Definition of contact homology]
Semi-global Kuranishi charts and the definition of contact homology

Erkao Bao
University of California, Los Angeles, Los Angeles, CA 90095
<EMAIL_ADDRESS>http://www.math.ucla.edu/126 bao

Ko Honda
University of California, Los Angeles, Los Angeles, CA 90095
<EMAIL_ADDRESS>http://www.math.ucla.edu/126 honda

[2010]Primary 53D10; Secondary 53D40.

We define the contact homology algebra for any contact manifold and show that it is an invariant of the contact manifold. More precisely, given a contact manifold $(M,\xi)$ and some auxiliary data $\mathcal{D}$, we define an algebra $HC(\mathcal{D})$.
If $\mathcal{D}_1$ and $\mathcal{D}_2$ are two choices of auxiliary data for $(M,\xi)$, then $HC(\mathcal{D}_1)$ and $HC(\mathcal{D}_2)$ are isomorphic. We use a simplified version of Kuranishi perturbation theory, consisting of semi-global Kuranishi charts.

§ INTRODUCTION

Symplectic field theory (SFT), proposed about 15 years ago by Eliashberg-Givental-Hofer [10], is a package which gives invariants of contact manifolds and symplectic manifolds with boundary as well as gluing formulas for Gromov-Witten invariants on closed symplectic manifolds. The transversality theory in the somewhere injective case and the Fredholm theory were carried out by Dragnev [8] and the SFT compactness was worked out by [4], both very early on in the development of the theory. However, the rigorous foundations of the full theory are still under development, although parts of the theory are gradually becoming more rigorous, thanks to the efforts of many authors, primarily by Hofer-Wysocki-Zehnder [18].  Partial work has been done by Hutchings-Nelson [20] and Bao-Honda [2] for cylindrical contact homology in dimension three and Bourgeois-Oancea [6] on $S^1$-equivariant symplectic homology.  Also the Kuranishi perturbation theory of Fukaya-Ono [11] and Fukaya-Oh-Ohta-Ono [12] (which is closest in spirit to this work) and the approaches of Liu-Tian [25] and [29] may be used to define SFT, although this has not been done yet.

The goal of this paper is to make a further contribution to this effort and define the full contact homology differential graded algebra (dga) for any closed contact manifold in any dimension and show that its isomorphism class is an invariant of the contact manifold. More precisely, given a closed cooriented contact $(2n+1)$-dimensional manifold $(M,\xi)$ and some auxiliary data $\mathcal{D}$ which includes a nondegenerate contact form $\alpha$ for $\xi$, we define a direct limit dga $\mathfrak{A}(\mathcal{D})$ whose homology is called the full contact homology algebra $HC(\mathcal{D})$. We then prove that:

If $\mathcal{D}_1$ and $\mathcal{D}_2$ are two choices of auxiliary data for $(M,\xi)$, then $HC(\mathcal{D}_1)$ and $HC(\mathcal{D}_2)$ are isomorphic. Hence the isomorphism class of the algebra $HC(\mathcal{D})$ is an invariant of $(M,\xi)$.

We denote the isomorphism class of $HC(\mathcal D)$ by $HC(\xi)$.  We also prove the following:

Let $W$ be a compact exact symplectic cobordism which restricts to $(M_+,\xi_+)$ at the positive boundary and to $(M_-,\xi_-)$ at the negative boundary.
Then there is an algebra homomorphism

$$\Phi_*:HC(\xi_+)\to HC(\xi_-).$$

In the case when $\xi_+=\xi_-$ and $W$ is the trivial cobordism, we have $\Phi_*=\op{id}$.

The construction of the direct limit dga $\mathfrak{A}(\mathcal{D})$ and the proofs of Theorems <ref> and <ref> use a particularly simple version of the Kuranishi multivalued perturbation theory of [11, 12], which we call semi-global Kuranishi structures.  The abstract perturbations that we use are supported near the ends of finite energy $J$-holomorphic curves and are described using the asymptotic eigenfunctions of the asymptotic operators.  For example, if a moduli space

$$\mathcal{M}^{\op{ind}=k}_J(\dot F,\widehat{W};\bs\gamma_+;\bs\gamma_-)$$

of Fredholm index $\op{ind}=k$ $J$-holomorphic maps $\dot F\to \widehat{W}$ which map to a completed exact symplectic cobordism $\widehat{W}$ and are asymptotic to collections $\bs\gamma_+$ and $\bs\gamma_-$ of Reeb orbits at the positive and negative ends is already compact (i.e., we do not need multiple-level curves in its symplectic field theory (SFT) compactification[Throughout this paper we will freely use the SFT compactness theorem of [4] without explicit mention.]), then we only need one Kuranishi chart, which we call a semi-global Kuranishi chart.

We emphasize that our simplification crucially uses the existence of at least one end limiting to a Reeb orbit and can be used to define SFT in the case when $\widehat{W}$ is an exact symplectic cobordism, although we only deal with contact homology in this paper.  Our simplified methods are not readily applicable to the case when there is a closed multiply-covered  $J$-holomorphic curve. However, we expect the combination of our methods and those of [11] and [12] to yield a rigorous definition of SFT.

Finally, during the preparation of this paper, Pardon [28] posted a paper proving the existence and invariance of contact homology in arbitrary dimensions.

Outline of the paper.  We will be using the language of orbifolds and multisections, following Adem-Leida-Ruan [1] and Fukaya-Ono [11]; this is reviewed in  Section <ref>. After some preparation in Sections <ref> and <ref>, we construct the semi-global Kuranishi chart in Section <ref>, with some modifications in Section <ref>.   The semi-global Kuranishi structure is constructed in Section <ref>, using the gluing results which are stated in Section <ref>; they are average specimens of the “gluing theorem” type and are proved in Section <ref>. Once the semi-global Kuranishi structure is constructed, the definition of contact homology, the definition of chain maps, and the invariance of contact homology (Theorems <ref> and <ref>) follow the usual lines of argument and are carried out in Section <ref>.

We note that the discussion until the end of Section <ref> is valid for curves of any genus; starting from Section <ref> we specialize to contact homology.

Acknowledgements. We thank Francis Bonahon, Tobias Ekholm, Michael Hutchings, Eleny Ionel, Gang Liu, and Kaoru Ono for very helpful discussions.  We also thank Tobias Ekholm and John Pardon for pointing out some errors.

§ ORBIFOLDS AND MULTISECTIONS

In this section we review the basics of (effective) orbifolds, orbibundles, and multisections from [1] and [11]. The definitions in this section, while elementary, are a bit cumbersome to state.  Also, at the end of the day, all we do is replace the words “manifold", “vector bundle”, and “section” by “orbifold”, “orbibundle”, and “multisection”, and treat them in exactly the same way for our purposes.  For the above reasons the reader may want to skip this section and return to it as the relevant concepts gradually start appearing in the rest of the paper.

§.§ Orbifolds and orbibundles

Given a topological space $X$, a triple $(V,\Gamma, \phi)$ is an orbifold chart of $X$ if

* $V$ is a connected open subset of $\mathbb R^n$;

* $\Gamma$ is a finite group that acts smoothly and effectively on $V$; and

* $\phi:V\to X$ is a $\Gamma$-invariant continuous map such that the quotient map $V/\Gamma \to U\subseteq X$ is homeomorphism, where  $U$ is some open subset of $X$.

If $(V_i,\Gamma_i,\phi_i)$ and $(V_j,\Gamma_j,\phi_j)$ are orbifold charts of $X$, we say that $(V_i,\Gamma_i,\phi_i)$ embeds into $(V_j,\Gamma_j,\phi_j)$ if there exists a smooth embedding $\psi_{ji}:V_i\hookrightarrow V_j$ such that $\phi_j\circ \psi_{ji}=\phi_i$.

If $(V_i,\Gamma_i,\phi_i)$ embeds into $(V_j,\Gamma_j,\phi_j)$, then there exists a monomorphism $\theta_{ji}: \Gamma_i\to \Gamma_j$ such that $\psi_{ji}$ is $\theta_{ji}$-equivariant. See [1] for details.

Let $X$ be a Hausdorff, second countable topological space.  A smooth orbifold structure on $X$ is a family $\mathcal{O}=\{(V_i,\Gamma_i, \phi_i)\}_{i\in\mathcal{I}}$ of orbifold charts such that

* $\{ U_i \}_{i\in\mathcal{I}}$ forms an open cover of $X$; and

* for any $x\in U_i \cap U_j \subseteq X$, there exist a neighborhood $U_k\subseteq U_i\cap U_j$ of $x$, a chart $(V_k,\Gamma_k,\phi_k)$ for $U_k$, and embeddings of orbifold charts $\psi_{ik}:(V_k,\Gamma_k,\phi_k) \hookrightarrow (V_i,\Gamma_i,\phi_i)$ and $\psi_{jk}:(V_k,\Gamma_k,\phi_k)\hookrightarrow (V_j,\Gamma_j,\phi_j)$.

The pair $\mathbb X=(X, \mathcal{O})$ is a smooth orbifold.

A smooth orbifold map

$$\lambda:(X,\mathcal{O})\to (X',\mathcal{O}')$$

is a continuous map $\lambda:X\to X'$ such that, for any $x\in X$, there exist orbifold charts $(V,\Gamma,\phi)$ around $x$ and $(V',\Gamma',\phi')$ around $\lambda(x)$, and a lift $\widehat \lambda : V\to V'$ of $\lambda|_{V/\Gamma}$ such that $\phi'\circ \widehat \lambda=\lambda \circ \phi$.

Let $ \pi: \mathbb E\to \mathbb X$ be a smooth orbifold map. The triple $(E\to V, \Gamma, \widetilde \phi)$ is an orbibundle chart of $\mathbb E \to \mathbb X$ of rank $n$, if

* $(V,\Gamma,\phi)$ and $(E,\Gamma,\widetilde\phi)$ are orbifold charts of $\mathbb X$ and $\E$;

* $E=V\times \mathbb R^n$ and $\widehat \pi:E \to V$ is the projection onto the first factor;

* $\Gamma$ acts diagonally on $E=V\times \R^n$, where the action on $\R^n$ is linear; and

* $\pi \circ \widetilde \phi=\phi \circ \widehat \pi$ and $\widetilde \phi$ induces a homeomorphism $E/\Gamma \stackrel\sim\to  \pi^{-1}(U)$.

We call $(E\to V,\Gamma)$ a local orbibundle model of rank $n$.

$$(E_i\stackrel{\widehat{\pi}_i}\to V_i,\Gamma_i,\widetilde\phi_i) \quad \mbox{and} \quad (E_j\stackrel{\widehat{\pi}_j}\to V_j,\Gamma_j,\widetilde\phi_j)$$

are orbibundle charts of $\mathbb E \to \mathbb X$ of rank $n$, we say $(E_i\to V_i,\Gamma_i,\widetilde\phi_i)$ embeds into $(E_j\to V_j,\Gamma_j,\widetilde\phi_j)$ if there exists an embedding of vector bundles $\widetilde \psi_{ji}:E_i\hookrightarrow E_j$ (i.e., an embedding which lies over an embedding $\psi_{ji}: (V_i,\Gamma_i,\phi_i)\hookrightarrow (V_j,\Gamma_j,\phi_j)$ of orbifold charts, takes vector space fibers to vector space fibers isomorphically, and satisfies $\widehat\pi_j\circ \widetilde \psi_{ji}= \psi_{ji}\circ \widehat{\pi}_i$) such that $\widetilde \phi_j \circ \widetilde \psi_{ji}=\widetilde \phi_i$.

Let $\pi: \mathbb E\to \mathbb X$ be a smooth orbifold map. A smooth rank $n$ orbibundle structure on $\mathbb E\to \mathbb X$ consists of a family of rank $n$ orbibundle charts $\{(E_i\to V_i,\Gamma_i,\widetilde\phi_i)\}_{i\in\mathcal{I}}$ that satisfy

* $\mathbb X=(X, \{(V_i,\Gamma_i, \phi_i)\}_{i\in\mathcal{I}})$; and

* for any $x\in U_i \cap U_j$ where $U_i=V_i/\Gamma_i$ and $U_j=V_j/\Gamma_j$, there exist a neighborhood $U_k\subseteq U_i\cap U_j$ of $x$, an orbibundle chart $(E_k\to V_k,\Gamma_k,\widetilde\phi_k)$ such that $U_k=V_k/\Gamma_k$, and embeddings of orbibundle charts

$$\widetilde \psi_{ik}:(E_k\to V_k,\Gamma_k,\widetilde\phi_k)\hookrightarrow (E_i\to V_i,\Gamma_i,\widetilde\phi_i)$$

$$\widetilde \psi_{jk}:(E_k\to V_k,\Gamma_k,\widetilde\phi_k)\hookrightarrow (E_j\to V_j,\Gamma_j,\widetilde\phi_j)$$

that lie over and are compatible with embeddings of orbifold charts

$$\psi_{ik}:(V_k,\Gamma_k,\phi_k) \hookrightarrow (V_i,\Gamma_i,\phi_i)\quad \mbox{and} \quad \psi_{jk}:(V_k,\Gamma_k,\phi_k)\hookrightarrow (V_j,\Gamma_j,\phi_j).$$

We refer to $\mathbb{X}$ as the base of the orbibundle and $\mathbb{E}$ as the total space of the orbibundle.

§.§ Multisections

Given a topological space $W$, we denote its $n$-fold symmetric product by

Here $S_n$ is the symmetric group of $n$ elements which acts on $W^n$ by

where $\sigma\in S_n$ and $(x_1,\dots,x_n)\in W^n$.
We denote the equivalence class of $(x_1,...,x_n)$ in $\op{Symp}^n(W)$ by $[x_1,...,x_n]$.
If a group $\Gamma$ acts on $W$, then $\Gamma$ also acts on $\op{Sym}^n(W)$ diagonally by

where $g\in \Gamma$ and $[x_1,\dots,x_n]\in \op{Sym}^n(W)$.
There is an inclusion

$$\iota^{n}:\op{Sym}^{m}(W)\to \op{Sym}^{mn}(W),$$

\[
[w_{1},\dots,w_{m}]\mapsto [\underbrace{w_{1},\dots,w_{1}}_{n\text{ copies}},\dots,\underbrace{w_{m},\dots,w_{m}}_{n\text{ copies}}].
\]

Let $(E\to V, \Gamma)$ be a local rank $n$ orbibundle model,
in particular, $E=\mathbb R^n \times V$.
Let $\op{Sym}^m (E)=\op{Sym}^m(\mathbb R^n) \times V$.
Then $\op{Sym}^m(E)\to V$ is a fiber bundle with an equivariant $\Gamma$-action.

A degree $m$ multisection (or $m$-multisection) of the local orbibundle model $(E\to V,\Gamma)$
is a $\Gamma$-equivariant section of $\op{Sym}^m(E)\to V$.

An $m$-multisection $\mathbf s$ of $(E\to V,\Gamma)$ is liftable if there exists a section $\widetilde {\mathbf s}=(s_1,s_2,\dots,s_m):V\to E^m$ which is not necessarily $\Gamma$-equivariant, such that $\mathbf s=p \circ \widetilde{\mathbf s}$, where $E^m=\mathbb R^{mn}\times V $ and $p:E^m\to \op{Sym}^m(E)$ is the quotient map. We call each such $s_i$ a branch of $\mathbf s$.

We assign the weight ${1\over m}$ to each branch $s_i$ of ${\bf s}$, if ${\bf s}$ is a degree $m$ multisection.

A liftable $m$-multisection $\mathbf s$ of $(E\to V,\Gamma)$ and a liftable $n$-multisection $\mathbf t$ of $(E\to V,\Gamma)$ are transverse (denoted by $\mathbf s \pitchfork \mathbf t$) if $s_i$ is transverse to $t_j$ for all $1\leq i \leq m$ and $1\leq j \leq n$.

In the above definition, for each $i\in\{1,\dots,m\}$ and $j\in\{1,\dots,n\}$, the space $s_i^{-1}(t_j)$ is a submanifold of $V$ (it may or may not invariant under $\Gamma$),
and we assign the weight ${1\over mn}$ to it.
In this way, $\mathbf s^{-1}(\mathbf t):=\cup_{i,j} s_i^{-1}(s_j)$ is an immersed weighted branched submanifold of $V$, and it is invariant under action of $\Gamma$.

An $m$-multisection of $(E\to V,\Gamma)$ can be mapped to an $\ell m$-multisection of $(E\to V,\Gamma)$ by postcomposing with $\iota^\ell$, which is interpreted as acting on each fiber separately.

An $\ell$-multisection $\mathbf s$ and an $m$-multisection $\mathbf s'$ of $(E\to V, \Gamma)$ are said to be equivalent if $\iota^m \circ \mathbf s=\iota^\ell \circ \mathbf s'$.

If ${\bf s}$ and ${\bf s}'$ are equivalent and ${\bf s}\pitchfork {\bf t}$, then $\mathbf s^{-1}(\mathbf t)$ and $\mathbf s^{'-1}(\mathbf t)$ represent the same weighted branched immersed submanifold of $V$.

A multisection $\mathbf s$ of an orbibundle $\mathbb E \to \mathbb X$ is a family $\{(E_i\to V_i,\Gamma_i,\widetilde\phi_i,\mathbf s_i)\}_{i\in\mathcal{I}}$ such that

* $\{(E_i\to V_i,\Gamma_i,\widetilde\phi_i)\}_{i\in\mathcal{I}}$ is a family of orbibundle charts of $\mathbb E \to \mathbb X$;

* $\mathbf s_i$ is an $m_i$-multisection of $(E_i\to V_i,\Gamma_i)$;

* for any embedding

$$\widetilde\psi_{ji}:(E_i\to V_i,\Gamma_i,\widetilde\phi_i)\hookrightarrow (E_j\to V_j,\Gamma_j,\widetilde\phi_j)$$

of orbibundle charts that lies over the embedding

$$\psi_{ji}:(V_i,\Gamma_i,\phi_i) \hookrightarrow (V_j,\Gamma_j,\phi_j)$$

of base orbifold charts, the $m_j$-multisection $\mathbf s'_{i}:=({\widetilde\psi_{ji}}^{-1})^{m_j} \circ \mathbf s_j\circ \psi_{ji}$ and the $m_i$-multisection $\mathbf s_i$ of $(E_i\to V_i,\Gamma_i)$ are equivalent.

The multisection ${\bf s}$ is liftable if each ${\bf s_i}$ is liftable.

In Definition <ref>,

$$\widetilde\psi_{ji}^{m_j}:\op{Sym}^{m_j}(E_i)\to \op{Sym}^{m_j}(E_j),$$

$$({\widetilde\psi_{ji}}^{-1})^{m_j}:\op{Sym}^{m_j}(E_j)|_{{\widetilde\psi_{ji}}^{m_j}(\op{Sym}^{m_j}(E_i))}\to \op{Sym}^{m_j}(E_i)$$

are the obvious maps induced by

$$\widetilde\psi_{ji}:E_i\to E_j \quad \mbox{and} \quad \widetilde \psi_{ji}^{-1}:E_j|_{\widetilde \psi_{ji}(E_i)}\to E_i.$$

A multisection $\mathbf s$ of an orbibundle $\mathbb E \to \mathbb X$ has bounded degree if, for any fixed $(E_j\to V_j, \Gamma_j, \widetilde \phi_j)$, the set of $m_i$ for which there exists an embedding $(E_i\to V_i, \Gamma_i, \widetilde \phi_i)\hookrightarrow (E_j\to V_j, \Gamma_j, \widetilde \phi_j)$ is bounded.

In this paper, we assume that our multisections have bounded degree and are liftable.

Let $\mathbf s$ and $\mathbf t$ be two multisections of $\mathbb E\to \mathbb X$ such that $\mathbf s \pitchfork \mathbf t$.
On each orbibundle chart $(E_i\to V_i, \Gamma_i, \widetilde \phi_i)$ of $\mathbb E\to \mathbb X$,
we know $\mathbf s_i^{-1}(t_i)$ is a weighted branched immersed submanifold of $V_i$ that is invariant under the action of $\Gamma_i$,
and we define $\mathbf s^{-1}(\mathbf t)$ to be the weighted branched immersed suborbifold of $\mathbb X$ obtained by patching together $\{\mathbf s_i^{-1}(\mathbf t_i)\}_{i\in \mathcal I}$.

A sequence of multisections $\{\mathbf s^n\}_{n\in \N}$ of $\mathbb E\to \mathbb X$ converges to a multisection $\mathbf s$ of $\mathbb E \to \mathbb X$ in the $C^k$-topology (where $k$ may be $\infty$), if for each fixed orbibundle chart $(E_i\to V_i,\Gamma_i,\widetilde\phi_i)$ of $\mathbb E\to \mathbb X$,
there exist $\ell_i$-multisections $\{\mathbf s'^n_i\}_{n\in \N}$ and $\mathbf s'_i$ of $(E_i\to V_i,\Gamma_i,\widetilde\phi_i)$ such that
$\mathbf s'^n_i$ is equivalent to $\mathbf s^n_i$, $\mathbf s'_i$ is equivalent to $\mathbf s_i$, and $\mathbf s'^n_i$ converges  to $\mathbf s'^n_i$ in the $C^k$-topology.

The following is proved in <cit.>.

Let $\mathbf s,\mathbf t$ be two multisections of an orbibundle $\mathbb E \to \mathbb X=(X,\mathcal{O})$. If the base $X$ is compact, then there exists a sequence $\{\mathbf s^n\}_{n\in \N}$ of multisections of $\mathbb E \to \mathbb X$ such that $\{\mathbf s^n\}_{n\in\N}$ converges to $\mathbf s$ in the $C^\infty$-topology and $\mathbf s^n$ is transverse to  $\mathbf t$.

