DIRAC SPECTRAL FLOW ON CONTACT THREE MANIFOLDS II: THURSTON–WINKELNKEMPER CONTACT FORMS
CHUNG-JUN TSAI
Abstract. Given an open book decomposition (Σ, τ ) of a three manifold Y , Thurston and Winkelnkemper [TW] construct a specific contact form a on Y . Given a spin-c Dirac operator D on Y , the contact form naturally associates a one parameter family of Dirac operators Dr= D −ir2 cl(a) for r ≥ 0. When r >> 1, we prove that the spectrum of Dr = D0−ir2 cl(a) within [−1 2r 1 2,1 2r 1
2] are almost uniformly distributed. With the result in Part I [Ts1], it implies
that the subleading order term of the spectral flow from D0to Dris of order r(log r)
9
2. Besides
the interests of the spectral flow, the method of this paper provide a tool to analyze the Dirac operator on an open book decomposition.
1. Introduction
Suppose that (Y, a) is a contact three manifold, and D is a spin-c Dirac operator on Y . It naturally associates a one parameter family of Dirac operators Dr= D −ir2 cl(a) for r ≥ 0. The
spectral flow from D0 to Dr appears in the study of contact geometry [T1]. In Part I [Ts1], we
analyze the eigensections of Dralong the Reeb vector field and on the contact hyperplane, then
relate the spectral flow to certain spectral asymmetry property of small eigenvalues of Dr.
This article focuses on the case when Y is given by an open book decomposition and a is the Thurston–Winkelnkemper contact form [TW]. We introduce another Dirac operator ˜Dr over a fibered three manifold. The Dirac operator ˜Dr captures the spectral properties of Dr, and the
spectrum of ˜Dr is easier to handle. It can be used to study the spectral asymmetry and the
spectral flow of Dr. Besides the interests of the spectral flow, our construction provides a tool
to analyze the Dirac equation on an open book decomposition.
1.1. Dirac spectral flow. Let (Y, ds2) be an oriented Riemannian three manifold. A spin-c structure on Y consists of a rank 2 Hermitian vector bundle S and a bundle map cl : T Y → End(S) such that:
• cl(v)2 = −|v|2 for any v ∈ T yY ;
• if |v| = 1, then cl(v) is unitary;
• if {e1, e2, e3} is an oriented orthonormal frame for TyY , then cl(e1) cl(e2) cl(e3) is the
identity endomorphism of S|y.
Date: February 28, 2015.
The bundle S is called the spinor bundle, and the map cl is called the Clifford action.
A spin-c connection on S is a Hermitian connection ∇ : C∞(Y ; S) → C∞(Y ; T∗Y ⊗ S) which is compatible with the Clifford action in the following sense: for any tangent vector field v and any section ψ of S,
∇(cl(v)ψ) = cl(∇LCv)ψ + cl(v)∇ψ
where ∇LC is the Levi-Civita connection. Given a spin-c connection, the associated spin-c Dirac operator D is defined to be the composition:
C∞(Y ; S)−→ C∇ ∞(Y ; T∗Y ⊗ S)metric dual−→ C∞(Y ; T Y ⊗ S)−→ Ccl ∞(Y ; S) .
Atiyah, Patodi and Singer [APS1, APS2, APS3] pioneered the study of the Dirac spectral flow. What follows is the basic idea. Suppose that D is a spin-c Dirac operator. Let {As}s∈[0,1]
be a one parameter family of real valued 1-forms. Consider the one parameter family of Dirac operators {DAs = D − i cl(As)}s∈[0,1]. In other words, they are Dirac operators associated to
the spin-c connections ∇ − iAsI, where I is the identity endomorphism. For simplicity, assume that DA0 and DA1 have trivial kernel. The Dirac spectral flow is the count of the total number
of zero eigenvalues of {DAs}s∈[0,1] with sign. More precisely, the eigenvalues near zero move in
a continuously differentiable manner if {As}s∈[0,1] is suitably generic. The Dirac spectral flow
is equal to the number of eigenvalues which cross zero with positive slope minus the number which cross zero with negative slope. The resulting count turns out to be path independent and so depends only on the ordered pair (A0, A1).
In particular, if there is a real valued 1-form a, we can consider the spectral flow from A0 = 0
to A1 = r2a for r >> 1. This spectral flow can be regarded as a function of r, and will be
denoted by fa(D, r). In [T1, §5] and [T2], Taubes studied this spectral flow function fa(D, r).
Theorem. ([T1, Proposition 5.5]) Suppose that Y is a compact, oriented three manifold with a Riemannian metric ds2. Suppose that D is a spin-c Dirac operator on Y . Then, there exist a universal constant δ ∈ (0,12) and a constant c1 is determined by ds2 and D such that
fa(D, r) − r2 32π2 Z Y a ∧ da ≤ c1r 3 2+δ
for any 1-form a with ||a||C3 ≤ 1 and any r ≥ c1.
This theorem specifies the leading order term of the spectral flow function, and gives a bound on the subleading order term. Recently, Savale [S] improved this theorem by showing that the subleading order term is ofO(r
3 2).
1.2. Contact three manifold. A 1-form a on an oriented three manifold is called a contact form if a∧da > 0. A contact form determines a vector field v by da(v, ·) = 0 and a(v) = 1. This vector field is called the Reeb vector field. A contact form also defines a two plane distribution by
ker(a) ⊂ T Y , which is called the contact hyperplane or the contact structure. By the Frobenius theorem, a ∧ da > 0 implies that the contact hyperplane is everywhere non-integrable.
Given a contact form, it is a convenient normalization to take an adapted metric to consider spin-c structures and Dirac operators. A Riemannian metric ds2 is said to be adapted to a if |a| = 1 and da = 2 ∗ a, where ∗ is the Hodge star operator. Chern and Hamilton [CH] proved that such a metric always exists.
Suppose that D is a spin-c Dirac operator on a contact three manifold (Y, a). It turns out that the zero eigensections of the Dirac operator Dr = D − ir2 cl(a) is closely related to the
geometry of the contact form:
• their derivative along the Reeb vector field is close to the multiplication by ir/2; • on the contact hyperplane, they almost solve certain Cauchy–Riemann equation. The precise statements can be found in [Ts1, §3]. The main goal is to understand more how Dr
is related to the geometry of the contact form. The following question is the first step in this direction: when a is a contact form, is the subleading order term of fa(D, r) of order r? Note
that it is impossible to obtain an estimate on the subleading order term beyond O(r) (see for instance [S, §5]).
In this paper, we confirm the answer for certain types of contact forms, with a slightly larger order.
Main Theorem. Suppose that a is a Thurston–Winkelnkemper contact form [TW] on an open book decomposition. Let D be a spin-c Dirac operator. Then, there exists a constant c2
determined by the contact form a, the adapted metric ds2 and the Dirac operator D such that fa(D, r) − r2 32π2 Z Y a ∧ da ≤ c2r(log r) 9 2 . for any r ≥ c2.
The Thurston–Winkelnkemper contact form will be explained momentarily. The celebrated Giroux correspondence [G] implies that each isotopy class of contact structures admits such a contact form. In other words, the theorem asserts that the subleading order term of the spectral flow function is of order r(log r)92 for certain types of contact forms in each isotopy
class of contact structures.
1.3. Open book decomposition. We now review the necessary background on the open book decomposition and the Thurston–Winkelnkemper contact form. The reader can find a complete discussion on the open book decomposition in [OS, ch.9] and [E]. The notations set up here will be used throughout the rest of this paper.
1.3.1. Open book. An (abstract ) open book consists of (Σ, τ ) where
• Σ is a Riemann surface with non-empty boundary, and ∂Σ is a finite union of circles;
• τ : Σ → Σ is a diffeomorphism such that τ is the identity map on a collar neighborhood of ∂Σ. The map τ is called the monodromy.
An open book (Σ, τ ) gives a three manifold Y as follows: Y = Σ ×τS1 ∪φ
a
|∂Σ|
S1× B
(1.1) where |∂Σ| is the number of boundary components and B is a two dimensional disk. The component Σ ×τ S1 is the mapping torus of τ ,
Σ ×τ S1=
Σ × [0, 2π] (τ (x), 0) ∼ (x, 2π) .
Since τ is the identity map on ∂Σ, the boundary of Σ ×τ S1 is (∂Σ) × S1. The gluing map φ
identifies the boundary of Σ ×τS1with the boundary of`|∂Σ|S1×B. It is determined uniquely
(up to isotopy) by the following properties: for each boundary component of Σ, • φ takes (∂Σ) × {y} to the longitude of ∂(S1× B) where y ∈ S1;
• φ takes {x} × S1 to the meridian of ∂(S1× B) where x ∈ ∂Σ.
Note that there is an S1 family of Σ in Y , which are called the pages. The cores`
|∂Σ|S1× {0}
of the attaching handles are called the bindings. The term ‘near the bindings’ refers to the attaching handles`
|∂Σ|S1× B. It is a particular tubular neighborhood of the bindings.
It is useful to describe the gluing map φ in terms of the local coordinate. Let {ρeit | 1 ≤ ρ < 1 + 50δ, eit ∈ S1} be a coordinate on a collar neighborhood of ∂Σ. By taking δ to be
sufficiently small, we may assume the monodromy τ to be the identity map on this chart. The mapping torus Σ ×τ S1 carries a canonical map to S1. Denote this map by eiθ. It follows
that {(ρeit, eiθ) | 1 ≤ ρ < 1 + 50δ} parametrize a collar neighborhood of (∂Σ) × S1. Let {(eit, ρeiθ) | ρ < 1 + 50δ} be the coordinate on S1 × B. The gluing map φ is defined by
identifying the corresponding coordinates.
