THE SPINORIAL τ-INVARIANT AND 0-DIMENSIONAL SURGERY.

B. Ammann and E. Humbert

Abstract. Let M be a compact manifold with a metric g and with a fixed spin structure χ. Let λ_{1}^{+}(g) be the first non-negative eigenvalue of the Dirac operator on (M, g, χ). We set

τ(M, χ) := sup inf λ_{1}^{+}(g) where the infimum runs over all metrics g of volume 1 in a conformal class [g_{0}] on M and where the supremum runs over all conformal classes [g_{0}] on M.

Let (M^{#}, χ^{#}) be obtained from (M, χ) by 0-dimensional surgery. We prove that

τ(M^{#}, χ^{#}) ≥ τ(M, χ).

As a corollary we can calculate τ(M, χ) for any Riemann surface M.

August 2006

MSC 2000: 53C27 (Primary) 58J05, 57R65 (Secondary) Keywords: connected sum, conformal geometry, Dirac operator, smooth Yamabe number, Schoen’s sigma-constant

Contents

1. Introduction 1 2. Preliminaries 4 2.1. Notation 4 2.2. Topological spin structures versus metric spin structures 4 2.3. Surgery, handles and spin structures 4 2.4. Identifying spinors for different metrics 5 2.5. The first eigenvalue of the Dirac operator in a conformal class 6 2.6. A conformal Sobolev inequality 7 3. Approximation by locally flat manifolds 9 4. Construction of the metrics on M^{#} 10 5. Proof of Theorem 1.1 11 6. The Dirac operator on Riemann surfaces 15 References 17

1. Introduction

We assume that (M, χ) is a compact spin manifold of dimension n ≥ 2. We will always use the terminology “spin manifold” in the sense of an oriented manifold together with a chosen spin structure. The open ball around x ∈ M with radius ε is denoted as B_{x}(ε). We choose p, q ∈ M, p ≠ q, and ε ∈ (0, min{d(p, q), injrad(M)}/2). Then we define

M^{#} := (M \ (B_{p}(ε) ∪ B_{q}(ε))) ∪ ([−1, 1] × S^{n−1})/ ∼

where ∼ indicates that we glue ∂(M \ (B_{p}(ε) ∪ B_{q}(ε))) ≅ S^{n−1} ˙∪S^{n−1} together with ∂([−1, 1] × S^{n−1}) ≅ S^{n−1} ˙∪S^{n−1} such that orientations are preserved. M^{#} carries a differential structure and a spin structure such that M \ (B_{p}(ε) ∪ B_{q}(ε)) ↪ M^{#} is smooth and preserves the spin structure. The spin structure on M^{#} is uniquely determined by the spin structure of M in the sense of Lemma 2.1. We say that M^{#} is obtained from M by 0-dimensional surgery. The connected sum construction is a special case of 0-dimensional surgery, namely the case when p and q are in different connected components. In some parts of the literature, 0-dimensional surgery is also called “adding a handle”. However, we will only use the term “adding a handle” in the sense of cobordism theory (see subsection 2.3 and [Kos93] for details).

For any metric g on M let λ_{+}^{1}(M, g, χ) be the first non-negative eigenvalue of the Dirac operator on (M, g, χ). We set

λ_{min}^{+}(M, [g], χ) := inf_{g̃∈[g]} λ_{+}^{1}(M, g̃, χ)Vol(M, g̃)^{1/n}

and

τ(M, χ) := sup_{[g]∈C(M)} λ_{min}^{+}(M, [g], χ) ∈ [0, (n/2)ω_{n}^{1/n}]

where [g] denotes the conformal class of g, where C(M) is the set of conformal classes on M, and where ω_{n} denotes the volume of the standard sphere. It follows from [Amm03b] that τ(M, χ) > 0, if and only if there exists a metric on M with invertible Dirac operator. Recall that the Atiyah-Milnor-Singer-invariant (see e.g. [LM89, II.7] for details) associates to any n-dimensional spin manifold (M, χ) an element α(M, χ) ∈ KO^{−n}(pt), where KO^{−n}(pt) ≅ ℤ if n is divisible by 4, where KO^{−n}(pt) ≅ ℤ/2ℤ if n ≡ 1, 2 mod 8 and where KO^{−n}(pt) = {0} in all other dimensions. The map α defines a surjective ring homomorphism from the spin cobordism ring Ω_{spin}^{∗} to ⨁_{n∈ℕ} KO^{−n}(pt). In particular, α(M, χ) is preserved under k-dimensional surgery on M, k ∈ {0, …, n}.

If α(M, χ) ≠ 0, then the Dirac operator has a nontrivial kernel for any metric, hence τ(M, χ) = 0. The converse statement, i.e. that α(M, χ) = 0 implies τ(M, χ) > 0 for connected M was proved in successive steps by [Hit74, Mai97, BD02, ADH06]. The essential step in [BD02] is that the positivity of τ(M, χ) is preserved under k-dimensional surgery for k ∈ {0, 1, …, n−3} and in [ADH06] it is shown that positivity of τ(M, χ) is also preserved under (n − 2)-dimensional surgery.

The goal of the present article is to compare the τ-invariants of M and M^{#} where M^{#} is obtained from M by 0-dimensional surgery.

Theorem 1.1. Let (M, g, χ) be a compact connected Riemannian spin manifold and let (M^{#}, χ^{#}) be obtained by 0-dimensional surgery on (M, χ). We assume that the Dirac operator D acting on (M, g, χ) is invertible. Then, there exists a sequence of Riemannian metrics (g_{ε}^{#})_{ε} on (M^{#}, χ^{#}) such that

lim_{ε→0} λ_{min}^{+}(M^{#}, [g_{ε}^{#}], χ^{#}) = λ_{min}^{+}(M, [g], χ).

As an immediate corollary, we get

Corollary 1.2. Let (M, χ) be a compact spin manifold and (M^{#}, χ^{#}) be obtained by 0-dimensional surgery on (M, χ). Then

τ(M^{#}, χ^{#}) ≥ τ(M, χ).

Theorem 1.1 and its corollary were already known in the special case M = S^{2} [AH03].

In the case n = 2 the corollary admits to calculate τ(M, χ).

Theorem 1.3. Let M be a compact oriented surface with spin structure χ.

τ(M, χ) = \begin{cases}0 & if α(M, χ) = 1 \\ 2\sqrt{π} & if α(M, χ) = 0 \end{cases}

For Riemann surfaces the α-invariant can be easily calculated: one calculates a quadratic form q_{χ} associated to the spin structure χ, then its Arf invariant Arf(q_{χ}) is related to α(M, χ) via Arf(q_{χ}) = (−1)^{α(M,χ)} (see section 6 for details).

If one replaces the first non-negative eigenvalue by the absolute value of the largest non-positive eigenvalue |λ_{1}^{−}(g)|, then one obtains an invariant τ_{−}(M, χ). Our results also hold for τ_{−}(M, χ). In dimensions n ≢ 3 mod 4 the spectrum of the Dirac operator is symmetric, and we have τ(M, χ) = τ_{−}(M, χ). However, in the case n ≡ 3 mod 4 these invariants are expected to be different.

Let us compare our results to various other results in the literature (see also [AH03]).

The τ-invariant is a spinorial analogue to the σ-constant on compact Riemannian manifolds [Kob87] (also [Sch89]) which is defined on (M, g) by

σ(M) := sup inf \frac{∫ Scalg̃ dvg̃}{Vol(M, g̃)^{\frac{n−2}{n}}}  (1)

where the infimum runs over all metrics in a conformal class g̃ ∈ [g], and where the supremum runs over all conformal classes. (In some parts of the literature σ(M) is called the Yamabe invariant of M.) When σ(M) is positive, the invariant σ(M) can be defined also in a way analogous to τ(M) where we use the smallest eigenvalue of the conformal Laplacian L_{g} := 4 \frac{n−1}{n−2} Δ_{g} + Scal_{g} instead of λ_{+}^{1}(g).

Unfortunately, there are only few examples for which σ(M) is known and different from 0, e.g. σ(S^{n}) = n(n − 1)ω_{n}^{2/n}, σ(S^{n−1} × S^{1}) = n(n − 1)ω_{n}^{2/n}, and σ(ℝP^{3}) = 6 (\frac{ω_{3}}{2} )^{2/3}. The reader might consult [BN04] for a very elegant and amazing calculation of σ(ℝP^{3}). In a recent article, Akutagawa and Neves [AN05] could strengthen these methods which lead to a complete classification of compact 3-manifolds M with σ(M) ≥ σ(ℝP^{3}). Gursky and LeBrun [GL98] could calculate the σ-constants of several Kähler surfaces using Seiberg-Witten theory or alternatively the Atiyah-Singer index theorem for twisted Dirac operators; in particular they obtain σ(ℂP^{2}#m(S^{1} × S^{3})) = 12\sqrt{2}π for any m ∈ ℕ.

