For any $\varepsilon > 0$ we define $\text{Thick}_\varepsilon(\dot{\mathcal{F}}_i, g_i)$ as the set of points $x \in \dot{\mathcal{F}}_i$ with injectivity radius $\rho_i(x) \ge \varepsilon$ and $\text{Thin}_\varepsilon(\dot{\mathcal{F}}_i, g_i)$ as the closure of the set of points $x \in \dot{\mathcal{F}}_i$ with injectivity radius $\rho_i(x) < \varepsilon$. If $i$ is sufficiently large and $\varepsilon$ is sufficiently small, then $\text{Thin}_\varepsilon(\dot{\mathcal{F}}_i, g_i)$ is a disjoint union of cusps and tubular neighborhoods of pinching geodesics. In this case, let $\nu$ be any pinching geodesic; we denote $\text{Thin}_\varepsilon(\dot{\mathcal{F}}_i, g_i, \nu)$ the connected component of $\text{Thin}_\varepsilon(\dot{\mathcal{F}}_i, g_i)$ containing $\nu$.

*Convergence on the thick part.* Fix $\varepsilon > 0$. Then the gradient bound (Equation (7.3.3)) implies that the sequence $\bar{u}_i$ converges in the $C_{\text{loc}}^\infty$ topology on the $\varepsilon$-thick part, after possibly passing to a subsequence and translating in the $s$-direction. In fact an elliptic bootstrapping argument propagates the gradient bound to bounds on the derivatives of any order and the Arzelà-Ascoli theorem produces a convergent subsequence. Now we take a sequence $\varepsilon_i \to 0$ and use a diagonal argument to find a limit finite energy holomorphic map $\bar{u}'_\infty$ defined on $F'_\infty$ (i.e., $\dot{\mathcal{F}}_\infty$ minus the double points). Depending on the connected component of $F'_\infty$, the restriction of $\bar{u}'_\infty$ can take values in $\overline{W}_-$, $\overline{W}$ or $\overline{W'}$. We observe that, although the maps $\bar{u}_i$ have boundary conditions in $L_{\bar{\mathfrak{a}}}^-$, the levels of the limit $\bar{u}'_\infty$ can have boundary conditions in $L_{\bar{\mathfrak{a}}}^-$ or in $L_{\bar{\mathfrak{a}}} \cup L_{\bar{\mu}(\bar{\mathfrak{a}})}$, depending on the connected component of $F'_\infty$.

We analyze the convergence of $\bar{u}'_\infty$ near the punctures in detail because the singularity in $L_{\bar{\mathfrak{a}}}^-$ is an unusual situation. We describe only the case of components mapping to $\overline{W}_-$; the behavior of components mapping to $\overline{W}$ or $\overline{W'}$ is analogous. The first step is to replace $\bar{\mathfrak{a}}$ with $\bar{\mathfrak{a}}$ (this is temporary notation for the union of $\bar{a}_i$ which have been extended by adding the $\bar{a}_{ij}$). This replacement has no practical effect on the curve $\bar{u}'_\infty$, but makes the geometric setting slightly less exotic: in fact the Lagrangian $L_{\bar{\mathfrak{a}}}^-$ has a *clean self-intersection* along $\partial B_- \times \{z_\infty\}$. We recall that a clean intersection is an intersection between two submanifolds which are nontransverse in a uniform way. Clean intersections in Lagrangian Floer homology — which represent the Morse-Bott case in that theory — have been treated in [Po] and [Schm].

There are four possibilities, depending whether the puncture is in the interior or on the boundary of $F'_\infty$ and whether $\bar{u}'_\infty$ is bounded or unbounded near the puncture. Interior punctures are treated in the standard way: if $\bar{u}'_\infty$ is bounded near an interior puncture, then the singularity is removable (i.e., $\bar{u}'_\infty$ can be extended as a holomorphic map across the puncture) by [MS, Theorem 4.1.2], and if $\bar{u}'_\infty$ is unbounded, then it is asymptotic to a closed orbit in the positive end of $\overline{W}_-$ by [HWZ1, Theorem 1.2]. (Note that this orbit can either be an orbit in $N$ or a multiple cover of the extra orbit $\delta_0$ over the point $z_\infty$.)

Next we consider boundary punctures of $\bar{u}'_\infty$. We restrict attention to the neighborhood $Z_- = (-\infty, 0] \times [0, 1] \subset \dot{\mathcal{F}}_\infty$ of a boundary puncture and prove the following lemma.

**Lemma 7.3.13.** — Let $w: Z_- \to \overline{W}_-$ be a finite energy $\bar{J}_-$-holomorphic map which maps $(-\infty, 0] \times \{0, 1\}$ to $L_{\bar{\mathfrak{a}}}^-$. Then there exists either

$$ (1) \text{ a point } p \in L_{\bar{\mathfrak{a}}}^- \text{ such that } \lim_{s \to -\infty} w(s, t) = p \text{ (uniformly in } t), \text{ or} $$
(2) an intersection point $y \in \bar{\mathbf{a}} \cap \bar{h}(\bar{\mathbf{a}})$ such that $w$ is asymptotic to the trivial strip over $y$ for $s \to -\infty$.

The removal of boundary singularities in the presence of clean intersections between Lagrangian submanifolds has been treated in [Frau, Section 3.7] and [Schm, Section 3]. For completeness we reprove it in a weak form (i.e., without reproving exponential convergence estimates) which suffices for our purposes, using ideas from [Ho].

*Proof.* — Consider the holomorphic map $f = \pi_{B_-} \circ w : Z_- \to B_-$. Since $f$ has finite energy, it either converges to a point $q \in \partial B_-$ (uniformly in $t$) as $s \to -\infty$ or for any $C \gg 0$ there exists $s_0 \gg 0$ such that $f|_{\{s \ge s_0\}}$ has image in the negative end $(-\infty, -C] \times [0, 1] \subset B_-$. Note that, in the latter case, the two components of $\partial Z_-|_{\{s \ge s_0\}}$ are mapped to the two boundary components of the end.

Fix a sequence $s_n \to +\infty$ and define $w_n : Z_- \to \bar{W}_-$ as $w_n(s, t) = w(s - s_n, t)$. The finiteness of the energy $E(w)$ implies the finiteness of the $\bar{\omega}$-area $E_{\bar{\omega}}(w)$. Therefore the sequence $w_n$ satisfies

$$ - E(w_n) < E(w), \text{ and} $$

$$ - \lim_{n \to \infty} E_{\bar{\omega}}(w_n) = 0. $$

(1) If $f$ converges to a point $q \in \partial B_-$, then the image of $w$ is contained in a compact subset of $\bar{W}_-$. Then by Lemma 7.3.4 there is a constant $C$ such that $\|\nabla w_n\|_{C^0} \le C$ for all $n$ and therefore we can extract a subsequence $w_{k_n}$ which converges in $C_{\text{loc}}^\infty$ to a $\bar{J}_-$-holomorphic map $w_\infty : Z_- \to \bar{W}_-$. The $\bar{\Omega}_-$-area

$$ E_{\bar{\Omega}_-}(w) = \int_{Z_-} w^* \bar{\Omega}_- $$

is finite because the energy of $w$ is finite and the image of $w$ is contained in a compact subset of $\bar{W}_-$. This implies that

$$ E_{\bar{\Omega}_-}(w_\infty) = \lim_{n \to \infty} E_{\bar{\Omega}_-}(w_{k_n}) = 0 $$

and thus $w_\infty$ is constant. Hence, for any sequence $s_n \to +\infty$, there is a subsequence $s_{k_n}$ and a point $p \in L_{\bar{\mathbf{a}}}^{-}$ such that

$$ \lim_{n \to \infty} w(s_{k_n}, t) = p $$

and the limit is uniform in $t$. Moreover, $\nabla w(s_{k_n}, t)$ also converges to 0 uniformly in $t$.

A different sequence $s'_n \to +\infty$ might a priori have a subsequence $s'_{k_n}$ such that $w(s_{k_n}, t)$ converges to a different $p' \in L_{\bar{\mathbf{a}}}^{-}$. We are then in Case (1) if we prove that $p = p'$. Up to extracting further subsequences we can assume that $s_{k_n} < s'_{k_n}$. We fix $\varepsilon$ which is strictly smaller than the distance between $p$ and $p'$. Then the Long Strip Lemma, Version
2 (Lemma 7.3.10) implies that, for $n$ sufficiently large, $w|_{[s_k, s'_k] \times [0, 1]}$ is $\epsilon$-close to a constant map, which is a contradiction.

(2) If $f = \bar{\pi}_{\mathbb{B}_-} \circ w$ has image in the negative end of $\mathbb{B}_-$ for $s$ sufficiently large, the sequence $w_n$ converges, after a translation of the image, to a trivial strip in $\bar{W}$ over some intersection point $y \in \bar{a} \cap \bar{h}(\bar{a})$. The uniqueness of the limit is proved as in the previous case using the Long Strip Lemma, Version 1 (Lemma 7.3.6). $\square$

*Convergence on the thin part.* The map $\bar{u}'_\infty$ defined above is not the SFT limit of the sequence $\bar{u}_i$ in general. In fact the $\bar{\omega}$-area of $\bar{u}'_\infty$ can be smaller than the $\bar{\omega}$-area of $\bar{u}_i$ and, when this happens, we have to look for the missing area in the thin part.

We analyze the behavior of the limit near a node of $\dot{F}_\infty$. Let $\upsilon$ be the pinching geodesics corresponding to a node $n$. We slightly change perspectives by identifying the neighborhoods $\text{Thin}_\epsilon(\dot{F}_i, g_i, \upsilon)$ with standard Euclidean cylinders $[-R_i^\epsilon, R_i^\epsilon] \times S^1$ or strips $[-R_i^\epsilon, R_i^\epsilon] \times [0, 1]$.

Suppose that $n$ is an interior node. For $i$ large and $\epsilon$ small, $u_i(\{\pm R_i^\epsilon\} \times S^1)$ is close to closed orbits $\gamma_\pm$ or points $p_\pm$ by the asymptotic properties of $u'_\infty$. The first observation is that $u_i([-R_i^\epsilon, R_i^\epsilon] \times S^1)$ cannot be close to a closed orbit on one side and to a point on the other side because there is no homotopically trivial closed Hamiltonian orbit in $N$. Similarly, if $n$ is a boundary node, then $u_i([-R_i^\epsilon, R_i^\epsilon] \times [0, 1])$ cannot be close to a chord on one side and to a point on the other side because all chords define nontrivial classes in $H_1(\bar{W}_-, L_{\bar{a}}^-)$.

**Lemma 7.3.14.** — Let $n \in \dot{F}_\infty$ be a node and let $e_\pm : [0, +\infty) \times S^1 \to F'_\infty$ or $e_\pm : [0, +\infty) \times [0, 1] \to F'_\infty$ be the two ends corresponding to $n$. If

$$ \lim_{s \to +\infty} (\bar{u}'_\infty \circ e_\pm)(s, t) = p_\pm \in \bar{W}_-, $$

then $p_+ = p_-.$

*Proof.* — Suppose that $n$ is a boundary node; the proof for an interior node is completely analogous. Arguing by contradiction, suppose that $p_- \neq p_+$. We choose $\eta < \frac{1}{2}d(p_-, p_+)$, where $d(p_-, p_+)$ is the distance between $p_-$ and $p_+$. Let $\eta' > 0$ be the constant which depends on $\eta$ and appears in the Long Strip Lemma, Version 2 and let $\bar{s}$ be a constant such that, for all $s > \bar{s}$, the distance from $p_\pm$ to $(\bar{u}'_\infty \circ e_\pm)(s, t)$ is less than $\eta'/2$. Let $\upsilon$ be the pinching geodesic that gives rise to $n$ and let $\text{Thin}_\epsilon(F'_\infty, g_\infty, \upsilon)$ be the union of the two connected components of $\text{Thin}_\epsilon(F'_\infty, g_\infty)$ corresponding to the node $n$. Then there is a constant $\bar{\epsilon} > 0$ such that, for all $\epsilon < \bar{\epsilon}$, the portions of the two ends of $F'_\infty$ corresponding to $n$ which are parametrized by $[0, \bar{s}] \times [0, 1]$ are contained in $\text{Thick}_\epsilon(F'_\infty, g_\infty, \upsilon)$. We then identify $\text{Thin}_\epsilon(\dot{F}_i, g_i, \upsilon) \cong [-R_i^\epsilon, R_i^\epsilon] \times [0, 1]$ with its standard complex structure. Then the $C_{\text{loc}}^\infty$-convergence of $\bar{u}_i$ to $\bar{u}'_\infty$ implies that, for $\epsilon < \bar{\epsilon}$ and $i \gg 0$, $\bar{u}_i(\{\pm R_i^\epsilon\} \times [0, 1])$ are contained in $\eta'$-neighborhoods of $p_\pm$. The Long Strip Lemma, Version 2 then implies that $p_+ = p_-$, a contradiction. $\square$
**Lemma 7.3.15.** — Let $n \in \dot{\mathbb{F}}_\infty$ be a node arising from pinching a geodesic in the homotopy class $\upsilon$. Assume that the two ends of $\mathbb{F}'_\infty$ corresponding to $n$ are asymptotic to closed orbits $\gamma_+$ and $\gamma_-$ or to chords $y_+$ and $y_-$ over intersection points in $\bar{a}_i \cap \bar{h}(\bar{a}_j)$. Then, for all $\epsilon$ sufficiently small, the restrictions $\bar{u}_i|_{\text{Thin}_\epsilon(\dot{\mathbb{F}}_i, g_i, \upsilon)}$ converge (in the SFT sense), as $i \to \infty$, to a stack of holomorphic cylinders or strips whose ends are asymptotic to closed orbits or chords, and whose successive components have matching ends, as in [BEHWZ, Figure 14].

*Proof.* — We consider the case of boundary nodes first. For $\epsilon > 0$ sufficiently small and $i$ sufficiently large (depending on $\epsilon$), the maps $\bar{u}_i|_{\text{Thin}_\epsilon(\dot{\mathbb{F}}_i, g_i, \upsilon)}$ define a homology class $A \in \mathbb{H}_2([0, 1] \times \bar{S}, (\{1\} \times \bar{a}) \cup (\{0\} \times \bar{h}(\bar{a})) \cup y_+ \cup y_-)$. There are two cases:

(1) $E_{\bar{\omega}}(A) = 0$, or

(2) $E_{\bar{\omega}}(A) > 0$.

We will identify $\text{Thin}_\epsilon(\dot{\mathbb{F}}_i, g_i, \upsilon) \cong [-\mathbb{R}_i^\epsilon, \mathbb{R}_i^\epsilon] \times [0, 1]$.

*Case (1).* We verify that $\bar{u}_i|_{\text{Thin}_\epsilon(\dot{\mathbb{F}}_i, g_i, \upsilon)}$ satisfies the conditions of the Long Strip Lemma, Version 1, for $i \gg 0$: Fix $\eta$ arbitrarily small, and consider $c_0$, $\mathbb{R}_0$, $\eta'$ and $\delta$ as in the Long Strip Lemma, Version 1. Let $\epsilon > 0$ be sufficiently small such that $\text{Thin}_\epsilon(\mathbb{F}'_\infty, g_\infty, \upsilon)$ is mapped by $\bar{u}'_\infty$ to $\frac{\eta'}{2}$-neighborhoods of trivial strips over $y_-$ and $y_+$. Then there exists $c > c_0$ such that for all $i \gg 0$,

$$ - \mathbb{R}_i^\epsilon - c > \mathbb{R}_0, $$

$$ - \bar{u}_i|_{[-\mathbb{R}_i^\epsilon, -\mathbb{R}_i^\epsilon + c] \times [0, 1]} \text{ is } \eta'\text{-close to a portion of the trivial strip over } y_-, $$

$$ - \bar{u}_i|_{[\mathbb{R}_i^\epsilon - c, \mathbb{R}_i^\epsilon] \times [0, 1]} \text{ is } \eta'\text{-close to a portion of the trivial strip over } y_+, \text{ and} $$

$$ - E_{\bar{\omega}}(\bar{u}_i|_{[-\mathbb{R}_i^\epsilon, \mathbb{R}_i^\epsilon] \times [0, 1)}) < \delta. $$

This follows from observing that every compact subset $K \subset \text{Thin}_\epsilon(\dot{\mathbb{F}}'_\infty, g_\infty, \upsilon)$ can be identified with some compact subset $K_i \subset \text{Thin}_\epsilon(\dot{\mathbb{F}}'_i, g_i, \upsilon)$ provided $i \gg 0$, and the sequence $\bar{u}_i|_{K_i}$ converges uniformly to $\bar{u}'_\infty|_K$ with respect to all derivatives.

We then apply the Long Strip Lemma, Version 1, to $\bar{u}_i|_{[-\mathbb{R}_i^\epsilon, \mathbb{R}_i^\epsilon] \times [0, 1]}$ and conclude that $y_+ = y_-$ and $\bar{u}_i|_{[-\mathbb{R}_i^\epsilon, \mathbb{R}_i^\epsilon] \times [0, 1]}$ is $\mathbb{C}^1$-close to a portion of the trivial strip over $y_+ = y_-$.

*Case (2).* Fix a constant $\hbar > 0$ such that, for every $A \in \mathbb{H}_2([0, 1] \times \bar{S}, (\{1\} \times \bar{a}) \cup (\{0\} \times \bar{h}(\bar{a})) \cup y' \cup y'')$ with $y'$ and $y''$ of action less than $E = E(\bar{u}_i)$, we have $\int_A \bar{\omega} > \hbar$. Such a constant exists by [BEHWZ, Lemma 10.9].

Fix $\epsilon > 0$ sufficiently small. Since $\bar{u}_i([-\mathbb{R}_i^\epsilon, \mathbb{R}_i^\epsilon] \times [0, 1])$ is contained in the strip end of $\bar{W}_-$, we may assume that the maps $\bar{u}_i$ take values in $\bar{W}$. Using the argument from the proof of the Long Strip Lemma, Version 1, if the image of $\bar{u}_i|_{[-\mathbb{R}_i^\epsilon, \mathbb{R}_i^\epsilon] \times [0, 1]}$ is contained in an $\eta$-neighborhood of the strip over $y_-$, then $\bar{u}_i|_{[-\mathbb{R}_i^\epsilon, \mathbb{R}_i^\epsilon] \times [0, 1]}$ is $\eta$-close in the $\mathbb{C}^0$-metric to a trivial strip, which contradicts (2). Hence there is a sequence of points $p_i \in [-\mathbb{R}_i^\epsilon, \mathbb{R}_i^\epsilon] \times [0, 1]$ such that (after passage to a subsequence) $\bar{u}_i(p_i)$ limits to a point $q_\infty$ which is a distance $> \eta$ from the strips over $y_\pm$, after suitable translations of $\bar{u}_i$ in the target.
Next consider the domain translations $\bar{u}_i^1(s, t) = \bar{u}_i(s - s(p_i), t)$. After passage to a subsequence the maps $\bar{u}_i^1$ converge in $C_{loc}^\infty$ to a $\bar{J}'$-holomorphic map $\bar{u}_\infty^1: \mathbf{R} \times [0, 1] \to \bar{W}$, after translations in the target. By construction, $E_{\bar{\omega}}(\bar{u}_\infty^1) > 0$. We repeat this process on $\bar{u}_i$ defined on $[-R_i^\epsilon, R_i^\epsilon] \times [0, 1]$ minus a suitable annulus $[K'_i, K''_i] \times [0, 1]$ until we obtain $\bar{J}'$-holomorphic maps $\bar{u}_\infty^1, \dots, \bar{u}_\infty^1$ such that $E_{\bar{\omega}}(\bar{u}_\infty^1) + \dots + E_{\bar{\omega}}(\bar{u}_\infty^1) = E_{\bar{\omega}}(\mathbf{A})$. An argument similar to that of Case (1) and based on the Long Strip Lemma, Version 1, shows that the ends of $\bar{u}_\infty^1, \bar{u}_\infty^1, \dots, \bar{u}_\infty^1$ match up with those of their adjacent curves. (Here we may need to reorder the $\bar{u}_\infty^1$ so they are adjacent.)

The case of interior nodes is completely analogous: the only difference is that we have to replace the Long Strip Lemma by the Long Cylinder Lemma. $\square$

This completes the proof of Proposition 7.3.2. $\square$

**Remark 7.3.16.** — Proposition 7.3.2 is only a preliminary result. We are still left with the task of analyzing the limit more precisely.

**7.4. Intersection numbers.** — In order to analyze the SFT limit $\bar{u}_\infty$, we use the intersection numbers $n^*(\bar{u}_i)$ and $n^*(\bar{v}_j)$ to constrain the behavior of holomorphic maps which are asymptotic to $\delta_0$ or to $[0, 1] \times \{z_\infty\}$.

We briefly recall the definition of the intersection numbers $n^*(\bar{u})$ given in Section 5. Let $\rho_0 > 0$ be sufficiently small. Consider the torus $T_{\rho_0} = \{\rho = \rho_0\} \subset \bar{N}$, oriented as the boundary of $\{\rho \le \rho_0\}$. We take an oriented identification $T_{\rho_0} \simeq \mathbf{R}^2/\mathbf{Z}^2$ such that the meridian has slope 0 and the closed orbits of $\bar{R}_0$ on $T_{\rho_0}$ have slope $m$. We pick a closed orbit $\delta_0^\dagger \subset T_{\rho_0}$ and consider the parallel sections $(\sigma_\infty^*)^\dagger$ determined by $\delta_0^\dagger$. We assume we have chosen $\delta_0^\dagger$ so that $(\sigma_\infty^*)^\dagger$ is disjoint from the relevant Lagrangian submanifold. Then $n^*(\bar{u}) = \langle \bar{u}, (\sigma_\infty^*)^\dagger \rangle$.

Since $n^-(\bar{u}_i) = m \gg 2g$ for an $\bar{W}_-$-curve $\bar{u}_i$ by Lemma 5.4.14, we have

$$ (7.4.1) \qquad \sum_{j=-b}^{a} n^*(\bar{v}_j) = m. $$

For each level $\bar{v}_j$ of $\bar{u}_\infty$ we have a decomposition

$$ \bar{v}_j = \bar{v}'_j \cup \bar{v}''_j $$

as in Equation (5.7.1), where $\bar{v}'_j$ is the union of the irreducible components of $\bar{v}_j$ which branch cover the section at infinity $\sigma_\infty^*$ and $\bar{v}''_j$ is the union of all other irreducible components.

**Lemma 7.4.1** (Intersection with $\bar{v}_j$, $j > 0$). — Suppose $j > 0$. Then the following hold for $\rho_0 > 0$ sufficiently small:
(1) If $\overline{v}_j''$ has a positive end $\mathcal{E}_+$ which converges to $\delta_0^\beta$, then

$$ (7.4.2) \qquad \langle \mathcal{E}_+, (\sigma_\infty')^\dagger \rangle \ge p. $$

(2) If $\overline{v}_j''$ has a negative end $\mathcal{E}_-$ which converges to $\delta_0^\beta$, then

$$ (7.4.3) \qquad \langle \mathcal{E}_-, (\sigma_\infty')^\dagger \rangle \ge m - p. $$

If $\overline{v}_j''$ has multiple ends at covers of $\delta_0$, then their contributions to $n'(\overline{v}_j)$ are summed.

*Proof.* — Let $\mathcal{E}_+$ be a positive end which converges to $\delta_0^\beta$. Let $\pi_{\overline{\mathbb{N}}} : \mathbf{R} \times \overline{\mathbb{N}} \to \overline{\mathbb{N}}$ be the projection onto the second factor. Provided $\rho_0$ is sufficiently small, $\pi_{\overline{\mathbb{N}}} (\mathcal{E}_+) \cap \mathrm{T}_{\rho_0}$ determines a homology class $(q, p) \in \mathrm{H}_1(\mathrm{T}_{\rho_0}) \simeq \mathbf{R}^2/\mathbf{Z}^2$. One can easily check that

$$ \langle \mathcal{E}_+, (\sigma_\infty')^\dagger \rangle = \det \begin{pmatrix} 1 & q \\ m & p \end{pmatrix} = p - qm. $$

Since $\langle \mathcal{E}_+, (\sigma_\infty')^\dagger \rangle > 0$ by the positivity of intersections in dimension four and $m \gg p$, we must have $q \le 0$. We then obtain:

$$ \langle \mathcal{E}_+, (\sigma_\infty')^\dagger \rangle \ge p. $$

Let $\mathcal{E}_-$ be a negative end which converges to $\delta_0^\beta$. As above, $\pi_{\overline{\mathbb{N}}} (\mathcal{E}_-) \cap \mathrm{T}_{\rho_0}$ determines a homology class $(q, -p) \in \mathrm{H}_1(\mathrm{T}_{\rho_0}) \simeq \mathbf{R}^2/\mathbf{Z}^2$ such that

$$ \langle \mathcal{E}_-, (\sigma_\infty')^\dagger \rangle = \det \begin{pmatrix} 1 & q \\ m & -p \end{pmatrix} = -p - qm. $$

Since $\langle \mathcal{E}_-, (\sigma_\infty')^\dagger \rangle > 0$ by the positivity of intersections in dimension four and $m \gg p$, we must have $q \le -1$. We then obtain:

$$ \langle \mathcal{E}_-, (\sigma_\infty')^\dagger \rangle \ge m - p. $$

Finally, if $\overline{v}_j''$ has multiple ends at covers of $\delta_0$, then the total intersection of $\overline{v}_j''$ with $(\sigma_\infty')^\dagger$ is bounded below by the sum of the contributions of each end. $\square$

Next we consider $\overline{v}_j$ when $j < 0$. Let $\pi_{\overline{\mathbb{S}}} : \mathbf{R} \times [0, 1] \times \overline{\mathbb{S}} \to \overline{\mathbb{S}}$ be the projection along the Hamiltonian vector field $\partial_t$. Also let $\mathcal{R}_{\phi_0}$ be the radial ray $\{\phi = \phi_0\} \subset D_\varepsilon^2 = \{\rho \le \varepsilon\}$, where $\varepsilon > 0$ is small. Under the projection $\pi_{\overline{\mathbb{S}}}$, each positive end $\mathcal{E}_+$ of $\overline{v}_j''$ which limits to $[0, 1] \times \{z_\infty\}$ maps to a sector $\mathcal{S}(\mathcal{E}_+)$ of $D_\varepsilon^2 = \{\rho \le \varepsilon\}$ going in the counterclockwise direction from $\mathcal{R}_{\phi_1} \subset \overline{a}_{i_1}$ to $\mathcal{R}_{\phi_2} \subset \overline{h}(\overline{a}_{i_2})$. The sector $\mathcal{S}(\mathcal{E}_+)$ is a thin wedge if $i_1 = i_2$ and the angle is $\frac{2\pi}{m}$. Similarly, each negative end $\mathcal{E}_-$ of $\overline{v}_j''$ which limits to $[0, 1] \times \{z_\infty\}$ maps to a sector $\mathcal{S}(\mathcal{E}_-)$ going counterclockwise from $\mathcal{R}_{\phi_1} \subset \overline{h}(\overline{a}_{i_1})$ to $\mathcal{R}_{\phi_2} \subset \overline{a}_{i_2}$.
**Lemma 7.4.2 (Intersection with $\bar{v}_j$, $j < 0$).** — Suppose $j < 0$. Then the following hold for $\rho_0 > 0$ sufficiently small:

(1) If $\bar{v}_j''$ has a positive end $\mathcal{E}_+$ which converges to $[0, 1] \times \{z_\infty\}$, then

$$ (7.4.4) \qquad \langle \mathcal{E}_+, \sigma_\infty^\dagger \rangle \ge 1, $$

and the relevant sector is a thin wedge if and only if equality holds. Moreover, if the sector is not a thin wedge, then

$$ \langle \mathcal{E}_+, \sigma_\infty^\dagger \rangle > 2g. $$

(2) If $\bar{v}_j''$ has a negative end $\mathcal{E}_-$ which converges to $[0, 1] \times \{z_\infty\}$, then

$$ (7.4.5) \qquad \langle \mathcal{E}_-, \sigma_\infty^\dagger \rangle > 2g. $$

*Proof.* — Suppose $\rho_0 < \varepsilon$. The restriction of $\delta_{\rho_0}$ to $[0, 1] \times \bar{S}$ consists of $m$ Reeb arcs $[0, 1] \times \{\phi_l\}$ on $\{\rho = \rho_0\}$, where $0 \le l < m$ and $\phi_l = \phi_0 + l(2\pi/m)$. Let $\mathcal{E}_+$ (resp. $\mathcal{E}_-$) be a positive (resp. negative) end of $\bar{v}_j''$ which converges to $[0, 1] \times \{z_\infty\}$ and let $\mathfrak{S}(\mathcal{E}_\pm)$ be the corresponding sector in $D_\varepsilon^2$.

