θ})|

= |∫_{t_{θ}}^{t} f′′(s) ds|

≤ r′ max_{ξ∈[t_{θ}−r′,t_{θ}+r′]} |f′′(ξ)|

→ 0

as θ → 0. Together with c = lim_{θ→0} f′(t_{θ}) we have

∥f(t) − f(t_{θ}) − c(t − t_{θ})∥_{C^{1}([t_{θ}−r′,t_{θ}+r′])} → 0.

Exponentiation of functions is a continuous map

C^{1}([t_{θ} − r′, t_{θ} + r′]) ∋ f̃ ↦ exp ∘f̃ ∈ C^{1}([t_{θ} − r′, t_{θ} + r′]).

Hence

∥\frac{α_{t}}{α_{t_{θ}}} − e^{c(t−t_{θ})}∥_{C^{1}([t_{θ}−r′,t_{θ}+r′])} = ∥e^{f(t)−f(t_{θ})} − e^{c(t−t_{θ})}∥_{C^{1}([t_{θ}−r′,t_{θ}+r′])} → 0

as θ → 0. We now write α_{t}^{2}h̃_{t} = α_{t}^{2}(h̃_{t} − h̃_{t_{θ}}) + \frac{α_{t}^{2}}{α_{t_{θ}}^{2}} α_{t_{θ}}^{2} h̃_{t_{θ}}. Using the fact that

lim_{θ→0} ∥h̃_{t} − h̃_{t_{θ}} ∥_{C^{1}(Bh̃_{t_{θ}} (y_{θ},α_{t_{θ}}^{−1}R))} = 0

uniformly for t ∈ [t_{θ} − r′, t_{θ} − r′] we get that the sequence \frac{α_{t}^{2}}{α_{t_{θ}}^{2}} (Θ_{θ}^{y})^{∗}(α_{t_{θ}}^{2} h̃_{t}) tends

to e^{2cs}ξ^{k} in C^{1} on B^{k}(r). Going back to Relation (47) this proves Lemma 4.1. □

We continue with the proof of Step 3. As in subcases I.1 and I.2 we apply Lemma 3.4 with (V, g) = (ℝ^{k+1} × S^{n−k−1}, G_{c}), α = θ, and g_{α} = Θ_{θ}^{∗}(g_{θ}) (we can apply this lemma since any compact subset of ℝ^{k+1} × S^{n−k−1} is contained in some B^{k}(r) × [−r′, r′] × S^{n−k−1}). We obtain a C^{1} spinor ψ which is a solution of

D^{G_{c}}ψ = λ̄|ψ|^{\frac{2}{n−1}} ψ

on (ℝ^{k+1} × S^{n−k−1}, G_{c}). From (17) it follows that

∫_{ℝ^{k+1}×S^{n−k−1}} |ψ|^{\frac{2n}{n−1}} dv^{G_{c}} ≤ 1.

From (16) it follows that ψ ∈ L^{∞}(ℝ^{k+1}×S^{n−k−1}), and from (16) and (45) it follows that ψ does not vanish identically. We want to show that ψ ∈ L^{2}(ℝ^{k+1} × S^{n−k−1}). From (17) we get that

\begin{aligned}∫_{B^{k}(r)×[−r′,r′]×S^{n−k−1}} |ψ|^{2} dv^{G_{c}} & = lim_{θ→0} ∫_{U_{θ}(r,r′)} |ψ_{θ}|^{2}dv^{g_{θ}} \\ & ≤ lim_{θ→0} ∫_{U^{N} (a)} |ψ_{θ}|^{2}dv^{g_{θ}} \end{aligned}  (48)

for some fixed number a > 0 independent of r, r′ and θ. Let χ be defined as in Step 1. Using the Hölder inequality, Lemma 3.5, and Equation (38) we see that

\frac{(n − k − 1)^{2}}{4}

≤ \frac{∫_{N} |D^{g_{θ}}(χψ_{θ})|^{2} dv^{g_{θ}}}{∫_{N} |χψ_{θ}|^{2} dv^{g_{θ}}}

= \frac{∫_{N} |dχ|_{g_{θ}}^{2}|ψ_{θ}|^{2} dv^{g_{θ}} + λ_{θ}^{2} ∫_{N} χ^{2}|ψ_{θ}|^{\frac{2(n+1)}{n−1}} dv^{g_{θ}}}{∫_{N} |χψ_{θ}|^{2} dv^{g_{θ}}}

≤ \frac{16 ∫_{U^{N} (2a)\U^{N} (a)} |ψ_{θ}|^{2} dv^{g_{θ}} + λ_{θ}^{2}∥ψ_{θ}∥L^{∞}(U^{N} (2a))^{\frac{2}{n−1}} ∫_{U^{N} (2a)} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}}}{∫_{U^{N} (a)} |ψ_{θ}|^{2} dv^{g_{θ}}} .