Kobayashi proved in [Kob87] that if M^{#} is obtained from M by 0-dimensional surgery, then σ(M^{#}) ≥ σ(M). A similar monotonicity formula for the σ-invariant was proved by [PY99]. Petean and Yun prove σ(M^{#}) ≥ min{σ(M), 0} if M^{#} is obtained from M by surgery of dimension 1, …, n − 3. See also [AB03] for another approach to this result. Clearly, this surgery result is particularly interesting in the case σ(M) ≤ 0, and it has many fruitful applications. In particular, any simply connected compact manifold of dimension at least 5 has σ(M) ≥ 0 [Pet03]. This result was generalized to some other simple fundamental groups in [BR02]. It also allows to rule out Einstein metrics on many spaces [Pet98]. However, in the case σ(M) > 0 it is still open under what conditions one has a monotonicity formula σ(M^{#}) ≥ σ(M). The article [Joy03] studies 0-dimensional surgery in more details, in particular he shows the non-uniqueness of minimizers of the infimum in (1) in the case σ(M) > 0. Some of these results have been recently generalized to the G-equivariant σ-invariant σ_{G} [Sung06].

The σ-invariant and the τ-invariant, are not only related by analogy, but also via Hijazi’s inequality [Hij86, Hij91] that implies

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

If M = S^{n} then equality is attained in this inequality. Upper bounds for τ(M, χ) may help to determine the σ-constant.

The structure of the article is as follows. In Section 2 we introduce some notations and recall some preliminaries. The aim of Section 3 is to show that we can assume without loss of generality that g is flat on small neighborhoods of p and q. The metrics g_{ε}^{#} are constructed in Section 4, and we devote Section 5 to the proof of Theorem 1.1. In the last section, namely in Section 6, the calculation of the τ-invariant for any Riemann surface with spin structure is explained.

Acknowledgement We thank Mattias Dahl for reading the manuscript profoundly and for many interesting comments. Thanks also to the referee for some good suggestions. The first named author also thanks the Albert-Einstein-Institute in Potsdam-Golm, Germany, in particular Gerhard Huisken for very generous support.

2. Preliminaries

In this section we want to introduce some notations and recall some preliminaries. We refer to [LM89, Fr00, Hij99, BG92] for more details.

2.1. Notation. The round metric on S^{n}, i.e. the metric of sectional curvature 1, will always be denoted by g_{round}^{n}. We also abbreviate 𝕊^{n} for the Riemannian manifold (S^{n}, g_{round}^{n}) equipped with the spin structure χ^{n} that arises as the boundary of the (n + 1)-dimensional disk. The open ball of radius 1 around 0 in ℝ^{n} is denoted by B_{0,ℝ^{n}}(1).

2.2. Topological spin structures versus metric spin structures. The bundle Gl_{+}(M) of positively oriented frames over an oriented manifold M of dimension n ≥ 2 is a Gl_{+}(n, ℝ)-principal bundle over M. The group Gl_{+}(n, ℝ) has fundamental group ℤ if n = 2 and fundamental group ℤ/2ℤ if n ≥ 3. We denote the unique connected double cover of Gl_{+}(n, ℝ) by G̃l_{+}(n, ℝ). A topological spin structure on M consists of a G̃l_{+}(n, ℝ)-principal bundle G̃l_{+}(M) together with a G̃l_{+}(n, ℝ) → Gl_{+}(n, ℝ) equivariant map χ : G̃l_{+}(M) → Gl_{+}(M) over the identity M → M. Two topological spin structures χ_{1} : G̃l_{+}(M)_{1} → Gl_{+}(M) and χ_{2} : G̃l_{+}(M)_{2} → Gl_{+}(M) are said to be equivalent if there is a G̃l_{+}(n, ℝ)-equivariant map H : G̃l_{+}(M)_{1} → G̃l_{+}(M)_{2} with χ_{1} = χ_{2} ∘ H. We denote a topological spin structure just by χ.

Note that Spin(n) is the preimage of SO(n) under the homomorphism G̃l_{+}(n, ℝ) → Gl_{+}(n, ℝ). A Spin(n)-principal bundle Spin(M, g) together with a Spin(n) → SO(n) equivariant map χ̂ : Spin(M, g) → SO(M, g) is called a metric spin structure on M. Two metric spin structures χ̂_{1} and χ̂_{2} are equivalent if there is a Spin(n)-equivariant map H between the Spin(n)-principal bundles such that χ̂_{1} = χ̂_{2} ∘ H.

If M carries a metric g and if SO(M, g) denotes the bundle of g-orthonormal positively oriented frames, then the restriction of a topological spin structure χ to Spin(M, g) := χ^{−1}(SO(M, g)) defines a metric spin structure on (M, g). This restriction yields a map from equivalence classes of topological spin structures on M to equivalence classes of metric spin structures on M, and one easily sees that this map is bijective. Working with topological spin structures has the advantage that it does not depend on a choice of metric. However, working with metric spin structures allows the definition of the spinor bundle.

Namely, the spinor bundle of a Riemannian manifold (M, g) with spin structure χ̂ is defined as the associated bundle Σ_{g,χ̂}M := Spin(M, g) ×_{σ} Σ_{n} where (σ, Σ_{n}) is the spinor representation of Spin(n). Sometimes we will just write Σ_{g}M or ΣM when the spin structure or the metric is clear from the context.

As restriction is a bijection from equivalence classes of topological spin structures to equivalence classes of metric spin structures, we will identify topological and metric spin structures from now on and just call them spin structures.

2.3. Surgery, handles and spin structures. In the introduction we introduced 0-dimensional surgery. In order to understand the behavior of spin structures under surgery it is useful to see it as a bordism.

An (n + 1)-dimensional (spin) manifold W with boundary −M_{1} ˙∪M_{2}, is called a (spin) bordism from M_{1} to M_{2} The manifold W carries an orientation and a spin structure, M_{2} carries the induced orientation and spin structure, and M_{1} carries the induced spin structure and the opposite of the induced orientation. This opposite orientation is indicated by the minus-sign before M_{1}. For example W := M ×[0, 1] is a spin bordism from M to M. If M^{#} is obtained from M by a k-dimensional surgery, then there is a bordism from M to M^{#}. This bordism is obtained by a construction called adding a k-handle to M × [0, 1]. We will explain this construction in the case k = 0, see e.g. [Kos93, VI.6] for details and the general case.

We start with the manifold W := M × [0, 1]. Choose two points p, q ∈ M and two diffeomorphisms φ_{p} : \overline{B_{0,ℝ^{n}}(1)} → \overline{B_{p}(ε)}⊂ M and φ_{q} : \overline{B_{0,ℝ^{n}}(1)} → \overline{B_{q}(ε)}⊂ M. For each x ∈ \overline{B_{0,ℝ^{n}}(1)} we identify

W ∋ (φ_{p}(x), 1) ∼ (x, −1) ∈ \overline{B_{0,ℝ^{n}}(1)} × [−1, 1] W ∋ (φ_{q}(x), 1) ∼ (x, 1) ∈ \overline{B_{0,ℝ^{n}}(1)} × [−1, 1].

The topological space

W^{#} := W ∪ (\overline{B_{0,ℝ^{n}}(1)} × [−1, 1])/ ∼

carries naturally the differential structure of a manifold with corners, but by smoothing out the corners one finds a differential structure on W such that the boundary ∂W^{#} = (−M) ˙∪M^{#} is smooth. [Kos93, Chapter VI].

We assume that M comes with a fixed orientation and spin structure. Furthermore we assume that φ_{p} preserves orientation and that φ_{q} reverses orientation. Then W^{#} and M^{#} also carry natural orientations.

In order to define the spin structures, we equip W with the product spin structure, denoted by κ. The spin structure on W extends to a spin structure on W^{#}. We choose the boundary spin structure on M^{#}. If p and q are in different connected components of M, then the spin structure on W^{#} (resp. M^{#}) is uniquely determined. However, if p and q are in the same connected component, then there are two non-equivalent choices of spin structures on W^{#} that extend the spin structure on W. These two spin structures are obtained from each other by a diffeomorphism as explained in the following lemma.

Lemma 2.1. Let n = dim M = dim W −1 ≥ 2. If κ_{1}^{#} and κ_{2}^{#} are two spin structures on W^{#} such that the inclusions (W, κ) ↪ (W^{#}, κ_{j}^{#}), j = 1, 2, are spin preserving embeddings, then there is a diffeomorphism f : W^{#} → W^{#} which is the identity on W and such that f^{∗}κ_{1}^{#} = κ_{2}^{#}. Similarly, if χ_{1}^{#} and χ_{2}^{#} are two spin structures on M^{#} such that the inclusions (M \ (B_{p}(ε) ∪ B_{q}(ε)), χ) ↪ (M^{#}, χ_{j}^{#}), j = 1, 2, are spin preserving embeddings, then there is a diffeomorphism f : M^{#} → M^{#} which is the identity on M \ (B_{p}(ε) ∪ B_{q}(ε)) and such that f^{∗}χ_{1}^{#} = χ_{2}^{#}.

