SURGERY AND HARMONIC SPINORS

BERND AMMANN, MATTIAS DAHL, AND EMMANUEL HUMBERT

Abstract. Let M be a compact spin manifold with a chosen spin structure. The Atiyah-Singer index theorem implies that for any Riemannian metric on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and the spin structure. We show that for generic metrics on M this bound is attained.

1. Introduction

We suppose that M is a compact spin manifold. By a spin manifold we will mean a smooth manifold equipped with an orientation and a spin structure. After choosing a metric g on M, one can define the spinor bundle Σ^{g}M and the Dirac operator D^{g} : Γ(Σ^{g}M) → Γ(Σ^{g}M), see [8, 16, 11].

Being a self-adjoint elliptic operator, D^{g} shares many properties with the Hodge-Laplacian Δ_{p}^{g} : Γ(Λ^{p}T^{∗}M) → Γ(Λ^{p}T^{∗}M). In particular, if M is compact, then the spectrum is discrete and real, and the kernel is finite-dimensional. Elements of ker Δ_{p}^{g} resp. ker D^{g} are called harmonic forms resp. harmonic spinors.

However, the relations of Δ_{p}^{g} and D^{g} to topology have a slightly different character. Hodge theory tells us that the Betti numbers dim ker Δ_{p}^{g} only depend on the topological type of M. The dimension of the kernel of D^{g} is invariant under conformal changes of the metric, however it does depend on the choice of conformal structure. The first examples of this phenomenon were constructed by Hitchin [12], and it was conjectured by several people that dim ker D^{g} depends on the metric for all compact spin manifolds of dimension ≥ 3.

On the other hand, the Index Theorem of Atiyah and Singer gives a topological lower bound on the dimension of the kernel of the Dirac operator. For M a compact spin manifold of dimension n this bound is [16], [3, Section 3],

dim ker D^{g} ≥ \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}  (1)

Here the Â-genus Â(M) ∈ ℤ and the α-genus α(M) ∈ ℤ/2ℤ are invariants of (the spin bordism class of) the differential spin manifold M, and g is any Riemannian metric on M.

Date: October 23, 2008.

2000 Mathematics Subject Classification. 53C27 (Primary) 55N22, 57R65 (Secondary).

Key words and phrases. Dirac operator, eigenvalue, surgery.

This work was initiated when the authors visited the Albert Einstein Institute, Golm, Germany, and was continued at the IECN Nancy, France. We wish to thank these institutes for their very kind hospitality and support.

It is hence natural to ask whether metrics exist for which equality holds in (1). Such metrics will be called D-minimal. In [17] it is proved that a generic metric on a manifold of dimension ≤ 4 is D-minimal. In [3] the same result is proved for manifolds of dimension at least 5 which are simply connected, or which have fundamental group of certain types. The argument in [3] utilizes the surgerybordism method which has proven itself very powerful in the study of manifolds allowing positive scalar curvature metrics. In a similar fashion we will use surgery methods to prove the following.

Theorem 1.1. Let M be a compact connected spin manifold without boundary. Then a generic metric on M is D-minimal.

Here “generic” means that the set of all D-minimal metrics is dense in the C^{∞}- topology and open in the C^{1}-topology on the set of all metrics. The essential step in the proof is to construct one D-minimal metric on a given spin manifold. Theorem 1.1 then follows from well-known results in perturbation theory.

Since dim ker D behaves additively with respect to disjoint union of spin manifolds while the Â-genus/α-genus may cancel it is easy to find disconnected manifolds with no D-minimal metric.

Let us also mention that if M is a compact Riemann surface of genus at most 2, then all metrics are D-minimal. The same holds for Riemann surfaces of genus 3 or 4 whose spin structures are not spin boundaries. However, if the genus is at least 5 (or equal to 3 or 4 with a spin structure which is a spin boundary), then there are also metrics with larger kernel, see [12], [4].

In order to explain the surgery-bordism method used in the proof of Theorem 1.1 we have to fix some notation.

Let M and N be spin manifolds of the same dimension. A smooth embedding f : N → M is called spin preserving if the pullback by f of the orientation and spin structure of M is the orientation and spin structure of N. If M is a spin manifold we denote by M^{−} the same manifold with the opposite orientation.

For l ≥ 1 we denote by B^{l}(R) the standard l-dimensional open ball of radius R and by S^{l−1}(R) its boundary. We abbreviate B^{l} = B^{l}(1) and S^{l−1} = S^{l−1}(1). The standard Riemannian metrics on B^{l}(R) and S^{l−1}(R) are denoted by g^{flat} and g^{round}. We equip S^{l−1}(R) with the bounding spin structure, i.e. the spin structure obtained by restricting the unique spin structure on B^{l}(R). (If l > 2 the spin structure on S^{l−1}(R) is unique, if l = 2 it is not.)

Let f : S^{k} × \overline{B^{n−k}} → M be a spin preserving embedding. We define

M̃ = (M \ f(S^{k} × \overline{B^{n−k}})) ∪ (\overline{B^{k+1}} × S^{n−k−1}) /∼,

where ∼ identifies the boundary of \overline{B^{k+1}} × S^{n−k−1} with f(S^{k} × S^{n−k−1}). The topological space M̃ carries a differential structure and a spin structure such that the inclusions M \f(S^{k}×B^{n−k}) ↪ M̃ and \overline{B^{k+1}}×S^{n−k−1} ↪ M̃ are spin preserving smooth embeddings. We say that M̃ is obtained from M by surgery of dimension k or by surgery of codimension n − k.

