Sub-leading asymptotics of ECH capacities

In previous work, the first author and collaborators showed that the leading asymptotics of the embedded contact homology (ECH) spectrum recovers the contact volume. Our main theorem here is a new bound on the sub-leading asymptotics.


Introduction
1.1. The main theorem. Let Y be a closed, oriented three-manifold. A contact form on Y is a one-form λ satisfying λ^dλ ą 0.
A contact form determines the Reeb vector field, R, defined by λpRq " 1, dλpR,¨q " 0, and the contact structure ξ :" Kerpλq. Closed orbits of R are called Reeb orbits.
If pY, λq is a closed three-manifold equipped with a nondegenerate contact form and Γ P H 1 pY q, then the embedded contact homology ECHpY, λ, Γq is defined. This is the homology of a chain complex freely generated over Z 2 by certain sets of Reeb orbits in the homology class Γ, relative to a differential that counts certain J-holomorphic curves in RˆY. (ECH can also be defined over Z, but for the applications in this paper we will not need this.) It is known that the homology only depends on ξ, and so we sometimes denote it ECHpY, λ, ξq. Any Reeb orbit γ has a symplectic action Apγq " ż γ λ and this induces a filtration on ECHpY, λq; we can use this filtration to define a number c σ pY, λq for every nonzero class in ECH, called the spectral invariant associated to σ; the spectral invariants are C 0 continuous and so can be extended to degenerate contact forms as well.
2000 Mathematics Subject Classification. 53D35, 57R57, 57R58 . D. C-G. is partially supported by NSF grant 1711976. N. S. is partially supported by the DFG funded project CRC/TRR 191. 1 We will review the definition of ECH and of the spectral invariants in §2. 1.
When the class c 1 pξq`2P.D.pΓq P H 2 pY ; Zq is torsion, then ECHpY, ξ, Γq has a relative Z grading, which we can refine to a canonical absolute grading gr Q by rationals [18], and which we will review in §2. 3. It is known that for large gradings the group is eventually 2-periodic and non-vanishing: ECH˚pY, ξ, Γq " ECH˚`2pY, ξ, Γq ‰ 0,˚" 0.
The main theorem of [11] states that in this case, the asymptotics of the spectral invariants recover the contact volume volpY, λq " ż Y λ^dλ.
The formula (1.1) has had various implications for dynamics. For example, it was a crucial ingredient in recent work [10] of the first author and collaborators showing that many Reeb vector fields on closed three-manifolds have either two or infinitely many distinct closed orbits, and it was used in [9] to show that every Reeb vector field on a closed three-manifold has at least two distinct closed orbits. It has also been used to prove C 8 closing lemmas for Reeb flows on closed three-manifolds and Hamiltonian flows on closed surfaces [1,17].
By (1.1), we can write where dpσ j q is opgr Q pσ j q 1{2 q as gr Q pσ j q tends to positive infinity. It is then natural to ask: Question 2. What can we say about the asymptotics of dpσ j q as gr Q pσ j q tends to positive infinity?
Previously, W. Sun has shown that dpσ j q is Opgr Q pσ j qq 125{252 [27,Thm. 2.8]. Here we show: Theorem 3. Let pY, λq be a closed, connected oriented three-manifold with contact form λ, and let Γ P H 1 pY q be such that c 1 pξq`2P.D.pΓq is torsion. Let tσ j u be any sequence of nonzero classes in ECHpY, λ, Γq with definite gradings tending to positive infinity. Define dpσ j q by (1.2). Then dpσ j q is Opgr Q pσ j qq 2{5 as gr Q pσ j q Ñ`8.
We do not know whether or not the Opgr Q pσ j qq 2{5 asymptotics here are optimal -in other words, we do not know whether there is some contact form on a three-manifold realizing these asymptotics. We will show in §4.3 that there exist contact forms with Op1q asymptotics for the dpσ j q. In Remark 13, we clarify where the exponent 2 5 comes from in our proof, and why the methods in the current paper can not improve on it.
Another topic which we do not address here, except in a very specific example, see §4.3, but which is of potential future interest, is whether the asymptotics of the dpσ j q carry interesting geometric information. In this regard, a similar question in the context of the spectral flow of a one-parameter family of Dirac operators was recently answered in [22]. This is particularly relevant in the context of the argument we give here, as our argument also involves estimating spectral flow, see Remark 13.