§ ALMOST COMPLEX STRUCTURES AND MODULI SPACES

Let $(M^{2n+1},\xi)$ be a contact manifold and $\alpha$ a nondegenerate contact form. Denote the Reeb vector field of $\alpha$ by $\mathbf R_\alpha$ and the set of closed orbits $\gamma$'s of $\mathbf R_\alpha$ with action $\mathcal A(\gamma)\leq L$ by $\mathcal P_\alpha^L$.

The discussion until the end of Section <ref> is valid for curves of any genus.

§.§ Almost complex structures

We will review some notions from [2].

A  contact form $\alpha$ is called $L$-simple[In [2] there is a notion called “$L$-supersimple", the difference between “$L$-simple" and “$L$-supersimple" is that “$L$-simple" allows the existence of elliptic orbits of action $\leq L$.]
if each simple orbit $\gamma\in \mathcal P_\alpha^L$ of $\mathbf R_\alpha$ is nondegenerate and around $\gamma$ there exists an embedding

$$\Phi_\gamma:\R/\mathcal{A}_\alpha(\gamma)\Z \times D_\delta \to M,$$

for some small $\delta>0,$ where

$$D_\delta=\Big\{(x_1,\dots,x_n,y_1,\dots,y_n)\in \R^{2n} ~|~\sum_i x_i^2+y_i^2\leq \delta^2 \Big\},$$
such that

$$\gamma=\Phi_\gamma(\R/\mathcal{A}_\alpha(\gamma)\Z \times\{0\})$$

$$\alpha=(c(\gamma)+Q)dt+\sum_{i=1}^n (x_idy_i-y_idx_i).$$

Here $c(\gamma)>0$ is a constant, $t$ is the coordinate for $\R/\mathcal{A}_\alpha(\gamma)\Z$, and $Q$ is a quadratic polynomial in $(x_1,\dots,x_n,y_1,\dots,y_n)$.

If we denote the Hamiltonian vector field of $H:=c(\gamma)+Q$ on $D_\delta$ with respect to the symplectic form $\sum_i dx_i\wedge dy_i$ by $X_H$,
then the vector field $\partial_t+X_H\in \ker d\alpha$. Hence there exists a positive function $g$ on $\Phi_\gamma(\R/\mathcal{A}_\alpha(\gamma)\Z \times D_\delta)$ such that

(g) $g \mathbf R_\alpha=\partial_t+X_H$.

Since $Q$ is quadratic, $X_H$ depends linearly on $(x_1,\dots,x_n,y_1,\dots,y_n)$, i.e.,

for some $X\in M_{2n\times 2n}(\R)$.

Given a contact form $\alpha$, an almost complex structure $J$ on $\R \times M$ is $\alpha$-tame if

* $J$ is independent of $s$, where $s$ is the $\R$-coordinate;

* $J(\partial_s)=g\mathbf R_\alpha$ for some positive function $g$ on $M$; and

* there exists a $2n$-plane field $\xi'$ on $M$ such that $J\xi'=\xi'$ and $d\alpha(v,Jv)>0$ for all nonzero $v\in \xi'$.

Given a Reeb orbit $\gamma$ of $\mathbf R_\alpha$, we write $\gamma^s$ for the simple orbit underlying $\gamma$ and $m(\gamma)$ for the multiplicity of $\gamma$ over $\gamma^s$. Choose a point $x_{\gamma^s}$ (a marker) on each $\gamma^s$.

Let $L>0$ and $\alpha$ be an $L$-simple contact form.

(1) An almost complex structure $J$ on $\R\times M$ is $L$-simple for $\alpha$ if $J$ is $\alpha$-tame and inside a sufficiently small neighborhood $\Phi_\gamma(\R/\mathcal{A}_\alpha(\gamma)\Z \times D_\delta)$ (as given in Definition <ref>) of each simple Reeb orbit $\gamma \in \mathcal P_\alpha^L$ we have

(i) $\xi'=\R\langle\frac{\partial}{\partial x_1},\cdots, \frac{\partial}{\partial x_n}, \frac{\partial}{\partial y_1},\cdots, \frac{\partial}{\partial y_n}\rangle$;

(ii) $J: \frac{\partial}{\partial x_i} \mapsto \frac{\partial}{\partial y_i}$ for $1\leq i \leq n$; and

(iii) the function $g$ satisfying $J(\partial_s)=g\mathbf R_\alpha$ is given by (g) above.

A pair $(\alpha,J)$ is said to be $L$-simple pair, if $\alpha$ is $L$-simple and $J$ is $L$-simple for $\alpha$.

(2) If $\gamma$ is simple, then the above coordinates

subject to the normalization

(N) $\Phi_\gamma(0,0)=x_{\gamma^s}$.

are called simple coordinates for $\gamma$.

(3) If $\gamma$ is not simple, then, by abuse of notation, the coordinates

on $\R/\mathcal{A}_\alpha(\gamma)\Z \times D$ are called simple coordinates for $\gamma$ where

$$\R/\mathcal{A}_\alpha(\gamma)\Z \times D \to \R/\mathcal{A}_\alpha(\gamma^s)\Z\times D$$

is the $m(\gamma)$-fold covering map.

Given an $L$-simple $\alpha$, we can construct an $L$-simple $J$ as follows: Let $\xi'\subset TM$ be a $2n$-plane field such that $\xi'=\R\langle\frac{\partial}{\partial x_1},\dots, \frac{\partial}{\partial x_n}, \frac{\partial}{\partial y_1},\dots, \frac{\partial}{\partial y_n}\rangle$ on a small neighborhood $V_\gamma$ of each $\gamma\in \mathcal P_\alpha^L$; $\xi'=\xi$ outside a slightly larger neighborhood $V'_\gamma$ of each $\gamma\in \mathcal P_\alpha^L$; and $\xi'$ interpolates between $\R\langle\frac{\partial}{\partial x_1},\dots, \frac{\partial}{\partial x_n}, \frac{\partial}{\partial y_1},\dots, \frac{\partial}{\partial y_n}\rangle$ and $\xi$ on the remaining annular regions.  Let $J$ be a complex structure of $\xi'$ such that $d\alpha(v,Jv)>0$ for all $0\neq v\in \xi'$ and $J(\frac{\partial}{\partial x_i})= \frac{\partial}{\partial y_i}$ for $1\leq i \leq n$ on $V_\gamma$. We then extend $J$ to $\mathbb R\times M$ by $J(\partial_s)=g\mathbf R_\alpha$, where $g$ is given by (g) on $V_\gamma$, $g=1$ outside $V'_\gamma$, and $g$ interpolates between then two in the remaining annular regions.

The main reason for using $L$-simple $J$ is the following: Let $u$ be a $J$-holomorphic curve in $\R \times M$ and suppose $u$ has an end which converges to $\R \times \gamma$ with coordinates $(s,t)$. Then, with respect to the supersimple coordinates, this end of $u$ admits a lift to $\R/\mathcal{A}_\alpha(\gamma)\Z \times D$ which can be written as $(s,t,\eta(s,t))$, where $\eta$ satisfies

$$\frac{\partial \eta }{\partial s}+j_0\frac{\partial \eta}{\partial t}+S\eta=0,$$

$j_0$ is the standard complex structure, and $S:=-j_0X$ is a $2n\times 2n$ constant symmetric matrix. In other words, the Cauchy-Riemann equations become linear near the ends for curves that are graphical over $\R\times \gamma$.

We are using the convention that when a matrix (e.g., $S$) acts on $\eta$ or its derivatives from the left, $\eta$ or its derivatives are viewed as column vectors.

More generally, let $(W,\alpha)$ be a $(2n+2)$-dimensional compact exact symplectic cobordism from $(M_+,\alpha_+)$ to $(M_-,\alpha_-)$, i.e., $\bdry W= M_+-M_-$, $\alpha$ is a $1$-form on $W$ such that $d\alpha$ is symplectic, and $\alpha_\pm=\alpha|_{M_\pm}$ is a contact form on $M_\pm$.  Let $(\widehat{W},\widehat\alpha)$ be the completion of $(W,\alpha)$, obtained by smoothly attaching the symplectization ends $[1,\infty)\times M_+$ and $(-\infty,-1]\times M_-$.  Let $s$ be the $[1,\infty)$- or $(-\infty,-1]$-coordinate at the positive and negative ends of $\widehat{W}$.

A contact form $\widehat \alpha$ on $\widehat W$ is said to be $(L_+,L_-)$-simple if it restricts to $L_\pm$-simple contact form $\alpha_\pm$ at the positive and negative ends.
Given an $(L_+,L_-)$-simple $\widehat \alpha$, an almost complex structure $J$ on $\widehat{W}$ said to be  $(L_+,L_-)$-simple for $\widehat \alpha$,
if it tames $\widehat{\alpha}$ and restricts to $J_\pm$ which is $L_\pm$-simple for $\alpha_\pm$ at the positive and negative ends.
A pair $(\widehat \alpha, J)$ is said to be $(L_+,L_-)$-simple, if $\widehat \alpha$ is $(L_+,L_-)$-simple and $J$ is $(L_+,L_-)$-simple for $\widehat \alpha$.

In this paper we will assume all almost complex structures $J$'s in symplectizations (resp. cobordisms) are $L$-simple (resp. $(L_+,L_-)$-simple)  for some appropriate $L$ (resp. $(L_+,L_-)$) and some appropriate form $\alpha$ (resp. $\widehat \alpha$).

Strictly speaking, it is not necessary to use simple $J$ in this paper, as most of our discussion carries over to the general case.  Simple almost complex structures allow for better control of the ends and also simplify gluing.

§.§ Riemann surfaces and holomorphic maps

Let $(F,j)$ be a closed connected Riemann surface.

For much of this paper we do not need to assume that the genus $g(F)$ of $F$ is zero.

A marked Riemann surface

is a quadruple which additionally consists of an ordered tuple $\mathbf{p}=\mathbf{p}_+ \sqcup \mathbf{p}_-$, $\mathbf{p}_\pm=(p_{\pm,1},\dots,p_{\pm,l_\pm})$, of marked points on $F$, where each point of $F$ is used at most once, and an ordered tuple $\mathbf{r} = \mathbf{r}_+ \sqcup \mathbf{r}_-$, $\mathbf{r}_\pm=(r_{\pm,1},\dots,r_{\pm,l_\pm})$ of asymptotic markers.
Here an asymptotic marker $r_{\pm,i}$ at a puncture $p_{\pm,i}$ is an element of $(T_{p_{\pm,i}} F-\{0\})/\R^+$.
We denote $\dot F=F-\mathbf{p}$.

Given two marked Riemann surfaces $\F=(F,j,\mathbf{p},\mathbf{r})$ and $\F'=(F',j',\mathbf{p}',\mathbf{r}')$,  we write $\F\sim_0\F'$ if there is a diffeomorphism $\phi:F\stackrel\sim \to F'$ satisfying $\phi_*(\F)=\F'$ (that is, $\phi_{*}j=j'$, $\phi(\mathbf{p}_\pm)=\mathbf{p}'_\pm,$ and $\phi_{*}(\mathbf{r}_\pm)=\mathbf{r}'_\pm$, where the latter two maps are maps of ordered tuples).

Let $(W,\alpha)$ be the exact symplectic cobordism from Section <ref>.

Given marked Riemann surfaces $\F=(F,j,\mathbf{p},\mathbf{r})$, $\F'=(F',j',\mathbf{p}',\mathbf{r}')$ and maps $u: \dot F\to \widehat{W}$, $u': \dot F'\to \widehat{W}$, we write $(\F,u)\sim (\F',u')$ if there is a diffeomorphism $\phi :  F\stackrel\sim \to F'$ satisfying $\phi_*(\F,u)=(\F',u')$ (that is, $\phi_{*}j=j'$, $\phi(\mathbf{p}_\pm)=\mathbf{p}'_\pm$ and $\phi_{*}(\mathbf{r}_\pm)=\mathbf{r}'_\pm$, again as ordered tuples, and $u' \circ \phi=u$).

The equivalence class of $(\F,u)$ under $\sim$ will be denoted by $[\F,u]$. If $\widehat{W}=\R\times M$, then the equivalence class of $(\F,u)$ under $\sim$ and $\R$-translations will be denoted by $\llbracket \F,u\rrbracket$.

Let $\boldsymbol\gamma_+=(\gamma_{+,1},\dots,\gamma_{+,l_+})$ and $\boldsymbol\gamma_-=(\gamma_{-,1},\dots,\gamma_{-,l_-})$ be ordered tuples of Reeb orbits for $\alpha_+$ and $\alpha_-$.  Then let $\mathcal{M}_J^{\op{ind}=k}(\dot F,\widehat{W};\boldsymbol\gamma_+;\boldsymbol\gamma_-)$ be the space of equivalence classes $[\F,u]$ of pairs, where $u$ is a $J$-holomorphic map $\dot F \to \widehat{W}$ of Fredholm index $\op{ind}(u)=k$ from $\boldsymbol\gamma_+$ to $\boldsymbol\gamma_-$ (i.e., $u$ is asymptotic to $\gamma_{+,i}$ at the positive end near $p_{+,i}$ and to $\gamma_{-,i}$ at the negative end near $p_{-,i}$) and $u$ “maps" the asymptotic markers $r_{\pm,i}$ to the markers $x_{\gamma^s_{\pm,i}}$.

If $\widehat{W}=\R\times M$, then $\mathcal{M}_J^{\op{ind}=k}(\dot F,\R\times M;\boldsymbol\gamma_+;\boldsymbol\gamma_-)/\R$ is the usual quotient of $\mathcal{M}_J^{\op{ind}=k}(\dot F,\R\times M;\boldsymbol\gamma_+;\boldsymbol\gamma_-)$ by $\R$-translations.

The $\alpha$-energy of a map $u$ from $\bs\gamma_+$ to $\bs\gamma_-$ is given by

where $\mathcal{A}_\alpha(\bs\gamma_\pm)=\sum_{i=1}^{l_\pm}\mathcal{A}_\alpha(\gamma_{\pm,i})$.

§.§ Sorting

The following definition is useful when keeping track of the combinatorics in the proof of $\bdry^2=0$ in Section <ref>.

Fix an ordering $\vartheta$ of $\mathcal P^{L}_\alpha$.  For example, if the action spectrum is discrete, we can order the orbits by increasing action.

An ordered tuple of Reeb orbits $\bs \gamma$ is sorted if

$$\bs \gamma=(\underbrace{\gamma_{1},\dots, \gamma_{1}}_{i_{1} \text{ copies }},\dots, \underbrace{\gamma_{k},\dots, \gamma_{k}}_{i_{k} \text{ copies }}),$$

where $\gamma_{i} \neq \gamma_{j}$ as long as $i\neq j$.  We also allow $k=0$, i.e., $\bs \gamma=\varnothing$.

If $\bs \gamma$ is sorted according to $\vartheta$, then $\bs \gamma$ is said to be $\vartheta$-sorted.

From now on we assume that all the positive (and likewise all the negative) ends of all the moduli spaces are $\vartheta$-sorted, unless stated otherwise.

§.§ (Symp) vs (Cob)

Consider the moduli space

$$\mathcal{M}=\mathcal{M}_J^{\op{ind}=k}(\dot F,\widehat W; \bs\gamma_+;\bs\gamma_-).$$

There will always be two cases to consider:

(Symp) $(\widehat W=\R\times M,J)$ is $\R$-invariant and is viewed as a symplectization.  This is the case when defining the differential $d$ and proving $d^2=0$. We usually take $\mathcal{M}/\R$.

(Cob) $(\widehat W,J)$ is viewed as a cobordism.  This is the case when defining chain maps.  We allow $(\widehat W,J)$ to be $\R$-invariant (e.g., when $(\widehat W,J)$ is the “identity cobordism”). In the case of a cobordism there is no quotienting by $\R$, even if there is an $\R$-action.

We will write $(\widehat W,J)\in \op{(Symp)}$ or $(\widehat W,J)\in\op{(Cob)}$ to indicate whether $(\widehat W,J)$ is viewed as a symplectization or a cobordism. There are two reasons for this distinction:

* We do not quotient by the $\R$-action for (Cob).

* Trivial cylinders are unperturbed (i.e., are more or less ignored) when considered in (Symp), but are perturbed (i.e., we construct an obstruction bundle $\E\to \V$ where the trivial cylinder is in $\V$) when considered in (Cob).

For the $\op{(Cob)}$ case, we also require $J$ to be

Let $J$ be an almost complex structure on $(\widehat W,\widehat \alpha)$ that is $(L_+,L_-)$-simple for $\widehat \alpha$.
We say $J$ is $(L_+,L_-)$-end-generic,
if for all $\bs\gamma_\pm$ with $\mathcal A_{\alpha_\pm}(\bs\gamma_\pm)\leq L_\pm$ and all $k\in \mathbb Z$,
there is no $J$-holomorphic curve $(\F, u)\in \mathcal M^{\op{ind}=k}_{J} (\dot F, \widehat W; \bs\gamma_+, \bs\gamma_-)$  that agrees with the trivial half cylinder near some puncture of $\dot F$ in the region $\widehat W_{s\geq T}$  or $ \widehat W_{s\leq -T}$, for some $T>1$.

We require $J$ to be $(L_+,L_-)$-end-generic, because with it we can add additional marked points to stabilize the domain of a $J$-holomorphic cylinder or plane in a more canonical way.

Given an $(L_+,L_-)$-simple pair $(\widehat \alpha, J)$, we can perturb $J$ to $J'$ such that $(\widehat \alpha, J')$ is still $(L_+,L_-)$-simple and $J'$ is $(L_+,L_-)$-end-generic.

For each simple $\gamma_+\in \mathcal P_{\alpha_+}^{L_+}$,
we first perturb $J$ in $[1,\infty) \times \gamma_+$.

Suppose $\bs\gamma_+$ is a tuple of Reeb orbits of $\alpha_+$ that has a component $\gamma_+$ and satisfies $\mathcal A_{\alpha_+}(\bs\gamma_+)\leq L_+$.
For each $\bs\gamma_-$, we consider $(\F, u)\in \mathcal M(\bs\gamma_+;\bs\gamma_-):=\sqcup_k \mathcal M^{\op{ind}=k}(\dot F, \widehat W;\bs\gamma_+;\bs\gamma_-)$. Suppose that near a puncture $p_+$ of $\dot F$, $u$ converges to $\gamma_+$.
Then near $p_+$, we have $u(s,t)=(s,t,\eta(s,t))$, where $\eta$ satisfies
$$\frac{\partial \eta}{\partial s}+j_0\frac{\partial \eta}{\partial t}+S\eta=0.$$

Define an asymptotic operator
$$A:W^{1,2}(S^1,\mathbb R^{2n})\to L^2(S^1,\mathbb R^{2n})$$
$$A:=-j_0\frac{\partial}{\partial t}-S.$$
Then $\eta$ satisfies
$$\frac{\partial \eta}{\partial s}=A \eta.$$

be the eigenfunctions of $A$ with respect to the eigenvalues
$$\cdots \theta_{-2},\theta_{-1},\theta_1,\theta_2, \dots$$

$$\cdots\leq \theta_{-2}<\theta_{-1}<0<\theta_1<\theta_2\leq \dots.$$

$$\eta=\sum_{i \leq -1}c_i e^{\theta_i s}g_i,$$
where the $c_i$ are constants.
There is a continuous evaluation map
$$ev_+:\mathcal M(\bs\gamma_+;\bs\gamma_-)\to \mathbb R^N$$
for any $N\in \mathbb N$, defined by
$$(\F,u)\mapsto (c_{-1},\dots,c_{-N}).$$

Note that $ev_+$ can be continuously extended to $\overline{\mathcal M}:=\bigsqcup \overline{\mathcal M}_{J}(\bs\gamma_+;\bs\gamma_-)$,
where the disjoint union is take over all tuples of Reeb orbits $\bs\gamma_\pm$ with $\mathcal A_{\alpha_\pm}(\bs\gamma_\pm)\leq L_\pm$ and $\bs\gamma_+$ having $\gamma_+$ as a component.

Since $\overline{\mathcal M}$ is compact, there exists an $N\gg 0$ such that the point $(0,\dots,0,1)\in \mathbb R^N$ does not lie in  $ ev_+(\overline{\mathcal M})$.
Now we can perturb $J$ in the region $[s_0,s_0+1]\times \gamma_+ \times D_\delta \subset \widehat W$ for some  $s_0\gg0$ and  small $\delta>0$ using the eigenfunction $g_{-N}$ as in the proof of Theorem 6.06 of [2] and show that there is no $J$-holomorphic curve that converges to $\gamma_+$ near some puncture $p_+$ and agrees with the trivial half cylinder over $\gamma_+$ near $p_+$.

We also need to deal with the case when $\bs\gamma_+$ has a component $\gamma'_+$ that is a $k$-fold cover of $\gamma_+$.
For this, we take the same perturbation to $J$ using $g_{-N}$, but we need to make $N$ even larger.
Now we explain how we pick such $N$.
As before, we can define an evaluation map $ev_+':\overline{\mathcal M}' \to \mathbb R^{N'}$ using $\gamma'_+$ in lieu of $\gamma_+$. Suppose $(\F, u)$ converges to $\gamma_+$ near puncture $p_+$, and $ev_+(\F, u)=(c_{-1},\dots,c_{-N})$.
Let $(\F',u')$ be a $k$-fold (branched) cover of $(\F, u)$,
and then
$$ev_+'(\F', u')=(\underbrace{c_{-1},0,\dots,0}_{k \text{ copies }},\dots,\underbrace{c_{-N},0,\dots,0}_{ k \text{ copies}}),$$

if we choose $N'=kN$.
By picking $N$ large, we can require
$$(\!\!\!\!\underbrace{0,\dots,0}_{(N-1)k \text{ copies}}\!\!\!\!\!,1,*,\dots,*)\notin ev_+'(\overline{\mathcal M}').$$

With such choice of $N$ one can follow Theorem 6.06 in [2] and show that there is no $J$-holomorphic curve that converges to $\gamma_+'$ near some puncture $p_+'$ and agrees with the half trivial cylinder over $\gamma_+'$ near $p_+'$.
We can pick a sufficiently large $N$, such that the above is true for all $k\in \mathbb N$ with $\mathcal A_{\alpha_+} (\gamma_+')\leq L_+$.