1.3.2. Contact form. Given an open book (Σ, τ ), Thurston and Winkelnkemper [TW] construct a contact form a on Y . To start, choose a 1-form ζ on Σ such that
• dζ defines an area form on Σ;
• 2ζ = (2 − ρ)dt on the collar neighborhood {1 ≤ ρ < 1 + 50δ} of ∂Σ. There always exists such a 1-form ζ; see [OS, p.141].
Let χ(θ) be a smooth, non-negative function of θ ∈ [0, 2π] such that χ(θ) = 1 near 0 and χ(θ) = 0 near 2π. For any θ ∈ [0, 2π],
ωθ= χ(θ)dζ + (1 − χ(θ))τ∗dζ
is an area form on Σ. Let V be a positive scalar such that 9 10 ≤ 1 + 2χ0(θ) V ζ ∧ τ∗ζ ωθ ≤ 10 9 (1.2) 4
on Σ and for any θ ∈ [0, 2π]. With χ(θ) and V chosen, the contact form on the mapping torus Σ ×τS1 is defined by
a = V dθ + 2χ(θ)ζ + 2(1 − χ(θ))τ∗ζ . (1.3) A direct computation shows that
da = 2 ωθ+ χ0(θ)dθ ∧ (ζ − τ∗ζ) , 1 2a ∧ da = V + 2χ 0(θ)ζ ∧ τ∗ζ ωθ dθ ∧ ωθ . It follows that the Reeb vector field is
V + 2χ0(θ)ζ ∧ τ ∗ζ ωθ −1 ∂ ∂θ − χ 0(θ)ω−1 θ (ζ − τ ∗ζ) . (1.4)
The above expression is written on Σ × [0, 2π], and ∂θ∂ is the coordinate vector field. The vector field ωθ−1(ζ − τ∗ζ) is defined by ωθ ωθ−1(ζ − τ∗ζ), · = (ζ − τ∗ζ)( · ).
To extend the contact form to the attaching handles `
|∂Σ|S1 × B, choose two smooth
functions f (ρ) and g(ρ) of ρ ∈ [0, 1 + 50δ) such that
• f (ρ) = V and g(ρ) = 2 − ρ when ρ ∈ [1 − 50δ, 1 + 50δ); • f (ρ) = ρ2 and g(ρ) = 2 − ρ2 when ρ ∈ [0, 50δ];
• for any ρ ∈ (0, 1 + 50δ), f0(ρ) ≥ 0 and g0(ρ) < 0.
It is not hard to see the existence of f and g. The contact form near the bindings is defined by
a = f (ρ)dθ + g(ρ)dt . (1.5)
When ρ < 50δ, it is equal to xdy − ydx + (2 − x2− y2)dt in terms of the rectangular coordinate
x + iy = ρeit. Therefore, a is a smooth 1-form on`
|∂Σ|S1× B.
1.3.3. Adapted metric. The Main Theorem requires a specific adapted metric. The metric is set to be
a2+ (dρ)2+1 4(f
0(ρ)dθ + g0(ρ)dt)2 (1.6)
near the bindings. Such an adapted metric always exists. Here is a parenthetical remark. The attaching handles`
|∂Σ|S1× B admits a S1× S1-action:
(S1× S1) × (S1× B) −→ (S1× B)
(eit0, eiθ0) , (eit, ρeiθ)
7→ (ei(t+t0), ρei(θ+θ0)) .
Near the boundary of the mapping torus Σ ×τS1, the first S1 factor rotates the boundary of
Σ, and the second S1 factor flips the pages. The method of this paper should apply to any adapted metric which is invariant under this S1× S1 action. However, adapted metrics are not
the main interests of this paper. We only considered the metric (1.6), and did not try to work out general (locally S1× S1 invariant) metrics.
1.4. Contents of this paper. Spin-c structures can be described more geometrically on a contact three manifold. §2 is a review of the construction. We also recall the results from Part I [Ts1] that will be used in this paper.
§3 contains the key geometric construction. Suppose that Y is given by an open book (Σ, τ ), and a is the Thurston–Winkelnkemper contact form. Let D be a spin-c Dirac operator on Y . Denote D − ir2 cl(a) by Dr. In §3 we construct another compactification ˜Y of Σ ×τS1, and a
one parameter family of Dirac operators { ˜Dr}r≥0 on ˜Y . They have the following significance.
• In contrast to the open book (1.1), ˜Y admits a canonical map to S1. • ˜Dr on Σ ×τS1 ⊂ ˜Y is exactly the same as Dr on Σ ×τ S1⊂ Y .
• The Bochner–Weitzenb¨ock formula of ˜Dr is very similar to that of Dr.
The canonical map ˜Y → S1 can be viewed as a gauge transform. With such a gauge transform, Vafa and Witten [VW] have a brilliant argument to estimate the gap of spectrum of a Dirac operator. By combining with the Bochner–Weitzenb¨ock formula, we prove that small spectrum of ˜Dr are almost uniformly distributed. This is done in §4.
In §5 we introduce two kinds of Dirac operators on S2 × S1. They mimic the behavior of
Dr on Y \(Σ ×τS1) and ˜Dr on ˜Y \(Σ ×τS1), respectively. The eigenvalues and eigensections of
these Dirac operators on S2× S1 can be solved fairly explicitly.
With ˜Dr on ˜Y and the Dirac operators on S2 × S1, one can imagine that their spectrum
approximate the spectrum of Dr on Y . §6 is devoted to compare these spectrum. The precise
statement is Theorem 6.3. It is proved by gluing eigensections.
In §7 we calculate the spectral flow from D0 to Dr with the help of the above models, and
prove the Main Theorem.
Remark 1.1. The constants c(·)in this paper are always independent of r. In other words, they only depend on the contact form a, the metric ds2 and the unperturbed spin-c Dirac operator. The subscript is simply to indicate that these constants might increase/decrease after each step. The subscript will be returned to 1 at the beginning of each section.
Acknowledgement. The author is very grateful to Cliff Taubes for bringing [VW] to his atten-tion, and for many helpful discussions. He would like to thank the Department of Mathematics at Harvard University, where part of this work was carried out. He would also like to thank the anonymous referees for providing useful references.
2. Dirac Operator on Contact Three Manifold
Suppose that (Y, a) is a contact three manifold with an adapted metric ds2. As in [T1, §2.1], the spin-c structures and spin-c connections can be described more geometrically. It works for a stable Hamiltonian structure as well, of which a contact form is a special case. We will also work with stable Hamiltonian structures in this paper.
2.1. Stable Hamiltonian structure. Suppose that Y is a compact, oriented three manifold. A stable Hamiltonian structure is a pair (b, ω) where b is a 1-form and ω is a 2-form such that
dω = 0 , b ∧ ω > 0 , ker(ω) ⊂ ker(db) .
This notion was identified in [BEH+, §2] and [CM, §2]. Given a contact form a, the pair (a, da) is a stable Hamiltonian structure. A stable Hamiltonian structure determines a vector field v by ω(v, ·) = 0 and b(v) = 1. We still call v the ‘Reeb vector field ’.
Such a structure needs not to come from a contact form. Here is a standard example. Suppose that N is a compact surface with a symplectic form ω, and τ is a symplectomorphism of (N, ω). Let dθ be the pull-back of the standard 1-form on S1 by the projection N ×
τS1 → S1. Since
τ is a symplectomorphism, the 2-form ω descends from N × R to N ×τ S1. Then, (dθ, ω) is
a stable Hamiltonian structure on the mapping torus N ×τ S1. Since ker(dθ) is everywhere
integrable, it is different from a contact form.
2.1.1. Conformally adapted metric. In Part I [Ts1], a slightly more general type of metric is also considered. Suppose that (b, ω) is a stable Hamiltonian structure on Y . A Riemannian metric ds2 is said to be conformally adapted to (b, ω) if
|b| = Ω−1 and ∗ ω = 2Ω−1b for some function Ω ∈ C∞(Y ; R) with
9
10 ≤ Ω ≤ 10
9 .
The particular bounds are just convenient normalizations; any other fixed bounds would do the job. The operator ∗ is the Hodge star operator of ds2. Note that Ω−2ds2 is an adapted metric1.
The function Ω is called the conformal factor. Such a metric always exists. The argument of Chern and Hamilton [CH] applies to a stable Hamiltonian structure as well. It is equivalent to an almost complex structure J on ker(b) such that Ω2ω( · , J ( · )) defines a metric on ker(b). 2.1.2. Spin-c structure. With the metric fixed, (b, ω) determines a canonical spin-c structure. The spinor bundle is given by C ⊕ K−1, where C is the trivial bundle and K−1 is isomorphic as an SO(2) bundle to ker(b) with the orientation defined by ω. More precisely, for any u ∈ ker(b), let J (u) be the metric dual of Ω2ω(u, · ). The local sections of K−1 consists of u − iJ (u) with
u ∈ ker(b).
The Clifford action is defined as follows. The Clifford action of the Reeb vector field v acts as iΩ on C and as −iΩ on K−1. In other words, C ⊕ K−1 is the eigenbundle splitting of cl(v).
1Namely, |b| = 1 and ∗ω = 2b with respect to Ω−2 ds2.
Let 1 be the depicted unit-normed section of C. For any unit vector u ∈ ker(b), the Clifford action of u is defined by cl(u) : 1 7→ √1 2(u − iJ (u)) 1 √ 2(u − iJ (u)) 7→ −1 , and the Clifford action of J (u) is defined by
cl(J (u)) : 1 7→ √i 2(u − iJ (u)) 1 √ 2(u − iJ (u)) 7→ i1 . It is straightforward to check that it does define a spin-c structure.
