$P'_{*,i} \cup P''_{*,i}, * = (L-, j_0), (R-, j_0),$ or $(+, j_0)$, as before so that $P'_{*,i}$ corresponds to $\bar{v}_*^{\circ, 1,'}$ and $P''_{*,i}$ corresponds to $\bar{v}_*^{\circ, 1,''}$.

The following definition does not assume (S).

**Definition 3.4.15.** — A boundary point $\mathbf{r} \in \partial F_{*0}, *_0 = (1, j_0)$, $0 \le j_0 \le a$, at $z_\infty$ falls into one of three (mutually exclusive) types:

(P$_1$) $\bar{v}'_{*0} = \emptyset$;

(P$_2$) $\bar{v}'_{*0} \neq \emptyset$ and all the vertices of $\mathcal{Z}$ lie on arcs of type $\bar{a}_{k,l}$ or all the vertices of $\mathcal{Z}$ lie on arcs of type $\bar{h}(\bar{a}_{k,l})$;

(P$_3$) $\bar{v}'_{*0} \neq \emptyset$ and $\mathcal{Z}$ has vertices that lie on arcs of both types $\bar{a}_{k,l}$ and $\bar{h}(\bar{a}_{k,l})$.

The boundary points $\mathbf{r} \in \partial F_{*0}, *_0 = (-1, j_0)$, $1 \le j_0 \le c + 1$, at $z_\infty$ are classified similarly.

The following is a strengthening of Lemma 3.4.7 when $\bar{v}'_{1,j} \cup \bar{v}^{\sharp}_{1,j} \neq \emptyset$ for some $j > 0$ and there are boundary points of type (P$_3$) at $z_\infty$:

**Lemma 3.4.16.** — Suppose $\bar{v}'_{1,j} \cup \bar{v}^{\sharp}_{1,j} \neq \emptyset$ for some $j > 0$. Let $\mathcal{E}_{-,i}, i = 1, \dots, q$, be the negative ends of $\cup_{j=1}^a \bar{v}^{\sharp}_{1,j}$ that converge to $z_\infty$, let $\mathcal{E}_{+,i}, i = 1, \dots, r$, be the positive ends of $\cup_{j=0}^{a-1} \bar{v}^{\sharp}_{1,j}$ that converge to $z_\infty$, and let $\mathcal{E}'_i, i = 1, \dots, s$, be the neighborhoods of the boundary points of type (P$_3$). Then

$$ (3.4.15) \qquad n^*((\cup_{i=1}^q \mathcal{E}_{-,i}) \cup (\cup_{i=1}^r \mathcal{E}_{+,i}) \cup (\cup_{i=1}^s \mathcal{E}'_i)) \ge m - p_+ $$

*Proof.* — Similar to that of Lemma 3.4.7. $\square$

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

**Lemma 3.4.17.** — If $\mathfrak{p}_1, \dots, \mathfrak{p}_s$ are the boundary points of type (P$_1$) and (P$_2$) and $\mathfrak{N}(\mathfrak{p}_i) \subset F_*$ is a small neighborhood of $\mathfrak{p}_i$, then $\sum_{i=1}^s n^*(\bar{v}_*(\mathfrak{N}(\mathfrak{p}_i))) \ge m$.

*Proof.* — (P$_2$) follows immediately from the description of $\mathcal{Z}$; (P$_1$) is similar. $\square$

<figure><img src="image_14.png" /></figure>

**3.4.4. Bounds on ECH indices, part II.** — In this subsection we give bounds on ECH indices in the presence of boundary points at $z_\infty$.

For simplicity we are still assuming (S). The ECH indices of $\bar{v}^{\circ, i}_{*0}$ are defined in a manner similar to that of Definition I.5.7.3.

**Lemma 3.4.18.** — If $\bar{v}'_{*0}, \bar{v}''_{*0} \neq \emptyset$, (S) holds, and the boundary point at $z_\infty$ is of type (P$_3$), then there exist contributions $I_+ \ge 0$, $I_{L-} = 2$, and $I_{R-} \ge 0$ from the ends that limit to $z_\infty$ at the positive, left negative, and right negative ends, such that

$$ (3.4.16) \qquad I(\bar{v}_{*0}) \ge I(\bar{v}^{\circ, 1, '}_{*0}) + I(\bar{v}^{\circ, 1, ''}_{*0}) + I_+ + I_{L-} + I_{R-}. $$

*Proof.* — The calculation of $I_{L-}$ is similar to the ECH index calculations of Section I.5.7 and in particular that of Lemma I.5.7.22.

<figure><img src="image_15.png" /></figure>
<figure><img src="image_16.png" /><figcaption>FIG. 10. — $c'_{L-} \cup c''_{L-}$ for $\mathfrak{r}$ of type $(P_3)$. Here $c''_{L-}$ is the strand in the front</figcaption></figure>

First observe that $P_{L-j_0, 1/2} = P_{L-j_0, 1}$. For simplicity suppose that the only chord of $\mathcal{Z}$ corresponding to a sector of type $\mathcal{S}(\bar{a}_{k,l}, \bar{a}_{k',l'})$ is $\mathfrak{z}'_1 \to \mathfrak{z}'_2$. Observe that there are no points of $\mathcal{Z}$ between $\mathfrak{z}'_1$ and $\mathfrak{z}'_2$, since otherwise $\mathcal{Z}$ winds more than once around $\mathbf{R}/2\pi\mathbf{Z}$. Hence $P''_{L-j_0, 1/2} = \{\mathfrak{z}'_1\}$, $P'_{L-j_0, 1/2} = \{\mathfrak{z}'_2, \mathfrak{z}''_1, \dots, \mathfrak{z}''_s\}$, $P''_{L-j_0, 1} = \{\mathfrak{z}'_2\}$, and $P'_{L-j_0, 1} = \{\mathfrak{z}'_1, \mathfrak{z}''_1, \dots, \mathfrak{z}''_s\}$, where $P_{L-j_0, 1/2} = P_{L-j_0, 1}$ is written as $\{\mathfrak{z}'_1, \mathfrak{z}'_2, \mathfrak{z}''_1, \dots, \mathfrak{z}''_s\}$ in cyclic order around $\mathbf{R}/2\pi\mathbf{Z}$, and the projection of the left negative end of $\bar{v}^{\circ, 1}_{*0}$ that limits to $z_\infty$ intersects $A_\varepsilon^{[1/2, 1]}$ along an arc $c''_{L-}$ with winding number $w(c''_{L-}) = 0$ or 1, depending on whether $\mathcal{S}(\bar{a}_{k,l}, \bar{a}_{k',l'})$ is a large sector; see Figure 10. The left negative ends of $\bar{v}^{\circ, 1, \prime}_{*0}$ that limit to $z_\infty$ give a grooming $c'_L$ on $A_\varepsilon^{[1/2, 1]}$ such that the winding number $w(c'_L) = 0$ or $-1$ and $c'_L$ connects $\mathfrak{z}''_i$ to $\mathfrak{z}''_i$, $i = 1, \dots, s$, by vertical arcs. Then the writhe of $c'_L \cup c''_L$ is $+1$ and resolving the positive crossing of $c'_L \cup c''_L$ yields a grooming $\tilde{c}_{L-}$ by vertical arcs from $P_{L-j_0, 1/2}$ to $P_{L-j_0, 1}$.

We now consider the disk $\tilde{\mathcal{D}}$ corresponding to resolving the positive crossing that we “append” to the left negative end of $\bar{v}^{\circ, 1}_{*0}$ as in the proof of Lemma I.5.7.22. The disk $\tilde{\mathcal{D}}$ contributes 1, 0, 1, and 0 to $\mathcal{Q}$, $c_1$, $\mu$, and the discrepancy: $\mathcal{Q}$ is equal to the writhe +1. The calculations for $\mu$ assume convention (R). Suppose $w(c'_{L-}) = w(c''_{L-}) = 0$ (the case $w(c'_{L-}) = -1$ and $w(c''_{L-}) = 1$ is similar and is left to the reader). Then $\mu$ of the positive ends of $\tilde{\mathcal{D}}$ are 1, 0, ..., 0 and $\mu$ of the negative ends are all 0. The discrepancy contribution at the negative end of $\tilde{\mathcal{D}}$ is zero since $\tilde{c}_{L-}$ is a grooming by (almost) vertical arcs. Hence $I_{L-} = 2$.

Finally, since $I(\bar{v}_{*0}) = I(\bar{v}^{\circ, 1}_{*0}) + I(\bar{v}^{\circ, 2}_{*0})$ and $I(\bar{v}^{\circ, 2}_{*0}) = 0$, we have $I(\bar{v}_{*0}) = I(\bar{v}^{\circ, 1}_{*0}) = I(\bar{v}^{\circ, 1, \prime}_{*0}) + I(\bar{v}^{\circ, 1, \prime\prime}_{*0}) + I_+ + I_{L-} + I_{R-}$. Lemma 3.4.14 implies that $I_+ \ge 0$ and $I_{R-} \ge 0$. $\square$

*Remark 3.4.19.* — In general, each collection of boundary points of type $(P_3)$ that map to the same point on the base contributes at least +2 towards $I$. The proof is similar to but slightly more complicated than that of Lemma 3.4.18, and is left to the reader.

The following is a strengthening of Lemma 3.4.14 in the presence of boundary points at $z_\infty$:

*Lemma 3.4.20.* — If fiber components are removed from $\bar{u}_\infty$ and the only boundary points are of type $(P_3)$, then
(1) the only components of $\bar{v}_*$ with negative ECH index are the branched covers of $\sigma_+^+$;

(2) the ECH index of each level $\bar{v}_* \neq \bar{v}_+$ is nonnegative; and

(3) For $0 \le j \le a$,

$$I(\bar{v}_{1,j}) \ge \begin{cases} I(\bar{v}'_{1,j}) + I(\bar{v}''_{1,j}) & \text{if } \mathfrak{b}\mathfrak{p}_{1,j} = 0; \\ I(\bar{v}'_{1,j}) + I(\bar{v}''_{1,j}) + 2 & \text{if } \mathfrak{b}\mathfrak{p}_{1,j} > 0. \end{cases}$$

Here $\mathfrak{b}\mathfrak{p}_*$ is the number of boundary points of type $(P_3)$ on $\bar{v}_*$.

*Proof.* — We explain the modifications that need to be made when there are boundary points at $z_\infty$ in view of Lemma 3.4.18 and Remark 3.4.19. We assume (S) for simplicity. The cycles $\mathcal{Z}^{(a-j)}$ are defined as in the proof of Lemma 3.4.13, for $j \ge j_0$. We define $\mathcal{Z}^{(a-j_0,+)} = \mathcal{Z}^{(a-j_0)}$ and $\mathcal{Z}^{(a-j_0,-)} = \mathcal{Z}^{(a-j_0,+)}$ with $\mathfrak{z}'_1 \to \mathfrak{z}'_2$ replaced by $\mathfrak{z}'_2$. Then $\mathcal{Z}^{(a-(j_0-1))}$ is obtained using $\mathcal{Z}^{(a-j_0,-)}$ instead of $\mathcal{Z}^{(a-j_0)}$. Also, $P^*_{R-,j_0,i}$, $\star = \emptyset, \emptyset, \emptyset$, is obtained from $P^*_{+,j_0,i}$ by replacing $\mathfrak{z}'_1$ by $\mathfrak{z}'_2$. The rest of the argument of Lemma 3.4.14 carries over. $\square$

**3.4.5.** *Case of $\bar{v}'_* \cup \bar{v}^*_* = \emptyset$ for all $\bar{v}_*$.*

*Lemma 3.4.21.* — If $\bar{u}_\infty \in \partial_{(+\infty)}\mathcal{M}$ and $\bar{v}'_* \cup \bar{v}^*_* = \emptyset$ for all levels $\bar{v}_*$ of $\bar{u}_\infty$, then $a = c = 0$; $I(\bar{v}_+) = 0$; $I(\bar{v}_-) = 2$; $\bar{v}_+$ is a $W_+$-curve; $\bar{v}_-$ is a $W_-$-curve; and there may be connectors $\bar{v}_{0,j}$ in between.

*Proof.* — Suppose that $\bar{u}_\infty \in \partial_{(+\infty)}\mathcal{M}$ and $\bar{v}'_* \cup \bar{v}^*_* = \emptyset$ for all levels $\bar{v}_*$ of $\bar{u}_\infty$. Then there is a point $q \in \text{int}(\mathbb{F}_-)$ and a sufficiently small neighborhood $N(q) \subset \dot{\mathbb{F}}_-$ of $q$ such that $\bar{v}_-(q) = \bar{m}(\infty)$ and $n^*(\bar{v}_-(N(q))) \ge m$. By Equation (3.4.2) and Lemma 3.4.8, there are no boundary points at $z_\infty$ and the only possible fiber component passes through $\bar{m}(\infty)$. Hence every level $\bar{v}_*$, $\star \neq -$, has image inside $W'$, $W$, or $W_+$ and $\bar{v}_-$ is a $\bar{W}_-$-curve or a degenerate $\bar{W}_-$-curve by the analog of Lemma I.7.5.2.

The ECH index of each level $\neq \bar{v}_+$ and which has no fiber component is non-negative by Lemma 3.4.14. Since $\bar{v}_+$ is a $W_+$-curve, $I(\bar{v}_+) \ge 0$. On other hand, by the previous paragraph, if there is a fiber component, then it is a component of $\bar{v}_-$. We claim that $I(\bar{v}_-) \ge 2$, with equality if and only if $\bar{v}_-$ is not a degenerate $\bar{W}_-$-curve. Indeed, if $\bar{v}_-$ is not degenerate, then $I(\bar{v}_-) \ge 2$ by the point constraint (this is the only place where we use the fact that $\bar{u}_\infty \in \partial_{(+\infty)}\mathcal{M}$) and the ECH index inequality, and if $\bar{v}_-$ is degenerate, then $I(\bar{v}_-) \ge 4$, as computed in the proof of Lemma I.7.5.5.

The lemma then follows from Equation (3.4.3). In particular, degenerate $\bar{W}_-$ curves are not allowed. $\square$

**3.4.6.** *Preliminary restrictions when $\bar{v}'_* \cup \bar{v}^*_* \neq \emptyset$ for some $\bar{v}_*$. — We now consider the case where $\bar{v}'_* \cup \bar{v}^*_* \neq \emptyset$ for some level $\bar{v}_*.$*
Lemma 3.4.22. — If $\bar{v}_*^\# \cup \bar{v}_*^{\#'} \neq \emptyset$ for some level $\bar{v}_*$ of $\bar{u}_\infty$, then:

(1) $p_- = \deg(\bar{v}_-) > 0$;

(2) $\bar{u}_\infty$ has no boundary point of type $(P_1)$ or $(P_2)$;

(3) $\bar{u}_\infty$ has no fiber components and no components of $\bar{v}_*^{\#'}$ that intersect the interior of a section at infinity;

(4) each component of $\bar{v}_{-1,j}^{\#'}$, $j = 1, \dots, c$, is a thin strip from $z_\infty$ to some $x_i$ or $x'_i$ with $I = 1$;

(5) each component of $\bar{v}_-^{\#'}$ is a section of $\bar{W}_- - W_-$ from $\delta_0$ to some $x_i$ or $x'_i$ with $I = 1$;

(6) each component of $\bar{v}_{0,j}^{\#'}$, $j = 1, \dots, b$, which has a positive end at a multiple of $\delta_0$ is a cylinder from $\delta_0$ to $h$ or $e$ with $I = 1$ or $2$;

(7) the only boundary points at $z_\infty$ are type $(P_3)$ points of $\bar{v}_{1,j}^{\#'}$, $j = 0, \dots, a$.

*Proof.* — The lemma is a consequence of Equation (3.4.2). Suppose $\bar{v}_*^\# \cup \bar{v}_*^{\#'} \neq \emptyset$ for some level $\bar{v}_*$. Then $\bar{v}_*^{\#'} \neq \emptyset$ for some $\bar{v}_*$ and each end of $\bar{v}_*^{\#'}$ that limits to some multiple of $z_\infty$ or $\delta_0$ contributes positively to $n^*$ by Lemma 3.4.2 and the proof of Lemma I.7.4.2 (also see Lemma 3.4.7(1)).

(1) If $\bar{v}_- = \emptyset$, then the restriction of $\bar{v}_-''$ to a neighborhood of $\bar{m}(+\infty)$ contributes $m$ towards $n^*(\bar{v}_-)$. This contradicts the discussion from the previous paragraph, proving (1).

Since $\bar{v}_- \neq \emptyset$, some $\bar{v}_{0,j}^{\#'}$, $j = 1, \dots, b+1$, or $\bar{v}_{1,j}^{\#'}$, $j = 1, \dots, a$, has a negative end at $z_\infty$ or a multiple of $\delta_0$. By Lemmas 3.4.2(1) and 3.4.16, it follows that:

$$ (3.4.17) \qquad \sum_{\bar{v}_* > \bar{v}_-} n^*(\bar{v}_*) \geq m - 2g, $$

where we are counting contributions from the ends and the boundary points of type $(P_3)$.

(2)–(7) are consequences of Equation (3.4.17). We explain (2) and (3), leaving the rest to the reader. By Lemma 3.4.17, the neighborhoods of the boundary points of type $(P_1)$ or $(P_2)$ contribute at least $m$ towards $n^*(\bar{v}_*)$ in total. Also, a non-ghost fiber component or a component of $\bar{v}_*^{\#'}$ that intersects the interior of a section at infinity contributes $m$ towards $n^*$. They both contradict Equation (3.4.17). $\square$

Lemma 3.4.23. — If $\bar{v}_*^\# \cup \bar{v}_*^{\#'} \neq \emptyset$ for some level $\bar{v}_*$, then the following alternative holds:

(a) either some $\bar{v}_{0,j_0}^{\#'}$, $j_0 = 1, \dots, b+1$, has a negative end at a multiple of $\delta_0$, in which case $\bar{v}_{0,j_0}^\# = \emptyset$, $\bar{v}_*^\# = \bar{v}_*^{\#'} = \emptyset$ for all levels $\bar{v}_* > \bar{v}_{0,j_0}$, and $\bar{v}_{0,j}^\# \neq \emptyset$ for all levels $\bar{v}_-^\# \le \bar{v}_{0,j}^\# < \bar{v}_{0,j_0}^\#$; or

(b) no $\bar{v}_{0,j}^{\#'}$, $j = 1, \dots, b+1$, has a negative end at a multiple of $\delta_0$, in which case $\bar{v}_*^\# \neq \emptyset$ for all levels $\bar{v}_-^\# \le \bar{v}_*^\# \le \bar{v}_+^\#$ and $\bar{v}_{1,j}^\# \cup \bar{v}_{1,j}^{\#'} \neq \emptyset$ for some $j > 0$.

*Proof.* — This follows from Lemmas 3.4.2(1) and 3.4.16 by observing that either case contributes at least $m - 2g$ towards $n^*$ and that it is not possible to have both since
$m \gg 2g$. It is also not possible to have multiple occurrences of negative ends of $\bar{v}_{0,j}^\sharp, j > 0$, that converge to some multiple of $\delta_0$. $\square$

**3.4.7.** *List of possibilities when $\bar{v}_*^\prime \cup \bar{v}_*^\sharp \neq \emptyset$ for some $\bar{v}_*$. — We first start with a useful definition.*

*Definition 3.4.24.*

(1) Let $X_1 \cup \cdots \cup X_r$ be an $r$-level split almost complex manifold with cylindrical ends, ordered from bottom to top, and let

$$v = v_1 \cup \cdots \cup v_r, \quad \mathrm{Im}(v_i) \subset X_i,$$

be a corresponding $r$-level holomorphic building, where each level has finite energy. If $\mathcal{E}$ is some collection of positive ends of $v_r$, then the holomorphic building hanging from $\mathcal{E}$ is the union of irreducible components $\tilde{v}_{i_0}$ for which there exist irreducible components $\tilde{v}_i$, $i = i_0 + 1, \dots, r$, where the limit of a positive end of $\tilde{v}_i$ agrees with the limit of a negative end of $\tilde{v}_{i+1}$ for all $i = i_0, \dots, r-1$ and a positive end of $\tilde{v}_r$ is contained in $\mathcal{E}$.

(2) If $X_1 \cup \cdots \cup X_{r+1}$ is an $(r+1)$-level split almost complex manifold, $v = v_1 \cup \cdots \cup v_{r+1}$ is a corresponding $(r+1)$-level holomorphic building and $\mathcal{E}$ is some collection of negative ends of $v_{r+1}$, then the holomorphic building hanging from $\mathcal{E}$ is defined similarly.

(3) If $X_1 \cup \cdots \cup X_r$ is an $r$-level split almost complex manifold, $v = v_1 \cup \cdots \cup v_r$ is a corresponding $r$-level holomorphic building and $\mathcal{E}$ is some collection of negative ends of $v_1$, then the holomorphic building sitting above $\mathcal{E}$ is the union of irreducible components $\tilde{v}_{i_0}$ for which there exist irreducible components $\tilde{v}_i$, $i = 1, \dots, i_0 - 1$, where the limit of a negative end of $\tilde{v}_i$ agrees with the limit of a positive end of $\tilde{v}_{i-1}$ for all $i = 2, \dots, i_0$ and a negative end of $\tilde{v}_1$ is contained in $\mathcal{E}$.

(4) If $X_0 \cup \cdots \cup X_r$ is an $(r+1)$-level split almost complex manifold, $v = v_0 \cup \cdots \cup v_r$ is a corresponding $(r+1)$-level holomorphic building and $\mathcal{E}$ is some collection of positive ends of $v_0$, then the holomorphic building sitting above $\mathcal{E}$ is defined similarly.