1.2.
A dynamical zeta function and a Weyl law. We now mention two corollaries of Theorem 3.
As another corollary, one may obtain information on the corresponding dynamical zeta function. To this end, first note that the ECH zeta function (1.5) ζ ECH ps; Y, λ, Γq :" ÿ c‰0PΣ pY,λ,Γq c´s converges for Re psq ą 2 by (1.1) and defines a holomorphic function of s in this region whenever c 1 pξq`2P.D.pΓq is torsion, by (1.4). In view of for example [13,14], one can ask if ζ ECH has a meromorphic continuation to C, and, if so, whether it contains interesting geometric information. The Weyl law (1.4) then shows: Corollary 5. The zeta function (1.5) continues meromorphically to the region Re psq ą 5 3 . The only pole in this region is at s " 2 which is further simple with residue Res s"2 ζ ECH ps; Y, λ, Γq " In §4.3, we give an example of a contact form for which ζ ECH has a meromorphic extension to all of C with two poles at s " 1, 2. The meromorphy and location of the poles of (1.5) would be interesting to figure out in general.
1.3. Idea of the proof and comparison with previous works. The method of the proof uses previous work by C. Taubes relating embedded contact homology to Monopole Floer homology. By using Taubes's results, we can estimate spectral invariants associated to nonzero ECH classes by estimating the energy of certain solutions of the deformed three-dimensional Seiberg-Witten equations. This is also the basic idea behind the proofs of Theorem 1 and the result of Sun mentioned above, and it was inspired by a similar idea in Taubes's proof of the three-dimensional Weinstein conjecture [24].
The essential place where our proof differs from these arguments involves a particular estimate, namely a key "spectral flow" bound for families of Dirac operators that appears in all of these proofs. This estimate bounds the difference between the grading of a Seiberg-Witten solution, and the "Chern-Simons" functional, which we review in §2.2, and is important in all of the works mentioned above. We prove a stronger bound of this kind than any previous bound, see Proposition 6 and the discussion about the eta invariant below, and this is the key point which allows us to prove Opgr Q pσ j qq 2{5 asymptotics. Spectral flow bounds for families of Dirac operators were also considered in [20,21,26]. The main difference here is that in those works the bounds were proved on reducible solutions where the connections needed to define the relevant Dirac operators were explicitly given. Here we must consider irreducible solutions, and so we rely on a priori estimates.
We have chosen to phrase this spectral flow bound in terms of a bound on the eta invariants of a family of operators. By the Atiyah-Patodi-Singer index theorem, the bound we need on the spectral flow is equivalent to a bound on the eta invariant, and we make the relationship between these two quantities precise in the appendix.
The paper is organized as follows. In §2, we review what we need to know about embedded contact homology, Monopole Floer cohomology and Taubes's isomorphism. §3 reviews the eta invariant, reviews the necessary estimates on irreducible solutions to the Seiberg-Witten equations, and proves the key Proposition 6. We then give the proof of Theorem 3 in §4 -while our argument in this section is novel, one could instead argue here as in [27], but we give our own argument here since it might be of independent interest, see Remark 15. The end of the paper reviews the sub-leading asymptotics and the dynamical zeta function in the case of ellipsoids, and an appendix rephrases the grading in Seiberg-Witten in terms of the eta invariant rather than in terms of spectral flow.

Floer homologies
We begin by reviewing the facts that we will need about ECH and Monopole Floer homology.

Embedded contact homology.
We first summarize what we will need to know about ECH. For more details and for definitions of the terms that we have not defined here, see [16].
Let pY, λq be a closed oriented three-manifold with a nondegenerate contact form. Fix a homology class Γ P H 1 pY q. As stated in the introduction, the embedded contact homology ECHpY, λ, Γq is the homology of a chain complex ECCpY, λ, Γq. To elaborate, the chain complex ECC is freely generated over Z 2 by orbit sets α " tpα j , m j qu where the α j 's are distinct embedded Reeb orbits while each m j P N; we further have the constraints that ř m j α j " Γ P H 1 pY q and m j " 1 if α j is hyperbolic. To define the chain complex differential B, we consider the symplectization pR tˆY , d pe t λqq, and choose an almost complex structure J that is R-invariant, rotates the contact hyperplane ξ :" kerλ positively with respect to dλ, and satisfies JB t " R. The differential on ECC pY, λ, Γq is now defined via Here M 1 pα, βq denotes the moduli space of J-holomorphic curves C of ECH index I pCq " 1 in the symplectization, modulo translation in the R-direction, and modulo equivalence as currents, with the set of positive ends given by α and the set of negative ends given by β. If J is generic, then the differential squares to zero B 2 " 0 and defines the ECH group ECH pY, λ; Γq . We will not review the definition of the ECH index here, see [16] for more details, but the key point is that the condition IpCq " 1 forces C to be (mostly) embedded and rigid modulo translation. As stated in the introduction, the homology ECH pY, λ; Γq does not depend on the choice of generic J, and only depends on the associated contact structure ξ; we therefore denote it ECH pY, ξ; Γq. (In fact, the homology only depends on the spin c structure determined by ξ, but we will not need that.) This follows from a canonical isomorphism between ECH and Monopole Floer homology [25], which we will soon review. The ECH index I induces a relative Z{dZ grading on ECH pY, ξ; Γq , where d is the divisibility of c 1 pξq`2P.D. pΓq P H 2 pY ; Zq mod torsion. In particular, it is relatively Z-graded when this second homology class is torsion Recall now the action of a Reeb orbit from the introduction. This induces an action on orbit sets α " tpα j , m j qu by The differential decreases action, and so we can define ECC L pY, λ, Γq to be the homology of the sub-complex generated by orbit sets of action strictly less than L. The homology of this sub-complex ECH L pY, λ, Γq is again independent of J but now depends on λ; there is an inclusion induced map ECH L pY, λ, Γq Ñ ECH pY, ξ, Γq . Using this filtration, we can define the spectral invariant associated to a nonzero class σ in ECH c σ pY, λq :" inf L | σ P image pECH L pY, λ, Γq Ñ ECH pY, ξ, Γq˘u.
As stated in the introduction, the spectral invariants are known to be C 0 continuous in the contact form, and so extend to degenerate contact forms as well by taking a limit over nondegenerate forms, see [15].