By the assumptions on the $\bar{a}_i$ given in Section 5.2.2, the angles of the thin wedges are $\frac{2\pi}{m}$ and the other sectors of $D_\varepsilon^2 - \cup_i \bar{a}_i - \cup_i \bar{h}(\bar{a}_i)$ have angles greater than $\frac{2\pi(2g)}{m}$. This implies that:

(i) $\langle \mathcal{E}_+, \sigma_\infty^\dagger \rangle \ge 1$;

(ii) $\langle \mathcal{E}_+, \sigma_\infty^\dagger \rangle = 1$ if and only if $\mathfrak{S}(\mathcal{E}_+)$ is a thin wedge; and

(iii) $\langle \mathcal{E}_-, \sigma_\infty^\dagger \rangle > 2g$.

The lemma follows. $\square$

<figure><img src="image_13.png" /></figure>

Similarly, we have the following lemma, whose proof is the same as those of Lemmas 7.4.1 and 7.4.2 and will be omitted.

**Lemma 7.4.3 (Intersection with $\bar{v}_0$).** — The following hold for $\rho_0 > 0$ sufficiently small:

(1) If $\bar{v}_0''$ has a positive end $\mathcal{E}_+$ which converges to $\delta^p$, then

$$ (7.4.6) \qquad \langle \mathcal{E}_+, (\sigma_\infty^-)^\dagger \rangle \ge p. $$

(2) If $\bar{v}_0''$ has a negative end $\mathcal{E}_-$ which converges to $[0, 1] \times \{z_\infty\}$, then

$$ (7.4.7) \qquad \langle \mathcal{E}_-, (\sigma_\infty^-)^\dagger \rangle > 2g. $$

*Definition 7.4.4 (Boundary point at $z_\infty$).* — Consider $\bar{v}_j'' : \dot{\mathbf{F}}_j'' \to \bar{W}_j$ for $j \le 0$. A point $p \in \partial \dot{\mathbf{F}}_j''$ will be called a boundary point at $z_\infty$ if $\bar{v}_j''(p) \in L_{\bar{a}}^- - L_{\bar{a}}^-$ for $j = 0$ and $\bar{v}_j''(p) \in (L_{\bar{h}(\bar{a})}^- - L_{\bar{h}(\bar{a})}^-) \cup (L_{\bar{a}}^- - L_{\bar{a}}^+) = \mathbf{R} \times \{0, 1\} \times \{z_\infty\}$ for $j < 0$.
**Lemma 7.4.5.** — Consider $\bar{v}_j'' : \dot{F}_j'' \to \bar{W}_j$, $j \le 0$. If $p \in \partial \dot{F}_j''$ is a boundary point at $z_\infty$, then, for any sufficiently small neighborhood $\nu(p) \subset F_j''$ of $p$, there exists $\rho_0 > 0$ such that $\langle \bar{v}_j''(\nu(p)), (\sigma_\infty^*)^\dagger \rangle \ge k_0 - 1 > 2g$.

Here the constant $k_0$ is as given in Section 5.2.2.

*Proof.* — We will prove the case $j < 0$, leaving $j = 0$ for the reader. Let $p$ be a boundary point at $z_\infty$ of $\bar{v}_j''$ which maps to a point on $\mathbf{R} \times \{1\} \times \{z_\infty\}$ (without loss of generality). We consider the projection $\pi_S : \mathbf{R} \times [0, 1] \times \bar{S} \to \bar{S}$. By Definition 5.3.2, the projection $\pi_S$ is holomorphic when restricted to $\pi_S^{-1}(D_\epsilon^2)$, where $\epsilon > 0$ is sufficiently small. Hence $\pi_S \circ \bar{v}_j''$ is holomorphic when restricted to $(\pi_S \circ \bar{v}_j'')^{-1}(D_\epsilon^2)$ and some sector of $D_\epsilon^2 - \bar{a}$ must be contained in $\mathrm{Im}(\pi_S \circ \bar{v}_j'')$ because holomorphic maps are open. This in turn shows that

$$ \langle \bar{v}_j''(\nu(p)), (\sigma_\infty^*)^\dagger \rangle \ge k_0 - 1 > 2g $$

by assumption, provided $\rho_0$ is sufficiently small. $\square$

**7.5.** *Some restrictions on $\bar{u}_\infty$.* — We now present some lemmas in preparation for Theorems 7.6.1 and 7.6.3. Lemmas 7.5.1–7.5.3 give restrictions on $\bar{u}_\infty$, which arise from intersection number calculations from Section 7.4, and Lemma 7.5.5 gives a lower bound on the ECH index of the levels $\bar{v}_j$.

In what follows, “component” is shorthand for “irreducible component”. We first discuss the “fiber components”, i.e., components $\tilde{v} : F \to \bar{W}_j$ of $\bar{v}_j$ which map to fibers of $\bar{W}_j$. There are three types of fiber components:

(i) ghosts, i.e., $\tilde{v}$ is constant;

(ii) “closed fiber components”, i.e., $F$ is closed and $\tilde{v}$ is nonconstant;

(iii) “boundary fiber components”, i.e., $\partial F \ne \emptyset$ and $\tilde{v}$ is nonconstant.

If $\tilde{v}$ is a closed fiber component, then $\tilde{v}$ is a branched cover of a fiber of $\bar{W}_j$. Next let $\tilde{v} : F \to \bar{W}_j$ be a boundary fiber component. If $j < 0$ and $\tilde{v}$ maps to $\pi_B^{-1}(p)$, $p \in \partial B$, then $\tilde{v}(F) \supset \pi_B^{-1}(p) - A$, where $A = L_{\bar{a}}$ or $L_{\bar{h}(\bar{a})}$. Similarly, if $j = 0$ and $\tilde{v}$ maps to $\pi_B^{-1}(p)$, $p \in \partial B_-$, then $\tilde{v}(F) \supset \pi_B^{-1}(p) - L_{\bar{a}}$.

**Lemma 7.5.1.** — The only possible non-ghost fiber component of $\bar{u}_\infty$ is a closed component $\pi_B^{-1}(p)$ which passes through $\bar{m}$. There is at most one such component.

*Proof.* — Let $\tilde{v}$ be a non-ghost fiber component. Then $\tilde{v}$ is a $d$-fold branched cover of a fiber if it is a closed fiber component and a $d$-fold branched cover of a fiber cut along $\bar{a}$ (or $\bar{h}(\bar{a})$) if it is a boundary fiber component. In either case, $n^*(\tilde{v}) = d \cdot m$. This implies that $d = 1$.
We now prove that $\tilde{v}$ is a closed fiber component passing through $\bar{m}$. Arguing by contradiction, if $\bar{m} \notin \text{Im}(\tilde{v})$, then there is a component $\hat{v}$ of $\bar{u}_\infty$ such that $\bar{m} \in \text{Im}(\hat{v})$. If $\hat{v}$ is a cover of $\sigma_\infty^-$, then there must be some component of $\bar{v}_j''$ for $j > 0$, which has a negative end at some $\delta_0'$. Therefore $n^-(\bar{v}_j'') > 0$ by Lemma 7.4.1(2). If $\hat{v}$ is not a cover of $\sigma_\infty^-$, then $\hat{v}$ has a nonzero intersection with $\sigma_\infty^-$ and hence $n^-(\hat{v}) > 0$. In either case we have $n^*(\bar{u}_\infty) > m$, which is a contradiction. This proves that $\bar{m} \in \text{Im}(\tilde{v})$.

Finally boundary fiber components are eliminated because they project to a point in $\partial B_-$, and therefore cannot pass through $\bar{m}$. $\square$

We recall the notation $\bar{v}_j = \bar{v}'_j \cup \bar{v}''_j$, where $\bar{v}'_j$ denotes the union of all components of $\bar{v}_j$ which cover the section at infinity $\sigma_\infty^*$ and $\bar{v}''_j$ denotes the union of all other components of $\bar{v}_j$. The covering degree of $\bar{v}'_j$ will always be denoted by $p_j$. We also define $\bar{v}_j^\sharp$ to be the union of the components of $\bar{v}''_j$ which are asymptotic to a multiple of $\delta_0$ or $z_\infty$ at either end and are not trivial cylinders or strips, $\bar{v}_j^\flat$ to be the union of the remaining non-fiber components of $\bar{v}''_j$, and $\bar{v}_j^\flat$ to be the union of the fiber components of $\bar{v}''_j$.

**Lemma 7.5.2.** — If $\bar{v}'_0 = \emptyset$, then, with the exception of ghost components:

(1) $\bar{v}_j = \emptyset$ and $\bar{v}_j^\sharp = \emptyset$ for all $j$;

(2) no $\bar{v}_j''$, $j \le 0$, has a boundary point at $z_\infty$;

(3) every level $\bar{v}_j$, $j \ne 0$, has image inside $W'$ or $W$; and

(4) $\bar{v}_0$ is a $\bar{W}_-$-curve or a degenerate $\bar{W}_-$-curve, i.e., $\bar{v}_0$ is the union of a $W_-$-curve and a fiber $\bar{\pi}_{\bar{B}_-}^{-1}(p)$ which passes through $\bar{m}$.

*Proof*. — If $\bar{v}'_0 = \emptyset$, then $\bar{v}_0 = \bar{v}''_0$. Since $\bar{u}_i$ passes through $\bar{m}$ for all $i$, the level $\bar{v}_0$: $\dot{F}_0 \to \bar{W}_-$ must also pass through $\bar{m}$. By Lemma 5.4.14 and the proof of Lemma 5.4.8, $n^-(\bar{v}_0) \ge m$ because $\bar{m} \in \sigma_\infty^-$. Equation (7.4.1) and the nonnegativity of $n^*$ then imply that $n^-(\bar{v}_0) = m$ and $n^*(\bar{v}_j) = 0$ for $j \ne 0$.

(1) If $\bar{v}'_j \ne \emptyset$ or $\bar{v}_j^\sharp \ne \emptyset$, then at least one of Equations (7.4.3)–(7.4.6) applies and $\sum_{j \ne 0} n^*(\bar{v}_j^\sharp) > 0$.

(2) This follows from $n^-(\bar{v}_0) = m$ and Lemma 7.4.5.

(3) Since $n^*(\bar{v}_j) = 0$ for $j \ne 0$, (1), combined with Lemma 5.3.20, implies that $\text{Im}(\bar{v}_j) \subset W'$ if $j > 0$, whereas (1) and (2), combined with Lemma 5.3.9, implies that $\text{Im}(\bar{v}_j) \subset W$ if $j < 0$.

(4) Since $n^-(\bar{v}_0) = m$, $\bar{v}_0$ intersects $\sigma_\infty^-$ only at $\bar{m}$ and the intersection is transverse by Lemma 5.4.14(2). By (1) and (2) and Lemmas 7.5.1 and 5.4.16, if $\bar{v}_0$ is not a $\bar{W}_-$-curve, then it must be a degenerate $\bar{W}_-$-curve.

This completes the proof of the lemma. $\square$

**Lemma 7.5.3.** — If $\bar{v}'_0 \ne \emptyset$, then, with the exception of ghost components:
(1) there is only one negative end of $\overline{v}_j^\sharp, j > 0$, (say $\overline{v}_{d'}^\sharp$) which is asymptotic to a multiple of $\delta_0$;

(2) $\bar{u}_\infty$ has no fiber components;

(3) no $\overline{v}_j'', j \le 0$, has a boundary point at $z_\infty$;

(4) $\overline{v}_j^\flat$ has image inside W, $W_-$, or W';

(5) $\overline{v}_j^\sharp, j < 0$, is a union of thin strips from $z_\infty$ to some $x_i$ or $x'_i$;

(6) $\overline{v}_0^\sharp$ has image inside $\overline{W}_-$ - int($W_-$) and has $\delta_0^{r_0}$, for some $r_0 > 0$, at the positive end and some subset of $\{x_1, \dots, x_{2g}, x'_1, \dots, x'_{2g}\}$ of cardinality $r_0$ at the negative end;

(7) $\overline{v}_j^\sharp, 0 < j \le a'$, has image inside $\overline{W}' - \text{int}(W')$ and has $\delta_0^{r_j}$, for some $r_j > 0$, at the positive end and $h_j^j e_j^{j'}$ with $r_j' + r_j'' = r_j$, at the negative end.

*Proof.* — (1) By Lemma 7.4.1, each negative end of $\overline{v}_j^\sharp$, for $j > 0$, which is asymptotic to a multiple of $\delta_0$ contributes at least $m - 2g$ to $\sum_{j=1}^a n^*(\overline{v}_j)$, where $m \gg 2g$, because the total multiplicity of $\delta_0$ is $\le 2g$. If there are at least two such negative ends of $\overline{v}_j^\sharp$, then the total contribution to $\sum_{j=1}^a n^*(\overline{v}_j)$ is at least $2(m - 2g) > m$, which is a contradiction. Suppose $\overline{v}_{d'}^\sharp, 0 < d' \le a$, has a negative end at a multiple at $\delta_0$.

(2) By Lemma 7.5.1, a fiber component of $\bar{u}_\infty$ is a fiber $\bar{\pi}_{B_-}^{-1}(p)$ which passes through $\bar{m}$ and additionally contributes $m$ to $\sum_{j=-b}^a n^*(\overline{v}_j)$, a contradiction.

(3) By Lemma 7.4.5, a boundary point at $z_\infty$ additionally contributes $\ge 2g$ to $\sum_{j=-b}^a n^*(\overline{v}_j)$. This is again a contradiction.

(4) By (3), $\overline{v}_j^\flat$ does not have any boundary point at $z_\infty$. Since the ends of $\overline{v}_j^\flat$ are contained in $\mathbb{N}$ or $[0, 1] \times \mathbb{S}$ and $n^*(\overline{v}_j^\flat) = 0$, we conclude that $\text{Im}(\overline{v}_j^\flat) \subset W, W_-$, or W by Lemmas 5.3.9, 5.4.16 and 5.3.20.

(5), (6) Let $p_j = \text{deg}(\overline{v}_j')$ be the covering degree of $\overline{v}_j'$ over $\sigma_\infty'$. Then $p_0 \le p_1 \le \dots \le p_{d'-1}$ and $p_d' = 0$. This follows from (1) since a negative end of $\overline{v}_{j+1}^\sharp$ at (a multiple of) $\delta_0$ is required for $p_j > p_{j+1}$.

Let $\bar{p}_d'$ be the multiplicity of $\delta_0$ at the negative end of $\overline{v}_{d'}^\sharp$. The negative end of $\overline{v}_{d'}^\sharp$ contributes at least $m - \bar{p}_d'$ to $n^*$ by Equation (7.4.3) and the positive ends of $\overline{v}_j^\sharp$, for $j = 0, \dots, d' - 1$, give a total contribution of at least $\bar{p}_d' - p_0$ to $n^*$ by Equations (7.4.2) and (7.4.6). Hence,

$$ (7.5.1) \qquad \sum_{j=0}^{a} n^*(\overline{v}_j^\sharp) \ge m - p_0. $$

On the other hand, by Equation (7.4.4), the contributions of the positive ends of $\overline{v}_j^\sharp$, $j = -b, \dots, -1$, add up to

$$ (7.5.2) \qquad \sum_{j=-b}^{-1} n^*(\overline{v}_j^\sharp) \ge p_0. $$
Equations (7.5.1) and (7.5.2) give:

$$ (7.5.3) \qquad \sum_{j=-b}^{a} n^*(\bar{v}_j^\sharp) \geq m. $$

Equality holds by Equation (7.4.1). This in turn implies that:

(i) equality holds in both Equations (7.5.1) and (7.5.2); and

(ii) $\bar{v}_j^\sharp, j \leq 0$, has no negative end which limits to $z_\infty$.

Since $p_0 \leq 2g$, each $\bar{v}_j^\sharp, j < 0$, must be a union of thin strips by Lemma 7.4.2. This gives (5). Moreover, $\bar{v}_0^\sharp$ has $\delta_0^{p_1-p_0}$ at the positive end and no negative ends at $z_\infty$ by (ii) and

$$ (7.5.4) \qquad n^-(\bar{v}_0^\sharp) = p_1 - p_0 $$

by Lemma 7.4.3 and (i).

The argument to prove (6) is similar to the proof of the blocking lemma in [0], and the presence of the Lagrangian boundary condition for $\bar{v}_0^\sharp$ does not change the proof in any essential way. We consider $C_\rho = \pi_\mathbb{N}(\bar{v}_0^\sharp) \cap T_\rho$, where $0 < \rho < 1$, as an element of $H_1(T_\rho, \pi_\mathbb{N}(L_{\bar{\mathbf{a}}}) \cap T_\rho)$; we are viewing $C_\rho$ as the boundary of the portion of $\pi_\mathbb{N}(\bar{v}_0^\sharp)$ outside the solid torus of radius $\rho$. We recall that $\pi_\mathbb{N}(L_{\bar{\mathbf{a}}}) \cap T_\rho$ consists of $2g$ parallel segments in $T_\rho$ which are tangent to the Hamiltonian flow. Suppose $p_1 > 0$ is small. Let $\cup_i \mathcal{E}_i$ be the union of positive ends of $\bar{v}_0^\sharp$ that limit to multiples of $\delta_0$, and let $C'_{\rho_1} = \pi_\mathbb{N}(\cup_i \mathcal{E}_i) \cap T_{\rho_1}$. We view $C'_{\rho_1}$ as an element of

$$ H_1(T_{\rho_1}) \hookrightarrow H_1(T_{\rho_1}, \pi_\mathbb{N}(L_{\bar{\mathbf{a}}}) \cap T_{\rho_1}). $$

Using “balanced coordinates” on $V = \mathbb{N} - \text{int}(N)$ (cf. Section 5.1.2), we obtain an identification $H_1(T_{\rho_1}) \cong \mathbb{Z}^2$ such that — writing vectors as rows — (1, 0) corresponds to the homology class of the meridian (i.e., the closed curve which bounds a disk in V) and (1, m) corresponds to the class of a closed Hamiltonian orbit.

With respect to this identification, $C'_{\rho_1} = (q, p_1 - p_0)$ for some $q \leq 0$ and we have

$$ p_1 - p_0 - qm = \det \begin{pmatrix} 1 & q \\ m & p_1 - p_0 \end{pmatrix} \geq p_1 - p_0 \geq 0. $$

By Equation (7.5.4), we obtain $q = 0$ and $C_{\rho_1} = C'_{\rho_1}$, since any intersections besides those coming from $\cup_i \mathcal{E}_i$ would give some extra contributions to $n^-(\bar{v}_0^\sharp)$.

Next consider $C_{\rho_2}$, where $\rho_2 = 1 - \varepsilon$, $\varepsilon > 0$ small. Since there are no ends of $\bar{v}_0^\sharp$ between $T_{\rho_1}$ and $T_{\rho_2}$, it follows that $C_{\rho_2}$ is the image of $(0, p_1 - p_0) \in H_1(T_{\rho_2})$ in $H_1(T_{\rho_2}, \pi_\mathbb{N}(L_{\bar{\mathbf{a}}}) \cap T_{\rho_2})$. Negative ends can approach $x_i$ or $x'_i$ either from the interior of V or from the exterior of V. A negative end of $\bar{v}_0^\sharp$ approaching $x_i$ or $x'_i$ from the interior traces a segment in $T_{\rho_2}$ with both endpoints in the same connected component of $\pi_\mathbb{N}(L_{\bar{\mathbf{a}}}) \cap T_{\rho_2}$
and whose relative homology class is the image of the class $(0, 1) \in H_1(T_{\rho_2})$. This means that at most $p_1 - p_0$ ends of $\bar{v}_0^\sharp$ approach $x_i$ or $x'_i$ from the interior.

Finally let $\rho_3 = 1 + \varepsilon$, $\varepsilon > 0$ small. The ends of $\bar{v}_0^\sharp$ between $T_{\rho_2}$ and $T_{\rho_3}$ are negative ends at $x_i$, $x'_i$ or positive ends at closed orbits on $T_1$. Let $C'_{\rho_3} = \pi_{\mathbb{N}}(\tilde{v}) \cap T_{\rho_3}$, where $\tilde{v}$ is obtained from $\bar{v}_0^\sharp$ by truncating the negative ends that limit to $x_i$ or $x'_i$ from the exterior of V. There are no positive ends of $\bar{v}_0^\sharp$ that limit to a closed orbit on $T_1$; this follows from asymptotic winding considerations of the positive ends and a comparison with $C_{\rho_2}$. Since there are no negative ends of $\bar{v}_0^\sharp$ that limit to a closed orbit on $T_1$, it follows that $[C'_{\rho_3}] = 0 \in H_1(T_{\rho_3}, \pi_{\mathbb{N}}(L_{\bar{a}}^-) \cap T_{\rho_3})$.

Finally, we eliminate the possibility of ends which limit to $x_i$ or $x'_i$ from the outside of V by observing that nothing else can contribute to $C_{\rho_2}$ but the ends at $x_i$ or $x'_i$. This implies that there are $p_1 - p_0$ of them, each representing the image of the class $(0, 1) \in H_1(T_{\rho_2})$, and leaving no room for ends limiting to $x_i$ or $x'_i$ from the outside. Then $C_{\rho_3}$ is homologically trivial, and therefore the positivity of intersections implies that it is empty. Since $\rho_3$ was arbitrarily close to 1, it follows that the image of $\bar{v}_0^\sharp$ cannot escape $\overline{W_-} - \text{int}(\overline{W_-})$, which implies (6).

(7) This is similar to (6) and is left to the reader. $\square$

**Remark 7.5.4.** — One can easily compute that $I(\bar{v}_0^\sharp) = r_0$ and $I(\bar{v}_j^\sharp) = r'_j + 2r''_j$ when $j > 0$.

**Lemma 7.5.5.** — The ECH index of each level $\bar{v}_j$, $j = -b, \dots, a$, is nonnegative if $\bar{J}_- \in \mathcal{J}_{\overline{W}_-}^{\text{reg}}$. Moreover, if $j \neq 0$, the ECH index of $\bar{v}_j$ is strictly positive.

*Proof.* — Recall that the restrictions $\bar{J}$ and $\bar{J}'$ are also regular by definition. Case $j > 0$. By [HT1, Proposition 7.15], $I_{\text{ECH}}(\bar{v}_j) \ge 1$ for $j > 0$ since $\bar{J}'$ is regular and no level $\bar{v}_j$ with $j > 0$ consists uniquely of trivial cylinders. Case $j < 0$. We write $\bar{v}_j = \bar{v}'_j \cup \bar{v}_j^\sharp \cup \bar{v}_j^\flat$.

Suppose that $\bar{v}'_0 = \emptyset$. Then $\bar{v}'_j = \emptyset$ and $\bar{v}_j^\sharp = \emptyset$ for all $j < 0$ by Lemma 7.5.2, and we are left with $\bar{v}_j^\flat$, which is simply-covered and has positive Fredholm index by regularity and translation invariance. By the index inequality (Theorem 4.5.13), $I(\bar{v}_j) = I(\bar{v}_j^\flat) \ge 1$.

Next suppose that $\bar{v}'_0 \neq \emptyset$. Then $\bar{v}_j^\sharp$ is a union of thin strips from $z_\infty$ to some $x_i$ or $x'_i$ by Lemma 7.5.3 and each thin strip has ECH index 1. We also have $I(\bar{v}'_j) = 0$ by Lemma 5.7.15 and $I(\bar{v}_j^\flat) \ge 0$ by the previous paragraph.

We claim that

$$ (7.5.5) \qquad I(\bar{v}_j) = I(\bar{v}'_j \cup \bar{v}_j^\sharp \cup \bar{v}_j^\flat) = I(\bar{v}'_j) + I(\bar{v}_j^\sharp) + I(\bar{v}_j^\flat). $$

Note that, although $\bar{v}'_j$, $\bar{v}_j^\sharp$ and $\bar{v}_j^\flat$ are disjoint, $\bar{v}'_j$ and $\bar{v}_j^\sharp$ are both asymptotic to (a multiple of) $z_\infty$ at the positive end and the additivity of the ECH indices of $\bar{v}'_j$ and $\bar{v}_j^\sharp$ needs to
be verified. For that purpose, recall from Section 5.7 that each $\bar{v}'_j$ comes equipped with data $\mathcal{D}'_j = ((\mathcal{D}')^t_j, (\mathcal{D}')^{from}_j)$ at the positive and negative ends, since $\bar{u}_\infty$ is the limit of the sequence $\{\bar{u}_i\}$. The key observation here is that $(\mathcal{D}')^t_j = (\mathcal{D}')^{from}_j$, since all the components of $\bar{v}^\sharp_j$, $j' < j$, are thin strips whose data $(\mathcal{D}^\sharp_+)^{t_j}$, $j' < j$, are thin strips whose data $(\mathcal{D}^\sharp_+)^{t_j} = ((\mathcal{D}^\sharp_+)^{t_j}, (\mathcal{D}^\sharp_+)^{from})$ at the positive end satisfies $(\mathcal{D}^\sharp_+)^{t_j} = (\mathcal{D}^\sharp_+)^{from}$. Hence we can choose a simultaneous grooming $\mathfrak{c}^+ = \{c_k^+\}$ for both $\bar{v}'_j$ and $\bar{v}^\sharp_j$ at the positive end $z_\infty$ such that $c_k^+$ has winding number $w(c_k^+) = 0$ (see Equation (5.7.4)) and connects $\bar{h}(\bar{a}_{ik,jk})$ to $\bar{a}_{ik,jk}$. If we choose a groomed multivalued trivialization $\tau$ compatible with $\mathfrak{c}^+$, then

$$I_{\tau}(\bar{v}'_j \cup \bar{v}^\sharp_j) = I_{\tau}(\bar{v}'_j) + I_{\tau}(\bar{v}^\sharp_j),$$

which immediately implies Equation (7.5.5).