The set of spin-c structures has a free transitive action of H2(Y ; Z), see [LM, Appendix D]. The action is given by tensoring with a complex line bundle E. It follows that a spinor bundle can be written as E ⊕ EK−1 for some Hermitian line bundle E. The Clifford action is induced from that on the canonical spinor bundle, and the splitting E ⊕ EK−1 is the eigenbundle splitting of cl(v).
2.1.3. Spin-c connection. The canonical spinor bundle C ⊕ K−1 carries a canonical spin-c con-nection, which we denote by ∇o. It is the unique spin-c connection whose associated Dirac
operator Do annihilates Ω−11, the depicted section 1 rescaled by Ω−1. The proof for its
exis-tence and uniqueness can be found in [H, Lemma 10.1].
The canonical connection can be written down explicitly in terms of a local trivialization. Let e1, e2, e3 be an oriented, orthonormal local frame for T Y , where e3 = Ω−1v is the Reeb
vector field multiplied by Ω−1. Using the trivialization 1 and √1
2(e1− ie2), the local sections
of C ⊕ K−1 are identified with C2 valued functions. With respect to this trivialization, the canonical connection is given by
∇oψ = dψ + 1 2 X j≤k θkj cl(ej) cl(ek)ψ + i 2 θ 2 1− Ω2∗ d(Ω −1b)ψ (2.1)
where θkj is the coefficient 1-form of the Levi-Civita connection, i.e. ∇LCej =Pkθkj ⊗ ek. We
leave it to the reader to check that the expression does define a spin-c connection (see also [LM, II.4]).
It follows from ker(ω) ⊂ ker(db) and the structure equation that
θ13(e3) − Ω−1e1(Ω) = 0 , and θ23(e3) − Ω−1e2(Ω) = 0 .
It follows from dω = 0 and the structure equation that
θ31(e1) + θ32(e2) + 2Ω−1e3(Ω) = 0 .
Using these relations, a direct computation shows that Dirac operator associated to (2.1) anni-hilates ψ = (Ω−1, 0). Hence, (2.1) defines the canonical connection.
With this understood, any spin-c connection on a spinor bundle E ⊕ EK−1 can be expressed as ∇o⊗ AE where AE is a unitary connection on E.
2.2. Results of Part I. We now recall the results of Part I [Ts1]. Suppose that (Y, a) is a contact three manifold with a conformally adapted metric ds2. Suppose that E → Y is a Hermitian line bundle with a unitary connection AE. For any r ≥ 0, consider the one parameter
family of Dirac operators on E ⊕ EK−1 defined by Dr= cl(∇o⊗ AE −
ir 2a) .
Denote by fa(r) the spectral flow from D0 to Dr. In Part I [Ts1], we obtained the following
estimate on fa(r).
Theorem 2.1. There exists a constant c1 determined by the contact form a, the conformally
adapted metric ds2 and the connection AE such that for any r ≥ 2c1,
fa(r) − r2 32π2 Z Y a ∧ da ≤ c1r(log r) 9 2 + | ˙η(r)| + c1 Z r 1 ¨ η(r)dr . The function ˙η(r) is defined by
˙ η(r) = r−12(log r) 1 2 X ψ∈Vr+ Z 1 3r 1 2 λψ e−20(r−1log r)u2du − X ψ∈Vr− Z λψ −13r12 e−20(r−1log r)u2du (2.2)
where Vr+consists of orthonormal eigensections of Dr whose eigenvalue belongs to (0,13r
1 2), Vr−
consists of orthonormal eigensections of Dr whose eigenvalue belongs to (−13r
1
2, 0), and λψ is
the corresponding eigenvalue. The function ¨η(r) is defined by ¨ η(r) = (r−32 log r) X ψ∈Vr (λψe−20(r −1log r)λ2 ψ) (2.3)
where Vr consists of orthonormal eigensections of Dr whose eigenvalue belongs to (−13r
1 2,1
3r
1 2).
Proof. It follows from Proposition 5.7 and Proposition 5.6 of [Ts1]. 2.3. Main result. Theorem 2.1 reduces the spectral flow estimate to ˙η(r) and ¨η(r). These two functions measure certain spectral asymmetry of Dr within (−13r
1 2,1
3r
1
2). The main goal of
this paper is to show that the spectrum within (−13r12,1
3r
1
2) are almost uniformly distributed
when a is the Thurston–Winkelnkemper contact form. The rest of this paper is devoted to the proof of the following theorem, and its conditions (i) and (ii) will be assumed throughout the rest of this paper.
Theorem 2.2. Let (Σ, τ ) be an open book. Denote the three manifold (1.1) by Y , and denote the Thurston–Winkelnkemper contact form by a. Suppose that
(i) ds2 is a conformally adapted metric which is equal to (1.6) near the bindings, and whose conformal factor Ω is equal to
1 +2χ 0(θ) V ζ ∧ τ∗ζ ωθ −1 ; (2.4)
this factor appears in (1.2) and (1.4), and is equal to 1 near the bindings;
(ii) the unitary connection AE on E → Y is gauge equivalent to the trivial connection near
the bindings; note that any bundle is topologically trivial near the bindings. Then, there exists a constant c2 determined by a, ds2 and AE such that
˙
η(r) ≤ c2r(log r)
1
2 and η(r) ≤ c¨ 2log r
for any r ≥ c2.
By combining this theorem with Theorem 2.1, there exists a constant c3 such that
fa(r) − r2 32π2 Z Y a ∧ da ≤ c3r(log r) 9 2 (2.5) for any r ≥ c3.
The technical conditions (i) and (ii) of Theorem 2.2 are not crucial for the spectral flow function fa(r). The reason goes as follows.
(i) As explained in [Ts1, §2.1], the spectral flow function fa(r) is invariant under the
conformal change of metric. It follows that the spectral flow estimate (2.5) works for any adapted metric that is equal to (1.6) near the bindings. Notice that we do not claim that ˙η(r) and ¨η(r) are invariant under the conformal change of metric.
(ii) According to [Ts1, Proposition 5.9], different choices of AE lead to a O(r) difference
of the spectral flow function.
With this understood, we conclude the following theorem.
Theorem 2.3. Suppose that a is a Thurston–Winkelnkemper contact form [TW]. Suppose that ds2 is an adapted metric which is equal to (1.6) near the bindings. Let D be a spin-c Dirac operator. Then, there exists a constant c4 determined by a, ds2 and D such that
fa(r) − r2 32π2 Z Y a ∧ da ≤ c4r(log r) 9 2 . for any r ≥ c4.
Remark 2.4. The conformal factor of Theorem 2.2(i) shows up naturally. In the construction of §1.3.2, there are two volume forms on Σ ×τS1:
1
2a ∧ da and dθ ∧ ωθ .
Their ratio defines a function which is equal to V on the boundary of Σ ×τS1. This particular
conformal factor plays a key role in the proof of Theorem 4.2 below.
3. From Open Book to Mapping Torus
Under the setting of Theorem 2.2, the main purpose of this section is to construct a model which captures the spectrum of Dr = cl(∇o⊗ AE −ir2a). For simplicity, assume from now on
that Σ has only one boundary component . If Σ has more than one boundary components, one simply needs to duplicate the construction and the argument.
The model consists of the following objects (which will be introduced momentarily): • another compactification ˜Y of Σ ×τS1, which is a surface bundle over S1;
• a stable Hamiltonian structure (˜a, 2˜ω) on ˜Y , and a conformally adapted metric ds2 on ˜
Y ;
• a spinor bundle ( ˜E ⊕ ˜E ˜K−1) ⊗ ˜Lr → ˜Y , and a Dirac operator ˜Dr on it.
For brevity, the model ( ˜Y → S1, ˜a, ˜ω, ds2, ( ˜E ⊕ ˜E ˜K−1) ⊗ ˜Lr, ˜Dr) will be denoted by ( ˜Y , ˜Dr).
The Dirac operator ˜Dr has the following salient features:
(i) on Σ ×τS1, ˜Dr is identically the same as Dr;
(ii) the “small” spectrum of ˜Dr is almost uniformly distributed; the precise statement
appears in Theorem 4.2.
3.1. The mapping torus. To start, compactify Σ by attaching a disk to its boundary. To be more precise, let {ρeit | ρ ≥ 1, eit ∈ S1} be the coordinate on a collar neighborhood of ∂Σ.
The attaching disk is given by B = {ρeit|0 ≤ ρ < 1 + 50δ}. The compactification is done by
identifying the coordinate. Denote the resulting closed surface by ˜Σ.
Since τ is the identity on a collar neighborhood of ∂Σ, it naturally extends to a monodromy ˜
τ of ˜Σ by ˜τ |B = IB. Let ˜Y be the mapping torus ˜Σ טτS1. Equivalently,
˜
Y = (Σ ×τS1) ∪
a
|∂Σ|=1
B × S1
where B is attached to Σ. It is a surface bundle over S1. Denote the fibration map ˜Y → S1 by eiθ.
3.2. The extension of the 1-form a. The 1-form a can be extended to ˜Y . The extension will be denoted by ˜a.
Near the boundary of Σ ×τ S1, the 1-form a (1.3) is equal to V dθ + (2 − ρ)dt. Choose a
smooth function ˜g(ρ) of ρ ∈ [0, 1 + 50δ) such that ˜
g(ρ) = 0 when ρ ≤ 50δ and ˜g(ρ) = 2 − ρ when ρ ≥ 1 − 50δ . On the attaching handle B × S1, the 1-form ˜a is defined by
˜
a = V dθ + ˜g(ρ)dt . (3.1)
It is clear that ˜a is a smooth 1-form on ˜Y . Notice that ˜a is no longer a contact form on ˜Y .