2.2.
Monopole Floer homology. We now briefly review what we need to know about Monopole Floer homology, referring to [18] for additional details and definitions.
Recall that a spin c structure on an oriented Riemannian three-manifold Y is a pair pS, cq consisting of a rank 2 complex Hermitian vector bundle and a Clifford multiplication endomorphism c : T˚Y bC Ñ End pSq satisfying c pe 1 q 2 "´1 and c pe 1 q c pe 2 q c pe 3 q " 1 for any oriented orthonormal frame pe 1 , e 2 e 3 q of T y Y . Let su pSq denote the bundle of traceless, skew-adjoint endomorphisms of S with inner product 1 2 tr pA˚Bq. Clifford multiplication c maps T˚Y isometrically onto su pSq. Spin c structures exist on any three-manifold, and the set of spin c structures is an Hence prescribing a spin c connection on S is the same as prescribing a unitary connection on det pSq. We let A pY, sq denote the space of all spin c connections on S. Given a spin c connection A, we denote by ∇ A the associated covariant derivative. We then define the spin c Dirac operator D A : Given a spin c structure s " pS, cq on Y , monopole Floer homology assigns three groups denoted by z HM pY, sq ,HM pY, sq and HM pY, sq. These are defined via infinite dimensional Morse theory on the configuration space C pY, sq " A pY, sqˆC 8 pSq using the Chern-Simons-Dirac functional L, defined as using a fixed base spin-c connection A 0 (we pick one with A t 0 flat in the case of torsion spin-c structures) and a metric g T Y .
The gauge group G pY q " Map pY, S 1 q acts on the configuration space C pY, sq by u. pA, Ψq " pA´u´1du b I, uΨq . The gauge group action is free on the irreducible part C˚pY, sq " tpA, Ψq P C pY, sq |Ψ ‰ 0u Ă C pY, sq and not free along the reducibles. The blow up of the configuration space along the reducibles C σ pY, sq " tpA, s, Φq | }Φ} L 2 " 1, s ě 0u then has a free G pY q action u¨pA, s, Φq " pA´u´1du b I, s, uΦq .
To define the Monopole Floer homology groups one needs to perturb the Chern-Simons-Dirac functional (2.1). First given a one form µ P Ω 1 pY ; iRq, one defines the functional e µ pAq :" 1 2 ş Y µ^F A t whose gradient is calculated to be˚dµ. To achieve non-degeneracy and transversality of configurations one uses the perturbed Chern-Simons-Dirac functional where µ is a suitable finite linear combination of eigenvectors of˚d with non-zero eigenvalue. Next let where a h denotes the harmonic part of a P Ω 1 pY, iRq. The gradient may be calculated p∇fq σ A "´p∇f q pA t q h , 0, 0¯. The Monopole Floer homology groups are now defined using solutions pA, s, Φq P C σ pY, sq to the three-dimensional Seiberg-Witten equa- where Λ pA, s, Φq " xD A Φ, Φy L 2 and pΦΦ˚q 0 :" Φ b Φ˚´1 2 |Φ| 2 defines a traceless, Hermitian endormophism of S. We denote by C the set of solutions to the above equations.
We first subdivide the solutions as follows: Next, we consider the free Z 2 modules generated by the three sets above The chain groups for the three versions of Floer homology mentioned above are defined by These chain groupsČ,Ĉ,C can be endowed with differentialsB,B,B with square zero; we do not give the precise details here, but the idea is to count Fredholm index one solutions of the four-dimensional equations, see [18,Thm. 22.1.4] for the details. The homologies of these three complexes are by definition the three monopole Floer homology groupsH M pY, sq , z HM pY, sq , HM pY, sq . They are independent of the choice of metric and perturbations µ, f.
Each of the above Floer groups has a relative Z{dZ grading where d is the divisibility of c 1 pSq P H 2 pY ; Zq mod torsion. This is defined using the extended Hessian p H pA,Ψq : The relative grading between two irreducible generators a i " pA i , (mod d) for some path of configurations pA t , Ψ t q starting at pA 2 , s 2 Φ 2 q and ending at pA 1 , s 1 Φ 1 q, where sf denotes the spectral flow.
In the case when the spin-c structure is torsion, the monopole Floer groups are further equipped with an absolute Q-grading, refining this relative grading. As we will review in the appendix, this is given via where Φ A k above denotes the kth positive eigenvector of D A (see §A), and η Y and η D A denote the eta invariant of the corresponding operator, which we will review in §3.