Finally, we carry out similar perturbation in the negative end of $\widehat W$.

In this paper, we always assume $J$ is $(L_+,L_-)$-end-generic for some appropriate $(L_+,L_-)$ when $(\widehat W, J)\in (\op{Cob})$.

§.§ Teichmüller space and mapping class group

In what follows we fix $(F,\mathbf{p})$ and hence $\dot F$.

Assume $\dot F$ has negative Euler characteristic.  Let $\op{Teich}(\dot F)$ be the Teichmüller space of $\dot F$.  Given $[j]\in \op{Teich}(\dot F)$, where $[j]$ is viewed as an equivalence class of complex structures on $\dot F$, let $\mathcal{U}_{[j]}\subset \op{Teich}(\dot F)$ be an open neighborhood of $[j]$. Given an open set $\mathcal{U}\subset \op{Teich}(\dot F)$, let $\widetilde{\mathcal{U}}$ be a Teichmüller slice over $\mathcal{U}$, i.e., a smooth choice of complex structure on $\dot F$ in each equivalence class $x\in \mathcal{U}$.

We make the following definition which is slightly nonstandard but adapted to our situation:

Let $\text{Mod}(F,{\bf p})$ be the quotient of the group $\op{Diff}(F,{\bf p})$ of diffeomorphisms of $F$ which fix ${\bf p}_\pm$ pointwise by the subgroup $\op{Diff}_0(F,{\bf p})$, which is the identity component of $\op{Diff}(F,{\bf p})$.

It is well-known that $\op{Mod}(F,{\bf p})$ acts properly discontinuously on $\op{Teich}(\dot F)$ and hence its quotient is an orbifold, the moduli space $\mathcal{M}(F,{\bf p})$ of complex structures on $F$ with positive marked points ${\bf p}_+$ and negative marked points ${\bf p}_-$. Let

$$\text{Mod}(F,{\bf p})_{[j]}=\{ [g]\in \text{Mod}(F,{\bf p})~|~ [g]([j])=[j]\}$$

be the stabilizer of $[j]$.  If we choose a representative $j\in[j]$, then it is well-known that the group $\op{Aut}(F,{\bf p}, j)$ of automorphisms of $(F,j)$ which take ${\bf p}_\pm$ to ${\bf p}_\pm$ pointwise satisfies

$$\op{Aut}(F,{\bf p}, j)\simeq \op{Mod}(F,{\bf p})_{[j]}.$$

An open set $\mathcal{U}\subset \op{Teich}(\dot F)$ is good if $\mathcal{U}=\mathcal{U}_{[j]}$ for some $[j]$ and, for any $[g]\in \text{Mod}(F,{\bf p})$, one of the following holds:

* $[g](\mathcal{U})\cap \mathcal{U}=\emptyset $;

* $[g]\in \text{Mod}(F,{\bf p})_{[j]}$ and $[g](\mathcal{U})=\mathcal{U}.$

The orbifold charts for $\mathcal{M}(F,{\bf p})$ are

$$\mathcal{U}_{[j]}\to \mathcal{U}_{[j]}/\op{Mod}(F,{\bf p})_{[j]},$$

where $\mathcal{U}_{[j]}$ is a sufficiently small good open set about $[j]$.

Also given $(\F,u)=((F,j,{\bf p},{\bf r}),u)$, let $\text{Mod}(F,{\bf p})_{[\F,u]}$ be the subgroup of $\text{Mod}(F,{\bf p})$ defined by

$$\text{Mod}(F,{\bf p})_{[\F,u]}=\{[g]\in \text{Mod}(F,{\bf p})~|~  g_*(\F,u)=(\F,u) \mbox{ for some $g\in[g]$}\}.$$

§ FREDHOLM THEORY

In this section we fix $(F,{\bf p})$ and hence $\dot F$.

§.§ Canonical cylindrical coordinates

Let $(\F,u)$ be a pair where $u:\dot F\to  \widehat{W}$ is a sufficiently differentiable map which is asymptotic to $\gamma_{+,i}$ at the positive end near $p_{+,i}$ and to $\gamma_{-,i}$ at the negative end near $p_{-,i}$. If $u$ can be written as a graph $(s,t)\mapsto (s,t,\eta_{\pm,i}(s,t))$ on $\R\times (\R/\mathcal{A}(\gamma_{\pm,i}))\times D$ near the end corresponding to the puncture $p_{\pm,i}\in \mathbf{p}$, then $(s,t)$ is said to be the canonical cylindrical coordinates on $\dot F$ near $p_{\pm,i}$ with respect to $u$.

By the supersimplicity of $J$, if $[\F,u]\in \mathcal{M}_J^{\op{ind}=k}(\dot F,\widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)$, then the canonical cylindrical coordinates for a representative $(\F,u)$ near the punctures are holomorphic coordinates on $\dot F$.  On the other hand, $(s,t)$ is not necessarily holomorphic if $u$ is not $J$-holomorphic.

If $\llbracket\F,u\rrbracket\in \mathcal{M}_J^{\op{ind}=k}(\dot F,\R\times M;\boldsymbol\gamma_+,\boldsymbol\gamma_-)/\R$, the coordinates $(s,t)$ for $(\F,u)$ are canonical up to translations in the $s$-direction.

§.§ Fredholm setup

Fix an $L$-supersimple $J$ on $\widehat{W}$ in the class $C^\infty$.

Recall the coordinate $s$ at the ends of $\widehat{W}$. Fix a Riemannian metric $g_0$ on $\widehat{W}$ which is $s$-invariant at the ends. For simplicity we assume that $g_0$ restricts to the standard flat metric

$$g_0=ds^2+dt^2+\sum_idx_i^2+\sum_i dy_i^2$$

on the $L$-supersimple coordinate charts (and hence $g_0=g_0^L$, i.e., depends on $L$). Let $\exp: T\widehat W \to \widehat{W}$ be the exponential map with respect to $g_0$, $\epsilon>0$ be a constant smaller than the injectivity radius of $g_0$, and

$$D_\epsilon\subset T\widehat W= \{(w,\xi)~|~w\in W, |\xi|_{g_0}< \epsilon\}$$

be the $\epsilon$-disk bundle of $T\widehat{W}$.

For the rest of the paper we assume that $\bs\gamma_+$ satisfies $\mathcal{A}_\alpha(\bs\gamma_+)<L$, unless indicated otherwise.

Let $\delta>0$ be a small positive number.  We define the “weighted Sobolev space” $W_\delta ^{k+1,p}( \dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)$, following Dragnev <cit.>:

Let $\mathcal{C}$ be the space of smooth maps $u:\dot F\to \widehat{W}$ that agree with holomorphic maps parametrizing trivial holomorphic half-cylinders $[1,+\infty)\times \gamma_{+,i}$ near $p_{+,i}$ and to $(-\infty,-1]\times\gamma_{-,i}$ near $p_{-,i}$.
Then we define the weighted Sobolev space of maps $\dot F\to\widehat{W}$ by

$$W_\delta ^{k+1,p}( \dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-):=\{\exp(u,\xi)~|~u\in \mathcal{C},\xi\in W_\delta^{k+1,p}(\dot F, u^*D_\epsilon)\},$$

where $W_\delta^{k+1,p}(\dot F, u^*D_\epsilon)$ is the usual weighted Sobolev space of sections of $u^*D_\epsilon$, defined using $g_0$.

Here in the definition of $W_\delta^{k+1,p}(\dot F, u^*D_\epsilon)$ we are using a smooth weight function $f_\delta:\widehat{W}\to \R^+$ which agrees with $e^{\delta |s|}$ at the ends of $\widehat{W}$. Note that the Sobolev norm depends on $g_0$ and $f_\delta$, but the topology on this Sobolev space does not depend on the particular choices of $g_0$ and $f_\delta$.

[Remark on values of $k$ and $p$]
By the Sobolev embedding theorem, if $k>r+{m\over p}$, then $W^{k,p}(\Omega)\subset C^r(\Omega)$ for any compact domain $\Omega$ of $\R^m$ with smooth boundary; see Remark <ref> for obtaining the Sobolev embedding theorem for $\dot F$.  In our case $m=\dim \dot F= 2$ and we take $r\geq 1$, since we want at least one continuous derivative (this will be used for defining $s_{\pm,i}$ in Step 1B in the proof of Theorem <ref>). Hence from now on we assume that  $k> r+1$ and $p= 2$.

The following is well-known:

We can use the Sobolev embedding theorem for compact domains to show that there exists a global constant $C>0$ such that for all

$$\xi\in W_\delta^{k+1,p}(\dot F, u^*D_\epsilon)$$

we have

\begin{equation}
|f_\delta\xi|_{C^r}\leq C\|f_\delta\xi\|_{W^{k+1,p}}=C \| \xi\|_{W^{k+1,p}_\delta}.
\end{equation}

Suppose for simplicity that $\dot F$ has only one end and $\dot F$ is written as the union $\Omega_0\cup\Omega_1\cup\Omega_2\dots$ of compact domains, where each $\Omega_i\cap \Omega_{i+1}$ is a circle, $\Omega_0$ is the “thick part”, and $\Omega_i$, $i>0$, is biholomorphic to the standard $S^1\times[0,1]$.  Then we apply the Sobolev embedding theorem for each $\Omega_i$, noting that

$$\| \xi\|_{W^{k+1,p}_\delta(\Omega_i)}\leq \| \xi\|_{W^{k+1,p}_\delta(\dot F)}$$

and that the Sobolev spaces $W^{k+1,p}_\delta(\Omega_i)$ and $W^{k+1,p}_\delta(\Omega_{i+1})$ can be identified for $i\gg 0$ because $u^*D_\epsilon$ is asymptotically cylindrical at the ends of $\dot F$.  Hence the same constant $C>0$ can be used when we apply the Sobolev embedding theorem to all $\Omega_i$.

Let $W_\delta^{k,p}(\dot F,\wedge^{0,1}u^{*}T\widehat{W})$ be the usual weighted Sobolev space of sections of

$$\wedge^{0,1}u^*T\widehat{W}=\wedge^{0,1}\dot F \otimes_J u^{*}T\widehat W\to\dot F$$

in the class $(k,p)$ with weight $f_\delta$.

In the functional analysis setup we use $W^{k+1,p}_\delta$ and $W^{k,p}_\delta$ spaces for large $k$ to extract a collection of finite-dimensional orbibundles $\E\to \V$. Once this is done we throw away the Sobolev space setup and work in the category of finite-dimensional orbifolds.  (This is analogous to the [11] and [12] setup and differs from that of [18].)

§.§.§ $\chi(\dot F)<0$

We first treat the case where $\dot F$ has negative Euler characteristic.
Given a Teichmüller slice $\widetilde{\mathcal{U}}$ of $\mathcal{U}$, define

\begin{align*}
\mathcal{B}_{\widetilde{\mathcal{U}}}= &\mathcal{B}_{\widetilde{\mathcal{U}}}(\dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)\\
=&\left\{ (\F,u)=((F,j,\mathbf{p},\mathbf{r}),u) \left|
\begin{array}{cc}
j\in \widetilde{\mathcal{U}},u\in W_\delta ^{k+1,p}( \dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-), \\  u(r_{\pm, i})=x_{\gamma_{\pm,i}^s}
\end{array}
\right.\right\},
\end{align*}

where $(F,\mathbf{p})$ are fixed.
We define the bundle

\begin{equation}
\pi_{\widetilde{\mathcal{U}}}:\mathcal{E}_{\widetilde{\mathcal{U}}}=\mathcal{E}_{\widetilde{\mathcal{U}}}(\dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)\to \mathcal{B}_{\widetilde{\mathcal{U}}},
\end{equation}

whose fiber over $(\F,u)$ is

$$(\mathcal{E}_{\widetilde{\mathcal{U}}})_{(\F,u)} = W_\delta^{k,p}(\dot F,\wedge^{0,1}u^{*}T\widehat{W}).$$

Then $\overline{\partial}_J^{\widetilde{\mathcal{U}}}$ is a section of $\mathcal{E}_{\widetilde{\mathcal{U}}}$ defined by

$$\overline{\partial}_J^{\widetilde{\mathcal{U}}} (\F, u)=(\F,u,\tfrac{1}{2}(du+J(u)du\circ j)).$$

Also let

\begin{equation} \label{eqn: proj to U}
\Pi_{\widetilde{\mathcal{U}}}: \mathcal{B}_{\widetilde{\mathcal{U}}} \to \widetilde{\mathcal{U}}, \quad (\F,u)=((F,j,{\bf p},{\bf r}),u)\mapsto j
\end{equation}

be the projection to $\widetilde{\mathcal{U}}$.
When the Teichmüller slice $\widetilde{\mathcal{U}}$ is understood, it will often be omitted from the notation, e.g., $\overline{\partial}_J$, $\mathcal{B}$, $\mathcal{E}$.

The following is immediate:

If $\mathcal{U}=\mathcal{U}_{[j]}$ is a good open set, then

$$\mathcal{M}_J(\dot F,\widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)\cap (\mathcal{B}_{\widetilde{\mathcal{U}}}/\sim ) =(\overline{\partial}_J^{\widetilde{\mathcal{U}}})^{-1}(0)/\op{Mod}(F,{\bf p})_{[j]}$$

as topological spaces (but not necessarily as orbifolds).

Let $\nabla$ be the Levi-Civita connection on $T\widehat W$ with respect to the metric $g_0$.  Let $L_{(\F,u)}$ be the differential

$$(\overline{\partial}_J)_*: T_{(\F,u)} \mathcal{B}\to T_{(\F,u,\overline\bdry_J u)}\mathcal{E}$$

postcomposed with the projection to $W_\delta^{k,p}(\dot F,\wedge^{0,1}u^{*}T\widehat{W}).$  Then $L_{(\F,u)}$ can be written as:

\begin{equation} \label{eqn: linearization}
L_{(\F,u)}: T\widetilde{\mathcal{U}}(j)\oplus\R^{2(l_++l_-)}\oplus W^{k+1,p}_\delta (\dot F,u^{*}T\widehat{W}) \to W^{k,p}_\delta (\dot F,\wedge ^{0,1} u^{*}T\widehat{W}),
\end{equation}

$$(\mathfrak{j},\mathfrak{a},\xi)\mapsto Y(\mathfrak{j})+D_u(\mathfrak{a}+\xi),$$

where $Y(\mathfrak{j})$ is induced from the variation of $j$ in the direction on $\mathfrak{j}$, $D_u$ is the usual linearized $\overline\bdry_J$-operator (which depends on the choice of $\nabla$ if $u$ is not $J$-holomorphic), and $\mathfrak{a}$ is obtained by damping out $\bdry_s$ and $\bdry_t$ at each end (away from $\pm \infty$).
Here $l_++l_-$ is the total number of punctures of $\dot F$.

§.§.§ $\chi(\dot F)=0$ or $1$

We now consider the case where $\chi(\dot F)=0$ or $1$.

Consider $(\F,u)$ where $u:(\dot F,j)\to (\widehat{W},J)$ is a finite energy holomorphic map. We assume that $u$ is not a trivial cylinder $\R\times \gamma$ over a Reeb orbit $\gamma$. (In our approach, we ignore trivial cylinders, since they will automatically be placed as “necks” during gluing.  The only case which requires further analysis is when $\widehat{W}=\R\times M$, viewed as a cobordism with an $\R$-invariant almost complex structure.) By Euler characteristic reasons, $u$ is also not a branched cover of a trivial cylinder and $E_\alpha(u)>0$.

Consider the positive end asymptotic to $\gamma\in \bs\gamma_+$ (actually there is only one positive end since $\dot F$ is a cylinder or a plane). Then on this end $u$ is graphical, i.e., can be written in canonical cylindrical coordinates on $\dot F$ and supersimple coordinates on the target as

after lifting to the $m(\gamma)$-fold cover $\R\times (\R/\mathcal{A}_\alpha(\gamma)\Z)\times D$. We assume that the ray $\{s\in [R,\infty), t=0\}\subset \dot F$ corresponds to the asymptotic marker $r_{+}$ which “maps” to $\gamma$.

For each representative of $(\F_0,u_0)$ of each $[\F_0,u_0]\in \mathcal{M}_J(\dot F,\widehat{W};\bs\gamma_+;\bs\gamma_-)$, we consider a small neighborhood $\mathcal{B}'_{(\F_0,u_0)}$ of $(\F_0,u_0)$ inside

\begin{align*}
\mathcal{B}'(\dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)=&\left\{ (\F,u)=((F,\mathbf{p},\mathbf{r}),u) \left|
\begin{array}{cc}
u\in W_\delta ^{k+1,p}( \dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-), \\  u(r_{\pm, i})=x_{\gamma_{\pm,i}^s}
\end{array}
\right.\right\}
\end{align*}

and define a map

$$I: \mathcal B'_{(\mathcal F_0, u_0)} \times [T',+\infty) \to \mathbb R$$

$$ I((\mathcal F, u),s)=\int_{s}^{+\infty} \int_0^{\mathcal A_\alpha(\gamma)} |\eta(s',t')|^2 dt'ds',$$

where $T'$ is a constant such that we can write $u(s,t)=(s,t,\eta(s,t))$ for any $(\mathcal F, u)\in \mathcal B'_{(\mathcal F_0, u_0)}$ and $s\geq T'$.
Clearly, $I$ is independent of $(\mathcal F_0, u_0)$, and $I((\mathcal F, u),\cdot)$ is strictly increasing if $u$ is not a (branched cover of) a trivial cylinder.

$I$ is a $C^1$-map.

Lemma <ref> will be proved in the Appendix.

In the $\op{(Symp)}$ case, suppose $\bs\gamma_+\neq \bs\gamma_-$. There exists $\varepsilon'>0$ small such that for each representative of $(\F,u)$ of each $[\F,u]\in \mathcal{M}_J(\dot F,\widehat{W};\bs\gamma_+;\bs\gamma_-)$ there exists a value

\begin{align*}
s'_+(\F,u) :=  \max\left\{ {s\in \mathbb R}\left|
%T'\geq 1, u(s,t)=(s,t,\eta(s,t)), \text{ for } s\geq T',\\
\int_s^{+\infty}\int_0^{\mathcal A_\alpha(\gamma)} |\eta(s',t')|^2 dt'ds'=\varepsilon'.
\right.\right\}.
\end{align*}

Moreover, if $u_T$ is $u$ translated up by $s=T$ units, then

$$s'_+(\F,u_T)= s'_+(\F,u)+T.$$

The same holds for $(\F',u')$ which is $C^1$-close to some representative $(\F,u)$ of $\mathcal{M}_J(\dot F,\widehat{W};\bs\gamma_+;\bs\gamma_-)$.

Arguing by contradiction, suppose there exist sequences $(\F_i,u_i)$ and $\varepsilon'_i\to 0$, $i\in \N$, such that:

* $(\F_i,u_i)$ represents $[\F_i,u_i]\in\mathcal{M}_J(\dot F,\widehat{W};\bs\gamma_+;\bs\gamma_-)$; and

* $s'_+(\F_i,u_i)$ does not exist for $\varepsilon_i'>0$.

Then one can verify that $(\F_i,u_i)$ limits to a possible branched cover of a trivial cylinder over a Reeb orbit.  In the (Symp) case a finite energy $J$-holomorphic map that is close to a possible branched cover of a trivial cylinder must be a possible branched cover of a trivial cylinder. This is impossible, because that $\bs\gamma_+\neq \bs\gamma_-$ and $\dot F$ is connected imply that $\mathcal{M}_J(\dot F,\widehat{W};\bs\gamma_+;\bs\gamma_-)$ contains no possible branched cover of a trivial cylinder.  The second and third assertions are immediate.

In the $\op{(Cob)}$ case, there exist a small compactly supported perturbation $J'$ of $J$ and $\varepsilon'>0$ small such that for each representative of $(\F,u)$ of each $[\F,u]\in \mathcal{M}_{J'}(\dot F,\widehat{W};\bs\gamma_+;\bs\gamma_-)$ there exists a value

$$s'_+(\F,u):= \max\Big\{ {s\in [1,+\infty)}~\Big|~\int_s^{+\infty}\hspace{-0.3cm}\int_0^{\mathcal A_\alpha(\gamma)} |\eta(s',t')|^2 dt'ds'=\varepsilon'\Big\}.$$

The same holds for $(\F',u')$ which is $C^1$-close to some representative $(\F,u)$ of $\mathcal{M}_{J'}(\dot F,\widehat{W};\bs\gamma_+;\bs\gamma_-)$.

We argue by contradiction as in Lemma <ref>. We only need to ensure that there are no holomorphic cylinders of $(\widehat{W},J')$ that agree with trivial half-cylinders at the positive end of $\widehat{W}$. This is done by taking an end-generic perturbation $J'$ of $J$ given in Lemma <ref>.

From now on we assume that in the (Cob) case $J$ is sufficiently generic so that the conclusion of Lemma <ref> holds.