3.3. The extension of the metric. Near the boundary of Σ×τS1, the metric (1.6) is equal to
a2+(dρ)2+(12dt)2. For any scalar σ ∈ (109,1110), choose a smooth function ˜hσ(ρ) of ρ ∈ [0, 1+50δ)
such that
• ˜hσ(ρ) = 12ρ2 when ρ ≤ 50δ;
• ˜hσ(ρ) = σ + 12(ρ − 2) when ρ ≥ 1 − 50δ;
• ˜h0σ(ρ) > 0 for any ρ ∈ (0, 1), where ˜h0σ(ρ) means the derivative of ˜hσ(ρ) in ρ.
Moreover, the functions {˜hσ(ρ)}9 10<σ<
11
10 have uniformly bounded C
k-norm for any non-negative
integer k. Namely, there exist constants ck such that
sup σ∈(9 10, 11 10) sup ρ∈[0,1+50δ) ∂k˜hσ(ρ) ∂ρk ≤ ck . (3.2)
The metric on the attaching handle B × S1 is taken to be
ds2 = ˜a2+ (dρ)2+ (˜h0σ(ρ)dt)2 , (3.3) and its volume form is
˜
h0σ(ρ) dθ ∧ dt ∧ dρ . (3.4)
It is clear that the construction gives a smooth extension of the Riemannian metric on Σ ×τS1
to ˜Y . The precise choice of σ will be made later.
3.4. The stable Hamiltonian structure. The 2-form 12da also admits an extension ˜ω by:
˜ ω = 1 2da on Σ ×τS 1 , ˜ h0σdt ∧ dρ on B × S1 .
It is straightforward to check that (˜a, 2˜ω) forms a stable Hamiltonian structure, and the metric defined in §3.3 is conformally adapted to it.
3.4.1. The canonical spin-c structure. As explained in §2.1.2, there is a canonical spin-c struc-ture determined by (˜a, 2˜ω) and the metric ds2. Denote the canonical spinor bundle by C ⊕K˜−1 → ˜Y .
It is convenient to fix a trivialization of ˜K−1 → B × S1 ⊂ ˜Y . Consider the unitary vector
field √ 2∂B= 1 √ 2e it ∂ ρ+ i ˜ h0 σ (∂t− ˜ g V∂θ) . (3.5)
The expression is smooth when ρ > 0. When ρ < 50δ, it is equal to √1
2(∂x− i∂y) in terms of
the rectangular coordinate x − iy = ρeit.
3.4.2. The extension of E. The Hermitian line bundle E → Y is assumed to be trivial near the bindings, and the connection AE is assumed to be the exterior derivative. It follows that
the bundle and the connection can naturally be regarded as being defined over ˜Y . Denote the bundle by ˜E → ˜Y , and the connection by AE˜.
3.5. The degree r bundle. The purpose of this subsection is to construct a Hermitian line bundle ˜Lr → ˜Y with a unitary connection ˜Ar for any r ≥ 20. The curvature of ˜Ar supports
only on the attaching handle B × S1, and is proportional to r. Since r > 20, 109 < [r]r < 1110. Set the constant σ to be
σ = [r]
r . (3.6)
Although σ depends on r, the function ˜hσ(ρ) is independent of r in the sense of (3.2).
To construct ˜Lr, consider the trivial bundle over Σ ×τS1 and B × S1. Let 1Σ and 1B be the
depicted unitary sections, respectively. On the overlap region {1 ≤ ρ < 1 + 50δ}, identify the bundles by the transition rule
ei[r]t· 1Σ= 1B . (3.7)
The unitary connection ˜Ar is defined as follows:
• over Σ ×τ S1, the connection ˜Ar is d with respect to 1Σ;
• over B × S1, the connection ˜A
r is d + ir(˜hσ+12g)dt with respect to 1˜ B.
When 1 ≤ ρ < 1 + 50δ,
ir(˜hσ +
1
2˜g)dt = i(rσ)dt = i[r]dt .
It follows that ˜Ar obeys the transition rule (3.7), and hence defines a connection on ˜Lr.
3.6. The Dirac operator on ( ˜E ⊕ ˜E ˜K−1) ⊗ ˜Lr. The bundle ( ˜E ⊕ ˜E ˜K−1) ⊗ ˜Lr → ˜Y is also a
spinor bundle. Let ∇o be the canonical connection on C ⊕ ˜K−1. The connection ∇o⊗ AE˜⊗ ˜Ar
is a spin-c connection on (C ⊕ ˜K−1) ⊗ ˜E ⊗ ˜Lr. Perturb the connection by −ir2˜a, and consider
the corresponding Dirac operator. Namely, ˜ ∇r = ∇o⊗ AE˜⊗ ˜Ar− ir 2˜a , ˜ Dr = cl ◦(∇o⊗ AE˜⊗ ˜Ar− ir 2˜a) . (3.8)
The Weitzenb¨ock formula for ˜Dr reads
˜
Dr2ψ = ˜∇∗r∇˜rψ + ˜κ(ψ) + cl( ˜Fr−
ir 2d˜a)(ψ)
where ˜Fr is the curvature of ˜Ar. Here ˜κ consists of the scalar curvature, the curvature of ∇o
and the curvature of AE˜; in particular, ˜κ is an operator independent of r. On Σ ×τ S1,
cl( ˜Fr−
ir
2d˜a) = irΩ
−1cl(˜a)
where Ω is the conformal factor (2.4). On B × S1,
cl( ˜Fr− ir 2d˜a) = cl ir(˜h 0 σ+ ˜ g0 2)dρ ∧ dt − ir ˜ g0 2dρ ∧ dt = ir cl(˜h0σdρ ∧ dt) = irΩ−1cl(˜a) . It follows that the Weitzenb¨ock formula becomes
˜
Dr2ψ = ˜∇∗r∇˜rψ + ˜κ(ψ) + irΩ−1cl(˜a)(ψ) . (3.9)
The operator i cl(˜a) acts diagonally on ( ˜E ⊕ ˜E ˜K−1)⊗ ˜Lr. It acts as −Ω−1on the ˜E ˜Lrsummand,
and acts as Ω−1 on the ˜E ˜K−1L˜r summand.
Remark 3.1. The spinor bundle ( ˜E ⊕ ˜E ˜K−1) ⊗ ˜Lr is topologically trivial over the attaching
handle B × S1. The bundle ˜E is trivialized by an AE˜-parallel, unit-normed section. The bundle
˜
Lr is trivialized by 1B as in §3.5. The bundle ˜K−1 is trivialized by
√
2∂B (3.5). They induce
a unitary trivialization of ( ˜E ⊕ ˜E ˜K−1) ⊗ ˜Lr over B × S1 ⊂ ˜Y , and the sections on B × S1 can
be identified with C2 valued functions.
4. Eigenvalue Distribution of ˜Dr
The main purpose of this section is to show that the “small eigenvalues” of ˜Dr are almost
“uniformly distributed”. The strategy here is learned from Vafa and Witten [VW]. They applied the Atiyah–Patodi–Singer index theorem [APS1, APS3] to prove that there cannot be large gaps in the Dirac spectrum.
The following proposition gives an integral estimate on the eigensections.
Proposition 4.1. There exists a constant c1 determined by the stable Hamiltonian structure
(˜a, ˜ω), the metric ds2 and the connection AE˜ such that the following holds. For any r ≥ c1,
suppose that ψ is an eigensection of ˜Dr whose eigenvalue λ satisfies |λ|2 ≤ 34r. Then
Z ˜ Y |β|2+ r−1 Z ˜ Y | ˜∇rβ|2 ≤ c1r−1 Z ˜ Y |α|2
where α is the ˜E ˜Lr component of ψ, and β is the ˜E ˜K−1L˜r component of ψ.
Proof. With the Weitzenb¨ock formula (3.9), the proof is exactly the same as that for [Ts1,
Proposition 2.2].
Here comes the main result about the spectrum distribution.
Theorem 4.2. There exist constants c2 and c3 determined by the stable Hamiltonian structure
(˜a, ˜ω), the metric ds2 and the connection AE˜ with the following significance. For any r ≥ c2, let
{λj}j∈Z be the spectrum of ˜Dr, which are arranged in ascending order. Then for any |λj| ≤ 12r
1 2, λj+j− λj− 1 V ≤ c2r− 1 2 where j = r(R ˜ Y dθ ∧ ˜ω) + c3 > 0.
Proof. Regard the fibration map ˜Y → S1 as a gauge transform. The Dirac operator eiθD˜re−iθ =
˜
Dr− i cl(dθ) is gauge equivalent to ˜Dr, and hence has the same spectrum as ˜Dr. Consider the
one parameter family of Dirac operators defined by Ds = ˜Dr− is cl(dθ)
for s ∈ [0, 1]. Arrange the eigenvalues λj(s) of Ds in ascending order,
−∞ < · · · ≤ λj−1(s) ≤ λj(s) ≤ λj+1(s) ≤ · · · < ∞ ,
and normalize the index so that at s = 0, λ1(0) is the smallest non-negative eigenvalue. Since
D0 is gauge equivalent to D1, there exists an integer j such that λj(1) = λj+j(0) for any j ∈ Z.
According to [APS3, section 7], the integer j is the spectral flow of the family {Ds}0≤s≤1, and
can be computed by the index formula [APS1, (4.3)].
(The spectral flow computation) Since D0 is gauge equivalent to D1, the boundary
contri-bution of the index formula at s = 0 cancels with that at s = 1. It follows that j = Z [0,1]× ˜Y 1 8c 2 1( ˜K −1˜ E2L˜2r) − 1 24p1([0, 1] × ˜Y ) .