The proof of Theorem 1.1 relies on the following surgery theorem.

Theorem 1.2. Let (M, g^{M}) be a compact Riemannian spin manifold of dimension n. Let M̃ be obtained from M by surgery of dimension k, where 0 ≤ k ≤ n − 2.

Then M̃ carries a metric gM̃ such that

dim ker D^{gM̃} ≤ dim ker D^{g^{M}} .

Theorem 1.2 fits nicely in a hierarchy of results on surgery invariance, all of which also hold in the trivial case of codim = 0 surgery. The α-genus (equal to the index of the Cℓ_{n}-linear Dirac operator 𝔇, [16, Chap. II, §7]) is invariant under all surgeries, the minimal dimension of ker D is non-increasing when codim ≥ 2 surgeries are performed, positivity of scalar curvature is preserved under codim ≥ 3 surgeries [10], and for p ≥ 1 positivity of p-curvature is preserved under codim ≥ p + 3 surgeries [15].

In the case n − k = codim ≥ 3, Theorem 1.2 is a special case of [3, Theorem 1.2], where the proof relies on the positive scalar curvature of the surgery cosphere S^{n−k−1} and the surgery result of [10]. In the codim = 2 case, the cosphere S^{1} does not allow positive scalar curvature. However, the spin structure on this cosphere is the one that bounds a disk. The fact that the Dirac operator on S^{1} with the bounding spin structure is invertible is used in an essential way in the proof of Theorem 1.2.

After the publication of the preprint version of the present article, the results and the techniques have been applied to ℤ-periodic manifolds, i.e. manifolds Ñ with a free ℤ-action such that N = Ñ/ℤ is a smooth compact spin manifold [19, 2].

2. Preliminaries

2.1. Spinor bundles for different metrics. Let M be a spin manifold of dimension n and let g and g′ be Riemannian metrics on M. The goal of this subsection is to identify the spinor bundles of (M, g) and (M, g′) using the method of Bourguignon and Gauduchon [6].

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

where (σ, Σ_{n}) is the complex spinor representation, and where [s, φ] ∈ Spin(M, g)×_{σ} 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} preserves fiberwise length of spinors.

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 [6, Thm. 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),  (2)

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

|A_{g′}^{g}| ≤ C|g − g′|_{g},  (3)

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

|B_{g′}^{g}| ≤ C(|g − g′|_{g} + |∇^{g}(g − g′)|_{g})  (4)

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 we have

gD^{g′}(F^{− \frac{n−1}{2}} ψ) = F^{− \frac{n+1}{2}} D^{g}ψ,  (5)

according to [12, 5, 11].

2.2. Notations for spaces of spinors. Throughout the article φ, ψ, and variants denote spinors, i.e. sections of the spinor bundle. If S is a closed or open subset of a manifold M we write C^{k}(S) both for the space of k times continuously differentiable functions on S and for the space of k times continuously differentiable spinors. On C^{k}(S) we have the norm

∥φ∥_{C^{k}(S)} = ∑_{l=0}^{k} sup_{x∈S} |∇^{l}φ(x)|.

We sometimes write ∥φ∥_{C^{k}(S,g)} instead of ∥φ∥_{C^{k}(S)} to indicate that the spinor bundle and the norm depend on g. The analogous notation is used for Schauder spaces C^{k,α}.

The spaces of square-integrable functions and spinors are denoted by L^{2}(S) = L^{2}(S, g) and equipped with the norm

∥φ∥_{L^{2}(S,g)}^{2} = ∫_{S} |φ|^{2} dv^{g}.

Further, H_{k}^{2}(S) = H_{k}^{2}(S, g) denotes the Sobolev spaces of functions and spinors equipped with the norm

∥φ∥_{H_{k}^{2}(S,g)}^{2} = ∑_{l=0}^{k} ∫_{S} |∇^{l}φ|^{2} dv^{g}.

Let U be an open subset of M. The set of locally C^{1}-spinors C_{loc}^{1}(U) carries a topology such that φ_{i} → φ in C_{loc}^{1}(U) if and only if φ_{i} → φ in C^{1}(K) for any compact subset K ⊂ U.

2.3. Regularity and elliptic estimates. In this section M is not assumed to be compact.

Lemma 2.1. Let (M, g) be a Riemannian manifold and let ψ be a spinor in L^{2}(M). If ψ is weakly harmonic, that is

∫_{M}⟨ψ, Dφ⟩ dv^{g} = 0

for all compactly supported smooth spinors φ, then ψ is smooth.

Lemma 2.2. Let (M, g) be a Riemannian manifold and let K ⊂ M be a compact subset. Then there is a constant C = C(K, M, g) such that

∥ψ∥_{C^{2}(K,g)} ≤ C∥ψ∥_{L^{2}(M,g)}

for all harmonic spinors ψ on (M, g).