Case $j=0$. Suppose that $\bar{v}'_0 = \emptyset$. Then, $\bar{v}_0$ is a $\bar{W}_-$-curve or a degenerate $\bar{W}_-$-curve by Lemma 7.5.2. If $\bar{v}_0$ is a $\bar{W}_-$-curve, then it is simply-covered and satisfies $I(\bar{v}_0) \ge \text{ind}(\bar{v}_0) \ge 0$. If $\bar{v}_0$ is a degenerate $\bar{W}_-$-curve, then $\bar{v}_0$ can be written as a union of a fiber C and a $\bar{W}_-$-curve $\bar{v}^\flat_0$. The Fredholm index of $\bar{v}^\flat_0$ is nonnegative since $\bar{v}^\flat_0$ is simply-covered and hence is regular. The Fredholm index of C is given by:

$$\begin{aligned} \text{ind}(C) &= -\chi(C) + 2\langle c_1(\text{TW}_-), C \rangle \\ &= (2g-2) + 2(2-2g) = 2-2g. \end{aligned}$$

The algebraic intersection number $\langle C, \bar{v}^\flat_0 \rangle$ is equal to $2g$ and

$$\begin{aligned} (7.5.6) \qquad I(\bar{v}_0) &\ge \text{ind}(C) + \text{ind}(\bar{v}^\flat_0) + 2\langle C, \bar{v}^\flat_0 \rangle \\ &\ge (2-2g) + 0 + 2(2g) = 2g+2, \end{aligned}$$

by Theorem 5.6.9.

Next suppose that $\bar{v}'_0 \ne \emptyset$. We have $I(\bar{v}'_0) = 0$ by Lemma 5.7.16. Next, by Lemma 7.5.3, if $\bar{v}^\sharp_0 \ne \emptyset$, then $\text{Im}(\bar{v}^\sharp_0) \subset \bar{W}_-$, $\text{int}(\bar{W}_-) \ne 0$, $p_1 - p_0 > 0$, at the positive end and some $(p_1 - p_0)$-element subset of $\{x_1, \dots, x_{2g}, x'_1, \dots, x'_{2g}\}$ at the negative end. Hence $I(\bar{v}^\sharp_0) = p_1 - p_0$ by Lemma 5.7.17. Finally, since $\text{Im}(\bar{v}^\flat_0) \subset \bar{W}_-$ by Lemma 7.5.3, $\bar{v}^\flat_0$ is simply-covered and $I(\bar{v}^\flat_0) \ge 0$. The ECH indices of $\bar{v}'_0$, $\bar{v}^\sharp_0$, and $\bar{v}^\flat_0$ are additive. This completes the proof of the lemma. $\square$

**7.6. Compactness theorem.** — Let $\bar{J}^\diamond(\varepsilon, \delta, p)$ be a generic almost complex structure which is $(\varepsilon, \text{U})$-close to $\bar{J}_- \in \mathcal{J}^{\text{reg}}_{\bar{W}_-}$, where U and $K_{\rho, 2\delta}$ are as in Convention 5.8.12. We write:

$$\text{(7.6.1)} \qquad \mathcal{M}_{\bar{m}}^{i}(\varepsilon, \delta, p) := \mathcal{M}_{\bar{J}^{\diamond}(\varepsilon, \delta, p)}^{l=i, n^* = m}(\gamma, \mathbf{y}; \bar{m}).$$
<figure><img src="image_14.png" /><figcaption>FIG. 4. — Schematic diagrams for the possible types of degenerations. Here • represents $\delta_0$, ◦ represents $z_\infty$, and × represents some $x_i$ or $x'_i$. A vertical line indicates a trivial cylinder or a restriction of a trivial cylinder, and a double vertical line indicates a branched double cover of a trivial cylinder or a restriction of a trivial cylinder. The labels on the graphs are ECH indices of each component</figcaption></figure>

As before, $\bar{u}_\infty \in \partial \mathcal{M}_{\bar{m}}^3(\varepsilon, \delta, \rho)$ is written as $\bar{u}_\infty = \bar{v}_{-b} \cup \dots \cup \bar{v}_a$ and each $\bar{v}_j$ is written as $\bar{v}_j = \bar{v}'_j \cup \bar{v}''_j = \bar{v}'_j \cup \bar{v}^{\sharp}_j \cup \bar{v}^{\flat}_j \cup \bar{v}'_j$, where $\bar{v}'_j = \emptyset$ for $j \neq 0$. There are no ghost components by an index argument as in Lemma 6.1.8.

We now prove the following compactness theorem, which is basically a consequence of two constraints: $I(\bar{u}_i) = 3$ and $n^*(\bar{u}_i) = m$. The list of possibilities should be viewed as a preliminary list, since we will subsequently eliminate Cases (2)–(6) in Theorem 7.10.1.

**Theorem 7.6.1.** — Let $\bar{J}^\diamond$ be $(\varepsilon, U)$-close to $\bar{J}_- \in \mathcal{J}_{\bar{W}_-}^{\text{reg}}$ and $K_{p,2\delta}$-regular with respect to $\bar{m}$, and let $\bar{u}_\infty \in \partial \mathcal{M}_{\bar{m}}^3(\varepsilon, \delta, \rho)$. If $\bar{v}'_0 \neq \emptyset$, then $\bar{u}_\infty$ contains one of the following subbuildings:

(1) A 3-level building consisting of $\bar{v}_1$ with $I = 2$ from $\mathbf{y}$ to $\delta_0 \mathbf{y}'; \bar{v}'_0 = \sigma^-_\infty$; and $\bar{v}^{\sharp}_{-1}$ which is a thin strip from $z_\infty$ to $x_i$ or $x'_i$.

(2) A 3-level building consisting of $\bar{v}_1$ with $I = 1$ from some $\mathbf{y}''$ to $\delta_0 \mathbf{y}'; \bar{v}'_0 = \sigma^-_\infty$; and $\bar{v}^{\sharp}_{-1}$ which is a thin strip.

(3) A 4-level building consisting of $\bar{v}_1$ with $I = 1$ from $\mathbf{y}$ to $\delta_0^2 \mathbf{y}'; \bar{v}'_0$ which is a branched double cover of $\sigma^-_\infty$; $\bar{v}'_{-1} = \sigma_\infty$; and $\bar{v}^{\sharp}_{-1}$ and $\bar{v}^{\sharp}_{-2}$ which are both thin strips.

(4) A 3-level building consisting of $\bar{v}_1$ with $I = 1$ from $\mathbf{y}$ to $\delta_0^2 \mathbf{y}'; \bar{v}'_0$ which is a branched double cover of $\sigma^-_\infty$; and $\bar{v}^{\sharp}_{-1}$ which is the union of two thin strips.

(5) A 3-level building consisting of $\bar{v}_1$ with $I = 1$ from $\mathbf{y}$ to $\delta_0^2 \mathbf{y}'; \bar{v}'_0 = \sigma^-_\infty$; $\bar{v}^{\sharp}_0$ with $I = 1$ from $\delta_0$ to $x_i$ or $x'_i$; and $\bar{v}^{\sharp}_{-1}$ which is a thin strip.

(6) A 4-level building consisting of $\bar{v}_2$ with $I = 1$ from $\mathbf{y}$ to $\delta_0^2 \mathbf{y}'; \bar{v}'_1 = \sigma'_\infty$; $\bar{v}^{\sharp}_1$ with $I = 1$ which is a cylinder from $\delta_0$ to $h$; $\bar{v}'_0 = \sigma^-_\infty$; and $\bar{v}^{\sharp}_{-1}$ which is a thin strip.

Here we are omitting levels which are connectors.

See Figure 4.
*Proof*. — By Proposition 7.3.2, $\bar{u}_i$ converges in the sense of SFT to a holomorphic building $\bar{u}_\infty = \bar{v}_a \cup \cdots \cup \bar{v}_{-b}$. We have three constraints:

(i) $\sum_{j=-b}^{a} I(\bar{v}_j) = 3,$

(ii) $I(\bar{v}_j) \ge 0$ for all $j$, and

(iii) $\sum_{j=-b}^{a} n^*(\bar{v}_j) = m.$

(i) comes from the additivity of ECH indices, (ii) comes from Lemma 7.5.5, and (iii) comes from Equation (7.4.1).

Suppose that $\bar{v}'_0 \neq \emptyset$, i.e., we are in the situation of Lemma 7.5.3. We have the following restrictions:

- the top level $\bar{v}_a$ is nontrivial and satisfies $I(\bar{v}_a) \ge 1$;

- $\cup_{j<0} \bar{v}_j^\sharp$ consists of $p_0 \ge 1$ thin strips and contributes $\sum_{j<0} I(\bar{v}_j^\sharp) = p_0 \ge 1$ to the total ECH index;

- $\cup_{j=-b}^{a} \bar{v}_j^\sharp$ consists of $p_{a'}^-$ components and each component has ECH index $\ge 1$, with exception of $\bar{v}_0^\sharp$, which has ECH index 0, where $a'$ and $p_{a'}^-$ are as in Lemma 7.5.3.

This immediately implies $p_{a'}^- \le 2$ because $p_{a'}^-$ is also the number of nontrivial curves with a positive end at $\delta_0$ and each of them has ECH index $I \ge 1$ by Lemma 7.5.3. We also have $p_0 \le p_{a'}^-$, and therefore we can divide the proof into three cases:

- Case I: $p_{a'}^- = p_0 = 1$.

- Case II: $p_{a'}^- = 2$ and $p_0 = 1$.

- Case II': $p_{a'}^- = 2$ and $p_0 = 2$.

*Case I.* In this case $\bar{v}'_0 = \sigma_\infty^-$ and $\bar{v}_{-1}^\sharp$ is a thin strip. This leaves two possibilities for $\bar{v}_1$: either $I(\bar{v}_1) = 2$ and we are in Case (1), or $I(\bar{v}_1) = 1$ and we are in Case (2).

*Case II'.* In this case we have $\bar{v}_{j_0}^\sharp \neq \emptyset$ and with a positive end at $\delta_0$ for some $j_0 \ge 0$. Since $I(\bar{v}_{j_0}^\sharp) = 1$ by Remark 7.5.4, $\bar{v}_{-1}$ consists of a single thin strip and there are no other levels with $j < 0$. If $j_0 = 0$ we are in Case (5), and if $j_0 > 0$ we are in Case (6). In Case (6), $\bar{v}_1^\sharp$ is a cylinder connecting $\delta_0$ with $h$ by Lemma 7.5.3 and Remark 7.5.4.

*Case II'*. In this case $\cup_{j<0} \bar{v}_j$ consists of two thin strips. If they are on the same level we are in Case (4) and if they are on different levels we are in Case (3).

This completes the proof of Theorem 7.6.1. $\square$

<figure><img src="image_15.png" /></figure>

*Remark 7.6.2.* — In Cases (3)–(6), the total number of branch points of $\cup_{j=-b}^{a} \bar{v}'_j$ is one, where we are not ignoring connector components that cover $\sigma_\infty^*$: Assume without loss of generality that the only nontrivial $\bar{v}'_j$ is $\bar{v}'_0$ and that $\bar{v}'_0$ double covers $\sigma_\infty^-$. Let $\mathfrak{b}$ be the number of branch points of $\bar{v}'_0$. Then $\mathrm{ind}(\bar{v}'_0) = \mathfrak{b} - 1$ by Proposition 5.5.5, the Riemann-Hurwitz formula, and the proof of Lemma 5.8.9. The index inequality, the additivity of the indices, and the condition $I(\bar{u}_i) = 3$ force $\mathfrak{b} = 1$.
The proof of the following theorem is similar and will be omitted.

**Theorem 7.6.3.** — Let $\bar{J}_{-}^{\diamond}$ be $(\varepsilon, U)$-close to $\bar{J}_{-} \in \mathcal{J}_{\bar{W}_{-}}^{\text{reg}}$ and $K_{\rho,2\delta}$-regular with respect to $\bar{\mathfrak{m}}$, and let $\bar{u}_{\infty} \in \partial \mathcal{M}_{\bar{\mathfrak{m}}}^{2}(\varepsilon, \delta, \rho)$. If $\bar{v}'_{0} \neq \emptyset$, then $\bar{u}_{\infty}$ contains a 3-level subbuilding consisting of $\bar{v}_{1}$ with $I = 1$ from $\gamma$ to $\delta_{0}\gamma'$; $\bar{v}'_{0} = \sigma_{\infty}^{\perp}$; and $\bar{v}_{-1}^{\parallel}$ which is a thin strip. Here it is possible to have $I_{\text{ECH}} = 0$ connectors in between.

**7.7. Asymptotic eigenfunctions.** — We now collect some facts about the asymptotic operator and asymptotic eigenfunctions, referring the reader to [HWZ1] and [HT2].

**7.7.1. The asymptotic operator.** — We study the local behavior of a holomorphic half-cylinder which converges to a degree $l \ge 1$ multiple cover of $\delta_0$, denoted by $\delta_0^l$. Let us denote $D_{\rho_0}^2 = \{\rho \le \rho_0\} \subset D^2$ with $\rho_0 > 0$ small (in particular $\le \frac{1}{2}$) and $K = [C, +\infty) \times (\mathbf{R}/2\mathbf{Z})$ with $l \in \mathbf{N}$. We will be using balanced coordinates on $(\mathbf{R}/2\mathbf{Z}) \times D_{\rho_0}^2$; see Section 5.1.2. A holomorphic half-cylinder which is asymptotic to $\delta_0^l$ at the positive end restricts to a holomorphic map

$$ \bar{u}: K \to \mathbf{R} \times (\mathbf{R}/2\mathbf{Z}) \times D_{\rho_0}^2, $$

which can be written as:

$$ (s, t) \mapsto (s, t, z(s, t)), $$

where $\lim_{s \to +\infty} z(s, t) = 0$.

**Lemma 7.7.1.** — The function $z: K \to D_{\rho_0}^2$ satisfies the equation

$$ (7.7.1) \qquad \partial_s z + i \partial_t z + \varepsilon z = 0, $$

where $\varepsilon = \frac{\pi}{m}$.

*Proof.* — The partial derivatives of $\bar{u}$ are: $\partial_s \bar{u} = (1, 0, \partial_s z)$ and $\partial_t \bar{u} = (0, 1, \partial_t z)$. Then $J(\partial_s \bar{u}) = \mathbf{R} + (0, 0, i \partial_s z)$, where $\mathbf{R}$ is the Hamiltonian vector field. We compute that $\mathbf{R} = \partial_t + \varepsilon \partial_\phi$. Hence $J(\partial_s \bar{u}) = (0, 1, i \partial_s z + i \varepsilon z)$. This gives Equation (7.7.1). $\square$

**Definition 7.7.2.** — We define the asymptotic operator

$$ A_l : L_1^2(\mathbf{R}/2\mathbf{Z}, \mathbf{C}) \to L^2(\mathbf{R}/2\mathbf{Z}, \mathbf{C}), $$

$$ f \mapsto -i\partial_t f - \varepsilon f. $$

We remark that the asymptotic operator which appears in [HWZ1] is $A_l$, whereas the asymptotic operator in [HT2] is $-A_l$. The eigenfunctions of $A_l$ are the *asymptotic eigenfunctions*, and are given by $c e^{\pi i n l / l}$, $c \in \mathbf{C}$, $n \in \mathbf{Z}$, with corresponding eigenvalues $\frac{\pi n}{l} - \varepsilon$.
An asymptotic eigenfunction $ce^{\pi int/l}$ is said to be *normalized* if $|c| = 1$. Let $E_{\pi n/l-\epsilon}$ be the eigenspace of $A_l$ corresponding to the eigenvalue $\frac{\pi n}{l} - \epsilon$.

For a strip-like end asymptotic to the intersection point $z_\infty$, the asymptotic operator is still $f \mapsto -i\partial f - \epsilon f$, but now acting on functions $f: [0, 1] \to \mathbf{C}$ with boundary conditions $f(0) \in e^{i(c_0+\epsilon)}\mathbf{R}$ and $f(1) \in e^{i(c_0}}\mathbf{R}$ for some real constant $c_0$. The eigenfunctions are $ce^{(\pi n-\epsilon)it+i(c_0+\epsilon)}$, $c \in \mathbf{R}$, $n \in \mathbf{Z}$, with corresponding eigenvalues $\pi n - 2\epsilon$. As before we say that an eigenfunction is *normalized* if $|c| = 1$.

**7.7.2.** The asymptotic eigenfunction at an end. — Let $\mathbf{y} \in \hat{\mathcal{O}}_{2g}$ and $\mathbf{y}' \in \hat{\mathcal{O}}_{2g-l}$. As before, the modifier $*$ is placed as in $M_{\bar{j}_-}^*(\mathbf{y}, \delta_0^l\mathbf{y}')$ to denote the subset of $M_{\bar{j}_-}(\mathbf{y}_0, \delta_0^l\mathbf{y}')$ satisfying $*$. The modifier $* = (l_1, \dots, l_\lambda)$ means $(l_1, \dots, l_\lambda)$ is a partition of $l$ and we restrict to curves with $\lambda$ ends at $\delta_0$ with covering multiplicities $l_1, \dots, l_\lambda$.

We consider the asymptotics of $\bar{u} \in M_{\bar{j}}^{(l_1, \dots, l_\lambda)}(\mathbf{y}, \delta_0^l\mathbf{y}')$ near the negative end $\delta_0^l$. Let

$$ \bar{\pi}_{D_{\rho_0}^2}: \mathbf{R} \times (\mathbf{R}/2\mathbf{Z}) \times D_{\rho_0}^2 \to D_{\rho_0}^2 $$

be the projection to $D_{\rho_0}^2$ with respect to balanced coordinates; we also write $\bar{\pi}$ for $\bar{\pi}_{D_{\rho_0}^2}$. Also let

$$ z_j = \bar{\pi}_{D_{\rho_0}^2} \circ \bar{u}: (-\infty, s_0] \times (\mathbf{R}/2l_j\mathbf{Z}) \to D_{\rho_0}^2 $$

be the projection of the negative end of $\bar{u}$ which corresponds to $\delta_0^l$. The following asymptotic result is due to Hofer-Wysocki-Zehnder [HWZ1]:

**Lemma 7.7.3.** — There exist constants $C_0, C_1 > 0$ such that the following holds: For any $\bar{u} \in M_{\bar{j}}^{(l_1, \dots, l_\lambda)}(\mathbf{y}, \delta_0^l\mathbf{y}')$ with a negative end at $\delta_0^l$, there exists an asymptotic eigenfunction $f_j: \mathbf{R}/2l_j\mathbf{Z} \to \mathbf{C}$ given by $f_j(t) = ce^{\pi it/l_j}$ such that

$$ (7.7.2) \qquad |z_j(s, t) - e^{(\pi/l_j - \epsilon)s} f_j(t)| < C_0 e^{(\pi/l_j - \epsilon + C_1)(s-s_0)}, $$

$$ (7.7.3) \qquad |\partial_t z_j(s, t) - e^{(\pi/l_j - \epsilon)s} f_j'(t)| < C_0 e^{(\pi/l_j - \epsilon + C_1)(s-s_0)}. $$

(We allow $c = 0$.)

Note that, because of the particularly simple form of the Hamiltonian flow around $\delta_0$, Lemma 7.7.3 could also be proved directly by standard Fourier series arguments.

**Definition 7.7.4.** — The function $f_j(t)$ satisfying Equation (7.7.2) is called the asymptotic eigenfunction for $\bar{u}$ at the negative end $\delta_0^l$.<fnref n="12" />

<footnote n="12">¹² This is the terminology from [HT2], which is slightly different from that of the Hofer school.</footnote>
The following is due to Wendl [We3] and Hutchings-Taubes [HT2, Prop. 3.2]. (The proof of [HT2, Prop. 3.2] essentially proves the following lemma, but is not quite stated in the same way.)

**Lemma 7.7.5.** — There exists an arbitrarily small perturbation $\tilde{J}'$ of $\tilde{J}'$ in $\mathcal{J}_{\overline{W}'}^{\text{reg}}$, which is supported on $\mathbf{R} \times \{\rho_1 < \rho < \rho_2\} \subset \mathbf{R} \times \overline{\mathbf{N}}$ for $0 < \rho_0 < \rho_1 < \rho_2$ arbitrarily small, such that the following holds:

(★) for all $\mathbf{\gamma}$ and $\mathbf{\gamma}'$, partitions $(l_1, \dots, l_\lambda)$ of $l, j \in \{1, \dots, \lambda\}$ and $r \in \mathbf{N} \cup \{0\}$, the set of elements of $M_{\tilde{J}'}^{\text{ind}=r, (l_1, \dots, l_\lambda)}(\mathbf{\gamma}, \delta_0^l \mathbf{\gamma}')/\mathbf{R}$ with vanishing asymptotic eigenfunction at $\delta_0^l$ is a real codimension 2 submanifold.

The real codimension 2 condition is due to the fact that $\dim_{\mathbf{R}} E_{\pi/l_i+\epsilon} = 2$. Let $\mathcal{J}_{\overline{W}'}^* \subset \mathcal{J}_{\overline{W}'}^{\text{reg}}$ be the (dense) subset of almost complex structures which satisfy (★).

**7.7.3. Definition of $\tilde{U}_{m-1}$.** — Let $\mathbf{\gamma} \in \hat{\mathcal{O}}_{2g}$, $\mathbf{\gamma}' \in \hat{\mathcal{O}}_{2g-1}$, and $\tilde{J}' \in \mathcal{J}_{\overline{W}'}^*$. Let $\bar{u} \in M_{\tilde{J}'}^{l=\text{ind}=2, n^*=m-1}(\mathbf{\gamma}, \delta_0 \mathbf{\gamma}')$. Since $\tilde{J}'$ satisfies (★), the asymptotic eigenfunction of $\bar{u}$ at $\delta_0$ is nonzero. Hence we can associate a normalized asymptotic eigenfunction at $\delta_0$ to $\bar{u}$.

Now let

$$ \langle \tilde{U}_{m-1}(\mathbf{\gamma}), \delta_0^p \mathbf{\gamma}' \rangle = \begin{cases} 0, & \text{if } p \neq 1; \\ \# \left( M_{\tilde{J}'}^{l=\text{ind}=2, n^*=m-1, \delta_0}(\mathbf{\gamma}, \delta_0 \mathbf{\gamma}')/\mathbf{R} \right), & \text{if } p = 1, \end{cases} $$

for a generic normalized asymptotic eigenfunction $f_{\delta_0}$. The modifier $f_{\delta_0}$ stands for “the normalized asymptotic eigenfunction at the negative end $\delta_0$ is $f_{\delta_0}$”.

**7.7.4. The limit $m \to \infty$.** — Suppose $m \gg 0$. Let $\bar{h}_m : \overline{S} \to \overline{S}$ be a smooth extension of $h : \overline{S} \to \overline{S}$ via $v_m$, as defined in Section 5.1.2, and let $\bar{h}_\infty : \overline{S} \to \overline{S}$ be the smooth extension of $h$ via $v_\infty$. We recall that $\bar{h}_\infty$ is the identity on the disk $D_{1/2}^2 \subset \overline{S}$ of radius $1/2$.

We will denote $\overline{N}_m$ (including $m = \infty$) the mapping torus of $\bar{h}_m$ and define $\overline{W}'_m = \mathbf{R} \times \overline{N}_m$. Of course $\overline{W}'_m$ is diffeomorphic to $\overline{W}'$ for all $m$ (including $m = \infty$); we will write $\overline{W}'_m$ instead of $\overline{W}'$ when it is necessary to keep track of $m$. This will happen especially when $m = \infty$. The Hamiltonian structure on $\overline{N}_m$ will be written as $(\alpha_{0,m}, \omega)$. We then define $\mathcal{J}_{\overline{W}'_m}$ as in Definition 5.3.14, with $\alpha_0$ replaced by $\alpha_{0,m}$. Similarly, we will use the notation $\overline{W}_{-,m}$ and $\mathcal{J}_{\overline{W}_{-,m}}$ instead of $\overline{W}_-$ and $\mathcal{J}_{\overline{W}_-}$ when we need to specify $m$.

Although the orbit $\delta_0$ is degenerate when $m = \infty$, we still define the set $\overline{\mathcal{P}} = \overline{\mathcal{P}} \cup \{\delta_0\}$ of orbits, the set $\hat{\mathcal{O}}_k$ (resp. $\hat{\mathcal{O}}_k$) of orbit sets constructed from $\overline{\mathcal{P}}$ (resp. $\overline{\mathcal{P}}$) which intersect $\overline{S} \times \{0\}$ exactly $k$ times, as in Section 5.3.2.

Let $\bar{J}'_\infty \in \mathcal{J}_{\overline{W}_\infty}$. In analogy with Definition 5.3.16, let $M_{\bar{J}'_\infty}(\mathbf{\gamma}, \delta_0^l \mathbf{\gamma}')$ be the moduli space of $\bar{J}'_\infty$-holomorphic maps in $\overline{W}'$ from $\mathbf{\gamma}$ to $\delta_0^l \mathbf{\gamma}'$ without fiber components. (We recall that $\mathbf{\gamma}$ and $\mathbf{\gamma}'$ consist of orbits in $\mathbf{N}$.)
The Hamiltonian vector field of $(\alpha_{0,\infty}, \omega)$ is degenerate, but all degenerate orbits are contained in the interior of V, and therefore transversality for the moduli spaces $M_{\bar{J}'_{\infty}}(\boldsymbol{\gamma}, \delta_0^{\prime} \boldsymbol{\gamma}')$ works in the usual way. An almost complex structure $\bar{J}'_{\infty} \in \mathcal{J}_{\overline{W}_{\infty}}$ is regular if $M_{\bar{J}'_{\infty}}^{s}(\boldsymbol{\gamma}, \delta_0^{\prime} \boldsymbol{\gamma}')$ is transversely cut out for all $0 < k \le 2g$, $l \ge 0$, $\boldsymbol{\gamma} \in \mathcal{O}_k$, and $\boldsymbol{\gamma}' \in \mathcal{O}_{k-l}$.