We have

λ_{θ}^{2}∥ψ_{θ}∥L^{∞}(U^{N} (2a))^{\frac{2}{n−1}} ∫_{U^{N} (2a)} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}} ≤ λ_{min}^{+}(S^{n}, σ^{n})^{2}C1^{\frac{2}{n−1}}

and

∫_{U^{N} (2a)\U^{N} (a)} |ψ_{θ}|^{2} dv^{g_{θ}}

≤ (∫_{U^{N} (2a)\U^{N} (a)} |ψ_{θ}|^{\frac{2n}{n−1}} dv^{g_{θ}})^{\frac{n−1}{n}} Vol (U^{N}(2a) \ U^{N}(a), g_{θ})^{\frac{1}{n}}
≤ Vol (U^{N}(2a) \ U^{N}(a), g_{θ})^{\frac{1}{n}} .

Since g_{θ} does not depend on θ on U^{N}(2a) \ U^{N}(a) for θ < a, we get the existence of a constant C such that

\frac{(n − k − 1)^{2}}{4} ≤ \frac{C}{∫_{U^{N} (a)} |ψ_{θ}|^{2} dv^{g_{θ}}} .

Together with (48) we obtain that

∫_{B^{k}(r)×[−r′,r′]×S^{n−k−1}} |ψ|^{2} dv^{G_{c}} ≤ C

where C is independent of r and r′. This proves that ψ ∈ L^{2}(ℝ^{k+1} × S^{n−k−1}). Since the spinor ψ is non-zero and

ψ ∈ L^{∞}(ℝ^{k+1} × S^{n−k−1}) ∩ C_{loc}^{1}(ℝ^{k+1} × S^{n−k−1}) ∩ L^{2}(ℝ^{k+1} × S^{n−k−1})

with

∫_{ℝ^{k+1}×S^{n−k−1}} |ψ|^{\frac{2n}{n−1}} dv^{G_{c}} ≤ 1

we get that λ̄ ≥ Λ_{n,k} by the definition of Λ_{n,k}. This ends the proof of this subcase and the proof of Theorem 1.2.

4.2. Proof of Theorem 1.1. We prove Theorem 1.1 by contradiction. Assume that there is a sequence c_{i} ∈ [−1, 1], i ∈ ℕ, for which

lim_{i→∞} λ̃_{min}^{+}(ℝ^{k+1} × S^{n−k−1}, G_{c_{i}}) = 0.

After removing the indices i for which λ̃_{min}^{+} is infinite we have for all i a solution of

D^{G_{c_{i}}} ψ_{i} = λ_{i}|ψ_{i}|^{\frac{2}{n−1}} ψ_{i}  (49)

where λ_{i} → 0 as i → ∞. Moreover, the spinors ψ_{i} are in L^{∞} ∩ L^{2} ∩ C_{loc}^{1} and

∫_{ℝ^{k+1}×S^{n−k−1}} |ψ_{i}|^{\frac{2n}{n−1}} dv^{G_{c_{i}}} ≤ 1.

Let m_{i} := ∥ψ_{i}∥_{L^{∞}}. We cannot assume that m_{i} is attained, but since (ℝ^{k+1} × S^{n−k−1}, G_{c_{i}}) is a symmetric space we can compose ψ_{i} with isometries so that |ψ_{i}(P)| > m_{i}/2 for some fixed point P ∈ ℝ^{k+1} × S^{n−k−1}. First we prove that

lim_{i→∞} m_{i} = ∞.  (50)

By Lemma 3.5 and Equation (49) we have

\frac{(n − k − 1)^{2}}{4} ≤ \frac{∫_{ℝ^{k+1}×S^{n−k−1}} |D^{G_{c_{i}}} ψ_{i}|^{2} dv^{G_{c_{i}}}}{∫_{ℝ^{k+1}×S^{n−k−1}} |ψ_{i}|^{2} dv^{G_{c_{i}}}}

≤ \frac{λ_{i}^{2} ∫_{ℝ^{k+1}×S^{n−k−1}} |ψ_{i}|^{\frac{2(n+1)}{n−1}} dv^{G_{c_{i}}}}{∫_{ℝ^{k+1}×S^{n−k−1}} |ψ_{i}|^{2} dv^{G_{c_{i}}}}

≤ λ_{i}^{2}mi^{\frac{4}{n−1}} .

