Perturbation theory of asymptotic operators of contact instantons and pseudoholomorphic curves on symplectization

1.1 Pseudoholomorphic curves in symplectization and contact instantons

By definition, we have dπ_Q du = dw. It was observed by Hofer [13] that u is J͠-holomorphic if and only if (w, f) satisfies

(We refer to Appendix B for the here unexplained notation ∂^π.) A contact instanton is a map w : Σ̇ → M that satisfies the system of nonlinear partial differential equations

on a contact triad (M, λ, J). The equation itself had been introduced by Hofer [14, p. 698]. Note that for any map u = (w, f) satisfying (1.6), w satisfies this equation with the additional property that the one-form w^*λ ∘ j is exact, not just closed.

In a series of papers, [37; 38] joined with Wang and in [27], the second-named author systematically developed the analysis of contact instantons (for the closed string case) without taking symplectization by the global covariant tensorial calculations using the notion of contact triad connection introduced by Wang and the second-named author in [36]. A relevant Fredholm theory has been developed by the second-named author in [27], [35] (for the closed string case). More recently the second-named author also studied its open string counterpart of the boundary value problem of (1.7) under the Legendrian boundary condition whose explanation is now in order. We mention that in the early 2000’s Abbas studied the Legendrian chord problem in [1], and more recently Cant [7] developed a detailed Fredholm theory on symplectization in the relative context.

Throughout the paper, we adopt the following notation.

For the simplicity and for the main purpose of the present paper, we will focus on the genus zero case so that Σ̇ is conformally the unit disc with boundary punctures z₀, …, z_k ∈ ∂ D² ordered counterclockwise, i.e.,

Then, for a (k + 1)-tuple R⃗ = (R₀, R₁, …, R_k) of Legendrian submanifolds, which we call an (ordered) Legendrian link, we consider the boundary value problem

as an elliptic boundary value problem for a map w : Σ̇ → M and derive the a priori coercive elliptic estimates. Here z_iz_i+1 ⊂ ∂ D² is the open arc between z_i and z_i+1.

Let ∇͠ := ∇^can be the canonical connection of this almost Hermitian manifold

i.e. the unique Riemannian connection whose torsion T satisfies T(X, J͠ X) = 0 for all X ∈ T(Q × ℝ); see [11; 22; 26] for its definition and basic properties. We note that this almost Hermitian structure on Q × ℝ is translational invariant in the radial s-direction. The usage of the canonical connection on the 𝔩𝔠𝔰-fication (Q × ℝ, J͠, ω_λ) (or equivalently via the contact triad connection on the triad (Q, λ, J)) plays an important role in the present authors’ analysis of the asymptotic operators of finite energy contact instantons in Section 5, and hence of finite energy pseudoholomorphic curves too. Roughly speaking, our coordinate-free approach enables us to compute the asymptotic operator associated to each isospeed Reeb orbit (γ, T) of a contact instanton w denoted by

simultaneously over all isospeed Reeb orbits in the covariant tensorial way in terms of the pull-back connection under the map w of the given contact triad connection or the Levi–Civita of the triad metric of the given triad (Q, λ, J). (See Definition 4.2 for the precise definition of the operator A(λ,J,∇)π .)

1.2 Asymptotic operators and their analysis

We first mention a few differences between the way how we study the asymptotic operators and those of [16] and of other literature such as [42, Appendix C], [43; 44], [45], [7].

In [42, Appendix E], [43; 44], [45, Section 3.3], there have been attempts to give a coordinate-free definition of the asymptotic operator along the associated asymptotic Reeb orbit for a pseudoholomorphic curve u = (w, f) on symplectization. However they fall short of a seamless definition of the ‘asymptotic operator’ of the Reeb orbits because the Reeb orbit lives on Q while the pseudoholomorphic curves live on the product Q × ℝ and the asymptotic limit of pseudoholomorphic curves lives at infinity Q × {± ∞}. (Cant studied the asymptotic operator in the relative context in [7, Section 6.3] by adapting Wendl’s.)

What the literature (e.g. [45, Section 3.3], [43; 44]) is describing, however, is actually the asymptotic operator of the contact instanton w but trying to describe it in terms of the pseudoholomorphic curves which prevents them from being able to give a seamless definition: Recall the decomposition

with respect to the splitting (1.4). (Compare their practices with our definition of the asymptotic operator of contact instantons given in Definition 4.2 and compare it therewith. See also [38, Section 11.2 & 11.5] for the precursor of our definition.)

In this regard, we will derive an explicit formula of the asymptotic operator in terms of the contact triad connection, which exhibits the way how the operator explicitly depends on J so it admits a perturbation theory under the change of J.