Proof of the lemmata. The assumptions of Lemma 2.1 imply that ∫_{M}⟨ψ, D^{2}φ⟩ dv^{g} = 0 for any compactly supported smooth spinor φ. Writing down the equation in local coordinates, one can use standard results on elliptic partial differential equations (see for example [9, Theorem 8.13]) to derive via recursion that ψ is contained in H_{k}^{2}(K, g) for any k ∈ ℕ and any compact K ⊂ M with smooth boundary. Further,

∥ψ∥_{H_{k}^{2}(K,g)} ≤ C∥ψ∥_{L^{2}(M,g)}.  (6)

The Sobolev embedding H_{k}^{2}(K, g) → C^{1}(K, g) for k > n/2+1 (see [1, Theorem 6.2]) tells us that ψ ∈ C^{1}(K, g) and gives us an estimate for ∥ψ∥_{C^{1}(K,g)} analogous to (6). Now one can use Schauder estimates (as in [9, Theorem 6.6]) to conclude that ψ is smooth on any compact set K′ contained in the interior of K, and similarly we derive the C^{2} estimate of Lemma 2.2. □

In the next lemma K ⊂ M again denotes a compact subset.

Lemma 2.3 (Ascoli’s theorem, [1, Theorem 1.30 and 1.31]). Let φ_{i} be a bounded sequence in C^{1,α}(K). Then a subsequence of φ_{i} converges in C^{1}(K).

2.4. Removal of singularities lemma. In the proof of Theorem 1.2 we will need the following lemma.

Lemma 2.4. Let (M, g) be an n-dimensional Riemannian spin manifold and let S ⊂ M be a compact submanifold of dimension k ≤ n − 2. Assume that φ is a spinor field such that ∥φ∥_{L^{2}(M)} < ∞ and D^{g}φ = 0 weakly on M \S. Then D^{g}φ = 0 holds weakly also on M.

Proof. Let ψ be a smooth spinor compactly supported in M. We have to show that

∫_{M}⟨φ, D^{g}ψ⟩ dv^{g} = 0.  (7)

Let U_{S}(δ) be the set of points of distance at most δ to S. For a small δ > 0 we choose a smooth function η : M → [0, 1] such that η = 1 on U_{S}(δ), |gradη| ≤ 2/δ, and η = 0 outside U_{S}(2δ). We rewrite the left hand side of (7) as

∫_{M}⟨φ, D^{g}ψ⟩ dv^{g} = ∫_{M}⟨φ, D^{g}((1 − η)ψ + ηψ)⟩ dv^{g}
= ∫_{M}⟨φ, D^{g}((1 − η)ψ)⟩ dv^{g}
+ ∫_{M}⟨φ, ηD^{g}ψ⟩ dv^{g} + ∫_{M}⟨φ, gradη · ψ⟩ dv^{g}.

As D^{g}φ = 0 weakly on M \ S the first term vanishes. The absolute value of the second term is bounded by

∥φ∥_{L^{2}(U_{S}(2δ))}∥D^{g}ψ∥_{L^{2}(U_{S}(2δ))}

which tends to 0 as δ → 0. Finally, the absolute value of the third term is bounded by

\begin{aligned}\frac{2}{δ}∥φ∥_{L^{2}(U_{S}(2δ))}∥ψ∥_{L^{2}(U_{S}(2δ))} & ≤ \frac{C}{δ} ∥φ∥_{L^{2}(U_{S}(2δ))}Vol(U_{S}(2δ) ∩ supp(ψ))^{\frac{1}{2}} \\ & ≤ C∥φ∥_{L^{2}(U_{S}(2δ))}δ^{\frac{n−k}{2} −1}. \end{aligned}

Since n − k ≥ 2, the third term also tends to 0 as δ → 0. □

2.5. Products with spheres.

Lemma 2.5. Let λ be an eigenvalue of (D^{g^{round}})^{2} on S^{l} for some l ≥ 1. Then λ ≥ l^{2}/4.

When l = 1 it is important that S^{1} is equipped with the spin structure which bounds the disk, for the non-bounding spin structure there are harmonic spinors and the estimate is not true.

Proof. First we look at the case l = 1. Since S^{1} carries the bounding spin structure it follows that the action of S^{1} on itself by multiplication does not lift to ΣS^{1}. However the action of the non-trivial double covering of S^{1}—which is again S^{1}— lifts to ΣM, turning Γ(ΣM) into an S^{1}-representation. The irreducible components have odd weight. From this one concludes that λ = (k + \frac{1}{2})^{2} for some k ∈ ℤ, and the lemma follows.

In the case l ≥ 2 the scalar curvature of (S^{l}, g^{round}) is scal^{g^{round}} = l(l − 1) and thus Friedrich’s bound [7] implies

λ ≥ \frac{l}{4(l − 1)} min_{x∈S^{l}} scal^{g^{round}} = l^{2}/4.

□

If (M, g) and (N, h) are compact Riemannian spin manifolds then the Dirac operator D^{g+h} on (M × N, g + h) can be written as D^{vert} + D^{hor} where D^{vert} is the part of D which only contains derivatives along M, and D^{hor} is the part which only contains derivatives along N. These two operators anticommute, and thus

(D^{g+h})^{2} = (D^{vert})^{2} + (D^{hor})^{2}.

It is easy to see that D^{vert} has the same spectrum as D^{g}, but with infinite multiplicities. The same holds for D^{hor} and D^{h}. We conclude the following.

Proposition 2.6. Let (M, g) be a compact spin manifold and l ≥ 1. Then the spectrum of (D^{g+g^{round}})^{2} on M × S^{l} is bounded from below by l^{2}/4.