ECH=HM.
We now state the isomorphism between the ECH and HM, proved in [25]. Given a contact manifold pY 3 , λq with dλcompatible almost complex structure J as before, we define a metric g T Y via g T Y | ξ " dλ p., J.q, |R| " 1 and R and ξ are orthogonal. This metric is adapted to the contact form in the sense˚dλ " 2λ, |λ| " 1. Decompose ξ bC " K lo omo on into the i,´i eigenspaces of J. The contact structure now determines the canonical spin-c structure s ξ via S ξ " C ' K´1 with Clifford multiplication c ξ given by Furthermore, there is a unique spin-c connection A c on S ξ with the property that D Ac " 1 0  " 0 and we call the induced connection A t c on K´1 " det`S ξ˘t he canonical connection. Tensor product with an auxiliary Hermitian line bundle The ECH/HM isomorphism is then (2.6)HM˚`´Y, s E˘" ECH˚pY, ξ; P.D.c 1 pEqq .
In the literature, this isomorphism is often stated with the left hand side given by the cohomology group z HM˚pY q instead; the point is that z HM˚pY q andHMp´Y q are canonically isomorphic, see [18,S 22.5,Prop. 28.3.4]. The isomorphism (2.6) allows us to define a Q-grading on ECH, by declaring that (2.6) preserves this Q-grading.
We now state the main ideas involved in the isomorphism (we restrict attention to the case when c 1`d et s E˘i s torsion, which is the case which is relevant here, and we sometimes state estimates that, while true, are stronger than those originally proved by Taubes). To this end, let σ PHM`´Y, s E˘. We use the perturbed Chern-Simons-Dirac functional (2.2) and its gradient flow (2.3) with µ " irλ, r P r0, 8q, in defining monopole Floer homology. (One also adds a small term η to µ to achieve transversality, see for example [11], but to simplify the notation we will for now suppress this term.) Giving a family of (isomorphic) monopole Floer groupsHM`´Y, s E˘, the class σ is hence representable by a formal sum of solutions to (2.3) corresponding to µ " irλ. Denote byČ r the µ " irλ version of the complexČ and note that its reducible generators are all of the form a " pA, 0, Φ k q where A " A 0´i rλ, A t 0 is flat and Φ k is the kth positive eigenvector of D A . An important estimate η pD A 0´i rλ q " O prq now gives that the grading of this generator gr Q ras " r 2 4π 2 ş λ^dλ`Oprq ą gr Q rσs by (2.5) for r " 0. Hence for r " 0 the class σ is represented by a formal sum of irreducible solutions to (2.3) with µ " irλ, and by a max-min argument, we may choose a family pA r , Ψ r q :" pA r , s r Φ r q satisfying gr Q rσs " gr Q rpA r , Ψ r qs .
Following a priori estimates on solutions to the Seiberg-Witten equations, one then proves another important estimate η´p H pAr,Ψrq¯" O`r 3{2˘u niformly in the class σ. This gives CS pA r q " O`r 3{2˘w hich in turn by a differential relation (see §4) leads to e λ pA r q " O p1q. The final step in the proof shows that for any sequence of solutions pA r , Ψ r q to Seiberg-Witten equations with e λ pA r q bounded, the E-component Ψr of the spinor Ψ r "

"
Ψr Ψŕ is a convergent analytic function of a complex variable s, as long as Repsq is sufficiently large; here, the sum is over the nonzero eigenvalues of D. Moreover, the function ηpD, sq has an analytic continuation to a meromorphic function on C of s, which we also denote by ηpD, sq, and which is holomorphic near 0. We now define ηpDq :" ηpD, 0q.
We should think of this as a formal signature of D, which we call the eta invariant of Atiyah-Patodi-Singer [2]. We will be primarily concerned with the case where D " D Ar , namely D is the spin-c Dirac operator for a connection A r solving (2.3). Another case of interest to us is where D is the odd signature operator on C 8 pY ; iT˚Y ' Rq sending pa, f q Þ Ñ p˚da´df, d˚aq , in which case we denote the corresponding η invariant by η Y . Now consider the Seiberg-Witten equations (2.3) corresponding to µ " irλ, for a torsion spin c structure as above, and note that an irreducible solution (after rescaling the spinor) corresponds to a solution pA r , Ψ r q to the Seiberg-Witten equations on C pY, sq given via

2)
A further small perturbation is needed to obtain transversality of solutions see [11, S 2.1]. We ignore these perturbation as they make no difference to the overall argument.
We can now state the primary result of this section: The purpose of the rest of the section will be to prove this.
3.1. Known estimates. We first collect some known estimates on solutions to the equations (3.2).