For proving this lemma, we can assume that p and q are in the same connected component. Let κ_{1}^{#} and κ_{2}^{#} be the two disjoint extensions of the spin structure κ on W. Let γ : [−1, 1] → SO(n) be a closed loop in SO(n) generating π_{1}(SO(n)). We can assume that γ(t) is the unit in SO(n) for t close to −1 or 1. We define the diffeomorphism f_{0} : [−1, 1]×\overline{B_{0,ℝ^{n}}(1)} → [−1, 1]×\overline{B_{0,ℝ^{n}}(1)}, f_{0}(t, x) := (t, γ(t)(x)). By extending with the identity map on W (and the usual smoothing procedure at the balls where [−1, 1] × \overline{B_{0,ℝ^{n}}(1)} is glued to W) one obtains a diffeomorphism f : W^{#} → W^{#}. One checks that f^{∗}(κ_{1}^{#}) = κ_{2}^{#}.

Similarly, the restriction of f to M^{#} yields a diffeomorphism of M^{#} to itself that pulls back χ_{1}^{#} to χ_{2}^{#}.

Also note a particularity in the case n = 2: S^{1} ×[−1, 1] carries 2 non-equivalent spin structures. However, it is clear from the construction above that the restriction of χ^{#} to S^{1}×[−1, 1] is the product spin structure of the bounding spin structure on S^{1} with the unique spin structure on [−1, 1].

Remark 2.2. Let χ_{1}^{#} and χ_{2}^{#} be two spin structures on M^{#} constructed from a spin structure χ on M. Let also g^{#} be a Riemannian metric on M^{#}. Since χ_{1}^{#} and χ_{2}^{#} may be non-equivalent, the spectra of the corresponding Dirac operators (defined for the same metric g^{#}) do not coincide in general. However, let f : M^{#} → M^{#} be the diffeomorphism given by Lemma 2.1 and let D_{1} (resp. D_{2}) be the Dirac operator associated to χ_{1}^{#} (resp. χ_{2}^{#}) in the metric g^{#} (resp. f^{∗}g^{#}). Lemma 2.1 then implies that D_{1} and D_{2} have the same spectrum. As a consequence, we get that τ(M^{#}, χ_{1}^{#}) = τ(M^{#}, χ_{2}^{#}).

2.4. Identifying spinors for different metrics. Throughout the paper, we need to identify spinor fields for a manifold M with a fixed (topological) spin structure χ, but two different metrics g and h. In order to recall this identification we follow [BG92].

Let x ∈ M. Since the metrics g and h are symmetric and positive definite bilinear forms, there is a unique symmetric b_{h}^{g} ∈ End(T_{x}M) such that for all v, w ∈ T_{x}M,

h(b_{h}^{g}v, b_{h}^{g}w) = g(v, w).

Note that b_{h}^{g} depends smoothly on x. Hence, the map

(b_{h}^{g})^{n} : SO(M, g) → SO(M, h)
(e_{1}, ⋯, e_{n}) ↦ (b_{h}^{g}(e_{1}), ⋯, b_{h}^{g}(e_{n}))

is an isomorphism of SO(n)-principal bundles between the oriented frame bundles of (M, g) and (M, h). This map commutes with the right action of SO_{n}. Furthermore, (b_{h}^{g})^{n} can be lifted to

[figure: Spin(M, g); Spin(M, h); SO(M, g); SO(M, h); M; M; β_{h}^{g}; (b_{h}^{g})^{n}; Id]

Hence, we obtain a map, also denoted by β_{h}^{g}, between the spinor bundles Σ_{g}M and Σ_{h}M in the following way:

Σ_{g}M = Spin(M, g) ×_{σ} Σ_{n} ⟶ Σ_{h}M = Spin(M, h) ×_{σ} Σ_{n}
ψ = [s, φ] ⟼ β_{h}^{g}ψ = [β_{h}^{g}(s), φ]  (3)

where (σ, Σ_{n}) is the complex spinor representation, and [s, φ] denotes the equivalence class of (s, φ) under the diagonal action of Spin(n). The identification β_{h}^{g} of spinors preserves the pointwise norm of spinors. Apparently, β_{h}^{g} ∘ β_{g}^{h} = id. However, in general, for three metrics g, k, h we have β_{g}^{h} ∘ β_{h}^{k} ∘ β_{k}^{g} ≠ id. From the construction of the bundle Spin(M, g) and the spinor bundle Σ_{g}M it follows directly that there is a section B_{h}^{g} : T M ⊗ Σ_{g}M → Σ_{g}M

β_{h}^{g}∇_{X}^{g}ψ − ∇_{X}^{h}β_{h}^{g}ψ = β_{h}^{g}(B_{h}^{g}(X ⊗ ψ))  (4)

The expression B_{h}^{g} only depends on g, h and their first derivatives. In particular, ∥B_{h}^{g}∥ → 0 if h converges to g in the C^{1}-topology. The Dirac operators with respect to g and h are locally defined as D_{g} = ∑_{i} e_{i}·∇_{e_{i}}^{g} and D_{h} = ∑_{i} b_{h}^{g}(e_{i}) · ∇_{b_{h}^{g}(e_{i})}^{h} where (e_{1}, …, e_{n}) ∈ SO(M, g). It follows that

|β_{h}^{g}D_{g}ψ − D_{h}β_{h}^{g}ψ| ≤ C_{h}^{g} (|ψ| + |∇^{g}ψ|)  (5)

where C_{h}^{g} → 0 if h → g in C^{1}.

In the special case when the metric h is conformal to g, i.e. when we have h = f^{2}g with f being a smooth positive function, it is shown in [Hit74, Hij86] that

D_{h}(f^{− \frac{n−1}{2}} β_{h}^{g}ψ) = f^{− \frac{n+1}{2}} β_{h}^{g}D_{g}ψ.  (6)

This equation implies, in particular, that the dimension of the kernel of the Dirac operator is constant on any conformal class.

2.5. The first eigenvalue of the Dirac operator in a conformal class. Let (M, g, χ) be a compact spin manifold of dimension n, ker D = {0}. For ψ ∈ Γ(ΣM) we define

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

Using techniques from [Lot86], Ammann proved in [Amm03b] that

λ_{min}^{+}(M, [g], χ) = inf_{ψ} J(ψ)  (7)

where the infimum is taken over the set of smooth spinor fields for which

(∫_{M}⟨Dψ, ψ⟩ dv_{g}) > 0.

We will need the following result:

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

λ_{min}^{+}(M, [g], χ) ≤ λ_{min}^{+}(𝕊^{n}) = \frac{n}{2} ωn^{\frac{1}{n}}  (8)

where ω_{n} stands for the volume of the standard sphere 𝕊^{n}.

The proposition was proven in [Amm03b] using geometric methods if n ≥ 3. In the case n = 2 the article [Amm03b] 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_{p}(ε) that verifies

J(ψ_{ε}) ≤ λ_{min}^{+}(𝕊^{n}) + o(ε)

(see [AHM03A] for details).

If inequality (8) holds even strictly, then one can show that the infimum in equation (7) is attained, say in φ. Then the infimum in the definition of λ_{min}^{+}(M, [g], χ) is attained in the generalized conformal metric g̃ := |φ|^{4/(n−1)}g (see [Amm03c] for details).

This fact, summarized in the following theorem will be a central ingredient in the proof of our main result Theorem 1.1.

Theorem 2.4 ([Amm03c, Amm03a]). Assume that Inequality (8) is strict. Then there exists a spinor field φ ∈ C^{2,α}(ΣM) ∩ C^{∞}(ΣM \ φ^{−1}(0)) on (M, g), α ∈ (0, 1) ∩ (0, 2/(n − 1)] such that

D(φ) = λ_{min}^{+}(M, [g], χ) |φ|^{\frac{2}{n−1}} φ, and ∥ φ ∥_{\frac{2n}{n−1}} = 1.  (9)

Furthermore, there is a generalized conformal metric g̃ (see [Amm03a] for the definition) such that λ_{1}^{+}(g̃)Vol(M, g̃)^{1/n} = λ_{min}^{+}(M, [g], χ).

2.6. A conformal Sobolev inequality. In some following estimates we have to compare ∫_{M} |Dψ|^{\frac{2n}{n+1}} and ∫_{M} |∇ψ|^{\frac{2n}{n+1}} . The pointwise identity

|Dψ| ≤ \sqrt{n}|∇ψ|

implies

\frac{∫_{M} |Dψ|^{\frac{2n}{n+1}} dv_{g}}{∫_{M} |∇ψ|^{\frac{2n}{n+1}}} ≤ n^{\frac{n}{n+1}} .

The fact that this fraction is also bounded from below is a standard result of elliptic regularity theory.

Proposition 2.5. Let (M, g, χ) be a compact spin manifold of dimension n ≥ 2. We assume that the Dirac operator D is invertible. Then for any p ∈ (1, ∞) there exists a constant C > 0 such that for all spinor fields ψ of class C^{1}, we have :

∫_{M} |∇ψ|^{p} dv_{g} ≤ C ∫_{M} |Dψ|^{p} dv_{g}.  (10)

Furthermore assume p′ ∈ (1, ∞), 1 − (n/p) ≥ −(n/p′). Then

(∫_{M} |ψ|^{p′} dv_{g})^{1/p′} ≤ C(∫_{M} |Dψ|^{p} dv_{g})^{1/p}.  (11)

Proof. Inequality (10) is a classical elliptic inequality for invertible elliptic operators on compact manifolds. Recall (e.g. [LM89, Corollary III.5.7]) that any elliptic differential operator P on a compact manifold has a Green operator G_{P} i.e. a classical pseudo-differential operator G such that

G_{P} P = P G_{P} = Id − π_{ker P} ,

where π_{ker P} denotes L^{2}-orthogonal projection to the kernel of P. Applied to an invertible Dirac operator D this means that D^{−1} is a well-defined classical pseudodifferential operator of order −1.