We now define the unordered set ${\bf q}$ of punctures:

(*) Identify $u^{-1}(\{s=s'_+(\F,u)\})\simeq \R/\mathcal{A}_\alpha(\gamma)\Z$ by projecting to the $t$-coordinate, where we are still using supersimple coordinates for $\gamma$. Pick the point $t=0\in \R/\mathcal{A}_\alpha(\gamma)\Z$ corresponding to the asymptotic marker and then define

$${\bf q}=\{(s,t)=(s'_+(\F,u),\tfrac{k}{2m(\gamma)}\mathcal A_\alpha(\gamma))~|~k=1,\dots,2m(\gamma)\}.$$

Let us write $\ddot F=\dot F-{\bf q}$; since the cardinality of $\bf q$ is $2m(\gamma)$, we have $\chi(\ddot F)<0$.

Change of notation. From now on, we write

$$(\F,u)=((F,j,{\bf p},{\bf q},{\bf r}),u),$$

with the understanding that ${\bf q}=\varnothing$ when $\chi(\dot F)< 0$, ${\bf q}$ is an unordered set, ${\bf p}$ and ${\bf q}$ are disjoint, and ${\bf q}$ are removable punctures.  We modify Definitions <ref> and <ref> so that the equivalence relations

$$\F=(F,j,{\bf p},{\bf q},{\bf r})\sim_0 \F'=(F,j',{\bf p},{\bf q}',{\bf r}')$$

and $(\F,u)\sim (\F',u')$ require $\phi({\bf q})={\bf q}'$ as ordered sets in addition. Also we modify the definitions of $\op{Mod}(F,{\bf p})$ and $\op{Aut}(F,{\bf p},{\bf q},j)$ to $\op{Mod}(F,{\bf p},{\bf q})$ and $\op{Aut}(F,{\bf p},{\bf q},j)$ so that the diffeomorphism of $F$ takes ${\bf q}$ to ${\bf q}$ setwise in addition.

The above uniform choice helps stabilize the automorphism group of $(\F,u)$. The following is a slight variation of the definition that was given for $\chi(\dot F)<0$:

\begin{align*}
\mathcal{B}= &\mathcal{B}(\dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)\\
=&\left\{ (\F,u)=((F,j,\mathbf{p},{\bf q}, \mathbf{r}),u) \left|
\begin{array}{cc}
j\in \widetilde{\mathcal{U}}, u\in W_\delta ^{k+1,p}( \dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-), u(r_{\pm, i})=x_{\gamma_{\pm,i}^s}, \\
u \mbox{ is $\varepsilon''$-close to $u'$ holomorphic in $W^{k+1,p}_\delta( \dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)$}\\
{\bf q} \mbox{ satisfies (*)}
\end{array}
\right.\right\},
\end{align*}

where $(F,j,{\bf p},{\bf q})$ and $\varepsilon''>0$ are fixed, and $\widetilde{\mathcal U}$ is a Teichmüller slice of $\mathcal U\subset \op{Teich}(\dot F)$ (not $\op{Teich}(\ddot F)$).

The fibers $\mathcal{E}_{(\F,u)}$ of the bundle $\pi: \mathcal{E}\to \mathcal{B}$ are defined by

$$\mathcal{E}_{(\F,u)}=W^{k,p}_\delta(\dot F, \wedge^{0,1}u^* T\widehat{W})$$

as before and $\overline\bdry_J$ is the same as before.  The linearization $L_{(\F,u)}$ is the same as that of Equation (<ref>) with the term $T\widetilde{\mathcal{U}}(j)$ removed (or viewed as the zero vector space).

§.§ Description of $D_u$

By McDuff-Salamon <cit.>,

$$D_u: W^{k,p}_{\delta} (\dot F, u^* T\widehat W)\to W^{k-1,p}_{\delta} (\dot F, \wedge^{0,1} u^* T\widehat W)$$

is given by

\begin{equation} \label{Du}
D_u \xi=\frac{1}{2}(\nabla \xi +J(u)\nabla \xi \circ j) -\frac{1}{2}J(u)(\nabla_\xi J)(u)\partial_J(u).
\end{equation}

Here, by abuse of notation, we do not distinguish between sections of $u^*T\widehat W$ and sections of $T\widehat W$ along $u$.

Suppose $(s,t)\in \R^\pm\times \R/\mathcal{A}_\alpha(\gamma)\Z $ is a holomorphic coordinate around a puncture $p\in {\bf p}_\pm$ of $\dot F$, $p$ corresponds to the Reeb orbit $\gamma$, and $u(s,t)=(s,t,\eta (s,t))$, written in supersimple coordinates for $\gamma$, is $J$-holomorphic. Then, near this puncture,

\begin{equation} \label{Dulocal}
D_u \xi=\frac{1}{2} (\nabla_s \xi +J(u)\nabla_t \xi + \widetilde S^\gamma \xi)\otimes (ds -i dt),
\end{equation}

0 & -1 & 0\\
1 & 0 & 0\\
X\eta & -j_{0}X\eta & j_{0}
\end{array}\right) \quad \text{and  }\quad \widetilde S^\gamma=\left(\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & -j_0X
\end{array}\right).$$

Note that if $u$ is $J$-holomorphic, then $D_u u_s=D_u u_t=0$.  If, instead of using the basis

$$\{\partial_s, \partial_t, \partial_{x_1},\dots,\partial_{x_n},\partial_{y_1},\dots,\partial_{y_n}\},$$

we use the basis

$$\{ u_s, u_t, \partial_{x_1},\dots,\partial_{x_n},\partial_{y_1},\dots,\partial_{y_n} \},$$

then Equation (<ref>) becomes

\begin{equation} \label{Dusimple}
D_u \xi=\frac{1}{2}(\partial_s \xi +J_0 \partial_t \xi+\widetilde S^\gamma\xi)\otimes (ds-idt),
\end{equation}

0 & -1 & 0\\
1 & 0 & 0\\
0 & 0 & j_{0}
\end{array}\right)$$

is the standard complex structure on $\R^{2n+2}$.

If $u$ is immersed, we denote by $N$ the normal bundle of $u$ such that

$$N=\R \langle\partial_{x_1},\dots,\partial_{x_{2n}},\partial_{y_1},\dots,\partial_{y_{2n}}\rangle$$

near the ends.  We define

$$D^N_u:W_{\delta}^{k,p}(\dot F, u^*N)\to W_{\delta}^{k-1,p}(\dot F, \wedge^{0,1}u^*N)$$

by projecting $D_u$ along the splitting $u^*T\widehat W=u^*N\oplus T\dot F$.

§.§ Asymptotic operator

Let $\gamma$ be a Reeb orbit of $\boldsymbol{R}_\alpha$ and $S^1=\R/\mathcal{A}_\alpha(\gamma)\Z$. Let

$$\widetilde A^\gamma=-J_0\frac{\partial}{\partial t}-\widetilde S^\gamma:W^{1,2}(S^1,\R^{2n+2})\to L^{2}(S^1,\R^{2n+2})$$

be the asymptotic operator for $\gamma$. Then $\widetilde A^{\gamma}$ is self-adjoint. Let

$$\cdots \leq \lambda_{-2}^\gamma < \lambda_{-1}^\gamma=0=\lambda_1^\gamma < \lambda_2^\gamma\leq \cdots$$
be eigenvalues of $\widetilde A^{\gamma}$ and

be the associated complete set of orthonormal eigenfunctions. Here the self-adjointness and orthonormality are defined with respect to the standard inner product in $\R^{2n+2}$.

We will write $\widetilde A$ and $\widetilde S$ for $\widetilde A^\gamma$ and $\widetilde S^\gamma$ if $\gamma$ is understood; similarly we may suppress the superscript $\gamma$ in $\lambda_i^\gamma, f_i^\gamma,\dots$ below, when they are understood.

$\widetilde A$ is an “enlargement” of the operator $A: W^{1,2}(S^1,\R^{2n})\to L^{2}(S^1,\R^{2n})$ in the proof of Lemma <ref> which takes into account only the transverse directions (i.e., the $\bdry_{x_i}$- and $\bdry_{y_i}$-directions).

Summarizing the above discussion, we have:

Suppose $(s,t)\in \R^\pm\times \R/\mathcal{A}_\alpha(\gamma)\Z $ is a holomorphic coordinate around a puncture $p\in {\bf p}_\pm$ of $\dot F$, $p$ corresponds to the Reeb orbit $\gamma$, and near $p$, $u$ admits a lift to $\R\times \R/\mathcal{A}_\alpha(\gamma)\Z\times D$ which can be written as $(s,t)\mapsto (s,t,\eta (s,t))$ with respect to supersimple coordinates for $\gamma$ and is $J$-holomorphic. Then, near $p$, we have

$$D_u \xi =\frac{1}{2}(\partial_s \xi -\widetilde A \xi )\otimes (ds-idt),$$

with respect to the basis

$$\frak{B}:=\{ u_s, u_t, \partial_{x_1},\dots,\partial_{x_n},\partial_{y_1},\dots,\partial_{y_n} \}.$$

(Here the tensor product $\otimes$ is taken to be $\C$-linear.)

Hence, for any $\xi \in \ker D_u$, near $p\in {\bf p}_+$ we have

$$\xi(s,t)=\sum_{\lambda_i<0} c_{+,i} e^{\lambda_i s} f_i(t),$$

and near $p\in {\bf p}_-$
we have

$$\xi(s,t)=\sum_{\lambda_i>0} c_{-,i} e^{\lambda_i s} f_i(t),$$

where the $c_{\pm,i}$'s are constant.

§.§ The adjoint operator

$$D_u^*: W^{k,p}_\delta (\dot F,\wedge ^{0,1} u^{*}T\widehat{W})\to W^{k-1,p}_\delta (\dot F,u^{*}T\widehat{W})$$

be the adjoint operator of $D_u$ defined by

\begin{equation} \label{Du*}
\int_{\dot F} \langle \zeta, D_u \xi \rangle_1 dvol_{\dot F} =\int_{\dot F} \langle D_u^* \zeta, \xi \rangle_0 dvol_{\dot F} ,
\end{equation}

where $\langle, \rangle_1$ is an inner product on $W_\delta^{k,p}(\dot F,\wedge^{0,1}u^{*}T\widehat{W})$, $\langle, \rangle_0$ is an inner product on $ W^{k+1,p}_\delta (\dot F,u^{*}T\widehat{W})$, and $dvol_{\dot F}$ is defined using a metric $g_{\dot F}$ on $\dot F$. We additionally assume that, near $p\in {\bf p}_\pm$:

(i) $g_{\dot F}$ restricts to $ds^2+dt^2$;

(ii)$\langle, \rangle_0$ is defined by requiring the basis ${\frak B}$ to be an orthonormal basis; and

(iii) $\langle, \rangle_1$ is induced by $\langle, \rangle_0$ and $g_{\dot F}$.

Recall that the adjoint operator $D_u^*$ involve some choices.  We are not using the Riemannian metric $g_0$ in the definition of $D_u^*$.

The following lemma is almost evident:

With the assumptions of Lemma <ref>, if $\zeta\in W^{k,p}_\delta(\dot F,\wedge^{0,1}u^*T\widehat{W})$ and $\zeta=\zeta^0\otimes (ds-idt)$ near $p\in {\bf p}_\pm$, then we have

$$D_u^* (\zeta^0\otimes (ds-idt)) =\frac{1}{2}(-\partial_s \zeta^0 -\widetilde A \zeta^0),$$

with respect to the basis ${\frak B}$. Hence, for any $\zeta\in \ker D_u^*$, near $p\in{\bf p}_+$ we have

$$\zeta^0(s,t)=\sum_{\lambda_i>0} c_{+,i} e^{-\lambda_i s} f_i(t),$$

whereas near $p\in{\bf p}_-$ we have

$$\zeta^0(s,t)=\sum_{\lambda_i<0} c_{-,i} e^{-\lambda_i s} f_i(t),$$

where the $c_{\pm,i}$'s are constant.

§ SEMI-GLOBAL KURANISHI CHARTS

Consider the moduli space

$$\mathcal{M}=\mathcal{M}_J^{\op{ind}=k}(\dot F,\widehat W; \bs\gamma_+;\bs\gamma_-).$$

As discussed in Section <ref>, there are two slightly different cases to consider: (Symp) and (Cob).

We would like to construct a quadruple

$$(\K,\pi_{\V}: \E\to \V,\overline\bdry_J,{\frak s}),$$

which we call a semi-global Kuranishi chart. In the (Symp) case this consists of:

* a large compact subset $\K\subset \mathcal{M}/\R$; if $\mathcal{M}/\R$ is compact, then we take $\K=\mathcal{M}/\R$;

* an orbibundle $\pi_{\V}: \E\to \V$ (called an obstruction orbibundle) and a section $\overline\bdry_J:\V\to \E$ such that

* $\V$ is an open set containing $\K$,

* elements of $\V$ are equivalence classes $\llbracket\F,u\rrbracket$ of maps

$$(\F,u)=((F,j,{\bf p},{\bf q},{\bf r}),u)$$

for the quadruple $(\dot F,\widehat W,\bs\gamma_+;\bs\gamma_-)$,

* the fibers $\E_{\llbracket\F,u\rrbracket}=\pi^{-1}_{\V}(\llbracket\F,u\rrbracket)$ are naturally identified with finite quotients of finite-dimensional subspaces
$$E_{(\F,u)}\subset W^{k,p}_\delta(\dot F,\wedge^{0,1}u^*T\widehat W)$$

that are transverse to the linearization $L_{(\F,u)}$,

* $\overline\bdry_J(\llbracket \F,u\rrbracket)\in \E_{\llbracket\F,u\rrbracket}$ is represented by $\overline\bdry_J(\F,u)\in E_{(\F,u)}$ and

$$\overline\bdry_J^{-1}(0)= \V\cap (\mathcal{M}/\R);$$

* a decomposition of $\bdry \V$ into the horizontal part $\bdry_h \V$ and the vertical part $\bdry_v\V$ which intersect along their mutual boundary and such that $\overline\bdry_J$ is never zero on $\bdry_h\V$;

* a multisection ${\frak s}: \V\to \E$ of bounded degree (called an obstruction multisection over $\V$) such that ${\frak s}=0$ on a neighborhood of $\bdry_h\V$, ${\frak s}$ is close to zero, and ${\frak s}\pitchfork \overline\bdry_J$.

In the (Cob) case, $\mathcal{M}/\R$ is replaced by $\mathcal{M}$ in (1)–(4).

In this section we do most of the construction, proving (1), (2), and (4) with “${\frak s}=0$ on a neighborhood of $\bdry_h\V$” replaced by “${\frak s}=0$ on a neighborhood of $\bdry \V$”, and leaving (3) and in particular the definitions of the horizontal and vertical boundary to Section <ref>.

§.§ Construction of $\overline{\bdry}_J$-transverse subbundles

Consider the Banach bundle $\pi: \mathcal{E}\to \mathcal{B}$ and the section $\overline{\bdry}_J: \mathcal{B}\to \mathcal{E}$ corresponding to the quadruple $(\dot F,\widehat W,\bs\gamma_+;\bs\gamma_-)$, as defined in Section <ref> or <ref>.

Let $\mathcal{V}$ be a subset of $\mathcal{B}$.
A subbundle $E\to \mathcal{V}$ of $\mathcal{E}|_{\mathcal{V}}$ is $\overline{\bdry}_J$-transverse if $E$ has finite rank and, for each $(\F,u)\in \mathcal{V}$,

\begin{equation} \label{eqn: transverse}
\op{Im} L_{(\F,u)} + E_{(\F,u)} =\mathcal{E}_{(\F,u)},
\end{equation}

where $E_{(\F,u)}$ and $\mathcal{E}_{(\F,u)}$ refer to the fibers of $E$ and $\mathcal{E}$ over $(\F,u)$. Here $+$ in Equation (<ref>) means span and does not mean direct sum.

Let $Z=\overline\bdry_J^{-1}(0)$.
The goal of this section is to prove the following:

Let $K\subset Z$ be a compact subset and $\mathcal{N}(K)$ be a sufficiently small open neighborhood of $K$ in $\mathcal{B}$.  Then there exist a positive integer $\ell_0=\ell_0(K)>0$ and a positive real number $\varepsilon_0=\varepsilon_0(K)>0$ (or $\ell_0=\ell_0(K,\varepsilon')$ and $\varepsilon_0=\varepsilon_0(K,\varepsilon')$ if $\chi(\dot F)=0$ or $1$ and $\varepsilon'>0$ is as in Section <ref>) such that for all integers $\ell\geq \ell_0$ and $0<\varepsilon\leq \varepsilon_0$, there exists a $\overline{\bdry}_J$-transverse subbundle

$$E=E^{\ell,\varepsilon}\to \mathcal{N}(K)$$

of $\mathcal{E}|_{\mathcal{N}(K)}$ satisfying:

* if $(\F,u), (\F',u')\in \mathcal{N}(K)$ and there exists a $C^\infty$-diffeomorphism $\phi: (\F,u)\stackrel\sim\to (\F',u')$, then there is a canonical identification $\phi_*E_{(\F,u)}= E_{(\F',u\circ \phi^{-1})}$ which is induced from

$$\phi_*:\wedge^{0,1} T\dot F\otimes u^* T\widehat{W} \stackrel\sim\longrightarrow  \wedge ^{0,1} T\dot F\otimes (u\circ\phi^{-1})^{*}T\widehat{W};$$

* if $(\widehat{W}=\R\times M,J)\in \op{(Symp)}$ and $(\F,u), (\F,u_T)\in \mathcal{N}(K)$, where $u_T=\Phi_T\circ u$ is $u$ translated up by $s=T$ and

$$\Phi_T: \R\times M\stackrel\sim\longrightarrow \R\times M, \quad (s,x)\mapsto (s+T,x)$$

is the $s=T$ translation map, then $(\Phi_T)_*E_{(\F,u)}= E_{(\F,u_T)}$.

When $\op{dim}(M)=3$, it is possible to obtain an effective bound on $\ell$ using the positivity of intersections and winding numbers (cf. [15, 17, 32]).

The most important feature of the construction of $E^{\ell,\varepsilon}$ is the following:

($\#$) given $(\F,u)\in\mathcal{N}(K)$, the essential choices only depend on $\varepsilon$, $\ell$, the image of $u$, and the hyperbolic geometry of a complete finite volume hyperbolic metric $g_{(\dot F,j)}$ on $\dot F$ which is compatible with $j$ (or on $g_{(\ddot F,j)}$, where $\ddot F=\dot F-{\bf q}$ and ${\bf q}$ depends on the image of $u$, in the case when $\chi(\dot F)=0$ or $1$).

A nonessential choice for example is a choice of a cutoff function $\beta:\R\to [0,1]$ which is fixed throughout. This implies (1) and (2).

The proof consists of the following five steps.

Step 1. (Definition of $s_{\pm,i}$)
Let $K\subset Z$ be a compact subset.  The goal of this step is to define continuous maps $s_{\pm,i}:K \to \R$
with differentiable extensions $s_{\pm,i}: \mathcal{N}(K)\to \R$, where the differentiability depends on the choice of $W^{k+1,p}$. Here each $s_{\pm,i}$ corresponds to $\gamma_{\pm,i}$.

The collection $\{s_{\pm,i}\}$ will be invariant under group actions as follows:

(a) If $(\F,u)=((F,j,{\bf p},{\bf q},{\bf r}),u)\sim (\F',u')=((F,j',{\bf p},{\bf q}',{\bf r}'),u')\in \mathcal{N}(K)$ via the automorphism $\phi:\dot F\stackrel\sim\to\dot F$, then  $\{s_{\pm,i}(\F,u)\}=\{s_{\pm,i}(\F',u')\}$.

(b) If $(\F,u), (\F,u_T)\in \mathcal{N}(K)$, then $\{s_{\pm,i}(\F,u_T)\}=\{s_{\pm,i}(\F,u)+T\}$.

Step 1A. Assume first that $\chi(\dot F)<0$.  Let $g=g_{(\dot F,j)}$.  For $\varepsilon>0$ sufficiently small, consider the $\varepsilon$-thin part $\op{Thin}_\varepsilon(\dot F,g)$ of $(\dot F,g)$, which is the union of cusps

\begin{equation}\label{equation: cusps}
\end{equation}

where each $\bdry C_{\pm,i}$ is a circle which corresponds to $\gamma_{\pm,i}$.
We assume that $\varepsilon=\varepsilon(K)$ is sufficiently small so that, for all $(\F,u)\in K$, $\op{Im}(u|_{C_{\pm,i}})$ is $\kappa$-close (in the Hausdorff metric derived from the Riemannian metric $g_0$ on $\widehat{W}$ which is $s$-invariant at the ends) to $[s_{+,i},\infty)\times \gamma_{+,i}^s$ or $(-\infty,s_{-,i}]\times \gamma_{-,i}^s$ for some $s_{\pm,i}$. (The same holds for all $(\F,u)\in \mathcal{N}(K)$, where $\mathcal{N}(K)$ is a sufficiently small neighborhood of $K$.) Here $\kappa>0$ is a small, predetermined constant which is independent of $K$.  The existence of $\varepsilon>0$ relies on the exponential decay estimates of the ends of $u$ from [16].

We then choose a specific $s_{\pm,i}=s_{\pm,i}(\F,u)$ as follows: First consider the $m(\gamma_{\pm,i})$-fold covering map
$\R/\mathcal{A}_\alpha(\gamma_{\pm,i})\Z\to \R/\mathcal{A}_\alpha(\gamma_{\pm,i}^s)\Z$. We then write $u$ on $C_{\pm,i}$ in terms of supersimple coordinates as

after lifting to the $m(\gamma_{\pm,i})$-fold cover $\R\times (\R/\mathcal{A}_\alpha(\gamma_{\pm,i})\Z)\times D$, subject to the condition that $t=0$ corresponds to the asymptotic marker $r_{\pm,i}$. Then we define $s_{\pm,i}$ as the $s$-coordinate of the point on $u|_{\bdry C_{\pm,i}}$ with $t=0$.