**Lemma 3.4.25.** — If $\bar{v}_*^\prime \cup \bar{v}_*^\sharp \neq \emptyset$ for some level $\bar{v}_*$, then $\bar{v}_-^\prime \neq \emptyset$, there are no boundary points at $z_\infty$, and $\bar{u}_\infty$ contains one of the following subbuildings:

(1) A 3-level building consisting of a component of $\bar{v}_{0,1}^\sharp$ with $I = 1$ from $\mathbf{y}$ to $\delta_0 \mathbf{y}'; \bar{v}_- = \sigma_\infty^-$; and a thin strip.

(2) A 3-level building consisting of a component of $\bar{v}_+^\sharp$ with $I = i$, $i = 0, 1$, from $\mathbf{y}$ or $\mathbf{y}''$ to $\delta_0 \mathbf{y}'; \bar{v}_- = \sigma_\infty^-$; and a thin strip.

(3) A 4-level building consisting of a component of $\bar{v}_+^\sharp$ with $I = 0$ from $\mathbf{y}$ to $\delta_0^2 \mathbf{y}'; \bar{v}_{0,1} = \sigma_\infty^-$; a component of $\bar{v}_{0,1}^\sharp$ which is an $I = 1$ cylinder from $\delta_0$ to $h$; $\bar{v}_- = \sigma_\infty^-$; and a thin strip.

(4) A 4-level building consisting of a component of $\bar{v}_+^\sharp$ with $I = 0$ from $\mathbf{y}$ to $\delta_0^2 \mathbf{y}'; \bar{v}_{0,1}^\prime$ with $I = 0$ and $\deg 2 = 2$; $\bar{v}_- = \sigma_\infty^-$; a component of $\bar{v}_-^\sharp$ which is an $I = 1$ curve from $\delta_0$ to some $x_i$ or $x_i^\prime$; and a thin strip.
<figure><img src="image_17.png" /><figcaption>FIG. 11. — Schematic diagrams for the possible types of degenerations. Here • represents $\delta_0$, ◦ represents $z_\infty$, □ represents a possible location of a branch point and × represents some $x_i$ or $x'_i$. A vertical line indicates the restriction of a trivial cylinder, a dotted vertical line indicates a trivial cylinder, a double vertical line indicates a degree 2 branched cover of a trivial cylinder or a restriction of a trivial cylinder, and a triple vertical line indicates a degree $p \ge 2$ branched cover of a trivial cylinder or a restriction of a trivial cylinder. The labels on the graphs are ECH indices of each component. All the thin strips of $\bar{v}_{-1,j}^\sharp$ are drawn on the same level for convenience</figcaption></figure>

(5) $A \ge 3$-level building consisting of a component of $\bar{v}_+^\sharp$ with $I=0$ from $\mathbf{y}$ to $\delta_0^2\mathbf{y}'; \bar{v}_{0,1}^\prime$ and $\bar{v}_-^\prime$ with $I=0$ and $\deg = 2$, where $\bar{v}_{0,1}^\prime = \emptyset$ is possible; and two thin strips.

(6i) $A \ge 4$-level building consisting of a component of $\bar{v}_{1,1}^\sharp$ with $I=i, i=1,2$, from $\mathbf{y}$ to $\{z_\infty^i\} \cup \mathbf{y}^\prime$; $0 < p_{0,0} \le p_{0,1} \le \dots \le p_{0,b+1} \le r$; $I(\bar{v}_+^\prime) = -p_{0,b+1}$; $I(\bar{v}_{0,j}^\prime) = 0$ for $j=1, \dots, b$; $\bar{v}_+^\sharp$ with $I=2-i$ which has no negative ends at a multiple of $\delta_0$; $p_{0,b+1} - p_{0,1}$ cylinders of $\bar{v}_{0,j}^\sharp$, $j \in \{1, \dots, b\}$, with $I=1$ each from $\delta_0$ to $h$; $p_{0,1} - p_{0,0}$ components of $\bar{v}_-^\sharp$ with $I=1$ each from $\delta_0$ to some $x_i$ or $x'_i$; and $p_{0,0}$ thin strips. The total ECH index of the building hanging from the negative end of $\bar{v}_+^\prime$ is $p_{0,b+1}$.

Here we are omitting levels which are connectors. If there is more than one thin strip, then the thin strips could be on the same level or on different levels.

See Figure 11.

*Proof.* — The lemma is a consequence of Equations (3.4.2) and (3.4.3). By Lemmas 3.4.20 and 3.4.22(2), (3), all the levels $\bar{v}_*$ besides $\bar{v}_+$ have nonzero ECH index, the only components of $\bar{u}_\infty$ which have negative ECH index are the branched covers of $\sigma_+^\infty$, and $I(\bar{v}_{1,j}) \ge I(\bar{v}_{1,j}^\prime) + I(\bar{v}_{1,j}^{\prime\prime})$ or $I(\bar{v}_{1,j}^\prime) + I(\bar{v}_{1,j}^{\prime\prime}) + 2$ for $0 \le j \le a$, depending on whether $\mathfrak{h}\mathfrak{p}_{1,j} = 0$ or $> 0$. In particular, all the components of $\bar{u}_\infty$ which are left out of the subbuildings in (1)–(6$_i$) and are not drawn in Figure 11 have $I \ge 0$.

By Lemma 3.4.22(1), $\bar{v}_-^\prime \ne \emptyset$. By Lemma 3.4.22(2), (7), there are no boundary points of types $(P_1)$ and $(P_2)$ and no boundary points of type $(P_3)$ except for $\bar{v}_{1,j}^{\prime\prime}$, $j=0, \dots, a$. Cases (1)–(5) have no boundary points at $z_\infty$ (since the ECH index would add up to more than 2) and Case (6$_i$) may have boundary points of type $(P_3)$.

First suppose that some $\bar{v}_{0,j_0}^\sharp$, $j_0 \in \{1, \dots, b\}$, has a negative end at $\delta_0^{\prime\prime}$ for some $p > 0$. This is the situation of Lemma 3.4.23(a) with $j_0 \in \{1, \dots, b\}$. In this case, all the
components and levels have nonnegative ECH index. Each component of $\bar{v}_*^\sharp < \bar{v}_{0,j_0}$ described in Lemma 3.4.22(4)–(6) satisfies $I \ge 1$ and there must be $p > 0$ such components since $\bar{v}_{0,j_0}$ has multiplicity $p$ at $\delta_0$. We then have the following contributions towards I:

(a) $\sum_{\bar{v}_*^\sharp < \bar{v}_{0,j_0}} I(\bar{v}_*^\sharp) \ge p$ and there is at least one thin strip.

(b) $I(\bar{v}_{0,j_0}^\sharp) \ge 1$ since $\bar{v}_{0,j_0}$ is nontrivial.

Then $p=1$ and all the components and levels besides those of (a) and (b) satisfy $I=0$. We are in Case (1).

Next suppose that $\bar{v}_{0,b+1}^\sharp = \bar{v}_+^\sharp$ has a negative end at $\delta_0^\beta$. This is the situation of Lemma 3.4.23(a) with $j=b+1$. Since $\bar{v}_+^\prime = \emptyset$, all the components and levels satisfy $I \ge 0$. Here $\bar{v}_+^\sharp$ cannot have any multiple of $z_\infty$ at the positive end, since otherwise we contradict Equation (3.4.2). Since $I(\bar{v}_+^\sharp) \ge 0$ and $\sum_{\bar{v}_*^\sharp < \bar{v}_+} I(\bar{v}_*^\sharp) \ge p$, either $p=1$ and we are in Case (2$_i$) or $p=2$ and we are in Cases (3)–(5).

Finally suppose that no $\bar{v}_{0,j}^\sharp, j=1, \dots, b+1$, has a negative end at a multiple of $\delta_0$. This is the situation of Lemma 3.4.23(b). We have the following contributions:

(α) $\deg \bar{v}_+^\prime = p_{0,b+1}$ and $I(\bar{v}_+^\prime) = -p_{0,b+1}$.

(β) By Lemma 3.4.22(4)–(6), the holomorphic building hanging from the negative end of $\bar{v}_+^\prime$ satisfies $\sum_{\bar{v}_*^\prime < \bar{v}_+} I(\bar{v}_*^\prime \cup \bar{v}_*^\sharp) \ge p_{0,b+1}$, and equality holds if and only if there is no cylinder from $\delta_0$ to $e$.

(γ) $I=2$ components of $\bar{v}_{0,j}^\sharp$ from $\delta_0$ to $e$ can be eliminated by observing that it is followed by an $I = -1$ component of $\bar{v}_-$ from $e$ to some $x_i$ or $x'_i$, which is a contradiction.<fnref n="10" />

(δ) Some $\bar{v}_{1,j_0}^\sharp, j_0 \ge 0$, must have an end at $z_\infty$ (or some boundary point of type (P$_3$) must have a neighborhood) which projects to a large sector of $D_{\rho_0}^2$. Since $\bar{v}_{1,j_0}^\sharp$ generically lies on a codimension 1 stratum of some ind($\bar{v}_{1,j_0}^\sharp$)-dimensional moduli space, the large sector contributes an additional +1 to the Fredholm and ECH indices, cf. Lemma I.5.7.21.

(ε) If $bp$ is the number of boundary points of type (P$_3$) and $bp > 0$, then the contribution to $I$ is at least 2.

(α), (β), and (γ) together give:

$$I(\bar{v}_+^\prime) + \sum_{\bar{v}_*^\prime < \bar{v}_+} I(\bar{v}_*^\prime \cup \bar{v}_*^\sharp) = 0.$$

Now, $I(\bar{v}_{1,j}^{\prime\prime}) \ge 1$ for each $j > 0$,<fnref n="11" /> (δ) contributes at least 1 to $I$, and (ε) contributes at least 2 to $I$ if $bp > 0$. Hence $bp = 0$, $a=1$, and we are in Case (6$_i$), $i=1, 2$. □

<footnote n="10">¹⁰ Strictly speaking, $\bar{J}_-$ is the restriction of a Morse-Bott $\bar{J}$. To give a proper treatment of regularity, we must perturb $\bar{J}$ using an arbitrarily small Morse function to obtain $(\bar{J})^\diamond$ and then restrict $(\bar{J})^\diamond$ to $\bar{W}_-$.</footnote>

<footnote n="11">¹¹ Recall that we are ignoring connectors.</footnote>
**3.4.8.** *Elimination of some cases when $\bar{v}_*^\prime \cup \bar{v}_*^\sharp \neq \emptyset$ for some $\bar{v}_*$. — The goal of this subsection is to eliminate all but one of the possibilities (namely Case (2₁)) given in Lemma 3.4.25 and to prove Lemma 3.3.2.*

*Definition 3.4.26.* — If $\pi : \Sigma \to \Sigma'$ is a branched cover and $\mathcal{B}(\pi)$ is the branch locus of $\pi$, then we define

$$ \mathfrak{b}(\pi) = \sum_{x \in \mathcal{B}(\pi)} (\deg(x) - 1), $$

where $\deg(x)$ is the degree of the branch point $x$. We also write $\mathfrak{b}(\bar{u}) = \mathfrak{b}(\bar{\pi}_* \circ \bar{u})$, where $\bar{\pi}_*$ is the projection to the base $\mathcal{B}$, $\mathcal{B}'$, etc., and refer to it informally as “the number of branch points of $\bar{u}$.”

Let $\mathcal{E}_{-,i}$, $i = 1, \dots, q$, be the negative ends of $\cup_{j=1}^a \bar{v}_{1,j}^\sharp$ that converge to $z_\infty$ and let $\mathcal{E}_{+,i}$, $i = 1, \dots, r$, be the positive ends of $\cup_{j=0}^{a-1} \bar{v}_{1,j}^\sharp$ that converge to $z_\infty$ as in Lemma 3.4.7.

**Lemma 3.4.27.** — Suppose we are in Case (6ᵢ) of Lemma 3.4.25. Then $\sum_{\bar{v}_*'} \mathfrak{b}(\bar{v}_*)'$, including branched points of connector components, is equal to the number of negative ends $\mathcal{E}_{-,i}$. Moreover, if $F$ is the surface obtained by gluing all the domains of $\bar{v}_*$', then $F$ is a connected planar surface whose compactification (after filling in the interior and boundary punctures) has a single positive boundary component, i.e., coming from levels $\bar{v}_{1,j}$, $j = 0, \dots, a$, together with some number of negative boundary components, i.e., coming from levels $\bar{v}_{-1,j}$, $j = 1, \dots, c + 1$.

*Proof.* — Suppose we are in Case (6ᵢ) of Lemma 3.4.25. We claim that we may make the following simplifying assumptions after rearranging some levels:

(1) $\bar{v}_{1,1}$ is a connector;

(2) all the ends $\mathcal{E}_{-,i}$ are negative ends of $\bar{v}_{1,2}^\sharp$;

(3) all the ends $\mathcal{E}_{+,i}$ are positive ends of $\bar{v}_+^\sharp$;

(4) all the branch points of $\bar{v}_*$' lie on $\bar{v}_{1,1}$.

Note that in Lemma 3.4.25 we omitted connectors, but in the present analysis we need to keep track of connectors which are branched covers. Since the lemma is of topological nature, we may assume that all the connectors between $\bar{v}_+$ and $\bar{v}_{1,1}$ have been merged and, after renaming, the connector level is $\bar{v}_{1,1}$ and the old $\bar{v}_{1,1}$ becomes $\bar{v}_{1,2}$. This gives (1). (2) is immediate and (3) can be arranged by topologically moving up/down the ends $\mathcal{E}_{+,i}$ if necessary. (4) can be obtained by pushing all the branched points of $\bar{v}_*$' from the lower levels to $\bar{v}_{1,1}$; this operation is possible because $\bar{v}_*^\sharp \le \bar{v}_+$ does not have a negative end that limits to $z_\infty$ in Case (6ᵢ). Note that this operation does not affect:

- the total Fredholm index $\sum_{\bar{v}_*'} \mathrm{ind}(\bar{v}_*')$; and

- the topological type of $F$.

As a consequence of (1)–(4), each component of $\bar{v}_*^\prime \le \bar{v}_+$ is an unbranched cover of the appropriate $\sigma_\infty^*$.
We now compute the Fredholm index of $\overline{v}'_{1,1} : \dot{\mathbb{F}}_{1,1} \to \overline{\mathbb{W}}$ using the Fredholm index formula

$$ \mathrm{ind}(\overline{v}'_{1,1}) = -\chi(\dot{\mathbb{F}}_{1,1}) + p_{1,1} + \mu_{\tau}(\overline{v}'_{1,1}) + 2c_1((\overline{v}'_{1,1})^*\mathrm{T}\overline{\mathbb{S}}, \tau); $$

see Equation (I.5.7.2). Recall we are writing $p_* = \deg(\overline{v}'_*)$. The groomed multivalued trivialization $\tau$ is defined as follows: Let $\mathcal{S}_{\pm,i}$ be the sector of $\mathrm{D}^2_{\rho_0}$ given by $\pi_{\mathrm{D}^2_{\rho_0}}(\mathcal{E}_{\pm,i})$. If (ii) and its analogs occur in the proof of Lemma 3.4.7, then we add the thin counterclockwise sectors $\mathcal{S}(\overline{a}_{k_2,l_2}, \overline{h}(\overline{a}_{k_2,l_2}))$, etc. to the set $\{\mathcal{S}_{+,i}\}$. The sets $\{\mathcal{S}_{+,i}\}$ and $\{\mathcal{S}_{-,i}\}$ correspond to the data $\overrightarrow{\mathcal{D}}_-$ and $\overrightarrow{\mathcal{D}}_+$ at the negative and positive ends of $\overline{v}'_{1,1}$<fnref n="12" /> and we let $\tau$ be the induced groomed multivalued trivialization.

By a calculation similar to that of Lemma I.5.7.15, $c_1((\overline{v}'_{1,1})^*\mathrm{T}\overline{\mathbb{S}}, \tau) = 1$ and the ends of $\overline{v}'_{1,1}$ contribute the following to $\mu_{\tau}(\overline{v}'_{1,1})$:

- 0 if $\mathcal{S}_{-,i}$ is a small sector;

- -1 if $\mathcal{S}_{-,i}$ is a large sector;

- -1 if $\mathcal{S}_{+,i}$ is a small sector; and

- -2 if $\mathcal{S}_{+,i}$ is a large sector.

Hence $\mu_{\tau}(\overline{v}'_{1,1}) = -p_{1,1} - 1$. Since $\chi(\dot{\mathbb{F}}_{1,1}) = p_{1,1} - \mathfrak{b}(\overline{v}'_{1,1})$, we obtain:

$$ \begin{aligned} \mathbf{(3.4.18)} \qquad \mathrm{ind}(\overline{v}'_{1,1}) &= (\mathfrak{b}(\overline{v}'_{1,1}) - p_{1,1}) + p_{1,1} + (-p_{1,1} - 1) + 2 \\ &= \mathfrak{b}(\overline{v}'_{1,1}) - (p_{1,1} - 1). \end{aligned} $$

Next we claim that $\partial F_{1,1}$ is connected. Indeed, if $\partial F_{1,1}$ is disconnected, then the method of Lemma 3.4.7 implies that the union of all the $\mathcal{S}_{\pm,i}$ covers $\mathrm{D}^2_{\rho_0}$ more than once; this contradicts Equation (3.4.2). The claim in turn implies that $\mathfrak{b}(\overline{v}'_{1,1}) \ge p_{1,1} - 1$, since otherwise $F_{1,1}$ is disconnected and $\partial F_{1,1}$ will have more than one component. Hence $\mathrm{ind}(\overline{v}'_{1,1}) \ge 0$; moreover, if $\mathrm{ind}(\overline{v}'_{1,1}) > 0$, then $\mathrm{ind}(\overline{v}'_{1,1}) \ge 2$.

We claim that $\mathrm{ind}(\overline{v}'_{1,1}) \ge 2$ is not possible. Indeed, we add up the Fredholm indices of all the remaining components as in the proof of Lemma 3.4.25:

(a) $\mathrm{ind}(\overline{v}''_{1,2}) \ge 1$ and $\mathrm{ind}(\overline{v}'_{+}) = -p_+$;

(b) $\sum_{\tilde{v}} \mathrm{ind}(\tilde{v}) \ge p_+$, where the summation is over all components $\tilde{v}$ of $\overline{u}_\infty$ that are hanging from the negative end of $\overline{v}'_+$;

(c) a large sector of $\mathrm{D}^2$ contributes an additional +1 to the Fredholm index; and

(d) all the other components of $\overline{u}_\infty$ have nonnegative Fredholm index.

(a), (b), (d) are clear, and (c) was explained in Lemma 3.4.25. The total of (a)–(d) is $\ge 2$, which is an index excess of +2, and the claim follows. The claim then implies that $\mathrm{ind}(\overline{v}'_{1,1}) = 0$, $\mathfrak{b}(\overline{v}'_{1,1}) = p_{1,1} - 1$, and $F_{1,1}$ is a disk.

<footnote n="12">¹² Note that the plus and minus signs are switched. This is due to the fact that, for example, $\{\mathcal{S}_{-,i}\}$ gives the data for the negative ends of $\overline{v}_{1,2}$ that limit to $z_\infty$, which agrees with the data for the positive ends of $\overline{v}_{1,1}$ that limit to $z_\infty$.</footnote>
Finally, since F is diffeomorphic to a surface obtained by gluing $\dot{\mathbb{F}}_{1,1}$ and $p_{1,1}$ surfaces which are diffeomorphic to $B_\tau$ or $B_+$, it must be connected and planar, and its compactification is as described in the lemma. □

**Lemma 3.4.28.** — Suppose $m \gg 0$. If $\bar{v}_* \cup \bar{v}_*^\perp \neq \emptyset$ for some level $\bar{v}_*$, then Cases (1), (2$_0$), (3)–(6$_i$) of Lemma 3.4.25 are not possible. This leaves us with Case (2$_1$).

*Proof.* — Arguing by contradiction, suppose there exist sequences $m_i \to \infty$, $\varepsilon_i, \delta_i \to 0$, and $\bar{u}_{ij} \to \bar{u}_{i\infty}$, where $\bar{u}_{ij} \in \mathcal{M}^{(m_i)}$, $\bar{u}_{i\infty} \in \partial_{(+\infty)}\mathcal{M}^{(m_i)}$, $\mathcal{M}^{(m_i)}$ is $\mathcal{M}$ with respect to the family $\{\mathbb{J}_\tau^\diamond(\varepsilon_i, \delta_i, \mathfrak{p}(\tau); m_i)\}$, and $\bar{u}_{i\infty}$ falls into one of Cases (1), (2$_0$), (3)–(6$_i$). We consider a diagonal subsequence $\bar{u}_i := \bar{u}_{j(i)}$.

Cases (1) and (2$_0$) are eliminated by an argument similar to that of Case (2) of Theorem I.7.10.1 and Cases (3)–(5) are eliminated by an argument similar to that of Cases (3)–(6) of Theorem I.7.10.1.

*Case* (6$_i$). Suppose for simplicity that $b = 1$, including connectors.