Here, p1 is the first Pontryagin class of the metric. It is constructed from the Weyl curvature
p1= 4π12(|W+|2− |W−|2), and hence vanishes on [0, 1] × ˜Y (see [APS2, p.421]). The first Chern
class of ˜K−1E˜2L˜2r is given by i
2π(−FK˜ + 2FE˜ + 2 ˜Fr− ird˜a − 2ids ∧ dθ) = 1
2π(−iFK˜ + 2iFE˜+ 2r ˜ω + 2ds ∧ dθ) . More precisely, the differential forms are pulled back by the projection map [0, 1] × ˜Y → ˜Y except ds. It follows that
j = r 4π2 Z ˜ Y dθ ∧ ˜ω + 1 8π2 Z ˜ Y dθ ∧ (2iFE˜ − iFK˜) . (4.1)
It is not hard to see that dθ ∧ ˜ω > 0. Thus, j > 0 provided r is sufficiently large.
(The spectral gap estimate) The second step is to estimates the difference between λj(0) and
λj(1). Since Ds is ˜Dr perturbed by a closed 1-form, Ds obeys a similar Weitzenb¨ock formula
as (3.9). As a result, Proposition 4.1 also holds for Ds for any s ∈ [0, 1]. 15
Let ψj(s) be the unit-normed eigensection of Ds with eigenvalue λj(s). Then λ0j(s) = Z ˜ Y h−i cl(dθ)ψj(s), ψj(s)i , (4.2)
and thus |λ0j(s)| ≤ c4 for c4 = supY˜ |dθ|. In particular, if |λj(0)|2 ≤ 14r, then |λj(s)|2 ≤ 34r for
any s ∈ [0, 1]. It follows that Proposition 4.1 applies to ψj(s) for all s ∈ [0, 1]. By (1.4), (3.1)
and (3.3),
dθ = V−1Ω ˜a + components in the metric dual of ker(˜a) . Therefore, h−i cl(dθ)ψj, ψji = 1 V(|αj| 2− |β j|2) + hb(αj), βji − hb†(βj), αji (4.3)
where αj and βj are the ˜E ˜Lr and ˜E ˜K−1L˜r components of ψj, respectively, and b and b† are
the off-diagonal components of −i cl(dθ). The endomorphisms b and b† are independent of r. According to (4.2), (4.3) and Proposition 4.1, there exists a constant c5 such that
if |λj(0)| ≤ 1 2r 1 2, then |λ0 j(s) − 1 V| ≤ c5r −1 2 for any s ∈ [0, 1] .
Integrating this inequality from s = 0 to s = 1 gives |λj+j(0) − λj(0) −
1
V| ≤ c5r
−1 2 .
The inequality and (4.1) complete the proof of theorem.
The following corollary is a direct consequence of the theorem.
Corollary 4.3. There exists a constant c6 determined by the stable Hamiltonian structure
(˜a, ˜ω), the metric ds2 and the connection AE˜ with the following significance. Suppose that
r ≥ c6, and λ−, λ+ ∈ [−12r
1 2,1
2r
1
2] are any two numbers with λ+− λ− ≥ 1
V. Then, the total
number of the eigenvalues (counting multiplicities) of ˜Dr within [λ−, λ+] is less than or equal
to c5r(λ+− λ−).
5. Two Local Models
The model ˜Y constructed in §3 is useful for analyzing eigensections of (Y, Dr) on the mapping
torus Σ ×τS1. The main purpose of this section is to introduce two Dirac operators on S2× S1.
They are useful for studying the eigensections of (Y, Dr) near the bindings, and the eigensections
of ( ˜Y , ˜Dr) on the attaching handle.
5.1. The local model for the open book. The main object of the first model is a contact form on S2× S1, which was introduced in [Ts, §4.3]. The model consists of the datum (S2×
S1, ˇa, ds2, C ⊕ ˇK−1, ˇDr). It will be denoted by ( ˇS = S1× S1, ˇDr) for brevity.
Let (ρ, eiθ) ∈ [0, 2] × S1 be the (re-parametrized) spherical2 coordinate for the S2 factor, and let eit be the coordinate for the S1 factor. The orientation is determined by the 3-form
dρ ∧ dθ ∧ dt for ρ ∈ (0, 2).
5.1.1. The contact form and the adapted metric. Choose two smooth functions ˇf (ρ) and ˇg(ρ) of ρ ∈ [0, 2] such that
• when 0 ≤ ρ < 1 + 50δ, the functions ˇf (ρ) and ˇg(ρ) coincide with the functions f (ρ) and g(ρ) constructed in §1.3.2;
• when 2 − 50δ ≤ ρ ≤ 2, ˇf (ρ) = (2 − ρ)2 and ˇg(ρ) = −2 + (2 − ρ)2; • for any ρ ∈ (0, 2), the functions ˇf and ˇf0g − ˇˇ f ˇg0 are positive. It is not hard to see that there always exist such ˇf and ˇg.
With these two functions chosen, the 1-form ˇ
a = ˇf (ρ)dθ + ˇg(ρ)dt (5.1)
is a contact form on S2× S1. The metric
ds2 = ˇa2+ (dρ)2+1 4( ˇf
0(ρ)dθ + ˇg0(ρ)dt)2 (5.2)
is adapted to the contact form ˇa.
5.1.2. The Dirac operator. As explained in §2.1.2 and §2.1.3, the contact form ˇa and the adapted metric ds2 determine a canonical spinor bundle C ⊕ ˇK−1 and a canonical spin-c Dirac operator
ˇ
Do on it. The bundle ˇK−1 is also a trivial bundle. It can be globally trivialized by the
unit-normed section e−iθ √ 2 ∂ρ− 2i ˇ f0g − ˇˇ f ˇg0(ˇg∂θ− ˇf ∂t) . (5.3)
Together with the depicted section 1C, the sections of C ⊕ ˇK−1 are identified with C2 valued functions on S2 × S1. With respect to this identification, let S
k,m be the space of smooth
sections whose C component have frequency k in eiθ and m in eit, and whose ˇK−1 component have frequency k + 1 in eiθ and m in eit. Namely,
Sk,m=ψ = (α, β) ∈ C∞(S2× S1; C ⊕ ˇK−1)
∂θψ = ikψ + i(0, β) and ∂tψ = imψ .
2To be more precise, choose a positive smooth function χ(ρ) of ρ ∈ [0, 2] such that χ(ρ) = 1 when ρ ≤ 1 10 or ρ ≥ 1910, andR2
0 χ(ρ)dρ = π
2. The parametrization of the standard sphere is given by x = sin( Rρ 0 χ(s)ds) cos θ, y = sin(Rρ 0 χ(s)ds) sin θ, z = cos( Rρ 0 χ(s)ds) cos θ. 17
Remark 5.1. When 0 ≤ ρ < 1 + 50δ, the contact form and the metric are the same as that near the bindings of the open book, S1× B ⊂ Y . It follows that 1Cand (5.3) also trivialize the canonical spinor bundle C ⊕ K−1 over S1× B ⊂ Y . Together with an AE-parallel, unit-normed section of E, the sections of E ⊕ EK−1 over S1× B ⊂ Y are identified with C2valued functions.
This provides an identification of sections of (E ⊕EK−1)|S1×B with sections of (C⊕ ˇK−1)|S1×B.
Moreover, the Dirac operator Dr is identified with ˇDr= ˇDo−ir2 cl(ˇa).
For any r > 0, consider the Dirac operator ˇDr = ˇDo−ir2 cl(ˇa). The following notion is useful
to describe the eigenvalues of ˇDr.
Definition 5.2. Observe that the function ˇg/ ˇf is monotone decreasing in ρ ∈ [0, 2]. For each positive integer k and integer m, there is a unique ˇρk,m∈ (0, 2) such that kˇg( ˇρk,m) = m ˇf ( ˇρk,m).
Let ˇγk,m be ˇ γk,m= 2m ˇf0( ˇρk,m) − 2kˇg0( ˇρk,m) ( ˇf0g − ˇˇ f ˇg0)( ˇρ k,m) = 2k ˇ f ( ˇρk,m) = 2m ˇ g( ˇρk,m) . (5.4)
The last equality only makes sense at where g( ˇρk,m) 6= 0. If k = 0 and m > 0, set ˇρk,m to be 0,
and set ˇγk,m to be m. If k = 0 and m < 0, set ˇρk,m to be 2, and set ˇγk,m to be −m.
The spectral properties of ˇDr are summarized in the following proposition.
Proposition 5.3. There exists a constant c1 determined by the contact form ˇa and the metric
ds2 with following significance.
(i) ˇDr(Sk,m) belongs to Sk,m for any k and m, and the eigenbasis of ˇDr can be chosen so
that each eigensection belongs to some Sk,m.
(ii) For any r ≥ c1, ˇDr has at most one eigenvalue λ within (−(13r)
1 2, (1
3r)
1
2) on each Sk,m.
If there does exist such an eigenvalue, then k > − min{1, |m|} and λ − r − ˇγk,m 2 ≤ c1 .
Moreover, the corresponding eigensection can be expressed as ˇϕk,m= ˇϕ (0) k,m+ ˇϕ
(ε) k,m with
the following properties.
(a) ˇϕk,m, ˇϕ(0)k,m and ˇϕ(ε)k,m are smooth sections. If 20δ < ˇρk,m < 2 − 20δ, the support
of ˇϕ(0)k,m is contained in {|ρ − ˇρk,m| ≤ 2δ}. If ˇρk,m ≤ 20δ, the support of ˇϕ(0)k,m is
contained in {ρ ≤ 40δ}. If ˇρk,m ≥ 2 − 20δ, the support of ˇϕ (0)
k,m is contained in
{ρ ≥ 2 − 40δ}.
(b) The L2 integrals satisfy Z ˇ S | ˇϕk,m|2 = 1 , Z ˇ S h ˇϕk,m, ˇϕ(ε)k,mi = 0 , Z ˇ S | ˇϕ(ε)k,m|2 ≤ c1r−7
where the integrals are against the volume form 12ˇa ∧ dˇa.