The $c_{\pm,i}$ will be called the point on $\bdry C_{\pm,i}$ corresponding to $r_{\pm,i}$.

Step 1B. Next assume that $\chi(\dot F)=0$ or $1$.  In view of the discussion from Section <ref> (with $\varepsilon'>0$ fixed but small), we take $\ddot F= \dot F-{\bf q}$ with $\chi(\ddot F)<0$. Let $g=g_{(\ddot F,j)}$ and apply the considerations from Step 1A with $\dot F$ replaced by $\ddot F$ to obtain $s_{\pm,i}=s_{\pm,i}(\F,u)$.

(a) and (b) are immediate from the construction, both in Steps 1A and 1B.

[Trivial cylinders]
We do not construct Kuranishi charts for trivial cylinders when $(\widehat W,J)$ is in the (Symp) case. On the other hand, when a symplectization is viewed in (Cob), we will construct Kuranishi charts around trivial cylinders.  We note that trivial (possibly multiply-covered) cylinders are automatically transverse.

Step 2. (Definition of $E^{\ell,\varepsilon}$) Let $\varepsilon=\varepsilon(K)$ be the constant from Step 1.

Let $\beta:\R\to [0,1]$ be a nondecreasing smooth cutoff function such that $\beta(s)=0$ for $s\leq 0$ and $\beta(s)=1$ for $s\geq 1$ and let $\beta^+_{s_0}(s)=\beta(s-s_0)$ and $\beta^-_{s_0}(s)=-\beta(s_0-s)$, where $s_0>0$ is a constant.

Given $(\F,u)\in K$ and $s_{\pm,i}=s_{\pm,i}(\F,u)$ from Step 1, let

\begin{equation} \label{f tilde}
\widetilde{f}^{\gamma_{\pm,i}}_p(s,t):=\frac{\partial \beta^\pm_{s_{\pm,i}}}{\partial s}(s)f^{\gamma_{\pm,i}}_p(t)\otimes \pi_j (ds-idt)\in \Gamma (\dot F ,\wedge^{0,1} u^{*}T\widehat{W}).
\end{equation}

Here $\pi_j:W^{k,p}(\dot F, \wedge ^1 T^{*} \dot F)\to W^{k,p}(\dot F, \wedge^{0,1}T^{*} \dot F)$ is the projection with respect to the hyperbolic metric and $\widetilde{f}^{\gamma_{\pm,i}}_j(s,t)$ has support on a single end of $\dot F$.

We then define

\begin{equation} \label{equation: def of each component of E}
E_{(\F,u),{\gamma_{\pm,i}}}^{\ell,\varepsilon}:=\R \langle \widetilde{f}^{\gamma_{\pm,i}}_1,\dots,\widetilde{f}^{\gamma_{\pm,i}}_\ell \rangle
\end{equation}

for ${\gamma_{\pm,i}}\in \bs\gamma_\pm$ and define

\begin{equation}
E_{(\F,u)}^{\ell,\varepsilon}:=\bigoplus_{{\gamma_{\pm,i}}\in\bs{\gamma}_+\cup\bs\gamma_-} E_{(\F,u),{\gamma_{\pm,i}}}^{\ell,\varepsilon}.
\end{equation}

This gives us a bundle $E^{\ell,\varepsilon} \to \mathcal{N}(K)$ with fiber $E_{(\F,u)}^{\ell,\varepsilon}$. The following will be proven in the Appendix:

$E^{\ell,\varepsilon}_{(\F,u)}\subset \mathcal{E}_{(\F,u)}$ for all $(\F,u)\in \mathcal{N}(K)$ and $E^{\ell,\varepsilon} \to \mathcal{N}(K)$ is of class $C^1$ if $k\geq 3$ and $p>2$.

Step 3. ($\overline{\bdry}_J$-transversality)  We first prove the following transversality lemma, which is applicable for a single $(\F,u)\in K$.

For each $(\F, u)\in K$, there exist $\ell_0 \in \N$ and $\varepsilon_0>0$ such that

\begin{equation} \label{eqn: trans lemma}
E_{(\F,u)}^{{\ell,\varepsilon}} + \op{Im} D_u=W^{k,p}_\delta (\wedge ^{0,1}\dot F \otimes u^{*}T\widehat W)
\end{equation}

for all $\ell\geq \ell_0$ and $0<\varepsilon\leq \varepsilon_0$.

Observe that we are actually proving a slightly stronger statement than

$$E_{(\F,u)}^{{\ell,\varepsilon}} + \op{Im} L_{(\F,u)}=W^{k,p}_\delta (\wedge ^{0,1}\dot F \otimes u^{*}T\widehat W).$$

Observe that, if Equation (<ref>) does not hold, then there exists $0\not=\zeta\in \ker D_u^*$ which is $L^2$-orthogonal to $E_{(\F,u)}^{\ell,\varepsilon}$. To see this, consider the projection of $E_{(\F,u)}^{\ell,\varepsilon}$ under the map $\pi:\ker D_u^*\oplus \op{Im} D_u\to \ker D_u^*$. If $\zeta\in \ker D_u^*$ is orthogonal to $\pi(E_{(\F,u)}^{\ell,\varepsilon})$, then $\zeta\perp E_{(\F,u)}^{\ell,\varepsilon}$ since $\zeta\perp \op{Im} D_u$.

Let $\varepsilon_0>0$ be sufficiently small so that on the $\varepsilon_0$-thin neighborhood of a puncture of $\dot F$ corresponding to the Reeb orbit $\gamma\in \bs\gamma_+$ ($\gamma\in \bs\gamma_-$ is similar), $\zeta\in \ker D_u^*$ can be written with respect to the basis $\frak{B}$ as

$$\zeta(s,t)=\sum_{\lambda_i>0} c_i e^{-\lambda_i^\gamma s} f_i^\gamma(t) \otimes (ds-idt),$$

and let $0<\varepsilon\leq \varepsilon_0$. The key point is that, when $\varepsilon_0$ is sufficiently small, $\frak{B}$ is arbitrarily close to $\{\bdry_s,\bdry_t,\bdry_{x_1},\dots,\bdry_{x_n},\bdry_{y_1},\dots,\bdry_{y_n}\}$ and, without loss of generality, we may assume $\zeta(s,t)$ is written with respect to the latter.  There exists a basis $\{\zeta_1,\dots, \zeta_k\}$ for $\ker D_u^*$ for which the coefficients $c_{ji}$ (this means $c_i$ for $\zeta_j$) form a matrix in row echelon form; in particular, there are no rows with all $c_{ji}=0$ since the corresponding $\zeta_j=0$ by unique continuation, which is a contradiction.

Let $\ell_0>0$ be larger than the smallest $i$ for which $c_{ki}\not=0$.  Now observe that if $\zeta\in \ker D_u^*$ is $L^2$-orthogonal to $E_{(\F,u)}^{\ell,\varepsilon}$, then its coefficients $c_i=0$ for all $i\leq \ell_0$, since the $L^2$-inner product of $\zeta$ and $\widetilde f_i^\gamma$ is nonzero if $c_i\not=0$.  This implies that $\zeta=0$.  The lemma then follows.

We then improve Lemma <ref> as follows:

There exist $\ell_0 \in \N$ and $\varepsilon_0>0$ such that

$$E_{(\F,u)}^{{\ell,\varepsilon}} + \op{Im} D_u=W^{k,p}_\delta (\wedge ^{0,1}\dot F \otimes u^{*}T\widehat W).$$

for all $(\F, u)\in \mathcal{N}(K)$, $\ell\geq \ell_0$, and $0<\varepsilon\leq \varepsilon_0$

This follows from Lemma <ref>, the compactness of $K$, and the following well-known property of Fredholm maps: Let $L,L':V\to W$ be Fredholm maps between Banach spaces. If $L$ and $L'$ are close in the operator norm and $\op{Im}(L)+W'=W$ for a finite-dimensional subspace $W'$, then $\op{Im}(L')+W'=W$.

For $(\F_0,u_0), (\F_1,u_1)\in K$ that are close, we identify the Banach spaces involved in $L_{(\F_i,u_i)}$, $i=0,1$, by defining an isomorphism $\mathcal P: u_0^*T\widehat W \stackrel\sim\to u_1^*T\widehat W$ as follows:

(i) on a large compact region of $\dot F$, $\mathcal P$ is given by parallel translation along the shortest geodesics between $u_0(z)$ and $u_1(z)$;

(ii) on the ends of $\dot F$ (assumed to be sufficiently small), $\mathcal P$ is given by identifying the bases

$$\{(u_0)_s,(u_0)_t, \partial_{x_1},\dots,\partial_{x_n},\partial_{y_1},\dots,\partial_{y_n}\}$$

$$\{(u_1)_s, (u_1)_t, \partial_{x_1},\dots,\partial_{x_n},\partial_{y_1},\dots,\partial_{y_n} \};$$

(iii) $\mathcal{P}$ interpolates between (i) and (ii) in the intermediate region.

Then $L_{(\F_0,u_0)}$ and $L_{(\F_1,u_1)}$ are close in the operator norm; the details are left to the reader.

Lemma <ref> then implies the proposition.

§.§ The bundles $\pi_V:E|_V\to V$

§.§.§ Specific choices for $K$

Recall the projection $\Pi=\Pi_{\widetilde{\mathcal{U}}}: \mathcal{B}_{\widetilde{U}}\to\widetilde{\mathcal{U}}$ given by $(\F,u)=((F,j,{\bf p},{\bf q},{\bf r}),u)\mapsto j$.

Let $\mathcal{K}=\mathcal{K}_{\varepsilon,L}\subset \widetilde{\mathcal{U}}$ be a compact domain with smooth boundary.  We assume that $K$ is

(i) the compact subset of $Z=\overline\bdry_J^{-1}(0)$ given by $Z\cap \Pi^{-1}(\mathcal{K})$ in the $\op{(Cob)}$ case;

(ii) a compact subset of $Z=\overline\bdry_J^{-1}(0)$ which projects onto $(Z\cap \Pi^{-1}(\mathcal{K}))/\R$ in the $\op{(Symp)}$ case.

§.§.§ Definition of $\pi_V:E|_V\to V$

With the bundle $E=E^{\ell,\varepsilon}\to \mathcal{N}(K)$ constructed in Theorem <ref>, we define

\begin{equation} \label{eqn: V and V prime}
V:= \overline\bdry_J^{-1}(E)\subset \mathcal{N}(K). %\quad \mbox{ and } \quad V^\flat:= V\cap \mathcal{N}^\flat(K).
\end{equation}

Since $E$ is $\overline\bdry_J$-transverse and of class $C^1$, $V\supset K$ is a finite-dimensional submanifold of $\mathcal{N}(K)$ of class $C^1$.  Once we have extracted $V$, since the elements of $V$ are smooth by elliptic regularity, $V$ can be “upgraded to" a smooth finite-dimensional manifold of the expected dimension.

Observe that if $(\F,u)\in V$, then $u:\dot F\to \widehat{W}$ is a smooth map whose ends are $J$-holomorphic and asymptotic to $\bs\gamma_+\cup \bs\gamma_-$. Let

\begin{equation}
\pi_V:E|_V\to V
\end{equation}

be the restriction of $E\to \mathcal{N}(K)$ to $V$ and let $\overline\bdry_J:  V\to E|_V$ be the restriction of $\overline\bdry_J$ to $V$. The vector bundle $\pi_V$ is a smooth vector bundle over $V$.

§.§.§ Patching together the bundles $\pi_V: E|_V\to V$

We would like to patch together the bundles $\pi_V: E|_V\to V$ to obtain an orbibundle.  To this end we prove Lemmas <ref>, <ref>, and <ref> below, which we can roughly state as:

* independence of the choice of Teichmüller slice;

* $G$-equivariance; and

* invariance under $\R$-translations.

We first consider (1). Suppose we are in the (Cob) case.  (An analogous result holds in the (Symp) case in view of (3) below, but will not be stated explicitly.)  Let $\widetilde{\mathcal{U}}_0$ and $\widetilde{\mathcal{U}}_1$ be two Teichmüller slices of $\mathcal{U}\subset \op{Teich}(\dot F)$. Then there is a smooth family of diffeomorphisms $\phi_x: \dot F\stackrel\sim\to \dot F$ isotopic to the identity parametrized by $x\in \mathcal{U}$ such that $(\phi_x)_* j_x = j'_x$, where $j_x\in\widetilde{\mathcal{U}}_0$ and $j_x'\in\widetilde{\mathcal{U}}_1$ are lifts of $x\in \mathcal{U}$.

Let $\pi_{V_i}:E_i|_{V_i}\to V_i$, $i=0,1$, be the bundle $\pi_V:E|_V\to V$ corresponding to $\widetilde{\mathcal{U}}_i$ and let $K_i$, $\mathcal{K}_i$ be $K$, $\mathcal{K}$ corresponding to $\widetilde{\mathcal{U}}_i$ such that $\mathcal{K}_0$, $\mathcal{K}_1$ are Teichmüller slices of the same compact subset of $\mathcal{U}$.

Suppose we are in the $\op{(Cob)}$ case. After possibly shrinking $V_i$ subject to the condition $V_i\supset K_i$, $V_0$ and $V_1$ can be made canonically diffeomorphic and the bundles $\pi_{V_i}:E_i|_{V_i}\to V_i$ can also be made canonically isomorphic.

Suppose $(\F,u)=(F,j,\bf p, \bf q, \bf r, u) \in V_0$, $(\F',u')=(F, j',\bf p, \bf q', \bf r', u') \in \mathcal{N}(K_1)$, and $\phi=\phi_x$ is the diffeomorphism satisfying

$$\phi_*((F,j,{\bf p},{\bf q},{\bf r}),u)= ((F,j',{\bf p},{\bf q}',{\bf r}'),u').$$

Then $\phi$ induces a natural identification $\phi_* E_{(\F,u)}\simeq E_{(\F',u')}$ by ($\#$) in the proof of Theorem <ref>, which applies equally well to our case, provided

$$E_i=E_i^{\ell,\varepsilon}\to \mathcal{N}(K_i), \quad i=0,1$$
are constructed as in Theorem <ref> with fixed $\ell$ and $\varepsilon$.  Hence $(\F',u')\in V_1$, provided $(\F,u)\in V_0$ is sufficiently close to $K_0$.

The family $\{\phi_x\}, x\in \mathcal{U},$ gives a smooth map $\Phi$ from $E_0|_{V_0}\to V_0$ to $E_1|_{V_1}\to V_1$, after possibly shrinking $V_0$.  Since $\Phi$ is a diffeomorphism onto its image and the image of $V_0$ contains $K_1$, by the invariance of domain, $\Phi$ is a diffeomorphism $E_0|_{V_0}\to V_0$ to $E_1|_{V_1}\to V_1$, after possibly shrinking $V_1$.

We pass to $\pi_{V_i}:E_i|_{V_i}\to V_i$ at an early stage to avoid the well-known “loss of derivative" issue when dealing with Sobolev spaces.  In particular, we avoid identifying $E_0\to \mathcal{N}(K_0)$ and $E_1\to \mathcal{N}(K_1)$.

(2) is stated more precisely as follows: Suppose we are in the $\op{(Cob)}$ case.  Let $\mathcal{U}=\mathcal{U}_{[j]}$ be a sufficiently small good neighborhood of $[j]$.

Let $\widetilde {\mathcal{U}}_0=\widetilde {\mathcal{U}}$ be a Teichmüller slice of $\mathcal{U}$ containing $j$. Then $\widetilde {\mathcal{U}}_1=g(\widetilde{\mathcal{U}})$ is also a Teichmüller slice of $\mathcal{U}$ where $g\in \op{Aut}(F,{\bf p},{\bf q}, j)$. As in (1), let $\{\phi_x\}_{x\in\mathcal{U}}$ be the family of diffeomorphisms taking $\widetilde{\mathcal{U}}_0$ to $\widetilde{\mathcal{U}}_1$. Also let $\pi_{V_i}:E_i|_{V_i}\to V_i$, $i=0,1$, be the bundle $\pi_V:E|_V\to V$ corresponding to $\widetilde{\mathcal{U}}_i$ and let $K_i$, $\mathcal{K}_i$ be $K$, $\mathcal{K}$ corresponding to $\widetilde{\mathcal{U}}_i$ such that $\mathcal{K}_0$, $\mathcal{K}_1$ are Teichmüller slices of the same $\op{Mod}(F,{\bf p})_{[j]}$-invariant compact subset of $\mathcal{U}$.

Suppose we are in the $\op{(Cob)}$ case. After possibly shrinking $V_i$ subject to the condition $V_i\supset K_i$, $V_0$ and $V_1$ can be made diffeomorphic via the automorphism $g$ and the family $\{\phi_x\}_{x\in\mathcal{U}}$ of diffeomorphisms and the bundles $\pi_{V_i}: E_i|_{V_i}\to V_i$ can be made isomorphic.

Similar to that of Lemma <ref>.

(3) is stated more precisely as follows:

Suppose we are in the $\op{(Symp)}$ case and $K_0$ and $K_1$ are compact subsets for $\mathcal{K}\subset \widetilde{\mathcal{U}}$ such that $(\F,u)\in \mathcal{N}(K_0)$ if and only if $(\F,u_T)\in \mathcal{N}(K_1)$, where $u_T$ is $u$ translated up by $s=T$. Then, after possibly shrinking $V_i$ subject to the condition $V_i\supset K_i$, the bundles $\pi_{V_i}: E_i|_{V_i}\to V_i$ are canonically isomorphic.

Similar to that of Lemma <ref>.

§.§ Semi-global Kuranishi chart

In this subsection we construct a semi-global Kuranishi chart

$$(\K,\pi_{\V}: \E\to \V,\overline\bdry_J,{\frak s})$$

for $\mathcal{M}=\mathcal{M}_J^{\op{ind}=k}(\dot F,\R\times M;\bs\gamma_+;\bs\gamma_-)$ or $\mathcal{M}/\R$ such that (1), (2), and (4) (with “${\frak s}=0$ on a neighborhood of $\bdry_h\V$” replaced by “${\frak s}=0$ on a neighborhood of $\bdry \V$”) from the beginning of this section hold.  The existence of the section $\overline\bdry_J$ is automatic from the construction.

Let $\K\subset \mathcal{M}/\R$ be a large compact subset; in particular, if $\mathcal{M}/\R$ is compact, then $\K=\mathcal{M}/\R$.  Consider a compact subset $K\subset Z=\overline\bdry_J^{-1}(0)$ for the appropriate Banach bundle $\mathcal{E}\to \mathcal B$ described in Section <ref>, such that the projection of $K$ to $\K$ is onto.  Theorem <ref> gives a $\overline\bdry_J$-transverse subbundle $E\to \mathcal{N}(K)$. If we let $V=\overline\bdry_J^{-1}(E)$, then $\pi_V: E|_V\to V$ is a finite rank vector bundle over a finite-dimensional manifold.

§.§.§ Construction of $\pi_{\V}:\E\to \V$

Let $\V$ be the quotient of $V$ by the equivalence relation $\sim$ and the $\R$-translations.  Also let $\E$ be the quotient of $E$ by the induced equivalences

$$((\F=(F,j,{\bf p},{\bf q},{\bf r}),u),\zeta)\sim ((\F'=(F,j',{\bf p},{\bf q},{\bf r}'),u'),\zeta'),$$

where $\phi: F\stackrel\sim\to F$ is a diffeomorphism such that $\phi_*j=j'$, $\phi({\bf p}_\pm)={\bf p}_\pm$, $\phi({\bf q})={\bf q}$, $\phi_*({\bf r}_\pm)={\bf r}'_\pm$, $u'\circ\phi=u$, and $\phi_*\zeta=\zeta'$, and

$$((\F,u),\zeta)\sim ((\F,u_T),\zeta_T),$$

where $u_T$ is $u$ translated up by $s=T$ and $\zeta_T$ is $\zeta$ translated up by $s=T$.

After possibly shrinking $\V$ subject to the condition $\V\supset \K$, there is an orbibundle structure on $\pi_{\V}:\E\to \V\supset \K$.

The proposition follows from Section <ref> and Lemmas <ref>, <ref>, and <ref>. The orbibundle charts are constructed by taking a sufficiently small good open set $\mathcal{U}_{[j]}$ about each $[j]\in \op{Teich}(\dot F)$, a slice $\widetilde{\mathcal{U}}_{[j]}$ of $\mathcal{U}_{[j]}$, and a compact domain $\mathcal{K}\subset \widetilde{\mathcal{U}}_{[j]}$ which is the closure of a good open set about $[j]\in\op{Teich}(\dot F)$. The group $\op{Mod}(F,{\bf p})_{[j]}$ can be made to act on the resulting $\pi_V: E\to V$; one may need to take a quotient if the action of $\op{Mod}(F,{\bf p})_{[j]}$ is not effective.

It remains to verify that $\V$ is Hausdorff: If there exist $x\not=y\in \V$ such that arbitrarily small neighborhoods $V_x$ and $V_y$ of $x$ and $y$ intersect nontrivially, then there exist lifts $\widetilde x\in V$ of $x$ and $\widetilde y\in V$ of $y$ and sequences $\widetilde x_i\to \widetilde x$ and $\widetilde y_i\to \widetilde y$ in $V$ such that $\widetilde x_i\sim \widetilde y_i$.  On the other hand, by Gromov-Hofer compactness, $\widetilde x\sim \widetilde y$, a contradiction.