Combining this with the basic properties of the triad connection and γ̇(t) = T R_λ(γ(t)), we can derive an explicit expression of the asymptotic operator purely in terms of λ, J and (γ, T) independent of the choice of the connection.

This study enables us to make precise statements on the spectral behavior of asymptotic operators under the perturbation of J’s. The ellipticity of A(λ,J,∇)π implies that it has a discrete set of eigenvalues, which we enumerate as

repeated with finite multiplicities allowed, where μ_k → ∞ (respectively μ_−k → − ∞) as k → ∞. It also has a uniform spectral gap (by [18, Theorem 6.29]) in that there exists some d > 0 with |μ_i+1 − μ_i| ≥ d > 0 for all i.

Our explicit formula of the asymptotic operator given in Proposition 1.5, which simultaneously applies to all closed Reeb orbits, enables us to provide a generic description of the spectral behavior of asymptotic operators under the change of λ-compatible CR almost complex structures J when λ is fixed. For example, we prove the following natural generic perturbation result of eigenvalues of the asymptotic operator.

See Section 5 for our derivation of the formula of the asymptotic operator.

1.3 Relationship with other literature and discussion

We have no doubt that a parameterized version of the proof of Theorem 5.1 will lead to the following spectral flow description of the index. Consider the nonlinear Fredholm map

whose domain and codomain are specified as in [27] or in [39] where this map was just written as Υ instead. Since we use the latter for a different purpose later in the present article, we use the different notation Υ(λ,J)cont here for the same operator appearing in [27] or in [39].

The main step of the proof is to establish the contact instanton version of Lemma 3.17 [45] which says that there is a perturbation (inside the space of abstract Fredholm operators) of the given one-parameter family of asymptotic operators such that all eigenvalue families of the perturbed operator are transversal in an explicit sense. In our case, such a transversality result can be obtained inside the smaller natural family of J perturbations by the parameterized version of the proof of Theorem 5.1. We omit the details of its proof leaving them to interested readers or to [45, Appendix C] which handles the case of pseudoholomorphic curves in symplectization the scheme of which can be now easily modified via the perturbations of J utilizing the scheme of our proof of Theorem 5.1.

Because the existing literature on the pseudoholomorphic curves on symplectization lacks an explicit formula of the asymptotic operator as given in Theorem 6.1, it has been the case that the general abstract perturbation theory of linear operators in Kato [18] is just quoted in their study of asymptotic operators which prevents one from making any statement on specific dependence on the compatible almost complex structures. (See [45, Section 3.2] especially see Lemma 3.17, Theorem 3.35 and Appendix C of [45] for example which in turn extends the exposition given in [15] to higher dimensions.)

As far as we can see, the aforementioned status of matter makes the existing description in the literature, at least, of the spectral flow related to the index formula of the linearized operator for the J͠-pseudoholomorphic curves on the fundamental case of ℝ × S¹ for the closed string case (or on ℝ × [0, 1] for the open string case) in symplectization or in SFT rather unsatisfying and incomplete. This is because the spectral flow definition over the real line ℝ ≅ (0, 1) starts from the requirement that the asymptotic operators

associated to the given J͠ have simple eigenvalues at the end points; note, however, that these operators may have eigenvalues with multiplicity before making some perturbation. (Such a requirement is vacuous over the closed circle S¹ considered as in the original article [3].) Furthermore one needs this description simultaneously over all asymptotic Reeb orbits for the given almost complex structure J͠. To achieve this requirement the existing studies of such a spectral flow representation in the literature had to use the much bigger perturbations of abstract Fredholm operators in the linearized level but not in the original off-shell nonlinear level. Furthermore the perturbations are made depending on each asymptotic orbit which is not a priori given, especially when even its existence is not known. (See [45, Lemma 3.17 & Appendix C], for example.) Under such current circumstances, it is very cumbersome to develop, for example, a gluing theory leading to the Kuranishi structures compatible over different stata of the compactified moduli space of pseudoholomorphic curves entering in the SFT compactification.

Our Theorem 1.7 cures at least this unsatisfying point on the spectral flow description of the index in the literature by considering the natural family

which connects the asymptotic operator at τ = ± ∞ for the given J, after one single perturbation thereof in the off-shell level (See (4.3) for the precise expression.) We believe that the kind of perturbation result stated in Theorem 5.1 and Theorem 1.8 will play some role in the construction of Kuranishi structures on the moduli space of finite energy contact instantons so that certain natural functors can be defined in our Fukaya-type category of contact manifolds [21], [20]. (See [5] for a construction of semi-global Kuranishi structures in their definition of contact homology, where the simpleness properties of the eigenvalues of the asymptotic operators is utilized in an essential way.)