3. Proof of Theorem 1.2

For Theorem 1.2 we have given a compact Riemannian spin manifold (M, g) of dimension n. The manifold M̃ is the result of surgery on M using an embedding f : S^{k} × \overline{B^{n−k}} → M. We assume n − k ≥ 2.

We will here use a slightly more detailed description of the surgery data which better suits our geometric constructions. Let i : S^{k} → M be an embedding and denote by S the image of i. Let π^{ν} : ν → S be the normal bundle of S in (M, g). We assume that a trivialization of ν is given in the form of a vector bundle map ι : S^{k} × ℝ^{n−k} → ν with (π^{ν} ∘ ι)(p, 0) = i(p) for p ∈ S^{k}. We also assume that ι is fiberwise an isometry when the fibers ℝ^{n−k} of S^{k} × ℝ^{n−k} are given the standard metric and the fibers of ν have the metric induced by g. We then get an embedding as above by setting f = exp^{ν} ∘ι : S^{k} × \overline{B^{n−k}(R)} → M for sufficiently small R. Here exp^{ν} denotes the normal exponential map of S. For small R we define open neighbourhoods U_{S}(R) of S by

U_{S}(R) = (exp^{ν} ∘ι)(S^{k} × B^{n−k}(R)).

3.1. Approximation by a metric of product form near S. In the following r(x) denotes the distance from the point x to S with respect to the metric g. We denote by h the restriction of g to the tangent bundle of S.

Lemma 3.1. For sufficiently small R > 0 there is a constant C > 0 so that

G = g − ((exp^{ν} ∘ι)^{−1})^{∗}(g^{flat} + h)

satisfies

|G(x)| ≤ Cr(x), |∇G(x)| ≤ C

on U_{S}(R).

Note that in this lemma the function r(x) is by definition the distance of x to S with respect to g but it coincides with the distance of x to S with respect to the metric ((exp^{ν} ∘ι)^{−1})^{∗}(g^{flat} + h).

Proof. Since x ↦ ∇G(x) is continuous on a neighborhood of S we can find a constant C such that |∇G(x)| ≤ C for sufficiently small R > 0. Now, let x ∈ S. First we notice that the spaces T_{x}S and ν_{x} are orthogonal with respect to both the scalar products g(x) and ((exp^{ν} ∘ι)^{−1})^{∗}(g^{flat} + h)(x). It is also clear that these scalar products coincide on T_{x}S. Since the differential d(exp^{ν} ∘ι) is an isometry, they coincide also on ν_{x}. This implies that g(x) = ((exp^{ν} ∘ι)^{−1})^{∗}(g^{flat} + h)(x) and hence that G(x) = 0. We obtain that G vanishes on S. Since G is C^{1}, |G| is 1-lipschitzian and thus there exists C > 0 such that |G(x)| ≤ Cr(x). □

The following proposition allows us to assume that the metric g has product form close to the surgery sphere S.

Proposition 3.2. Let (M, g) and S be as above. Then there is a metric g̃ on M and δ_{0} > 0 such that d^{g}(x, S) = dg̃(x, S), g̃ has product form on U_{S}(δ_{0}), and

dim ker Dg̃ ≤ dim ker D^{g}.

For δ > 0 let η be a smooth cut-off function such that 0 ≤ η ≤ 1, η = 1 on U_{S}(δ), η = 0 on M \ U_{S}(2δ), and |dη|_{g} ≤ 2/δ. We set

g_{δ} = η((exp^{ν} ∘ι)^{−1})^{∗}(g^{flat} + h) + (1 − η)g.

Then g_{δ} has product form on U_{S}(δ) and d^{g}(x, S) = d^{g_{δ}}(x, S) = r(x). Through a series of lemmas we will prove Proposition 3.2 for g̃ = g_{δ} with sufficiently small δ.

In the following estimates C denotes a constant whose value might vary from one line to another, which is independent of δ and η but might depend on M, g, S. Terms denoted by o_{i}(1) tend to zero when i → ∞.

Lemma 3.3. Let δ_{i} be a sequence with δ_{i} → 0 as i → ∞. Let φ_{i} be a sequence of spinors on (M, g_{δ_{i}}) such that D^{g_{δ_{i}}} φ_{i} = 0 and ∫_{M} |φ_{i}|^{2} dv^{g_{δ_{i}}} = 1. Then the sequence β_{g}^{g_{δ_{i}}} φ_{i} is bounded in H_{1}^{2}(M, g).

Proof. As ∫ |β_{g}^{g_{δ_{i}}} φ_{i}|^{2} dv^{g} = 1 + o_{i}(1) we have to show that

α_{i} = \sqrt{∫_{M} |∇^{g}(β_{g}^{g_{δ_{i}}} φ_{i})|_{g}^{2} dv^{g}}

is bounded. Suppose that the opposite is true, that is α_{i} → ∞, and set ψ_{i} = α_{i}^{−1}β_{g}^{g_{δ_{i}}} φ_{i}. Then we have ^{g}D^{g_{δ_{i}}} ψ_{i} = 0 since β_{g_{δ_{i}}}^{g} ∘ β_{g}^{g_{δ_{i}}} = Id, so the Schrödinger-Lichnerowich formula [16, Theorem 8.8, p. 160] together with (2) tells us that

1 = ∫_{M} |∇^{g}ψ_{i}|_{g}^{2} dv^{g}

= ∫_{M}(|D^{g}ψ_{i}|^{2} − \frac{1}{4}scal^{g}|ψ_{i}|^{2}) dv^{g}

= ∫_{M}(|A_{g_{δ_{i}}}^{g} (∇^{g}ψ_{i}) + B_{g_{δ_{i}}}^{g} (ψ_{i})|^{2} − \frac{1}{4}scal^{g}|ψ_{i}|^{2}) dv^{g}

≤ ∫_{M}(2|A_{g_{δ_{i}}}^{g} (∇^{g}ψ_{i})|^{2} + 2|B_{g_{δ_{i}}}^{g} (ψ_{i})|^{2} − \frac{1}{4}scal^{g}|ψ_{i}|^{2}) dv^{g}.