§.§.§ Construction of ${\frak s}$

Next we choose a generic smooth multisection ${\frak s}$ of $\pi_{\V}: \E\to \V$ which is a perturbation of the zero (multi-)section, subject to the conditions ${\frak s}=0$ on a neighborhood of $\bdry \V$ and ${\frak s}\pitchfork \overline\bdry_J$.   By the multisection perturbation lemma (Lemma <ref>) or the relative version thereof, there exists ${\frak s}$ which satisfies the desired conditions. Note that each (local) branch of ${\frak s}$ is assigned a weight by the discussion in Section <ref>.

We then consider $\overline\bdry_J^{-1}({\frak s})$. Since $\overline\bdry_J(\F,u)=0$ if and only if $\llbracket\F,u\rrbracket\in \mathcal{M}/\R$,  we have $\overline\bdry_J^{-1}({\frak s})\cap \bdry \V=\varnothing$, i.e., $\overline\bdry_J^{-1}({\frak s})$ stays away from $\bdry \V$.

§.§ Dependence on $\varepsilon$ and $\ell$

In this subsection we briefly discuss the effect of increasing $\ell>0$ and shrinking $\varepsilon>0$.

§.§.§ Stabilizations

Let $E\to \mathcal{N}(K)$ be a $\overline\bdry_J$-transverse subbundle. A subbundle $E'\subset \mathcal{E}|_{\mathcal{N}(K)}\to \mathcal{N}(K)$ is a stabilization of $E$ if $E'\simeq E\oplus \R^a$ for some $a\in \Z^+$.

An example of a stabilization of $E^{\ell,\varepsilon}\to \mathcal{N}(K)$ is $E^{\ell+1,\varepsilon}\to \mathcal{N}(K)$.

Given a stabilization $E'$ of $E\to\mathcal{N}(K)$, we have the following diagram:

\begin{equation} \label{equation: diagram}
\begin{diagram}
E & \rInto^{j} & E'=E\oplus \R^a\\
\dTo^{\pi_{V}} &  & \dTo_{\pi_{V'}} \\
V & \rInto^{i} & V'
\end{diagram}
\end{equation}

where $V=\overline\bdry_J^{-1}(E)$ and $V'=\overline\bdry_J^{-1}(E')$.

After shrinking $V'$ subject to $V'\supset K$ if necessary, we may assume that:

(StV) $i:V\hookrightarrow V'$ extends to a diffeomorphism

$$i: V\times (-1,1)^a\stackrel\sim \longrightarrow V'$$

so that $\overline{\bdry}_J(i((\F,u),x))=(\overline{\bdry}_J(\F,u),x)$, where $x$ is viewed both as an element of $(-1,1)^a$ and as an element of $\R^a$.

A bundle $\pi_{V'}:E'|_{V'}\to V'$ is a stabilization of $\pi_V:E|_V\to V$ if

* $V=\overline\bdry_J^{-1}(E)$ and $V'=\overline\bdry_J^{-1}(E')$, where $E'\to \mathcal{N}(K)$ is a stabilization of $E\to \mathcal{N}(K)$;

* $\pi_{V'}$ and $\pi_V$ satisfy Diagram (<ref>); and

* $V$ and $V'$ are related by (StV).

We also say that “$(\pi_{V'},\pi_V)$ is a stabilization”.

An orbibundle $\pi_{\V'}: \E'\to \V'$ is a stabilization of the orbibundle $\pi_{\V}: \E\to \V$ if

* for each $p\in \V$ there exist orbibundle charts $\pi_{V'}:E'|_{V'}\to V'$ and $\pi_{V}:E|_V\to V$ for $\pi_{\V'}$ and $\pi_{\V}$ about $p$ such that $(\pi_{V'},\pi_V)$ is a stabilization;

* given any two stabilizations $(\pi_{V'_1},\pi_{V_1})$ and $(\pi_{V'_2},\pi_{V_2})$ that overlap and any $p\in \V$ in the overlap, there exists a stabilization $(\pi_{V'_0},\pi_{V_0})$ about $p$ such that, for $i=1,2$, $\pi_{V'_0}$ embeds in $\pi_{V'_i}$, $\pi_{V_0}$ embeds in $\pi_{V_i}$, and we have a commuting cube consisting of two copies of Diagram (<ref>) (for $(\pi_{V'_0},\pi_{V_0})$ and $(\pi_{V'_i},\pi_{V_i})$) and the embedding maps between them.

Given a stabilization $\pi_{V'}:E'|_{V'}\to V'$ of $\pi_V:E|_V\to V$ and a section $s: V\to E$, we construct the stabilization $s': V'\to E'|_{V'}$ of $s$ as follows: Assume $E'=E\oplus \R$. (The case of $E'=E\oplus \R^a$ can be done inductively.)
Let $g:(-1,1)\to [0,1]$ be a smooth even function such that $g(x)=1$ on $0\leq x\leq {1\over 3}$ and $g(x)=0$ on $x\geq {2\over 3}$.
We then define

\begin{equation} \label{eqn: stabilization}
s'(i((\F,u),x))= (g(x)s(\F,u),0).
\end{equation}

The pair $(E',s')$ consisting of a stabilization $E'$ of $E$ and a stabilization $s'$ of $s$ is called a stabilization of $(E,s)$.

If $(E',s')$ is a stabilization of $(E,s)$, then $\overline\bdry^{-1}_J(s)=\overline\bdry^{-1}_J(s')$.

Immediate from the construction of $(E',s')$.

Analogously, if ${\frak s}$ is a $\overline\bdry$-transverse multisection of $\pi_{\V}: \E\to \V$ constructed in Section <ref>, then we can construct the stabilization ${\frak s}'$ of ${\frak s}$ which is a multisection of $\pi_{\V'}: \E'\to \V'$ such that ${\frak s}'$ is $\overline\bdry$-transverse, ${\frak s}'=0$ on $\bdry \V'$, and $\overline\bdry^{-1}_J({\frak s})=\overline\bdry^{-1}_J({\frak s}')$ as weighted branched manifolds.

§.§.§ Changing $\varepsilon$

The effect of changing $\varepsilon>0$ can be summarized as follows:

[Changing $\varepsilon>0$]
There exist $\varepsilon_0>0$ and $\ell_0>0$ such that Theorem <ref> holds for all $\varepsilon>0$ satisfying $\varepsilon_0\geq\varepsilon$ and $\ell$ satisfying $\ell\geq \ell_0$.  If we fix $\ell$, then for any two $0<\varepsilon_1,\varepsilon_2<\varepsilon_0$, there exists a $1$-parameter family of $\overline\bdry_J$-transverse bundles $E^{\ell,\varepsilon_t}\to \mathcal{N}(K)$, $t\in[1,2]$.  In the process of extracting weighted branched manifolds in later sections, it follows that the weighted branched manifolds corresponding to $\varepsilon_1$ and $\varepsilon_2$ are cobordant.

§ GLUING

The goal of this section is to define the terms that appear in the gluing theorems and also state the gluing theorems that we need.

Suppose we are considering (Symp) case; in this case we need to quotient by the $\R$-translations. A key observation is that $E_{(\F,u)}$ is equivariant under $\R$-translations by the construction in Theorem <ref>.  The necessary modifications can be made for the (Cob) case.

§.§ The setup

We will explain a simplified situation, but the general case is similar. Let

$$\mathcal{M}=\mathcal{M}^{\op{ind}=k}_J(\dot F,\R\times M;\bs\gamma_+;\bs\gamma_-).$$

Let $\overline{\mathcal{M}/\R}$ be the compactification of $\mathcal{M}/\R$ and let $\bdry (\mathcal{M}/\R)$ be the boundary of $\mathcal{M}/\R$.

We assume the following:

There exist

$$\mathcal{M}_1=\mathcal{M}^{\op{ind}=k_1}_J(\dot F_1,\R\times M;\bs\gamma;\bs\gamma_-),$$

$$\mathcal{M}_2=\mathcal{M}^{\op{ind}=k_2}_J(\dot F_2,\R\times M;\bs\gamma_+;\bs\gamma),$$

with $k_1+k_2=k$ such that $\dot F_1$ and $\dot F_2$ are connected, $\mathcal{M}_1/\R$ and $\mathcal{M}_2/\R$ are compact, and

$$\bdry (\mathcal{M}/\R)\subset \mathcal{M}_1/\R\times \mathcal{M}_2/\R.$$

Here and in the rest of the paper will use the convention that the lower subscript corresponds to the lower SFT level.

For $i=1,2$, $(F_i,{\bf p}_i,{\bf q}_i)$ and hence $\ddot F_i$ are fixed. Let $\varepsilon_i'>0$ be the constant that appears in the definition of $s'_+(\F_i,u_i)$ in Section <ref>, in case $\chi(\dot F)=0$ or $1$. We take a Teichmüller slice $\widetilde{\mathcal{U}}_i$ such that each $[\F_i,u_i]\in \mathcal{M}_i$ admits a representative $(\F_i,u_i)=((F_i,j_i,{\bf p}_i,{\bf q}_i,{\bf r}_i),u_i)$ with $j_i\in \widetilde{\mathcal{U}}_i$. Define

$$\overline\bdry^{\widetilde{\mathcal{U}}_i}_J: \mathcal{E}_{\widetilde{\mathcal{U}}_i}\to \mathcal{B}_{\widetilde{\mathcal{U}_i}}$$

as before.  Let $Z_i=(\overline\bdry_J^{\widetilde{\mathcal{U}}_i})^{-1}(0)$, $K_i\subset Z_i$ be a compact subset which projects onto $\mathcal{M}_i/\R$, and let $\mathcal{N}(K_i)$ be a sufficiently small neighborhood of $K_i$ in $\mathcal{B}_{\widetilde{\mathcal{U}_i}}$.

We are also assuming that

$${\bf p}=({\bf p}_1\cup {\bf p}_2)-({\bf p}_{+,1}\cup{\bf p}_{-,2}),\quad {\bf r}=({\bf r}_1\cup {\bf r}_2)-({\bf r}_{+,1}\cup{\bf r}_{-,2}),$$

and $F=F_1\# F_2$, where $\#$ is a multiple connected sum and the gluing occurs along subsets ${\bf p}_{+,1}\subset {\bf p}_1$, ${\bf p}_{-,2}\subset {\bf p}_2$ that correspond to ${\bf r}_{+,1}\subset {\bf r}_1$, ${\bf r}_{-,2}\subset {\bf r}_2$.

We apply Theorem <ref> to $K_i\subset Z_i$ to obtain

$$E_i=E^{\ell,\varepsilon}_i\to V_i/\R,$$

where $V_i=(\overline\bdry^{\widetilde{\mathcal{U}}_i}_J)^{-1}(E_i)\subset \mathcal{N}(K_i)$ as in Equation (<ref>).
By abuse of notation, we are still denoting the total spaces of the vector bundles by $E_i$ since the fibers are still the same.

In what follows $\varepsilon>0$ is a very small constant which we may shrink (but never let be zero) from step to step.

We fix a slice $V_i/\R\to V_i$ and write $(\F_i,u_i)\in V_i/\R$ to indicate the representative of its equivalence class in $V_i/\R$ in that slice. We also assume that

$$\min_j\{s_{-,j}(\F_2,u_2)\}\gg 0, \quad \max_j \{s_{+,j}(\F_1,u_1)\}\ll 0$$

for all $(\F_i,u_i)\in V_i/\R$.

§.§ Close to breaking

Let $\delta>0$ be a small constant[Unrelated to the weight $\delta$ used in the Sobolev spaces earlier.] satisfying $\varepsilon_i'\gg \delta$, $i=1,2$. Given a constant $T\in \R$, let $u_{i,T}$ (resp. $u_T$) be $u_i$ (resp. $u$) translated up by $s=T$ units.

We define the notion of close to breaking in the spirit of <cit.> as follows:

A map $(\F,u)=((F,j,{\bf p},{\bf q},{\bf r}),u)$ is $\delta$-close to a building

$$(\F_1,u_1)\cup (\F_2,u_2),\quad (\F_i,u_i)\in V_i/\R,\quad i=1,2,$$

if the following hold:  after possibly translating $u$,

* there exist $T_2> T_2'>0$ such that the domain $\ddot F_2'$ of $u_{2,T_2}|_{s\geq T_2'}$ is obtained from $\ddot F_2$ by removing ends $C_\gamma$ corresponding to $\gamma\in\bs\gamma$; each end $C_\gamma$ is contained in the $\varepsilon$-thin part of $g_{(\ddot F_2,j_2)}$ and admits a biholomorphism

$$(C_\gamma,j_2)\simeq ((-\infty,T_2']\times \R/\mathcal{A}_\alpha(\gamma)\Z,j_{12})$$

with coordinates $(s,t)$ and standard complex structure $j_{12}:\bdry_s\mapsto \bdry_t$, such that $u_{2,T_2}|_{C_\gamma}$ is $\delta$-close in the $C^0$-norm to the map

$$C_\gamma\to \R\times (\R/\mathcal{A}_\alpha(\gamma)\Z)\times D\subset \R\times M,\quad (s,t)\mapsto (s,t,0),$$

where we are using supersimple coordinates for $\gamma$;

* there exist $T_1> T_1'>0$ such that the domain $\ddot F_1'$ of $u_{1,-T_1}|_{s\leq -T_1'}$ and the ends satisfy analogous conditions;

* let $r(\bdry C_\gamma)\in \bdry C_\gamma$ be the point corresponding to the asymptotic marker for $C_\gamma$ as in Definition <ref>; writing

$$A_\gamma=([-T_1,T_2']\times \R/\mathcal{A}_\alpha(\gamma)\Z,j_{12})$$

with $\gamma\in \bs\gamma$, coordinates $(s,t)$, and standard complex structure $j_{12}:\bdry_s\mapsto \bdry_t$, the Riemann surface $(\ddot F,j)$ is $\delta$-close in the Teichmüller metric to the Riemann surface $(\ddot F,j')$[We are identifying $\ddot F$ with the surface obtained by gluing $\ddot F'_1$, $\sqcup_\gamma([-T_1,T_2']\times \R/\mathcal{A}_\alpha(\gamma)\Z)$, and $\ddot F'_2$.] obtained by gluing

$$(\ddot F'_1,j_1), \quad \sqcup_{\gamma\in \bs \gamma} A_\gamma, \quad \mbox{and} \quad (\ddot F'_2,j_2),$$

along their appropriate boundaries such that each $r(\bdry C_\gamma)$ is mapped to $(s,t)=(-T_1,0)$ or $(T_2',0)$;

* $u_{2,T_2}|_{\ddot F'_2}$ is $\delta$-close in the $C^0$-norm to the map $u|_{\ddot F'_2}$;

* $u_{1,-T_1}|_{\ddot F'_1}$ is $\delta$-close in the $C^0$-norm to the map $u|_{\ddot F'_1}$; and

* for each $\gamma\in\bs\gamma$, $u|_{A_\gamma}$ is $\delta$-close in the $C^0$-norm to the map

$$A_\gamma\to \R\times (\R/\mathcal{A}_\alpha(\gamma)\Z)\times D\subset \R\times M,\quad (s,t)\mapsto (s,t,0),$$

where we are using supersimple coordinates for $\gamma$.

We make our choice of ${\bf q}$ more precise for $(\F,u)=((F,j,{\bf p},{\bf q},{\bf r}),u)$ which is $\delta$-close to $(\F_1,u_1)\cup (\F_2,u_2)$, following Section <ref>: Let us write the ends (or truncated ends) of $u|_{s\geq T_2'}$ and $u|_{s\leq -T_1'}$ by $u(s,t)=(s,t,\eta(s,t))$ using supersimple coordinates. For each $s\in (-T'_1, T'_2)$, we have

$$\eta(s)=\sum_{i}c_ie^{\theta_i s}g_i(t),$$
as in the proof of Lemma <ref>. Hence,

$$\int_0^{\mathcal A_{\alpha}(\gamma)}|\eta(s,t)|^2dt=\sum_{i} c_i^2e^{2\theta_i s}.$$

Since $\sum_{\theta_i>0} c_i^2e^{2\theta_i s}$ is strictly increasing with respect to $s$ and $\sum_{\theta_i<0} c_i^2e^{2\theta_i s}$ is strictly decreasing with respect to $s$, there exists a unique $s=s_0\in (-T'_1,T'_2)$ that minimizes $\int_0^{\mathcal A_{\alpha}(\gamma)}|\eta(s,t)|^2dt$.
As in Lemma <ref> let us define

$$s'_+(u|_{s\geq T_2'},\gamma)=\min\Big\{ {s\in [s_0,+\infty)}~\Big|~\int_{s_0}^{s}\int_0^{\mathcal A_\alpha(\gamma)} |\eta(s',t')|^2 dt'ds'=\varepsilon_2'\Big\},$$

$$s'_+(u|_{s\leq -T_1'},\gamma)=\max\Big\{ {s\in (-\infty,s_0]}~\Big|~\int_s^{s_0}\int_0^{\mathcal A_\alpha(\gamma)} |\eta(s',t')|^2 dt'ds'=\varepsilon'_1\Big\},$$

where $\gamma\in\bs\gamma_+$ in the former case and $\gamma\in\bs\gamma$ in the latter case.  Then ${\bf q}={\bf q}'_2\cup {\bf q}'_1$ is an unordered collection of removable punctures where ${\bf q}'_2$ and ${\bf q}'_1$ are lists of punctures defined using $s'_+(u|_{s\geq T_2'},\gamma)$ and $s'_+(u|_{s\leq -T_1'},\gamma)$ and (*) in Section <ref>.

If $\chi(\dot F_i)<0$, then $\bf q_i'=\varnothing$, where $i=1,2$.
Note that ${\bf q}'_2$ and ${\bf q}'_1$ are close to ${\bf q}_2$ and ${\bf q}_1$ for $(\F_2,u_{2,T_2})$ and $(\F_1,u_{1,-T_1})$ under the identifications of $\ddot F_2'$ and $\ddot F_1'$ with subsurfaces of $\ddot F$ as in Definition <ref>.

We observe that for $(\F,u)$ close to breaking into $(\F_1,u_1)\cup (\F_2,u_2)$, the data $(\F_1,u_1)\cup (\F_2,u_2)$ and $((F,j,{\bf p},{\bf r}),u)$ are sufficient to recover ${\bf q}$. In other words, the forgetting map

$$((F,j,{\bf p},{\bf q},{\bf r}),u)\mapsto ((F,j,{\bf p},{\bf r}),u)$$

presents no loss of information.

Let $\mathcal{G}_\delta(V_1/\R,V_2/\R)$ be the set of equivalence classes of maps

$$(\F,u)=((F,j,{\bf p},{\bf q}, {\bf r}),u)$$

modulo $\R$-translations which have representatives that are $\delta$-close to some building $(\F_1,u_1)\cup (\F_2,u_2)$ with $(\F_i,u_i)\in V_i/\R$ and let $\widetilde{\mathcal{G}}_\delta(V_1/\R,V_2/\R)$ be a slice of $\mathcal{G}_\delta(V_1/\R,V_2/\R)$ whose elements are maps which are $\delta$-close to breaking.

§.§ The bundle $E'$

We define the bundle

$$E'\to \widetilde{\mathcal{G}}_\delta(V_1/\R,V_2/\R)$$

for $\delta>0$ small as follows:  Given $(\F,u)=((F,j,{\bf p},{\bf q},{\bf r}),u)\in \widetilde{\mathcal{G}}_\delta(V_1/\R,V_2/\R)$, consider Step 1 of the proof of Theorem <ref>.  The $\varepsilon$-thin part $\op{Thin}_\varepsilon(\ddot F,g)$ for $g=g_{(\ddot F,j)}$ is now a union of cusps (as in Equation (<ref>)) and annuli. We now apply Step 1 of the proof of Theorem <ref> and use the boundaries of the cusps to obtain $\{s_{+,i}(\F,u)\}$ and $\{s_{-,i}(\F,u)\}$ (corresponding to the $\pm$ ends of $(\F,u)$) and the boundaries of the annuli to obtain $\{s_{-,i}'(\F,u)\}$ (corresponding to the negative ends of the top level) and $\{s_{+,i}'(\F,u)\}$ (corresponding to the positive ends of the bottom level). This gives us $E^{\ell,\varepsilon}_{(\F,u),\gamma}$ as in Equation (<ref>), where $\gamma$ is viewed as an element of

$$\bs\gamma_+\sqcup \bs \gamma \sqcup \bs \gamma\sqcup \bs \gamma_-.$$

We then set

\begin{equation} \label{equation: defn of E prime}
E'_{(\F,u)}:=\bigoplus_{\gamma\in\bs\gamma_+\sqcup \bs \gamma \sqcup \bs \gamma\sqcup \bs \gamma_-}E^{\ell,\varepsilon}_{(\F,u),\gamma}.
\end{equation}

§.§ Statements of gluing theorems

In this subsection we collect the gluing theorems that we need to construct the semi-global Kuranishi structures.  They are standard gluing results and will be proven in the next section.

In general, the $2$-level building condition in Assumption <ref> is replaced by an analogous $m$-component condition: We are gluing $m$ non-trivial (not trivial cylinder) components, one from each $V_i/\R$, and we can analogously define

$$E'\to \widetilde{\mathcal{G}}_\delta(V_1/\R,\dots, V_m/\R).$$

Let us define

$$\widetilde{\mathcal{G}}^{E'}_\delta(V_1/\R,\dots,V_m/\R):= \overline\bdry_J^{-1}(E')\subset \widetilde{\mathcal{G}}_\delta(V_1/\R,\dots,V_m/\R).$$

We also introduce the notation $V_i^{\text{sh}}$ (resp. $V_i^{\text{en}}$) to denote a slight $\R$-invariant shrinking (resp. enlargement) of $V_i$ which contains $K_i$ and is still open.