(c) ˇϕ(0)k,m is an approximate eigensection in the sense that Z
ˇ S
| ˇDrϕˇ(0)k,m− λ ˇϕ(0)k,m|2≤ c1r−6 .
(iii) On the other hand, for any positive integer k and integer m with |r − ˇγk,m|2≤ 13r, ˇDr
on Sk,m does admit an eigenvalue λ satisfying
λ − r − ˇγk,m 2 ≤ c1 .
This proposition was proved in [Ts, §5]. We will give the precise reference of each assertion in §A.1.
5.2. The local model for the mapping torus. The main object of the second model is a sta-ble Hamiltonian structure on S2× S1. The model consists of the datum (S2× S1, ˆa, ˆω, ds2, ˆL
r⊕
ˆ
LrKˆ−1, ˆDr). It will be denoted by ( ˆS = S2× S1, ˆDr) for brevity.
Let (ρ, eit) ∈ [0, 2] × S1 be the (re-parametrized) spherical coordinate for the S2 factor. Let eiθ be the coordinate for the S1 factor of S2× S1. Note that the roles of θ and t are switched
from the convention in §5.1.
5.2.1. Geometric structures. Choose a smooth function ˆg(ρ) such that
• when ρ < 1 + 50δ, ˆg(ρ) coincides with the function ˜g(ρ) defined in §3.2; • when ρ ≥ 2 − 50δ , ˆg(ρ) is equal to 0.
For any σ ∈ (109,1110), choose a smooth function ˆhσ(ρ) such that
• when ρ < 1 + 50δ, ˆhσ(ρ) coincides with the function ˜hσ(ρ) defined in §3.3;
• when ρ ≥ 2 − 50δ, ˆhσ(ρ) = 2σ − 12(ρ − 2)2;
• its derivative in ρ is positive ˆh0σ(ρ) > 0, at any ρ ∈ (0, 2).
With these two functions chosen, the Riemannian metric on S2× S1 is taken to be
ds2 = ˆa2+ (dρ)2+ (ˆh0σ(ρ)dt)2 (5.5) where
ˆ
a = V dθ + ˆg(ρ)dt . The orientation on S2 is determined by
ˆ
ω = ˆh0σ(ρ)dt ∧ dρ , and the orientation on S2× S1 is determined by ˆω ∧ dθ = ˆh0
σ(ρ)dρ ∧ dθ ∧ dt.
The pair (ˆa, 2ˆω) constitutes a stable Hamiltonian structure. The metric ds2 is adapted to it. According to §2.1.2 and §2.1.3, they determine a canonical spinor bundle C ⊕ ˆK−1 with a canonical connection ∇o.
The symmetric 2-tensor (dρ)2 + (ˆh0
σ(ρ)dt)2 defines a metric on S2. The metric with the
symplectic form ˆω determines a complex structure on S2. Let KS−12 be the anti-canonical
bundle. It is not hard to see that ˆK−1 is isomorphic to the pull-back of KS−12 by the projection
map.
5.2.2. The degree 2r bundle. Suppose that r > 20. Set σ to be [r]r as before. In order to study the Dirac operator introduced in §3.6, consider the Hermitian line bundle ˆLr over S2 defined
as follows. Take the trivial bundles over {0 ≤ ρ < 1 + 50δ} and {1 < ρ ≤ 2}. Let 1−and 1+ be
the depicted unitary sections, respectively. On the overlap region {1 < ρ < 1 + 50δ}, identify the bundles by the transition rule
e2i[r]t· 1+= 1− .
Define a unitary connection ˆAr on ˆLr as the following:
• on {1 ≤ ρ < 1 + 50δ}, the connection ˆAr is d + ir(ˆhσ(ρ) +12ˆg(ρ))dt with respect to the
trivialization 1−;
• on {1 < ρ ≤ 2}, the connection ˆAr is d + ir(ˆhσ(ρ) + 12ˆg(ρ) − 2σ)dt with respect to the
trivialization 1+; note that the term −2σ guarantees the smoothness of the connection
near ρ = 2.
On the overlap region {1 < ρ < 1 + 50δ}, the first expression is d + i[r]dt, and the second expression is d−i[r]dt. They obey the transition rule of ˆLr, and hence ˆAris a smooth connection
on ˆLr. It is easy to see that the first Chern number of ˆLris 2[r], either from the transition rule
or the curvature computation.
5.2.3. The Dirac operator on ˆLr⊕ ˆLrKˆ−1. Consider the connection and the Dirac operator
ˆ ∇r= ∇o⊗ ˆAr− ir 2ˆa ˆ Dr= cl ◦(∇o⊗ ˆAr− ir 2ˆa) on ˆLr⊕ ˆLrKˆ−1. Here ˆLr is pulled back as a bundle over S2× S1.
To study the Dirac spectrum, consider the S1 action on the S1 factor of S2× S1. Namely,
eiθ0· (ρ, eit, eiθ) 7→ (ρ, eit, ei(θ0+θ)) .
This action preserves the stable Hamiltonian structure and the metric. It does not change the projection map onto the S2 factor.
With the help of the S1action, the Dirac equation is reduced to a Cauchy–Riemann equation on S2. Suppose that ˆ ϕ = √1 2πe ikθ( ˆα k, ˆβk) (5.6) 20
is an eigensection of ˆDr with eigenvalue λ, where ˆαkis a section of ˆLr→ S2 and ˆβkis a section
of ˆLrKS−12 → S2. To derive the equation for ˆαk and ˆβk, trivialize the bundles ˆLr and ˆLrKˆ
−1
over {0 ≤ ρ < 2} by the sections
1− and 1−⊗√1 2e it ∂ ρ+ i ˆ h0 σ (∂t− ˆ g V∂θ) , (5.7)
respectively. The latter expression can be identified with √1 2e
it(dρ + iˆh0
σdt), and defines a local
section of KS−12. With respect to this trivialization, the Dirac operator reads
ˆ Dr= i V∂θ −e it ∂ ρ+hˆi0 σ (∂t−Vˆg∂θ) e−it ∂ρ−ˆhi0 σ (∂t− Vˆg∂θ) −i V∂θ + r 1 2 eit ˆ h0 σ (ˆhσ+12g)ˆ e−it ˆ h0 σ (ˆhσ+12ˆg) −12 + 1 2ˆh0σ " 0 2eit(1 − 2ˆh00σ) 0 ˆg0 # . (5.8)
It follows that the Dirac equation ˆDrϕ = λ ˆˆ ϕ reduces to the following equations on S2:
(r 2 − k V) ˆαk+ √ 2 ¯∂r,k∗ βˆk= λ ˆαk , √ 2 ¯∂r,kαˆk− ( r 2 − k V − ˆ g0 2ˆh0 σ ) ˆβk = λ ˆβk . (5.9)
Here, ¯∂r,k : ˆLr→ ˆLrKS−12 is the usual Cauchy–Riemann operator associated with
ˆ
∇r,k= ˆAr− i
k Vˆg dt .
In other words, perturb the connection ˆArby the globally defined 1-form −iVkg dt. The operatorˆ
¯
∂r,k∗ is the L2-adjoint operator of ¯∂r,k.
With this reduction, the eigenvalues of ˆDr can be found by the Riemann–Roch formula and
the vanishing argument.
Proposition 5.4. There exists a constant c2 determined by the stable Hamiltonian structure
(ˆa, ˆω) and the metric ds2 such that the following holds.
(i) For any r ≥ c2, the spectrum of ˆDr on ˆLr⊕ ˆLrKˆ−1 that lies within [−(13r)
1 2, (1 3r) 1 2] consists of r 2− k V k ∈ Z , and | r 2 − k V| ≤ ( 1 3r) 1 2 .
(ii) For any r ≥ c2 and any integer k satisfying |r2 − Vk| ≤ (13r)
1
2, the corresponding
eigenspace has dimension 2[r] + 1, and is isomorphic to ker ¯∂r,k via the identification
ˆ αk∈ ker ¯∂r,k7→ 1 √ 2πe ikθ( ˆα k, 0) . 21
Proof. Suppose that ˆψ = 2π1 eikθ( ˆα
k, ˆβk) is an eigensection of ˆDr whose eigenvalue λ lies within
[−(13r)12, (1
3r)
1
2]. Integrating the Bochner–Weitzenb¨ock formula for βk gives
2 Z S2 | ¯∂r,k∗ βˆk|2= Z S2 | ˆ∇r,kβˆk|2+ Z S2 ( iFAˆr,k ˆ ω + κ)| ˆβk| 2
where κ is the Gaussian curvature. The left hand side is equal to 2R
S2h ¯∂r,k∂¯r,k∗ βˆk, ˆβki. Using
(5.9) to replace ¯∂r,k∂¯r,k∗ βˆk, the equation becomes
0 = Z S2 | ˆ∇r,kβˆk|2+ Z S2 (r 2 − k V) 2+ (r − λ2+ ˆg0 ˆ h0 σ λ) + κ| ˆβk|2 .
Since |λ| ≤ (13r)12, there exists a constant c3 such that ˆβk vanishes for any r ≥ c3. It follows
from ˆβk ≡ 0 and (5.9) that
|r 2 − k V| ≤ ( 1 3r) 1 2 , ¯ ∂r,kαˆk= 0 .
According to the Riemann–Roch formula, it suffices to show that the kernel of ¯∂r,k∗ is trivial to conclude that the dimension of ker ¯∂r,k is 2[r] + 1. The vanishing of ker ¯∂r,k∗ follows form the
same Bochner–Weitzenb¨ock formula and the condition |r2−Vk| ≤ (13r)12. This finishes the proof
of the lemma.
We need the following notion to describe the eigensections of ˆDr.