We will now explain how to show that such an operator is a continuous map

D^{−1} : L^{p}(ΣM) → L_{1}^{p}(ΣM), 1 < p < ∞,

where L_{1}^{p}(ΣM) denotes the Sobolev space of spinors to the norm ∥ψ∥_{L_{1}^{p}} := ∥ψ∥_{L^{p}} + ∥∇ψ∥_{L^{p}}. At first, we choose a finite system of coordinate charts (α_{i} : U_{i} → V_{i}) covering M and a corresponding partition of unity (χ_{i}), i.e. smooth functions χ_{i} : M → [0, 1] with K_{i} := suppχ_{i} ⊂ U_{i} and ∑ χ_{i} = 1. We can assume that the spinor bundle is trivialized on U_{i} via the bundle isomorphisms A_{i} : ΣM|_{U_{i}} → V_{i} × ℂ^{N} =: ΣV_{i}, where N = 2^{[n/2]} is the complex rank of the spinor bundle. If φ is a spinor on M (or U_{i}), then (A_{i})_{∗}φ := A_{i} ∘ φ ∘ α_{i}^{−1} is spinor on V_{i}. As the symbol σ(χ_{i}D^{−1}) = χ_{i}σ(D^{−1}) has support in K_{i}, it is clear from the definition of pseudo-differential operators on compact manifolds (e.g. [LM89, III, Def. 3.15 and following statements]) that we can decompose

χ_{i}D^{−1} = (A_{i})_{∗}^{−1} ∘ P_{i} ∘ (A_{i})_{∗} + S_{i}

where P_{i} : Γ(ΣV_{i}) → Γ(ΣV_{i}) is a pseudodifferential operator of order −1 and where S_{i} : Γ(ΣM) → χ_{i}Γ(ΣM) is a smoothing operator. Obviously S_{i} is bounded from L^{p} to L_{1}^{p}. The operator ∇ ∘ P_{i} : Γ(ΣV_{i}) → Γ(ℝ^{n} ⊗ ΣV_{i}) is a pseudodifferential operator of order 0, and such an operator is a continuous operator from L^{p} to L^{p} [Tay81, Theorem XI.2.1] for all p ∈ (1, ∞). It follows that χ_{i}D^{−1} and hence D^{−1} = ∑ χ_{i}D^{−1} is continuous from L^{p} to L_{1}^{p}. This might be equivalently expressed as

∥∇D^{−1}φ∥_{L^{p}} + ∥D^{−1}φ∥_{L^{p}} ≤ C∥φ∥_{L^{p}}  (12)

and this obviously implies (10) by substituting ψ := D^{−1}φ.

Inequality (11) then follows from (12) together with the classical bounded Sobolev empedding L_{1}^{p} → L^{p′}.

Corollary 2.6. Let (M, g, χ) be a compact Riemannian spin manifold with invertible Dirac operator. Let (Ω, h) be conformal to an open subset of (M, g). Then there is a constant C = C(M, g, χ) such that any compactly supported spinor field ψ ∈ Γ(Σ(Ω, h, χ)) of class C^{1} satisfies

(∫_{Ω} |ψ|^{\frac{2n}{n−1}} dv_{h})^{\frac{n−1}{n}} ≤ C(∫_{Ω} |D_{h}ψ|^{\frac{2n}{n+1}} dv_{h})^{\frac{n+1}{n}} .

Proof. Let f be the conformal factor, i.e. h = f^{2}g. With the notation of subsection 2.4, we let φ := f^{\frac{n−1}{2}} β_{g}^{h}ψ ∈ Γ(Σ(Ω, g, χ)). As φ is compactly supported we can extend it with zero to a spinor of class C^{1} on M. Since the map β_{g}^{h} : Σ_{h}M → Σ_{g}M is a pointwise isometry, and since dv_{h} = f^{n} dv_{g} and by (6), we get that

∫_{Ω} |ψ|^{\frac{2n}{n−1}} dv_{h} = ∫_{Ω} |φ|^{\frac{2n}{n−1}} dv_{g}

and

∫_{Ω} |D_{h}ψ|^{\frac{2n}{n+1}} dv_{h} = ∫_{Ω} |D_{g}φ|^{\frac{2n}{n+1}} dv_{g}.

The result is then an immediate consequence of inequality (11) for p := \frac{2n}{n+1} and p′ := \frac{2n}{n−1}. □

Example 2.7. Let M = 𝕊^{n} be the sphere. Then the Mercator projection is a conformal embedding F : ℝ × 𝕊^{n−1} → 𝕊^{n}

F : (\begin{matrix}t \\ x_{1} \\ ⋯ \\ x_{n} \end{matrix} ) ↦ (tanh(t) ; \frac{x_{1}}{cosh(t)} ; ⋯; \frac{x_{n}}{cosh(t)}) .

The image of F is 𝕊^{n} with the North and South pole removed. Recall from the preliminaries that we always equip 𝕊^{m} with the spin structure χ^{m} that arises as restriction of the unique spin structure on \overline{B_{0,ℝ^{m+1}}(1)} to the boundary. We also remind that on S^{m}, m ≥ 2 this is the only spin structure, whereas there is a second spin structures on S^{1}. One easily verifies that F^{∗}(χ^{n}) is the product structure of the unique spin structure on ℝ and of χ^{n−1}.

Hence, any compactly supported spinor ψ on the cylinder ℝ × 𝕊^{n−1} satisfies

(∫_{ℝ} ∫_{𝕊^{n−1}} |ψ|^{\frac{2n}{n−1}} dv_{g_{round}^{n−1}} dt)^{\frac{n−1}{n}} ≤ C(∫_{ℝ} ∫_{𝕊^{n−1}} |Dψ|^{\frac{2n}{n+1}} dv_{g_{round}^{n−1}} dt)^{\frac{n+1}{n}} .  (13)

3. Approximation by locally flat manifolds

The first step in the proof of Theorem 1.1 is to approximate (M, g) by metrics that are flat in a neighborhood of p and q, and then show that λ_{min}^{+} does not change much under this approximation. The results of this section will allow us to assume that g is flat in a neighborhood of p and q.

Lemma 3.1. Let (M, g, χ) be a compact spin manifold of dimension n ≥ 2. We assume that the Dirac operator D is invertible. Let p, q ∈ M. There is a family of Riemannian metrics (g_{ε})_{ε∈(0,α)} on M, such that each g_{ε} is flat in a neighborhood of p and q and such that g_{ε} → g in C^{1} when ε → 0.

Proof. Let η : [0, ∞) → [0, 1] be a smooth function that equals to 1 on [0, 1] and whose support is contained in [0, 2). Choose a small number ε > 0. On M \ (B_{p}(2ε) ∪ B_{q}(2ε)) we define g_{ε} = g. In normal coordinates centered in p and defined on B_{p}(2ε) we write g(x) = ∑_{ij} g_{ij}(x) dx^{i} dx^{j}. We define on B_{p}(2ε)

g_{ε}(x) := ∑_{ij}(η(ε^{−1}|x|) δ_{ij} + (1 − η(ε^{−1}|x|)) g_{ij}(x)) dx^{i} dx^{j},

and similarly on B_{q}(2ε). Then one calculates that g_{ε} → g in C^{1} when ε → 0. □

Proposition 3.2. Let (M, g, χ) be a compact spin manifold of dimension n ≥ 2 with invertible Dirac operator D. If g_{ε} is a sequence of Riemannian metrics converging to g in C^{1}, then

lim_{ε→0} λ_{min}^{+}(M, [g_{ε}], χ) = λ_{min}^{+}(M, [g], χ).

Proof. Since D is invertible, and as the spectrum of D seen as of function of g depends continuously on g, it follows that the Dirac operator on (M, g_{ε}, χ), denoted by D_{ε}, is invertible as well. Let J (resp. J_{ε}) be the functional (see Paragraph 2.5) associated to (M, [g], χ) (resp. (M, [g_{ε}], χ)). Let ψ ∈ Γ(Σ_{g}M) be a smooth spinor field on (M, g, χ). We denote by D_{ε} the Dirac operator acting on (M, g_{ε}, χ). We can choose ψ such that J(ψ) ≤ λ_{min}^{+}(M, [g], χ) + δ where δ > 0 is small. By (5), (10) and(11) for p′ = 2n/(n − 1) and p = 2n/(n + 1) one easily gets that

lim sup_{ε→0} λ_{min}^{+}(M, [g_{ε}], χ) ≤ lim sup_{ε→0} J_{ε}(β_{g_{ε}}^{g}ψ) = J(ψ) ≤ λ_{min}^{+}(M, [g], χ) + δ.