Since λ_{i} tends to zero this proves (50). Restricting to subsequence we can assume that lim_{i→∞} c_{i} exists and we denote this limit by c ∈ [−1, 1]. We apply Lemma 3.3 with α = 1/i, (V, γ_{α}) = (ℝ^{k+1} × S^{n−k−1}, G_{c_{i}}), (V, γ_{0}) = (ℝ^{k+1} × S^{n−k−1}, G_{c}),

p_{α} = p = P, and b_{α} = mi^{\frac{2}{n−1}} . For r > 0 we obtain a diffeomorphism

Θ_{i} : B^{n}(r) → B^{G_{c_{i}}} (P, mi^{\frac{2}{n−1}} r)

such that Θ_{i}^{∗}(mi^{\frac{4}{n−1}} (G_{c_{i}})) tends to the Euclidean metric ξ^{n} on B^{n}(r). Proceeding as in Subcase I.1 of Theorem 1.2 we construct a non-zero spinor ψ belonging to L^{\frac{2n}{n−1}} (ℝ^{n}) such that

D^{ξ^{n}}ψ = lim_{i→∞} λ_{i}|ψ|^{\frac{2}{n−1}} ψ = 0.

Again as in Subcase I.1 of Theorem 1.2 we get 0 ≥ λ_{min}^{+}(S^{n}, σ^{n}), which is false. This proves Theorem 1.1.

References

[1] B. Ammann, The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions, Preprint, 2003.

[2] , A spin-conformal lower bound of the first positive Dirac eigenvalue, Diff. Geom. Appl. 18 (2003), 21–32.

[3] , A variational problem in conformal spin geometry, Habilitationsschrift, Universität Hamburg, 2003.

[4] B. Ammann, M. Dahl, and E. Humbert, Surgery and harmonic spinors, Preprint, ArXiv math.DG/0606224, 2006, to appear in Adv. Math. 220 (2009) no. 2, 523–539.

[5] , Smooth Yamabe invariant and surgery, Preprint, ArXiv 0804.1418, 2008.

[6] B. Ammann, J. F. Grosjean, E. Humbert, and B. Morel, A spinorial analogue of Aubin’s inequality, Math. Z. 260 (2008), 127–151.

[7] B. Ammann and E. Humbert, The spinorial τ-invariant and 0-dimensional surgery, J. reine angew. Math., 624 (2008), 27–50.

[8] H. Baum, Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten, Teubner Verlag, 1981.

[9] J.-P. Bourguignon and P. Gauduchon, Spineurs, opérateurs de Dirac et variations de métriques, Comm. Math. Phys. 144 (1992), 581–599.

[10] T. Friedrich, Dirac Operators in Riemannian Geometry, Graduate Studies in Mathematics 25, AMS, Providence, Rhode Island, 2000.

[11] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Grundlehren der mathematischen Wissenschaften, no. 224, Springer-Verlag, 1977.

[12] O. Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys. 104 (1986), 151–162.

[13] , Première valeur propre de l’opérateur de Dirac et nombre de Yamabe, C. R. Acad. Sci. Paris t. 313, Série I (1991), 865–868.

[14] , Spectral properties of the Dirac operator and geometrical structures., Ocampo, Hernan (ed.) et al., Geometric methods for quantum field theory. Proceedings of the summer school, Villa de Leyva, Colombia, July 12-30, 1999. Singapore: World Scientific. 116–169 , 2001.

[15] N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1–55.

[16] R. C. Kirby, The topology of 4-manifolds, Lecture Notes in Mathematics, vol. 1374, Springer-Verlag, Berlin, 1989.

[17] O. Kobayashi, Scalar curvature of a metric with unit volume, Math. Ann. 279 (1987), no. 2, 253–265.

[18] A. A. Kosinski, Differential manifolds, Pure and Applied Mathematics, vol. 138, Academic Press Inc., Boston, MA, 1993.

[19] H. B. Lawson and M.-L. Michelsohn, Spin geometry, Princeton University Press, Princeton, 1989.

[20] J. Lott, Eigenvalue bounds for the Dirac operator, Pacific J. of Math. 125 (1986), 117–126.

[21] R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations, Lect. 2nd Sess., Montecatini/Italy 1987, Lect. Notes Math. 1365, 120-154, 1989.

[22] R. E. Stong, Notes on cobordism theory, Mathematical notes, Princeton University Press, Princeton, N.J., 1968.

NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germany

E-mail address: bernd.ammann@mathematik.uni-regensburg.de

Institutionen för Matematik, Kungliga Tekniska Högskolan, 100 44 Stockholm, Sweden

E-mail address: dahl@math.kth.se

Institut Élie Cartan, BP 239, Université de Nancy 1, 54506 Vandoeuvre-lès-Nancy Cedex, France

E-mail address: humbert@iecn.u-nancy.fr