We also refer readers to the arXiv version [19] of the present paper for some application to the finer study of asymptotic convergence of contact instantons which simplifies the exposition given in the literature on the pseudoholomorphic curves on symplectization via the systematic tensorial calculus.

Finally we would like to just mention that the same asymptotic study can be made by now in a straightforward way by incorporating the boundary condition as done in [39], [28], [33].

The present work has been first announced in the survey paper [34] submitted for MATRIX Annals for the IBS-CGP and MATRIX workshop on Symplectic Topology held for December 5–16, 2022.

2.1 Fundamental equation of contact Cauchy–Riemann maps

The following fundamental identity is derived in [37].

The following elegant expression of the Fundamental Equation in isothermal coordinates (x, y), i.e. one such that z = x + iy provides a complex coordinate of (Σ̇, j) such that h = dx² + dy², will be extremely useful for the study of higher a priori C^k,α Hölder estimates.

The fundamental equation in cylindrical (or strip-like) coordinates is nothing but the linearization equation of the contact Cauchy–Riemann equation in the direction ∂∂τ . This plays an important role in the derivation of the exponential decay of the derivatives at cylindrical ends; see [38, Part II].

2.2 Generic nondegeneracy of Reeb orbits and of Reeb chords

Nondegeneracy of closed Reeb orbits or of Reeb chords is fundamental in the Fredholm property of the linearized operator of contact instanton equations as well as of pseudoholomorphic curves on symplectization.

2.3 Exponential asymptotic C^∞ convergence of contact instantons

In this section, we recall the asymptotic behavior of contact instantons on the Riemann surface (Σ̇, j) associated with a metric h with cylinder-like ends for the closed string context and with strip-like ends for the open string context.

We assume that there exists a compact set K_Σ ⊂ Σ̇, such that Σ̇-Int(K_Σ) is a disjoint union of interior-punctured disks each of which is isometric to the half cylinder [0, ∞) × S¹ or the half strip (−∞, 0] × S¹, where the choice of positive or negative strips depends on the choice of analytic coordinates at the punctures. We denote by {pi+}i=1,…,l+ the positive punctures, and by {pj−}j=1,…,l− the negative punctures. Here l = l⁺ + l⁻. The case of boundary-punctured disks is similar. Then we denote by ϕi± such cylinder-like or strip-like coordinates depending on whether they are boundary or interior punctures.

We separately describe the cases of interior punctures and of boundary punctures. We first mainly state our assumptions for the study of the behavior of boundary punctures. (The case of interior punctures is treated in [37, Section 6] and will be briefly mentioned at the end of this section.)

Throughout this section, we work locally near one boundary puncture p, i.e., on a punctured semi-disc D^δ(p) ∖ {p}. By taking the associated conformal coordinates ϕ⁺ = (τ, t) : D^δ(p) ∖ {p} → [0, ∞) × [0, 1] such that h = dτ² + dt², we need only look at a map w defined on the semi-strip [0, ∞) × [0, 1] without loss of generality.

Under the nondegeneracy hypothesis from Definition 2.8 and the transversality hypothesis, the exponential C^∞ convergence is derived from the subsequence convergence and the charge-vanishing result in [37], [39]. For readers’ convenience and for the self-containedness of the paper, we recall the subsequence and charge in Appendix C, and the exponential convergence result in this subsection.

Suppose that the tuple R⃗ = (R₀, …, R_k) is transversal in the sense all pairwise Reeb chords are nondegenerate. In particular we assume that the entries of the tuple are pairwise disjoint. Firstly, we state the following C⁰-exponential convergence from [38] for the closed string and from [39] for the open string case.

Once the above C⁰-exponential decay is established, the C^∞-exponential convergence w(τ, ⋅) → γ follows by establishing the C^∞-exponential decay of

The proof of the latter decay is carried out in [39] by an alternating bootstrap argument by decomposing

as follows. Let z = x + i y be any isothermal coordinates on (D₂, ∂ D₂) ⊂ (Σ̇, ∂ Σ̇) adapted to the boundary, i.e. satisfying that ∂∂x is tangent to ∂ Σ̇. We set

Then we show that the fundamental equation (2.2) is transformed into the following system of equations for the pair (ζ, α)

and

for some i = 0, …, k. With this coupled system of equations for (ζ, χ) at our disposal, the proof follows the alternating bootstrap between ζ and χ similarly as in the proof of higher regularity results carried out by the alternating bootstrap argument in [29; 39].