**Lemma 7.7.6.** — A generic $\bar{J}'_{\infty} \in \mathcal{J}_{\overline{W}_{\infty}}$ is regular and satisfies (⋆).

*Proof*. — The Fredholm theory for holomorphic curves with Morse-Bott asymptotics uses Sobolev spaces with exponential weights. The regularity of simply-covered moduli spaces in this setting is treated in Wendl [We3]. □

As before, we write $\mathcal{J}_{\overline{W}_{\infty}}^{\text{reg}} \subset \mathcal{J}_{\overline{W}_{\infty}}$ for the dense subset of regular almost complex structures and $\mathcal{J}_{\overline{W}_{\infty}}^{\star} \subset \mathcal{J}_{\overline{W}_{\infty}}^{\text{reg}}$ for the dense subset of almost complex structures which satisfy (⋆). We claim that if $\bar{J}'_{\infty} \in \mathcal{J}_{\overline{W}_{\infty}}^{\star}$, then nearby almost complex structures $\bar{J}'_m \in \mathcal{J}_{\overline{W}'_m}$ for $m \gg 0$ are in $\mathcal{J}_{\overline{W}'_m}^{\star}$. Indeed, using the same exponential weights at $\delta_0^l$ for both $\bar{J}'_{\infty}$ and $\bar{J}'_m$, if $\bar{J}'_{\infty} \in \mathcal{J}_{\overline{W}_{\infty}}$ is regular, then so is a nearby $\bar{J}'_m \in \mathcal{J}_{\overline{W}'_m}$ by continuity. We can then turn off the exponential weights for $0 \ll m < \infty$ without affecting the regularity. The condition (⋆) also carries over by continuity.

On the other hand, compactness is more subtle. In fact the Hamiltonian vector field of $(\alpha_{0,\infty}, \omega)$ has a Morse-Bott family of orbits corresponding to $\text{int}(\mathcal{D}_{1/2}^2)$ and a circle of degenerate orbits corresponding to $\partial\mathcal{D}_{1/2}$, and the standard asymptotic convergence theorems do not work for degenerate orbits. For this reason, we will need extra work to be able to apply the SFT compactness theorem; namely we will need to show that a family of degenerating $\bar{J}'_m$-holomorphic curves satisfying the appropriate topological constraints cannot approach an orbit of $\mathcal{D}_{1/2}^2$ besides a multiple of $\delta_0$.

Given a $\bar{J}'_{\infty}$-holomorphic map $u: \dot{\mathcal{F}} \to \overline{W}'$, we say that an end of $\dot{\mathcal{F}}$ accumulates to a Hamiltonian orbit $\gamma$ if, after parametrizing the end by $(s_0, +\infty) \times S^1$ or $(-\infty, -s_0) \times S^1$, there is a sequence $s_k \to +\infty$ such that $\lim_{k \to \infty} u(\pm s_k, t) = \gamma(t)$ uniformly in $t$.

**Lemma 7.7.7.** — Let $u: \dot{\mathcal{F}} \to \overline{W}'$ be a finite energy $\bar{J}'_{\infty}$-holomorphic map. Then every end of $\dot{\mathcal{F}}$ accumulates to a Hamiltonian orbit $\gamma$. If an end of $\dot{\mathcal{F}}$ does not accumulate to any degenerate orbit, than it is asymptotic to a (Morse-Bott) nondegenerate orbit.

*Proof*. — The accumulation to a Hamiltonian orbit follows from [Ho]. If an end does not accumulate to any degenerate orbit, then there is a neighborhood of the degenerate orbits which does not intersect the image of the end, as the set of degenerate orbits is compact. Then we can repeat the arguments of [HWZ1, HWZ2] in the complement of that neighborhood. □
Let $\gamma_z$ be the orbit corresponding to $z \in D_{1/2}^2$ for the stable Hamiltonian structure $(\alpha_{0,\infty}, \omega)$ on $\overline{\mathbb{N}}$. We will use intersection arguments to constrain the behavior of a sequence of $\overline{J}_m$-holomorphic maps that accumulate to orbits $\gamma_z$ for some $z \in D_{1/2}^2$. (Recall that all other closed orbits in $\text{int}(\mathbb{V})$ intersect a fiber more than $2g$ times for $m \gg 0$, and therefore are automatically excluded.)

**Lemma 7.7.8.** — Let $F$ be a compact Riemann surface with boundary and let

$$ \partial F = B_1^+ \sqcup \dots \sqcup B_{l_+}^+ \sqcup B_1^- \sqcup \dots \sqcup B_{l_-}^-, \quad l_\pm \ge 1 $$

be a decomposition of $\partial F$ into connected components. We parametrize the components $B_j^-, j = 1, \dots, l_-$, by $\beta_j : S^1 \to B_j^-$ in an orientation-reversing manner. Let $\eta$ be a positive real number and $v : F \to \overline{\mathbb{W}}$ a $\overline{J}_m$-holomorphic map with $m \gg 0$, such that

(i) each $\pi_{\overline{\mathbb{N}}} \circ v \circ \beta_j$ is $\eta$-close to a cover of an orbit $\gamma_{z_j}$ with $z_j \in D_{1/2}^2$ and $\eta < \frac{|z_j|}{2}$, and

(ii) each $\pi_{\overline{\mathbb{N}}} \circ v(B_j^+)$ is contained in a small neighborhood of $\overline{N} = \overline{\mathbb{N}} - \text{int}(\mathbb{V})$.

Then $n'(v) \ge m$.

**Proof.** — Consider subsets $V_\eta = D_\eta^2 \times S^1$ and $V_{1/2+\eta} = D_{1/2+\eta}^2 \times S^1$ of $\overline{\mathbb{N}}$. Both $\partial V_\eta$ and $\partial V_{1/2+\eta}$ are foliated by Hamiltonian orbits. We choose coordinates on $\partial V_\eta$ and $\partial V_{1/2+\eta}$ such that the meridian has slope $(1, 0)$, the orbits on $\partial V_\eta$ have slope $(0, 1)$ and the orbits on $\partial V_{1/2+\eta}$ have slope $(1, k)$ with $k \gg 0$ (here we are assuming that $\eta$ has been perturbed so that the foliation on $\partial V_{1/2+\eta}$ has rational slope).

Perturb $\eta$ if necessary so that $\pi_{\overline{\mathbb{N}}} \circ v$ is transverse to $\partial V_\eta$ and $\partial V_{1/2+\eta}$ and let

$$ C_- = \pi_{\overline{\mathbb{N}}} \circ v(F) \cap \partial V_\eta \quad \text{and} \quad C_+ = \pi_{\overline{\mathbb{N}}} \circ v(F) \cap \partial V_{1/2+\eta}, $$

where $C_-$ and $C_+$ are oriented as the boundaries of $\text{Im}(\pi_{\overline{\mathbb{N}}} \circ v) \cap V_\eta$ and $\text{Im}(\pi_{\overline{\mathbb{N}}} \circ v) \cap V_{1/2+\eta}$. Since $\pi_{\overline{\mathbb{N}}} \circ v$ maps no $B_j^+$ inside $V_{1/2+\eta}$ and $C_+$ is positively transverse to the orbits on $\partial V_{1/2+\eta}$, it follows that $C_+$ represents some homology class $(q, p)$ with $p, q \ge 1$. Then a simple homology computation implies that $C_-$ represents the homology class $(q, 0)$. Hence $n'(v) = qm \ge m$. $\square$

**Lemma 7.7.9.** — Let $\overline{J}'_{m_i} \in \mathcal{J}_{\overline{W}^*_{m_i}}^*$ be a sequence such that $\overline{J}'_{m_i} \to \overline{J}_\infty \in \mathcal{J}_{\overline{W}^*_{\infty}}^*$ as $i \to \infty$. Given a partition $(l)$ of $l$, a sequence of curves

$$ \bar{u}_i \in \mathcal{M}_{\overline{J}'_{m_i}}^{1=(l), n'=m-l}(\boldsymbol{\gamma}, \delta_0^l \boldsymbol{\gamma}') $$

with $i \to \infty$ converges (up to extracting a subsequence) to a curve

$$ \bar{u}_\infty \in \mathcal{M}_{\overline{J}'_\infty}^{1=(l), \tilde{n}=0}(\boldsymbol{\gamma}, \delta_0^l \boldsymbol{\gamma}') $$

where $\tilde{n}(\bar{u}_\infty)$ is the intersection number of $\bar{u}_\infty$ with the section at infinity.
*Proof.* — The first steps of the compactness theorem are unchanged by the presence of degenerate orbits. Let $\dot{F}_i$ be the domains of the maps $\bar{u}_i$, which we assume to have the same topological type and be hyperbolic. (Otherwise we can remove one point from $\dot{F}_i$, without changing the name.) We claim that $\bar{u}_i$ satisfies the uniform gradient bound (Equation (7.3.3)). Indeed, if there is a sequence of points in $\dot{F}_i$ with gradient blow up, then we obtain a nonconstant holomorphic plane that accumulates to an orbit, contradicting the noncontractibility of all the orbits.

After passing to a subsequence, the punctured surfaces $\dot{F}_i$ converge to a punctured nodal surface $\bar{F}_\infty$ in the Deligne-Mumford moduli space. We denote by $F'_\infty$ the punctured Riemann surface obtained by removing the nodes from $\bar{F}_\infty$. By the gradient bound, we can extract a subsequence from $\bar{u}_i$ which converges in $C_{\text{loc}}^\infty$ to a $\bar{J}'_\infty$-holomorphic map $\bar{u}'_\infty : F'_\infty \to \bar{W}'$. (The holomorphic map denoted by $\bar{u}'_\infty$ in this proof should not be confused with the holomorphic map $\bar{u}'_\infty$ introduced in Section 5.7.2). We assume that no component of $\bar{u}'_\infty$ is a connector over $\delta_0$. The modifications required by the presence of a connector over $\delta_0$ are easy and left to the reader.

Fix $\varepsilon > 0$ small and suppose that $i \gg 0$. Consider the connected component of $\text{Thin}_\varepsilon(\dot{F}_i)$ with a puncture that limits to $\delta'_0$; we identify it with $(-\infty, s_i) \times S^1$. We claim that there is no $s < s_i$ such that $\pi_{\bar{N}} \circ \bar{u}_i|_{[s] \times S^1}$ is sufficiently close to an orbit $\gamma_z$ with $z \in D_{1/2}^2 - \{0\}$. Arguing by contradiction, if there is such an $s$, then we consider $C_0 := (-\infty, s] \times S^1$ and $C_1 := \dot{F}_i - (-\infty, s] \times S^1$. Since $C_0$ limits to $\delta'_0$, we have $n'(\bar{u}_i|_{C_0}) \ge m_i - l$. On the other hand, there exists a compact surface with boundary $F \subset C_0$ such that $\bar{u}_i|_F$ satisfies the conditions of Lemma 7.7.8 and $n'(\bar{u}_i|_F) \ge m_i$, a contradiction. We can similarly argue that, for any other component $(s_-, s_+)$ $\times S^1$ of $\text{Thin}_\varepsilon(\dot{F}_i)$, there is no $s \in (s_-, s_+)$ such that $\pi_{\bar{N}} \circ \bar{u}_i|_{[s] \times S^1}$ is sufficiently close to an orbit $\gamma_z$ with $z \in D_{1/2}^2 - \{0\}$; here $s_- = -\infty$ or $s_+ = +\infty$ are allowed.

We now apply the usual compactness argument (in view of Lemma 7.7.7) to obtain a limit $\bar{J}'_\infty$-holomorphic building $\bar{u}_\infty$ such that no level of the building has an end that accumulates to an orbit $\gamma_z$ for $z \in D_{1/2}^2 - \{0\}$. Finally, since $\bar{J}'_\infty \in \mathcal{J}_{\bar{W}^\gamma_\infty}^*$, every level of $\bar{u}_\infty$ has ind $\ge 1$. On the other hand, since $\bar{u}_\infty$ has total Fredholm index $1$, it is a single-level building. $\square$

Let $\bar{J}'_\infty \in \mathcal{J}_{\bar{W}^\gamma_\infty}^*$. By the compactness and regularity of moduli spaces, there are only finitely many curves $C_1, \dots, C_r$, modulo $\mathbf{R}$-translation, such that

$$C_i \in \mathcal{M}_{\bar{J}'_\infty}^{1=1, \tilde{n}=0, (l_i)}(\gamma_i, \delta_0^{l_i} \gamma_i')$$

for some orbit sets $\gamma_i, \gamma_i' \in \tilde{\mathcal{O}}_*$ and partition $(l_i)$ of $l_i$. Here if $\bar{u}$ is a curve with an end at $\delta_0^p$, then its ECH index is computed using the Conley-Zehnder index of $\delta_0^p$ with respect to $\bar{h}_m, m \gg 0$, and $\tilde{n}(\bar{u})$ is defined as the intersection number of $\bar{u}$ and the section at infinity. Let

$$f_i : \mathbf{R}/2l_i\mathbf{Z} \to \mathbf{C}, \quad t \mapsto c_i e^{\pi i t / l_i},$$
be the asymptotic eigenfunction corresponding the end $\delta_0^{l_i}$ of $C_i$. The condition $c_i \neq 0$ follows from Lemma 7.7.6. We may therefore assume without loss of generality that all the $f_i$ are normalized.

**7.7.5. Radial rays.**

**Definition 7.7.10.** — A bad radial ray is a radial ray $\mathcal{R}_{\phi_0} = \{\phi = \phi_0, \rho \ge 0\}$ in $\mathbf{C}$ which passes through a point in

$$ \{f_i(t) \mid i = 1, \dots, r; 0 < t < 2l_i; t \equiv 3/2 \mod 2\}. $$

A radial ray $\mathcal{R}_{\phi_0}$ which is not bad is said to be good.

A good radial ray must exist and, as it is explained in the following remark, we can assume it is $\mathcal{R}_{\pi}$.

**Remark 7.7.11.** — The set of bad radial rays is determined by $(\overline{W}'_{\infty}, \overline{J}'_{\infty})$. Strictly speaking, we should choose the set of endpoints $E \subset \partial D^2$ as in Section 5.2.2 such that $\mathcal{R}_{\phi_0+\pi}$ is a good radial ray and $\phi_0 < \phi(y_j(m)) < \phi_0 + c(m)$ where $c(m) \to 0$ as $m \to \infty$. After a rotational coordinate change of $D^2$, we may assume that $\mathcal{R}_{\pi}$ is a good radial ray and $0 < \phi(y_j(m)) < c(m)$.

**7.8. The rescaled function.** — In this subsection we will use limiting arguments in which $m \to \infty$ and $\bar{h}_m \to \bar{h}_{\infty}$; see Section 7.7.4. Hence many of the almost complex structures and moduli spaces will have an additional subscript $m$, where $m = \infty$ is also a possibility. Let $\bar{J}_{\infty} \in \mathcal{J}_{W'_{\infty}}^{\star}$ and let $\bar{J}_m \in \mathcal{J}_{W'_{m}}^{\star}$ be a nearby almost complex structure with respect to the integer $m \gg 0$. Let $\bar{J}_{-,m} \in \mathcal{J}_{\overline{W}_{-,m}}^{\text{reg}}$ be an almost complex structure which restricts to $\bar{J}_m$ and let $\bar{J}_{-,m}^{\diamond}$ be $(\varepsilon, \text{U})$-close to $\bar{J}_{-,m}$.

Let $m_i$, $i \in \mathbf{N}$, and $\bar{u}_{ij}$, $i, j \in \mathbf{N}$, be sequences satisfying the following properties:

(S1) $\lim_{i \to \infty} m_i = \infty$;

(S2) $\bar{u}_{ij} \in \mathcal{M}_{\bar{J}_{-,m_i}}^{I=3,n^*=m_i}(\mathbf{y}, \mathbf{y}; \overline{\mathbf{m}})$, where $\mathbf{y} = \{x_l\} \cup \mathbf{y}'$ or $\{x'_l\} \cup \mathbf{y}'$ for some $l$; and

(S3) for all $i \in \mathbf{N}$ and $\kappa, \nu > 0$, there exists $j_{i,\kappa,\nu}$ such that, if $j \ge j_{i,\kappa,\nu}$, then $\bar{u}_{ij}$ is $(\kappa, \nu)$-close (cf. Definition 6.5.2) to a $\bar{J}_{-,m_i}^{\diamond}$-holomorphic building $\bar{u}_{i\infty} = \cup_l \bar{v}_{l,i}$ with $\bar{v}'_{0,i} \neq \emptyset$.

Moreover, assume that one of the following holds:

(S4') the first negative end at $\delta_0$ in the building $\bar{u}_{i\infty}$ has multiplicity one, or

(S4'') the first negative end at $\delta_0$ in the building $\bar{u}_{i\infty}$ has multiplicity two.

Property (S3) is a consequence of the fact that the sequence $\bar{u}_{ij}$ converges in the SFT sense to the building $\bar{u}_{i\infty}$ for each fixed $i$. Property (S4') corresponds to Cases (1) and (2) in Theorem 7.6.1 and Property (S4'') corresponds to the remaining cases.
**7.8.1. Truncation.** — Recall the projection $\bar{\pi} = \bar{\pi}_{D_{p_0}^2}$ using balanced coordinates.

**Lemma 7.8.1.** — Let $\bar{v}$ be a $\bar{J}_m$-holomorphic curve in $\bar{W}'_m$ with ECH index $I \le 2$ for $\bar{J}'_m \in \mathcal{J}_{\bar{W}'_m}^*$ and let $\tilde{v} : (-\infty, s_0] \to \bar{W}'$ be a negative end of $\bar{v}$ which is asymptotic to a multiple cover of $\delta_0$. Then for every $\eta, \kappa > 0$ there exist positive constants $\bar{R}$ and $\kappa' < \eta$ such that $\frac{\kappa'}{\kappa} > \frac{1}{\eta}$ and

$$|\bar{\pi} \circ \tilde{v}(-\bar{R}, t) - \kappa' f(t)| \le \frac{\kappa}{2},$$

$$|\partial_t(\bar{\pi} \circ \tilde{v})(-\bar{R}, t) - \kappa' f'(t)| \le \frac{\kappa}{2}$$

for all $t$, where $f$ is the normalized asymptotic eigenfunction for $\tilde{v}$.

**Proof.** — This is a slightly weaker rephrasing of Lemma 7.7.3, in view of Lemma 7.7.5. $\square$

Let $F$ be a Riemann surface and $\bar{u}: F \to (\bar{W}_-, \bar{J}_-)$ a holomorphic map. We denote by $p: F \to B_-$ the map $\bar{\pi}_{B_-} \circ \bar{u}$ and by $s: F \to \mathbf{R}$, $t: F \to \mathbb{S}^1$ the functions obtained by composing $p$ with the coordinates $(s, t)$ on $B_-$. The functions $(s, t)$ give local coordinates on $F$ outside the critical points of $p$.

Let $\bar{g}$ be the restriction of an $s$-invariant Riemannian metric on $\bar{W}'$ to $\bar{W}_-$ and let $d$ be the distance induced by $\bar{g}$ on $\bar{W}_-$.

**Definition 7.8.2 (Truncation).** — A truncation $(\tilde{u}, \tilde{F}, \{R^{(l)}\}_{l=1}^{l_0}, \tilde{\mathfrak{m}}, \mathfrak{e}^{\pm}, \kappa')$ of a holomorphic map $\bar{u}: F \to (\bar{W}_-, \bar{J}_-)$ is a tuple where:

- $\tilde{F}$ is a subsurface of $F$ and $\tilde{u}$ is the restriction of $\bar{u}$ to $\tilde{F}$;

- $p(\tilde{F}) = B_- \cap \{R^{(1)} \le s \le R^{(l_0)}\}$;

- for each $l = 1, \dots, l_0 - 1$, the restriction

$$p: \tilde{F} \cap p^{-1}(\{R^{(l)} \le s \le R^{(l+1)}\}) \to B_- \cap \{R^{(l)} \le s \le R^{(l+1)}\}$$

is a branched cover;

- $\sup_{x \in \tilde{u}(\tilde{F})} d(x, \sigma_{\infty}^{-}) \le 2\kappa'$;

- if $d(\tilde{u}(y), \sigma_{\infty}^{-}) \le \frac{\kappa'}{2}$, then $y \in \tilde{F}$;

- $\tilde{\mathfrak{m}} \in \tilde{F}$ is the unique point such that $\tilde{u}(\tilde{\mathfrak{m}}) = \bar{\mathfrak{m}}$;

- $\mathfrak{e}^+$ (resp. $\mathfrak{e}^-$) is the union of components of $\partial \tilde{F} - p^{-1}(\partial B_-)$ for which $ds(\mathfrak{n}) > 0$ (resp. $< 0$), where $\mathfrak{n}$ is the outward normal vector field along $\partial \tilde{F}$.

The map $p$ is a branched cover only restricted to portions of $\tilde{F}$, and not globally, because there can be portions of $\partial \tilde{F}$ which are mapped to the interior of $B_- \cap \{R^{(1)} \le s \le R^{(l_0)}\}$.
*Definition 7.8.3.* — A good truncation

$$ (\tilde{u}, \tilde{F}, \{R^{(l)}\}_{l=1}^{l_0}, \tilde{m}, \mathfrak{e}^{\pm}, \kappa, \kappa') $$

is a truncation $(\tilde{u}, \tilde{F}, \{R^{(l)}\}_{l=1}^{l_0}, \tilde{m}, \mathfrak{e}^{\pm}, \kappa')$, together with a constant $\kappa > 0$, such that $\frac{\kappa'}{\kappa} > 2$ and:

$$ (G) \quad \| \pi \circ \tilde{u} |_{\mathfrak{e}^{\pm}} - f \|_{C^1} \le \kappa, \text{ where } f \text{ is an asymptotic eigenfunction of } \delta_0 \text{ or } z_{\infty} \text{ (as appropriate) on each component of } \mathfrak{e}^{\pm} \text{ and } \|f\|_{C^0} \ge \kappa'. $$

**Lemma 7.8.4.** — Let $m_i$ and $\bar{u}_{ij}$ be sequences satisfying (S1)–(S3) and either (S4') or (S4"). Then there is a sequence $j(i)$ such that the following hold for all $j \ge j(i)$:

(1) $\bar{u}_{ij}$ admits a good truncation

$$ (\tilde{u}_{ij}, \tilde{F}_{ij}, \{R_{ij}^{(l)}\}_{l=1}^{l_0}, \tilde{m}_{ij}, \mathfrak{e}_{ij}^{\pm}, \kappa_i, \kappa_i'); $$

(2) $\lim_{i \to \infty} \kappa_i' = \lim_{i \to \infty} \kappa_i = 0$ and $\lim_{i \to \infty} \frac{\kappa_i'}{\kappa_i} = +\infty$;

(3) $\lim_{i \to \infty} R_{ij(i)}^{(l_0)} = +\infty$ and $\lim_{i \to \infty} R_{ij(i)}^{(1)} = -\infty$.

*Proof.* — We prove the lemma in Case (S4'), where the notation is simpler, i.e., we can use $(s, t)$ as global coordinates on $\tilde{F}_{ij}$. In this case $l_0 = 2$. Case (S4") is conceptually the same.

Fix a sequence $\eta_i \to 0$. Then, by Lemma 7.8.1, for each $i$ there exist $\kappa_i'$, $\kappa_i$ and $\bar{R}_i$ which satisfy:

- $\kappa_i', \kappa_i < \eta_i$ and $\frac{\kappa_i'}{\kappa_i} > \frac{1}{\eta_i}$; and

- $|\pi \circ \tilde{v}_i(-\bar{R}_i, t) - \kappa_i' f_i(t)| \le \frac{\kappa_i}{2}$ for all $t$, where $f_i$ is the normalized asymptotic eigenfunction for a negative end $\tilde{v}_i$ of a $\bar{J}_{m_i}$-holomorphic curve $\hat{v}_i$ of ECH index $I \le 2$ that limits to $\delta_0$. The curve $\hat{v}_i$ is a component of the SFT limit of the sequence $\{\bar{u}_{ij}\}_j$.

On the other hand, by (S3) and Definition 6.5.2, for all $i$ there exists $j(i)$ such that if $j > j(i)$ then there exists $R_{ij}$ such that

$$ |\pi \circ \bar{u}_{ij}(R'_{ij} - \bar{R}_i, t) - \pi \circ \tilde{v}_i(-\bar{R}_i, t)| \le \frac{\kappa_i}{2} $$

for all $t$. Setting $R_{ij}^{(1)} = R_{ij(i)}' - \bar{R}_i$, we obtain:

$$ \begin{aligned} |\pi \circ \bar{u}_{ij}(R_{ij}^{(1)}, t) - \kappa_i' f_i(t)| &\le |\pi \circ \tilde{u}_{ij}(R_{ij}^{(1)}, t) - \pi \circ \tilde{v}_i(-\bar{R}_i, t)| \\ &\quad + |\pi \circ \tilde{v}_i(-\bar{R}_i, t) - \kappa_i' f_i(t)| \le \kappa_i. \end{aligned} $$

Similar considerations also hold for the $t$-derivative and at the negative truncated end. $\square$
**Remark 7.8.5.** — As $i \to \infty$, the curves $\hat{v}_i$ approach one of the $C_{i0}$ from Section 7.7.4, modulo $\mathbf{R}$-translation, by Lemma 7.7.9. Hence the normalized asymptotic eigenfunctions $f_i$ limit to the normalized eigenfunction $f_{i0j0}$ for $C_{i0}$ at the negative end as $i \to \infty$.

**7.8.2. Ansatz.** — We define $\tilde{z}_i = \bar{\pi} \circ \tilde{u}_{ij(i)}$ and, to simplify computations, we will make the ansatz

$$ (7.8.1) \qquad \tilde{z}_i = e^{-\epsilon_{i,s}} \tilde{w}_i, $$

where $\epsilon_i = \frac{\pi}{m_i}$.

**Lemma 7.8.6.** — The functions $\tilde{w}_i : \tilde{F}_i \to \mathbf{C}$ defined by Equation (7.8.1) are holomorphic with respect to the standard complex structure on $\mathbf{C}$.