As in the proof of Theorem I.7.10.1, for $i \gg 0$, we consider a truncation $\tilde{u}_i : \Sigma_i \to \bar{W}_{\tau_i}$ of $\bar{u}_i$ (i.e., a restriction of $\bar{u}_i$ to a neighborhood of $\sigma_\infty^{\tau_i}$) such that there exist real numbers

$$R_{0,i} < R_{1,i} < R_{2,i} < R_{3,i}, \quad R_{0,i} \ll -l(\tau_i), \quad R_{3,i} \gg l(\tau_i)$$

and a map

$$\pi_i : \Sigma_i \to B_{\tau_i} \cap \{R_{0,i} \le s \le R_{3,i}\},$$

such that the restriction of $\pi_i$ to $\pi_i^{-1}(\{R_{j,i} \le s \le R_{j+1,i}\})$, $j = 0, 1, 2$, is a degree $p_{0,j}$ branched cover. Here $p_{0,0} \le p_{0,1} \le p_{0,2}$. We project $\tilde{u}_i$ to $D_{\rho_0}^2 \subset \bar{S}$ for $\rho_0 > 0$ small using balanced coordinates and then apply the ansatz from Equation (I.7.8.1) to obtain $w_i : \Sigma_i \to \mathbf{C}$.

Applying the method of Section I.7.8, we rescale $w_i$ by a positive real constant and take the limit $m_i \to \infty$ to obtain a 3-level holomorphic building

$$w_\infty = w_- \cup w_{0,1} \cup w_+.$$

We write $w_* : \Sigma_* \to \mathbf{CP}^1$ for the components of the building $w_\infty$ and $\pi_* : \Sigma_* \to \text{cl}(B_*)$ for the corresponding branched covers, where $* = +, -, $ or $(0, 1)$, and $B_{0,1} = \mathbf{R} \times \mathbb{S}^1$. We may also use subscripts $(0, 0)$ and $(0, 2)$ to mean $-$ and $+$. Note that $\deg \pi_{0,j} = p_{0,j}$. By Lemma 3.4.27, the surface $\Sigma_\infty$, obtained by gluing the $\Sigma_*$, is connected and planar. The next few paragraphs are devoted to the description of $w_*$ and $\pi_*$.

Suppose for simplicity that $\Sigma_-$, $\Sigma_{0,1}$, and $\Sigma_+$ are connected. Starting from the bottom, $w_-$ and $\pi_-$ satisfy the following:

$$(i_-) w_-(\partial \Sigma_-) \subset \{\phi = 0, \rho > 0\};$$
(ii_) $\pi_-^{-1}(+\infty)$ is a single point and $w_-(\pi_-^{-1}(+\infty)) = \infty$;

(iii_) $w_-(z_0) = 0$ for one of the points $z_0 \in \pi_-^{-1}(\bar{m}^b(\infty))$; and

(iv_) $w_{-|_{int}(\Sigma_-)}$ is a biholomorphism onto its image.

(i_) and (iii_) are clear, (ii_) follows from the fact that $\Sigma_\infty$ (and hence $\Sigma_-$) is connected and planar, and (iv_) is a consequence of Equation (3.4.2). Let us write $f_* = \pi_* \circ w_*^{-1}$, where defined. Then $f_-$ maps the asymptotic marker $\dot{\mathcal{R}}_\pi(\infty)$ for $\infty \in \mathbf{CP}^1$, corresponding to the radial ray $\mathcal{R}_\pi$, to the asymptotic marker $\dot{\mathcal{L}}_{3/2}(+\infty)$ for $+\infty \in \mathcal{cl}(\mathbf{B}_-)$, corresponding to the half-line $\mathcal{L}_{3/2}$, by the Involution Lemma I.7.9.3.

We then move up to $w_{0,1}$, which satisfies the following:

(i$_{0,1}$) $w_{0,1}(\pi_{0,1}^{-1}(-\infty)) = \{d_1 = 0, d_2, \dots, d_k\}$, where $d_i \in \mathbf{R}^{\ge 0}$ and $d_i < d_{i+1}$;

(ii$_{0,1}$) $\pi_{0,1}^{-1}(+\infty)$ is a single point and $w_{0,1}(\pi_{0,1}^{-1}(+\infty)) = \infty$;

(iii$_{0,1}$) $w_{0,1}$ is a biholomorphism; and

(iv$_{0,1}$) $f_{0,1}: \mathbf{CP}^1 \to \mathcal{cl}(\mathbf{B}_{0,1})$ maps $\dot{\mathcal{R}}_\pi(0)$ to $\dot{\mathcal{L}}_{3/2}(-\infty)$.

The placement of the points $d_2, \dots, d_k$ in (i$_{0,1}$) follows from observing that $\bar{v}_-^\sharp$ consists of $I=1$ components from $\delta_0$ to $x_i$ or $x'_i$. In particular, the asymptotic eigenfunctions of $\bar{v}_-^\sharp$ at the positive end are close to constant functions with values on $\mathbf{R}^+ \subset \mathbf{C}$. (ii$_{0,1}$) follows from the fact that $\Sigma_\infty$ is connected and planar and (iii$_{0,1}$) follows from Equation (3.4.2). (iv$_{0,1}$) is a consequence of the fact that $f_-$ maps $\dot{\mathcal{R}}_\pi(\infty)$ to $\dot{\mathcal{L}}_{3/2}(+\infty)$. Then $f_{0,1}$ maps $\dot{\mathcal{R}}_\pi(\infty)$ to $\dot{\mathcal{L}}_{3/2}(+\infty)$ by the Involution Lemma I.7.9.4.

We now describe the construction of $w_+$ in some detail. By translating

$$ \pi_i^{-1}(\mathbf{B}_{\tau_i} \cap \{\mathbf{R}_{2,i} \le s \le \mathbf{R}_{3,i}\}) $$

down by $T_i$, we obtain the branched cover

$$ \pi_i^+ : \Sigma_i^+ \to \mathbf{B}_+ \cap \{\mathbf{R}_{2,i} - T_i \le s \le \mathbf{R}_{3,i} - T_i\} $$

and the holomorphic map $w_i^+ : \Sigma_i^+ \to \mathbf{C}$. We may assume that $\mathbf{R}_{2,i} - T_i \to -\infty$ and $\mathbf{R}_{3,i} - T_i \to +\infty$ have been chosen so that there is no sequence of branch points of $\pi_i^+$ that limits to $s = \pm\infty$ as $i \to \infty$. Indeed, the branch points that “escape to $s = \pm\infty$” properly belong to a different level. Then $w_+$ is the limit of $w_i^+$, after suitably rescaling by positive real constants.

Let $\vec{D} = \{(i'_k, j'_k) \to (i_k, j_k)\}_{k=1}^\beta$ be the data at the positive end of $w_+$. For $i \gg 0$, the component of $w_i^+ |_{(\pi_i^+)^{-1}(\mathcal{S}=\mathbf{R}_{3,i}-T_i)}$ corresponding to $(i'_k, j'_k) \to (i_k, j_k)$ is arbitrarily close to a normalized asymptotic eigenfunction $\phi_k$ from $\bar{h}(\bar{a}_{i'_k, j'_k})$ to $\bar{a}_{i_k, j_k}$, after multiplying by some positive real constant. (Here $\phi_k$ is an eigenfunction of an asymptotic operator and we are not making any *a priori* assumptions on the corresponding eigenvalues.) Now we claim that some $\phi_{k_0}$ must sweep out a large sector of $D^2$, since otherwise $\mathrm{Im}(w_i^+)$ cannot pass through the origin, contradicting the “continuity” to $w_{0,1}$. In the limit $i \to \infty$, $\phi_{k_0}$ will sweep out an angle of $2\pi$.

The top level $w_+$ satisfies the following:
(i$_{+}$) $w_{+}(\pi_{+}^{-1}(-\infty)) = \{d_{1}^{+} = 0, d_{2}^{+}, \dots, d_{k_{+}}^{+}\}$, where $d_{i}^{+} \in \mathbb{R}^{\ge 0}$ and $d_{i}^{+} < d_{i+1}^{+}$;

(ii$_{+}$) $w_{+}(\partial\Sigma_{+}) \subset \{\phi = 0, 0 < \rho \le \infty\}$;

(iii$_{+}$) $w_{+}$ maps some point of $\pi_{+}^{-1}(+\infty)$ to $\infty$;

(iv$_{+}$) $f_{+}$ maps $\dot{\mathcal{R}}_{\pi}(0)$ to $\dot{\mathcal{L}}_{3/2}(-\infty)$; and

(v$_{+}$) $w_{+}$ is a biholomorphism onto its image.

The placement of the points $d_2^+, \dots, d_{k_+}^+$ in (i$_+$) follows from observing that $\bar{v}_{0,1}^\sharp$ consists of $I=1$ components from $\delta_0$ to $h$ and that $\mathcal{R}_{\phi_h} \to \mathcal{R}_0$ as $m \to \infty$ by our choice of $h$ from Section 3.2.3. (ii$_+$) and (iii$_+$) are immediate consequences of the construction of $w_+$ from the previous paragraphs. (iv$_+$) is a consequence of the fact that $f_{0,1}$ maps $\dot{\mathcal{R}}_\pi(\infty)$ to $\dot{\mathcal{L}}_{3/2}(+\infty)$. We now apply the Involution Lemma I.7.9.5 to conclude that $f_+(\infty) = \mathcal{L}_{3/2} \cap \partial B_+$. Note that the Lemma I.7.9.5 applies because the compactification of $\Sigma_+$ is a closed disk by Lemma 3.4.27. This contradicts (ii$_+$). We have eliminated Case (6$_i$). $\square$

We are now in a position to prove Lemma 3.3.2.

*Proof of Lemma 3.3.2.* — Suppose $\bar{u}_{\infty} \in \partial_{(+\infty)} \mathcal{M}$. By Lemma 3.4.21, if $\bar{v}'_* \cup \bar{v}^\sharp_* = \emptyset$ for all levels $\bar{v}_*$ of $\bar{u}_\infty$, then $\bar{u}_\infty \in A_1$. If $\bar{v}'_* \cup \bar{v}^\sharp_* \neq \emptyset$ for some level $\bar{v}_*$, then $\bar{u}_\infty$ is as given in Lemma 3.4.25. Now, by Lemma 3.4.28, the only possibility left is Case (2$_1$), which implies that $\bar{u}_\infty \in A_2$. $\square$

**3.5. Degeneration at $-\infty$.** — In this subsection we study the limit of holomorphic maps to $\bar{W}_\tau$ as $\tau \to -\infty$, i.e., when $\bar{W}_\tau$ degenerates into $\bar{W}_{-\infty,1} \cup \bar{W}_{-\infty,2}$. This will prove Lemma 3.3.5.

We assume that $m \gg 0$; $\varepsilon, \delta > 0$ are sufficiently small; and $\{\bar{J}_\tau\} \in \bar{\mathcal{I}}^{\mathrm{reg}}$ and $\{\bar{J}_\tau^\diamond(\varepsilon, \delta, \mathfrak{p}(\tau))\}$ satisfy Lemma 3.2.22. Fix $\mathbf{y} \in \mathcal{S}_{\mathbf{a},\mathfrak{h}(\mathbf{a})}$, $\mathbf{y}' \in \mathcal{S}_{\mathbf{b},\mathfrak{h}(\mathbf{b})}$ and let

$$ \mathcal{M} = \mathcal{M}_{\{\bar{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathfrak{p}(\tau))\}}^{I=2, n^*=m} (\mathbf{y}, \mathbf{y}'; \bar{\mathbf{m}}), \quad \mathcal{M}_{\tau} = \mathcal{M}_{\{\bar{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathfrak{p}(\tau))\}}^{I=2, n^*=m} (\mathbf{y}, \mathbf{y}'; \bar{\mathbf{m}}(\tau)). $$

We will analyze $\partial_{(-\infty)} \mathcal{M}$.

Let $\bar{u}_i$, $i \in \mathbb{N}$, be a sequence of curves in $\mathcal{M}$ such that $\bar{u}_i \in \mathcal{M}_{\tau_i}$ and $\lim_{i \to \infty} \tau_i = -\infty$, and let

$$ \text{(3.5.1)} \qquad \bar{u}_{\infty} = \bar{v}_2 \cup (\bar{v}_{L,1} \cup \dots \cup \bar{v}_{L,a}) \cup \bar{v}_1 \cup (\bar{v}_{R,1} \cup \dots \cup \bar{v}_{R,b}) \\ \qquad \qquad \cup (\bar{v}_{B,1} \cup \dots \cup \bar{v}_{B,c}) \cup (\bar{v}_{T,1} \cup \dots \cup \bar{v}_{T,d}), $$

be the limit holomorphic building, where each $\bar{v}_*$ is an SFT-type level, $\bar{v}_j$ maps to $\bar{W}_{-\infty,j}$, $j = 1, 2$; $\bar{v}_{L,j}$ and $\bar{v}_{R,j}$ map to $[-2, 2] \times \mathbf{R} \times \bar{S}$; $\bar{v}_{B,j}$ and $\bar{v}_{T,j}$ map to $\mathbf{R} \times [0, 1] \times \bar{S}$. The levels in the first row, called the *horizontal levels*, are arranged in cyclic order from left to right, the levels in the second row, called the *vertical levels*, are arranged in order from bottom to top, and $\bar{v}_1$ is between $\bar{v}_{B,c}$ and $\bar{v}_{T,1}$. The terms “horizontal” and “vertical” refer
to the positions of the levels in Figure 5, and do not imply a difference in the geometry. Both refer to fairly standard SFT degenerations and the existence of the limit $\bar{u}_{\infty}$ is a consequence of the SFT compactness discussion from Section I.7.3 with minimal change. For notational convenience we refer to $\bar{v}_1$ as $\bar{v}_{L,a+1}$, $\bar{v}_{R,0}$, $\bar{v}_{B,c+1}$, or $\bar{v}_{T,0}$, and $\bar{v}_2$ as $\bar{v}_{L,0}$ or $\bar{v}_{R,b+1}$. As usual, we write $p_* = \deg \bar{v}'_*$.

*Terminology.* Thin counterclockwise sectors in $\mathbf{C}$ from $\bar{b}_{i,j}$ to $\bar{a}_{i,j}$ and from $\bar{h}(\bar{b}_{i,j})$ to $\bar{h}(\bar{a}_{i,j})$ will also be referred to as thin sectors.

*Outline of proof of Lemma 3.3.5.* The initial steps of the proof are similar to those of Section I.7.4–I.7.11 and Section 3.4. However, the authors were unable to prove that $I(\bar{v}_{L,j})$ and $I(\bar{v}_{R,j})$ were nonnegative when $\bar{v}'_1 \neq \emptyset$, since we could not sufficiently control the groomings in order to apply the ECH index inequality (Lemma I.5.7.21). Note that Lemma 3.5.2(4) specifically excludes the case $\bar{v}'_1 \neq \emptyset$. Sections 3.5.3–3.5.6 are intended as a substitute, and involve ideas from tropical geometry (see for example Parker [Pa]).

**3.5.1. Continuation argument.** — We discuss the continuation argument in the current case; this is similar to but more complicated than those of Sections 3.4.1 and 3.4.3. Suppose that $\bar{v}'_* \cup \bar{v}^*_* \neq \emptyset$ for some level $\bar{v}_*$ of $\bar{u}_\infty$. For simplicity we assume that there are no boundary points at $z_\infty$; we leave it to the reader to make the appropriate modifications when there are boundary points at $z_\infty$.

*Case 1.* Suppose that $\bar{v}'_{T,j} \cup \bar{v}^*_{T,j} \neq \emptyset$ for some $j > 0$. We start at a nontrivial negative end $\mathcal{E}_1$ of some $\bar{v}_{T,j1}$, $j_1 > 0$, limiting to $z_\infty$. We consider the continuation

$$g_{j_1-1,1}^1, \dots, g_{1,1}^1, g_{-a-1,1}, \dots, g_{b+1,1}, g_{1,1}^0, \dots, g_{d,1}^0$$

of the $t=1$ boundary of $\mathcal{E}_1$ in the direction of $\partial_+ B_\tau$. Here Definition 3.4.5 needs to be adapted in the obvious way to the breaking of a component of $\bar{u}_i|_{\partial \check{G}_i}$ as $\tau_i \to -\infty$, where $\check{G}_i$ is the domain of $\bar{u}_i$. The components $g_{-a-1,1}$ and $g_{b+1,1}$ correspond to $\bar{v}_1$ and the components $g_{j,1}$, $j = -a, \dots, b$, correspond to $\bar{v}_{L,j}$, $j = a, a-1, \dots, 0$, and $\bar{v}_{R,j}$, $j = b, b-1, \dots, 1$, in that order. There are three possibilities:

(i) there is some $g_{j,1}^1$, $0 \le j \le j_1 - 1$, which is nontrivial;

(ii) all the $g_{j,1}^1$ are trivial but some $g_{j,1}$ is nontrivial;

(iii) all the $g_{j,1}^1$ and $g_{j,1}$ are trivial.

Cases (i) and (iii) have already been treated in the proof of Lemma 3.4.7. If we are in Case (ii), then the nontrivial component $g_{j,1}$ contains the $s=2$ boundary of a right end $\mathcal{E}_2$ that limits to $z_\infty$. We then consider the continuation of the $s=-2$ boundary of $\mathcal{E}_2$ in the direction of $\partial_- B_\tau$. The details of the continuation are left to the reader, but in the end the sectors $\pi_{D_{\bar{p}_0}^2}(\mathcal{E}_i)$ will sweep out a neighborhood of $z_\infty$ with the exception of thin sectors.
Case 2. Suppose that $\bar{v}'_{T,j} \cup \bar{v}^\sharp_{T,j} = \emptyset$ for all $j > 0$. Then $\bar{v}'_1 = \emptyset$. Consider the horizontal levels:

$$ (3.5.2) \qquad \bar{v}_1 = \bar{v}_{R,0}, \dots, \bar{v}_{R,b}, \bar{v}_2, \bar{v}_{L,1}, \dots, \bar{v}_{L,a+1} = \bar{v}_1. $$

Suppose that $\bar{v}'_* \cup \bar{v}^\sharp_* \neq \emptyset$ for some horizontal level $\bar{v}_*$. Let $\bar{v}_*$ be the leftmost level in Equation (3.5.2) such that $\bar{v}^\sharp_*$ has a right end $\mathcal{E}_1$ at $z_\infty$. The sector $\pi_{D_0^2}(\mathcal{E}_1)$ is not a thin sector and we apply the usual continuation argument to the levels of Equations (3.5.2).

Case 3. The case where $\bar{v}'_{T,j} \cup \bar{v}^\sharp_{T,j} = \emptyset$ for all $j > 0$, $\bar{v}'_* \cup \bar{v}^\sharp_* = \emptyset$ for all horizontal levels $\bar{v}_*$, and $\bar{v}'_{B,j} \cup \bar{v}^\sharp_{B,j} \neq \emptyset$ for some $j \le c$ is treated similarly.

Let $\mathcal{Z} = (\zeta_1 \to \dots \to \zeta_k \to \zeta_1)$ be the cycle constructed using the above continuation argument. Boundary points of type $(P_1)$, $(P_2)$, and $(P_3)$ are defined in the same way: for $(P_2)$ all the vertices of $\mathcal{Z}$ lie on arcs of the same type (Type $\bar{a}$, Type $\bar{b}$, Type $\bar{h}(\bar{a})$, or Type $\bar{h}(\bar{b})$), and for $(P_3)$ the vertices of $\mathcal{Z}$ must lie on arcs of more than one type.

**3.5.2.** *Some restrictions on $\bar{u}_\infty$. — The following are analogs of Lemmas 3.4.16, 3.4.20, 3.4.21 and 3.4.22.*

**Lemma 3.5.1.** — Suppose that $\bar{v}'_* \cup \bar{v}^\sharp_* \neq \emptyset$ for some level $\bar{v}_*$ of $\bar{u}_\infty$. If $\mathcal{E}_i$, $i = 1, \dots, q$, are the ends of all the $\bar{v}^\sharp_*$ that converge to $z_\infty$ and $\mathcal{E}'_i$, $i = 1, \dots, r$ are the neighborhoods of the boundary points of type $(P_3)$, then

$$ \sum_{i=1}^{q} n^*(\mathcal{E}_i) + \sum_{i=1}^{r} n^*(\mathcal{E}'_i) \geq m - 2g. $$

**Lemma 3.5.2.** — If fiber components are removed from $\bar{u}_\infty$ and the only boundary points at $z_\infty$ are of type $(P_3)$, then the following hold:

(1) the ECH index of each $\bar{v}''_*$ is nonnegative;

(2) the only components of $\bar{u}_\infty$ which have negative ECH index are those of $\bar{v}'_1$, i.e., the branched covers of $\sigma_\infty^{-\infty,1}$;

(3) the ECH index of each level $\bar{v}_{T,j}, \bar{v}_{B,j} \neq \bar{v}_1$ is nonnegative;

(4) if $\bar{v}'_1 = \emptyset$, then the ECH index of each $\bar{v}_{L,j}, \bar{v}_{R,j} \neq \bar{v}_1, \bar{v}_2$ is nonnegative;

(5) there is an additional contribution of $\mathfrak{b}\mathfrak{p}_* + 1$ towards I, where $\mathfrak{b}\mathfrak{p}_*$ is the number of boundary points of type $(P_3)$ on $\bar{v}_*$.

*Proof.* — (1) and (2) are consequences of the index inequality. (3) and (4) follow from the proof of Lemma 3.4.14. In the proof of (4), the vertical levels $\bar{v}_{1,a}, \dots, \bar{v}_{1,0}$ and the arcs $\bar{a}, \bar{h}(\bar{a})$ are replaced by the horizontal levels given by Equation (3.5.2) and the arcs $\bar{b}, \bar{a}$. (5) is argued in the same way as Lemma 3.4.20. $\square$
**Lemma 3.5.3.** — If $\bar{u}_{\infty} \in \partial_{(-\infty)} \mathcal{M}$ and $\bar{v}_{*}' \cup \bar{v}_{*}^{\sharp} = \emptyset$ for all levels $\bar{v}_{*}$ of $\bar{u}_{\infty}$, then $\bar{u}_{\infty}$ satisfies the following: $a = b = c = d = 0$; $I(\bar{v}_{1}) = \text{ind}(\bar{v}_{1}) = 0$ and $I(\bar{v}_{2}) = \text{ind}(\bar{v}_{2}) = 2$; and $\bar{v}_{1}$ is a $W_{-\infty,1}$-curve and $\bar{v}_{2}$ is a $\bar{W}_{-\infty,2}$-curve.