Using (3), (4), Lemma 3.1, and the fact that g and g_{δ_{i}} coincide outside U_{S}(2δ_{i}) we get

1 ≤ Cδ_{i}^{2} ∫_{U_{S}(2δ_{i})} |∇^{g}ψ_{i}|_{g}^{2} dv^{g} + C ∫_{U_{S}(2δ_{i})} |ψ_{i}|^{2} dv^{g} + C ∫_{M} |ψ_{i}|^{2} dv^{g}

≤ Cδ_{i}^{2} + C ∫_{U_{S}(2δ_{i})} |ψ_{i}|^{2} dv^{g} + α_{i}^{−2}(1 + o_{i}(1))

≤ C ∫_{U_{S}(2δ_{i})} |ψ_{i}|^{2} dv^{g} + o_{i}(1).

≤ Cα_{i}^{−2} \underbrace{∫_{M} |φ_{i}|^{2} dv^{g}}_{=1+o_{i}(1)} +o_{i}(1) → 0

which is obviously a contradiction. □

Lemma 3.4. Again let δ_{i} be a sequence with δ_{i} → 0 as i → ∞ and let φ_{i} be a sequence of spinors on (M, g_{δ_{i}}) such that D^{g_{δ_{i}}} φ_{i} = 0 and ∫_{M} |φ_{i}|^{2} dv^{g_{δ_{i}}} = 1. Then, after passing to a subsequence, β_{g}^{g_{δ_{i}}} φ_{i} converges weakly in H_{1}^{2}(M, g) and strongly in L^{2}(M, g) to a harmonic spinor on (M, g).

Proof. According to Lemma 3.3 the sequence β_{g}^{g_{δ_{i}}} φ_{i} is bounded in H_{1}^{2}(M, g) and hence a subsequence converges weakly in H_{1}^{2}(M, g). After passing to a subsequence once again we obtain strong convergence in L^{2}(M, g). We denote the limit spinor by φ.

0 < ρ << r_{0} < r_{1}/2 << R_{max}

Figure 1. Hierarchy of variables

Lemma 2.2 implies that β_{g}^{g_{δ_{i}}} φ_{i} is bounded in C^{2}(M \ U_{S}(ε)) for any ε > 0, and Lemma 2.3 then implies that a subsequence converges in C^{1}(M \U_{S}(ε)). Hence the limit φ is in C_{loc}^{1}(M \ S) and satisfies D^{g}φ = 0 on M \ S. Since φ is in L^{2}(M, g) it follows from Lemma 2.4 that φ is a weak solution of Dφ = 0 on (M, g). By Lemma 2.1 it then follows that φ is a strong solution and a harmonic spinor on (M, g). □

Proof of Proposition 3.2. Let m = lim inf_{δ→0} dim ker D^{g_{δ}}. For sufficiently small δ let φ_{δ}^{1}, …, φ_{δ}^{m} ∈ ker D^{g_{δ}} be spinors such that

∫_{M}⟨φ_{δ}^{j}, φ_{δ}^{k}⟩ dv^{g_{δ}} = \begin{cases}1, & if j = k; \\ 0, & if j ≠ k. \end{cases}  (8)

According to Lemma 3.4 there are spinors φ^{1}, …, φ^{m} ∈ ker D^{g} and a sequence δ_{i} → 0 such that β_{g}^{g_{δ_{i}}} φ_{δ_{i}}^{j} converges to φ^{j} weakly in H_{1}^{2}(M, g) and strongly in L^{2}(M, g) for j = 1, …, m. Because of strong L^{2}-convergence the orthogonality relation (8) is preserved in the limit so dim ker D^{g} ≥ m. Hence there is a δ_{0} > 0 so that dim ker D^{g_{δ_{0}}} = m ≤ dim ker D^{g} and the Proposition is proved with g̃ = g_{δ_{0}}. □

3.2. Proof for metrics of product form near S. We now assume that g is a product metric on U_{S}(R_{max}) for some R_{max} > 0. This we may do by Proposition 3.2. In polar coordinates (r, Θ) ∈ (0, R_{max}) × S^{n−k−1} on B^{n−k}(R_{max}) we have

g = g^{flat} + h = dr^{2} + r^{2}g^{round} + h.

Let ρ > 0 be a small number which we will finally let tend to 0 (see also Figure 1). We decompose M into three parts

(1) M \ U_{S}(R_{max}),

(2) U_{S}(R_{max}) \ U_{S}(ρ/2) = [ρ/2, R_{max}) × S^{n−k−1} × S^{k},
(3) U_{S}(ρ/2) = B^{n−k}(ρ/2) × S^{k}.