Since δ > 0 is arbitrary, we obtain that

lim_{ε→0} λ_{min}^{+}(M, [g_{ε}], χ) ≤ λ_{min}^{+}(M, [g], χ).  (14)

Now, let ψ_{ε} ∈ Γ(Σ_{g}M) be a smooth spinor field on (M, g, χ) such that

J_{ε}(β_{g_{ε}}^{g}ψ_{ε}) ≤ λ_{min}^{+}(M, [g_{ε}], χ) + δ.

In order to abbreviate, we set \overline{ψ}_{ε} := β_{g_{ε}}^{g}ψ_{ε}. Without loss of generality, we can assume that

∫_{M}⟨D_{ε}\overline{ψ}_{ε}, \overline{ψ}_{ε}⟩ dv_{g_{ε}} = 1.

Then, since J_{ε}(\overline{ψ}_{ε}) is bounded (by Relation (8)), then ∫_{M} |D_{ε}\overline{ψ}_{ε}|^{\frac{2n}{n+1}} dv_{g_{ε}} is also bounded. We have

|∥ D_{ε}\overline{ψ}_{ε} ∥_{L^{\frac{2n}{n+1}} (M,g_{ε})} − ∥ Dψ_{ε} ∥_{L^{\frac{2n}{n+1}} (M,g)}| ≤ |∥ D_{ε}\overline{ψ}_{ε} ∥_{L^{\frac{2n}{n+1}} (M,g_{ε})} − ∥ D_{ε}\overline{ψ}_{ε} ∥_{L^{\frac{2n}{n+1}} (M,g)}|

+ |∥ D_{ε}\overline{ψ}_{ε} ∥_{L^{\frac{2n}{n+1}} (M,g)} − ∥ Dψ_{ε} ∥_{L^{\frac{2n}{n+1}} (M,g)}| .  (15)

By definition of g_{ε}, it is clear that (1 − o(ε)) dv_{g} ≤ dv_{g_{ε}} ≤ (1 + o(ε)) dv_{g}, and hence for ε → 0

|(∫_{M} |D_{ε}\overline{ψ}_{ε}|^{\frac{2n}{n+1}} dv_{g_{ε}})^{\frac{n+1}{2n}} − (∫_{M} |D_{ε}\overline{ψ}_{ε}|^{\frac{2n}{n+1}} dv_{g})^{\frac{n+1}{2n}} | ≤ o(ε) \underbrace{(∫_{M} |D_{ε}\overline{ψ}_{ε}|^{\frac{2n}{n+1}} dv_{g_{ε}})^{\frac{n+1}{2n}}}_{J_{ε}(\overline{ψ}_{ε})} → 0.

In addition, we get from relation (5) that

|(∫_{M} |D_{ε}\overline{ψ}_{ε}|^{\frac{2n}{n+1}} dv_{g})^{\frac{n+1}{2n}} − (∫_{M} |Dψ_{ε}|^{\frac{2n}{n+1}} dv_{g})^{\frac{n+1}{2n}} | ≤ a_{ε} (∫ (|ψ_{ε}| + |∇_{g}ψ_{ε}|)|^{\frac{2n}{n+1}} dv_{g} )^{\frac{n+1}{2n}} .

where lim_{ε→0} a_{ε} = 0. Using the triangle inequality and Hölder’s inequality, we see the existence of a constant C > 0 independent of ε such that

|(∫_{M} |D_{ε}\overline{ψ}_{ε}|^{\frac{2n}{n+1}} dv_{g})^{\frac{n+1}{2n}} − (∫_{M} |Dψ_{ε}|^{\frac{2n}{n+1}} dv_{g})^{\frac{n+1}{2n}} | ≤ Ca_{ε} ((∫ |ψ_{ε}|^{\frac{2n}{n−1}} dv_{g} )^{\frac{n−1}{2n}} + (∫ |∇_{g}ψ_{ε}|^{\frac{2n}{n+1}} dv_{g} )^{\frac{n+1}{2n}} ) .

By inequalities (10) and (11), we obtain that

|(∫_{M} |D_{ε}\overline{ψ}_{ε}|^{\frac{2n}{n+1}} dv_{g})^{\frac{n+1}{2n}} − (∫_{M} |Dψ_{ε}|^{\frac{2n}{n+1}} dv_{g})^{\frac{n+1}{2n}} | ≤ Ca_{ε} (∫ |∇_{g}ψ_{ε}|^{\frac{2n}{n+1}} dv_{g} )^{\frac{n+1}{2n}} .

Coming back to (15), this clearly implies that ∫_{M} |Dψ_{ε}|^{\frac{2n}{n+1}} dv_{g} is bounded and that ∫_{M} |ψ_{ε}|^{\frac{2n}{n−1}} dv_{g} is bounded, too. Since lim_{ε→0} a_{ε} = 0, this also implies that

lim_{ε→0} |∫_{M} |D_{ε}\overline{ψ}_{ε}|^{\frac{2n}{n+1}} dv_{g_{ε}} − ∫_{M} |Dψ_{ε}|^{\frac{2n}{n+1}} dv_{g}| = 0.  (16)

Similarly, using relation (5) we have

lim_{ε→0} |∫_{M}⟨D_{ε}\overline{ψ}_{ε}, \overline{ψ}_{ε}⟩ dv_{g_{ε}} − ∫_{M}⟨Dψ_{ε}, ψ_{ε}⟩ dv_{g}| = 0.  (17)

Relations (16) and (17) imply that

λ_{min}^{+}(M, [g_{ε}], χ) + δ ≥ J_{ε}(\overline{ψ}_{ε}) ≥ J(ψ_{ε}) + o(1) ≥ λ_{min}^{+}(M, [g], χ) + o(1).

Since δ is arbitrary, we get that

lim inf_{ε→0} λ_{min}^{+}(M, [g_{ε}], χ) ≥ λ_{min}^{+}(M, [g], χ).  (18)

Together with (14), this proves Theorem 3.2.

4. Construction of the metrics on M^{#}

The aim of this section is to construct the sequence of metrics g_{ε}^{#} of Theorem 1.1. Using the result of the previous section, we can assume from now on and in the rest of the article that the metric g is flat on B_{p}(ρ) and B_{q}(ρ) for a small ρ > 0. For 0 < α < β < ρ we introduce the notation

B_{p,q}(α) = B_{p}(α) ∪ B_{q}(α) and C_{p,q}(α, β) = B_{p,q}(β) \ B_{p,q}(α).

Let ε > 0 be small. We explained in the introduction that M^{#} is obtained (as a topological space) by gluing a cylinder [−1, 1] × S^{n−1} with M \ B_{p,q}(ε) along ∂B_{p}(ε) on one side and along ∂B_{q}(ε) on the other side. Evidently this can also be expressed by saying

M^{#} = M \ B_{p,q}(ε)/ ∼

where ∼ indicates that we glue ∂B_{p}(ε) with ∂B_{q}(ε) via an orientation preserving diffeomorphism. M^{#} is equipped with a differential structure and a spin structure such that I_{ε} : M \ B_{p,q}(ε) → M^{#} is smooth and compatible with the spin structures. We also introduce for all a ∈ (ε, ρ) the notation

H(a, ε) := I_{ε}(C_{p,q}(ε, a)) = C_{p}(ε, a) ∪ C_{q}(ε, a)/ ∼ .

Hence M^{#} is the disjoint union of M \ B_{p,q}(a) and H(a, ε).

Let us denote by d_{p} (resp. d_{q}) the distance in (M, g) to the point p (resp. q). We define a function f_{ε} on (M \ B_{p,q}(3ε)) ∪ C_{p,q}(ε, 2ε) by setting

f_{ε}(x) = \{ \begin{matrix}1 & if & x ∈ M \ B_{p,q}(3ε) \\ d_{p}^{−1} & if & x ∈ C_{p}(ε, 2ε) \\ d_{q}^{−1} & if & x ∈ C_{q}(ε, 2ε). \end{matrix}

We can extend f_{ε} smoothly and positively to M \ B_{p,q}(ε) such that f_{ε} satisfies on C_{p,q}(2ε, 3ε):

|df_{ε}| ≤ \frac{2}{ε}.

As (C_{p}(ε, 2ε), f_{ε}^{2}g) and (C_{q}(ε, 2ε), f_{ε}^{2}g) are isometric to ([0, log 2) × S^{n−1}, dt^{2} + g_{round}^{n−1}), f_{ε}^{2}g defines a metric g_{ε}^{#} on M^{#}, or more precisely: there is a unique metric g_{ε}^{#} on M^{#} such that

f_{ε}^{2}g = I_{ε}^{∗}g_{ε}^{#} holds on M \ B_{p,q}(ε).

Note that (H(a, ε), g_{ε}^{#}) is then conformal (but in general not isometric) to ((− log(a/ε), log(a/ε)) × S^{n−1}, dt^{2} + g_{round}^{n−1}).

5. Proof of Theorem 1.1

In this section we will prove that the metrics (g_{ε}^{#})_{ε} are the desired metrics. We denote by D_{ε} the Dirac operator acting on (M^{#}, g_{ε}^{#}, χ^{#}). We set λ = λ_{min}^{+}(M, [g], χ) and λ_{ε} = λ_{min}^{+}(M^{#}, [g_{ε}^{#}], χ^{#}). We denote by J (resp. J_{ε}) the functional associated to λ (resp. λ_{ε}) (see Paragraph 2.5).