*Proof.* — Similar to that of Lemma 3.4.21 and is omitted. $\square$

**Lemma 3.5.4.** — If $\bar{v}_{*}' \cup \bar{v}_{*}^{\sharp} \neq \emptyset$ for some level $\bar{v}_{*}$ of $\bar{u}_{\infty}$, then:

(1) $p_{2} = \text{deg}(\bar{v}_{2}') > 0$;

(2) there is no boundary point of type $(P_{1})$ or $(P_{2})$, i.e., the only boundary points at $z_{\infty}$ are of type $(P_{3})$;

(3) $\bar{u}_{\infty}$ has no fiber components and no components $\bar{v}_{*}''$ that intersect the interior of a section at infinity;

(4) each of $\bar{v}_{L,j}, j = 1, \dots, a$, and $\bar{v}_{R,j}, j = 1, \dots, b+1$, consists of thin strips and trivial strips; in particular, $\bar{v}_{L,j}, j = 1, \dots, a$, and $\bar{v}_{R,j}, j = 1, \dots, b+1$ have no boundary points at $z_{\infty}$.

*Proof.* — (1), (2), (3) We have the following contributions towards $n^{*}$: (1) the restriction of $\bar{v}_{2}''$ to a neighborhood of $\bar{m}(-\infty)$ contributes $m$ if $\bar{v}_{2}' = \emptyset$; (2) boundary points of type $(\bar{P}_{1})$ or $(\bar{P}_{2})$ contribute at least $m$ in total; and (3) a fiber component or a component of $\bar{v}_{*}''$ that intersects a section at infinity contributes at least $m$. All cases contradict Lemma 3.5.1.

(4) Arguing by contradiction, suppose that some level $\bar{v}_{R,j_{0}}$, $1 \le j_{0} \le b$, has a component which is not a thin strip or a trivial strip. (The case of $\bar{v}_{L,j_{0}}$, $0 \le j_{0} \le a$, only differs in notation.) Here $\bar{v}_{R,j_{0}}$ may have a boundary point of type $(\bar{P}_{3})$.

We claim that the following hold:

(a) $n^{*}(\bar{v}_{R,j_{0}}) = m$ and $n^{*,alt}(\bar{v}_{R,j_{0}}) \ge m - 2g$;

(b) $\bar{v}_{R,j_{0}}^{\sharp} \neq \emptyset$ and some end of $\bar{v}_{R,j_{0}}^{\sharp}$ corresponds to a large sector;

(c) $\bar{v}_{1}' = \emptyset$ and $\bar{v}_{T,j}^{\sharp} = \emptyset$ for all $j > 0$;

(d) no left end of $\bar{v}_{R,j_{0}}$ limits to a multiple of $z_{\infty}$; in particular $\bar{v}_{R,j_{0}}^{\prime} = \emptyset$;

(e) there are no boundary points of type $(\bar{P}_{3})$;

(f) the ECH index of each $\bar{v}_{L,j}$, $1 \le j \le a$, and $\bar{v}_{R,j}$, $1 \le j \le b+1$ is nonnegative and $I(\bar{v}_{R,j_{0}}) \ge 2$.

Recall the definition of $n^{*,alt}$ from Section 3.2.3.

We first prove (a). Consider the projection

$$ \pi_{\bar{S}} : [-2, 2] \times \mathbf{R} \times \bar{S} \to \bar{S}. $$

Since we are only dealing with compactness issues, we may assume without loss of generality that $\bar{J}_{-\infty,2}^{\Delta}(\epsilon, \delta, \mathfrak{p}(-\infty))$ is a product complex structure and the projection $\pi_{\bar{S}}$ is holomorphic. Since $\bar{v}_{R,j_{0}}$ has a component which is not a thin strip or a trivial strip,
$\mathrm{Im}(\pi_S \circ \bar{v}_{R,j_0})$ must contain the complement of all the thin strips between the $\bar{b}_i$ and the $\bar{a}_i$. This proves (a).

We now prove (b)–(f). If (c) does not hold, then there is some negative end $\mathcal{E}$ of $\cup_{j=1}^a \bar{v}_{T,j}^\sharp$ that limits to $z_\infty$ and $n^*(\mathcal{E}) \ge k_0 - 1 \gg 2g$. This is a contradiction of (a). If (d) does not hold, i.e., a left end of $\bar{v}_{R,j_0}$ limits to a multiple of $z_\infty$, then in view of (c) there is some right end $\mathcal{E}$ of $\bar{v}_*^\sharp$ to the left of $\bar{v}_{R,j_0}$ that limits to $z_\infty$ and satisfies $n^*(\mathcal{E}) \ge k_0 - 1 \gg 2g$. This is a contradiction of (a). (e) is a consequence of (d). (b) follows from (a) and (e) by excluding some possibilities using (2), (3). Finally we prove (f). By Lemma 3.5.2(4) and (c), the ECH indices of $\bar{v}_{L,j}$, $j = 1, \dots, a$, and $\bar{v}_{R,j}$, $j = 1, \dots, b+1$, are nonnegative. Since $\bar{v}_{R,j_0}^\sharp$ has a large sector by (b), its ECH index is increased by one. Hence $I(\bar{v}_{R,j_0}) \ge 2$.

We now return to the proof of (4). By (d), the right end of $\bar{v}_{R,j_0}$ limits to a multiple of $z_\infty$. By (a), each component of $\bar{v}_*^\sharp \ne \bar{v}_1^\sharp$ to the right of $\bar{v}_{R,j_0}$ must be a thin strip and the projection to $\bar{S}$ of the union of all the ends of $\bar{v}_1^\sharp$ limiting to $z_\infty$ is a union of thin wedges of type $\mathfrak{S}(\bar{b}_{i,j}, \bar{a}_{i,j})$. Using Figure 6 one can verify that such a curve $\bar{v}_1^\sharp$ does not exist. This implies that $\bar{v}_1^\sharp = \emptyset$ and that the ECH index of $\bar{v}_1$ is $\ge 0$. Finally, since the ECH index of each thin strip is 1, the analog of Equation (3.4.3) for our case, Lemma 3.5.2, and (f) together imply that no component of $\bar{v}_* \ne \bar{v}_1$ to the right of $\bar{v}_{R,j_0}$ can be a thin strip. This is a contradiction and (4) follows. $\square$

**3.5.3. Truncations.** — In the rest of Section 3.5 until the proof of Lemma 3.5.13, we consider the case where $\bar{v}_*^\prime \cup \bar{v}_*^\sharp \ne \emptyset$ for some $\bar{v}_*$. We have $\bar{v}_2^\prime \ne \emptyset$ in view of Lemma 3.5.4(1).

In Sections 3.5.3–3.5.5 we consider the case $\bar{v}_1^\prime = \emptyset$. By Lemma 3.5.4(4), $\bar{u}_\infty$ has no boundary points at $z_\infty$.

**Definition 3.5.5.** — Let $X$ be set and $\varepsilon > 0$ be a positive real number. Then two functions $f, g: X \to \mathbf{C}^\times$ are $\varepsilon$-approximate if

$$|f(x) - g(x)| < \varepsilon \cdot |g(x)| \quad \text{and} \quad |f(x) - g(x)| < \varepsilon \cdot |f(x)|$$

for all $x \in X$.

We now define the following sequence of truncations, analogous to those which appear in the proof of Theorem I.7.10.1.

**Definition 3.5.6.** — With respect to the above assumptions on $\bar{u}_i : \dot{G}_i \to \bar{W}_{\tau_i}$, an $\varepsilon_i$-truncation of $\bar{u}_i$ with $0 < \varepsilon_i < \frac{\rho_0}{2}$ is the restriction of $\bar{u}_i$ to a subsurface $\Sigma_i \subset \dot{G}_i$ which satisfies the following: First write

$$\partial_v \Sigma_i = \partial \Sigma_i - (\pi_{B_{\tau_i}} \circ \bar{u}_i)^{-1}(\partial B_{\tau_i})$$
$$ \partial_h \Sigma_i = \partial \Sigma_i \cap (\pi_{B_{\tau_i}} \circ \bar{u}_i)^{-1}(\partial B_{\tau_i}). $$

Then

(T1) $\bar{u}_i(\Sigma_i)$ is contained in a $2\epsilon_i$-neighborhood of $\sigma_{\infty}^{\tau_i}$;

(T2) if $\bar{u}_i(x)$ is contained in an $\frac{\epsilon_i}{2}$-neighborhood of $\sigma_{\infty}^{\tau_i}$, then $x \in \Sigma_i$;

(T3) $\pi_{B_{\tau_i}} \circ \bar{u}_i$ maps each component $c$ of $\partial_v \Sigma_i$ to some $t = \text{const}$ and $\partial_h \Sigma_i$ to $\{s = \pm 2\}$;

(T4) there exist constants $r(\tau_i) - 2 > R_i^{(1)} > \dots > R_i^{(t+1)} > 2$ and a decomposition $\Sigma_i = \Sigma_i^{(1)} \cup \dots \cup \Sigma_i^{(t)}$ such that

$$ \Sigma_i^{(j)} = (\pi_{B_{\tau_i}} \circ \bar{u}_i)^{-1}([-2, 2] \times [R_i^{(j+1)}, R_i^{(j)}]) \cap \Sigma_i $$

and each

$$ \pi_{B_{\tau_i}} \circ \bar{u}_i : \Sigma_i^{(j)} \to [-2, 2] \times [R_i^{(j+1)}, R_i^{(j)}] $$

is a branched cover with possibly empty branch locus;

(T5) for each component $c$ of $\partial_v \Sigma_i$, $\pi_{D_{\rho_0}^2} \circ \bar{u}_i|_c$ is $\frac{\epsilon_i}{10}$-approximate to a positive multiple $\epsilon_i$ of a normalized eigenfunction of $\bar{v}_{L,j}^{\sharp}, j = 0, \dots, a$, or $\bar{v}_{R,j}^{\sharp}, j = 0, \dots, b$; moreover, for all $c$, $\max_c |\pi_{D_{\rho_0}^2} \circ \bar{u}_i| = \epsilon_i$.

Here

$$ \pi_{D_{\rho_0}^2} : \pi_{B_{\tau_i}}^{-1}([-2, 2] \times [R_i^{(t+1)}, R_i^{(1)}]) \cap \{\rho \le \rho_0\} \to D_{\rho_0}^2 $$

is obtained by projecting out the $\partial_s$- and $\bar{R}_{\tau_i}$-directions.

From now on we assume that $\bar{u}_i|_{\Sigma_i}$ is a sequence of $\epsilon_i$-truncations, where $\epsilon_i \to 0$ and $R_i^{(1)} - R_i^{(t+1)} \to \infty$ as $i \to \infty$.

Let $\tilde{\pi}_i$ be the map obtained by postcomposing

$$ \pi_{B_{\tau_i}} \circ \bar{u}_i : \Sigma_i \to [-2, 2] \times [R_i^{(t+1)}, R_i^{(1)}] $$

with a $-\frac{r(\tau_i)+1}{2}$-translation in the $t$-direction, and let $\tilde{w}_i = \pi_{D_{\rho_0}^2} \circ \bar{u}_i|_{\Sigma_i}$. Also let

$$ R_i^+ = R_i^{(1)} - \frac{r(\tau_i) + 1}{2}, \quad R_i^- = R_i^{(t+1)} - \frac{r(\tau_i) + 1}{2}. $$

**Notation.** When we want to distinguish the $t$-coordinates for $B_{-\infty,1}$ and $B_{-\infty,2}$, we write $t_i$ for the $t$-coordinate for $B_{-\infty,i}$. Note that $\tilde{\pi}_i$ can be viewed as a map to $B_{-\infty,2}$ with $(s, t_2)$-coordinates.

**Lemma 3.5.7.** — If $i \gg 0$, then $\Sigma_i$ is a disk with $\ge 2p_2$ boundary punctures.

*Proof.* — This is a consequence of (T5) and the following:
(i) $n^*(\bar{u}_i|_{\Sigma_i}) = m$;

(ii) $\tilde{w}_i(z_0) = 0$ for some $z_0 \in (\pi_{B\tau_i} \circ \bar{u}_i)^{-1}(\bar{m}^b(\tau_i))$;

(iii) $\tilde{w}_i$ maps each component of $\tilde{\pi}_i^{-1}(\{s=2\})$ to a different $\mathcal{R}_{\phi(\bar{a}_*)}$ and each component of $\tilde{\pi}_i^{-1}(\{s=-2\})$ to a different component of $\mathcal{R}_{\phi(\bar{b}_*)}$;

(iv) $\tilde{w}_{i|_{\text{int}(\Sigma_i)}}$ is a biholomorphism onto its image.

Here $\phi(\bar{a}_*)$ is the $\phi$-coordinate for $\bar{a}_*$, etc. $\square$

**3.5.4. Large-scale behavior of $\Xi_i$.** — When taking the limits of $\tilde{\pi}_i$ and $\tilde{w}_i$, we are simultaneously stretching in two directions $t_2$ and $\log \rho$. In this subsection we study the large-scale behavior of the map

$$ \Xi_i = (t_2 \circ \tilde{\pi}_i, \log \rho \circ \tilde{w}_i) : \Sigma_i \to [R_i^-, R_i^+] \times [-\infty, \infty) $$

as $i \to \infty$. We use coordinates $(x' = t_2, y')$ on $[R_i^-, R_i^+] \times [-\infty, \infty)$. The goal is to construct a “tropical curve”

$$ \Xi_i : \Gamma_i \to [-1, 1] \times [0, d_i], $$

with some $d_i \ge 1$ which approximates $\Xi_i$ when viewed from “far away”; see Figure 12. Here $\Gamma_i$ is a finite graph whose topological type is independent of $i \gg 0$. The analysis is of the same type as that of Parker [Pa].

**Step 1.** We start with the following lemma, which is a consequence of Gromov compactness and which describes the behavior of $\tilde{w}_i$ and $\tilde{\pi}_i$ for large $i$.

**Lemma 3.5.8.** — Given $\varepsilon > 0$ small, after passing to a subsequence and possibly shrinking $\varepsilon_i > 0$ subject to the condition $R_i^{(1)} - R_i^{(t+1)} \to \infty$, there exist constants $L > 0, \kappa, \kappa' \in \mathbb{Z}^+$ such that for each $i$ there exist:

- disjoint compact subsurfaces $K_{i1}, \dots, K_{ik} \subset \Sigma_i$ and

- components $C_{i1}, \dots, C_{ik'}$ of $\Sigma_i - \cup_j K_{ij}$ which are strips

such that:

(1) $\tilde{\pi}_i|_{K_{ij}}$ is a branched cover over $[-2, 2] \times [\tau_{ij}, \tau_{ij} + L]$ for some $\tau_{ij}$ and $\tilde{\pi}_i$ has no branch points outside $K_{i1}, \dots, K_{ik}$;

(2) $K_{ij}$ is disjoint from $\partial_v \Sigma_i$;

(3) there is a component $K_{ij0}$ such that $(0, -\infty) \in \Xi_i(K_{ij0})$;

(4) for each $j$, the sequence $\{\tilde{w}_i|_{K_{ij}}\}_{i=1}^{\infty}$, after rescaling by positive constants, limits to some $\tilde{w}_{\infty j} : K_{\infty j} \to \mathbf{C}$;

(5) $\tilde{w}_i|_{C_{ij}}$ is $\varepsilon$-approximate to a multiple of $e^{\lambda_{ij}(t-is)}$, where $\lambda_{ij} \in \mathbb{R} - \{0\}$ is $\pm \frac{1}{4}$ times the angle of a sector of type $\mathfrak{S}(\bar{b}_{k,l}, \bar{a}_{k',l'})$ (here $\pm \frac{1}{4}$ comes from the fact that $s \in [-2, 2]$);

(6) $\Xi_i(C_{ij})$ is $\varepsilon$-close to a line segment $\{y' = \lambda_{ij}x' + \beta_{ij} | x' \in t_2 \circ \tilde{\pi}_i(C_{ij})\}$, where $\beta_{ij}$ is a constant;
(7) there exists $d'_i \in \mathbf{R}$ such that, for each component $c$ of $\partial_v \Sigma_i$, $y' \circ \Xi_i(c)$ is $\varepsilon$-close to $d'_i$ and $\max_c y' \circ \Xi_i(c) = d'_i$.

*Proof.* — (1)–(5) follow from Gromov compactness, once we observe using the ECH compactness theorem (cf. Section I.3.4) that there is an upper bound on the number of branch points of $\tilde{\pi}_i$ which is independent of $i$ and $m \gg 0$. (6) follows from (5). (7) is a consequence of (6) and (T5) in Definition 3.5.6. $\square$

*Step 2.* We now construct the “tropical curve” $\overline{\Xi}_i : \Gamma_i \to [-1, 1] \times [0, d_i]$, where $(x, y)$ are coordinates on $[-1, 1] \times [0, d_i]$. We first define the map

$$ \Xi'_i : \Gamma_i \to [\mathbb{R}_i^-, \mathbb{R}_i^+] \times [-\infty, \infty), $$

where $\Gamma_i = (V_{\Gamma_i}, E_{\Gamma_i})$, $V_{\Gamma_i} = V_{\Gamma_{i,i}} \cup V_{\Gamma_{i,e}}$ is the set of vertices, $E_{\Gamma_i}$ is the set of edges, and the following hold:

(1) $V_{\Gamma_{i,i}}$ is in one-to-one correspondence with the set $\{K_{i1}, \dots, K_{ik}\}$ and $V_{\Gamma_{i,e}}$ is in one-to-one correspondence with the components of $\partial_v \Sigma_i$;

(2) the set $E_{\Gamma_i}$ is in one-to-one correspondence with the set $\{C_{i1}, \dots, C_{ik'}\}$;

(3) $\Xi'_i$ maps the vertices corresponding to $K_{ij}$ and the component $c$ of $\partial_v \Sigma_i$ to

$$ (x', y') = (\tau_{ij}, \max(\log \rho \circ \tilde{w}_i|_{K_{ij}})) \text{ and } (t_2 \circ \tilde{\pi}_i(c), \max(\log \rho \circ \tilde{w}_i|_c)); $$

and each $e \in E_{\Gamma_i}$ to a straight line segment.

Note that $\Xi'_i$ has image in $\{y'_{ij0} \le y' \le d'_i\}$, where $y'_{ij0} = \max(\log \rho \circ \tilde{w}_i|_{K_{ij0}})$. The map $\overline{\Xi}_i$ is obtained by postcomposing $\Xi'_i : \Gamma_i \to [\mathbb{R}_i^-, \mathbb{R}_i^+] \times [y'_{ij0}, d'_i]$ by an affine transformation

$$ [\mathbb{R}_i^-, \mathbb{R}_i^+] \times [y'_{ij0}, d'_i] \xrightarrow{\sim} [-1, 1] \times [0, d_i], $$

where $d_i > 0$ is chosen so that $\max_{e \in E_{\Gamma_i}} |\lambda'_i(e)| = 1$, where $\lambda'_i : E_{\Gamma_i} \to \mathbf{R}$ maps $e$ to the slope of $\overline{\Xi}_i(e)$.

**Lemma 3.5.9.** — Fix $\varepsilon, \delta > 0$ small. Then, after passing to a subsequence, the map $\overline{\Xi}_i$ satisfies the following:

(1) $\max_{e \in E_{\Gamma_i}} |\lambda'_i(e)| = 1$;

(2) if $\lambda_i : E_{\Gamma_i} \to \mathbf{R}$ maps $e \mapsto \lambda_{ij}$, where $e$ corresponds to $C_{ij}$ and $\lambda_{ij}$ is as in Lemma 3.5.8(5), then $\lambda'_i$ and a constant multiple of $\lambda_i$ are $\varepsilon$-approximate;

(3) if $E'_{\Gamma_i} \subset E_{\Gamma_i}$ is the set of edges $e$ such that $|\lambda'_i(e)| < 1 - \delta$, then $|\lambda'_i(e)| \le K/m$ for some constant $K > 0$ which is independent of $m \gg 0$ and $i$;

(4) each vertex of $V_{\Gamma_{i,i}}$ has the same number $\ge 1$ of adjacent edges whose interiors have larger x-coordinate and whose interiors have smaller x-coordinate;
<figure><img src="image_18.png" /><figcaption>FIG. 12. — The top figure represents $[-2, 2] \times [R_i^-, R_i^+]$, where $\times$ indicates the location of $\bar{m}^t(\tau_i)$ after translation and the shaded regions are $K_{ij}$. The bottom is a schematic diagram for the image of $\Xi_i$ which becomes the image of $\bar{\Xi}_i$ “when seen from far away”. The dots along the line $y' = d'_i$ correspond to the endpoints of $V_{\Gamma_i, e}$ and the shaded regions are $t_2(K_{ij}) \times \log \rho(\tilde{w}_i \circ \tilde{\pi}_i^{-1}(K_{ij}))$. When viewed in $[-1, 1] \times [0, d_i]$, the edges with the largest slope (in absolute value) satisfy $|\lambda'_i| \approx 1$ and the remaining edges have much smaller slope when $m \gg 0$. (Color figure online)</figcaption></figure>

$$
(5) \quad y \circ \bar{\Xi}_i(p) = d_i \text{ for all } p \in V_{\Gamma_i, e};
$$

$$
(6) \text{ there is a vertex } q_0 \in V_{\Gamma_i, i} \text{ corresponding to } K_{ij_0} \text{ (cf. Lemma 3.5.8(3)) such that} \\
y \circ \bar{\Xi}_i(q_0) = 0;
$$

$$
(7) \text{ there is a path } e_0 e_1 \dots e_{l-1} \text{ consisting of edges such that } \lambda'_i(e_j) \ge 1 - \delta \text{ for } j = 0, \dots, l-1, \\
e_0 \text{ starts at } q_0, \text{ and } e_{l-1} \text{ ends near } (1, d_i) \text{ or } (-1, d_i), \text{ and } d_i > 1 - \delta.
$$

Proof. — (1), (4), (5), and (6) follow from the construction. (2) is a consequence of Lemma 3.5.8(5) and (3) follows from (2). (7) follows from the construction, Lemma 3.5.8(5), and the fact that $\overline{v_1^\sharp}$ must have a large sector by Lemma 3.5.4. $\square$

See Figure 12 for an example. The following lemma is immediate from the con-
struction.