The manifold M̃ is obtained by removing part (3) and by gluing in S^{n−k−1} ×B^{k+1}, that is M̃ is the union of

(1) M \ U_{S}(R_{max}),

(2) U_{S}(R_{max}) \ U_{S}(ρ/2) = [ρ/2, R_{max}) × S^{n−k−1} × S^{k},

(3’) S^{n−k−1} × B^{k+1}.

We now define a sequence of metrics g_{ρ} on M̃. The metrics g_{ρ} will coincide with g on part (1), but will be modified on part (2) in order to close up nicely on part (3’).

Let r_{0}, r_{1} be fixed such that 2ρ < r_{0} < r_{1}/2 < R_{max}/2. Define g_{ρ} on M̃ by

(1) g_{ρ} = g on M \ U_{S}(R_{max}),

(2) g_{ρ} = F^{2}(dr^{2} + r^{2}g^{round} + f_{ρ}^{2}h) on (ρ/2, R_{max}) × S^{n−k−1} × S^{k}, where F and f_{ρ} satisfy

F (r) = \begin{cases}1, & if r_{1} < r < R_{max}; \\ 1/r, & if r < r_{0}, \end{cases} and f_{ρ}(r) = \begin{cases}1, & if r > 2ρ; \\ r, & if r < ρ . \end{cases}

Figure 2. The metric g_{ρ}.

(3’) g_{ρ} = g^{round} + γ_{ρ} on S^{n−k−1} × B^{k+1} where γ_{ρ} is some metric so that g_{ρ} is

smooth. In order to visualize the metric g_{ρ} two projections are drawn in Figure 2. In

both projections the horizontal direction represents − log r. In the first projection the vertical direction indicates the size of the cosphere S^{n−k−1}. In the second projection the vertical direction indicates the size of S which is fiberwise homothetic to (S ≅ S^{k}, h).

Our goal is to prove that

dim ker D^{g_{ρ}} ≤ dim ker D^{g}  (9)

for small ρ > 0. Before we are able to do that we need some estimates. For α ∈ (0, R_{max}) let Ũ(α) = M̃ \ (M \ U_{S}(α)) so that M \ U_{S}(α) = M̃ \ Ũ(α).

Proposition 3.5. Let s ∈ (0, r_{0}/2). Let ψ_{ρ} be a harmonic spinor on (M̃, g_{ρ}). Then for 2ρ ∈ (0, s) it holds that

∫_{Ũ(s)\Ũ(2ρ)} |F^{\frac{n−1}{2}} ψ_{ρ}|^{2} dv^{g} ≤ 32 ∫_{Ũ(2s)\Ũ(s)} |F^{\frac{n−1}{2}} ψ_{ρ}|^{2} dv^{g}.

Proof. Let η ∈ C^{∞}(M̃) be a cut-off function with 0 ≤ η ≤ 1, η = 1 on Ũ(s), η = 0 on M̃ \ Ũ(2s), and

|dη|_{g} ≤ \frac{2}{s}.  (10)

The spinor ηψ_{ρ} is compactly supported in Ũ(2s). Moreover, the metric g_{ρ} can be written as g_{ρ} = g^{round}+h_{ρ} on Ũ(2s) where the metric h_{ρ} is equal to r^{−2}dr^{2}+r^{−2}f_{ρ}^{2}h on Ũ(2s)\Ũ(ρ/2) and is equal to γ_{ρ} on S^{n−k−1}×B^{k+1} = Ũ(ρ/2). Hence (Ũ(2s), g_{ρ}) is isometric to an open subset of a manifold of the form S^{n−k−1} × N equipped with a product metric g^{round} + g_{N}, where N is compact. By Proposition 2.6 the squared eigenvalues of the Dirac operator on this product manifold are greater than or equal to (n − k − 1)^{2}/4 ≥ 1/4. Writing the Rayleigh quotient of ηψ_{ρ} we obtain

\frac{1}{4} ≤ \frac{∫_{Ũ(2s)} |D^{g_{ρ}}(ηψ_{ρ})|^{2} dv^{g_{ρ}}}{∫_{Ũ(2s)} |ηψ_{ρ}|^{2} dv^{g_{ρ}}} .  (11)

Since D^{g_{ρ}}ψ_{ρ} = 0 we have D^{g_{ρ}}(ηψ_{ρ}) = grad^{g_{ρ}}η · ψ_{ρ} so

|D^{g_{ρ}}(ηψ_{ρ})|^{2} = |grad^{g_{ρ}}η · ψ_{ρ}|^{2} = |dη|_{g_{ρ}}^{2}|ψ_{ρ}|_{g_{ρ}}^{2}.  (12)

By definition dη is supported in Ũ(2s) \ Ũ(s). On M̃ \ Ũ(2ρ) we have g_{ρ} = F^{2}g. Moreover, by Relation (10) and since F = 1/r on the support of dη, we have

|dη|_{g_{ρ}}^{2} = r^{2}|dη|_{g}^{2} ≤ \frac{4r^{2}}{s^{2}}

and hence

|D^{g_{ρ}}(ηψ_{ρ})|^{2} ≤ \frac{4r^{2}}{s^{2}} |ψ_{ρ}|^{2}.