Lemma 7.
For some constants c q , q " 0, 1, 2, . . ., we have Proof. We first note that we have the estimates:ˇΨrˇˇď The first two of these estimates are proved in [24,Lem. 2.2]. The third and fourth are proved in [24,Lem. 2.3].
The lemma now follows by combining the above estimates with the equation (3.2).
In (4), we will also need: Lemma 8. One has the bound where the constant c 0 only depends on the metric contact manifold.
3.2. The η invariant of families of Dirac operators. In this section, we prove the key Proposition 6. The main point that we need is the following fact concerning the η invariant: Proposition 9. Let A r be a solution to (3.2). Then η pD Ar q is Opr 3{2 q as r Ñ 8.
Before giving the proof, we first explain our strategy.
The first point is that we have the following integral formula for the η invariant: (3.5) ηpD Ar q " 1 ? πt where the right hand side is a convergent integral. This is proved in [7, S 2], by Mellin transform it is equivalent to the fact that the eta function ηpD Ar , sq in (3.1) is holomorphic for Repsq ą´2.
We therefore have to estimate the integral in (3.5). To do this, we will need the following estimates: Once we have proved Lemma 10, Proposition 9 will follow from a short calculation, which we will give at the end of this section.
The proof of Lemma 10 will require two auxiliary lemmas, see Lemma 11 and Lemma 12 below, and some facts about the heat equation associated to a Dirac operator that we will now first recall. Let D be a Dirac operator on a Clifford bundle V over a closed manifold Y . The heat equation associated to D is the equation Bs Bt`D 2 s " 0 for sections s, and nonnegative time t; the operator e´t D 2 is the solution operator for this equation. The heat equation has an associated heat kernel H t px, yq which is a (time-dependent) section of the bundle V bV over YˆY whose fiber over a point px, yq is V x b Vẙ ; it is smooth for t ą 0. For any smooth section s of V and t ą 0, the heat kernel satisfies Also, where D x denotes the Dirac operator applied in the x variables. Moreover, Hence, we can prove Lemma 10 by bounding |H t | along the diagonal. The operator De´t D 2 has a kernel L t px, yq as well, and the analogous results hold. A final fact we will need is Duhamel's principle: this says that the inhomogeneous heat equation Bs Bt`D 2s t " s t has a unique solution tending to 0 with t, given by (3.9)s t pxq " as long as s t is a smooth section of S, continuous in t. Now let D be D Ar , and V the spinor bundle for the spin c structure S, and let H r t and L r t be defined as above, but with D " D Ar . Let ρpx, yq the Riemannian distance function. Define an auxiliary function In the case of Y " R 3 , with ρ the standard Euclidean distance, the function h t px, yq is precisely the ordinary heat kernel. In our case, the kernel H r t px, yq has an asymptotic expansion as t Ñ 0, (3.10) H r t px, yq " h t px, yq pb r 0 px, yq`b r 1 px, yqt`b r 2 px, yqt 2`. . .`q, that is studied in detail in [ Lemma 11. There exists for all i " 0, 1, 2, . . . sections b r i px, yq such that: ‚ The b r i are supported in any neighborhood of the diagonal. ‚ The asymptotic expansion (3.10) may be formally differentiated to obtain asymptotic expansions for the derivative. In particular, there is an asymptotic expansion (3.11) L r t px, yq " h t px, yq pb r 0 px, yq`b r 1 px, yqt`b r 2 px, yqt 2`. . .`q whereb r n px, yq " pD Ar`c pρdρ{2tqqb r n px, yq. ‚ For any n, t ą 0, is Opt i´1{2 q, in the C 0 -norm on the product, as t Ñ 0. ‚ Proof. The lemma summarizes those parts of the proof of [6, Thm. 2.30] that we will soon need; the arguments in [6, Thm. 2.30] provide the proof. The idea behind the first bullet point is that h t px, yq is on the order of t 8 away from the diagonal. The reason for the i´1{2 exponent in the third bullet point is that h t has a t´3 {2 term. For the fourth bullet point, the point is that the coefficients b r i are constructed so as to satisfy (3.7) when formally differentiating (3.10) and equating powers of t; this gives a recursion which is relevant for our purposes because it implies that when we truncate the expansion at a finite n, the inhomogeneous equation (3.12) is satisfied.
In view of the first bullet point of the above lemma, we only have to understand the coefficients b r i in a neighborhood of the diagonal. To facilitate this, let i g T Y denote the injectivity radius of the Riemannian metric g, and given y P Y , let B y´i g T Y 2¯d enote a geodesic ball of radius i g T Y 2 centered at y, and let y denote a choice of co-ordinates on this ball. Define G k y Ă C 8´B y´i g T Y 2¯¯t o be the subspace of (rdependent) functions f satisfying the estimate B α y f " O´r k`| α| 2¯a s r Ñ 8, @α P N 3 0 , and for each j P 1 2 N 0 , further define the subspace Finally, given y P Y , we choose a convenient frame for S ξ y and E over B y´i g T Y 2¯, which we will call a synchronous frame; specifically, choose an orthonormal basis for each of S ξ y , E y , and parallel transport along geodesics with A c , A r to obtain local orthonormal trivializations bp¨, yq " 2 ÿ k,l"1 f y b,kl p.q ps k b eq p.q ps l b eq˚pyq .