*Lemma 3.5.10.* — After passing to a subsequence, we may assume that the following do not depend on the choice of *i*:

- the graph $\Gamma_i$, the function $\lambda_i$, and the set $E_{\Gamma_i}'$;

- given an edge $E_{\Gamma_i}$ with endpoints $p, q$, whether $x \circ \bar{\Xi}_i(p) \ge x \circ \bar{\Xi}_i(q)$ and whether $y \circ \bar{\Xi}_i(p) \ge y \circ \bar{\Xi}_i(q)$.

In view of Lemma 3.5.10, we may write $\Gamma = (V_\Gamma, E_\Gamma)$ for $\Gamma_i = (V_{\Gamma_i}, E_{\Gamma_i})$.
<figure><img src="image_19.png" /><figcaption>FIG. 13. — Schematic diagrams for the possible types of degenerations corresponding to Lemmas 3.5.12 and 3.5.13. Here ◦ represents $z_∞$ or $z'_∞$, □ represents a branch point, and △ represents an end with a large sector. Double dotted lines indicate multiple covers of $σ'_∞$. The labels on the graphs indicate the components and their ECH indices. If there is more than one □ or △ in a diagram, then we interpret it as one of the possible locations for □ or △</figcaption></figure>

Finally, we orient the edges $p \xrightarrow{\epsilon} q \in E_Γ$ so that $y \circ \overline{\Xi}_i(q) - y \circ \overline{\Xi}_i(p) > 0$; we denote the corresponding orientation by $\mathbf{o}$. Given $p, q \in V_Γ$, we write $p \ge q$ (resp. $p \simeq q$) to mean $y \circ \overline{\Xi}_i(p) - y \circ \overline{\Xi}_i(q) \ge 0$ (resp. $= 0$) for all $i$.

**3.5.5.** *The case $\overline{v}'_1 = \emptyset, \overline{v}'_2 \neq \emptyset$. — We continue to assume that $\overline{v}'_1 = \emptyset, \overline{v}'_2 \neq \emptyset$. See Figure 13(1).* We write $\approx$ to mean “is close to”.

We start with the following useful lemma:

**Lemma 3.5.11 (Comparison Lemma).** — Suppose $m \gg 0$, $i = i(m) \gg 0$, and $\epsilon = \epsilon(m) > 0$ is small. Let $q \xrightarrow{\epsilon} q'$ be an edge of $\Gamma$ such that $|\lambda'_i(e)| \approx 1$ and let

$$ \delta' = (p_1 \xrightarrow{f_1} \dots \xrightarrow{f_k} p_{k+1}) $$

be an oriented path of $\Gamma$ from $p_1$ to $p_{k+1}$ such that $p_1 \le q \le q' \le p_{k+1}$ and $0 < |\lambda'_i(f_j)| \le K/m$ for $j = 1, \dots, k$. Then

$$ |x \circ \overline{\Xi}_i(q') - x \circ \overline{\Xi}_i(q)| \le K'/m, $$

where $K'$ is independent of $m$ and $i$.

Note that, by Lemma 3.5.9(3), $0 < |\lambda'_i(f_j)| \le K/m$ if and only if $|\lambda'_i(f_j)|$ is not close to 1.

*Proof.* — The horizontal variation $\sum_{j=1}^k |x \circ \overline{\Xi}_i(p_{j+1}) - x \circ \overline{\Xi}_i(p_j)|$ is bounded above by 2 times the maximal covering degree of $\overline{v}'_{L,j}, j = 1, \dots, a$, and $\overline{v}'_{R,j}, j = 1, \dots, b+1$, which we denote by $\mathcal{K}$. (Here the 2 comes from the width of the interval $[-1, 1]$.) On the other hand, since $p_1 \le q \le q' \le p_{k+1}$, we have:

$$ |x \circ \overline{\Xi}_i(q') - x \circ \overline{\Xi}_i(q)| \approx |y \circ \overline{\Xi}_i(q') - y \circ \overline{\Xi}_i(q)| $$
$$
\begin{align*}
&\le \sum_{j=1}^{k} |\lambda'_{i}(f_j)| \cdot |x \circ \bar{\Xi}_{i}(p_{j+1}) - x \circ \bar{\Xi}_{i}(p_j)| \\
&\le \left( \max_{j=1, \dots, k} |\lambda'_{i}(f_j)| \right) (2\mathcal{K}) \le \mathbf{K}'/m,
\end{align*}
$$

where $\mathbf{K}' = 2\mathcal{K}\mathbf{K}$.

<figure><img src="image_20.png" /></figure>

Given $(x_0, y_0) \in [-1, 1] \times [0, d_i]$, let

$$
l(x_0, y_0) = \{x = x_0, 0 \le y \le y_0\}
$$

be a line segment oriented in the positive y-direction. Next choose an orientation $\tilde{o}$ of
$\Gamma$ so that the oriented edges of $\bar{\Xi}_i(\Gamma)$ are pointing in the positive x-direction and write
$\tilde{\Gamma} = (\Gamma, \tilde{o})$. (This is different from the orientation $o$ used previously.)

Consider the *weight function*

$$
W_i : ([-1, 1] \times [0, d_i]) - \bar{\Xi}_i(\Gamma) \to \mathbf{Z}^{\ge 0},
$$

which is defined as follows:

(1) $W_i$ is locally constant;

(2) $W_i(x,y)$ is the signed intersection number $\langle \bar{\Xi}_i(\tilde{\Gamma}), l_{(x,y)} \rangle$ for generic $(x,y)$, with respect to the orientations $(\partial_x, \partial_y)$ for $[-1, 1] \times [0, d_i]$ and $\tilde{o}$.

The function $W_i$ is well-defined by Lemma 3.5.9(4).

**Lemma 3.5.12.** — If $m \gg 0$, then there is no $\bar{u}_\infty \in \partial_{\{-\infty\}} \mathcal{M}$ such that $\bar{v}'_1 = \emptyset$ and $\bar{v}'_2 \neq \emptyset$.

*Proof*. — The proof we give is not the most efficient, but carries over more easily to other situations. We choose $m \gg 0$ so that $0 < \mathbf{K}'/m \ll 1$.

Let $\delta = (q_0 \xrightarrow{e_0} \cdots \xrightarrow{e_{l-1}} q_l)$ be the oriented path from $q_0$ to $q_l \in V_{\Gamma,e}$ given by Lemma 3.5.9(7), i.e., such that $|\lambda'_i(e_j)| \approx 1$ for each $i$ and each edge $e_j, j = 0, \dots, l-1$. We may assume that $x \circ \bar{\Xi}_i(q_l) = \pm 1$ in view of Lemma 3.5.4(4), since the only end $\mathcal{E}$ of $\bar{v}_{L,j}^{\sharp}, j = 1, \dots, a+1$, or $\bar{v}_{R,j}^{\sharp}, j = 1, \dots, b+1$, whose $\pi_{D_{\rho_0}^2}$-projection is a sector with angle $> \pi$ is an end of $\bar{v}_1^{\sharp} = \bar{v}_{L,a+1}^{\sharp}$.

We claim that, for each $l_0 = 0, \dots, l-1$, there is an oriented path

$$
\delta' = (p_1 \xrightarrow{f_1} \cdots \xrightarrow{f_k} p_{k+1})
$$

such that $p_1 \le q_{l_0} \le q_{l_0+1} \le p_{k+1}$ and $0 < |\lambda'_i(f_j)| < \mathbf{K}'/m$ for $j = 1, \dots, k$. Arguing by contradiction, if the claim does not hold, then $\bar{\Xi}_i(\Gamma) \cap \{y=y_0\}$ consists of only one point
$(x_0, y_0) \in \bar{\Xi}_i(e_{l_0}) \cap \{y = y_0\}$ for any constant $y_0$ in the interval $(y \circ \bar{\Xi}_i(q_{l_0}), y \circ \bar{\Xi}_i(q_{l_0+1}))$. This means that there is an integer $\kappa$ such that $W_i(x, y_0) = \kappa$ for $-1 \le x < x_0$ and $W_i(x, y_0) = \kappa \pm 1$ for $1 \ge x > x_0$. This contradicts $W_i(-1, y_0) = W_i(1, y_0) = 0$, which is due to the fact that all the exterior vertices $p \in V_{\Gamma_{i,e}}$ satisfy $y \circ \bar{\Xi}_i(p) = d_i$ by Lemma 3.5.9(5).

The claim, together with the Comparison Lemma, implies that

$$|x \circ \bar{\Xi}_i(q_{k+1}) - x \circ \bar{\Xi}_i(q_k)| \le K'/m$$

for each $k = 0, \dots, l-1$. Hence,

$$\sum_{k=0}^{l-1} |x \circ \bar{\Xi}_i(q_{k+1}) - x \circ \bar{\Xi}_i(q_k)| \le lK'/m \ll 1$$

for $m \gg 0$. This contradicts Lemma 3.5.9(5) with $d_i \ge 1$. $\square$

<figure><img src="image_21.png" /></figure>

**3.5.6.** The case $\bar{v}'_1 \ne \emptyset, \bar{v}'_2 \ne \emptyset$. — In this subsection we consider the case $\bar{v}'_1 \ne \emptyset, \bar{v}'_2 \ne \emptyset$. For simplicity assume that there are no boundary points of type (P$_3$). Some of the possibilities are given by Figure 13(2) and (3).

We outline the necessary modifications in the current case:

(1) We consider the truncation $\bar{u}_i : \Sigma_i \to \bar{W}_{\tau_i}$ of $\bar{u}_i$ so that (T1) and (T2) in Definition 3.5.6 hold. (T4) becomes:

(T4') there exists a decomposition $\Sigma_i = \Sigma_i^{(1)} \cup \dots \cup \Sigma_i^{(t)}$ such that each $\pi_{B_{\tau_i}} \circ \bar{u}_i|_{\Sigma_i^{(j)}}$ is a branched cover with possible empty branch locus over a component of $B_{\tau_i}$ which is cut up by (possibly multiple) arcs of type $t = \mathbb{R}$ with $2 < R < r(\tau_i) - 2$ and $s = \mathbb{R}'$ with $R' < -3$ or $R' > 3$.

(T5) is slightly modified so that the normalized eigenfunction is that of $\bar{v}_*^\sharp$ for any $*$.

(2) Given $\varepsilon > 0$ small, there exists $L > 0$ so that the analog of Lemma 3.5.8 holds; here we restrict $\Sigma_i$ to $|s| \le L$, while keeping the same notation. To the list of compact subsets $K_{ij}$ of Lemma 3.5.8 (viewed as subsets of $B_{\tau_i}$ instead of $[-2, 2] \times [R_i^-, R_i^+]$), we add the compact subset $K_{ij}$ of the following type, which we call “type $\bar{v}_1$”:

$$K_{ij} = B_{\tau_i} \cap \{|s| \le L, |t - 1/2| \le L\}.$$

(3) After suitable translations, contractions of components of $\pi_{B_{\tau_i}}^{-1}(K_{ij})$ to points, and rescalings, we obtain the “tropical curves”

$$\bar{\Xi}_i : \Gamma \to [-1, 1]/(-1 \sim 1) \times [0, d_i],$$

where the equivalence relation $\sim$ is consistent with contracting each component of $\pi_{B_{\tau_i}}^{-1}(K_{ij})$ of type $\bar{v}_1$ to a point and the graph $\Gamma = (V_\Gamma, E_\Gamma)$ is independent of $i$.

(4) The set $V_\Gamma$ of vertices admits the decomposition $V_{\Gamma,i} \sqcup V_{\Gamma,e}$, where $V_{\Gamma,i}$ is in one-to-one correspondence with the set of components of $\pi_{B_{\tau_i}}^{-1}(K_{ij})$ and $V_{\Gamma,e}$ is in one-to-one correspondence with the set of left and right ends of $\bar{v}_{L,j}^\sharp, j = 1, \dots, a+1$, and $\bar{v}_{R,j}^\sharp$,
$j=1, \dots, b+1$, that limit to $z_\infty$. There is a subset $V'_{\Gamma,i} \subset V_{\Gamma,i}$ consisting of vertices which are not initial points of any oriented edge; $V'_{\Gamma,i}$ is obtained by contracting components of $\pi_{B_{\Gamma,i}}^{-1}(K_{ij})$ of type $\bar{v}_1$ to points.

(5) In Lemma 3.5.9, (1)-(3) and (5) still hold, in (4) we replace $V_{\Gamma,i}$ by $V_{\Gamma_i} - V'_{\Gamma,i}$, and (6) becomes: there is a vertex $q_0 \in V_{\Gamma,i}$ corresponding to $K_{ij_0}$ which satisfies $\bar{\Xi}_i(q_0) = (0, 0)$.

(6) The weight function $W_i$ is now a function

$$W_i : (([-1, 1]/\sim) \times [0, d_i]) - \bar{\Xi}_i(\Gamma) \to \mathbb{Z}^{\ge 0},$$

i.e., it is periodic in the x-direction.

**Lemma 3.5.13.** — If $m \gg 0$, then there is no $\bar{u}_\infty \in \partial_{(-\infty]} \mathcal{M}$ such that $\bar{v}'_1 \neq \emptyset$ and $\bar{v}'_2 \neq \emptyset$.

*Proof.* — The proof is similar to that of Lemma 3.5.12 and uses the Comparison Lemma.

Suppose that there are no boundary points of type (P$_3$). Let $\delta = (q_0 \xrightarrow{e_0} \dots \xrightarrow{e_{l-1}} q_l)$ be a maximal oriented path which starts from $q_0$, has $|\lambda_i(e)| \approx 1$ for each edge, and ends at some $q_l$ with $x \circ \bar{\Xi}_i(q_l) = \pm 1$. We claim that, for each $l_0 = 0, \dots, l-1$, there is an oriented path

$$\delta' = (p_1 \xrightarrow{f_1} \dots \xrightarrow{f_k} p_{k+1})$$

such that $p_1 \le q_{l_0} \le q_{l_0+1} \le p_{k+1}$ and $0 < |\lambda'_i(f_j)| \le K'/m$ for $j = 1, \dots, k$. Indeed, using the same notation as that of Lemma 3.5.12, there is a point $(x_0, y_0)$ and an integer $\kappa$ such that $W_i(x, y_0) = \kappa$ for $-1 \le x < x_0$ and $W_i(x, y_0) = \kappa \pm 1$ for $x_0 \le x \le 1$, which is impossible by the x-periodicity. The claim gives a contradiction as in the proof of Lemma 3.5.12.

Suppose there are boundary points of type (P$_3$). Then, by Lemma 3.5.4(4), there are no boundary points of type (P$_3$) on $\bar{v}_{l,j}, j = 1, \dots, a$, and $\bar{v}_{R,j}, j = 1, \dots, b+1$. The maximal oriented path $\delta$ from the previous paragraph has $y \circ \bar{\Xi}_i(q_l)$ which is much larger than $y \circ \bar{\Xi}_i$ of the endpoint of another path. This is a contradiction. $\square$

*Proof of Lemma 3.3.5.* — Suppose $\bar{u}_\infty \in \partial_{(-\infty]} \mathcal{M}$. If $\bar{v}'_* \cup \bar{v}^\sharp_* = \emptyset$ for all levels $\bar{v}_*$ of $\bar{u}_\infty$, then, by Lemma 3.5.3, $\bar{u}_\infty$ is a 2-level building $\bar{v}_1 \cup \bar{v}_2$, where $\bar{v}_1$ is a $W_{-\infty,1}$-curve with $I=0$ and $\bar{v}_2$ is a $\bar{W}_{-\infty,2}$-curve with $I=2$ which passes through $\bar{m}(-\infty)$. By Lemma 3.2.15, $\mathbf{y}_2$ and $\mathbf{y}_4$ satisfy the conditions of A$_3$.

On the other hand, it is not possible that $\bar{v}'_* \cup \bar{v}^\sharp_* \neq \emptyset$ for some level $\bar{v}_*$ by Lemmas 3.5.12 and 3.5.13. $\square$

**3.6. Breaking in the middle.** — In this subsection we study the limit of holomorphic maps to $\bar{W}_\tau$ as $\tau \to T'$ for some $T' \in (-\infty, \infty)$. This will prove Lemma 3.3.8.
We assume that $m \gg 0$; $\varepsilon, \delta > 0$ are sufficiently small; and $\{\bar{J}_{\tau}\} \in \bar{\mathcal{I}}^{\text{reg}}$ and $\{\bar{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))\}$ satisfy Lemma 3.2.22. Fix $\mathbf{y} \in \mathcal{S}_{\mathbf{a},\mathbf{h}(\mathbf{a})}$, $\mathbf{y}' \in \mathcal{S}_{\mathbf{b},\mathbf{h}(\mathbf{b})}$ and let

$$ \mathcal{M} = \mathcal{M}_{\{\bar{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))\}}^{I=2, n^*=m}(\mathbf{y}, \mathbf{y}'; \bar{\mathbf{m}}), \quad \mathcal{M}_{\tau} = \mathcal{M}_{\{\bar{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))\}}^{I=2, n^*=m}(\mathbf{y}, \mathbf{y}'; \bar{\mathbf{m}}(\tau)). $$

We will analyze $\partial_{(-\infty, \infty)} \mathcal{M}$.

Let $\bar{u}_i$, $i \in \mathbf{N}$, be a sequence of curves in $\mathcal{M}$ such that $\bar{u}_i \in \mathcal{M}_{\tau_i}$ and $\lim_{i \to \infty} \tau_i = T'$, and such that its limit

$$ \bar{u}_{\infty} = (\bar{v}_{-1,1} \cup \dots \cup \bar{v}_{-1,c}) \cup \bar{v}_0 \cup (\bar{v}_{1,1} \cup \dots \cup \bar{v}_{1,a}) $$

is a holomorphic building in $\partial_{[T']}(\mathcal{M})$, ordered from bottom to top, where each $\bar{v}_*$ is an SFT level, $\bar{v}_{-1,j}, j = 1, \dots, c$, and $\bar{v}_{1,j}, j = 1, \dots, a$, map to $\bar{W}$ and $\bar{v}_0$ maps to $\bar{W}_{T'}$. Sometimes we will refer to $\bar{v}_0$ as $\bar{v}_{-1,c+1}$ or $\bar{v}_{1,0}$.

The following are analogs of Lemmas 3.4.20–3.4.22, stated without proof.

**Lemma 3.6.1.** — If fiber components are removed from $\bar{u}_{\infty}$ and the only boundary points at $z_{\infty}$ are of type $(P_3)$, then the ECH index of each level $\bar{v}_* \neq \bar{v}_0$ is nonnegative,

$$ I(\bar{v}_{1,j}) \ge \begin{cases} I(\bar{v}'_{1,j}) + I(\bar{v}''_{1,j}) & \text{if } \mathfrak{b}\mathfrak{p}_{1,j} = 0; \\ I(\bar{v}'_{1,j}) + I(\bar{v}''_{1,j}) + 2 & \text{if } \mathfrak{b}\mathfrak{p}_{1,j} > 0, \end{cases} $$

for $0 \le j \le a$, and the only components of $\bar{u}_{\infty}$ which have negative ECH index are the following:

(1) branched covers of $\sigma_{\infty}^{T'}$; and

(2) at most one component $\tilde{v}$ of $\bar{v}_0''$ with $I(\tilde{v}) = -1$.

Here (2) occurs when $T' \in \mathcal{T}_1$ and $\tilde{v} \in \mathcal{M}_{\{\bar{J}_{T'}^{\diamond}(\varepsilon, \delta, \mathbf{p}(T'))\}}^{\dagger, s, \text{irr}, \text{ind}=-1}(\mathbf{z}, \mathbf{z}')$, as described in Lemma 3.2.22(1).

**Lemma 3.6.2.** — If $\bar{u}_{\infty} \in \partial_{(-\infty, \infty)} \mathcal{M}$ and $\bar{v}'_* \cup \bar{v}''_* = \emptyset$ for all levels $\bar{v}_*$ of $\bar{u}_{\infty}$, then $\bar{u}_{\infty}$ is one of the following:

(1) $a = 0, c = 1; \bar{v}_0$ is a $\bar{W}_{T'}$-curve with $I = 1$ which passes through $\bar{\mathfrak{m}}(T')$; and $\bar{v}_{-1,1}$ is a W-curve with $I = 1$; or

(2) $a = 1, c = 0; \bar{v}_{1,1}$ is a W-curve with $I = 1$; and $\bar{v}_0$ is a $\bar{W}_{T'}$-curve with $I = 1$ which passes through $\bar{\mathfrak{m}}(T')$.