Since g_{ρ} = r^{−2}g on Ũ(2s) \ Ũ(s) we have dv^{g_{ρ}} = r^{−n} dv^{g}. Using equation (12) it follows that

\begin{aligned}∫_{Ũ(2s)} |D^{g_{ρ}}(ηψ_{ρ})|^{2} dv^{g_{ρ}} & ≤ \frac{4}{s^{2}} ∫_{Ũ(2s)\Ũ(s)} r^{2+(n−1)−n}|r^{− \frac{n−1}{2}} ψ_{ρ}|^{2} dv^{g} \\ & ≤ \frac{8}{s} ∫_{Ũ(2s)\Ũ(s)} |F^{\frac{n−1}{2}} ψ_{ρ}|^{2} dv^{g}, \end{aligned} (13)

where we also use that r ≤ 2s on the domain of integration. Since η ∈ [0, 1] on Ũ(2s) \ Ũ(s), since η = 1 on Ũ(s), and since g_{ρ} = r^{−2}g on Ũ(s) \ Ũ(2ρ), we have

\begin{aligned}∫_{Ũ(2s)} |ηψ_{ρ}|^{2} dv^{g_{ρ}} & ≥ ∫_{Ũ(s)\Ũ(2ρ)} |ψ_{ρ}|^{2} dv^{g_{ρ}} \\ & = ∫_{Ũ(s)\Ũ(2ρ)} r^{(n−1)−n}|r^{− \frac{n−1}{2}} ψ_{ρ}|^{2} dv^{g} \\ & ≥ \frac{1}{s} ∫_{Ũ(s)\Ũ(2ρ)} |F^{\frac{n−1}{2}} ψ_{ρ}|^{2} dv^{g}, \end{aligned}  (14)

where we use that r ≤ s in the last inequality. Plugging (13) and (14) into (11) we get

\frac{1}{4} ≤ \frac{\frac{8}{s} ∫_{Ũ(2s)\Ũ(s)} |F^{\frac{n−1}{2}} ψ_{ρ}|^{2} dv^{g}}{\frac{1}{s} ∫_{Ũ(s)\Ũ(2ρ)} |F^{\frac{n−1}{2}} ψ_{ρ}|^{2} dv^{g}}

and hence Proposition 3.5 follows. □

Proof of Theorem 1.2. We are now ready to prove (9). For a contradiction suppose that this relation does not hold. Then there is a strictly decreasing sequence ρ_{i} → 0 such that dim ker D^{g} < dim ker D^{g_{ρ_{i}}} for all i. To simplify the notation for subsequences we define E = \{ρ_{i} : i ∈ ℕ\}. We have 0 ∈ \overline{E} and passing to a subsequence of ρ_{i} means passing to a subset E′ ⊂ E with 0 ∈ \overline{E′}.

Let m = dim ker D^{g} + 1. For all ρ ∈ E we can find D^{g_{ρ}}-harmonic spinors ψ_{ρ}^{1}, …, ψ_{ρ}^{m} on (M̃, g_{ρ}) such that

∫_{M\U_{S}(s)}⟨φ_{ρ}^{j}, φ_{ρ}^{k}⟩ dv^{g} = ∫_{M̃\Ũ(s)}⟨φ_{ρ}^{j}, φ_{ρ}^{k}⟩ dv^{g} = \begin{cases}1, & if j = k; \\ 0, & if j ≠ k, \end{cases}  (15)

where φ_{ρ}^{j} = F^{\frac{n−1}{2}} ψ_{ρ}^{j} and s ≤ r_{0} < r_{1}/2 is fixed as above. The spinor fields φ_{ρ}^{j} are defined on M \ U_{S}(2ρ) and by (5) they are D^{g}-harmonic there.

Step 1. Let δ ∈ (0, R_{max}). For j = 1, …, m and ρ ∈ E small enough we have

∫_{M\U_{S}(δ)} |φ_{ρ}^{j}|^{2} dv^{g} ≤ 33.  (16)

By Proposition 3.5 we have

∫_{U_{S}(s)\U_{S}(2ρ)} |φ_{ρ}^{j}|^{2} dv^{g} ≤ 32 ∫_{U_{S}(2s)\U_{S}(s)} |φ_{ρ}^{j}|^{2} dv^{g},

and hence if 2ρ ≤ δ it follows that

∫_{U_{S}(s)\U_{S}(δ)} |φ_{ρ}^{j}|^{2} dv^{g} ≤ 32 ∫_{M\U_{S}(s)} |φ_{ρ}^{j}|^{2} dv^{g}.

We conclude that

\begin{aligned}∫_{M\U_{S}(δ)} |φ_{ρ}^{j}|^{2} dv^{g} & = ∫_{M\U_{S}(s)} |φ_{ρ}^{j}|^{2} dv^{g} + ∫_{U_{S}(s)\U_{S}(δ)} |φ_{ρ}^{j}|^{2} dv^{g} \\ & ≤ (1 + 32) ∫_{M\U_{S}(s)} |φ_{ρ}^{j}|^{2} dv^{g}, \end{aligned}

and (16) follows from (15).

Step 2. There exists E′ ⊂ E with 0 ∈ \overline{E′} and spinors Φ^{1}, …, Φ^{m} ∈ C^{1}(M \ S), D^{g}-harmonic on M \ S such that φ_{ρ}^{j} tend to Φ^{j} in C_{loc}^{1}(M \ S) as ρ → 0, ρ ∈ E′.