*Proof.* — We give the proof for Case (S4”); Case (S4’) is similar but simpler. The functions $(s, t)$ give local conformal coordinates on $\tilde{F}_i$ outside the branch locus because $p_i : \tilde{F}_i \to \mathbb{B}_-$ is holomorphic. Then $\tilde{z}_i : \tilde{F}_i \to \mathbf{C}$ satisfies Equation (7.7.1):

$$ \partial_s \tilde{z}_i + i \partial_t \tilde{z}_i + \epsilon_i \tilde{z}_i = 0. $$

If we plug in Equation (7.8.1) into Equation (7.7.1) we obtain:

$$ -\epsilon_i e^{-\epsilon_{i,s}} \tilde{w}_i + e^{-\epsilon_{i,s}} \partial_s \tilde{w}_i + i e^{-\epsilon_{i,s}} \partial_t \tilde{w}_i + \epsilon_i e^{-\epsilon_{i,s}} \tilde{w}_i = 0. $$

Hence $\partial_s \tilde{w} + i \partial_t \tilde{w}_i = 0$, so $\tilde{w}_i$ is a holomorphic map to the standard complex line $(\mathbf{C}, i)$ in the complement of the branch point. Then it is holomorphic everywhere by the removal of singularities. $\square$

**7.8.3. Rescaling.** — Consider the diagonal subsequence $\bar{u}_i = \bar{u}_{ij(i)}$, $i = 1, 2, \dots$. We abbreviate $\tilde{F}_i = \tilde{F}_{ij(i)}$, $p_i = p_{ij(i)}|_{\tilde{F}_{ij(i)}}$, $\tilde{m}_i = \tilde{m}_{ij(i)}$, etc. Fix $R'_0$, $R_0 > 0$ and let $K_i$ be the connected component of $p_i^{-1}(\mathbb{B}_- \cap \{-R'_0 \le s \le R_0\})$ containing $\tilde{m}_i$. By passing to a subsequence we may assume that the topologies of $K_i$ and $\tilde{F}_i - K_i$ are independent of $i$. In order to simplify the exposition, we will assume that this is the case for all $i$.

**Definition 7.8.7.** — We define $C_i = \sup_{z \in K_i} |\tilde{w}_i(z)|$ and $w_i = \tilde{w}_i/C_i$.

The holomorphic maps $w_i : \tilde{F}_i \to \mathbf{C}$ satisfy the following properties:

(1) $\sup_{z \in K_i} |w_i(z)| = 1$;

(2) there is a unique point $\tilde{m}_i \in \tilde{F}_i$ such that $w_i(\tilde{m}_i) = 0$; the zero $\tilde{m}_i$ is a simple zero;
(3) if $x \in \partial \tilde{F}_i$ and $p_i(x) = (s, t) \in \partial B_-$, then $w_i(x) \in e^{i(\varepsilon_i t + \phi_i)} \mathbf{R}^+$ with $\varepsilon_i = \frac{\pi}{m_i}$ and
$\lim_{i \to \infty} \phi_i = 0$ for some $\phi_i$ depending on the boundary component of $\partial \tilde{F}_i$ contain-
ing $x$;

(4) for every truncated end $\mathfrak{c}$ (i.e., a connected component of $\mathfrak{c}^+$ or $\mathfrak{c}^-$) there are constants $\tilde{\kappa}_i$ and $\tilde{\kappa}_i'$ such that the inequalities

$$
(7.8.2) \quad \|w_i|_{\mathfrak{c}} - f_i\|_{C^1} \le \tilde{\kappa}_i, \quad \|f_i\|_{C^0} \ge \tilde{\kappa}_i'
$$

holds, where $f_i$ is an asymptotic eigenfunction of $\delta$ or $z_\infty$ (as appropriated) on $\mathfrak{c}$, and $\tilde{\kappa}_i$ and the constants $\tilde{\kappa}_i'$ satisfy $\tilde{\kappa}_i'/\tilde{\kappa}_i \to \infty$ as $i \to \infty$.

**Remark 7.8.8.** — The loops of $w_i|_{\epsilon_i^\pm}$ have winding number 1 around the origin when $i$ is sufficiently large: in fact $\tilde{\kappa}_i' > \tilde{\kappa}_i$, so the linear homotopy between $w_i|_{\epsilon_i^\pm}$ and $f_i(t)$ is contained in $\mathbf{C}^\times$.

**Lemma 7.8.9.** — After passing to a subsequence, $w_i|_{K_i}$ converges to a nonconstant holomorphic function $w_\infty|_{K_\infty}$.

*Proof*. — After passing to a subsequence we may assume that $K_i$ converges to a Riemann surface $K_\infty$ with boundary. Since $w_i|_{K_i}$ is uniformly bounded, the lemma follows from Montel's theorem. $\square$

From now on we assume that we have passed to a subsequence so that Lemma 7.8.9
holds. In the following subsections we analyze the convergence of $w_i$, not just on the uni-
formly bounded part $K_i$. This involves techniques of SFT compactness.

**7.8.4. Energy bound.** — Following Hofer [Ho], we define an *energy* for holomorphic functions on Riemann surfaces with boundary and punctures.

**Definition 7.8.10 (Energy).** — Let $\mathcal{C}$ be the set of smooth functions $\varphi : [0, \infty) \to [0, 1]$ such that $\varphi(0) = 0$ and $\varphi'(r) \ge 0$ for all $r \in [0, \infty)$. If $\mathcal{F}$ is a Riemann surface and $u : \mathcal{F} \to \mathbf{C}$ a holomorphic function, we define the energy of $u$ as

$$
E(u) = \sup_{\varphi \in \mathcal{C}} \int_{\mathcal{F}} u^* d(\varphi(r) d\theta),
$$

where $(r, \theta)$ are polar coordinates on $\mathbf{C}$.

**Remark 7.8.11.** — If we identify $\mathbf{C}^\times \to \mathbf{R} \times \mathbf{S}^1$ using the log map, then

$$
\sup_{\varphi \in \mathcal{C}} \int_{\mathcal{F}-u^{-1}(0)} u^* d(\varphi(r) d\theta)
$$
agrees with the usual expression for the Hofer energy of

$$ \log \circ u : \mathbf{F} - u^{-1}(0) \to \mathbf{R} \times \mathbf{S}^{1}. $$

**Lemma 7.8.12.** — The sequence $w_i$ has uniformly bounded energy.

*Proof.* — By Stokes' theorem,

$$ \int_{\tilde{F}_i} w_i^*(d\varphi(r)d\theta) = \int_{\partial \tilde{F}_i} w_i^*(\varphi(r)d\theta). $$

The boundary $\partial \tilde{F}_i$ is the union of $e_i^\pm$, $f_{ik} = p_i^{-1}(\{t=k\}) \cap \partial \tilde{F}_i$, $k=0, 1$, and $f_{i2} = p_i^{-1}(\{1<t<2\}) \cap \partial \tilde{F}_i$, all oriented using the boundary orientation. If $c$ is a component of $e_i^\pm$, then $\int_c w_i^* d\theta = 2\pi \deg(w_i|_c)$ if $c$ is a circle and $\int_c w_i^* d\theta \approx \frac{\pi}{m}$ if $c$ is an arc because $w_i|_c$ is $C^l$-close to an asymptotic eigenfunction with $l \ge 1$ (compared to the $C^0$ norm of the eigenfunction). This follows from the exponential decay estimates of [HWZ1]; also see [HT2, Lemma 2.3]. Moreover, for $k=0, 1$, $\int_{f_{ik}} w_i^*(\varphi(r)d\theta) = 0$ since $w|_{f_{ik}}$, $k=0, 1$, projects to a radial ray. Finally, $\int_{f_{i2}} w_i^*(\varphi(r)d\theta) < 0$ since $w|_{f_{i2}}$ always has a component in the negative $\theta$-direction; here it is important to remember that we are projecting using balanced coordinates. This proves the lemma. $\square$

**Remark 7.8.13.** — Observe that $E(cu) = E(u)$ where $c \in \mathbf{C}^\times$.

**7.8.5. Bubbling.** — The goal of this subsection is to eliminate certain types of bubbling.

Let $(\tilde{F}^i, \tilde{g}_i)$, $i \in \mathbf{N}$, be a sequence of Riemannian surfaces which are compatible with the complex structures and have injectivity radii which are uniformly bounded below. For example, one could obtain the metrics $\tilde{g}_i$ by scaling the compatible hyperbolic metrics by a conformal factor so that the thick parts remain hyperbolic, while the thin parts become flat cylinders. We will use these metrics to compute the pointwise norm of the differential of the functions $w_i$ which are denoted by $|w'_i|$.

**Lemma 7.8.14.** — For every compact set $K \subset \mathbb{B}_-$, there is a constant $C_K > 0$ such that $|w_i(z)| < C_K$ and $|w'_i(z)| < C_K$ for all $z \in p_i^{-1}(K)$ and all $i \in \mathbf{N}$.

*Proof.* — The lemma holds for $K \cap K_i$ by Lemma 7.8.9. Hence it suffices to consider $K \cap K_i^c$, where $K_i^c := \tilde{F}_i - \text{int}(K_i)$. Since $w_i(K_i^c) \subset \mathbf{C}^\times$, we compose with $\log : \mathbf{C}^\times \to \mathbf{R} \times (\mathbf{R}/2\pi\mathbf{Z})$ and consider $\zeta_i = \log \circ w_i|_{K_i^c}$. The sequence $\zeta_i$ has uniformly bounded energy by Lemma 7.8.12. If $|\zeta'_i|$, $i \in \mathbf{N}$, is not uniformly bounded on $K_i^c$, then the usual bubbling analysis yields a nonconstant finite energy plane $\tilde{v}_\infty^+ : \mathbf{C} \to \mathbf{R} \times (\mathbf{R}/2\pi\mathbf{Z})$ or half-plane $\tilde{v}_\infty^+ : \mathbf{H} \to \mathbf{R} \times (\mathbf{R}/2\pi\mathbf{Z})$; see the proof of [Ho, Theorem 31]. If $\tilde{v}_\infty^+$ is a half-plane, then $\tilde{v}_\infty^+(\partial \mathbf{H})$ is contained in a line $\mathbf{R} \times \{pt\} \subset \mathbf{R} \times (\mathbf{R}/2\pi\mathbf{Z})$ as a consequence
of the boundary conditions for the maps $w_i$, and we can double $\tilde{v}_\infty^+$ to obtain a nonconstant finite energy plane by the Schwarz reflection principle. On the other hand, by [Ho, Lemma 28], there are no nonconstant finite energy planes in $\mathbf{R} \times (\mathbf{R}/2\pi\mathbf{Z})$, a contradiction. Hence $|\zeta'_i|$, $i \in \mathbf{N}$, is uniformly bounded on $K_i^c$. This in turn implies uniform bounds on $|w_i|$ and $|w'_i|$ on $K \cap K_i^c$. $\square$

**7.8.6. Case (S4')**. — In Cases (S4') and (S4'') we often pass to a subsequence without explicit mention.

Suppose we are in Case (S4'). Recall the compactification $cl(\mathbf{B}_-)$ of $\mathbf{B}_-$ defined in Section 5.1.1, obtained by adjoining the points at infinity $\mathfrak{p}_+$, $\mathfrak{p}_-$. Here $cl(\mathbf{B}_-)$ is isomorphic to the closed unit disk, $\mathfrak{p}_+$ is a marked point in the interior and $\mathfrak{p}_-$ is a marked point on the boundary under this identification.

**Theorem 7.8.15.** — The sequence $(w_i)_{i \in \mathbf{N}}$ converges uniformly on compact subsets to a holomorphic map $w_\infty : \mathbf{B}_- \to \mathbf{C}$ such that:

(1) $w_\infty(\partial \mathbf{B}_-) \subset \mathbf{R}^+$ and $w_\infty(\bar{\mathfrak{m}}^b) = 0$;

(2) $\lim_{s \to +\infty} |w_\infty(s, t)| = +\infty$ and $\lim_{s \to +\infty} \frac{w_\infty(s, t)}{|w_\infty(s, t)|} = f_{i_0 j_0}(t)$, for some $i_0, j_0$;

(3) $\lim_{s \to -\infty} w_\infty(s, t) = c \in \mathbf{R}^+$;

(4) $w_\infty|_{\mathrm{int}(\mathbf{B}_-)}$ is a biholomorphism onto its image. In particular $\bar{\mathfrak{m}}^b$ is the unique zero of $w_\infty|_{\mathrm{cl}(\mathbf{B}_-)}$ and is simple;

(5) $w_\infty$ extends to a holomorphic map $cl(\mathbf{B}_-) \to \mathbf{CP}^1$, still called $w_\infty$, such that $w_\infty(\partial cl(\mathbf{B}_-)) = [a_1, a_2] \subset \mathbf{R}^+$ and $w_\infty(\mathfrak{p}_+) = \infty$.

Recall that $f_{i_0 j_0}(t)$ is a normalized asymptotic eigenfunction of the curve $C_{i_0}$ with a negative end asymptotic to a multiple of $\delta_0$.

*Proof*. — The uniform convergence on compact subsets is a consequence of the bounds from Lemma 7.8.14. (1) is immediate from the convergence, together with the observation that the angles between the arcs in $\bar{\mathfrak{a}}$ tend to 0 as $m_i \to \infty$.

(2) We can either use SFT compactness and analyze the levels, or argue directly as follows: Let $K_i^+$ (resp. $K_i^-$) be the component of $\tilde{F}_i - \mathrm{int}(K_i)$ with positive (resp. negative) s-coordinates. We expand $w_i$ and $w_\infty$ in Fourier series on $K_i^+$:

$$w_i(s, t) = \sum_{n=-\infty}^{+\infty} a_n^i e^{\pi n(s+it)}, \quad w_\infty(s, t) = \sum_{n=-\infty}^{+\infty} a_n^\infty e^{\pi n(s+it)}.$$

By the uniform convergence of $w_i$ to $w_\infty$, $\lim_{i \to \infty} a_n^i = a_n^\infty$ for all $n$. Then (2) is equivalent to the conditions:

(i) $\lim_{i \to \infty} \frac{a_n^i}{|a_1^i|} e^{i\pi t} = f_{i_0 j_0}(t)$; and
$$
(ii) \lim_{i \to \infty} a_n^i = 0 \text{ when } n \ge 2.
$$

Since we are in Case (S4'), $\{\mathbb{R}_i^{(l)}\}_{l=1}^{l_0} = \{\mathbb{R}_i^{(1)}, \mathbb{R}_i^{(2)}\}$, i.e., $l_0 = 2$. By Equation (7.8.2),

$$
\left| \frac{1}{2} \int_{0}^{2} \left( w_{i}(\mathbb{R}_{i}^{(2)}, t) - f_{i}(t) \right) e^{-\pi i n t} dt \right| \le \tilde{\kappa}_{i},
$$

for all $n \in \mathbb{Z}$. On the other hand, if we write $f_i(t) = c_i e^{\pi i t}$, where $f_i(t)$ is not necessarily normalized, then

$$
\frac{1}{2} \int_{0}^{2} \left( w_{i}(\mathbb{R}_{i}^{(2)}, t) - f_{i}(t) \right) e^{-\pi i n t} dt = a_{n}^{i} e^{\pi n \mathbb{R}_{i}^{(2)}} - c_{i} \delta_{1n},
$$

where $\delta_{1n}$ is the Kronecker delta. Hence

$$
(a) |a_n^i| \cdot e^{\pi n \mathbb{R}_i^{(2)}} \le \tilde{\kappa}_i \text{ for all } n \ne 1; \text{ and}
$$

$$
(b) |a_1^i e^{\pi R_1^{(2)}} - c_1| \le \tilde{\kappa}_1.
$$

We prove (ii). Arguing by contradiction, suppose that $\lim_{i \to \infty} |a_n^i| \ne 0$ for some $n \ge 2$.
By (a), there exists $C > 0$ such that $C e^{2\pi i \mathbb{R}_i^{(2)}} < \tilde{\kappa}_i$ for all $i$. Since $\mathbb{R}_i^{(2)} \to \infty$ and $\lim_{i \to \infty} a_1^i$
exists, we obtain $|a_1^i e^{\pi \mathbb{R}_i^{(2)}}| < \tilde{\kappa}_i$ for all $i$. On the other hand, $|c_i| \ge \tilde{\kappa}_i'$ and $\lim_{i \to \infty} \frac{\tilde{\kappa}_i'}{\tilde{\kappa}_i} = +\infty$,
which contradicts (b).

Next we prove (i). We claim that $\lim_{i \to \infty} a_1^i \ne 0$ for topological reasons. Indeed, if
$\lim_{i \to \infty} a_1^i = 0$, then $\lim_{i \to \infty} a_n^i = 0$ for all $n \ge 1$ by (ii) and the curve $w_i|_{s=\mathbb{R}_i^{(2)}}$ has nonpositive
winding number around 0 when $i \gg 0$, a contradiction. This proves the claim. Finally,
$$
\lim_{i \to \infty} \left| a_1^i \cdot \frac{e^{\pi \mathbb{R}_i^{(2)}}}{|c_i|} - \frac{c_i}{|c_i|} \right| = 0, \text{ since } \lim_{i \to \infty} \frac{\tilde{\kappa}_i}{\tilde{\kappa}_i'} = \lim_{i \to \infty} \frac{\tilde{\kappa}_i}{|c_i|} = 0. \text{ Hence}
$$

$$
\lim_{i \to \infty} \frac{a_1^i}{|a_1^i|} e^{\pi i t} = f_{ij0}(t),
$$

which proves (i).

(3) This is similar to (2). We expand $w_i$ in Fourier series on $K_i^-:$

$$
w_i(s, t) = \sum_{-\infty}^{+\infty} a_n^i e^{\epsilon_i i} e^{(\pi n - \epsilon_i)(s+it)}.
$$

By the uniform convergence, $\lim_{s \to -\infty} a_n^i = a_n^\infty$ and we can similarly prove that $a_n^\infty = 0$ for all $n < 0$. This implies (3) because the normalized eigenfunctions converge to a constant as $i \to +\infty$.
(4) Since $H_2(cl(B_-), \partial cl(B_-)) \cong H_2(CP^1, [a_1, a_2]) \cong \mathbb{Z}$, we have a well-defined notion of degree for $w_\infty$. Moreover, as in the closed case, the degree is equal to the cardinality of the inverse image of a regular value in $CP^1 - [a_1, a_2]$. Hence $\deg w_\infty|_{int(B_-)} = 1$ because $w_\infty^{-1}(\infty) = \{0\}$ and $0$ is a simple pole. This implies that $w_\infty : int(B_-) \to \mathbb{C} - [a_1, a_2]$ is a biholomorphism.

(5) follows from (2) and (3). $\square$

<figure><img src="image_16.png" /></figure>

**7.8.7. Case (S4' ).** — We give a brisk treatment of the construction of the limit, mostly pointing out the differences with (S4'). The main difference is that the holomorphic maps $\bar{u}_i$ admit good truncations with a component which is a branched double cover over its image with a single branch point $b_i \in B_-$, and we must analyze different cases depending on the behavior of the branch point as $i \to \infty$.

There are five cases:

(a) $\lim_{i \to \infty} b_i = b_\infty \in int(B_-)$,

(b) $\lim_{i \to \infty} b_i = b_\infty \in \partial B_-$,

(c) $\lim_{i \to \infty} s(b_i) = +\infty,$

(d') $\lim_{i \to \infty} s(b_i) = -\infty$ and $d(b_i, \partial B_-) \ge C$, where $C > 0$ is a constant, or

(d'') $\lim_{i \to \infty} s(b_i) = -\infty$ and $d(b_i, \partial B_-) \to 0.$

The sequence $p_i : \tilde{F}_i \to B_-$ converges in the SFT sense to a 1- or 2-level holomorphic building $p_\infty = p_\infty^0, p_\infty^1 \cup p_\infty^2$, or $p_\infty^{-1} \cup p_\infty^0$, where $p_\infty^{-1}, p_\infty^0, p_\infty^1$ are branched covers of $B$, $B_-$, $B'$, respectively. We have a 1-level building in Cases (a) and (b) and a 2-level building in Cases (c), (d'), and (d''). Let $p_\infty^m : \tilde{F}_\infty^m \to B_-$ be the restriction of $p_\infty^0$ to the component containing the limit $\tilde{m}_\infty$ of $\tilde{m}_i$ and let $p_\infty^b : \tilde{F}_\infty^b \to B$ or $B'$ be the restriction of $p_\infty^{-1}, p_\infty^0$, or $p_\infty^1$ to the component containing the limit $b_\infty$ of $b_i$. If $p_\infty^m = p_\infty^b$ (i.e., in Cases (a) and (b)) we drop the superscripts.

Near all the punctures of $\tilde{F}_\infty^*, \star \in \{\mathfrak{m}, b\}$, we use cylindrical or rectangular coordinates $(s, t)$ induced by the coordinates $(s, t)$ on $B_-$ by pullback, after a possible translation in the $s$-direction.

For $\star \in \{\emptyset, \emptyset, -\}$, we denote the compactification of $B^*$, obtained by adjoining the points at infinity $\mathfrak{p}_\pm^*$, by $cl(B^*)$. Similarly, let $cl(\tilde{F}_\infty^*)$, $\star \in \{\mathfrak{m}, b\}$, be the compactification of $\tilde{F}_\infty^*$, obtained by adjoining $\mathfrak{q}_{\pm,j}^*$, where $j \in \{1\}$ or $\{1, 2\}$, depending on the number of ends. (If there is only one end, we suppress the index.) The maps $\mathfrak{p}_\infty^* : \tilde{F}_\infty^* \to B^*$, can be compactified to $\mathfrak{p}_\infty^* : cl(\tilde{F}_\infty^*) \to cl(B^*)$, where $\mathfrak{q}_{\pm,j}^*$ is mapped to $\mathfrak{p}_\pm^*$.

In Case (a), $\tilde{F}_\infty$ is an annulus with a puncture $\mathfrak{q}_+$ in the interior and a puncture $\mathfrak{q}_{-,j}$ on each boundary component. In Case (b), $\tilde{F}_\infty$ is a disk with one puncture $\mathfrak{q}_+$ in the interior, two punctures $\mathfrak{q}_{-,j}, j = 1, 2$, on the boundary, and two boundary points $b_{\infty,j}$, $j = 1, 2$, which are glued together to give $b_\infty$. The points $\mathfrak{q}_{-,j}$ and $b_{\infty,j}$ alternate along the boundary. In Case (c), $\tilde{F}_\infty^m$ is a disk with a puncture $\mathfrak{q}_+^m$ in the interior and a puncture $\mathfrak{q}_-^m$
on the boundary, and $\tilde{F}_{\infty}^b$ is a sphere with three punctures $q_+^b, q_{-j}^b, j = 1, 2$. In Case (d'), $\tilde{F}_{\infty}^m$ is a disk with a puncture $q_+^m$ in the interior and two punctures $q_{-j}^m$ on the boundary and $\tilde{F}_{\infty}^b$ is a disk with four punctures $q_{\pm,j}^b$ on the boundary. In Case (d''), $\tilde{F}_{\infty}^m$ is a disk with a puncture $q_+^m$ in the interior and two punctures $q_{-j}^m$ on the boundary and $\tilde{F}_{\infty}^b$ is a disk with four punctures $q_{\pm,j}^b$ on the boundary and two boundary points $b_{\infty,j}, j = 1, 2$, identified.

Cases (a) and (b) are similar to Case (S4'), while the situation in Cases (c), (d') and (d'') is complicated by the fact that the limit is a 2-level holomorphic building.

Cases (a) and (b).

**Theorem 7.8.16.** — Suppose $\lim_{i \to \infty} b_i = b_{\infty} \in B_-$. Then $(w_i)_{i \in \mathbb{N}}$ converges to a holomorphic map $w_{\infty} : \tilde{F}_{\infty} \to \mathbf{C}$ such that:

(1) $w_{\infty}(\partial \tilde{F}_{\infty}) \subset \mathbf{R}^{+}$;

(2) at the positive puncture $q_+$,

$$ \lim_{s \to +\infty} |w_{\infty}(s, t)| = +\infty, \quad \lim_{s \to +\infty} \frac{w_{\infty}(s, t)}{|w_{\infty}(s, t)|} = f_{i_0 j_0}(t); $$

(3) at the negative punctures $q_{-j}, j = 1, 2$, $\lim_{s \to -\infty} w_{\infty}(s, t) = c_j \in \mathbf{R}^{+}$;

(4a) in Case (a), $w_{\infty}$ extends to a holomorphic map $w_{\infty} : cl(\tilde{F}_{\infty}) \to \mathbf{CP}^1$ such that $w_{\infty} : int(cl(\tilde{F}_{\infty})) \to \mathbf{CP}^1 - ([a_1, a_2] \cup [a_3, a_4])$ is a biholomorphism;

(4b) in Case (b), $w_{\infty}$ extends to a holomorphic map $w_{\infty} : cl(\tilde{F}_{\infty}) \to \mathbf{CP}^1$ such that $w_{\infty} : int(cl(\tilde{F}_{\infty})) \to \mathbf{CP}^1 - [a_1, a_2]$ is a biholomorphism;

(5) $\tilde{m}_{\infty}$ is the unique zero of $w_{\infty}|_{int(cl(\tilde{F}_{\infty}))}$ and is simple; $p_{\infty}(\tilde{m}_{\infty}) = \overline{\tilde{m}}^b$.

**Proof.** — The proof of Theorem 7.8.15 goes through without modification to give (1)–(3).

(4a) As in the proof of Theorem 7.8.15(4), we can define the degree for maps of pairs $(cl(\tilde{F}_{\infty}), \partial cl(\tilde{F}_{\infty})) \to (\mathbf{CP}^1, \mathbf{R}^+)$.

The degree of $w_{\infty}$ is 1 because it has a unique pole of order 1. The order of the pole at the positive puncture can be computed from the winding number of $f_{i_0 j_0}$, which is the smallest one for a positive eigenvalue by Lemma 7.7.5.

Then $w_{\infty}$ can have no branch points in the interior of $cl(\tilde{F}_{\infty})$. (4b) is similar. (5) follows from (4a) and (4b). $\square$

Cases (c), (d') and (d''). When the sequence $\{b_i\}$ is unbounded, there are surfaces $\tilde{F}_i^*, \star \in \{\mathbf{m}, \mathbf{b}\}$, with embeddings $\iota_i^* : \tilde{F}_i^* \to \tilde{F}_i$, such that $\tilde{F}_i^*$ converges to $\tilde{F}_{\infty}^*$. Let $p_i^m : \tilde{F}_i^m \to B_-$ be the restriction of $p_i$ and let $p_i^b : \tilde{F}_i^b \to B^*, \star = \emptyset$ or $'$, be the composition of an $s$-translation and $p_i|_{\tilde{F}_i^b}$ so that $s$-coordinate of $p_i^b(b_i)$ is zero.