Here either $T' \in \mathcal{T}_2$ and there is a component of $\bar{v}_0$ which is in

$$ \mathcal{M}_{\{\bar{J}_{T'}^{\diamond}(\varepsilon, \delta, \mathbf{p}(T'))\}}^{\dagger, s, \text{irr}, \text{ind}=1, n^*=m}(\mathbf{z}, \mathbf{z}', \bar{\mathfrak{m}}(T')) $$

from Lemma 3.2.22(2), for some $\mathbf{z}, \mathbf{z}'$; or $T' \in \mathcal{T}_1$ and there is a component of $\bar{v}_0$ which does not pass through $\bar{\mathfrak{m}}(T')$ but is in

$$ \mathcal{M}_{\{\bar{J}_{T'}^{\diamond}(\varepsilon, \delta, \mathbf{p}(T'))\}}^{\dagger, s, \text{irr}, \text{ind}=-1, n^*=0}(\mathbf{z}, \mathbf{z}') $$

from Lemma 3.2.22(1).
**Lemma 3.6.3.** — If $\bar{v}_*^\sharp \cup \bar{v}_*^{\sharp\prime} \neq \emptyset$ for some level $\bar{v}_*$ of $\bar{u}_\infty$, then:

(1) $p_0 = \deg(\bar{v}'_0) > 0$;

(2) $\bar{u}_\infty$ has no boundary point of type $(P_1)$ or $(P_2)$;

(3) $\bar{u}_\infty$ has no fiber components and no components of $\bar{v}_*^{\prime\prime}$ that intersect the interior of a section at infinity;

(4) each component of $\bar{v}_{-1,j}^{\sharp}$, $j = 1, \dots, c$, is a thin strip from $z_\infty$ to some $x_i$ or $x'_i$ with $I = 1$;

(5) the only boundary points at $z_\infty$ are type $(P_3)$ points of $\bar{v}_{1,j}^{\prime\prime}$, $j = 0, \dots, a$.

The following is the analog of Lemma 3.4.25:

**Lemma 3.6.4.** — If $\bar{v}_*^\prime \cup \bar{v}_*^{\prime\prime} \neq \emptyset$ for some level $\bar{v}_*$, then there are no boundary points at $z_\infty$ and $\bar{u}_\infty$ contains a subbuilding consisting of $\bar{v}_{1,a}^{\sharp}$ with $I \ge 1$; $\bar{v}_{1,j}^\prime$, $1 \le j < a$, which branch cover $\sigma_\infty$; $\bar{v}'_0$ with $I = -p$ which is a degree $p$ branched cover of $\sigma_\infty^{\mathrm{T'}}$; $\cup_{j=1}^c \bar{v}_{-1,j}^{\sharp}$ which is a union of $p$ thin strips; and possibly the following:

— $\bar{v}_0^{\sharp}$ with positive ends at multiples of $z_\infty$ and no negative ends at multiples of $z_\infty$; and

— $\bar{v}_{1,j}^{\sharp}$, $1 \le j < a$, with $I(\bar{v}_{1,j}^{\sharp}) \ge 1$ and (positive or negative) ends at multiples of $z_\infty$.

Here at most one component of $\bar{v}_0^{\prime\prime}$ satisfies $I = -1$ and the remaining components of $\bar{v}_*^{\prime\prime}$ satisfy $I \ge 0$.

*Proof.* — We explain why there are no boundary points of type $(P_3)$; the rest of the proof is similar to that of Lemma 3.4.25. A boundary point of type $(P_3)$ contributes +2 towards $I$ by Lemma 3.4.20, there is a large sector which contributes +1, and some $\bar{v}_{1,j}$, $j > 0$, contributes +1, for a total of $I = 4$. Since there is at most one component which contributes negatively to $I$, namely a component of $\bar{v}_0^{\prime\prime}$ with $I = -1$, we have a total of $I \ge 3$, a contradiction. $\square$

See Figure 14 for some possibilities. Observe that in the current case there is at most one component of $\bar{v}_0^{\prime\prime}$ with $I = -1$ whereas there are none in Lemma 3.4.25.

**Lemma 3.6.5.** — For each interval $[-T, T]$, there exists $m \gg 0$ such that there is no sequence of curves $\bar{u}_i \in \mathcal{M}_{\tau_i}$, $\tau_i \to T' \in [-T, T]$, that limits to $\bar{u}_\infty$ for which $\bar{v}_*^\prime \cup \bar{v}_*^{\prime\prime} \neq \emptyset$ for some level $\bar{v}_*$.

*Proof.* — This is similar to Case (6)$_i$ of Lemma 3.4.28. We apply the usual rescaling argument with $m \to \infty$ and obtain $w_0 : \Sigma_0 \to \mathrm{CP}^1$ and a branched cover $\pi_0 : \Sigma_0 \to \mathrm{cl}(\mathrm{B}_{T'})$ such that:

(i) $w_0(\partial \Sigma_0) \subset \{\phi = 0, \rho > 0\} \cup \{\infty\}$;

(ii) $w_0(z_0) = \infty$ for some point $z_0 \in \pi_0^{-1}(+\infty)$;

(iii) $w_0(z_1) = 0$ for some point $z_1 \in \pi_0^{-1}(\bar{m}^b(T'))$;

(iv) $w_0|_{\mathrm{int}(\Sigma_0)}$ is a biholomorphism onto its image.

Let us write $f_0 = \pi_0 \circ w_0^{-1}$ where defined. We now apply the Involution Lemma I.7.9.6. Using the notation of Lemma I.7.9.6, let $\bar{\Sigma}_1$ be the compact Riemann surface with
<figure><img src="image_22.png" /><figcaption>FIG. 14. — Schematic diagrams for some possible types of degenerations. Here ◦ represents $z_∞$, □ represents one or more branch points and × represents some $x_i$ or $x'_i$. A vertical line indicates a trivial cylinder or a restriction of a trivial cylinder, and a triple vertical line indicates a degree $p$ branched 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>

boundary whose interior is biholomorphic to $w_0(\mathrm{int}(\Sigma_0))$ and let $\bar{\Sigma}_2 = \mathrm{cl}(\mathbb{B}_{T'})$. We then extend $f_0$ to a holomorphic map $\bar{\Sigma}_1 \to \bar{\Sigma}_2$. By the Involution Lemma I.7.9.6 and (i), (iii) and (iv), $f_0$ maps the point on $\bar{\Sigma}_1$ which corresponds to $\infty \in \mathbf{CP}^1$ to $\mathcal{L}_{(r(T')+1)/2} \cap \partial \mathrm{cl}(\mathbb{B}_{T'})$. This contradicts (ii). □

We are now in a position to prove Lemma 3.3.8.

*Proof of Lemma 3.3.8.* — Suppose $\bar{u}_\infty \in \partial_{(-\infty, +\infty)} \mathcal{M}$. By Lemma 3.6.2, if $\bar{v}'_* \cup \bar{v}^\sharp_* = \emptyset$ for all levels $\bar{v}_*$ of $\bar{u}_\infty$, then $\bar{u}_\infty \in A_4$ or $A_5$. By Lemma 3.6.5, for any $T > 0$, there exists $m \gg 0$ such that if $\bar{u}_\infty \in \partial_{[-T, T]} \mathcal{M}$, then $\bar{v}'_* \cup \bar{v}^\sharp_* = \emptyset$ for all $\bar{v}_*$.

It remains to consider the case where there exist sequences $m_i \to \infty$, $\varepsilon_i$, $\delta_i \to 0$, $T_i \to \infty$, and $\bar{u}_{ij} \to \bar{u}_{i\infty}$, where $\bar{u}_{ij} \in \mathcal{M}^{(m_i)}$, $\bar{u}_{i\infty} \in \partial_{\{\pm T_i\}} \mathcal{M}^{(m_i)}$, $\mathcal{M}^{(m_i)}$ is $\mathcal{M}$ with respect to the family $\{\bar{J}_\tau^\diamond (\varepsilon_i, \delta_i, \mathfrak{p}(\tau); m_i)\}$, and $\bar{u}_{i\infty}$ satisfies $\bar{v}'_* \cup \bar{v}^\sharp_* \neq \emptyset$ for some $*$. We then take a diagonal subsequence $\bar{u}_{ij(i)}$. The proofs of Lemmas 3.4.28 and 3.5.13 carry over to give a contradiction. □

**3.7.** *Degeneration at +∞, part II.* — In Sections 3.7–3.9 we study the limit of holomorphic maps to $\bar{W}_\tau$ whose positive end is of the form $\mathbf{z} = \{z_{\infty,i}\}_{i \in \mathcal{I}} \cup \mathbf{y}$, i.e., we are in Step 4 of the proof of Theorem 3.3.1.

Let us write

$$ \mathcal{M} := \mathcal{M}^{1=2, n^* \le m+|\mathcal{I}|} (\mathbf{z}, \mathbf{y}'; \bar{\mathbf{m}}), \text{ and} $$

$$ \mathcal{M}_{\tau} := \mathcal{M}^{1=2, n^* \le m+|\mathcal{I}|} (\mathbf{z}, \mathbf{y}'; \bar{\mathbf{m}}), $$

where $|\mathcal{I}| \ge 1$.
Let $\bar{u}_{\infty} \in \partial_{+\infty} \mathcal{M}$ be the limit of $\bar{u}_i \in \mathcal{M}_{\tau_i}$, where $\tau_i \to +\infty$. We use the notation from Section 3.4 for the levels and components of $\bar{u}_{\infty}$ and write $p_* = \deg(\bar{v}_*)$ as before.
We now have two constraints

$$ (3.7.1) \qquad n^*(\bar{u}_i) = \sum_{\bar{v}_*} n^*(\bar{v}_*) = m + |\mathcal{I}|; $$

$$ (3.7.2) \qquad I(\bar{u}_i) = \sum_{\bar{v}_*} I(\bar{v}_*) = 2, $$

where the summations are over all the levels $\bar{v}_*$ of $\bar{u}_{\infty}$.

*Outline of proof of Lemma 3.3.10.* The proof is similar to that of Lemma 3.3.2, with the following differences: The proof of Lemma 3.7.10, which is the analog of Lemma 3.4.22, is more involved and the proof of Lemma 3.7.4, which is the analog of Lemma 3.4.14, uses a slightly more complicated notion of an “almost alternating” pair $(P_{*,0}, P_{*,1})$.

**3.7.1. Continuation argument.** — First observe that Lemma 3.4.2 carries over verbatim. The analog of Lemmas 3.4.7 and 3.4.16 hinge on the continuation argument: Let $\mathcal{E}_{-,i}$, $i = 1, \dots, q$, be the negative ends of $\cup_{j=1}^a \bar{v}_{1,j}^{\sharp}$ that converge to $z_{\infty}$ and let $\mathcal{E}_{+,i}$, $i = 1, \dots, r$, be the positive ends of $\cup_{j=0}^{a-1} \bar{v}_{1,j}^{\sharp}$ that converge to $z_{\infty}$. Suppose that $q \neq 0$ (i.e., some $\mathcal{E}_{-,i}$ exists) or not all $\mathcal{E}_{+,i}$ project to thin sectors. Also for simplicity we assume that there are no boundary points of type $(P_3)$. By assumption, we may start with an end $\mathcal{E}_{-,i}$ or $\mathcal{E}_{+,i}$ that projects to a non-thin sector. The continuation argument (i.e., the proof of Lemma 3.4.7) carries over with one modification: When we are considering the continuation

$$ g_{j_2+1,2}^0, \dots, g_{a,2}^0 $$

of $g_{j_2,2}^0$, it is possible that $g_{j_2}^0$ is trivial for all $j_2 + 1 \le j \le a$. In other words, there is no nontrivial negative end $\mathcal{E}_{-,2}$ such that

$$ \pi_{D_{\rho_0}^2}(\mathcal{E}_{-,2}) = \mathfrak{S}(\bar{h}(\bar{a}_{k_3,l_3}), \bar{a}_{k_4,l_4}) $$

for some $(k_4, l_4)$. This happens when $(k_3, l_3) \to (k_3, l_3)$ belongs to the data $\overrightarrow{D}$ at the positive end of $\bar{v}'_{1,a}$. In this case we set $j_3 = a$ and $\bar{a}_{k_3,l_3} = \bar{a}_{k_4,l_4}$ and skip $\mathfrak{S}(\bar{h}(\bar{a}_{k_3,l_3}), \bar{a}_{k_3,l_3})$. The rest of the argument is the same.

**3.7.2. Bounds on ECH indices.** — The goal of this subsection is to show the non-negativity of $I(\bar{v}_*)$ except when $\bar{v}_* = \bar{v}_+$, under the assumption that there are no boundary points at $z_{\infty}$.

Let $A_{\epsilon} = \partial D_{\epsilon}^2 \times [0, 1]$ for $0 < \epsilon < \rho_0$ small and let $\pi_{[0,1]\times\bar{S}}$ be the projection of $\bar{W}$ or the positive end of $\bar{W}_{\tau}$ to $[0, 1] \times \bar{S}$. Let $\epsilon'$ be the grooming on $A_{\epsilon}$ corresponding
<figure><img src="image_23.png" /><figcaption>FIG. 15. — The rectangular box, with the sides identified, is $A_ε$. The grooming $c'$ connects $Q_0$ to $Q_1$ and the grooming $c''$ connects $R_0$ to $R_1$. The vertical lines corresponding to $Q_1$ (resp. $Q_0$) are the intersections $(\bar{a}_{i,j} \cap \partial D_\varepsilon^2) \times [0, 1]$ (resp. $(\bar{h}(\bar{a}_{i,j}) \cap \partial D_\varepsilon^2) \times [0, 1]$). The dotted lines, together with some solid lines, give the main cycle $Z_{\text{main}}$. There are also two auxiliary cycles of $Z_{\text{aux}}$ corresponding to the middle and right arcs of $c'$</figcaption></figure>

to the data $\vec{D}'_{+,a}$ at $z_\infty$ for the positive end of $\vec{v}'_{1,a}$, such that the winding number $w(c')$ is zero. Here the data $\vec{D}'_{+,a}$ satisfies $(\vec{D}'_{+,a})^{\text{to}} = (\vec{D}'_{+,a})^{\text{from}}$. Let $c'' = \pi_{[0,1]\times\bar{S}}(\cup_i \mathcal{E}_{-,i}) \cap A_\varepsilon$. By Section 3.7.1, $c''$ is groomed and the sets of initial and terminal points of $c''$ alternate along $(0, 2\pi)$. We then define $P_0$ and $P_1$ as the initial and terminal points of $c' \cup c''$.

**Remark 3.7.1.** — It is possible for initial/terminal points of $c'$ to also appear as initial/terminal points of $c''$.

**Definition 3.7.2.** — A pair of points $(q_0, q_1) \subset \partial D_\varepsilon^2$ is a thin pair if $q_0 = \bar{h}(\bar{a}_{i,j}) \cap \partial D_\varepsilon^2$ and $q_1 = \bar{a}_{i,j} \cap \partial D_\varepsilon^2$ for the same $i, j$. A pair $(P_{*,0}, P_{*,1})$ consisting of disjoint finite subsets of $(0, 2\pi)$ with the same cardinality is almost alternating along $(0, 2\pi)$ if each $P_{*,i}$, $i = 0, 1$, admits a splitting $P_{*,i} = Q_{*,i} \sqcup R_{*,i}$ such that the pair $(R_{*,0}, R_{*,1})$ is alternating along $(0, 2\pi)$ and there is a partition of $Q_{*,0} \cup Q_{*,1}$ into thin pairs $(q_0, q_1)$, $q_i \in Q_{*,i}$.

By definition, $(P_0, P_1)$ is almost alternating along $(0, 2\pi)$. See Figure 15 for an example.

Section 3.7.1 gives the main cycle

$$Z_{\text{main}} = (\mathfrak{z}_0 \to \mathfrak{z}_1 \to \dots \to \mathfrak{z}_{k-1} \to \mathfrak{z}_0),$$

where $\{\mathfrak{z}_1, \dots, \mathfrak{z}_k\}$ can be decomposed into an almost alternating pair and the cycle winds around $\mathbf{R}/2\pi\mathbf{Z}$ once. If we apply the continuation method to the positive ends of $\vec{v}'_{1,a}$, then we also obtain a union $Z_{\text{aux}}$ of auxiliary cycles of the form $(\mathfrak{z}_0 \to \mathfrak{z}_1 \to \mathfrak{z}_0)$, where $(\mathfrak{z}_0, \mathfrak{z}_1)$ is a thin pair and the chords are short chords from $\mathfrak{z}_i$ to $\mathfrak{z}_{1-i}$, $i = 0, 1$. The sets of initial and terminal points of $Z_{\text{main}} \cup Z_{\text{aux}}$ are $P_0$ and $P_1$.

Let $\vec{D}_{\pm,j}$ be the data at $z_\infty$ for the $\pm$ end of $\bar{v}_{1,j}$ and let $P_{\pm,j,0}$ and $P_{\pm,j,1}$ be the initial and terminal points on $A_\varepsilon$ determined by $\vec{D}_{\pm,j}$. Then we write

$$P_{\pm,j,i} = P'_{\pm,j,i} \cup P''_{\pm,j,i},$$
where $P'_{\pm,j,i}$ corresponds to $\bar{v}'_{1,j}$ and $P''_{\pm,j,i}$ corresponds to $\bar{v}''_{1,j}$. For convenience we write $P_{-,a+1,i} = P'_{-,a+1,i}$, $i = 0, 1$, for the endpoints of $c'$. Also let $Q_*$ and $R_*$ be the thin and alternating parts of $P_*$ as given in Definition 3.7.2.

The following lemmas are analogs of Lemma 3.4.13 and 3.4.14:

**Lemma 3.7.3.** — If $\bar{u}_\infty$ has no boundary point at $z_\infty$, then, for each $*= (\pm, j)$, $P_{*,0} \subset P_0$ and $P_{*,1} \subset P_1$ and the pair $(P_{*,0}, P_{*,1})$ is almost alternating along $(0, 2\pi)$.

*Proof.* — Similar to but slightly more complicated than that of Lemma 3.4.13. We set $P_i^{(0)} = P_i$ and $Z_*^{(0)} = Z_*$, where $*= \text{main or aux}$. Then $P_{-,a+1,i} = P'_{-,a+1,i} \subset P_i^{(0)}$ and the pair $(P_{-,a+1,0}, P_{-,a+1,1})$ is almost alternating. In (j0)–(j4) we replace “alternating” by “almost alternating”, $j_0$ by $a+1$, and (j2) and (j4) by:

(j2) there is a partition of $P_{-,a-j+1,0} \cup P_{-,a-j+1,1}$ into pairs of type $\{\mathfrak{p}_0, \mathfrak{p}_1\}$, $\mathfrak{p}_i \in P_{-,a-j+1,i}$, such that $\mathfrak{p}_0 \ll_{R_0^{(j)} \cup R_1^{(j)}} \mathfrak{p}_1$ or $\{\mathfrak{p}_0, \mathfrak{p}_1\}$ is a thin pair of $Q_0^{(j)} \cup Q_1^{(j)}$; in particular, the points of $P_{-,a-j+1,0}$ and $P_{-,a-j+1,1}$ almost alternate along $(0, 2\pi)$;

(j4) there is a partition of $P'_{+,a-j,0} \cup P'_{+,a-j,1}$ into pairs of type $\{\mathfrak{p}_0, \mathfrak{p}_1\}$, $\mathfrak{p}_i \in P'_{+,a-j,i}$, such that $\mathfrak{p}_0 \ll_{R_0^{(j+1)} \cup R_1^{(j+1)}} \mathfrak{p}_1$ or $\{\mathfrak{p}_0, \mathfrak{p}_1\}$ is a thin pair of $Q_0^{(j+1)} \cup Q_1^{(j+1)}$; in particular, the points of $P'_{+,a-j,0}$ and $P'_{+,a-j,1}$ almost alternate along $(0, 2\pi)$.

We inductively define $P_i^{(j)}$ and $Z_*^{(j)}$ as follows: For each pair $\{\mathfrak{p}_0, \mathfrak{p}_1\}$ of $P''_{+,a-j,0} \cup P''_{+,a-j,1}$, if $(\mathfrak{p}_1 \to \mathfrak{p}_0)$ is a chord of $Z_{aux}^{(j-1)}$, then we remove $(\mathfrak{p}_1 \to \mathfrak{p}_0 \to \mathfrak{p}_1)$ from $Z_{aux}^{(j-1)}$; otherwise, in $Z_{main}^{(j-1)}$, we replace $\mathfrak{q} \to \mathfrak{p}_1 \to \mathfrak{p}_0 \to \mathfrak{q}'$ by $\mathfrak{q} \to \mathfrak{q}'$, given by concatenation. Then $P_0^{(j)}$ and $P_1^{(j)}$ are the sets of endpoints of $Z_{main}^{(j)} \cup Z_{aux}^{(j)}$.

The verification of (j0)–(j4) is left to the reader. $\square$

**Lemma 3.7.4.** — If $\bar{u}_\infty$ has no fiber components and there are no boundary points at $z_\infty$, then the ECH index of each level $\bar{v}_* \neq \bar{v}_+$ is nonnegative, the only components of $\bar{u}_\infty$ which have negative ECH index are branched covers of $\sigma_+^\infty$, and $I(\bar{v}_{1,j}) \ge I(\bar{v}'_{1,j}) + I(\bar{v}''_{1,j})$ for $0 \le j \le a$.

Before embarking on the proof of Lemma 3.7.4 we encourage the reader to review Section I.5.7.10 and in particular Lemmas I.5.7.22 and I.5.7.23.