Lemma 12.
There is a constant c 0 independent of r such that for any t ą 0, r ě 1, we have To prove the second bullet point, we use the fact that the terms b r j in the heat kernel expression (3.10) are known to satisfy a recursion, as alluded to above, and explained in the proof of [6, Thm. 2.30]. Specifically, fix y P Y , choose geodesic coordinates y around y, mapping 0 to y, and choose a synchronous frame as in (3.14). Then, in these coordinates, we have where g " det pg jk q. Moreover, if use these coordinates to identify sections of S b Sẙ with a vector of functions, then we have where on the right hand side of this equation, we mean that we are integrating this vector of functions component by component. Now recall the Bochner-Lichnerowicz-Weitzenbock formula for the Dirac operator: where κ denotes the scalar curvature; we will want to combine this with (3.17). In coordinates, we have where each Γ i is the i th Christoffel symbol for A r . We also have where the Γ i j,k are the Christoffel symbols of the Riemannian metric. Since we have A r " 1bA`A c b1, where A c is the canonical connection on S ξ , we can decompose each Christoffel symbol where the c i are Christoffel symbols for A c and the a i are Christoffel symbols for A. The c i are independent of r. To understand the a j , first write the defining equations for the curvature F kj " B k a j´Bj a k . Now write the coordinate x " px 1 , x 2 , x 3 q, and consider ř 3 k"1 x k pB k a jB j a k q. Reintroducing the radial coordinate ρ, we have On the other hand, since the frame e is parallel, we have ∇ x 1 Bx 1`x 2 Bx 2`x 3 Bx 3 e " ∇ ρBρ e " 0, hence ř 3 k"1 x k a k " 0. Thus, we have In particular, it follows from the a priori estimate (3.23) and ( . Hence, by (3.18), (3.19), (3.20), and (3.23), we have that the square of the Dirac operator has the schematic form where P j P W 1 2 and Q P W 1 y . The Lemma now follows by induction, using (3.16) and (3.17).
We now give the promised: Proof of Lemma 10. The second bullet point follows by combining (3.8) and (3.15).
To prove the first bullet point, our strategy will be to bound the pointwise size of the kernel L r t py, yq and appeal to the version of (3.8) for L r t . To do this, consider the asymptotic expansion (3.11). By a theorem of Thm. 2.4]), for any y P Y , trL r t py, yq is O`t 1{2˘a s t Ñ 0. So we have trL r t py, yq " tr R r t py, yq for the remainder R r t px, yq :"L r t´D Ar rh t pb 0`t b 1 qs . By (3.12), R r t satisfies the inhomogeneous heat equatioǹ and by the third bullet point of Lemma 11, R r t Ñ 0 as t Ñ 0. We can then apply Duhamel's principle (3.9) to write (3.25)

ds.
We can then apply the key property of the heat kernel (3.6) to write By the first bullet point of Lemma 11, we can assume that K s pz, yq is yu. Thus, we just have to bound c 0 rpt´sq h s py, 0qh 2pt´sq py, 0qsK s py, 0qdyds, where y are geodesic coordinates centered at y. To do this, choose a synchronous frame for the spinor bundle, as we have been doing above. Then, following (3.21), (3.22), in these coordinates the Dirac operator is seen to have the form for r´independent w jk and K P W 1 2 y , in the geodesic coordinates and orthonormal frame introduced before. Combining this with the second bullet point of Lemma 12 gives that the term K s P W y se c 0 rt h 2pt´sq py, 0q h s py, 0q y α r k , k ď 5 2`| α| 2 .
On B y´i g T Y 2¯, we have y I h t py, 0q ď c 1 t 1 2 |I| h 2t py, 0q , for some constant c 1 . Hence, we can bound the above integral by (3.27) c 1 r k e c 0 rt y s 1`| α| 2 h 2pt´sq py, 0q h 2s py, 0q dy.
We also have ż Y h t px, yqh t 1 py, zqdy ď c 2 h 4pt`t 1 q px, zq as proved in [20, Sec. A]. So, we can bound (3.27) by We know that k´1`| α| We can finally give: Proof of Proposition 9. Define Epxq :" signpxqerfcp|x|q " signpxq¨2 ? π ş 8 |x| e´s 2 ds ă e´x 2 . This is a rapidly decaying function, so the function EpD Ar q is defined, and its trace is a convergent sum where λ is an eigenvalue of D Ar . The eta invariant in unchanged under positive rescaling η pD Ar q " ηˆ1 ? r D Ar˙. Now use (3.5) to rewrite the right hand side of the above equation ašˇˇˇż The absolute value of the first summand in the above expression is bounded from above by a constant multiple of r 3{2 , by the first bullet point in Lemma 10. The absolute value of the second summand in the same expression is bounded from above by tr e´1 r D 2 Ar , which by the second bullet point in Lemma 10 is bounded by a constant multiple of r 3{2 as well.
Proof of Proposition 6. An application of the Atiyah-Patodi-Singer index theorem as in §A gives The spectral flow term above is estimated to be O`r 3{2˘a s in [24, S 5.4] while η pD Ar q " O`r 3{2˘b y Proposition 9.
We also note that the constant in Proposition 6 above is only a function of pY, λ, Jq and independent of the class σ " rpA r , Ψ r qs P HM`´Y, s E˘d efined by the Seiberg-Witten solution.
Remark 13. The reason that we can not improve upon gr Q asymptotics is because we do not know how to strengthen the O`r 3{2˘s pectral flow estimate on the irreducible solutions of Propositions 6 or 9. A better Oprq estimate does however exist [20,22] for reducible solutions for which one understands the connection precisely in the limit r Ñ 8. However, the a priori estimates (3.23) are not strong enough to carry out the same for irreducibles.