Let $w_i$ be as in Definition 7.8.7. Let

$$ K_i^b = (p_i \circ \iota_i^b)^{-1}(B_- \cap \{s(b_i) - 1 \le s \le s(b_i) + 1\}) \subset \tilde{F}_i^b $$
and $C_i^b = \sup_{z \in K_i^b} |w_i(i_i^b(z))|$. Then we set $w_i^m = w_i \circ i_i^m$ and $w_i^b = (w_i \circ i_i^b)/C_i^b$.

**Lemma 7.8.17.** — For every compact set $K \subset B^*$, there is a constant $C_K > 0$ such that $|w_i^*(z)| < C_K$, and $|(w_i^*)'(z)| < C_K$ for all $z \in (p_i^*)^{-1}(K)$ and all $i \in \mathbf{N}$. Here $\star \in \{\mathbf{m}, \mathbf{b}\}$ and $\star \in \{\emptyset, \emptyset, -\}$, as appropriate.

*Proof.* — Similar to Lemma 7.8.14. $\square$

<figure><img src="image_17.png" /></figure>

Lemma 7.8.17 implies that the limits $w_\infty^m$ and $w_\infty^b$ exist. The following lemma gives the behavior of $w_\infty^m$ and $w_\infty^b$ near the punctures.

**Lemma 7.8.18.** —

(1) Let $u : \mathbf{R}^+ \times (\mathbf{R}/\pi\mathbf{Z}) \to \mathbf{C}^\times$ be a finite energy holomorphic map. If the map $t \mapsto u(s, t)$ has degree one for some (and therefore all) $s \in \mathbf{R}^+$, then $\lim_{s \to +\infty} |u_\infty(s, t)| = +\infty$ and

$$ \lim_{s \to +\infty} \frac{u_\infty(s, t)}{|u_\infty(s, t)|} = ce^{\pi it} \text{ with } c \neq 0. $$

(2) Let $u : \mathbf{R}^- \times (\mathbf{R}/\pi\mathbf{Z}) \to \mathbf{C}^\times$ be a finite energy holomorphic map. If the map $t \mapsto u(s, t)$ has degree one for some (and therefore all) $s \in \mathbf{R}^-$, then $\lim_{s \to -\infty} |u_\infty(s, t)| = 0$.

*Proof.* — (1) Let us view $u$ as a map $\mathbf{R}^+ \times (\mathbf{R}/\pi\mathbf{Z}) \to \mathbf{R} \times (\mathbf{R}/\pi\mathbf{Z})$. As in the proof of Lemma 7.8.14, since $u$ has finite energy, it has bounded derivative. Let $u_n(s, t) = u(s + k_n, t)$, where $k_n \in \mathbf{R}^+$ and $\lim_{n \to +\infty} k_n = +\infty$. The sequence $u_n$ has uniformly bounded derivative and converges to a finite energy holomorphic map

$$ u_\infty : \mathbf{R} \times (\mathbf{R}/\pi\mathbf{Z}) \to \mathbf{R} \times (\mathbf{R}/\pi\mathbf{Z}). $$

Such a holomorphic map is of the form $u_\infty(s, t) = (s + a, t + b)$, where $a, b$ are constants. This implies (1). (2) is similar and is left to the reader. $\square$

<figure><img src="image_18.png" /></figure>

Case (c).

**Theorem 7.8.19.** — Suppose $\lim_{i \to \infty} s(b_i) = +\infty$. Then $(w_i^m)_{i \in \mathbf{N}}$ converges to a holomorphic map $w_\infty^m : \tilde{\mathbf{F}}_\infty^m \to \mathbf{C}$ such that:

(1) $w_\infty^m(\partial \tilde{\mathbf{F}}_\infty^m) \subset \mathbf{R}^+$;

(2) at the positive puncture $\mathbf{q}_+^m$,

$$ \lim_{s \to +\infty} |w_\infty^m(s, t)| = +\infty, \quad \lim_{s \to +\infty} \frac{w_\infty^m(s, t)}{|w_\infty^m(s, t)|} = f(t), $$

where $f$ is a normalized eigenfunction of the asymptotic operator at $\delta_0$ with winding number one;
(3) at the negative puncture $\mathfrak{q}_{-}^{m}$, $\lim_{s \to -\infty} w_{\infty}^{m}(s, t) = c \in \mathbf{R}^{+}$;

(4) $w_{\infty}^{m}$ extends to a holomorphic map $w_{\infty}^{m}:cl(\tilde{\mathbb{F}}_{\infty}^{m}) \to \mathbf{CP}^{1}$ such that $w_{\infty}^{m}:\mathrm{int}(cl(\tilde{\mathbb{F}}_{\infty}^{m})) \to \mathbf{CP}^{1} - [a_1, a_2]$ is a biholomorphism; and

(5) $\tilde{\mathfrak{m}}^b$ is the unique zero of $w_{\infty}^m|_{\mathrm{int}(cl(\tilde{\mathbb{F}}_{\infty}^{m}))}$ and is simple.

Also $(w_i^b)_{i \in \mathbb{N}}$ converges to a holomorphic map $w_{\infty}^b: \tilde{\mathbb{F}}_{\infty}^b \to \mathbf{C}$ such that:

(6) at the positive puncture $\mathfrak{q}_{+}^{b}$,

$$ \lim_{s \to +\infty} |w_{\infty}^{b}(s, t)| = +\infty, \quad \lim_{s \to +\infty} \frac{w_{\infty}^{b}(s, t)}{|w_{\infty}^{b}(s, t)|} = f_{i_0 j_0}(t); $$

(7) at the negative puncture $\mathfrak{q}_{-,1}^{b}$ that connects to $\mathfrak{q}_{+}^{m}$, $\lim_{s \to -\infty} w_{\infty}^{b}(s, t) = 0$;

(8) at the other negative puncture $\mathfrak{q}_{-,2}^{b}$, $\lim_{s \to -\infty} w_{\infty}^{b}(s, t) = c \in \mathbf{R}^{+}$;

(9) at the punctures $\mathfrak{q}_{+}^{m}$ and $\mathfrak{q}_{-,1}^{b}$, $\lim_{s \to +\infty} \frac{w_{\infty}^{m}(s, t)}{|w_{\infty}^{m}(s, t)|} = \lim_{s \to -\infty} \frac{w_{\infty}^{b}(s, t)}{|w_{\infty}^{b}(s, t)|}$;

(10) $w_{\infty}^{b}$ extends to a biholomorphism $w_{\infty}^{b}: \mathbf{CP}^{1} \to \mathbf{CP}^{1}$; and

(11) $\mathfrak{q}_{-,1}^{b}$ is the unique zero of $w_{\infty}^{b}$ and is simple.

*Proof*. — The proof of Theorem 7.8.15 goes through with little modification, in view of Lemma 7.8.18. We remark that (8) is a consequence of Convention 6.6.4 and the proof technique of Lemma 6.6.5. $\square$

Cases (d') and (d'').

**Theorem 7.8.20.** — Suppose $\lim_{i \to \infty} s(b_i) = -\infty$. Then $(w_i^m)_{i \in \mathbb{N}}$ converges to a holomorphic map $w_{\infty}^m: \tilde{\mathbb{F}}_{\infty}^m \to \mathbf{C}$ such that:

(1) $w_{\infty}^{m}(\partial \tilde{\mathbb{F}}_{\infty}^{m}) \subset \mathbf{R}^{+}$;

(2) at the positive puncture $\mathfrak{q}_{+}^{m}$,

$$ \lim_{s \to +\infty} |w_{\infty}^{m}(s, t)| = +\infty, \quad \lim_{s \to +\infty} \frac{w_{\infty}^{m}(s, t)}{|w_{\infty}^{m}(s, t)|} = f_{i_0 j_0}(t); $$

(3) at the negative punctures $\mathfrak{q}_{-,j}^{m}, j = 1, 2$, $\lim_{s \to -\infty} w_{\infty}^{m}(s, t) = c_j \in \mathbf{R}^{+}$;

(4) $w_{\infty}^{m}$ extends to a holomorphic map $w_{\infty}^{m}:cl(\tilde{\mathbb{F}}_{\infty}^{m}) \to \mathbf{CP}^{1}$ such that $w_{\infty}^{m}:\mathrm{int}(cl(\tilde{\mathbb{F}}_{\infty}^{m})) \to \mathbf{CP}^{1} - [a_1, a_2]$ is a biholomorphism;

(5) $\tilde{\mathfrak{m}}_{\infty}$ is the unique zero of $w_{\infty}^{m}|_{\mathrm{int}(cl(\tilde{\mathbb{F}}_{\infty}^{m}))}$ and is simple.

Also $(w_i^b)_{i \in \mathbb{N}}$ converges to a constant map $w_{\infty}^b: \tilde{\mathbb{F}}_{\infty}^b \to \mathbf{C}$.
**7.9. Involution lemmas.** — In this subsection we collect some lemmas on holomorphic maps between Riemann surfaces with anti-holomorphic involutions. These lemmas, collectively referred to as the *involution lemmas*, will play an important role in Section 7.10 and in the sequel [II].

Our starting point is the following observation, whose proof is straightforward.

**Observation 7.9.1.** — Let $\Sigma_1, \Sigma_2$ be Riemann surfaces with anti-holomorphic involutions $\iota_1, \iota_2$, respectively. If $f : \Sigma_1 \to \Sigma_2$ is a holomorphic map, then $\tilde{f} := \iota_2 \circ f \circ \iota_1$ is also holomorphic. Moreover, if $f = \tilde{f}$, then $f(\text{Fix}(\iota_1)) \subset \text{Fix}(\iota_2)$, where $\text{Fix}(\iota_i)$ is the fixed point set of $\iota_i$.

There are four versions of the involution lemma; the first two will be used in this paper and the last two only in the sequel [II]. We start by introducing some common notation which will be used in all four versions: For $i = 1, 2$, the Riemann surface $\Sigma_i$ is an open subset of $\mathbb{CP}^1$ which is invariant under complex conjugation and has finitely generated fundamental group; moreover no component of $\mathbb{CP}^1 - \Sigma_i$ is a single point. The complex conjugation on $\mathbb{CP}^1$ restricts to an anti-holomorphic involution $\iota_i : \Sigma_i \to \Sigma_i$. On each $\Sigma_i$ we fix “radial rays”

$$ \mathcal{R}_i = \Sigma_i \cap (\mathbb{R}^{\le 0} \cup \{\infty\}). $$

The asymptotic marker $\dot{\mathcal{R}}_i(0)$ is the connected component of $T_0\mathbb{RP}^1 - \{0\}$ (i.e., a tangent half-line) consisting of vectors with negative $\partial_\alpha$-component; similarly, the asymptotic marker $\dot{\mathcal{R}}_i(\infty)$ is the component of $T_\infty\mathbb{RP}^1 - \{0\}$ that is mapped to $\mathcal{R}_i(0)$ under the inversion $z \mapsto \frac{1}{z}$. The radial rays $\mathcal{R}_i$ and their related asymptotic markers are invariant under the involution $\iota_i$. In this section we will use the notation $\mathbf{D}$ for the open unit disk in $\mathbf{C}$, considered as a Riemann surface.

**Lemma 7.9.2.** — Given $\Sigma_i$ as above, there is a compact Riemann surface with boundary $\bar{\Sigma}_i$ with a biholomorphism $\Sigma_i \to \text{int}(\bar{\Sigma}_i)$. Moreover there is an anti-holomorphic involution $\iota_i : \bar{\Sigma}_i \to \bar{\Sigma}_i$ such that the diagram

<figure><img src="image_19.png" /></figure>

commutes.

*Sketch of proof.* We outline the proof of the first statement. Use the uniformization theorem to identify the universal cover of $\Sigma = \Sigma_1$ or $\Sigma_2$ with the open upper half space $\mathbf{H}$. Let $G \subset \text{PSL}(2, \mathbb{R})$ be the deck transformation group of $\mathbf{H}$ such that $\mathbf{H}/G = \Sigma$. If $G$ is finitely generated, then $(\partial\mathbf{H} - L)/G$ is a collection of boundary circles of $\Sigma$, where $L \subset \partial\mathbf{H}$ is the limit set of $G$. Hence $\bar{\Sigma} = (\bar{\mathbf{H}} - L)/G$. $\square$
<figure><img src="image_20.png" /><figcaption>FIG. 5. — The map $f|_{\Sigma_1}$ in Lemma 7.9.3. The left-hand side is $\Sigma_1$, where the point at infinity is represented by the dotted circle, and the right-hand side is $\Sigma_2$. The thicker lines on left-hand side are the slits $[a_{2i-1}, a_{2i}]$ and the thinner lines are the asymptotic markers. The cross on the left is the origin and the cross on the right is $f(0) \in \mathcal{R}_2$. Lemma 7.9.3 states that $f$ maps the asymptotic marker $\mathcal{R}_1(\infty)$ at $\infty \in \Sigma_1$ to the asymptotic marker $\mathcal{R}_2(0)$ at $0 \in \Sigma_2$</figcaption></figure>

Note that $\bar{\Sigma}_i$ will not necessarily be the closure of $\Sigma_i$ in $\mathbf{CP}^1$. However, when referring to points in the interior of $\bar{\Sigma}_i$, we denote them by the corresponding point in $\Sigma_i$. In the same way we view the radial rays $\mathcal{R}_i$ as subsets of $\bar{\Sigma}_i$, and the asymptotic markers as tangent half-lines to $\bar{\Sigma}_i$ (provided they make sense). Lemma 7.9.2 implies that they are invariant by the involution on $\bar{\Sigma}_i$.

The image of an asymptotic marker by a holomorphic function is defined by the differential at the regular points. At the singular points the local behavior of a holomorphic map still allows us to define the image of a tangent ray.

For the first version of the involution lemma, let

$$ \Sigma_1 = \mathbf{CP}^1 - ([a_1, a_2] \cup \dots \cup [a_{2p-1}, a_{2p}]), $$

where $a_i \in \mathbf{R}^+$ and $a_1 < \dots < a_{2p}$, and let $\Sigma_2 = \mathbf{D}$. We write $\partial \bar{\Sigma}_1 = \partial_1 \bar{\Sigma}_1 \sqcup \dots \sqcup \partial_p \bar{\Sigma}_1$, where $\partial_i \bar{\Sigma}_1$, $i = 1, \dots, p$, corresponds to the slit $[a_{2i-1}, a_{2i}]$.

**Lemma 7.9.3 (Involution Lemma, Version 1).** — Let $f: \bar{\Sigma}_1 \to \bar{\Sigma}_2$ be a holomorphic map which is a $q$-fold branched cover with $q \ge p$, such that:

(i) $f(\partial_i \bar{\Sigma}_1) = \partial \bar{\Sigma}_2$, $i = 1, \dots, p;$

(ii) $f^{-1}(0) = \{\infty\}; \text{ and}$

(iii) $f(0) \in \mathcal{R}_2$.

Then $f$ maps $\mathrm{Fix}(\iota_1)$ to $\mathrm{Fix}(\iota_2)$ and $\dot{\mathcal{R}}_1(\infty)$ to $\dot{\mathcal{R}}_2(0)$. (See Figure 5.)

*Proof.* — We claim that $f = \tilde{f} = \iota_2 \circ f \circ \iota_1$. To compare $f$ and $\tilde{f}$, we consider their quotient $Q = f/\tilde{f}$. We observe three facts:

(1) $Q$ has no poles, since $f$ and $\tilde{f}$ have zeros only at $\infty$ and their orders agree.

(2) $|Q(z)| = 1$ for all $z \in \partial \bar{\Sigma}_1$, so the maximum modulus theorem implies that $Q(\bar{\Sigma}_1) \subset \bar{\mathbf{D}} = \{|z| \le 1\}$.

(3) The degree of $Q|_{\partial_i \bar{\Sigma}_1}$, viewed as a map to $\mathbb{S}^1 = \partial \mathbf{D}$, is zero, since

$$ \deg(f|_{\partial_i \bar{\Sigma}_1}) = \deg(\tilde{f}|_{\partial_i \bar{\Sigma}_1}). $$
If $\mathcal{Q}$ is not constant, then by (3) there must be a branch point of $\mathcal{Q}$ along $\partial_i \bar{\Sigma}_1$. In particular, $\mathcal{Q}(\bar{\Sigma}_1) \not\subset \mathbf{D}$, which contradicts (2). Hence $\mathcal{Q}$ is a constant map and $f = \tilde{f}$ for some $c \in \mathbf{C} - \{0\}$. Now (iii) implies that $c = 1$, and we have $f = \tilde{f}$.

Finally we apply Observation 7.9.1 to conclude that $f$ maps $\mathrm{Fix}(\iota_1)$ to $\mathrm{Fix}(\iota_2)$ and, by (iii), maps $\dot{\mathcal{R}}_1(\infty)$ to $\dot{\mathcal{R}}_2(0)$. $\square$

The proofs of the other versions of the involution lemma are similar, and will be omitted.

**Lemma 7.9.4 (Involution Lemma, Version 2).** — Let $\Sigma_1 = \Sigma_2 = \mathbf{CP}^1$ and $a_i \in \mathbf{R}^{\ge 0}$ with $a_1 = 0 < a_2 < \dots < a_p$. Assume $f : \Sigma_1 \to \Sigma_2$ is a holomorphic map which is a $q$-fold branched cover with $q \ge p$, such that:

(i) $f^{-1}(\infty) = \{a_1, \dots, a_p\}$;

(ii) $f^{-1}(0) = \{\infty\}$; and

(iii) $f$ maps $\dot{\mathcal{R}}_1(0)$ to $\dot{\mathcal{R}}_2(\infty)$.

Then $f$ maps $\mathrm{Fix}(\iota_1)$ to $\mathrm{Fix}(\iota_2)$ and $\dot{\mathcal{R}}_1(\infty)$ to $\dot{\mathcal{R}}_2(0)$.

**Lemma 7.9.5 (Involution Lemma, Version 3).** — Let $\Sigma_1 = \Sigma_2 = \mathbf{D}$ and $a_i \in \mathbf{R}^{\ge 0}$ with $a_1 = 0 < a_2 < \dots < a_p < 1$. Assume $f : \bar{\Sigma}_1 \to \bar{\Sigma}_2$ is a holomorphic map which is a $q$-fold branched cover with $q \ge p$, such that:

(i) $f^{-1}(0) = \{a_1, \dots, a_p\}$;

(ii) $f^{-1}(\partial \bar{\Sigma}_1) = \partial \bar{\Sigma}_2$; and

(iii) $f$ maps $\dot{\mathcal{R}}_1(0)$ to $\dot{\mathcal{R}}_2(0)$.

Then $f$ maps $\mathrm{Fix}(\iota_1)$ to $\mathrm{Fix}(\iota_2)$. Moreover $f(-1) = -1$.

For the fourth version of the involution lemma, we consider $\Sigma_1 = \mathbf{D} - ([a_1, a_2] \cup \dots \cup [a_{2p-1}, a_{2p}])$, where $a_i \in (0, 1)$ with $a_1 < \dots < a_{2p}$, and $\Sigma_2 = \{\mathbb{R} < |z| < 1\}$, where $0 < \mathbb{R} < 1$. We write $\partial \bar{\Sigma}_1 = \partial_0 \bar{\Sigma}_1 \cup \dots \cup \partial_p \bar{\Sigma}_1$, where $\partial_0 \bar{\Sigma}_1$ is the boundary component which can be identified with $\{|z|=1\}$ and $\partial_i \bar{\Sigma}_1$, $i=1, \dots, p$, corresponds to the slit $[a_{2i-1}, a_{2i}]$.

**Lemma 7.9.6 (Involution Lemma, Version 4).** — Let $f : \bar{\Sigma}_1 \to \bar{\Sigma}_2$ be a holomorphic map which is a $q$-fold branched cover with $q \ge p$, and $\mathcal{I} \subset \{1, \dots, p\}$, such that:

(i) $f(\partial_i \bar{\Sigma}_1) = \{|z|=1\}$ when $i \in \mathcal{I}$;

(ii) $f(\partial_i \bar{\Sigma}_1) = \{|z|=R\}$ when $i=0$ or $i \notin \mathcal{I}$; and

(iii) $f(0) \in \mathcal{R}_2$.

Then $f$ maps $\mathrm{Fix}(\iota_1)$ to $\mathrm{Fix}(\iota_2)$. Moreover $f(-1) = -R$.

See Figure 6.
<figure><img src="image_21.png" /><figcaption>FIG. 6. — The map $f|_{\Sigma_1}$ in Lemma 7.9.6. The left-hand side is $\Sigma_1$ and the right-hand side is $\Sigma_2$. The thick lines on the left-hand side are the slits $[a_{2i-1}, a_{2i}]$. The cross on the left is the origin and the cross on the right is $f(0) \in \mathbb{R}_2$. The dots represent $-1 \in \partial\Sigma_1$ and $f(-1) \in \partial\Sigma_2$. Lemma 7.9.6 states that $f(-1) = -\mathbb{R}$</figcaption></figure>

**7.10. Elimination of some cases.** — We are now in a position to eliminate some of possibilities that appear in Theorem 7.6.1:

**Theorem 7.10.1.** — Suppose $m \gg 0$, $\varepsilon, \delta > 0$ are sufficiently small, $\bar{u}_\infty \in \partial \mathcal{M}_{\bar{m}}^i(\varepsilon, \delta, p)$, and $\bar{v}'_0 \neq \emptyset$.

(i) If $i=2$, then the 3-level subbuilding described in Theorem 7.6.3 does not occur.

(ii) If $i=3$, then Cases (2)–(6) of Theorem 7.6.1 do not occur.

We briefly sketch the idea of the proof. In all the cases that are eliminated by Theorem 7.10.1, the (unique) component of $\cup_{j=1}^a \bar{v}_j^\mu \subset \bar{u}_\infty$ which is asymptotic to a multiple of $\delta_0$ at the negative end has ECH index $I=1$.

Suppose that, for $m \gg 0$, there is a sequence of holomorphic curves which converges to a configuration $\bar{u}_\infty$ that we want to exclude. In Section 7.8, we applied a rescaling argument to construct a holomorphic building which keeps track of how the limit $\bar{u}_\infty$ is approached; this is similar to the layer structures of Ionel-Parker [IP, Section 7]. In the simplest case, this building is a holomorphic map $w_\infty : \mathbb{B}_- \to \mathbb{C}$ which satisfies the asymptotic condition

$$ \lim_{s \to +\infty} \frac{w_\infty(s, t)}{|w_\infty(s, t)|} = f_{i_0 j_0}(t), $$

where $f_{i_0 j_0}(t)$ is a normalized asymptotic eigenfunction of an $I=1$ curve with a negative end asymptotic to $\delta_0$. The condition $I=1$ is used as follows: since there are only finitely many $I=1$ curves with negative ends asymptotic to $\delta_0$, we may assume that $-1 \notin \{f_{i_0 j_0}(\frac{3}{2})\}$ as explained in Section 7.7.5 and Remark 7.7.11.

**Remark 7.10.2.** — In this subsection we identify $cl(\mathbb{B}_-) \simeq \bar{\mathbb{D}}$ so that $\mathfrak{p}_+$ corresponds to 0 and $\mathfrak{p}_-$ corresponds to 1. There is an anti-holomorphic involution $\iota$ on $\mathbb{B}_-$ that fixes the half-line $\{t = \frac{3}{2}\}$, and $\{t = \frac{3}{2}\}$ corresponds to the radial ray $\mathcal{R} = \bar{\mathbb{D}} \cap \mathbb{R}^{\le 0}$ by Observation 7.9.1. In particular, $\bar{\mathfrak{m}}^b$ is mapped to a point on $\mathcal{R}$.

Similarly we identify $cl(\mathbb{B}') \simeq \mathbb{C}\mathfrak{P}^1$ so that $\mathfrak{p}_+$ corresponds to 0, $\mathfrak{p}_-$ corresponds to $\infty$, and $\{t = \frac{3}{2}\}$ corresponds to the radial ray $\mathbb{R}^{\le 0}$,
We now use the involutions lemmas from Section 7.9 to obtain a contradiction. By the involution lemmas and the symmetric placement of the basepoint $\bar{\mathfrak{m}}^b$, we obtain that

$$w_{\infty} \circ \iota = i \circ w_{\infty}. \text{ Hence } \lim_{s \to +\infty} \frac{w_{\infty}(s, \frac{3}{2})}{|w_{\infty}(s, \frac{3}{2})|} = -1, \text{ which contradicts } -1 \notin \{f_{ij0}(\frac{3}{2})\}.$$

**7.10.1.** *Proof of Theorem 7.10.1.* — In this subsection we use limiting arguments in which $m \to \infty$ and $\bar{h}_m \to \bar{h}_\infty$; see Section 7.7.4. Hence many of the almost complex structures and moduli spaces will have an additional subscript $m$, where $m = \infty$ is also a possibility. For example, we will use the notation $\mathcal{J}_{\bar{W}'_m}$ and $\mathcal{J}_{\bar{W}_{-,m}}$ introduced in Section 7.7.4 to refer to $\mathcal{J}_{\bar{W}}$ and $\mathcal{J}_{\bar{W}_-}$ with respect to $m$.

Let $\bar{J}_{\infty} \in \mathcal{J}_{\bar{W}'_{\infty}}^{*}$ and let $\bar{J}'_{m} \in \mathcal{J}_{\bar{W}'_{m}}^{*}$ be a nearby almost complex structure with respect to the integer $m \gg 0$. Let $\bar{J}_{-,m} \in \mathcal{J}_{\bar{W}_{-,m}}^{\text{reg}}$ be an almost complex structure which restricts to $\bar{J}'_{m}$ and let $\bar{J}_{-,m}^{\diamond}$ be $(\varepsilon, \text{U})$-close to $\bar{J}_{-,m}$.

We will treat Theorem 7.6.1 in detail and leave Theorem 7.6.3 to the reader. Suppose that, for a sequence $m_i \to \infty$, there sequences $\bar{u}_{ij}$ of $\bar{J}_{-,m_i}^{\diamond}$-holomorphic curves which converge to a $\bar{J}_{-,m_i}^{\diamond}$-holomorphic building $\bar{u}_{i\infty}$ falling into one of Cases (2)–(6).

*Elimination of Case (2).* Suppose for each $i$ the sequence $\bar{u}_{ij}$ converges to a building $\bar{u}_{i\infty}$ satisfying Case (2). By Theorem 7.8.15, we obtain a holomorphic map $w_{\infty} : \text{cl}(\mathbb{B}_{-}) \to \mathbf{CP}^1$, whose restriction to $\text{int}(\text{cl}(\mathbb{B}_{-}))$ is a biholomorphism onto its image.

We apply the Involution Lemma 7.9.3 to obtain a contradiction: Let $\bar{\Sigma}_1$ be the compactification of $\Sigma_1 = \mathbf{CP}^1 - [a_1, a_2]$ and let $\bar{\Sigma}_2 = \text{cl}(\mathbb{B}_{-})$ be identified with $\bar{\mathbf{D}}$ as in Remark 7.10.2. Let $f: \bar{\Sigma}_1 \to \bar{\Sigma}_2$ be the extension of $(w_{\infty}|_{\text{int}(\text{cl}(\mathbb{B}_{-}))})^{-1}$. Such an extension exists because $\Sigma_1$ is biholomorphic to the open unit disk and biholomorphisms of the open unit disk extend continuously to the boundary.