*Proof.* — Similar to the proof of Lemma 3.4.14. The levels $\bar{v}_{0,j}$, $1 \le j \le b$, satisfy $I(\bar{v}_{0,j}) \ge 0$ by [HT1, Proposition 7.15(a)].

In order to treat the case of $\bar{v}_{1,j}$, $0 \le j \le a$, we use Lemma 3.7.3 and Lemmas I.5.7.22 and I.5.7.23 to compute $I(\bar{v}_{1,j})$. Let

$$ \pi_{[0,1]\times\bar{S}} : \bar{W} = [-1, 1] \times [0, 1] \times \bar{S} \to [0, 1] \times \bar{S} $$

be the projection along $[-1, 1]$.
We claim that there exist representatives $\check{C}_1, \check{C}_2, \check{C}_3 \subset \check{W}$ such that the following hold:

(0) $\check{C}_1 \cup \check{C}_2$ is a representative of $\overline{v}'_{1,j}$, $\check{C}_3$ is a representative of $\overline{v}''_{1,j}$, $\check{C}_1 \subset [-1, 1] \times [0, 1] \times D_{\varepsilon/2}^2$, and $\check{C}_2 \subset [-1, 1] \times [0, 1] \times D_{\varepsilon}^2$;

(1) $\mathbf{c}_1^{\pm} := \pi_{[0,1]\times\bar{S}}(\check{C}_1|_{s=\pm 1}) \subset A_{\varepsilon/2}$, $\mathbf{c}_1^{+} = \mathbf{c}_1^{-}$ is groomed, and the endpoints of each component of $\mathbf{c}_1^{\pm}$ forms a thin pair;

(2) the components of $\pi_{[0,1]\times\bar{S}}(\check{C}_i|_{s=\pm 1})$, $i = 2, 3$, corresponding to $z_{\infty}$ are contained in $A_{\varepsilon}$; let us write $\mathbf{c}_i^{\pm}$ for their unions;

(3) $\mathbf{c}_2^+ \cup \mathbf{c}_3^+$ and $\mathbf{c}_2^- \cup \mathbf{c}_3^-$ are groomed;

(4) $w(\mathbf{c}_2^+ \cup \mathbf{c}_3^+) = 0$ or $-1$, and $w(\mathbf{c}_2^- \cup \mathbf{c}_3^-) = 0$ or $1$;

(5) $\mathbf{c}_2^+ \cup \mathbf{c}_3^+$ satisfies the first condition of $(G_3')$ from Section I.5.7.10 and $\mathbf{c}_2^- \cup \mathbf{c}_3^-$ satisfies $(G_3)$ from Section I.5.7.10.

We first choose a representative $\check{C}_3$ of $\overline{v}''_{1,j}$ such that (2) holds for $i=3$ and

$$ \mathbf{c}_3^{\pm} = \pi_{[0,1]\times\bar{S}}(\cup_k \mathcal{E}_{\pm,k}) \cap A_{\varepsilon}, $$

where $\mathcal{E}_{\pm,k}$ are the $\pm$ ends of $\overline{v}''_{1,j}$ that limit to $z_{\infty}$. We then choose the representative $\check{C}_1 \cup \check{C}_2$ of $\overline{v}'_{1,j}$ as follows: Let $\check{C}_1 = [-1, 1] \times \mathbf{c}_1^+$ so that (1) holds and the endpoints of $\mathbf{c}_1^+ = \mathbf{c}_1^-$ are partitioned into thin pairs of $(P'_{+,j,0}, P'_{+,j,1})$. Let $\check{C}_2 \subset [-1, 1] \times [0, 1] \times D_{\varepsilon}^2$ be a disk of the type used in the proof of Lemma I.5.7.15, such that (2) holds for $i=2$, $\check{C}_2|_{s=\pm 1} = \mathbf{c}_2^{\pm}$, $\mathbf{c}_2^{\pm}$ are groomed, $w(\mathbf{c}_2^+) = 0$ or $-1$, $w(\mathbf{c}_2^-) = 0$ or $1$, and the endpoints of $\mathbf{c}_2^{\pm}$ are alternating along $(0, 2\pi)$. This is possible by (j4) in the proof of Lemma 3.7.3. (3)-(5), in particular the fact that adding $\mathbf{c}_3^+$ to $\mathbf{c}_2^+$ — and similarly adding $\mathbf{c}_3^-$ to $\mathbf{c}_2^-$ — leaves the grooming property invariant, follow from the proof of Lemma 3.7.3.

We now prove that $I(\overline{v}_{1,j}) \ge 0$ for $j > 0$; the verification of $I(\overline{v}_{1,j}) \ge I(\overline{v}'_{1,j}) + I(\overline{v}''_{1,j})$ for $0 \le j \le a$ is similar and is left to the reader. First observe that $I(\check{C}_1) = 0$ and $I(\check{C}_2) = 0$, where the latter follows from the proof of Lemma I.5.7.15. Also $I(\check{C}_3) \ge 0$ by the ECH index inequality. Next observe that $\langle \check{C}_1, \check{C}_2 \rangle = -l$, where $l$ is the degree of $\check{C}_1$. Hence the contribution of $\check{C}_1 \cap \check{C}_2$ to $I(\overline{v}_{1,j})$ is $-2l$. We now apply Lemmas I.5.7.22 and I.5.7.23, where we split into $\check{C}_1$ and $\check{C}_2 \cup \check{C}_3$ instead of $\overline{v}'_{1,j}$ and $\overline{v}''_{1,j}$. Observe that $q_2 = 0$ (resp. $q_2 = 1$) in Lemma I.5.7.22 corresponds to $q_2 = -1$ (resp. $q_2 = 0$) in Lemma I.5.7.23. One can verify that the extra contributions to $I$ from the rightmost terms of Equations (I.5.7.10) and (I.5.7.11) add up to at least $2l$. Hence $I(\overline{v}_{1,j}) \ge 0$ for $j > 0$.

The case of $\overline{v}_{-1,j}$, $1 \le j \le c+1$, is easier and will be omitted. $\square$

<figure><img src="image_24.png" /></figure>

**3.7.3. Boundary points at $z_{\infty}$**. — In this subsection we describe the necessary modifications when $\overline{u}_{\infty}$ has boundary points at $z_{\infty}$. We use the notation from Section 3.4.3, with some modifications: The continuation method gives rise to a main cycle $Z_{\text{main}}$ which winds around $\mathbf{R}/2\pi\mathbf{Z}$ once and a union $Z_{\text{aux}}$ of auxiliary cycles. Also, in Definition 3.4.15 we replace $Z$ by $Z_{\text{main}}$.
<figure><img src="image_25.png" /><figcaption>FIG. 16. — $c'_{L-} \cup c''_{L-}$ for $\mathfrak{r}$ of (i) type $(P_3)$ and (ii) type $(P_2)$. Here $c''_{L-}$ is the strand in the front</figcaption></figure>

**Lemma 3.7.5.** — If $\bar{u}_{\infty}$ has no fiber components, then the ECH index of each level $\bar{v}_{*} \neq \bar{v}_{+}$ is nonnegative, the only components of $\bar{u}_{\infty}$ which have negative ECH index are branched covers of $\sigma_{\infty}^{+}$, and there exist constants $\delta_{1,j}$ such that

$$ (3.7.3) \qquad I(\bar{v}_{1,j}) = I(\bar{v}'_{1,j}) + I(\bar{v}''_{1,j}) + \delta_{1,j} $$

for $0 \le j \le a$, where:

(i) if there is a boundary point of type $(P_3)$ on $\bar{v}_{1,j}$, then $\delta_{1,j} \ge 2$;

(ii) if there is a boundary point of type $(P_2)$ on $\bar{v}_{1,j}$, then $\delta_{1,j} \ge 2p_{1,j}$, where $p_{*} = \deg(\bar{v}'_{*})$;

(iii) otherwise, $\delta_{1,j} = 0$.

We remark that we have not tried to obtain the best lower bound, just one that suffices for our purposes of eliminating boundary points at $z_{\infty}$.

*Proof.* — Let $\mathfrak{r}$ be a boundary point at $z_{\infty}$ on $\bar{v}_{*0}$ with $*_{0} = (1, j_0)$. We compute the extra contribution $I_{L-}$ to $I(\bar{v}_{*0}^{\circ, 1})$ that comes from the left negative end at $z_{\infty}$ as in Lemma 3.4.18.

(i) Suppose $\mathfrak{r}$ is of type $(P_3)$. We assume (S) from Section 3.4.3 for simplicity. First observe that $P_{L-j_0, 1/2} = P_{L-j_0, 1}$. By (S), $\mathcal{Z}_{\text{main}}$ has only one chord $\mathfrak{z}'_1 \to \mathfrak{z}'_s$ corresponding to a sector of type $\mathcal{S}(\bar{a}_{k,l}, \bar{a}_{k',l'})$. Hence we have:

$$ P''_{L-j_0, 1/2} = \{\mathfrak{z}'_1\}, \quad P'_{L-j_0, 1/2} = \{\mathfrak{z}'_2, \dots, \mathfrak{z}', \mathfrak{z}''_1, \dots, \mathfrak{z}''_l\}, $$

$$ P''_{L-j_0, 1} = \{\mathfrak{z}'_s\}, \quad P'_{L-j_0, 1} = \{\mathfrak{z}'_1, \dots, \mathfrak{z}'_{s-1}, \mathfrak{z}''_1, \dots, \mathfrak{z}''_l\}, $$

where $P_{L-j_0, 1/2} = P_{L-j_0, 1}$ is written as $\{\mathfrak{z}'_1, \dots, \mathfrak{z}', \mathfrak{z}''_1, \dots, \mathfrak{z}''_l\}$ in cyclic order around $\mathbf{R}/2\pi\mathbf{Z}$; see Figure 16(i). Note that $\mathfrak{z}'_2, \dots, \mathfrak{z}'_{s-1}$ are points of $\mathcal{Z}_{\text{aux}}$ but not $\mathcal{Z}_{\text{main}}$; otherwise $\mathcal{Z}_{\text{main}}$ winds more than once around $\mathbf{R}/2\pi\mathbf{Z}$. The projection of the left negative end of $\bar{v}_{*0}^{\circ, 1, 1}$ that limits to $z_{\infty}$ intersects $A_{g}^{[1/2, 1]}$ along an arc $c''_{L-}$ with winding number
$w(c''_{L-}) = 0$ or 1, depending on whether $\mathcal{S}(\bar{a}_{k,l}, \bar{a}_{k',l'})$ is a large sector. The left negative ends of $\bar{v}_{*0}^{0,1'}$ that limit to $z_\infty$ can be modified to give a grooming $c'_{L-}$ on $A_{\varepsilon/2}^{1/2,1}$ such that the winding number $w(c'_{L-}) = 0$ or $-1$ and $c'_{L-}$ connects $\mathfrak{z}''_i$ to $\mathfrak{z}''_i$, $i = 1, \dots, t$, by vertical arcs.

We now consider the disk $\tilde{D}$ that we “append” to the left negative end of $\bar{v}_{*0}^{0,1}$ as in the proof of Lemma 1.5.7.22. The writhe of $c'_{L-} \cup c''_{L-}$ is equal to $s-1$. Resolving the (positive) crossings of $c'_{L-} \cup c''_{L-}$ yields a grooming with vertical arcs from $\mathfrak{z}'_i$ to $\mathfrak{z}'_i$; this corresponds to a disk $\tilde{D}$ whose contributions to $\mathcal{Q}$, $c_1$, $\mu$ are $s-1, 0, s-1$. [The calculations for $\mu$ assume (R) from Section 3.4.3. For example, if $w(c'_{L-}) = w(c''_{L-}) = 0$, then $\mu$ of the positive ends are 1 for $s-1$ arcs and 0 for the rest and $\mu$ of the negative ends are all 0.] The discrepancy contribution to $I$ is $\ge 0$. Hence $I_{L-} = 2(s-1) \ge 2$.

In general, each collection of boundary points of type $(P_3)$ that map to the same point on the base contributes at least +2 towards $I$; this is analogous to Remark 3.4.19.

The cycles $Z_{\text{main}}^{(a-j)}$ and $Z_{\text{aux}}^{(a-j)}$ are defined as before, for $j \ge j_0$. We define $Z_*^{(a-j_0,+)} = Z_{\text{aux}}^{(a-j_0,+)}$, $Z_{\text{main}}^{(a-j_0,-)} = Z_{\text{aux}}^{(a-j_0,-)}$, and $Z_{\text{main}}^{(a-j_0,+)}$ as $Z_{\text{main}}^{(a-j_0,+)}$ with $\mathfrak{z}'_1 \to \mathfrak{z}'_s$ replaced by $\mathfrak{z}'_s$. Also, $P_{R_{-j_0,i}}^* \star = \emptyset, \emptyset, \emptyset$ is obtained from $P_{+j_0,i}^*$ by replacing $\mathfrak{z}'_1$ by $\mathfrak{z}'_s$. The rest of the arguments of Lemmas 3.7.3 and 3.7.4 carry over.

(ii) Suppose $\mathfrak{r}$ is the only boundary point of type $(P_2)$. In this case $\mathfrak{z}'_1 = \mathfrak{z}'_s$ and $Z_{\text{main}} = (\mathfrak{z}'_1 \to \mathfrak{z}'_1)$. Then:

$$ 
\begin{aligned}
P''_{L_{-j_0,1/2}} &= \{\mathfrak{z}'_1\}, & P'_{L_{-j_0,1/2}} &= \{\mathfrak{z}''_1, \dots, \mathfrak{z}''_{p_{*0}}\}, \\
P''_{L_{-j_0,1}} &= \{\mathfrak{z}'_1\}, & P'_{L_{-j_0,1}} &= \{\mathfrak{z}''_1, \dots, \mathfrak{z}''_{p_{*0}}\}.
\end{aligned}
$$

Also $c''_{L-}$ is an arc from $\mathfrak{z}'_1$ to itself with $w(c''_{L-}) = 1$ and $c'_{L-}$ consists of $p_{*0}$ vertical arcs from $\mathfrak{z}''_i$ to itself; see Figure 16(ii).

In order to groom $c'_{L-} \cup c''_{L-}$ so that the result $c$ satisfies $w(c) = 0$, we switch the (positive) crossings of $c'_{L-} \cup c''_{L-}$ while keeping the same endpoints. This gives rise to $\tilde{D}$ which is a union of $p_{*0} + 1$ disks. The total contributions to $\mathcal{Q}$, $c_1$, $\mu$, and the discrepancy, are $2p_{*0} + 1, 1, -2, 0$. [Again recall (R) from Section 3.4.3. The writhe of $c'_{L-} \cup c''_{L-}$ is $2p_{*0}$ which contributes $2p_{*0}$ towards $\mathcal{Q}$. The writhe of $c''_{L-}$ and its pushoff is 1, which contributes an additional 1 towards $\mathcal{Q}$. $\mu$ of the positive (resp. negative) ends are 0, ..., 0, -2 (resp. all 0).] Hence $I_{L-} = 2p_{*0}$.

We also obtain a lower bound of $2p_{*0}$ in the general case of more boundary points of type $(P_2)$; the details are left to the reader. $\square$

The following analog of Lemma 3.4.16 is a consequence of Section 3.7.1:

**Lemma 3.7.6.** — Suppose $\bar{v}'_{1,j} \cup \bar{v}''_{1,j} \ne \emptyset$ for some $j > 0$. Let $\mathcal{E}_{-,i}$, $i = 1, \dots, q$, be the negative ends of $\cup_{j=1}^a \bar{v}''_{1,j}$ that converge to $z_\infty$, let $\mathcal{E}_{+,i}$, $i = 1, \dots, r$, be the positive ends of $\cup_{j=0}^{a-1} \bar{v}''_{1,j}$
that converge to $z_\infty$, and let $\mathcal{E}'_i$, $i=1, \dots, s$, be the neighborhoods of the points of type $(P_3)$. Then

$$ (3.7.4) \qquad n^+(\mathcal{E}_{-,i}) \ge k_0 - 1 \gg 2g $$

for each $i$, where the constant $k_0$ is as given in Section I.5.2.2. If $q \ne 0$ (i.e., some $\mathcal{E}_{-,i}$ exists) or not all $\mathcal{E}_{+,i}$ project to thin sectors, then

$$ (3.7.5) \qquad n^+((\cup_{i=1}^q \mathcal{E}_{-,i}) \cup (\cup_{i=1}^r \mathcal{E}_{+,i}) \cup (\cup_{i=1}^s \mathcal{E}'_i)) \ge m - p_+, $$

where $p_+ = \deg(\bar{v}'_+)$. Moreover,

$$ D_{\rho_0}^2 - (\cup_{i=1}^q \pi_{D_{\rho_0}^2}(\mathcal{E}_{-,i})) \cup (\cup_{i=1}^r \pi_{D_{\rho_0}^2}(\mathcal{E}_{+,i})) \cup (\cup_{i=1}^s \pi_{D_{\rho_0}^2}(\mathcal{E}'_i)) $$

consists of at most $p_+$ thin sectors.

The following is the analog of Lemma 3.4.17:

**Lemma 3.7.7.** — If $\mathfrak{p}_1, \dots, \mathfrak{p}_s$ are the boundary points of type $(P_1)$ and $(P_2)$ and $\mathrm{N}(\mathfrak{p}_i) \subset \mathrm{F}_*$ is a small neighborhood of $\mathfrak{p}_i$, then $\sum_{i=1}^s n^*(\bar{v}_*(\mathrm{N}(\mathfrak{p}_i))) \ge m$.

**3.7.4.** *Asymptotic eigenfunctions.* — Fix $m \gg 0$ and let $\bar{u}_i \to \bar{u}_\infty \in \partial_{\{+\infty\}} \mathcal{M}$.

**Lemma 3.7.8.** — There is no $\bar{u}_\infty \in \partial_{\{+\infty\}} \mathcal{M}$ such that:

(a) for all $\bar{v}_* \ge \bar{v}_+$, $\deg(\bar{v}') = p_+ \ge 1$, $\bar{v}_*^\# = \emptyset$, and $\bar{v}'_* \cap \bar{v}_*'' = \emptyset$; and

(b) $\bar{v}_{0,b}^\#$ is a union of $p_+$ cylinders from $\delta_0$ to $h$.

*Proof.* — Arguing by contradiction, we restrict $\bar{u}_i$ to a neighborhood of $\sigma_\infty^{\tau_i}$ and further restrict it to the components that are close to $\bar{v}_'$. After projecting to $D_{\rho_0}^2$ using balanced coordinates, applying the ansatz given by Equation (I.7.8.1), rescaling, and taking the SFT limit with $m \gg 0$ fixed, we obtain a holomorphic map $w_+ : \Sigma_+ \to \mathbb{C}P^1$ together with the branched cover $\pi_+ : \Sigma_+ \to \mathrm{cl}(B_+)$. By (a), $w_+$ maps each component of $\partial \Sigma_+$ to a distinct thin sector $\mathcal{S}(\{\phi = \phi(\bar{a}_{i,j}) + \frac{\pi_i}{m}\}, \bar{h}(\bar{a}_{i,j}))$.<fnref n="13" /> By (a) and (b), the image of $w_+$ cannot contain $\infty \in \mathbb{C}P^1$. Hence, by the open mapping property, $w_+$ maps each component of $\Sigma_+$ to a distinct $\mathcal{S}(\{\phi = \phi(\bar{a}_{i,j}) + \frac{\pi_i}{m}\}, \bar{h}(\bar{a}_{i,j}))$.

From now on assume without loss of generality that $p_+ = 1$ and $\Sigma_+ = B_+$. We identify $\mathrm{cl}(B_+)$ with the closed unit disk $\bar{D} = \{|z| \le 1\}$ by sending $+\infty$ to 1 and $-\infty$ to 0.

The Lagrangian boundary condition for $\bar{u}_i$, when projected to $D_{\rho_0}^2$ using balanced coordinates, descends to the boundary condition $w_+(e^{i\theta}) \in \mathcal{R}_{\varphi(e^{i\theta})}$, where the map $\varphi : \partial \bar{D} - \{1\} \to S^1 = \mathbb{R}/2\pi\mathbb{Z}$ satisfies the following:

<footnote n="13">¹³ This complicated expression is the result of using balanced coordinates instead of $\pi_{D_{\rho_0}^2}$.</footnote>
- $\lim_{\theta \to 0^+} \varphi(e^{i\theta}) = \varphi(\bar{a}_{i_0,j_0}) + \frac{\pi}{m}$ and $\lim_{\theta \to 0^-} \varphi(e^{i\theta}) = \varphi(\bar{h}(\bar{a}_{i_0,j_0})) = \varphi(\bar{a}_{i_0,j_0}) + \frac{2\pi}{m}$ for some $(i_0,j_0)$;

- $\varphi(e^{i\theta}) \in (0, 2\pi)$ is a nondecreasing function of $\theta \in (0, 2\pi)$; and

- $\varphi \circ i_0 = i_1 \circ \varphi$, where $i_0 : \partial \mathbf{D} - \{1\} \to \partial \mathbf{D} - \{1\}$ is the reflection across the x-axis and $i_1 : S^1 \to S^1$ is the reflection across $\mathcal{R}_{\phi_0}$ where $\phi_0 = \varphi(\bar{a}_{i_0,j_0}) + \frac{3\pi}{2m}$.

**Lemma 3.7.9.** — The holomorphic map $w_+ : \bar{\mathbf{D}} \to \mathbf{CP}^1$ satisfies the following:

(i) $w_+(1) = 0$ and $w_+(e^{i\theta}) \in \mathcal{R}_{\varphi(e^{i\theta})}$ for $e^{i\theta} \neq 1$;

(ii) $w_+(0) \in \mathcal{R}_{\phi_0}$ and $w_+$ maps $\mathcal{R}_0 \cup \mathcal{R}_{\pi}$ to $\mathcal{R}_{\phi_0}$;

(iii) $\mathrm{Im}(w_+) \subset \mathcal{S}(\{\varphi = \varphi(\bar{a}_{i_0,j_0}) + \frac{\pi}{m}, \bar{h}(\bar{a}_{i_0,j_0})\})$;

(iv) $w_+$ is a biholomorphism onto its image.