Lemma 5.1.

lim sup_{ε→0} λ_{ε} ≤ λ.  (19)

Proof. Let γ > 0 be small and let ψ be a smooth spinor field such that J(ψ) ≤ λ + γ. Clearly, for each small number α > 0, one can construct a cut-off function η_{α} ∈ C^{∞}(M), such that 0 ≤ η_{α} ≤ 1, equal to 1 on M \ B_{p,q}(2α), equal to 0 on B_{p,q}(α) and which satisfies |dη_{α}| ≤ \frac{2}{α}. As easily seen, we have lim_{α→0} J(η_{α}ψ) = J(ψ). We choose α small enough such that J(η_{α}ψ) ≤ λ + 2γ. Now, if 3ε < α, then η_{α}ψ is supported on the common part of (M, g, χ) and (M^{#}, g_{ε}^{#}, χ^{#}) and hence can be seen as a spinor field on (M^{#}, g_{ε}^{#}, χ^{#}). We have

λ_{ε} ≤ J_{ε}(η_{α}ψ) ≤ λ + 2γ

for all small ε > 0. Since γ is arbitrary, we proved the lemma. □

It remains to prove that

lim inf_{ε→0} λ_{ε} ≥ λ.  (20)

This inequality is more involved than Equation (19) and will occupy the rest of this section.

We set λ_{0} := lim inf_{ε→0} λ_{ε} and pass to a sequence of ε_{i} → 0 with lim_{i→∞} λ_{ε_{i}} = λ_{0}. To simplify notation we write λ_{i} := λ_{ε_{i}}, D_{i} := D_{ε_{i}}, g_{i}^{#} := g_{ε_{i}}^{#}, dv_{i} := dv_{g_{ε_{i}}^{#}} , and so on. In the following arguments we will frequently pass to subsequences. Slightly abusing the notation, we will continue with the same index notation λ_{i}, D_{i}, g_{i}^{#}, and so on.

In the case λ_{0} = λ_{min}^{+}(𝕊^{n}) Equation (20) follows directly from (8). Hence, after possibly passing to a subsequence, we can assume that λ_{i} < λ_{min}^{+}(𝕊^{n}) for all i ∈ ℕ. By Theorem 2.4, there exists a spinor field ψ_{i} of class C^{1} defined on (M^{#}, g_{i}^{#}, χ^{#}) that satisfies

D_{i}ψ_{i} = λ_{i}|ψ_{i}|^{\frac{2}{n−1}} ψ_{i}  (21)

with the normalization

∫_{M^{#}} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i} = 1.  (22)

Lemma 5.2. For any δ > 0 and a_{0} > 0 there is an a ∈ (0, a_{0}) such that after possibly passing to a subsequence

lim inf_{i→∞} ∫_{C_{p,q}(a,2a)} |ψ_{i}|^{\frac{2n}{n−1}} dv_{g} < δ  (23)

and such that g is flat on C_{p,q}(a, 2a).

Note that the integral above has a meaning for ε_{i} ≤ a, since C_{p,q}(a, 2a) ⊂ M^{#}. If in addition, 3ε_{i} < a, we have g_{i} = g on C_{p,q}(a, 2a). For the proof of Lemma 5.2 we proceed by contradiction. Let N_{0} be such that g is flat on B_{p,q}(2^{−N_{0}}). If (23) is false, then for any N > N_{0} we can find a sequence i_{j} → 0 such that for all k ∈ {N_{0}, ⋯, N} we have

lim_{i→∞} ∫_{C_{p,q}(2^{−(k+1)},2^{−k})} |ψ_{i_{j}}|^{\frac{2n}{n−1}} dv_{g} ≥ δ.

Since

∫_{M^{#}} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i} = 1

and since the cylinders C_{p,q}(2^{−(k+1)}, 2^{−k}), k ∈ {N_{0}, …, N} are disjoint, we obtain a contradiction if N − N_{0} > \frac{1}{δ} for a = 2^{−(N+1)}. This proves Relation (23).

We fix a ∈ (0, a_{0}) for which Relation (23) is verified. After possibly extracting a subsequence of (ψ_{i}), we can assume that the limit infimum in (23) is a limit. In other words, we have found a number a such that

∫_{C_{p,q}(a,2a)} |ψ_{i}|^{\frac{2n}{n−1}} dv_{g} < δ  (24)

for all i in a suitable subsequence.

Since

∫_{M\B_{p,q}(2a)} |ψ_{i}|^{\frac{2n}{n−1}} dv_{g} + ∫_{C_{p,q}(a,2a)} |ψ_{i}|^{\frac{2n}{n−1}} dv_{g} + ∫_{H(a,ε)} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i} = 1

since for ε small

∫_{C_{p,q}(a,2a)} |ψ_{i}|^{\frac{2n}{n−1}} dv_{g} ≤ δ

and since δ can be chosen lower than \frac{1}{3}, we have:

lim sup_{i→∞} ∫_{M\B_{p,q}(2a)} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i} ≥ \frac{1}{3}  (25)

or

lim sup_{i→∞} ∫_{H(a,ε_{i})} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i} ≥ \frac{1}{3}.  (26)

where H(a, ε_{i}) is defined as in section 4.

The theorem now follows from the following two lemmata.

Lemma 5.3. Relation (25) implies λ_{0} ≥ λ.

Lemma 5.4. Relation (26) implies λ_{0} ≥ λ_{min}^{+}(𝕊^{n}).

Proof of Lemma 5.4. We suppose that Inequality (26) holds.

Step 1. {i ∈ ℕ | λ_{i} = 0} is finite.

We prove this step by contradiction and assume that after passing to a subsequence λ_{i} = 0 for all i.

This means that the spinors ψ_{i} are harmonic on (M^{#}, g_{i}^{#}).

By Example 2.7, the manifold (H(2a, ε), g_{i}^{#}) is conformal to a subdomain of the sphere 𝕊^{n}. By Proposition 2.5 there exists a constant C which does not depend on i and δ such that for all spinor field θ of class C^{1} and whose support is included in H(2a, ε_{i}), we have

(∫_{M^{#}} |θ|^{\frac{2n}{n−1}} dv_{i})^{\frac{n−1}{n}} ≤ C(∫_{M^{#}} |D_{i}θ|^{\frac{2n}{n+1}} dv_{i})^{\frac{n+1}{n}} .  (27)

Let η ∈ C^{∞}(M^{#}), 0 ≤ η ≤ 1 be a cut-off function equal to 1 on H(a, ε_{i}) supported in H(2a, ε_{i}), and which satisfies |dη| ≤ \frac{2}{a}. Then we apply Inequality (27) with θ = ηψ_{i}. Since ψ_{i} is harmonic and by (26), we get that

3^{− \frac{n−1}{n}} ≤ C(∫_{M^{#}} |(grad η) · ψ_{i}|^{\frac{2n}{n+1}} dv_{i})^{\frac{n+1}{n}} .

Using the Hölder inequality, we get that

3^{− \frac{n−1}{n}} ≤ C(∫_{M^{#}} |dη|^{n} dv_{i})^{\frac{2}{n}} (∫_{Supp(dη)} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i})^{\frac{n−1}{n}} .

Since |dη| ≤ \frac{2}{a}, since dη is supported in C_{p,q}(a, 2a) whose volume for g is bounded by Ca^{n} and since g_{i} is constant equal to g on C_{p,q}(a, 2a), there exists a constant C′ independent of i and δ such that

∫_{M} |dη|^{n} dv_{i} ≤ C′.  (28)

Using (24), we get that

∫_{Supp(dη)} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i} ≤ δ.

We obtain

\frac{1}{3^{\frac{n−1}{n}}} ≤ CC′^{\frac{2}{n}} δ^{\frac{n−1}{n}} .

If δ is small enough, we get a contradiction. This proves Relation step 1.

Step 2. Proof of the inequality

lim_{i→∞} λ_{i} ≥ λ_{min}^{+}(𝕊^{n}).  (29)

Let η be the same function as above. Since the manifold (H(2a, ε), g_{i}^{#}) is conformal to a subdomain of the sphere 𝕊^{n} and since ηψ_{i} is supported in H(2a, ε_{i}), we have

λ_{min}^{+}(𝕊^{n}) ≤ J(ηψ_{i}).  (30)

By Equation (21) we have

∫_{M^{#}}⟨ηD_{i}ψ_{i}, ηψ_{i}⟩ dv_{i} = λ_{i} ∫_{M^{#}} η^{2}|ψ_{i}|^{\frac{2n}{n−1}} dv_{i} ∈ ℝ.  (31)

On the other hand

\underbrace{∫_{M^{#}}⟨D_{i}(ηψ_{i}), ηψ_{i}⟩ dv_{i}}_{∈ℝ} = \underbrace{∫_{M^{#}}⟨grad η · ψ_{i}, ηψ_{i}⟩ dv_{i}}_{∈iℝ} + \underbrace{∫_{M^{#}} η^{2}⟨D_{i}ψ_{i}, ψ_{i}⟩ dv_{i}}_{∈ℝ},

implies that we can replace the left hand side of (31) by ∫_{M^{#}}⟨D_{i}(ηψ_{i}), ηψ_{i}⟩. Since η ≡ 1 on H(a, ε_{i}), we obtain that

∫_{M^{#}}⟨D_{i}(ηψ_{i}), ηψ_{i}⟩ dv_{i} ≥ λ_{i} ∫_{H(a,ε_{i})} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i}.  (32)

We also have by Equation (21)

(∫_{M^{#}} |D(ηψ_{i})|^{\frac{2n}{n+1}} dv_{i})^{\frac{n+1}{n}} = (∫_{M^{#}} |grad η · ψ_{i} + η λ_{i}|ψ_{i}|^{\frac{2}{n−1}} ψ_{i}|^{\frac{2n}{n+1}} dv_{i})^{\frac{n+1}{n}} .