Asymptotics of capacities
4.1. The main theorem. In this section we now prove our main theorem Theorem 3 on ECH capacities.
Proof of Theorem 3. Let 0 ‰ σ j P ECH pY, λ, Γq, j " 0, 1, 2, . . ., be a sequence of non-vanishing classes with definite gradings gr Q pσ j q tending to positive infinity. As in §2.3, we use the perturbed Chern-Simons-Dirac functional (2.2) L µ and its gradient flow (2.3) with µ " irλ, r P r0, 8q, in defining monopole Floer homology. Hence for each r P r1, 8q, the class σ j may be represented by a formal sum of solutions to (2.3) with µ " irλ. As noted in §2.3, this solution is eventually irreducible. Without loss of generality we may assume where q is a fixed rational number and j P 2N.
We now estimate r 1 pjq, the infimum of the values of r such that each solution ra j s r to (2.3) representing σ j is irreducible. For this note that a reducible solution is of the form a " pA, 0, Φ k q where A " A 0´i rλ, A t 0 flat and Φ k the kth positive eigenvector of D A . The grading of such a reducible is given by (2.5). The important estimateˇˇη D A 1`r λˇď c 0 r, ([21, Thm. 1.2]) now shows gr Q rpA, 0, Φ k qs ą gr Q pσ j q " q`j for r ąr 1 pjq :" sup " r| r 2 4π 2 vol pY, λq ă c 0 r`q`j * .
In addition, by combining (2.5) and Proposition 9, we have (4.3)ˇˇˇˇ1 2π 2 CS pA r q´pq`jqˇˇˇˇď c 0 r 3{2 , with the constant c 0 ą 0 being independent of the grading j. We also have the differential relation r de λ dr " dCS dr between the two functionals, away from the discrete set of points where derivatives are undefined, see [11,Lem. 2.5]. Now define F prq " 1 2 r 2 1 vol pY, λq`ş r r 1 e λ pA s q ds. This is a continuous function, and v is continuous as well, so we may integrate the above equation to conclude that CS prq " rF 1´F valid for all r away from the above discrete set; here, we have used [27,Property 2.3.(i)], together with the computation in [11,Lem. 2.3] in the computation of the terms at r 1 .
On account of (4.3), F is then a super/subsolution to the ODEś for r ě r 1 . This gives Next the estimate (4.2) in terms of F is We let ρ 0 be the smallest positive root of 1 3´r ρ`ρ 2`ρ3`ρ4 s " 0 and definer which is finite on account of (4.4). Further with c 3 " 1`3`2 c 1 and set r 2 pjq :" max tr 2 pjq ,r 2 pjqu. We note that r 2 pjq " O`j 1{2˘. We now have the following lemma.
Remark 15. One could replace the arguments in this subsection with the arguments in Sun's paper [27], if desired -the key reason why we have a stronger bound than Sun is because of our stronger bound on the Chern-Simons functional, and not because of anything we do in this subsection. We have chosen to include our argument here, which we developed independently of the arguments in [27], for completeness, and because it might be of independent interest, although we emphasize that we do use the result of Sun establishing [27, Property 2.3.(i)].
On the other hand, the arguments in [11] are not quite strong enough for Theorem 3, even with the improved bound in Proposition 9.

Proofs of Corollaries.
Here we prove the two corollaries Corollary 4 and Corollary 5, both following immediately from the capacity formula Theorem 3.
Proof of Corollary 5. As in the previous corollary, the ECH zeta function is given, modulo a finite and holomorphic in s P C, sum by With ζ R psq denoting the Riemann zeta function, we may using Theorem 3 comparěˇˇˇˇˇˇˇˇˇˇˇ8 hence the difference is holomorphic for Re psq ą 5 3 . The corollary now follows on knowing s " 2 to be the only pole of the Riemann zeta function ζ R`s 2˘w ith residue 1.