Definition 5.5. There exists a constant c4 such that for any r ≥ c4 and |r2− Vk| ≤ (13r)
1 2, rˆh0σ + (r 2 − k V)ˆg 0 > 0
for any ρ ∈ (0, 2). For any r ≥ c4 and any integer n ∈ (−2[r], 0), let ˆρk,n∈ (0, 2) be the unique
solution of r(ˆhσ( ˆρk,n) + 1 2g( ˆˆ ρk,n)) − k Vˆg( ˆρk,n) + n = 0 . For n = 0, set ˆρk,n to be 0. For n = −2[r], set ˆρk,n to be 2.
Proposition 5.6. There exists a constant c5 determined by the stable Hamiltonian structure
(ˆa, ˆω) and the metric ds2 with the following significance. For any r ≥ c5 and any integer k with
|r2−Vk| ≤ (13r)12, ker ¯∂r,k has an orthonormal basis { ˆαk,n= ˆα(0)
k,n+ ˆα (ε)
k,n}−2[r]≤n≤0 satisfying the
following properties.
(i) With respect to the trivialization 1−, ∂tαˆk,n= in ˆαk,n, so do ˆα(0)k,n and ˆα(ε)k,n.
(ii) If 20δ < ˆρk,n < 2 − 20δ, the support of ˆα (0)
k,n is contained in {|ρ − ˆρk,n| ≤ 2δ}. If
ˆ
ρk,n ≤ 20δ, the support of ˆα(0)k,n is contained in {ρ ≤ 40δ}. If ˇρk,n ≥ 2 − 20δ, the support
of ˆα(0)k,m is contained in {ρ ≥ 2 − 40δ}.
(iii) ˆα(0)k,n almost solves ¯∂r,k in the sense that Z S2 | ¯∂r,k( ˆα (0) k,n)| 2 ≤ c 5e −r c5 < c5r−6 .
(iv) The remainder term ˆα(ε)k,n obeys Z S2 | ˆα(ε)k,n|2≤ c5e− r c5 < c5r−7 and Z S2 h ˆαk,n, ˆα (ε) k,ni = 0 .
(v) If |n −Vk + [r]| < Vk(48δ), the support of ˆαk,n(0) is contained in {|ρ − 1| < 50δ}, and thus can be regarded as a smooth function on S2 (with respect to 1−). It is proportional to
the approximate eigensection ˇϕ(0)k,n+[r] given by Proposition 5.3. More precisely, ˇ
ϕ(0)k,n+[r]= √1 2πe
i(kθ+[r]t)(c
k,nαˆ(0)k,n, 0)
for some scalar ck,n with |ck,n− 1| ≤ c5e −r
c5. Here ˇϕ(0)
k,n+[r]is identified with a C
2 valued
function by (5.3).
The condition of (iv) |n − Vk + [r]| < (48kδ)/V transforms to |m − Vk| < (48kδ)/V by m = n + [r]. They are equivalent to that | ˆρk,n− 1| < 48δ and |ˇρk,m− 1| < 48δ, respectively.
The proof of the proposition is basically by solving ordinary differential equations with integral factor. The proof appears in §A.2.
6. Gluing Eigensections
The main purpose of this section is to prove that the “small eigenvalues” of (Y, Dr) is almost
the same as that of ( ˜Y , ˜Dr). The strategy is to divide [−12r
1 2,1
2r
1
2] into sub-intervals about of
unit length, and to show that the total number of eigenvalues of Dr and ˜Dr are same within
each sub-interval.
Lemma 6.1. There exist constants c1 and c2 determined by the contact form a, the metric ds2,
and the connection AE with the following property. For any r ≥ c1, there exists a sequence of
numbers {νj : −[12r 1 2] < j < [1 2r 1 2]} such that
(i) for any j, |νj− j| ≤ 101, and thus −12r
1
2 < · · · < νj < νj+1< · · · < 1
2r
1 2;
(ii) for any j, there is no spectrum within [νj− c2r−1, νj+ c2r−1] for ( ˜Y , ˜Dr), ( ˇS, ˇDr) and
( ˆS, ˆDr);
(iii) for any j, there is no spectrum within [νj− c2r−
3
2, νj+ c2r− 3
2] for (Y, Dr).
Proof. For any j ∈ {−[12r12] + 1, −[1
2r
1
2] + 2, · · · , [1
2r
1
2] − 1}, consider the interval
Uj = [j −
1 15, j +
1 15] .
• According to Corollary 4.3 and Proposition 5.4, there exists a constant c3 such that
the total number of eigenvalues of ( ˜Y , ˜Dr) and ( ˆS, ˆDr) within Uj is less than c3r. 23
• Since ( ˇS, ˇDr) is constructed from a contact form with an adapted metric (5.2), [Ts1,
Corollary 3.4] implies that the total number of eigenvalues of ( ˇS, ˇDr) within Uj is less
than c4r for some constant c4.
• Due to [Ts1, Corollary 3.3(i)], there exists a constant c5 such that the total number
of eigenvalues of (Y, Dr) within [−12r
1 2,1 2r 1 2] is less than c5r 3
2. It follows that the total
number of eigenvalues within Uj is bounded by c5r
3
2. The metric is only conformally
adapted, and [Ts1, Corollary 3.4] cannot apply to (Y, Dr).
Let c6 = max{c3, c4}. Divide Uj into sub-intervals of length between (60c6r)−1 and (30c6r)−1.
There are at least 4c6r sub-intervals. Let {Uj,k}1≤k≤K be the sub-intervals which do not contain
any eigenvalues of ( ˜Y , ˜Dr), ( ˇS, ˇDr) and ( ˆS, ˆDr). It follows from the pigeonhole principle that
K ≥ 3c6r.
Let 12Uj,k be the sub-interval of Uj,k with the same midpoint and of half length. Further
divide each 12Uj,k into sub-intervals of length between (60c5)−1r−
3
2 and (50c5)−1r− 3
2. The total
number of sub-intervals is at least (3c6r) × 1 120c6r × (50c5r 3 2) > c5r 3 2 .
Hence, there exists some sub-interval which does contain any eigenvalue of (Y, Dr). Choose any
one of such a sub-interval, and set νj to be its midpoint. It follows from the construction that
νj satisfies the assertion of the lemma with c2 = 5001 (max{c5, c6})−1.
Item (ii) of the lemma guarantees spectral gaps of 2c2r−1. This allows us to invert the Dirac
operator. It plays a key role for gluing eigensections.
Definition 6.2. Let c1 be the same constant of Lemma 6.1. For any r ≥ c1, let {νj : −[12r
1 2] <
j < [12r12]} be the sequence given by this same lemma. With this sequence, introduce the
following sets of eigenvalues for any r ≥ c and −[12r12] < j < [1
2r 1 2] − 1: Ij =λ ∈ spec(Dr) νj < λ < νj+1 , ˜ Ij =λ ∈ spec( ˜Dr) νj < λ < νj+1 .
The definition abuses3 the notation: the multiplicity of eigenvalues is counted. Also introduce the following index sets:
ˇ Ij =(k, m) ∈ Z≥0× Z spec( ˇDr|Sk,m) ∩ (νj, νj+1) 6= ∅, and k < mV , ˆ Ij =(k, n) ∈ Z × Z νj < r 2 − k V < νj+1, and k V − [r] < n ≤ 0 .
The condition that k < mV is equivalent to ˇρk,m < 1. The condition that k < (n + [r])V is
equivalent to ˆρk,n< 1.
3The set-theoretically correct definition is(λ, k) ∈ R × N
λ ∈ spec(Dr), νj< λ < νj+1, k ≤ dim ker(Dr− λI) .
As explained in the proof of Lemma 6.1, there exists a constant c7 such that
#˜Ij+ #ˇIj+ #ˆIj ≤ c7r and #Ij ≤ c7r
3
2 (6.1)
for any r ≥ c7 and −[12r
1
2] < j < [1
2r
1 2] − 1.
Theorem 6.3. There exists a constant c8 > c1 determined by the contact form a, the metric
ds2 and the connection A
E with the following property. For any r ≥ c8, let {νj : −[12r
1
2] < j <
[12r12]} be the sequence given by Lemma 6.1. Then,
#Ij − #ˇIj = #˜Ij − #ˆIj
for any r ≥ c8 and −[12r
1
2] < j < [1
2r
1
2] − 1. As a consequence, #Ij ≤ 2c7r.
The proof of this theorem occupies the rest of this section. It is organized as follows. §6.1 is devoted to construct approximate eigensections of (Y, Dr) corresponding to #Ij− #ˇIj. In §6.2
we prove that these approximate eigensections have small L2-norm near the bindings, and hence are approximate eigensections of ( ˜Y , ˜Dr). §6.3 contains a linear algebra lemma which gives a
precise estimate on the difference between genuine eigenvalues and approximate eigenvalues. In §6.4 we combine the above results to prove that (#Ij− #ˇIj) + #ˆIj ≤ #˜Ij. Another direction, (#˜Ij − #ˆIj) + #ˇIj ≤ #Ij, can be proved by the same argument.
6.1. Approximate eigensections for #Ij− #ˇIj. The first step is to construct approximate
eigensections of Dr corresponding to #Ij− #ˇIj.
Lemma 6.4. There exist constants c9 > c1 and c10 determined by the contact form a, the
metric ds2 and the connection AE with the following significance. For any r ≥ c9, let {νj :
−[1 2r 1 2] < j < [1 2r 1
2]} be the sequence given by Lemma 6.1. Let
Vj = span{ψ | Drψ = λψ for some λ ∈ Ij} and
ˇ
Vj = span{ ˇϕ(0)k,m| (k, m) ∈ ˇIj}
where ˇϕ(0)k,m is the approximate eigensection given by Proposition 5.3. Since the elements of ˇVj
only support on {ρ ≤ 1}, they can be regarded as smooth sections of E ⊕EK−1→ Y by (5.3) and Remark 5.1. Then, Vj and ˇVj satisfy the following properties for any −[12r
1
2] < j < [1
2r
1 2] − 1.