By Lemma 7.9.3, $f$ maps $\dot{\mathcal{R}}_1(\infty)$ to $\dot{\mathcal{R}}_2(0)$ and, conversely, $w_{\infty}$ maps $\dot{\mathcal{R}}_2(0)$ to $\dot{\mathcal{R}}_1(\infty)$. Since the asymptotic marker $\dot{\mathcal{R}}_2(0)$ in $\bar{\Sigma}_2$ corresponds to the asymptotic marker $\{t = \frac{3}{2}\}$ for $\mathfrak{p}_{+} \in \text{cl}(\mathbb{B}_{-})$ by Remark 7.10.2, $\dot{\mathcal{R}}_1(\infty)$ is a bad radial ray (in the sense of Definition 7.7.10) by Theorem 7.8.15(2). This contradicts Remark 7.7.11, so we have eliminated Case (2).

*Elimination of Cases (3) and (4).* We will treat Case (4); Case (3) is almost identical. Suppose for each $i$ the sequence $\bar{u}_{ij}$ converges to a building $\bar{u}_{i\infty}$ satisfying Case (4). By Remark 7.6.2, for each $i$, the total number of branched points of $\cup_{j=-b}^{a} \bar{v}'_{j,i}$ is one. If we exercise some care in choosing the diagonal sequence in Lemma 7.8.4, we can divide the argument for Case (4) further into Subcases (a), (b), (c), (d') and (d'') as in Section 7.8.7, depending on the behavior of the branch points of the maps $\bar{\pi}_{\mathbb{B}_{-}} \circ \bar{v}'_{0,i}$.

*Subcases (a) and (b).* By Theorem 7.8.16, we obtain holomorphic maps

$$w_{\infty} : \text{cl}(\tilde{\mathbb{F}}_{\infty}) \to \mathbf{CP}^1, \quad p_{\infty} : \tilde{\mathbb{F}}_{\infty} \to \mathbb{B}_{-},$$

where $w_{\infty}|_{\text{int}(\text{cl}(\tilde{\mathbb{F}}_{\infty}))}$ is a biholomorphism onto its image $\Sigma_1 = \mathbf{CP}^1 - ([a_1, a_2] \cup [a_3, a_4])$ and $p_{\infty}$ is a branched double cover with one branch point. Let $\bar{\Sigma}_1$ be the compactification
of $\Sigma_1$ as in Lemma 7.9.2, and let $\bar{\Sigma}_2 = cl(B_−)$ be identified with $\bar{D}$ as in Remark 7.10.2. We define $f: \bar{\Sigma}_1 \to \bar{\Sigma}_2$ as the extension of $p_∞ \circ (w_∞|_{int(cl(\tilde{F}_∞))})^{-1}$. Such an extension exists because $\Sigma_1$ is biholomorphic to an open annulus, and biholomorphisms of the annulus always extend to the boundary. At this point we apply Lemma 7.9.3 as in Case (2) to obtain a contradiction. Case (b) is completely analogous and can be excluded in the same way.

**Subcase (c).** By Theorem 7.8.19, we obtain pairs of holomorphic maps

$$w_{\infty}^{b}: cl(\tilde{F}_{\infty}^{b}) \to \mathbf{CP}^{1}, \quad p_{\infty}^{b}: \tilde{F}_{\infty}^{b} \to B',$$

$$w_{\infty}^{m}: cl(\tilde{F}_{\infty}^{m}) \to \mathbf{CP}^{1}, \quad p_{\infty}^{m}: \tilde{F}_{\infty}^{m} \to B.$$

The map $w_∞^m$ restricts to a biholomorphism of $int(cl(\tilde{F}_∞^m))$ with $\Sigma_1^m = \mathbf{CP}^1 - [a_1, a_2]$ and the map $p_∞^m$ is a biholomorphism because it has degree 1. Let $\bar{\Sigma}_1^m$ be the compactification of $\Sigma_1^m$. Identify $\bar{\Sigma}_2^m = cl(B_−)$ with $\bar{D}$ as in Remark 7.10.2. Let $f^m: \bar{\Sigma}_1^m \to \bar{\Sigma}_2^m$ be the extension of $p_∞^m \circ (w_∞^{|_{int(cl(\tilde{F}_∞^m))}})^{-1}$. As in Case (2), Lemma 7.9.3 implies that $f^m$ maps $\dot{\mathcal{R}}_1^m(\infty)$ to $\dot{\mathcal{R}}_2^m(0)$ and, conversely, $w_∞^m$ maps $\dot{\mathcal{R}}_2^m(0)$ to $\dot{\mathcal{R}}_1^m(\infty)$.

Next we consider the “upper level” $(w_∞^b, p_∞^b)$. The map $w_∞^b: cl(\tilde{F}_∞^b) \to \mathbf{CP}^1$ is a biholomorphism and the map $p_∞^b: \tilde{F}_∞^b \to B'$ is a branched double cover with one branch point. We define $f^b = p_∞^b \circ (w_∞^b)^{-1}: \mathbf{CP}^1 \to \mathbf{CP}^1$, using the identification $cl(B') \simeq \mathbf{CP}^1$ from Remark 7.10.2. By Theorem 7.8.19(9) and the previous paragraph, $f^b$ maps $\dot{\mathcal{R}}_1(0)$ to $\dot{\mathcal{R}}_2(\infty)$. Then, by Lemma 7.9.4 and Theorem 7.8.19(7),(8), $f^b$ maps $\dot{\mathcal{R}}_1(\infty)$ to $\dot{\mathcal{R}}_2(0)$. As in Case (2), this is a contradiction because $\dot{\mathcal{R}}_1(\infty)$ is a good radial ray.

**Subcases (d') and (d'')**. By Theorem 7.8.20, we obtain a holomorphic map $w_∞^m: cl(\tilde{F}_∞^m) \to \mathbf{CP}^1$ which restricts to a biholomorphism of $int(cl(\tilde{F}_∞^m))$ with $\mathbf{CP}^1 - [a_1, a_2]$. Then the proof proceeds as in Case (2).

**Elimination of Cases (5) and (6).** The limit configurations of Cases (5) and (6) must contain a connector over $\delta_0$. This implies that Theorem 7.8.19 applies, and we obtain pairs of holomorphic maps

$$w_{\infty}^{b}: cl(\tilde{F}_{\infty}^{b}) \to \mathbf{CP}^{1}, \quad p_{\infty}^{b}: \tilde{F}_{\infty}^{b} \to B,$$

$$w_{\infty}^{m}: cl(\tilde{F}_{\infty}^{m}) \to \mathbf{CP}^{1}, \quad p_{\infty}^{m}: \tilde{F}_{\infty}^{m} \to B.$$

Then the argument of Case (4c) applies.

This completes the proof of Theorem 7.10.1.

**7.11.** *Proof of Lemma 7.2.3.* — We begin with the following corollary of Theorem 7.10.1. Recall the notation from Convention 5.8.12, Step 2 of Section 7.2 and Equation (7.6.1).

**Corollary 7.11.1.** — Suppose $m \gg 0$ and $\varepsilon, \delta > 0$ are sufficiently small constants. If $\bar{u} \in \mathcal{M}_{\bar{m}}^{2}(\varepsilon, \delta, \rho)$ or $\mathcal{M}_{\bar{m}}^{3,(r_0)}(\varepsilon, \delta, \rho)$, then $\mathrm{Im}(\bar{u}) \cap \mathbf{K}_{p,2\delta} \neq \emptyset$. In the latter case, $r_0 \gg 0$ and $\varepsilon, \delta > 0$ are sufficiently small constants which depend on $r_0$.
*Proof.* — Arguing by contradiction, suppose there are sequences $\varepsilon_i, \delta_i \to 0$ and $\bar{u}_i \in \mathcal{M}_{\bar{m}}^{3,(\bar{v}_0)}(\varepsilon_i, \delta_i, p)$ such that $\mathrm{Im}(\bar{u}_i) \cap K_{p,2\delta_i} = \emptyset$. Then the limit $\bar{u}_\infty \in \partial \mathcal{M}_{\bar{m}}^3(0, 0, p)$ of $\bar{u}_i$ has a nontrivial $\bar{v}'_0$ component. By Theorems 7.6.1 and 7.10.1(ii), $\bar{u}_\infty$ satisfies Case (1) of Theorem 7.6.1. Hence, for $i \gg 0$, $\bar{u}_i \in G(\mathfrak{P}_{(\bar{v}_0)})$, which is a contradiction. The case of $\bar{u} \in \mathcal{M}_{\bar{m}}^2(\varepsilon, \delta, p)$ is easier and is a consequence of Theorems 7.6.3 and 7.10.1(i). $\square$

*Proof of Lemma 7.2.3.* — Suppose that $\bar{u}_\infty \in \partial_1 \mathcal{M}_{\bar{m}}^{3,(\bar{v}_0)}$. We are in the situation of Lemma 7.5.2. If $\bar{v}_0$ is a degenerate $\bar{W}_-$-curve, then $\mathrm{I}(\bar{v}_0) \ge 4$ by Equation (7.5.6) since $g \ge 1$. Therefore degenerate $\bar{W}_-$-curves are ruled out by Constraint (i) in the proof of Theorem 7.6.1. Hence $\bar{v}_0$ is a $\bar{W}_-$-curve and the other levels $\bar{v}_j, j \ne 0$, are multisections of $\bar{W}'$ or $\bar{W}$. In this case, the curve $\bar{v}_0$ is simply-covered. Corollary 7.11.1 implies that the component through $\bar{m}$ also intersects $K_{p,2\delta}$. Hence passing through $\bar{m}$ is a generic condition and we must have $\mathrm{ind}(\bar{v}_0) \ge 2$ and $\mathrm{I}(\bar{v}_0) \ge 2$. This gives us two options for $\bar{u}_\infty = \bar{v}_{-b} \cup \cdots \cup \bar{v}_a$: either

(α) $b=0$, $\mathrm{I}(\bar{v}_a) = 1$, and $\mathrm{I}(\bar{v}_j) = 0, j=1, \dots, a-1$; or

(β) $a=0, b=1$, and $\mathrm{I}(\bar{v}_{-1}) = 1$.

Ghost components are not possible since each ghost component takes up $\mathrm{ind} \ge 2$ by Lemma 6.1.8. Hence $\bar{u}_\infty \in A_1 \cup A_2$. $\square$

**7.12. Gluing.** — The goal of this subsection is to prove Theorem 7.2.2.

**7.12.1. The moduli space $\mathcal{N}$**. — Let $\varepsilon = \frac{\pi}{m}$, where $m$ is a sufficiently large integer and let $\eta_\varepsilon : [-\pi, \pi] \to \mathbf{R}$ be a smooth function such that:

- $\eta_\varepsilon(\theta) = \varepsilon$ for $-\pi \le \theta \le \theta_1$;

- $\eta_\varepsilon(\theta) = 0$ for $\theta_2 \le \theta \le \pi$; and

- $\eta_\varepsilon$ is monotonically decreasing for $\theta_1 \le \theta \le \theta_2$;

for some $-\pi < \theta_1 < \theta_2 < \pi$.

We use two coordinate systems on $B_-$: (i) the usual coordinates $(s, t) \in \mathbf{R} \times [0, 2]/(0 \sim 2)$ and (ii) complex coordinates via an identification of $\mathrm{cl}(B_-)$ with $\bar{D}$ such that $\mathfrak{p}_+$ is mapped to 0 and $\mathfrak{p}_-$ is mapped to $-1$. Then $\partial B_-$ is parametrized in an orientation-preserving manner by a coordinate $e^{i\theta}$, $\theta \in (-\pi, \pi)$.

**Definition 7.12.1.** — Let $\mathcal{N} = \mathcal{N}_{\eta_\varepsilon}$ be the space of holomorphic maps $w : B_- \to \mathbf{C}$ such that the following properties hold:

(N$_1$) $w(e^{i\theta}) \in \mathbf{R}^+ \cdot e^{i\eta_\varepsilon(\theta)}$ for all $\theta \in (-\pi, \pi)$;

(N$_2$) $\lim_{s \to -\infty} |w(s, t) - c_1 e^{-\varepsilon(s+it-i)}| < \infty$ for some $c_1 \in \mathbf{R}^+$; and

(N$_3$) $\lim_{s \to +\infty} |w(s, t) - c_2 e^{\pi(s+it)}| < \infty$ for some $c_2 \in \mathbf{C}^*$.

In particular:
(N₄) deg(w) = 1 away from the sector {0 ≤ φ ≤ ε} ⊂ CP¹; and

(N₅) after composing with the chosen identification cl(B_) ≅ D, w extends continuously to D so that w(0) = w(-1) = ∞.

Multiplication by a real constant gives an **R**<sup>+</sup>-action on **N**.

Even though $\mathcal{N}$ is the space we are interested in, it will be convenient for technical reasons to regard $\mathcal{N}$ as an open subset of a vector space $\tilde{\mathcal{N}}$ obtained by relaxing properties $(N_1)-(N_3)$.

**Definition 7.12.2.** — Let $\tilde{\mathcal{N}} = \tilde{\mathcal{N}}_{\eta_\epsilon}$ be the space of holomorphic maps $w : B_- \to \mathbf{C}$ such that the following properties hold:

$$ (\tilde{\mathcal{N}}_1) \quad w(e^{i\theta}) \in \mathbf{R} \cdot e^{i\eta_\epsilon(\theta)} \text{ for all } \theta \in (-\pi, \pi); $$

$$ (\tilde{\mathcal{N}}_2) \quad \lim_{s \to -\infty} |w(s, t) - c_1 e^{-\epsilon(s+it-i)}| < \infty \text{ for some } c_1 \in \mathbf{R}; \text{ and} $$

$$ (\tilde{\mathcal{N}}_3) \quad \lim_{s \to +\infty} |w(s, t) - c_2 e^{\pi(s+it)}| < \infty \text{ for some } c_2 \in \mathbf{C}. $$

In order to compute the dimension of $\tilde{\mathcal{N}}$, we identify $B_-$ with $\overline{D} - \{-1, 0\}$ and $\tilde{\mathcal{N}}$ with the space of the holomorphic sections of a holomorphic line bundle $E \to \overline{D}$ with values in a real rank one subbundle F along $\partial \overline{D} - \{-1\}$.

We construct the bundles E and F as follows. Consider a cover of $\overline{D}$ by three open sets

$$ U_0 = \overline{D} - \{0, 1\}, \quad U_1 = \{z \in \mathbf{D} \mid |z| < 1/3\}, \quad U_2 = \{z \in \overline{D} \mid |z+1| < 1/3\}. $$

Over each open set we take a trivial line bundle $E_i = \mathbf{C} \times U_i \to U_i$ and define the bundle E by gluing the bundles $E_i$ via the transition maps

$$ \psi_1 : E_0|_{U_0 \cap U_1} \to E_1|_{U_0 \cap U_1}, \quad \psi_1(z, v) = (z, zv), $$

$$ \psi_2 : E_0|_{U_0 \cap U_2} \to E_2|_{U_0 \cap U_2}, \quad \psi_2(z, v) = \left(z, i \left( \frac{z+1}{-z+1} \right) v\right). $$

Let $\pi_{E_i} : E_i = \mathbf{C} \times U_i \to \mathbf{C}$ be the projections corresponding to the trivializations. If we parametrize $\partial \overline{D} - \{-1\}$ by $\theta \in (-\pi, \pi)$, the subbundle F is given, as a subbundle of $E_0$, by $F(\theta) = \mathbf{R} \cdot e^{i\eta_\epsilon(\theta)}$.

It is convenient to view $\Sigma = \overline{D} - \{-1, 0\}$ as a surface with a negative strip-like end and a positive cylindrical end and E as a line bundle over $\Sigma$. Let $(-\infty, -R) \times [0, 1]$ be a strip with coordinates $(s, t)$. We identify the strip-like end Z of $\Sigma$ with $(-\infty, -R) \times [0, 1]$ via the map

$$ \phi : (-\infty, -R) \times [0, 1] \to \Sigma, \quad \phi(s, t) = \frac{e^{\pi(s+it)} - i}{e^{\pi(s+it)} + i}. $$

Here the fractional linear transformation $B(\zeta) = \frac{\zeta - i}{\zeta + i}$ maps the upper half plane $\mathbf{H}$ to the unit disk $\mathbf{D}$ and $0$ to $-1$. Observe that $A(\zeta) = i(\frac{\zeta + 1}{-\zeta + 1})$ which appears in the definition of
$\psi_2$ is the inverse of $\mathbf{B}(\zeta)$. Hence the gluing map $\psi_2$ becomes

$$ \psi_2((s, t), v) = ((s, t), e^{\pi(s+it)}), $$

with respect to coordinates $(s, t)$.

The linear Cauchy-Riemann operator

$$ D : W^{1,p}(E, F) \to L^p(T^{0,1}\Sigma \otimes_{\mathbb{C}} E) $$

is Fredholm for $p > 2$ and its kernel consists of smooth holomorphic functions. We denote by $H^0(E, F)$ its kernel and by $H^1(E, F)$ its cokernel.

**Lemma 7.12.3.** — There is an identification $H^0(E, F) \cong \tilde{\mathcal{N}}$ for every choice of $\eta_\epsilon$.

*Proof.* — The isomorphism $H^0(E, F) \cong \tilde{\mathcal{N}}$ associates to a holomorphic section $\xi \in H^0(E, F)$ the holomorphic function $\pi_{E_0} \circ \xi : \Sigma \to \mathbb{C}$, i.e., $\pi_{E_0} \circ \xi$ is obtained by writing $\xi|_{U_0}$ with respect to the trivialization of $E_0$. On the negative end Z we can write

$$ \pi_{E_2} \circ \xi(s, t) = \sum_{n \ge 1} c_n e^{(n\pi - \epsilon)(s+it) + i\epsilon}, $$

since $\pi_{E_2} \circ \xi$ is holomorphic. By applying the transition function $\psi_2^{-1}$, we obtain that the leading term of $\pi_{E_0} \circ \xi$ on Z is $e^{-\epsilon(s+it)+i\epsilon}$, which is condition $(\tilde{\mathbb{N}}_2)$ in Definition 7.12.2. For a similar reason $\pi_{E_0} \circ \xi$ has a pole at 0 of order at most 1. $\square$

We will consider also the compactified surface $\tilde{\Sigma}$ obtained by adding the “segment at infinity” to the strip-like end Z. Alternatively, $\tilde{\Sigma}$ admits an identification with the truncated surface $\Sigma - Z$. Let $\tilde{E} \to \tilde{\Sigma}$ be the line bundle obtained by extending $E \to \Sigma$.

**Lemma 7.12.4.** — $\mathrm{ind}D = 3$ for every choice of $\eta_\epsilon$.

*Proof.* — We decompose $\partial\tilde{\Sigma} = cl(\partial\Sigma) \cup (\partial\tilde{\Sigma} - \partial\Sigma)$, where $\partial\tilde{\Sigma} - \partial\Sigma$ corresponds to the “segment at infinity” of the negative strip-like end Z and $cl(\partial\Sigma)$ is the closure of $\partial\Sigma$ in $\partial\tilde{\Sigma}$. We parametrize $cl(\partial\Sigma)$ by $\theta \in [-\pi, \pi]$ and $\partial\tilde{\Sigma} - \partial\Sigma$ by $\theta' \in [0, 1]$ in a manner compatible with the orientation of $\tilde{\Sigma}$ induced by the complex structure on $\Sigma$. In particular, $\theta = -\pi$ is identified with $\theta' = 1$ and $\theta = \pi$ is identified with $\theta' = 0$.

We define a trivialization $\tau$ of $\tilde{E}|_{\partial\tilde{\Sigma}}$ by:

$$ \left\{ \begin{array}{ll} \tau(\theta) = e^{i\eta_\epsilon(\theta)} & \text{along } cl(\partial\Sigma), \text{ with respect to the trivialization of } E_0, \\ \tau(\theta') = -e^{i(\pi+\epsilon)\theta'} & \text{along } \partial\tilde{\Sigma} - \partial\Sigma, \text{ with respect to the trivialization of } E_2. \end{array} \right. $$
We also define a Lagrangian subbundle $\check{\mathcal{F}} \subset \check{\mathcal{E}}|_{\partial\check{\Sigma}}$ by $\check{\mathcal{F}}|_{\partial\Sigma} = \mathcal{F}$ and rotating it in the counterclockwise direction by the minimal amount on $\partial\check{\Sigma} - \partial\Sigma$. This means:

$$ 
\left\{ 
\begin{array}{l} 
F(\theta) = \mathbf{R} \cdot e^{i\eta_{\epsilon}(\theta)} \\ 
\quad \text{along } cl(\partial\Sigma), \text{ with respect to the trivialization of } E_0, \\ 
F(\theta') = \mathbf{R} \cdot e^{i\epsilon\theta'} \\ 
\quad \text{along } \partial\check{\Sigma} - \partial\Sigma, \text{ with respect to the trivialization of } E_2. 
\end{array} 
\right. 
$$

By the doubling argument of Theorem 5.5.1, Lemma 5.5.3 (or, rather, its proof) and the formula for the index of the Cauchy-Riemann operator on line bundles over punctured surfaces (see for example [We2, Formula 2.1]), the index of D is

$$ \mathrm{ind} D = \chi(\mathbf{D}) + \mu_{\tau}(\check{\mathcal{F}}) + 2c_1(\check{\mathcal{E}}, \tau) - 1. $$

From the explicit definitions of $\tau$ and $\check{\mathcal{F}}$ one computes that $\mu_{\tau}(\check{\mathcal{F}}) = -1$ and $c_1(\check{\mathcal{E}}, \tau) = 2$. Hence $\mathrm{ind} D = 3$. $\square$

**Lemma 7.12.5.** — The operator D is surjective for every choice of $\eta_{\epsilon}$.

*Proof.* — In view of Theorem 5.5.1, the surjectivity of D follows from [We2, Formula 2.5] and [We2, Proposition 2.2(2)] applied to the double of D. $\square$

**Corollary 7.12.6.** — The real dimension of $\tilde{\mathcal{N}}$ is 3 for every choice of $\eta_{\epsilon}$.

**7.12.2.** *The maps $\mathcal{E}$ and $\mathfrak{F}$.* — We define an $\mathbf{R}$-linear map:

$$ 
\begin{aligned}
\mathcal{E} : \tilde{\mathcal{N}}_{\eta_{\epsilon}} &\rightarrow \mathbf{R} \times \mathbf{C}, \\
w &\mapsto (c_1, c_2),
\end{aligned} 
$$

where $c_1, c_2$ are the coefficients from $(\tilde{\mathcal{N}}_1)$ and $(\tilde{\mathcal{N}}_2)$ of Definition 7.12.2.

**Lemma 7.12.7.** — The map $\mathcal{E}$ is an isomorphism.

Hence $\mathcal{E}(\tilde{\mathcal{N}})$ is an open positive cone contained in $\mathbf{R}^{+} \times \mathbf{C}^{\times}$.

*Proof.* — Since $\mathcal{E}$ is a linear map, it suffices to check that $\mathcal{E}$ is injective. First observe that $\ker \mathcal{E}$ consists of continuous functions $w : \bar{\mathbf{D}} \rightarrow \mathbf{C}$ such that $w(-1) = 0$, $w(e^{i\theta}) \in \mathbf{R} \cdot e^{i\eta_{\epsilon}(\theta)}$ for all $\theta \in (-\pi, \pi)$, and $w$ is holomorphic on $\mathbf{D}$.

If $w \neq 0$, then we can apply [Se2, Lemma 11.5]; referring to the equation on the second-to-last line of [Se2, p.156], the left-hand side of the equation is positive while the right-hand side is zero, a contradiction. $\square$

Next we prove that $\tilde{\mathcal{N}}$ is nonempty for any choice of $\eta_{\epsilon}$.
**Lemma 7.12.8.** — Let $w_0 = \mathbb{C}^{-1}(1, 0)$. Then $w_0 : \bar{\mathbf{D}} - \{-1\} \to \mathbf{C}$ is holomorphic, $w_0(e^{i\theta}) \in \mathbf{R}^+ \cdot e^{in_e(\theta)}$, $w_0$ has no zeros, and $\lim_{s \to -\infty} e^{\epsilon(s+it-i)}u_0(s, t) = 1$ for coordinates $(s, t) \in (-\infty, 0) \times [0, 1]$ on a negative strip-like end around $-1$.

*Proof.* — The only nontrivial part of the statement is that $w_0$ has no zeros in the interior and along the boundary — this follows from the equation on the second-to-last line of [Se2, p.156], where the right-hand side of the equation is zero. $\square$

**Corollary 7.12.9.** — $\mathcal{N}$ is nonempty for any choice of $\eta_\epsilon$.

*Proof.* — Take any $w \in \tilde{\mathcal{N}}$ with $c_2 \neq 0$. Then $w + cw_0 \in \mathcal{N}$ for any $c$ sufficiently large. $\square$

**Lemma 7.12.10.** — If $w \in \mathcal{N}$, then $w$ has a unique (simple) zero in $\mathbf{D} - \{0\}$.

*Proof.* — The lemma follows from a straightforward winding number argument, since $w$ maps $\partial\mathbf{B}_-$ to the sector $\{0 \le \theta \le \epsilon, \rho > 0\}$ and has a pole at 0. $\square$

We denote $\mathbf{PN} = \mathcal{N}/\mathbf{R}^+$, where $\mathbf{R}^+$ acts on $\mathcal{N}$ by multiplication and we define the maps

$\tilde{\mathfrak{F}} : \mathcal{N} \to \mathbf{D} - \{0\}$, $\mathfrak{F} : \mathbf{PN} \to \mathbf{D} - \{0\}$

by $\tilde{\mathfrak{F}}(w) = w^{-1}(0)$ and $\mathfrak{F}([w]) = \tilde{\mathfrak{F}}(w)$.

**Lemma 7.12.11.** — The map $\tilde{\mathfrak{F}} : \mathbf{PN} \to \mathbf{D} - \{0\}$ is a diffeomorphism.

*Proof.* — We first prove the injectivity of $\tilde{\mathfrak{F}}$. Let $w_0$ and $w_1$ be maps in $\mathcal{N}$ such that $w_0^{-1}(0) = w_1^{-1}(0)$. Then $\omega = \frac{w_0}{w_1}$ is a holomorphic map on $\bar{\mathbf{D}}$ such that $\omega(\partial\bar{\mathbf{D}}) \subset \mathbf{R}$, so it is constant. Then $w_0$ and $w_1$ represent the same element in $\mathbf{PN}$.