For sufficiently large $R>0$, there exists a gluing map

\begin{equation}
G_{(1,\dots,m)}:(V_1/\R) \times\dots \times (V_m/\R) \times [R,\infty)^{m-1}\to \widetilde{\mathcal{G}}^{E'}_\delta(V_1/\R,\dots, V_m/\R)
\end{equation}

which is a $C^1$-diffeomorphism onto its image and whose image contains

$$\widetilde{\mathcal{G}}^{E'}_{\delta'}(V_1^{\text{\em sh}}/\R,\dots,V_m^{\text{\em sh}}/\R)$$

for sufficiently small $\delta'>0$.

Writing $V_{(i,\dots,j)}/\R=\widetilde{\mathcal{G}}^{E'_{(i,\dots,j)}}_{\delta'}(V_i^{\text{sh}}/\R,\dots,V_j^{\text{sh}}/\R)$, where $E'_{(i,\dots,j)}$ is $E'$ for $V_i/\R,\dots,$ $V_j/\R$, we also have the following:

For sufficiently large $R>0$, there is a gluing map

\begin{align*}
G_{(1,\dots, (i,\dots,j),\dots m)}:&  (V_1/\R)\times\dots\times (V_{i-1}/\R)\times (V_{(i,\dots,j)}/\R) \\
& \times (V_{j+1}/\R)\times  \dots\times (V_m/\R) \to \widetilde{\mathcal{G}}^{E'_{(1,\dots,m)}}_\delta(V_1/\R,\dots, V_m/\R),
\end{align*}

such that $G_{(1,\dots, (i,\dots,j),\dots m)}\circ (id,\dots, G_{(i,\dots,j)},\dots, id)$ is $C^1$-close to the gluing map $G_{(1,\dots,m)}$.

§ DETAILS OF GLUING

We prove Theorem <ref> with $m=2$, following the general outline of [22] and [2].  The case for general $m$ only differs in notation.  The proof of Theorem <ref> also uses the same type of estimates as in Theorem <ref>  and is left to the reader.

We are assuming that $\varepsilon>0$ is small, that $(\alpha,J)$ is an $L$-simple pair (cf. Definition <ref>), and that $J$ is smooth.

We write $E_+\to V_+$ for $E_2\to V_2$ and $E_-\to V_-$ for $E_1\to V_1$.  Let $\llbracket\F_\pm,u_\pm \rrbracket \in \V_\pm/\R$, where $\F_\pm=(F_\pm,j_\pm,{\bf p}_\pm,{\bf r}_\pm)$. For simplicity we assume that $(\F_+,u_+)$ is a curve from $\gamma_+$ to $\bs\gamma=(\gamma_1,\dots,\gamma_k)$ and $(\F_-,u_-)$ is a curve from $\gamma_1$ to $\bs\gamma_-=(\gamma_{-,1},\dots,\gamma_{-,l})$, i.e., we are gluing the first negative end of $[(\F_+,u_+)]$ to the positive end of $[(\F_-,u_-)]$.  We fix representatives $(\F_\pm,u_\pm)$ of $\V_\pm/\R$ such that $s_{\mp,1}(\F_\pm,u_\pm)=0$. Here $s_{\pm,i}(\F,u)$ is as defined in the proof of Theorem 5.1.2.  We write “$(\F_\pm,u_\pm)\in \V_\pm/\R$” to mean $(\F_\pm,u_\pm)$ is the chosen representative in its equivalence class.  Note that the elements of $(E_\pm)_{(\F_\pm,u_\pm)}$ are supported on $-1\leq s\leq 0$ for $(\F_+,u_+)$ and on $0\leq s\leq 1$ for $(\F_-,u_-)$.

In general, if we decorate $(\mathcal{F},u)$ with subscripts and superscripts as in $(\mathcal{F}^*_\star,u^*_\star)$, then $\mathcal{F}^*_\star=(F^*_\star, j^*_\star, {\bf p}^*_\star,{\bf r}^*_\star)$ and $\dot F^*_\star=F^*_\star-{\bf p}^*_\star$.

In the next several subsections we retrace the steps of Sections 8.1–8.4 of [2], adapted to the current setting.

§.§ Pregluing

Fix a constant $T_0\gg 0$.  Also let $T\gg T_0$, which is allowed to vary.

If $\tau_T: \R\times M \stackrel\sim\to \R\times M$ is  given by the translation $(s,x)\mapsto (s+T,x)$, then let $u_{\pm,T}=\tau_{\pm (T+T_0)}\circ u_\pm$.  In general a subscript or superscript $+,T$ (resp. $-,T$) indicates the result of translating up (resp. down) by $s=T+T_0$.

Recall the simple coordinates $(s,t,x=(x_1,\dots,x_n),y=(y_1,\dots,y_n))$ on a sufficiently small neighborhood $\R\times\R/\mathcal{A}_\alpha(\gamma_1)\Z \times D$ of $\R\times \gamma_1$; here we are slightly abusing notation and passing to a finite cover if $\gamma_1$ is not simple. We assume that $T_0>0$ is sufficiently small so that $u_-|_{s\geq -T_0}$ (resp. $u_+|_{s\leq T_0}$) can be written as a graph $(s,t,\eta_-(s,t))$ (resp. $(s,t,\eta_+(s,t))$) over $[0,\infty)\times \gamma_1$ (resp. $(-\infty,0]\times\gamma_1$).

Fix constants $0<h<1$ and $r\gg h^{-1}$.  We take $T_0> 5r$.
Choose a cutoff function $\beta: \mathbb{R} \to [0,1]$ such that $\beta(s)=0$ for $s\leq 0$ and $\beta(s)=1$ for $s\geq 1$.  Let $\beta_{-,T}(s)=\beta({T-s\over hr})$ and $\beta_{+,T}(s)=\beta({T+s\over hr})$.

For $T'\leq T+T_0$, let $\mathcal{E}_{+, T'}$ be the component of

$$u_{+,T}^{-1}(\{s< T'\})\simeq u_+^{-1}(\{s< T'-(T+T_0)\})$$

corresponding to $\gamma_1$, let $\mathcal{E}_{-,T'}$ be

$$u_{-,T}^{-1}(\{s> -T'\})\simeq u_-^{-1}(\{s> -T'+(T+T_0)\}),$$

and let $A_{[T_1,T_2]}$ be the annulus $[T_1,T_2]\times \R/\mathcal{A}_\alpha(\gamma_1)\Z$ with coordinates $(s,t)$ and the standard complex structure.

The cutoff functions $\beta_{\pm,T}$.  The little bump functions indicate the support of $(E_\pm)_{(\F_\pm,u_{\pm,T})}$

We define the pregluing of $u_+$ and $u_-$ with gluing parameter $T$ as follows: Let $F^\circ_+:=\dot F_+ -\mathcal{E}_{+,T}$ and $F^\circ_-:=\dot F_- -\mathcal{E}_{-,T}$.  The domain $(\dot F =F-{\bf p},j)$ is then obtained by gluing $(F^\circ_+,j_+|_{F^\circ_+})$, $(F^\circ_-,j_-|_{F^\circ_-})$, and $A_{[-T,T]}$ in the expected way. The preglued map $u_*=u_*(u_+,u_-,T)$ agrees with $u_{+,T}$ on $F^\circ_+$ and with $u_{-,T}$ on $F^\circ_-$, and is given by

\begin{equation}\label{eqn: pregluing}
u_*(s,t)= (s,t, \beta_{+,T}(s)\eta_{+,T}(s,t)+\beta_{-,T}(s)\eta_{-,T}(s,t) )
\end{equation}

on $A_{[-T,T]}$.

§.§ Gluing of domain complex structures

We consider the following domain gluing, which is close to (but not quite the same as) the gluing of the domains in Section <ref>.  The reason for the slight discrepancy is that each of $F^\circ_+$ and $F^\circ_-$ is obtained from $\dot F_+$ and $\dot F_-$ by removing a cusp that is close to but not necessarily the same as a $\widetilde \varepsilon_\pm$-thin cusp for some $\widetilde \varepsilon_\pm$.

Fix $\dot F_-$, $\dot F_+$, and $\varepsilon''>0$ small. We are gluing $\dot F_-$ and $\dot F_+$ along one “positive end” of $\dot F_-$ and one “negative end” of $\dot F_+$. Let $\widetilde{\mathcal{U}}_\pm$ be a Teichmüller slice for a small open set $\mathcal{U}_\pm\subset \op{Teich}(\dot F_\pm)$ and let $g(j_\pm)$ be the complete finite-volume hyperbolic metric compatible with $j_\pm\in \widetilde{\mathcal{U}}_\pm$. We also fix $j^0_\pm\in \widetilde{\mathcal{U}}$. Let $\mathcal{E}_\pm^{\varepsilon''}(j_\pm)$ be the (interior of the) $\varepsilon''$-thin component of $g(j_\pm)$ corresponding to the end that is glued and let $F_\pm^\bullet(j_\pm):= \dot F_\pm - \mathcal{E}_\pm^{\varepsilon''}(j_\pm)$.  Also choose $r_\pm(j_\pm)\in \bdry F_\pm^\bullet(j_\pm)$ smoothly with respect to $j_\pm$.

Given $(T,\tau)\in [R,\infty)\times \R$, we glue

$$(F_-^\bullet(j_-),j_-) \cup ([0,T]\times \R/\Z,j_{std}) \cup (F_+^\bullet(j_+),j_+)$$

so that $r_-(j_-)$ is identified with $(0,0)\in [0,T]\times \R/\Z$ and $r_+(j_+)$ is identified with $(T,\tau)\in [0,T]\times \R/\Z$.
We denote the resulting Riemann surface by $(\dot F(j_-,j_+,T,\tau),j(j_-,j_+,T,\tau))$. We also write $\dot F=\dot F(j_-^0,j_+^0,R,0)$; this is our reference surface.  (Note that when $(T,\tau_1), (T,\tau_2)\in [R,\infty)\times \R$ and $\tau_2-\tau_1\in \Z$, the complex structures are diffeomorphic, but differ by a Dehn twist about a separating surface and correspond to distinct points in Teichmüller space.)

This defines a map

$$\kappa:\widetilde{\mathcal{U}}_-\times \widetilde{\mathcal{U}}_+\times[R,\infty)\times \R\to \op{Teich}(\dot F),$$

$$(j_-,j_+,T,\tau)\mapsto (\dot F(j_-,j_+,T,\tau),j(j_-,j_+,T,\tau)).$$

For $R\gg 0$, $\kappa$ is a $C^1$-embedding.

This follows from Kazdan-Warner [23, 24]. For $R\gg 0$, we can down a metric with negative curvature on $\dot F(j_-,j_+,T,\tau)$ which is conformally equivalent to $j(j_-,j_+,T,\tau)$, agrees with $g(j_\pm)$ on $F_\pm^\bullet(j_\pm)$, and whose curvature is very close to $-1$ on $[0,T]\times \R/\Z$. In order to find a hyperbolic metric in the same conformal class we solve an equation of type $\Delta u = 1-he^u$, where $u:\dot F\to \R$, $\Delta$ is a Laplacian, and $h:\dot F\to \R$ is close to $1$ (and approaches $1$ as $R\to \infty$); see <cit.>.  When $h$ is close to $1$, this can be solved using the inverse function theorem; see [24].  We remark that negative curvature implies that the linearized operator is invertible and also that solving the equation can be done in a differentiable family.

The above discussion implies that the Fenchel-Nielsen coordinates for $j_\pm\in \widetilde{\mathcal{U}}_\pm$ are very close to the corresponding Fenchel-Nielsen coordinates for $\kappa(j_-,j_+,T,\tau)\in \op{Teich}(\dot F)$ when $R\gg 0$, implying the theorem.

§.§ Banach spaces

The function spaces that we use are Morrey spaces, following <cit.>.[This is rather nonstandard and we chose to adopt it to avoid redoing some work in [22] for $W^{k,p}$-spaces.] Let $u:\dot F\to \R\times M$ be a finite energy holomorphic curve. On $\dot F$ we choose a Riemannian metric so that the ends are cylindrical and  we use the $\R$-invariant Riemannian metric on $\R\times M$.

The Morrey space $\mathcal{H}_0(\dot F, \wedge^{0,1} T^*\dot F\otimes  u^*T(\R\times M))$ is the Banach space which is the completion of the compactly supported sections of $\wedge^{0,1} T^*\dot F\otimes u^*T(\R\times M)$ with respect to the norm

$$\| \xi\|= \left( \int_{\dot F} |\xi|^2  \right)^{1/2} + \left( \sup_{x\in \dot F} \sup_{\rho\in(0,1]} \rho^{-1/2} \int_{B_\rho(x)} |\xi|^2  \right)^{1/2},$$

where $B_\rho(x)\subset \dot F$ is the ball of radius $\rho$ about $x$.  Similarly, $\mathcal{H}_1(\dot F, u^*T(\R\times M))$ is the completion of the compactly supported sections of $u^*T(\R\times M)$ with respect to
$$\| \xi \|_*= \|\nabla \xi\| + \|\xi\|.$$

The analog of the usual Sobolev embedding theorem is the following:

There is a bounded linear map

$$\mathcal{H}_1(\dot F, u^*T(\R\times M))\to C^{0,1/4}(\dot F, u^*T(\R\times M)), \quad \xi\mapsto \xi,$$

where $C^{0,1/4}$ denotes the space of Hölder continuous functions with exponent ${1\over 4}$.

We also define the weighted Morrey spaces $\mathcal{H}_{i,\delta}(*)=\mathcal{H}_{0,\delta}(\dot F_\pm, \wedge^{0,1} T^*\dot F_\pm\otimes  (u_\pm)^*T(\R\times M))$ and $\mathcal{H}_{1,\delta}(\dot F_\pm, (u_\pm)^*T(\R\times M))$ with weight $\delta>0$ as the space of sections $\psi$ such that $\psi\cdot f_\delta^\pm\in \mathcal{H}_i (*)$. Here $f_\delta^\pm: \R\times M\to \R^+$ is the weight function for $u_\pm$ which agrees with $e^{\delta |s|}$ at the ends and is normalized so that $f_\delta^\pm(s,x)= e^{\delta|s|}$ for $\mp s\geq 0$. Their norms are denoted by $\|\cdot \|_\delta$ and $\|\cdot\|_{*,\delta}$.  We will also write $\|\cdot\|_\delta^{\pm,T}$ and $\|\cdot\|_{*,\delta}^{\pm,T}$ with respect to the translates $f_\delta^{\pm,T}$.

§.§ The setup

Let $\psi_{\pm}$ be sections of $u_{\pm}^*T(\R\times M)$ and let $\psi_{\pm,T}=\tau_{\pm (T+T_0)}\circ \psi_\pm$ be sections of $u^*_{\pm,T} T(\R\times M)$.  The exponential maps $\exp_{u_{\pm,T}}$ can be chosen such that

$$\exp_{u_{\pm,T}}\psi_{\pm,T}= (s,t,\eta_{\pm,T})+\psi_{\pm,T}$$

on $\mathcal{E}_{\pm,T+T_0}$, by taking the Riemannian metric $g_0$ on $\R\times M$ to be $\R$-invariant and equal to the standard flat metric on $\R\times\R/\mathcal{A}_\alpha(\gamma_1)\Z \times D$.

We then deform $u_*$ to

\begin{equation}\label{equation: deform u*}
u=\exp_{u_*} (\beta_{+,T}\psi_{+,T}+\beta_{-,T}\psi_{-,T}),
\end{equation}

where $\exp_{u_*}$ agrees with $\exp_{u_{\pm,T}}$ on $\dot F_\pm -\mathcal{E}_{\pm,T+T_0}$ and with

$$\exp_{u_*} (\beta_{+,T}\psi_{+,T}+\beta_{-,T}\psi_{-,T})=(s,t,\eta_*)+\beta_{+,T}\psi_{+,T}+\beta_{-,T}\psi_{-,T}$$

on $A_{[-(T+T_0), T+T_0]}$.
Here $\eta_*=\beta_{+,T}\eta_{+,T} +\beta_{-,T}\eta_{-,T}$.  We want to solve for $\psi_+$ and $\psi_-$ in the equation

\begin{equation}
\overline{\partial}u\in E'_{(\F,u)},
\end{equation}

with $(\F,u)=(F,j,{\bf p}, u)$.  Here $J$ is understood and $E'$ is as given in Equation (6.3.1).

We identify

$$\Phi=\Phi_T:\wedge^{0,1}u^*  T(\mathbb R\times M)\stackrel\sim\longrightarrow\wedge^{0,1}(u_*)^*T(\R\times M)$$

as follows: away from the neck region $A_{[-(T+2T_0), T+2T_0]}$ we parallel transport using a complex linear connection induced by the Levi-Civita connection of an adapted Riemannian metric (cf. <cit.>); on $A_{[-(T+T_0),T+T_0]}$ we map $\eta\otimes (ds-idt)\mapsto \eta\otimes (ds-idt)$; and on $A_{[-(T+2T_0), -(T+T_0)]}$ and $A_{[T+T_0,T+2T_0]}$ we interpolate between the two identifications.

Similarly, we identify

$$\Phi_\pm=\Phi_{\pm}^{\psi_{\pm,T}}: \wedge^{0,1}(\exp_{u_{\pm,T}}\psi_{\pm,T})^{*}T(\R\times M)\stackrel\sim\longrightarrow\wedge^{0,1}u_{\pm,T}^{*}T(\R\times M).$$

The equation $\overline{\partial} u\in E'_{(\F,u)}$, or more precisely the equation

\begin{equation}\label{d-bar2}
{\Phi} \overline{\partial} u\in {\Phi } E'_{(\F,u)},
\end{equation}

can then be written as

\begin{equation} \label{eqn: d-bar rewritten}
\beta_{-,T} \left(D_{-,T} \psi_{-,T} + e_{-,T} +\mathcal{L}_- +\mathcal{R}_-\right)+\beta_{+,T}\left(D_{+,T}\psi_{+,T} + e_{+,T}+\mathcal{L}_+ +\mathcal{R}_+\right)\in {\Phi} E_{(\mathcal F, u)}',
\end{equation}

where $D_\pm$ (resp. $D_{\pm,T}$) is the linearization of $\overline{\partial}$ for $u_{\pm}$ (resp. $u_{\pm,T}$), $e_\pm:=\overline{\partial} u_\pm$ (resp. $e_{\pm,T}:=\overline{\partial} u_{\pm,T}$), and $\mathcal{L}_\pm$ and $\mathcal{R}_\pm$ are described below. (The descriptions of $\mathcal{L}_\pm$ and $\mathcal{R}_\pm$ on $A_{[-T,T]}$ are obtained by expanding $\tfrac{\bdry u}{\bdry s} +J(u) \tfrac{\bdry u}{\bdry t}$, where we write

$$u(s,t)=\begin{pmatrix} s \\ t \\ \beta_{+,T}\eta_{+,T} +\beta_{-,T}\eta_{-,T}\end{pmatrix}+\beta_{+,T}\psi_{+,T}+\beta_{-,T}\psi_{-,T}$$

\begin{pmatrix}
0 & -1 & 0\\
1 & 0 & 0\\
X\eta' & -j_0 X\eta'  & j_0
\end{pmatrix},$$

* we are using simple coordinates $(s,t,(x,y))$ written as column vectors;

* $\eta'= \beta_{+,T}(\eta_{+,T}+\psi_{+,T}^N) +\beta_{-,T}(\eta_{-,T}+\psi_{-,T}^N)$;

* $X$ is the constant matrix defined in the paragraph after Definition 3.1.1;

* $j_0$ is the standard complex structure on $\mathbb R^{2n}$;

* $\psi^N_{\pm,T}$ is the $(x,y)$-component of $\psi_{\pm,T}$.

The details are left to the reader.)

1. $\mathcal L_\pm=\mathcal L_\pm(\psi_{-,T},\psi_{+,T})$ has support on $A_{[-T,T]}$ and is given by

\begin{align} \label{equation: mathcal L}
2\mathcal L_\pm= &
\left(
\begin{array}{ccc}
0\\0\\ \frac{\partial \beta_{\mp,T}}{\partial s}\eta_{\mp,T}
\end{array}
\right)+\frac{\partial \beta_{\mp,T}}{\partial s}\psi_{\mp,T}\\
\nonumber &
\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0 \\
X \eta_{\pm,T} &  -j_0 X \eta_{\pm,T} & 0
\end{array}
\right)\frac{\partial \psi_{\mp,T}}{\partial t}
\\
\nonumber &+\widetilde\beta_{\pm,T}\beta_{\mp,T}\left(
\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0 \\
X \psi^N_{\pm,T} &  -j_0 X \psi^N_{\pm,T} & 0
\end{array}
\right)\frac{\partial \psi_{\mp,T}}{\partial t}\\
\nonumber &+
\widetilde\beta_{\pm,T}(\beta_{\pm,T}-1)\left(
\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0 \\
X\eta_{\pm,T} & -j_0X\eta_{\pm,T} & 0
\end{array}
\right)
\frac{\partial \psi_{\pm,T}}{\partial t}.
\end{align}

* the factor $\otimes (ds-idt)$ is omitted from each term;

* $\widetilde\beta_{-,T}(s)=\beta({T+hr-s\over hr})$ and $\widetilde\beta_{+,T}(s)=\beta({T+hr+s\over hr})$;

* We say that a function $F(\psi)$ is “linear”  if there exists a constant $c>0$ such that $|F(\psi)(x)|<c|\psi(x)|$ at every point $x$ of the domain of $\psi$.  We will use the shorthand ${\frak l}(\psi)$ to denote an unspecified “linear" map.