Again since η ≡ 1 on H(a, ε_{i}), we can write that

∫_{M^{#}} |grad η · ψ_{i} + η λ_{i}|ψ_{i}|^{\frac{2}{n−1}} ψ_{i}|^{\frac{2n}{n+1}} dv_{i} ≤

∫_{C_{p,q}(a,2a)} |grad η · ψ_{i} + η λ_{i}|ψ_{i}|^{\frac{2}{n−1}} ψ_{i}|^{\frac{2n}{n+1}} dv_{i} + λi^{\frac{2n}{n+1}} ∫_{H(a,ε_{i})} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i}.

Using the fact that for s, t ∈ ℝ, we have |s + t|^{\frac{2n}{n+1}} ≤ |2s|^{\frac{2n}{n+1}} + |2t|^{\frac{2n}{n+1}} and using Hölder inequality, we get that

∫_{C_{p,q}(a,2a)} |grad η · ψ_{i} + η λ_{i}|ψ_{i}|^{\frac{2}{n−1}} ψ_{i}|^{\frac{2n}{n+1}} dv_{i}

≤ 2^{\frac{2n}{n+1}} ∫_{C_{p,q}(a,2a)} |dη|^{\frac{2n}{n+1}} |ψ_{i}|^{\frac{2n}{n+1}} dv_{i} + 2^{\frac{2n}{n+1}} λi^{\frac{2n}{n+1}} ∫_{C_{p,q}(a,2a)} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i}

≤ 2^{\frac{2n}{n+1}} (∫_{C_{p,q}(a,2a)} |dη|^{n} dv_{i})^{\frac{2}{n+1}} (∫_{C_{p,q}(a,2a)} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i})^{\frac{n+1}{n−1}} + 2^{\frac{2n}{n+1}} λi^{\frac{2n}{n+1}} δ.

Using again relation (28), we get that

∫_{C_{p,q}(a,2a)} |grad η · ψ_{i} + η λ_{i}|ψ_{i}|^{\frac{2}{n−1}} ψ_{i}|^{\frac{2n}{n+1}} dv_{i} ≤ C(δ^{\frac{n+1}{n−1}} + δ)

where C is a constant independent of i and δ. Finally, we obtain that

(∫_{M^{#}} |Dηψ_{i}|^{\frac{2n}{n+1}} dv_{i})^{\frac{n+1}{n}} ≤ (C(δ^{\frac{n+1}{n−1}} + δ) + λi^{\frac{2n}{n+1}} ∫_{H(a,ε_{i})} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i})^{\frac{n+1}{n}} .  (33)

Plugging (33) and (32) into (30), we get that

λ_{min}^{+}(𝕊^{n}) ≤ \frac{(C(δ^{\frac{n+1}{n−1}} + δ) + λi^{\frac{2n}{n+1}} ∫_{H(a,ε_{i})} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i})^{\frac{n+1}{n}}}{λ_{i} ∫_{H(a,ε_{i})} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i}} .

Since

∫_{H(a,ε_{i})} |ψ_{i}|^{\frac{2n}{n−1}} dv_{i} ≥ \frac{1}{3}

and letting δ go to 0, we obtain Relation (29). Since by (8), we have λ ≤ λ_{min}^{+}(𝕊^{n}), we get (20).

This proves the lemma.

Proof of Lemma 5.3. We just sketch the proof of Lemma 5.3 because the method is the same as the proof of Lemma 5.4, At first, we prove step 1. We proceed by contradiction and assume that λ_{i} = 0 for infinitely many i ∈ ℕ. This means that the spinors ψ_{i} are harmonic on (M^{#}, g_{i}^{#}). By Proposition 2.5, there exists C > 0 such that for all spinor θ of class C^{1} defined on M, we have

(∫_{M} |θ|^{\frac{2n}{n−1}} dv_{g_{i}^{#}})^{\frac{n−1}{n}} ≤ C(∫_{M} |Dθ|^{\frac{2n}{n+1}} dv_{g})^{\frac{n+1}{n}} .

We apply this inequality with θ = (1 − η)ψ_{i} where η is defined as in the proof of Lemma 5.4 and we obtain a contradiction if δ is small enough. This proves step 1. Then, we say that

λ ≤ J((1 − η)ψ_{i}).

As in the proof of Lemma 5.4, we obtain (20) in the limit δ → 0. This proves Theorem 1.1.

6. The Dirac operator on Riemann surfaces

In this section we want to prove Theorem 1.3. In fact, we use Theorem 1.1 to calculate the τ-invariant for all compact connected Riemann surfaces equipped with spin structures.

As already mentioned in the introduction, the Atiyah-Milnor-Singer invariant α is a ring homomorphisms from the ring of spin-cobordism classes into KO^{−∗}(pt). It allows to define an index theorem for the Dirac operator that is non-trivial in dimensions 0, 1, 2, 4 mod 8 (see e.g. [LM89, II.7]).

Let us to recall an equivalent definition of the α-invariant in the special case n = 2. In this case α(M) ∈ KO^{−2}{pt} = ℤ_{2} := ℤ/(2ℤ) = {0, 1}. Some more details can also be found in [KS96] and [AB02, Section 2 and 3]. We will also recall the index theorem in dimension 2, Theorem 6.3.

A quadratic form is a map q : H_{1}(M, ℤ_{2}) → ℤ_{2} such that

q(a + b) = q(a) + q(b) + a ∩ b

holds for all a, b ∈ H_{1}(M, ℤ_{2}). Here ∩ denotes the intersection form H_{1}(M, ℤ_{2}) × H_{1}(M, ℤ_{2}) → ℤ_{2} which is a non-degenerate (anti-)symmetric bilinear form on H_{1}(M, ℤ_{2}).

The difference of two such quadratic forms is a linear map H_{1}(M, ℤ_{2}) → ℤ_{2}, and vice versa if one adds a linear map H_{1}(M, ℤ_{2}) → ℤ_{2} to a quadratic form, one easily sees that one obtains a quadratic form again. The space of quadratic forms on H_{1}(M, ℤ_{2}) is an affine spaces modelled on the space H^{1}(M, ℤ_{2}) := Hom(H_{1}(M, ℤ_{2}), ℤ_{2}) = Hom(H_{1}(M, ℤ), ℤ_{2}).

We will now associate to any spin structure χ on a Riemann surface M a quadratic form q_{χ} : H_{1}(M, ℤ_{2}) → ℤ_{2}. This association will define a bijection from the set of equivalence classes of spin structures to the set of quadratic forms.

For simplicity of notation, we fix a Riemannian metric g on M. Let SO(M, g) denote the S^{1}-principal bundle of positively oriented orthonormal frames on M. If one specializes the description of a spin structure (Subsection 2.2) to dimension 2, then a spin structure χ consists of a principal S^{1}-bundle Spin(M, g, χ) → M and a double covering μ_{χ} : Spin(M, g) → SO(M, g) with μ_{χ}(φ · z) = μ_{χ}(φ) · z^{2} for all φ ∈ Spin(M, g) and all z ∈ S^{1}.

Any homology class in H_{1}(M, ℤ_{2}) can be represented by a closed embedded loop γ : ℝ/(Lℤ) → M, L > 0, parameterized by arclength. If γ̇ : ℝ/(Lℤ) → SO(M, g) lifts to a map ℝ/(Lℤ) → Spin(M, g), then we define q_{χ}(γ) := 1, otherwise we define q_{χ}(γ) := 0. One checks that if γ_{1} and γ_{2} represent the same homology class in H_{1}(M, ℤ_{2}), then q_{χ}(γ_{1}) = q_{χ}(γ_{2}), hence q_{χ} defines a map q_{χ} : H_{1}(M, ℤ_{2}) → ℤ_{2}. One checks that this map is in fact a quadratic map.

For any quadratic map q : V → ℤ_{2} associated to a non-degenerate symmetric bilinear form on a finitedimensional ℤ_{2}-vector space V one defines one defines the Arf-invariant

Arf(q) := \frac{1}{\sqrt{#V}} ∑_{a∈V}(−1)^{q_{χ}(a)}.