Combining this and the elliptic C^k,α-estimates given in [37; 39], the proof of the C^∞-convergence of w(τ, ⋅) → γ as τ → ∞ is completed.

So far we have recollected various foundational analytic results on contact instantons both in the closed and in the open string context, which will be used later in our study of asymptotic operators. Since the same arguments can be applied to the open string case with by now straightforward incorporation of the boundary condition exercised in [39], [33], [28] and [40], we will focus on the case of closed strings to streamline our study of asymptotic operators and to highlight the main points of our approach in the rest of the paper.

2.4 Exponential convergence of pseudoholomorphic curves on symplectization

Finally we make a brief mention on how the exponential convergence for pseudoholomorphic curves on symplectization follows from that of contact instantons now. We consider the symplectization

of the contact manifold (Q, ξ) equipped with contact form λ.

On Q, the Reeb vector field R_λ associated to the contact form λ is the unique vector field X =: R_λ satisfying

We call (y, s) the cylindrical coordinates. On the cylinder [0, ∞) × Q ⊂ (− ∞, ∞) × Q, we have the natural splitting (1.4) of the tangent bundle TM.

Now we describe a special family of almost complex structure compatible to the given cylindrical structure of M.

For our purpose, we will need to consider a family of symplectic forms to which the given J is compatible and their associated metrics. For any λ-compatible J, the J-compatible metric associated to ω is expressed as

on ℝ × Q. Now we regard the triple (ω, J, g_(ω,J)) as an almost Hermitian manifold near the level surface s = 1. We then fix the canonical connection ∇ associated to (ω, J, g_(ω,J)); see Appendix A.

The following is a general property of the canonical connection.

Consider the decomposition (1.4) and the canonical connection ∇͠ on Q × ℝ, which in particular is J-linear. Recalling the expression u = (w, f) with f = s ∘ u and w = π_Q ∘ u for each map u : Σ̇ → Q × ℝ, we know that if it is J-holomorphic, it satisfies

on Σ̇. In particular w is a contact instanton. Then we have already shown the exponential convergence w to the Reeb orbit w(⋅ , t).

For the convergence of f, we use the equation

which is equivalent to

By taking the differential, we obtain

Firstly, this equation shows that f is a subharmonic function. More explicitly, we derive that

where we know from Theorem C.5 that the convergence 12 |d^π w|² → T² is exponentially fast. This immediately gives rise to the following exponential convergence of the radial component, which will complete the study of asymptotic convergence property of finite energy pseudoholomorphic planes in symplectization.

4.1 Asymptotic operator in contact triad connection

Now we find the formula for the above limit operator with respect to the contact triad connection As T(R_λ, ⋅ ) = 0, ∂w∂τπ=−J∂w∂tπ   and  ∂w∂t(τ,⋅)↦TRλ exponentially fast, we obtain

On the other hand, we have

This converges to T2J∇Rλ since w^* λ → T dt as τ → ∞.

This immediately gives rise to the following simple explicit formula for the asymptotic operator.

Proposition 4.3 enables us to derive the following.

We note that the right hand side formula in (4.6) is canonically defined depending only on λ and J, independent of the choice of the connection. In other words, we can derive the first variation of the operator ∇_Rλ|_ξ associated to the triad connection of (Q, λ, J) with respect to the compatible pair (λ, J) or with respect to J when λ is fixed.

Now in the context of pseudoholomorphic curves on symplectization, we proceed by decomposing the full asymptotic operator A(λ,J,∇)π of the pseudoholomorphic curves

into

Here ℂ stands for the pull-back bundle

which is canonically trivialized, and hence may be regarded as a vector bundle over the curves w_τ on Q. (Compare this with [41, Definition 2.28].)

4.2 Asymptotic operator in Levi–Civita connection

Up until now, we have emphasized the usage of the triad connection which gives rise to an optimal form of tensorial expressions. We recall that ∇_Y J = 0 for all Y ∈ ξ for the contact triad connection ∇ by one of the defining axioms of the contact triad connection. We have ∇_Rλ J ≠ 0 for the connection (in fact, ∇_Rλ J = 0 if and only if R_λ is a Killing vector field, i.e. ∇ R_λ = 0 with respect to ∇; see [37, Remark 2.4]). On the other hand, while ∇YLCJ ≠ 0 for the Levi–Civita connection ∇^LC in general, the Levi–Civita connection has the following surprising property

see Lemma 6.1 in [6], Proposition 4 in [36]. Although it will not be used in the present paper, we also convert the formula of the asymptotic operator A_(λ,J,∇) into one written in terms of the Levi–Civita connection for a possible future purpose.

For this purpose, the following lemma is crucial.

Therefore we have derived the simple relationship