* Following <cit.> we say that $F(\psi)$ is type $1$ quadratic if it can be written as

$$F(\psi)=P(\psi) + Q(\psi)\cdot \nabla\psi,$$

where there exists a constant $c>0$ such that $|P(\psi)(x)|\leq c|\psi(x)|^2$ and $|Q(\psi)(x)|\leq c|\psi(x)|$ at every point $x$ of the domain of $\psi$.

2. The remaining terms of ${\Phi} \overline{\partial}u$ are grouped into $\beta_-\mathcal{R}_-$ and $\beta_+\mathcal{R}_+$.

One can see that $\mathcal{R}_{\pm}= \mathcal{R}_{\pm} (\psi_{\pm,T})$ is type $1$ quadratic. On $A_{[-T,T]}$, we can explicitly write

\begin{equation}
\mathcal R_\pm={1\over 2} \beta_{\pm,T}
\left(
\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0\\
X\psi_{\pm,T}^N & - j_0X\psi_{\pm,T}^N & 0
\end{array}
\right)
\frac{\partial \psi_{\pm,T}}{\partial t}.
\end{equation}

We also write $E'_{(\F,u)}=E'_{\psi_\pm,T}$ and decompose

$$E_{\psi_\pm,T}'=(E'_{\psi_\pm,T})_+\oplus (E'_{\psi_\pm,T})_-,$$

where $(E'_{\psi_\pm,T})_\pm$ corresponds to $u_{\pm,T}$. In order to solve Equation (<ref>) we solve for $(\psi_-,\psi_+)$ in the pair of equations

\begin{equation} \label{theta+-}
D_{\pm,T} \psi_{\pm,T} + e_{\pm,T} + \mathcal{L}_{\pm}(\psi_{-,T},\psi_{+,T}) + \mathcal{R}_{\pm}(\psi_{\pm,T})\in \Phi_{\pm}^{\psi_{\pm,T}} (E'_{\psi_\pm,T})_\pm.
\end{equation}

$$\Pi_\pm^{\psi_\pm,T}:\mathcal{H}_{0,\delta}(\dot F, \wedge^{0,1}u_{\pm,T}^{*}T(\R\times M))\to  (\Phi_\pm^{\psi_{\pm,T}} (E'_{\psi_\pm,T})_\pm)^\perp$$

be the projection onto the $L^2$-orthogonal complement of $\Phi_\pm^{\psi_{\pm,T}} (E'_{\psi_\pm,T})_\pm$.

Then Equation (<ref>) is equivalent to

\begin{equation} \label{Theta pm}
\Theta_\pm(\psi_-,\psi_+) :=\Pi_\pm^{\psi_\pm,T}(D_{\pm,T} \psi_{\pm,T} + e_{\pm,T} + \mathcal L_\pm(\psi_{-,T},\psi_{+,T}) + \mathcal{R}_\pm(\psi_{\pm,T}))=0.
\end{equation}

Since $D_{\pm,T}$ is transverse to $E_\pm$ and $\Phi_{\pm}^{\psi_{\pm,T}} (E'_{\psi_\pm,T})_\pm$ is close to $E_\pm$ when $T\gg 0$ and $\|\psi_\pm\|_{*,\delta}$ is small, it follows that $D_{\pm,T}$ is transverse to $\Phi_{\pm}^{\psi_{\pm,T}} (E'_{\psi_\pm,T})_\pm$, i.e.,

$$\Pi_\pm^{\psi_\pm,T} D_{\pm,T}: \mathcal{H}_{1,\delta}(\dot F, u_{\pm,T}^{*}T(\R\times M))\to (\Phi_\pm^{\psi_{\pm,T}} (E'_{\psi_\pm,T})_\pm)^\perp$$

is surjective.

§.§ Definition of gluing map

In this subsection we make some estimates and define the gluing map at the end.  We use the convention that constants such as $C$ may change from line to line when making estimates.

Let $\lambda= \min \{\lambda_1, |\lambda_{-1}|\}$. Suppose that $\delta$ satisfies $0<100\cdot\delta<\lambda$.  Let $\mathcal{H}_\pm =\mathcal{H}_{1,\delta}(\dot F,u_\pm^{*}T(\R\times M))$, let $\mathcal{H}_\pm^{\perp,\psi_{\pm},T}$ be the $L^2$-orthogonal complement of

$$\ker \left( \Pi_\pm^{\psi_\pm,T}\circ D_{\pm,T}\circ \tau_{\pm(T+T_0)}\right)$$

in $\mathcal{H}_{\pm}$, where $\tau_{\pm(T+T_0)}$ is the map induced by $\tau_{\pm(T+T_0)}$, and let $\mathcal{B}_{\pm}$ be the closed ball of radius $\widetilde\varepsilon$ in $\mathcal{H}_{\pm}$ centered at $0$.

The following lemma is modeled on <cit.>, but the estimates are slightly different.

There exist $r\gg 0$ and $C,\widetilde\varepsilon>0$ such that for $T\gg 0$ and $\|e_+\|_{*,\delta}$ small the following holds:

* There is a map $P_+:\mathcal{B}_-\to \mathcal{H}_+$ such that $\Theta_+(\psi_-,P_+(\psi_-))=0$ and $\psi_+=P_+(\psi_-)\in \mathcal{H}_+^{\perp,\psi_+,T}$.

* $\| P_+(\psi_-)\|_{*,\delta}\leq C (\|e_{+}\|_{*,\delta}+ r^{-1/2}e^{-2(\lambda-\delta) T}+(e^{-2\delta T}+r^{-1})\|\psi_-\|_{*,\delta})$.

* $\| P_+(\psi_-) - P_+(\psi'_-)\|_{*,\delta} \leq C \| \psi_- - \psi_-'\|_{*,\delta} (e^{-2\delta T} + r^{-1}  +\widetilde\varepsilon)$.

Let $\mathcal{I}_+: \mathcal{B}_-\times \mathcal{B}_+\to \mathcal{H}_+$ be the map given by:

\begin{align} \label{def of I}
\mathcal{I}_+(\psi_-,\psi_+) =&  -\tau_{+(T+T_0)}^{-1}(\Pi_+^{\psi_+,T}D_{+,T})^{-1}\Pi_+^{\psi_+,T}(e_{+,T}  + \mathcal L_+(\psi_{-,T},\psi_{+,T}) + \mathcal{R}_+(\psi_{+,T})).
\end{align}

Here $(\Pi_+^{\psi_+,T}D_{+,T})^{-1}$ is the bounded inverse of $\Pi_+^{\psi_+,T}D_{+,T}$.

(1) We are trying to solve for $\psi_+$ in

$$\Pi_+^{\psi_+,T} (D_{+,T} \psi_{+,T}+e_{+,T}+\mathcal L_+(\psi_{-,T},\psi_{+,T})+\mathcal R_+(\psi_{+,T}))=0.$$

Using the function $\mathcal{I}_+$ we define $P(\psi_-)$ as the unique fixed point, i.e., $\psi_+$ satisfying

\begin{equation} \label{fixed point}
\mathcal{I}_+(\psi_-,\psi_+)=\psi_+.
\end{equation}

The unique fixed point is guaranteed by the contraction mapping theorem, which follows from Claims <ref> and <ref>.

\begin{align*}
\|\mathcal{I}_+(\psi_-,\psi_+)\|_{*,\delta}  \leq & (\widetilde C_1(\|\psi_+\|_{*,\delta})+\widetilde C_2(T))\|e_{+}\|_{*,\delta} \\
\nonumber & + \widetilde C(\|\psi_+\|_{*,\delta}) ( r^{-1/2}e^{-2(\lambda-\delta) T}+(e^{-2\delta T}+r^{-1})\|\psi_-\|_{*,\delta}\\
\nonumber &\quad  +\|\psi_-\|_{*,\delta}\|\psi_+\|_{*,\delta}  +e^{-2\lambda T}\|\psi_+\|_{*,\delta}+\|\psi_+\|^2_{*,\delta}),
\end{align*}

where $\widetilde C(\sharp)$, $\widetilde C_1(\sharp)$, $\widetilde C_2(\sharp)$ are continuous functions of $\sharp$;  $\widetilde C_1(\sharp)\to 0$ as $\sharp\to 0$; and $\widetilde C_2(\sharp)\to 0$ as $\sharp\to\infty$.

We will be using Lemma <ref> (more precisely $|\zeta|_{C^0}\leq C \|\zeta\|_{*,\delta}$) several times without explicit mention.

We first obtain

\begin{align} \label{one}
\left\| \mathcal L_+(\psi_{-,T},\psi_{+,T}) \right\|_\delta^{+,T}\leq &  C(r^{-1/2}e^{-2(\lambda-\delta) T}+r^{-1}\|\psi_-\|_{*,\delta}+e^{-2\delta T}\|\psi_-\|_{*,\delta}\\
\nonumber & \quad +\|\psi_-\|_{*,\delta}\|\psi_+\|_{*,\delta} +e^{-2\lambda T}\|\psi_+\|_{*,\delta}),
\end{align}

where each term on the right-hand side corresponds to the terms on the right-hand side in Equation (<ref>).  The first term has bound

\begin{align*}
\| \tfrac{\bdry \beta_{-,T}}{\bdry s}& \eta_{-,T}\|_\delta^{+,T}  = \| \tfrac{\bdry \beta_{-,T}}{\bdry s} \eta_{-,T} f_\delta^{+,T}\|
=\| \tfrac{\bdry \beta_{-,T}}{\bdry s} \eta_{-,T} e^{-\delta(s-(T+T_0))}\|  \\
& \leq C(r^{-1}\cdot e^{-\lambda\cdot 2T}\cdot e^{\delta\cdot 2T_0}\cdot \sqrt{hr}\cdot  + r^{-1}\cdot e^{-\lambda \cdot 2T}\cdot e^{\delta\cdot 2T_0})\\
& \leq C(r^{-1/2}e^{-2(\lambda-\delta) T}),
\end{align*}

where we are using the bound $|{\bdry\beta_-\over \bdry s}|< cr^{-1}$ for some $c>0$ and exponential decay bounds for $\eta_-$. The second term has bound

\begin{align*}
\|\tfrac{\bdry\beta_{-,T}}{\bdry s}\psi_{-,T} \cdot f_\delta^{+,T}\|& = \|\tfrac{\bdry\beta_{-,T}}{\bdry s}\psi_{-,T} \cdot e^{-\delta (s-(T+T_0))} \| \leq\|\tfrac{\bdry\beta_{-,T}}{\bdry s}\psi_{-,T} \cdot e^{2\delta T}\| \\
& \leq Cr^{-1} \|\psi_-\|_\delta\leq Cr^{-1}\|\psi_-\|_{*,\delta}.
\end{align*}

The third term has bound

\begin{align*}
C\|\widetilde\beta_{+,T}\beta_{-,T} & e^{\lambda (s-(T+T_0))}\tfrac{\bdry \psi_{-,T}}{\bdry t} e^{-\delta (s-(T+T_0))}\| \\
& = C \|\widetilde\beta_{+,T}\beta_{-,T} e^{ (\lambda-2\delta)s-\lambda(T+T_0)}  \tfrac{\bdry \psi_{-,T}}{\bdry t}e^{\delta( s+ (T+T_0))}  \| \\
& \leq  C  e^{-2\delta(T+T_0)}\|\psi_-\|_{*,\delta} \leq C e^{-2\delta T}\|\psi_-\|_{*,\delta}.
\end{align*}

The fourth term has bound

\begin{align*}
C\|\widetilde\beta_{+,T}\beta_{-,T} & \psi_{+,T} \psi_{-,T} e^{-\delta(s-(T+T_0))}\| \\
& \leq  C |\psi_{+,T} e^{-\delta(s-(T+T_0))}|_{C^0}\cdot \|\widetilde\beta_{+,T}\beta_{-,T} \psi_{-,T}\|\\
& \leq C \|\psi_+\|_{*,\delta} \|\psi_-\|_{\delta}\leq C \|\psi_+\|_{*,\delta} \|\psi_-\|_{*,\delta}.
\end{align*}

The last term has bound

\begin{align*}
C\| (\widetilde \beta_{+,T})(\beta_{+,T}-1)\eta_{+,T} \tfrac{\bdry\psi_{+,T}}{\bdry t} e^{-\delta(s-(T+T_0))}\| & \leq C e^{-2\lambda T} \|\psi_+\|_{*,\delta}.
\end{align*}

Also we obtain

\begin{align}\label{two}
\| \mathcal{R}_+(\psi_{+,T})\|^{+,T}_\delta \leq C\|\psi_+\|^2_{*,\delta},
\end{align}

since $\mathcal{R}_+(\psi_{+,T})$ is type 1 quadratic.

Next we consider the $L^2$-projection $\Pi_+$ to $(\Phi_+ E'_+)^\perp$, where we are suppressing ${\psi_\pm,T}$ from the notation. Let $e_1(\psi_+,T),\dots,e_k(\psi_+,T)$ be an orthonormal basis for $\Phi_+ E'_+$; it is not hard to see that $e_i(\psi_+,T)$ can be taken to be continuous with respect to $\psi_+\in \mathcal{B}_+$.

\begin{align*}
\Pi_+(\zeta)=\zeta - \sum_i \langle e_i(\psi_+,T),\zeta\rangle e_i(\psi_+,T),
\end{align*}

where $\langle\cdot,\cdot\rangle$ is the $L^2$-inner product.
This allows us to estimate

\begin{align} \label{three}
\|\Pi_+(\zeta)\|_\delta \leq   \|\zeta\|_\delta +\sum_i\|\zeta\|_\delta\cdot \|e_i(\psi_+,T)\|_\delta\leq \widetilde C(\|\psi_+\|_{*,\delta}) \|\zeta\|_\delta.
\end{align}

The first inequality follows from $|\langle e_i(\psi_+,T),\zeta\rangle|\leq \|e_i(\psi_+)\|_{L^2} \cdot\|\zeta\|_{L^2} \leq \|\zeta\|_\delta$, where $\| \cdot \|_{L^2}$ is the $L^2$-norm, and the second inequality follows from the continuity of $e_i(\psi_+,T)$ with respect to $\psi_+$.

$(\Pi_+ D_+)^{-1}$ depends continuously on $\psi_+\in\mathcal{B}_+$.

Observe that the domain of $(\Pi_+ D_+)^{-1}$ uses the norm $\|\cdot \|_\delta$ and the range of $(\Pi_+ D_+)^{-1}$ uses $\|\cdot \|_{*,\delta}$.

$\Pi_+$ depends continuously on $\psi_+\in\mathcal{B}_+$ and hence so does $\Pi_+ D_+$.  The inverse hence also depends continuously on $\psi_+$.

We also obtain

\begin{align} \label{four}
\|\tau_{+(T+T_0)}^{-1}(\Pi_+^{\psi_+,T}D_{+,T})^{-1}\Pi_+^{\psi_+,T}(e_{+,T})\|_\delta \leq (\widetilde C_1(\|\psi_+\|_{*,\delta})+\widetilde C_2(T))\|e_{+}\|_{*,\delta}.
\end{align}

Claim <ref> then follows from combining Estimates (<ref>), (<ref>), (<ref>), (<ref>), and Claim <ref>.

If $r,T\gg 0$, $\widetilde\varepsilon>0$ is small, and $\|e_+\|_{*,\delta}$ is small, then $\|\mathcal I (\psi_-,\psi_+)\|_{*,\delta}<\widetilde\varepsilon$ whenever $\|\psi_-\|_{*,\delta}, \|\psi_+\|_{*,\delta}<\widetilde\varepsilon$. Hence $\mathcal{I}_+(\psi_-,\cdot)$ maps a radius $\widetilde\varepsilon$ ball into itself.

\begin{align*}
\|\mathcal{I}_+(&\psi_-,\psi_+) -\mathcal{I}_+(\psi_-, \psi_+')\|_{*,\delta}\\
\leq &  C ( \|\psi_-\|_{*,\delta}+e^{-2\lambda T}+\|\psi_+\|_{*,\delta} +\|\psi_+'\|_{*,\delta})\cdot \| \psi_+-\psi'_+\|_{*,\delta}\\
& + C\|\psi_+-\psi_+'\|_{*,\delta}\cdot (\|e_{+}\|_{*,\delta} + r^{-1/2}e^{-2(\lambda-\delta) T} +(e^{-2\delta T}+r^{-1})\|\psi_-\|_{*,\delta}\\
& \qquad\qquad \qquad +\|\psi_-\|_{*,\delta}\|\psi_+\|_{*,\delta}  +e^{-2\lambda T}\|\psi_+\|_{*,\delta}+\|\psi_+\|^2_{*,\delta}).
\end{align*}

This follows from the estimates

\begin{align*}
\| \mathcal{L}_+(\psi_{-,T},\psi_{+,T})-\mathcal{L}_+(\psi_{-,T},\psi_{+,T}')\|_\delta^{+,T} & \leq C (\|\psi_-\|_{*,\delta}+e^{-2\lambda T})\|\psi_+-\psi_+'\|_{*,\delta},\\
\|\mathcal{R}_+(\psi_{+,T})-\mathcal{R}_+(\psi_{+,T}')\|_\delta^{+,T} & \leq C( \|\psi_+\|_{*,\delta} +\|\psi_+'\|_{*,\delta}) \|\psi_+ - \psi_+'\|_{*,\delta},
\end{align*}

as well as the calculations from Claim <ref>.

Hence if $r,T\gg 0$, $\widetilde\varepsilon>0$ is small, and $\|e_+\|_{*,\delta}$ is small, then $\mathcal{I}_+(\psi_-,\cdot)$ gives a contraction mapping, provided $\|\psi_+\|_{*,\delta}, \|\psi_+'\|_{*,\delta}\leq \widetilde\varepsilon$. This proves (1).  We also prove the following, which is used in (3):

\begin{align*}
\|\mathcal{I}_+(\psi_-,\psi_+') &-\mathcal{I}_+(\psi_-',\psi_+')\|_{*,\delta} \\
& \leq   \widetilde C(\|\psi_+'\|_{*,\delta} ) (e^{-2\delta T} + r^{-1} + \|\psi_+'\|_{*,\delta})\|\psi_- - \psi_-'\|_{*,\delta}.
\end{align*}

This follows from the estimate

\begin{align*}
\| \mathcal{L}_+(\psi_{-,T},\psi_{+,T}')-\mathcal{L}_+(\psi_{-,T}',\psi_{+,T}')\|^{+,T}_\delta & \leq C (e^{-2\delta T}+ r^{-1} + \|\psi_+'\|_{*,\delta})\|\psi_- - \psi_-'\|_{*,\delta}.
\end{align*}

as well as the calculations from Claim <ref>.

Now we continue the proof of Lemma <ref>.

Since $\psi_+=P_+(\psi_-)$ satisfies $\mathcal{I}_+(\psi_-,\psi_+)=\psi_+$, Claim <ref> gives:

\begin{align*}
\|\psi_+\|_{*,\delta}  \leq & \widetilde C(\| \psi_+\|_{*,\delta}) (\|e_{+}\|_{*,\delta} + r^{-1/2}e^{-2(\lambda-\delta) T} +(e^{-2\delta T}+r^{-1})\|\psi_-\|_{*,\delta}\\
&+\|\psi_-\|_{*,\delta}\|\psi_+\|_{*,\delta} +e^{-2\lambda T}\|\psi_+\|_{*,\delta}+\|\psi_+\|^2_{*,\delta}).
\end{align*}

By moving the last three terms on the right-hand side to the left, we obtain (2) since $\widetilde C(\| \psi_+\|_{*,\delta})(\|\psi_-\|_{*,\delta}+e^{-2\lambda T}+ \|\psi_+\|_{*,\delta})<1 $ for $\widetilde\varepsilon>0$ small and $r>0$ large.

(3) Letting $\psi_+=P_+(\psi_-)$ and $\psi_+'= P_+(\psi_-')$,

\begin{align*}
\| P_+(\psi_-)- & P_+(\psi_-')\|_{*,\delta} =  \| \I (\psi_-,\psi_+)- \I (\psi_-',\psi_+')\|_{*,\delta}\\
\leq & \| \I (\psi_-,\psi_+)- \I (\psi_-,\psi'_+)\|_{*,\delta} + \| \I (\psi_-,\psi'_+) - \I (\psi_-',\psi_+')  \|_{*,\delta},
\end{align*}

and the two terms on the right-hand side are bounded using Claims <ref> and <ref>.  The terms from the right-hand side of the inequality in Claim <ref> can be moved to the left.  $\|\psi_-\|_{*,\delta}\leq \widetilde \varepsilon$ by assumption. If $r,T\gg 0$, $\widetilde\varepsilon>0$ is small, and $\|e_{+}\|_{*,\delta}$ is small, then $\|\psi_+\|_{*,\delta}=\|P_+(\psi_-)\|_{*,\delta}\leq \widetilde \varepsilon$ by (2). (3) then follows.

Analogously, we have the following for $P_-$:

There exist $r\gg 0$ and $C, \widetilde\varepsilon>0$ such that for $T\gg 0$ and $\|e_-\|_{*,\delta}$ small the following holds:

* There is a map $P_-:\mathcal{B}_+\to \mathcal{H}_-$ such that $\Theta_-(P_-(\psi_+),\psi_+))=0$ and $\psi_-=P_-(\psi_+)\in \mathcal{H}_-^{\perp,\psi_-,T}$.

* $\| P_-(\psi_+)\|_{*,\delta}\leq C (\|e_{-}\|_{*,\delta}+ r^{-1/2}e^{-2(\lambda-\delta) T}+(e^{-2\delta T}+r^{-1})\|\psi_+\|_{*,\delta})$.

* $\| P_-(\psi_+) - P_-(\psi'_+)\|_{*,\delta} \leq C \| \psi_+ - \psi_+'\|_{*,\delta} (e^{-2\delta T} + r^{-1} +\widetilde\varepsilon)$.