One can check (e.g. [AB02, Section 3]) that the sum is either +1 or −1. We now define α(M, χ) via (−1)^{α(M,χ)} = Arf(q_{χ}).

Example 6.1. Let M = S^{2} with the spin structure χ^{2}. Then q_{χ^{2}} : {0} → ℤ_{2}, q_{χ^{2}}(0) = 0, α(S^{2}, χ^{2}) = 0.

Example 6.2. Let M be of genus 1, i.e. diffeomorphic to T^{2}, with a metric g. After performing a conformal change (which does not affect neither the quadratic form nor the dimension of the kernel) we can assume that g is flat, i.e. (M, g) is isometric to ℝ^{2}/Γ equipped with the euclidean metric for a lattice Γ ⊂ ℝ^{2}, Γ = π_{1}(M) = H_{1}(M, ℤ). Then SO(M, g) is a trivial S^{1} bundle, where a trivialization is given by a parallel frame (e_{1}, e_{2}). We can also write SO(M, g) = (ℝ^{2}/Γ) × SO(2). Any group homomorphism γ : Γ → {−1, +1} ⊂ ker(Spin(2) → SO(2)) ⊂ Spin(2) = S^{1} defines a diagonal action of Γ on ℝ^{2} ×Spin(2), and we obtain a Spin(2)-principal bundle by factoring out this action

Spin(M, g)_{Γ} := ℝ^{2} ×_{γ} Spin(2).

This principal bundle together with the natural map

χ_{γ} : Spin(M, g) = ℝ^{2} ×_{Γ} Spin(2) → SO(M, g) = (ℝ^{2}/Γ) × SO(2)
[(x, z)]_{γ} ↦ (x + Γ, z^{2})

defines a spin structure χ_{γ} on M. Note that in the sense of Spin(2)-principal bundles, the possible Spin(M, g) are all equivalent. However, χ_{γ_{1}} and χ_{γ_{2}} are not equivalent in the sense of spin structures if γ_{1} ≠ γ_{2}. Furthermore any spin structure on M is obtained in this way. Denote the image of v ∈ Γ in H_{1}(M, ℤ_{2}) = Γ ⊗_{ℤ} ℤ_{2} by \overline{v}. The quadratic form q of the spin structure associated to γ then fulfills

\begin{matrix}q(\overline{v}) = γ(v) + 1 & for \overline{v} ≠ 0, \\ q(\overline{v}) = γ(v) = 0 & for \overline{v} = 0. \end{matrix}

This implies α(M, χ) = 1 if χ is the spin structure associated to the trivial map γ, and α(M, χ) = 0 in all other cases. The boundary of a solid torus has a map γ which is non-trivial, hence α(M, χ) = 0 in this case.

On the other hand we can also calculate the dimension of the kernel of D. As g is flat, Dφ = 0 is equivalent to ∇φ = 0.

If γ is the trivial map, then the spinor bundle is trivialized by parallel spinors, i.e. dim ker D_{g} = 2. If γ is non-trivial, then dim ker D_{g} = 0.

This terminates the example, and we return to the general case.

We now recall the index theorem for Dirac operators on compact Riemann surfaces. Note that the spinor bundle ΣM → M carries the structure of a quaternionic vector bundle over M, and the quaternionic multiplication commutes with the Dirac operator. Hence, the complex dimension of any eigenspace of the Dirac operator is divisible by 2.

Theorem 6.3 (Index theorem). For any compact surface with Riemannian metric g and spin structure χ we have

(dim_{ℂ} ker D)/2 ≡ α(M, χ) mod 2.

In the examples above, we have verified this relation if M is diffeomorphic to S^{2} or T^{2}. We will sketch a short proof of the theorem in Remark 6.7.

Remark 6.4. According to [Hit74] the (complex) dimension k_{g} of the kernel of the Dirac operator on a compact Riemann surface (M, g) of genus γ is at most γ + 1. Hence, k_{g} is already determined by α(M, χ) and γ if γ ≤ 2, or if γ = 3, α(M, χ) = 1. However, in all other cases, k_{g} depends on the conformal class of g (see [Hit74], [BS92]). The spectrum of the Dirac operator depends continuously on g in the C^{1}-topology. Hence, k_{g} ≥ lim sup_{h→g} k_{h}. If g_{i} → g with k_{g_{i}} < k_{g} then due to the symmetric of the spectrum of D, some positive and some negative eigenvalue converges to 0, both having the same, even multiplicity. Hence, if k_{g} jumps then by a multiple of 4, and k_{g} mod 4 is therefore independent of g.

Proof of Theorem 1.3. In the case α(M, χ) = 0, the index theorem implies that the Dirac operator has a kernel for any given metric on M. This immediately implies τ(M, χ) = 0 which yields the first case in Theorem 1.3.

In order to derive the second case, we have to study the behavior of the Arf-invariant on sums of quadratic forms and the α-invariant under connected sum.

If q_{i} : V_{i} → ℤ_{2} are quadratic maps for i = 1, 2, then on V_{1} ⊕ V_{2} we have a product quadratic map defined via q_{1}⊕q_{2} : V_{1}⊕V_{2} → ℤ_{2}, (q_{1}⊕q_{2})(v_{1}, v_{2}) = q_{1}(v_{1})+q_{2}(v_{2}). One checks that Arf(q_{1}⊕q_{2}) = Arf(q_{1})Arf(q_{2}).

The quadratic form of a disjoint union of (M_{1}, χ) and (M_{2}, χ_{2}) is just the product quadratic map q_{χ_{1}} ⊗q_{χ_{2}}. Furthermore, one easily sees that the quadratic forms of (M_{1}#M_{2}, χ_{1}#χ_{2}) and (M_{1}, χ_{1}) ˙∪(M_{2}, χ_{2}) can be identified.

It follows that

α(M_{1}#M_{2}, χ_{1}#χ_{2}) ≡ α((M_{1}, χ_{1}) ˙∪(M_{2}, χ_{2})) ≡ α(M_{1}, χ_{1}) + α(M_{2}, χ_{2}) mod 2

As α(S^{2}, χ_{2}) = 0, it follows that α(M, χ) is a spin-cobordism invariant. Hence, α defines a map from the 2-dimensional spin-cobordism group to ℤ_{2}.

Inversely, it can be shown (e.g. with the statements of [AB02, section 3]):

Lemma 6.5. Let M be an compact oriented surface of genus γ with spin structure χ, and α(M, χ) = 0. Then (M, χ) is spin-diffeomorphic to the connected sum of γ 2-tori

T^{2}#T^{2}# ⋯ #T^{2}

where each 2-torus carries a spin structure that is associated to a non-trivial homomorphism.

Corollary 6.6. Let M be an compact oriented surface of genus γ with spin structure χ, and α(M, χ) = 0. Then (M, χ) is obtained from (S^{2}, χ^{2}) by a sequence of 0-dimensional surgeries.

Hence, Theorem 1.1 implies that τ(M, χ) ≥ τ(S^{2}, χ^{2}) = 2\sqrt{π}, but as τ(M, χ) ≤ 2\sqrt{π} (Proposition 8), we obtain the conclusion in the second case of Theorem 1.3. □

Remark 6.7. With the methods provided in this section Theorem 6.3 can be proved in a simple and geometric way. From the construction of α(M, χ) out of the quadratic form it is clear that α(M, χ) is preserved under 0-dimensional surgery, in particular it is additive under connected sum. Together with α(S^{2}, χ^{2}) = 0 it follows that it is a spin-cobordism invariant. We have already seen that dim ker D mod 4 is invariant on the metric. The following Proposition implies that dim ker D might only jump by multiples of 4 when a 0-dimensional surgery is performed that introduces a (long and thin) cylinder. Hence, dim ker D mod 4 is spin-cobordant as well. As we have already checked the index theorem for tori, the index theorem Theorem 6.3 follows in general.

Proposition 6.8. Let (M, g, χ) be a compact spin manifold with Dirac operator D, and let (M^{#}, χ^{#}) be obtained from M by 0-dimensional surgery. Then there is a sequence of metrics g_{i}^{#}, i → ∞, on M^{#} such that the Dirac operator D_{i} on (M^{#}, g_{i}^{#}, χ^{#}) satisfies

(a) dim ker D_{i} is independent on i,

(b) dim ker D_{i} ≤ dim ker D,

(c) If dim ker D_{i} < dim ker D, then there are positive and negative eigenvalues converging to 0. Their combined multiplicity is equal to dim ker D − dim ker D_{i}.

Part (a) and (b) of the Proposition are proved in [ADH06]. Part (c) is not proved in there explicitly, but the arguments and constructions in [ADH06] can be adapted such that we obtain (c).

The metrics g_{i} are unchanged outside the attached cylinders, and the cylinders equipped with the metrics g_{i} get longer and thinner when i tends to ∞.

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Authors’ addresses:

Bernd Ammann NWF I – Mathematik Universität Regensburg 93040 Regensburg Germany

email:

bernd.ammann at
mathematik.uni-regensburg.de

Emmanuel Humbert

Institut Élie Cartan BP 239

Université de Nancy 1

54506 Vandoeuvre-lès -Nancy Cedex France

email:

                                            humbert at iecn.u-nancy.fr