Let Z ∈ ℕ be an integer, Z > 1/s. By (16) the sequence {φ_{ρ}^{j}}_{ρ∈E} is bounded in L^{2}(M \ U_{S}(1/Z)). By Lemma 2.2 it follows that {φ_{ρ}^{j}}_{ρ∈E} is bounded in C^{2}(M \ U_{S}(2/Z)) for all sufficiently large Z. For a fixed Z_{0} > 1/s we apply Lemma 2.3 and conclude that for any j there is a subsequence {φ_{ρ}^{j}}_{ρ∈E_{0}} of {φ_{ρ}^{j}}_{ρ∈E} that converges in C^{1}(M \ U_{S}(2/Z_{0})) to a spinor Φ_{0}^{j}. Similarly we construct further and further subsequences {φ_{ρ}^{j}}_{ρ∈E_{i}} converging to Φ_{i}^{j} in C^{1}(M \ U_{S}(2/(Z_{0} + i))) with E_{i} ⊂ E_{i−1} ⊂ ⋯ ⊂ E_{0} ⊂ E, 0 ∈ \overline{E_{i}}. Obviously Φ_{i}^{j} extends Φ_{i−1}^{j}. Define E′ ⊂ E as consisting of one ρ_{i} from each E_{i} chosen so that ρ_{i} → 0 as i → ∞. Then the sequence {φ_{ρ}^{j}}_{ρ∈E′} converges in C_{loc}^{1}(M \ S) to a spinor Φ^{j}. As φ_{ρ}^{j} is D^{g}-harmonic on (M \ U_{S}(2ρ)) the C_{loc}^{1}(M \ S)-convergence implies that D^{g}Φ^{j} = 0 on M \ S. We have proved Step 2.

Step 3. Conclusion.

By (16) we conclude that

∫_{M\S} |Φ^{j}|^{2} dv^{g} ≤ 33,

and hence Φ^{j} ∈ L^{2}(M) for j = 1, …, m. By Lemmas 2.4 and 2.1 we then conclude that Φ^{j} is harmonic and smooth on all of (M, g). Since M \ U_{S}(s) is a relatively compact subset of M \ S the normalization (15) is preserved in the limit ρ → 0 and hence

∫_{M\U_{S}(s)}⟨Φ^{j}, Φ^{k}⟩ dv^{g} = \begin{cases}1, & if j = k; \\ 0, & if j ≠ k. \end{cases}

This proves that Φ^{1}, …, Φ^{m} are linearly independent harmonic spinors on (M, g) and hence dim ker D^{g} ≥ m which contradicts the definition of m. This proves Relation (9) and Theorem 1.2. □

4. Proof of Theorem 1.1

The proof will follow the argument of [3] and we introduce notation in accordance with that paper. For a compact spin manifold M the space of smooth Riemannian metrics on M is denoted by R(M) and the subset of D-minimal metrics is denoted by R_{min}(M).

From standard results in perturbation theory it follows that R_{min}(M) is open in the C^{1}-topology on R(M) and if R_{min}(M) is not empty then it is dense in R(M) in all C^{k}-topologies, k ≥ 1, see for example [17, Prop. 3.1]. We define the word generic to mean these open and dense properties satisfied by R_{min}(M) if non-empty. Theorem 1.1 is then equivalent to the following.

Theorem 4.1. Let M be a compact connected spin manifold. Then there is a D-minimal metric on M.

Before we start the proof we note the following consequence of Theorem 1.2.

Proposition 4.2. Let N be a compact spin manifold which has a D-minimal metric and suppose that M is obtained from N by surgery of codimension ≥ 2. Then M has a D-minimal metric.

Proof. This follows from Theorem 1.2 since the left hand side of (1) is the same for M and N while the right hand side may only decrease. □

From the theory of handle decompositions of bordisms we have the following.

Proposition 4.3. Suppose that M is connected, dim M ≥ 3, and that M is spin bordant to a manifold N. Then M can be obtained from N by a sequence of spincompatible surgeries of codimension ≥ 2.

Proof. If dim M = 3 the statement follows from [13, VII Theorem 3]. If n = dim M ≥ 4, then we can do surgery in dimensions 0 and 1 on a given spin bordism between M and N and obtain a connected, simply connected spin bordism between M and N. It then follows from [14, VIII Proposition 3.1] that one can obtain M from N by surgeries of dimensions 0, …, n − 2. □

Proof of Theorem 4.1. From the solution of the Gromov-Lawson conjecture by Stolz [20] together with knowledge of some explicit manifolds with D-minimal metrics one can show that any compact spin manifold is spin bordant to a manifold with a D-minimal metric. This is worked out in detail in [3, Prop. 3.9]. We may thus assume that the given manifold M is spin bordant to a manifold N equipped with a D-minimal metric. If dim M ≥ 3 the Theorem then follows from Propositions 4.2 and 4.3.

Next we assume dim M = 2. If α(M) = 0 then M is obtained by surgeries of dimension 0 on S^{2}. We let the 2-torus T^{2} be equipped with the spin structure for which α(T^{2}) ≠ 0, then if α(M) ≠ 0 we know that M is obtained by surgeries of dimension 0 on T^{2}. The canonical metrics on S^{2} and T^{2} are D-minimal, so Theorem 4.1 follows from Proposition 4.2. □

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

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

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

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

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

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