4.3.
The ellipsoid example. We close by presenting an example with Op1q asymptotics, and where the corresponding ζ ECH function extends meromorphically to all of C.
Consider the symplectic ellipsoid Epa, bq :" The symplectic form on R 4 has a standard primitive This restricts to BEpa, bq as a contact form, and the ECH spectrum of pBEpa, bq, λq is known. Specifically, let Npa, bq be the sequence whose j th element (indexed starting at j " 0) is the pj`1q st smallest element in the matrix pma`nbq pm,nqPZ ě0ˆZě0 . Then, the ECH spectrum S BEpa,bq is precisely the values in Npa, bq. Moreover, the homology ECH˚pBEpa, bqq has a canonical Z-grading, such that the empty set of Reeb orbits has grading 0, and it is known to have one generator σ j in each grading 2j, see [16]. The spectral invariant associated to σ j is precisely the j th element in the sequence Npa, bq.
With this understood, we now have: Proposition 16. Let σ j be any sequence of classes in ECHpBEpa, bqq with grading tending to infinity. Then, the dpσ j q are Op1q. In fact, if a{b is irrational, then lim jÑ8 dpσ j q j " a`b 2 .
Proof. Assume that a{b is irrational. If t " c σ j , then by the above description, the grading of σ j is precisely twice the number of terms in Npa, bq that have value less than t. With this understood, the example follows from [12, Lem. 2.1]. When a{b is rational, a similar argument still works to show Op1q asymptotics. Namely, if t " c σ j , then by above, the grading of σ j is precisely twice the number of terms in Npa, bq that have value less than t, up to an error no larger than some constant multiple of ? j. Now apply [5,Thm. 2.10].
Thus ζ ECH ps; Y, λ, Γq (4.14) is known to possess a meromorphic continuation to the entire complex plane in this example. Its only two poles are at s " 1, 2 with residues Res s"2 ζ ECH ps; Y, λ, Γq " 1 ab Res s"1 ζ ECH ps; Y, λ, Γq " 1 2ˆ1 a`1 b˙ The APS signature theorem for the manifold X with boundary also gives´1 where η Y is the eta invariant of the odd signature operator on C 8 pY ; T˚Y ' Rq sending pa, f q Þ Ñ p˚da´df,´d˚aq . Combining the above we have A reducible generator ra k s " "`A , 0, Φ A k˘‰ P C s however has 1 2 F A t " dµ and Φ A k the kth eigenvector of D A . Hence, where D As is a family of Dirac operators, associated to a family of connections starting at the flat connection and ending at one satisfying 1 2 F A t "´dµ. Hence, by interpreting this spectral flow as an index through another application of Atiyah-Patodi-Singer [3, p. 95], and applying [2, eq. 4.3] to compute this index, we get gr Q "`A , 0, Φ A k˘‰ " 2k´η pD A q`1 4 η Y´1 2π 2 CSpAq (A.1) as the absolute grading of a reducible generator.
The absolute grading of an irreducible generator ra 1 s " pA 1 , s, Φ 1 q, s ‰ 0, is then given by gr Q ra 1 s " gr Q ra 0 s´2 sf in terms of spectral flow of the Hessians (2.4) for a path pA ε , Ψ ε q P A pY, sqˆC 8 pSq, ε P r0, 1s starting at ra 0 s " rpA 0 , 0qs and ending at pA 1 , sΦ 1 q. As above, we can interpret this spectral flow as an index; this time, to compute the relevant index, we need to apply ([2, Thm. 3.10]), which gives that the above is equal to gr Q ra 1 s "´η´p H pA,sΦ 1 q¯`5 4 η Y´2 ż Yˆr0,1s ε ρ 0 .
Here ρ 0 is the usual Atiyah-Singer integrand, namely the local index density defined as the constant term in the small time expansion of the local supertrace str´e´t D 2¯w ith and where pA ǫ , Ψ ǫ q is the chosen path of configurations. To compute the index density we choose a path of the form pA ε , Ψ ε q " # pA`2ε pA 1´A q , 0q ; 0 ď ε ď 1 2 , pA 1 , p2ε´1q Ψq ; 1 2 ď ε ď 1. On the interval " 0, 1 2 ‰ , the integral of the local density is given by the usual local index theorem: as above, we havé 2 ż Yˆr0, 1 2 s ε ρ 0 "´1 2π 2 CS pAq .
On the other hand, for the calculation on Yˆ" Duhamel's principle then gives that the coefficients in the small time heat kernel expansion of the difference above are of the form » -0 00 00 fi fl with respect to the decomposition iT˚Y ' R ' S. Hence we have in summary: gr Q ras " # 2k´η pD A q`1 4 η Y´1 2π 2 CS pAq ; a "`A, 0, Φ A k˘P C s , η´p H pA,sΦq¯`5 4 η Y´1 2π 2 CS pAq ; a " pA, s, Φq P C o , s ‰ 0.