(i) Let prj be the L2-orthogonal projection onto Vj. The dimension of prj( ˇVj) is the same
as the dimension of ˇVj, i.e. prj : ˇVj → Vj is injective.
(ii) Let ˇVj\ be the L2-orthogonal complement of prj( ˇVj) in Vj. The space ˇVj\ admits a
L2-orthonormal basis {ψ
j,`}1≤`≤#Ij−#ˇIj such that
Drψj,`= µj,`ψj,`+ ψj,`(ε) 25
for some scalar µj,`∈ R and ψj,`(ε)∈ prj( ˇVj). They obey the estimate: νj+ c2r− 3 2 < µj,`< νj+1− c2r− 3 2 , and Z Y |ψj,`(ε)|2≤ c10r−6 for any 1 ≤ ` ≤ #Ij− #ˇIj.
(iii) Moreover, let pr−1j : prj( ˇVj) → ˇVj be the inverse map of that in item (i). Then
Z
Y
| pr−1j (ψ(ε)j,`) − ψ(ε)j,`|2 ≤ c10r−8 .
Proof. (Assertion (i): pr( ˇVj)) For any (k, m) ∈ ˇIj, let λk,m be the corresponding eigenvalue of
ˇ
Dr on ˇϕk,m. According to Lemma 6.1(ii), νj + c2r−1 < λk,m < νj+1− c2r−1. Due to Remark
3.1 and (5.3), Drϕˇ(0)k,m on Y is the same as ˇDrϕˇ(0)k,m on ˇS. By Proposition 5.3(ii.b) and (ii.c),
there exists a constant c11 such that
Z Y |Drϕˇ(0)k,m− λk,mϕˇ(0)k,m|2 ≤ c11r−6 , and 1 − Z Y | ˇϕ(0)k,m|2 ≤ c11r−7 . (6.2)
In terms of the spectral decomposition induced by Dr, write ˇϕ(0)k,m as
ˇ
ϕ(0)k,m= prj( ˇϕ(0)k,m) + ˇϕ+k,m+ ˇϕ−k,m (6.3) where prj( ˇϕ(0)k,m) is the L2-orthogonal projection of ˇϕ(0)k,m onto Vj, ˇϕ+k,m is the L2-orthogonal
projection of ˇϕ(0)k,m onto the space spanned by eigensections whose eigenvalue is greater than νj+1, and ˇϕ−k,mis the L2-orthogonal projection of ˇϕ(0)k,monto the space spanned by eigensections
whose eigenvalue is less than νj. It follows from Lemma 6.1(iii) that
c2r− 3 2 Z Y | ˇϕ+k,m|2 ≤ Z Y h(Dr− λk,m) ˇϕ(0)k,m, ˇϕ+k,mi ≤ 1 2c2 r32 Z Y |(Dr− λk,m) ˇϕ (0) k,m| 2+ 1 2c2r −32 Z Y | ˇϕ+k,m|2 . Then appeal to (6.2) to conclude that
Z Y | ˇϕ+k,m|2≤ c11 (c2)2 r−3 . (6.4) Similarly,R Y | ˇϕ −
k,m|2 has the same upper bound.
Proposition 5.3(ii) implies that { ˇϕ(0)k,m| (k, m) ∈ ˇIj} are mutually orthogonal to each other with respect to the L2-inner product. It together with (6.3) and (6.4) finds a constant c12 so
that 1 − Z Y | pr( ˇϕ(0)k,m)|2 ≤ c12r−3 , Z Y hpr( ˇϕ(0)k,m), pr( ˇϕ(0)k0,m0)i = X +,− Z Y h ˇϕ±k,m, ˇϕ±k0,m0i ≤ c12r−3 (6.5) 26
for any (k, m), (k0, m0) ∈ ˇIj and (k, m) 6= (k0, m0). On the other hand, the dimension of
ˇ
Vj is no greater than c7r (6.1). Based on these facts, a linear algebra argument shows that
{prj( ˇϕ(0)k,m) | (k, m) ∈ ˇIj} still forms a linearly independent set. This proves Assertion (i) of the
lemma.
(Assertion (ii): ˇVj\) It follows from the construction that Dr(Vj) ⊂ Vj. Let Dr,j\ : ˇV \ j → ˇV
\ j
be the restriction of Dr on ˇVj\ composing with the L2-orthogonal projection onto ˇV \ j. The
L2 self-adjointness of Dr implies the L2 self-adjointness of Dr,j\ . The construction of Dr,j\ is
done within Vj, and it is simply a finite-dimensional linear algebra. Hence, there exists a L2
-orthonormal eigenbasis {ψj,`}` of D\r,j on ˇVj\. Denote the corresponding eigenvalue by µj,`. By
(6.3), ψj,`∈ ˇVj\⊂ Vj is not only L2-orthogonal to pr( ˇϕ (0)
k,m), but also L2-orthogonal to ˇϕ (0) k,m.
Let ψj,`(ε) = Drψj,`− µj,`ψj,`. It follows from the construction that any ψ (ε)
j,` belongs to the
L2-orthogonal complement of ˇVj\ in Vj, which is prj( ˇVj). To show that νj + c2r−
3
2 < µj,` <
νj+1− c2r−
3
2, express µj,` as R
YhDrψj,`, ψj,`i. An elementary linear algebra argument shows
that inf Ij ≤ sup ψ∈Vj\{0} R YhDrψ, ψi R Y |ψ|2 ≤ sup Ij ,
and the desired bound on µj,` follows from Lemma 6.1(ii).
It remains to estimate the L2-norm of ψ(ε)j,`. Suppose that P
ˇ
Ijck,mprj( ˇϕ
(0)
k,m) is a smooth
section in prj( ˇVj) with unit L2-norm. Due to (6.2), (6.3), (6.4), (6.1) and (6.5),
1 =X ˇ Ij |ck,m|2 Z Y | prj( ˇϕ(0)k,m)|2− X (k,m)∈ ˇIj X (k0,m0) 6=(k,m) ck,m¯ck0,m0 Z Y hprj( ˇϕ(0)k,m), prj( ˇϕ(0)k0,m0)i ≥X ˇ Ij |ck,m|2 Z Y | prj( ˇϕ(0)k,m)|2−1 2 X (k,m)∈ ˇIj X (k0,m0) 6=(k,m) (|ck,m|2+ |ck0,m0|2) Z Y hprj( ˇϕ(0)k,m), prj( ˇϕ(0)k0,m0)i ≥X ˇ Ij |ck,m|2 Z Y | prj( ˇϕ(0)k,m)|2− X (k,m)∈ ˇIj |ck,m|2 X (k0,m0) 6=(k,m) Z Y hprj( ˇϕ(0)k,m), prj( ˇϕ(0)k0,m0)i ≥ (1 − c13r−2) X ˇ Ij |ck,m|2
for some constant c13> 0. It follows that
X
ˇ Ij
|ck,m|2≤ 1 + 2c13r−2 . (6.6)
Consider the following L2-inner product: Z Y hψj,`(ε),X ˇ Ij ck,mprj(ϕ(0)k,m)i ≤ X ˇ Ij |ck,m| Z Y hDrψj,`− µj,`ψj,`, prj(ϕ (0) k,m)i =X ˇ Ij |ck,m| Z Y hDrψj,`, ϕ(0)k,mi = X ˇ Ij |ck,m| Z Y hψj,`, Drϕ(0)k,mi =X ˇ Ij |ck,m| Z Y hψj,`, (Dr− λk,m)ϕ(0)k,mi . By (6.2) and (6.6), Z Y hψ(ε)j,`,X ˇ Ij ck,mprj(ϕ(0)k,m)i 2 ≤ X ˇ Ij |ck,m|2c11r−6 X ˇ Ij Z ρ≤1 | prk,m(ψj,`)|2 ≤ c11r−6(1 + 2c13r−2) Z Y |ψj,`|2 .
Here, ψj,`|ρ≤1 is regarded as a local section of C ⊕ ˇK−1 → ˇS by Remark 3.1, and prk,m is the
projection onto Sk,m defined in §5.1.2. Since the estimate holds for any unit-normed section in
prj( ˇVj), we conclude thatR Y |ψ (ε) j,`| 2 ≤ 2c 11r−6.
(Assertion (iii): pr−1(ψj,`(ε))) Proposition 5.3(ii.b) and (6.6) imply that Z Y | pr−1j (ζ)|2 ≤ (1 + 2c13r−2) Z Y |ζ|2 for any ζ ∈ prj( ˇVj). Since
R
Y |ζ| 2=R
Yhpr
−1(ζ), ζi for any ζ ∈ pr j( ˇVj), Z Y | pr−1j (ψ(ε)j,`) − ψj,`(ε)|2= Z Y | pr−1j (ψj,`(ε))|2− |ψj,`(ε)|2 ≤ 2c 13r−2 Z Y |ψj,`(ε)|2≤ 4c 11c13r−8 .
This completes the proof of the lemma.
6.2. Almost vanishing near the bindings. The main purpose of this subsection is to prove that the approximate eigensections constructed by Lemma 6.4 have small L2-integral near the bindings.
Lemma 6.5. There exists a constant c15determined by the contact form a, the metric ds2 and
the connection AE such that
Z ρ≤ρ0−5δ |ψj,`|2 ≤ c15(r−6+ r−1 Z ρ≤ρ0 |ψj,`|2)
for any ψj,` produced by Lemma 6.4(ii) and ρ0 ∈ [12, 1]. (ρ is the coordinate near the bindings
as in §1.3.1.)