Next we prove surjectivity. Fix an element $w_0 \in \mathcal{N}$ and let $z_0 \in \mathbf{D}$ such that $w_0(z_0) = 0$. For any $z_1 \in \mathbf{D} - \{0, z_0\}$ we look for a holomorphic function $\omega_{z_1} : \bar{\mathbf{D}} \to \mathbf{CP}^1$ such that $\omega_{z_1}(z_1) = 0$, $\omega_{z_1}(z_0) = \infty$, $\omega_{z_1}(\partial\bar{\mathbf{D}}) \subset \mathbf{R}^+$, and $\omega_{z_1}|_{\mathbf{D}}$ is a biholomorphism onto its image. By the argument of the previous paragraph, if such $\omega_{z_1}$ exists, it is unique up to multiplication by a positive real constant. If we set $w = \omega_{z_1}w_0$, then $w \in \mathcal{N}$, $w(z_1) = 0$, and $\tilde{\mathfrak{F}}(w) = z_1$.

The function $\omega_{z_1}$ with the desired properties is given by

$$ \omega_{z_1}(\zeta) = f(g(\zeta)^2), $$

where $g : \bar{\mathbf{D}} \to \bar{\mathbf{H}}$ is a fractional linear transformation such that $g(z_0) = i$, $\mathrm{Re}(g(z_1)) = 0$, and $0 < \mathrm{Im}(g(z_1)) < 1$, and $f : \mathbf{CP}^1 \to \mathbf{CP}^1$ is an element of PSL(2, $\mathbf{R}$) (i.e., $f$ fixes the
real axis) such that $f((g(z_1))^2) = 0, f((g(z_0))^2) = f(-1) = \infty$, and $f(0) = 1$. Note that $f$ and $g$ are uniquely determined by the above conditions. This proves the surjectivity of $\mathfrak{F}$.

Finally, we claim that the $d\mathfrak{F}([w])$ is an isomorphism for all $[w] \in \mathbf{PN}$. To this end, let us define $\Xi(z) = \omega_z w_0 \in \mathcal{N}$. Then $\mathfrak{F}^{-1}(z)$ is the class of $\Xi(z)$ in $\mathbf{PN}$. The map $\Xi$ is smooth because the maps $f, g$, and hence $\omega_z$ are rational maps whose coefficients depend smoothly on $z$. In order to prove the claim it suffices to prove that $d\Xi(z)$ is injective for any $z \in \mathbf{D} - \{0\}$. Here, in order to distinguish the differential of $\Xi$ at $z \in \mathbf{D} - \{0\}$ from the differential of the function $\Xi(z) : \mathbf{D} - \{0, -1\} \to \mathbf{C}$, we will use the notation $d\Xi(z)$ for the former and $\frac{\partial \Xi(z)}{\partial \zeta}$ for the latter. We define the map

$$F: \mathcal{N} \times (\mathbf{D} - \{0\}) \to \mathbf{C}, \quad F(w, z) = w(z).$$

By differentiating the identity $F(\Xi(z), z) = 0$ we obtain

$$\frac{\partial F}{\partial w} d\Xi(z) + \frac{\partial F}{\partial z} = 0.$$

We observe that $\frac{\partial F}{\partial z}\big|_{(\Xi(z),z)} = \frac{\partial \Xi(z)}{\partial \zeta}\bigg|_{\zeta=z}$, which is invertible because $z$ is a simple zero of $\Xi(z)$. Hence $d\Xi(z)$ is injective. $\square$

**7.12.3.** *Proof of Theorem 7.2.2.* — We refer to Section 7.2 for the definition of the moduli spaces $M_1, M_0$, and $M_{-1}$, and of the gluing parameter space $\mathfrak{P}$. In order to simplify the exposition we will assume (without loss of generality) that all the multiplicities of $\boldsymbol{y}'$ are 1 and that each of $M_1/\mathbf{R}, M_0$, and $M_{-1}/\mathbf{R}$ is connected; in particular, both $M_0$ and $M_{-1}/\mathbf{R}$ are single points.

We choose a smooth slice $\tilde{M}_1$ of the $\mathbf{R}$-action on $M_1$ such that the following hold for some $\kappa'_0, \kappa_0 > 0, K_0 > \pi - \varepsilon$, and for all $\bar{v}_1 \in \tilde{M}_1$:

- each component of $\bar{v}_1|_{s \le 0}$ is $(\kappa'_0 + \kappa_0, 0)$-close to a cylinder over a component of $\delta_0 \boldsymbol{y}'$; and

- the component $\tilde{v}_1$ of $\bar{v}_1|_{s \le 0}$ which is close to $\sigma'_\infty$ satisfies

$$(7.12.1) \qquad \left| \bar{\pi} \circ \tilde{v}_1 - \kappa'_0 e^{(\pi - \varepsilon)s} (c e^{\pi i t}) \right|_{C^0} \le \kappa_0 e^{K_0 s},$$

where $c e^{\pi i t}$ is the normalized asymptotic eigenfunction corresponding to the negative end $\delta_0$ of $\bar{v}_1$ and $\bar{\pi} = \bar{\pi}_{D_{\rho_0}^2}$ is the projection to $D_{\rho_0}^2$ with respect to balanced coordinates. By a slight abuse of notation, we will refer to $\kappa'_0 e^{(\pi - \varepsilon)s} (c e^{\pi i t})|_{s=0} = \kappa'_0 c e^{\pi i t}$ as the asymptotic eigenfunction of $\bar{v}_1$ at $\delta_0$.

Similarly, we choose a smooth slice $\tilde{M}_{-1}$ of the $\mathbf{R}$-action on $M_{-1}$ such that the following hold for some $\kappa'_1, \kappa_1 > 0, K_1 > 2\varepsilon$:

- each component of $\bar{v}_{-1}|_{s \ge 0}$ is $(\kappa'_1 + \kappa_1, 0)$-close to a strip over a component of $\{z_\infty\} \cup \boldsymbol{y}'$; and
– the component $\tilde{v}_{-1}$ of $\overline{v}_{-1}|_{s \ge 0}$ which is close to $\sigma_{\infty}$ satisfies

$$ (7.12.2) \qquad \left| \pi \circ \tilde{v}_{-1} - \kappa'_1 e^{-2\epsilon s} (de^{\epsilon i t}) \right|_{C^0} \le \kappa_1 e^{-K_1 s}, $$

where $de^{-\epsilon i t}$ is a normalized asymptotic eigenfunction corresponding to the positive end $z_{\infty}$ of $\overline{v}_{-1}$. Note that $d$ is completely determined by the data $\{(i, j) \to (i, j)\}$ at the positive end $z_{\infty}$. Without loss of generality, we assume that $\bar{a}_{i,j} = \mathbf{R}^+$, $\bar{h}(\bar{a}_{i,j}) = \mathbf{R}^+ \cdot e^{i\epsilon}$, so that $d = e^{i\epsilon}$. Similarly, we refer to $\kappa'_1 e^{-2\epsilon s} de^{-\epsilon i t}|_{s=0} = \kappa'_1 de^{-\epsilon i t}$ as the asymptotic eigenfunction of $\overline{v}_{-1}$ at $z_{\infty}$.

In the rest of this section, when we write $\overline{v}_i$, $i = 1, -1$, we will assume that $\overline{v}_i$ is in the slice $\tilde{M}_i$. Also, for $T \in \mathbf{R}$ and $i = \pm 1$, let $\overline{v}_{i,T}$ be the T-translates of $\overline{v}_i$ in the $\mathbf{R}$-direction; i.e., if $s: \overline{W}^* \to \mathbf{R}$, $s = \emptyset,'$, is the $\mathbf{R}$-coordinate, then $s \circ \overline{v}_{i,T} = s \circ \overline{v}_i + T$. Let $f(t) = \beta_0 e^{\pi i t}$ be the asymptotic eigenfunction of $\overline{v}_1$ at $\delta_0$ with eigenvalue $\pi - \epsilon$. Then the asymptotic eigenfunction of $\overline{v}_{1,T}$ at $\delta_0$ is $e^{-(\pi-\epsilon)T}f$. Similarly, let $g(t) = \alpha_0 e^{i\epsilon(1-t)}$ be the asymptotic eigenfunction of $\overline{v}_{-1}$ at $z_{\infty}$ with eigenvalue $2\epsilon$. Then the asymptotic eigenfunction of $\overline{v}_{-1,T}$ at $z_{\infty}$ is $e^{2\epsilon T}g$.

Also, for the rest of this section, we fix $\boldsymbol{\gamma} \in \tilde{\mathcal{O}}_{2g}$ and $\boldsymbol{y} \in S_{a, \hbar(\boldsymbol{a})}$ once and for all. Since there is no risk of confusion, we will write

$$ \mathcal{M} = \mathcal{M}_{\bar{j}_{-}}^{I=3,n^{-}=m}(\boldsymbol{\gamma}, \boldsymbol{y}), \quad \mathcal{M}^{\text{ext}} = \mathcal{M}_{\bar{j}_{-}}^{I=3,n^{-}=m,\text{ext}}(\boldsymbol{\gamma}, \boldsymbol{y}). $$

After identifying the quotient $\mathcal{M}_i/\mathbf{R}$ with the slice $\tilde{M}_i$ for $i = \pm 1$ we write

$$ \mathfrak{P} = (5r, \infty)^2 \times \tilde{M}_1 \times \mathcal{M}_0 \times \tilde{M}_{-1}; $$

compare with Equation (7.2.2).

We describe the gluing map

$$ G: \mathfrak{P} \to \mathcal{M}^{\text{ext}}, \quad \mathfrak{D} = (T_{\pm}, \bar{v}_1, \bar{v}_0, \bar{v}_{-1}) \mapsto \bar{u}(\mathfrak{D}), $$

defined in a manner similar to that of Section 6.5.2. First we form the preglued curve $\bar{u}^{\#}(\mathfrak{D})$ from the data $\mathfrak{D} = (T_{\pm}, \bar{v}_1, \bar{v}_0, \bar{v}_{-1})$ by patching together

$$ (\bar{v}_{1,2T_+})|_{s \ge T_+}, \quad \bar{v}_0|_{-T_- \le s \le T_+}, \quad (\bar{v}_{-1,-2T_-})|_{s \le -T_-}. $$

Then we choose cutoff functions $\tilde{\beta}_i$, $i = -1, 0, 1$, on the domains of $\overline{v}_i$. If $\psi_{-1}, \psi_0$, and $\psi_1$ are deformations of $\overline{v}_{-1, -2T_-}$, $\overline{v}_0$, and $\overline{v}_{1, 2T_+}$, the condition for the deformation $\tilde{\beta}_{-1} \psi_{-1} + \tilde{\beta}_0 \psi_0 + \tilde{\beta}_1 \psi_1$ of $\bar{u}^{\#}(\mathfrak{D})$ to be holomorphic is given by the following equation:

$$ (7.12.3) \qquad \tilde{\beta}_{-1} \Theta_{-1}(\psi_{-1}, \psi_0) + \tilde{\beta}_0 \Theta_0(\psi_{-1}, \psi_0, \psi_1) + \tilde{\beta}_1 \Theta_1(\psi_0, \psi_1) = 0, $$

Equation (7.12.3) is analogous to Equation (6.5.1) and is solved as in Step 3 of Section 6.5.2. The situation considered here is simpler than that of Section 6.5.2: in fact there is no obstruction bundle because $\overline{v}_0$ is regular. For $r$ sufficiently large, $G$ is a homeomorphism onto its image.
Given $\delta > 0$ small, let $D_\delta(\bar{m}^b)$ be the closed ball of radius $\delta$ and center $\bar{m}^b$ in $B_-$ with respect to some fixed metric. Then let $M_{(\kappa, \nu)}^\delta \subset M^{\text{ext}}$ be the subset of curves $\bar{u}$ that pass through $D_\delta(\bar{m}^b) \times \{z_\infty\} \subset \bar{W}_-$ and are $(\kappa, \nu)$-close to breaking.

**Claim 7.12.12.** — $M_{(\kappa, \nu)}^\delta \subset \mathcal{M}$.

*Proof.* — Something stronger is actually true: if the image of $\bar{u} \in \mathcal{M}^{\text{ext}}$ intersects the section at infinity $\sigma_\infty^-$ at a point in the interior of $\sigma_\infty^-$, then $\bar{u} \in \mathcal{M}$. This is a consequence of the intersection theory developed in Section 7.4: in fact $n^-(\bar{u}) = m$ and the intersection point with $\sigma_\infty^-$ at the interior already contributes $m$ to $n^-(\bar{u})$. However, if $\bar{u} \in \mathcal{M}^{\text{ext}} - \mathcal{M}$, then the image of $\bar{u}$ also intersects $\sigma_\infty^-$ at some boundary points, which give some extra contribution to $n^-(\bar{u})$. This is a contradiction. $\square$

Let $\mathfrak{d} = (T_\pm, \bar{v}_1, \bar{v}_0, \bar{v}_{-1}) \in \mathfrak{P}$. Writing $\bar{u}^\# = \bar{u}^\#(\mathfrak{d})$ and $\bar{u} = \bar{u}(\mathfrak{d})$, we define $w^\# = w^\#(\mathfrak{d})$ and $w = w(\mathfrak{d})$ as follows:

$$ (7.12.4) \qquad w^\# = e^{\epsilon s} \cdot (\bar{\pi} \circ \bar{u}^\#|_{-2T_- \le s \le 2T_+}) \quad \text{and} \quad w = e^{\epsilon s} \cdot (\bar{\pi} \circ \bar{u}|_{-2T_- \le s \le 2T_+}). $$

The map $w$ is holomorphic by Lemma 7.8.6.

**Lemma 7.12.13.** — Let $\mathfrak{d}_i = (T_{\pm,i}, \bar{v}_{1,i}, \bar{v}_0, \bar{v}_{-1}) \in \mathfrak{P}$ be a sequence such that $\lim_{i \to \infty} T_{\pm,i} = +\infty$ and $\lim_{i \to \infty} \bar{v}_{1,i} = \bar{v}_{1,\infty} \in \widetilde{M}_1$. Then the sequence $\bar{u}_i = \bar{u}(\mathfrak{d}_i)$ has a subsequence which converges to $(\bar{v}_{1,\infty}, \bar{v}_0, \bar{v}_{-1})$.

*Proof.* — The lemma is a consequence of the following standard gluing result (cf. [HT2, Theorem 7.3(a)]): Given $(\kappa, \nu)$, there exists $r \gg 0$ such that $\bar{u}(\mathfrak{d})$ is $(\kappa, \nu)$-close to $(\bar{v}_1, \bar{v}_0, \bar{v}_{-1})$ for each $\mathfrak{d} = (T_\pm, \bar{v}_1, \bar{v}_0, \bar{v}_{-1}) \in \mathfrak{P} = (5r, \infty)^2 \times \widetilde{M}_1 \times \mathcal{M}_0 \times \widetilde{M}_{-1}$. $\square$

Recall that $f(t) = \beta_0 e^{\pi it}$ is the asymptotic eigenfunction of $\bar{v}_1$ at the negative end $\delta_0$ and $g(t) = \alpha_0 e^{it(1-t)}$ is the asymptotic eigenfunction of $\bar{v}_{-1}$ at the positive end $z_\infty$. We define the function

$$ \Pi : \mathfrak{P} \to \mathbf{R}^+ \times \mathbf{C}^\times, \\ \Pi(T_\pm, \bar{v}_1, \bar{v}_0, \bar{v}_{-1}) = (\alpha, \beta), $$

where $\alpha e^{it(1-t)}$ is the asymptotic eigenfunction of $\bar{v}_{-1, -2T_-}$ at $z_\infty$ and $\beta e^{i\pi t}$ is the asymptotic eigenfunction of $\bar{v}_{1, 2T_+}$ at $\delta_0$.

**Lemma 7.12.14.** — Let $\mathfrak{d}_i = (T_{\pm,i}, \bar{v}_{1,i}, \bar{v}_0, \bar{v}_{-1}) \in \mathfrak{P}$ be a sequence such that $\lim_{i \to +\infty} T_{\pm,i} = +\infty$. Let $\bar{m}_i = \bar{u}(\mathfrak{d}_i)^{-1}(\sigma_\infty^-)$ and let $f_i(t) = \beta_i e^{\pi it}$ be the asymptotic eigenfunction of $\bar{v}_{1,i}$ at the negative end $\delta_0$. Then, after passing to a subsequence, $\lim_{i \to +\infty} \beta_i = \beta_\infty$, and, after rescaling
by positive real numbers in addition, the sequence $w_i = w(\mathfrak{d}_i)$ of holomorphic functions defined as in Equation (7.12.4) converges in the $C_{\text{loc}}^{\infty}$-topology to a holomorphic map $w_{\infty} \in \tilde{\mathcal{N}}$ which satisfies the following:

(1) $\mathfrak{E}(w_{\infty}) = (\lambda_1\alpha_0, \lambda_2\beta_{\infty})$ for some $\lambda_1, \lambda_2 \ge 0$, $\lambda_1, \lambda_2$ not both zero;

(2) $\Pi(\mathfrak{d}_i)$, after rescaling by positive constants, limits to $\mathfrak{E}(w_{\infty})$;

(3) if $\lim_{i \to +\infty} \bar{m}_i = \bar{m}_{\infty} \in \text{int}(B_-)$, then $w_{\infty}(\bar{m}_{\infty}) = 0$ and $w_{\infty} \in \mathcal{N}$; and

(4) if $w_{\infty}(\bar{m}_{\infty}) = 0$ with $\bar{m}_{\infty} \in \text{int}(B_-)$, then $\lim_{i \to +\infty} \bar{m}_i = \bar{m}_{\infty}$.

*Proof.* — By Lemma 7.12.13 we can extract a subsequence (which we still call $\mathfrak{d}_i$) such that $\bar{u}(\mathfrak{d}_i)$ converges to $\bar{u}_{\infty} = (\bar{v}_{1,\infty}, \bar{v}_0, \bar{v}_{-1})$. By the SFT convergence of $\bar{u}(\mathfrak{d}_i)$ to $\bar{u}_{\infty}$ we obtain a sequence of good truncations satisfying the estimates of Lemma 7.8.4. The proof of Theorem 7.8.15 then goes through essentially unchanged to yield the limit $w_{\infty}$. (1) is a consequence of the nontriviality of $w_{\infty}$ and (2)–(4) are consequences of the convergence in $C_{\text{loc}}^{\infty}$. $\square$

Let $w_0 \in \mathcal{N}$ be a map such that $w_0(\bar{m}^b) = 0$ and let $\mathcal{V}_\delta \subset \mathcal{N}$ be the subset consisting of maps $w$ such that $w(\bar{m}) = 0$ for some $\bar{m} \in D_\delta(\bar{m}^b)$. Note that $\mathcal{V}_\delta$ is an $\mathbf{R}^+$-invariant neighborhood of $w_0$, whose slice under the $\mathbf{R}^+$-action is compact in view of Lemma 7.12.11. We now have the following diagram:

<figure><img src="image_22.png" /></figure>

Let us write $\mathfrak{P}_\delta := \Pi^{-1}(\mathfrak{E}(\mathcal{V}_\delta))$. Recall that $\mathfrak{E}|_{\tilde{\mathcal{N}}}$ is a diffeomorphism.

*Lemma 7.12.15.* — If $r$ is sufficiently large, then the following hold:

(1) $M_{(\kappa, v)}^{\delta/3} \cap \text{Im}(G) \subset G(\mathfrak{P}_{\delta/2})$.

(2) $G(\mathfrak{P}_{2\delta/3}) \subset \text{int}(M_{(\kappa, v)}^{\delta})$.

*Proof.* — (1) We show that, for $r$ sufficiently large, if

$$ \bar{u}(\mathfrak{d})^{-1}(\sigma_{\infty}^{-}) \in D_{\delta/3}(\bar{m}^b), $$

then $\mathfrak{d} \in \mathfrak{P}_{\delta/2}$. Arguing by contradiction, suppose there exist sequences $r_i \to \infty$, $\mathfrak{d}_i = (\Gamma_{\pm,i}, \bar{v}_{1,i}, \bar{v}_0, \bar{v}_{-1}) \notin \mathfrak{P}_{\delta/2}$, and $\bar{m}_i \in D_{\delta/3}(\bar{m}^b)$ such that $T_{\pm,i} > 5r_i$ and $\bar{u}(\mathfrak{d}_i)^{-1}(\sigma_{\infty}^{-}) = \bar{m}_i$. By passing to a subsequence, we may assume that $\lim_{i \to \infty} \bar{m}_i = \bar{m}_{\infty} \in D_{5\delta/12}(\bar{m}^b)$ and that the asymptotic eigenfunctions of $\bar{v}_{1,i}$ converge to $\beta_{\infty}e^{\pi i t}$ for $i \to \infty$. We now apply
Lemma 7.12.14 to the sequence $\bar{u}(\mathfrak{d}_i)$ to obtain a holomorphic map $w_\infty \in \mathcal{N}$ satisfying Lemma 7.12.14(1)–(3). In particular, $\Pi(\mathfrak{d}_i)$, after rescaling, limits to $\mathfrak{E}(w_\infty)$ by Lemma 7.12.14(2) and $w_\infty(\bar{\mathfrak{m}}_\infty) = 0$ by Lemma 7.12.14(3). Since $\bar{\mathfrak{m}}_\infty \in D_{5\delta/12}(\bar{\mathfrak{m}}^b)$, this contradicts $\mathfrak{d}_i \notin \mathfrak{P}_{\delta/2}$ (or equivalently $\Pi(\mathfrak{d}_i) \in \mathfrak{E}(\mathcal{V}_{\delta/2})$).

(2) Suppose there exist sequences $r_i \to \infty$ and $\mathfrak{d}_i = (T_{\pm,i}, \bar{v}_{1,i}, \bar{v}_0, \bar{v}_{-1}) \in \mathfrak{P}_{2\delta/3}$ such that $T_{\pm,i} > 5r_i$ and $\bar{u}(\mathfrak{d}_i) \notin \text{int}(\mathcal{M}_{(\kappa,\nu)}^\delta)$. By passing to a subsequence and applying Lemma 7.12.14 to $\bar{u}(\mathfrak{d}_i)$, we obtain a holomorphic map $w_\infty \in \tilde{\mathcal{N}}$ such that $\mathfrak{E}(w_\infty) \in \mathfrak{E}(\mathcal{V}_{2\delta/3})$ by Lemma 7.12.14(2). On the other hand, $w_\infty^{-1}(0) \notin D_\delta(\bar{\mathfrak{m}}^b)$ since $\bar{u}(\mathfrak{d}_i)^{-1}(\sigma_\infty^-) \notin D_\delta(\bar{\mathfrak{m}}^b)$ by Lemma 7.12.14(4). This is a contradiction. $\square$

Fix $\delta > 0$ and take $r_0 > 5r$ sufficiently large. We write

$$ \partial\mathfrak{P}_{2\delta/3,(r_0)} := \mathfrak{P}_{2\delta/3} \cap \partial\mathfrak{P}_{(r_0)}, $$

where $\mathfrak{P}_{(r_0)} = \{T_+ \ge r_0\} \subset \mathfrak{P}$ and $\partial\mathfrak{P}_{(r_0)} = \{T_+ = r_0\}$. By Lemma 7.12.15(2), we can define

$$ (7.12.5) \qquad \Upsilon' : \partial\mathfrak{P}_{2\delta/3,(r_0)} \to D_\delta(\bar{\mathfrak{m}}^b), $$

which maps $\mathfrak{d}$ to $G(\mathfrak{d})^{-1}(\sigma_\infty^-)$. Since $\partial\mathfrak{P}_{2\delta/3,(r_0)}$ is compact (note that restricting to $\{T_+ = r_0\}$ is the same as taking a slice under the $\mathbf{R}^+$-action on $\mathcal{V}_{2\delta/3}$), the map $\Upsilon'$ is proper and the local degree is well-defined.

Also define

$$ (7.12.6) \qquad \Upsilon'' : \partial\mathfrak{P}_{2\delta/3,(r_0)} \to D_\delta(\bar{\mathfrak{m}}^b), $$

which maps $\mathfrak{d}$ to the zero of $\mathfrak{E}^{-1} \circ \Pi(\mathfrak{d})$. The map $\Upsilon''$ is well-defined by the definitions of $\mathfrak{P}_{2\delta/3}$ and $\mathcal{V}_\delta$. It is also proper and the local degree is well-defined.

The maps $\Upsilon'$ and $\Upsilon''$ are sufficiently close in the following sense.

**Lemma 7.12.16.** — For any $k$, $\Upsilon'$ and $\Upsilon''$ can be made arbitrarily $\mathcal{C}^0$-close by choosing $r_0$ sufficiently large.

*Proof.* — This follows from Lemma 7.12.14. $\square$

<figure><img src="image_23.png" /></figure>

In particular, the local degrees of $\Upsilon'$ and $\Upsilon''$ near $\bar{\mathfrak{m}}^b$ agree. The local degree of $\Upsilon'$ is equal to the left-hand side of Equation (7.2.3), while the local degree of $\Upsilon''$ is equal to the right-hand side of Equation (7.2.3). (We were assuming that each of $M_1/\mathbf{R}$, $M_0$, and $M_{-1}/\mathbf{R}$ is connected.) This completes the proof of Theorem 7.2.2.

## Acknowledgements

We are indebted to Michael Hutchings for many helpful conversations and for our previous collaboration which was a catalyst for the present work. We also thank Denis
Auroux, Tobias Ekholm, Dusa McDuff, Ivan Smith and Jean-Yves Welschinger for illuminating exchanges. Part of this work was done while KH and PG visited MSRI during the academic year 2009–2010. We are extremely grateful to MSRI and the organizers of the “Symplectic and Contact Geometry and Topology” and the “Homology Theories of Knots and Links” programs for their hospitality; this work probably would never have seen the light of day without the large amount of free time which was made possible by the visit to MSRI. KH also thanks the Simons Center for Geometry and Physics for their hospitality during his visit in May 2011. Finally we thank the referees for a large number of corrections and suggestions.

**Declarations:**

**Competing Interests**

The authors declare no competing interests.

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V. C.
CNRS, Laboratoire de Mathématiques Jean Leray,
Nantes Université,
Nantes, F-44000, France
Vincent.Colin@univ-nantes.fr

P. G.
CNRS, IF,
Univ. Grenoble Alpes,
Grenoble, 38000, France
paolo.ghiggini@univ-grenoble-alpes.fr

K. H.
University of California, Los Angeles,
Los Angeles, CA, 90095, USA
honda@math.ucla.edu

Manuscrit reçu le 7 août 2012

Version révisée le 10 avril 2017

Manuscrit accepté le 28 février 2024

publié en ligne le 2 avril 2024.