Moreover, $w_+$ is the unique holomorphic map $\bar{\mathbf{D}} \to \mathbf{CP}^1$ which satisfies (i), (iii) and (iv), up to multiplication by a positive real constant.

*Proof.* — (i), (iii), and (iv) are immediate from the construction. To prove the uniqueness statement, suppose there exists $\tilde{w}_+$ satisfying (i), (iii) and (iv). Then we consider $\frac{\tilde{w}_+}{w_+}$ as in Section I.7.9. It is easy to see that $\frac{\tilde{w}_+}{w_+}$ is a positive real constant on $\partial \mathbf{D}$ and has no zeros or poles on $\mathbf{D}$, hence is a constant map. (ii) then follows from the uniqueness, Observation I.7.9.1, and the fact that $i_0$ and $i_1$ extend to involutions of $\mathbf{D}$ and $\mathbf{CP}^1$. Here we are viewing $S^1$ as the unit circle in $\mathbf{CP}^1$. $\square$

The proof of Lemma 3.7.8 now follows from the first part of Lemma 3.7.9(ii): Suppose $\tilde{v}_{0,b}^\mu$ is a cylinder from $\delta_0$ to $h$. Then by the choice of $h$ from Section 3.2.3 we have a contradiction, since the asymptotic eigenfunction of $\tilde{v}_{0,b}^\mu$ at the positive end $\delta_0$ is a constant $c \in \mathcal{R}_{\phi=-2\pi/m}$. $\square$

**3.7.5. Preliminary restrictions.**

**Lemma 3.7.10.** — If $m \gg 0$ and $\bar{u}_\infty \in \partial_{\{+\infty\}} \mathcal{M}$, then the following hold:

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

(2) there is no level $\bar{v}_*$ such that $\bar{v}_*' \cap \bar{v}_*'' \neq \emptyset$ and $\bar{v}_*' \cap \bar{v}_*'' \subset \mathrm{int}(\bar{v}_*')$; and

(3) $\bar{u}_\infty$ has no boundary point at $z_\infty$.

*Proof.* — First observe that “not (1)”, “not (2)”, and “not (3)” are mutually exclusive by considerations of $n^*$.

(1) Arguing by contradiction, suppose that $\bar{u}_\infty$ has a fiber component $\tilde{v} : \dot{\mathbf{F}} \to \bar{W}_*$. The fiber component $\tilde{v}$ satisfies $n^*(\tilde{v}) \ge m$.

We first claim the following:

(a) The only ends of $\bar{v}_*^\mu$ that limit to multiples of $z_\infty$ or $\delta_0$ are positive ends.
(b) All the positive ends $\mathcal{E}_+$ of $\bar{v}_{0,j}^\sharp, j=0, \dots, b$, that limit to $\delta_0^r$ satisfy $n^*(\mathcal{E}_+) = r$.

(c) All the positive ends of $\bar{v}_{1,j}^\sharp, j=0, \dots, a$, and $\bar{v}_{-1,j}^\sharp, j=1, \dots, c$, that limit to $z_\infty$ project to thin sectors under $\pi_{D_{\rho_0}^2}$.

(d) $p_{*1} \le p_{*2}$ if $\bar{v}_{*1} \le \bar{v}_{*2}$.

(a) Arguing by contradiction, let $\mathcal{E}_-$ be a negative end of some $\bar{v}_*^\sharp$ that limits to a multiple of $z_\infty$ or $\delta_0$. Then $n^*(\mathcal{E}_-) \gg 2g$ by Lemmas 3.4.2(1) and 3.7.6. Since $n^*(\tilde{v}) \ge m$, such a negative end $\mathcal{E}_-$ cannot exist by Equation (3.7.1). The arguments for (b) and (c) are similar and (d) is a consequence of (a)–(c).

Next we have the following contributions towards I:

$$ (\beta_1) \quad \mathrm{I}(\bar{v}_+') = -p_{0,b+1}. $$

$$ (\beta_2) \quad \cup_{j=1}^b \bar{v}_{0,j}^\sharp \text{ is a union of } p_{0,b+1} - p_{0,1} \text{ cylinders from } \delta_0 \text{ to } h \text{ or } e \text{ and} $$

$$ \sum_{j=1}^{b} \mathrm{I}(\bar{v}_{0,j}^{\sharp}) \geq p_{0,b+1} - p_{0,1}. $$

$$ (\beta_3) \quad \bar{v}_-^\sharp \text{ is a union of } p_{0,1} - p_{0,0} \text{ sections from } \delta_0 \text{ to } x_i \text{ or } x_i' \text{ and} $$

$$ \mathrm{I}(\bar{v}_-^\sharp) = p_{0,1} - p_{0,0}. $$

$$ (\beta_4) \quad \cup_{j=1}^c \bar{v}_{-1,j}^\sharp \text{ is a union of } p_{0,0} \text{ trivial strips and} $$

$$ \sum_{j=1}^{c} \mathrm{I}(\bar{v}_{-1,j}^{\sharp}) = p_{0,0}. $$

(β_5) If the fiber $\tilde{v}$ is a component of $\bar{v}_*$, then $\tilde{v}$ and the intersection $\tilde{v} \cap (\bar{v}_* - \tilde{v})$ contribute $2g + 2 \ge 4$ towards I.

(β_6) After removing any fiber components, all the levels $\ne \bar{v}_+$ have nonnegative ECH index, $\mathrm{I}(\bar{v}_+^\prime) \ge 0$, and $\mathrm{I}(\bar{v}_+) \ge \mathrm{I}(\bar{v}_+) + \mathrm{I}(\bar{v}_+^\prime)$.

$(\beta_1)-(\beta_4)$ are clear.

We now prove $(\beta_5)$. Suppose that $\tilde{v}$ is a component of $\bar{v}_{*0}$. If $\bar{v}_{*0}^{\prime\prime}$ has a boundary point at $z_\infty$, then a neighborhood of the boundary point contributes $n^* \gg 2g$. Since $n^*(\tilde{v}) \ge m$, we have a contradiction of Equation (3.7.1). Hence $\bar{v}_{*0}^{\prime\prime}$ does not have a boundary point at $z_\infty$.

When $\tilde{v}$ is an interior fiber, $(\beta_5)$ follows from Equation (I.7.5.6).

Next suppose that $\tilde{v}$ is a boundary fiber, i.e., $\tilde{v}(\tilde{F}) \subset \{p\} \times \bar{S}$, where $\tilde{F}$ is the domain of $\tilde{v}$ and $p \in \partial B_{*0}$. If $(p, z_\infty) \notin \tilde{v}(\partial \tilde{F})$, i.e., $\tilde{v}(\partial \tilde{F})$ consists of $2g$ slits along $\hat{a}$, for example, then the Fredholm index of $\tilde{v}$ is:

$$ \mathrm{ind}(\tilde{v}) = -\chi(\tilde{F}) + \mu_\tau(\partial \tilde{F}) + 2c_1(\tilde{v}^*T\bar{S}, \tau) $$
$$ = -(2 - 4g) + 0 + 2(2 - 2g) = 2, $$

where $\tau$ is a partially defined trivialization of $T\bar{S}$ along $\hat{\mathfrak{a}}$ which directs $T\hat{\mathfrak{a}}$. Since there are $2g$ arcs of $\bar{u}_i$ that are pinched to points when taking the limit, $\tilde{v}$ and the “pinched points” contribute a total of $2g + 2$ towards I.

Now suppose that $(p, z_\infty) \in \tilde{v}(\partial\tilde{F})$. Suppose that $\bar{v}''_{*0} = \emptyset$ and that $*_{0} = (1, j)$ with $j \ge 1$. By considerations of $n^*$,

(*) all the components of the data $\vec{D}_\pm$ of the positive and negative ends of $\bar{v}_{*0}$ are of the form $(i,j) \to (i,j)$.

Let $\tau_0$ be a groomed multivalued trivialization which is compatible with $\vec{D}_\pm$ and let $c_+ = c_- \subset A_\epsilon = [0, 1] \times \partial D_\epsilon^2$ be the corresponding groomings. Then a $\tau_0$-trivial representative $\check{C}$ of $\tilde{v} \cup \bar{v}'_{*0}$ satisfies

$$ [\check{C}] = [[-1, 1] \times c] + [\bar{S}] \in \pi_2(\mathbf{z}_+, \mathbf{z}_-, \tau_0), $$

where $\mathbf{z}_\pm = \{z_\infty^{2g}(\vec{D}_\pm)\}$. ($\beta_5$) then follows from the calculation for a closed interior fiber $\tilde{v}$ when $\bar{v}''_{*0} = \emptyset$ and $*_{0} = (1, j)$ with $j \ge 1$. The general case follows by incorporating considerations of the previous paragraph and is left to the reader.

($\beta_6$) follows from Lemma 3.7.4 by observing that the proof carries over when fibered components are removed from $\bar{u}_\infty$: Removing interior fibers and boundary fibers with $(p, z_\infty) \notin \tilde{v}(\partial\tilde{F})$ do not affect groomings. If $\bar{u}_\infty$ has a boundary fiber with $(p, z_\infty) \in \tilde{v}(\partial\tilde{F})$, then (*) holds, which also is sufficient.

Summing $(\beta_1)-(\beta_6)$, the total ECH index is $\ge 4$, which contradicts Equation (3.7.2). Hence (1) follows.

(2) Arguing by contradiction, suppose there exist sequences $m_l \to \infty$ and $\bar{u}_{li} \to \bar{u}_{l\infty}$, where $\bar{u}_{li} \in \mathcal{M}^{(m_l)}$, $\bar{u}_{l\infty} \in \partial_{(l+\infty)} \mathcal{M}^{(m_l)}$, and $\mathcal{M}^{(m_l)}$ is $\mathcal{M}$ with respect to $m_l$, such that each $\bar{u}_{l\infty}$ has some $\bar{v}_{l,*_0}$ such that $\bar{v}'_{l,*_0} \cap \bar{v}''_{l,*_0} \ne \emptyset$. Unless indicated otherwise, we fix $l \gg 0$ and suppress $l$ from the notation.

The same argument as in (1) implies that (a)-(d) hold. Since $n^*(\bar{u}_i) = m + |\mathcal{I}| \le m + 2g$, there is only one intersection point of $\bar{v}'_{*0}$ and $\bar{v}''_{*0}$, which we denote by $\mathfrak{r} = (\mathfrak{r}^b, z_\infty)$.

(2A) Suppose that $*_{0} = +$. As in the proof of (1), the holomorphic building hanging from the positive end of $\bar{v}'_+$ has total ECH index $\ge 0$ — this is obtained by adding all the ECH index contributions from $(\beta_1)-(\beta_4)$. We also have $(\beta'_5)$ instead of $(\beta_5)$:

($\beta'_5$) $\bar{v}'_{*0} \cap \bar{v}''_{*0}$ contributes $2 \cdot m(\mathfrak{r})$ towards I, where $m(\mathfrak{r}) \ge 1$ is the multiplicity of $\mathfrak{r}$. Hence $\sum_{\bar{v}_*} I(\bar{v}_*) \ge 2 \cdot m(\mathfrak{r})$. This implies that $m(\mathfrak{r}) = 1$ and $p_+ = \deg \bar{v}'_+ = 1$. The sum of the ECH indices from $(\beta_1)-(\beta_4)$, $(\beta'_5)$, $(\beta_6)$, and $I(\bar{v}''_+) \ge 0$ is at least $+2$.

We claim that $p_- \ge 1$ by considerations of $n^*$. Indeed, if $\bar{v}'_- = \emptyset$, then there exist a point $\mathfrak{q} \in \text{int}(\tilde{F}_-)$ and a sufficiently small neighborhood $N(\mathfrak{q}) \subset \vec{F}_-$ of $\mathfrak{q}$ such that $\bar{v}_-(\mathfrak{q}) = \bar{m}(+\infty)$ and $n^*(\bar{v}_-(N(\mathfrak{q}))) \ge m$. This is a contradiction.
Since $p_+ = 1$ and $p_- \ge 1$, it follows that $p_{0,j} = 1$ for all $j = 0, \dots, b+1$ by (d).
For the purposes of applying the rescaling argument, we may assume that $b=0$. By
restricting $\bar{u}_{li}$ to a neighborhood of $\sigma_{\infty}^{\tau_{li}}$, taking the limit of a diagonal subsequence $\bar{u}_{li(l)}$,
and applying the rescaling argument, we obtain a 2-level holomorphic building $w_+ \cup w_-$,
where $w_{\pm} : cl(\mathbf{B}_{\pm}) \to \mathbf{CP}^1$ satisfy the following:

(i) $w_-(\partial B_-) \subset \{\phi = 0, \rho > 0\}$;

(ii) $w_-(\overline{m}^b(+\infty)) = 0$ and $w_-(+\infty) = \infty$;

(iii) $w_+(\partial B_+) \subset \{\phi = 0, \rho > 0\}$;

(iv) $w_+(-\infty) = 0$ and $w_+(\mathbf{r}^b) = \infty$;

(v) $w_{\pm}|_{\text{int}(\mathbf{B}_{\pm})}$ is a biholomorphism onto its image.

Then, by the Involution Lemma I.7.9.3, $w_-$ maps the marker $\mathcal{L}_{3/2}(+\infty)$ to $\mathcal{R}_{\pi}(\infty)$ and $w_+$ maps the ray $\mathcal{L}_{3/2}$ to the ray $\mathcal{R}_{\pi}$. This gives rise to two constraints for $\bar{v}_+''$: $\mathbf{r}^b$ is restricted on $\mathcal{L}_{3/2}$ and the derivative of $\bar{v}_+''$ at $\mathbf{r}$ in the $D_{\rho_0}^2$-direction is also restricted. Hence $I(\bar{v}_+'') \ge 2$ and the total ECH index is at least 4, a contradiction.

(2B) Suppose that $*_0 = (1, j_0)$ for some $1 \le j_0 \le a$. Summing the ECH indices using $(\beta_1)-(\beta_4)$, $(\beta'_5)$, $(\beta_6)$, and $I(\bar{v}''_{*0}) \ge 1$, the total ECH index is at least 3, a contradiction. However, we want to do slightly better so that the argument carries over in the case of Lemma 3.9.2: The rescaling argument with $m_l \to \infty$ gives $w_+ : cl(\mathbf{B}_+) \to \mathbf{CP}^1$ satisfying the following:

(i) $w_+(\partial B_+) \subset \{\phi = 0, \rho > 0\}$;

(ii) $w_+(-\infty) = 0$ and $w_+(+\infty) = \infty$;

(iii) $w_+$ maps the marker $\mathcal{L}_{3/2}(-\infty)$ to $\mathcal{R}_{\pi}(0)$;

(iv) $w_+|_{\text{int}(\mathbf{B}_+)}$ is a biholomorphism onto its image.

By the Involution Lemmas, $w_+^{-1}$ maps $\mathcal{R}_0(0)$ to $\mathcal{L}_{3/2}(-\infty)$. This contradicts (iii).

(2C) Suppose that $*_0 = (0, j_0)$ for some $1 \le j_0 \le b$. We replace $(\beta_2)$ by:

(β'_2) If $j \ge j_0$, then each positive end of $\bar{v}_{0,j}^{\sharp}$ that limits to $\delta_0$ contributes an additional
+1 towards I.

(β''_2) All the other components of $\bar{v}_{0,j}^{\sharp}$, $1 \le j < j_0$, are cylinders from $\delta_0$ to $h$ or $e$
with $I=1$ or $2$.

$(\beta'_2)$ is a consequence of the argument of Lemma 3.7.8: there is an independent condition imposed on the asymptotic expansion of each positive end of $\bar{v}_{0,j}^{\sharp}$ with $j \ge j_0$ that limits to $\delta_0$. $(\beta''_2)$ is clear. Summing the ECH indices using $(\beta_1)$, $(\beta'_2)$, $(\beta''_2)$ $(\beta_3)$, $(\beta_4)$, $(\beta'_5)$, $(\beta_6)$, and $I(\bar{v}''_{*0}) \ge 1$, the total ECH index is at least 3, a contradiction.

(2D) Suppose that $*_0 = -$. In a manner similar to (2C) we replace $(\beta_3)$ by:

$(\beta'_3)$ Each positive end of $\bar{v}_-$ that limits to $\delta_0$ and is not the end of a section from $\delta_0$ to $x_i$ or $x'_i$ contributes an additional +1 towards I.
($\beta_3''$) All the other components of $\bar{v}_{-}^{\sharp}$ are sections from $\delta_0$ to $x_i$ or $x'_i$ with $I = 1$.

Summing the ECH indices using $(\beta_1)$, $(\beta_2)$, $(\beta'_3)$, $(\beta_3'')$, $(\beta_4)$, $(\beta'_5)$, $(\beta_6)$, and $I(\bar{v}_{-}^{\prime\prime}) \ge 0$, the
total ECH index is at least 2.

Suppose that $\mathbf{r}^b \neq \bar{\mathbf{m}}^b(+\infty)$. Then the rescaling argument gives $w_- : cl(B_-) \to \mathbf{CP}^1$ which satisfies the following:

(i) $w_-(\partial B_-) \subset \{\phi = 0, \rho > 0\}$;

(ii) $w_-(\bar{\mathbf{m}}^b(+\infty)) = 0$ and $w_-(\mathbf{r}^b) = \infty$;

(iii) $w_-(+\infty) \subset \{\phi = 0, \rho > 0\}$; and

(iv) $w_{-|\mathrm{int}(B_-)}$ is a biholomorphism onto its image.

Here (iii) follows from considering the continuation of $w_{-,m_l}$ to upper levels $w_{*,m_l}$ and $\pi_{*,m_l}$ (here $w_{*,m_l}$ is the limit of $\bar{u}_{l_i}$ for $m_l \gg 0$, without taking $m_l \to \infty$) and the fact that $w_{+,m_l} : \Sigma_+ \to \mathbf{CP}^1$ must map $\partial \Sigma_+$ to a thin sector near $\{\phi = 0, \rho > 0\}$.

By the Involution Lemma I.7.9.3 and (i)–(iv), $\mathbf{r}^b$ must lie on $\mathcal{L}_{1/2} \cup \mathcal{L}_{3/2}$ and the map $w_-^{-1}$ maps the marker $\dot{\mathcal{R}}_\pi(\infty)$ to the marker $\dot{\mathcal{L}}_{1/2}$ or $\dot{\mathcal{L}}_{3/2}$ at $\mathbf{r}^b$, as appropriate. This gives rise to two constraints for $\bar{v}''$. Hence $I(\bar{v}''_-) \ge 2$ and the total ECH index is $\ge 4$, a contradiction.

Now suppose that $\mathbf{r}^b = \bar{\mathbf{m}}^b(+\infty)$. Since passing through $\bar{\mathbf{m}}^b(+\infty)$ is a codimension 2 condition, we have $I(\bar{v}''_-) \ge 2$. The total ECH index is $\ge 4$, a contradiction.

(2E) Suppose that $*_0 = (-1, j_0)$ for some $1 \le j_0 \le c$. Then $(\beta_1)-(\beta_3)$ hold and the rescaling argument with $m_l \to \infty$ gives $w_- : \Sigma_- \to \mathbf{CP}^1$ and $\pi_- : \Sigma_- \to cl(B_-)$ such that:

(i) $w_-(\partial \Sigma_-) \subset \{\phi = 0, \rho > 0\}$;

(ii) $w_-(z_0) = 0$ for some $z_0 \in \pi_-^{-1}(\bar{\mathbf{m}}^b(+\infty))$;

(iii) $w_-(z_1) = \infty$ for some $z_1 \in \pi_-^{-1}(-\infty)$;

(iv) $w_-(\pi_-^{-1}(+\infty)) \subset \{\phi = 0, \rho > 0\}$;

(v) $w_{-|\mathrm{int}(\Sigma_-)}$ is a biholomorphism onto its image.

Here (iv) follows from the considerations similar to those of Lemma 3.7.8, together with
(a)–(d). By the Involution Lemmas, $\pi_- \circ w_-^{-1}$, where defined, maps the component of
$\{\mathrm{Im}(z) = 0\}$ passing through 0 to $\mathcal{L}_{3/2}$. This contradicts (iii).

(3) Suppose there is a boundary point $\mathbf{r} \in \partial F_{*0}$ at $z_\infty$. Then the neighborhood of $\bar{v}_{*0}''(\mathbf{r})$ contributes at least $k_0 - 1 \gg 2g$ towards $n^*$ by Lemma 3.4.8. We remark that a priori there could be more than one boundary point at $z_\infty$.

(3A) Suppose that $\bar{v}'_- = \emptyset$. Then there is a point $\mathbf{q} \in \mathrm{int}(F_-)$ and a sufficiently small neighborhood $N(\mathbf{q}) \subset \dot{F}_-$ of $\mathbf{q}$ such that $\bar{v}_-(\mathbf{q}) = \bar{\mathbf{m}}^+(+\infty)$ and $n^*(\bar{v}_-(N(\mathbf{q}))) \ge m$. This is a contradiction.

(3B) Suppose that $\bar{v}'_- \neq \emptyset$ and $\bar{v}_{*0}' = \emptyset$, i.e., $\mathbf{r}$ is of type $(\mathbf{P}_1)$. Then there exist restrictions $\tilde{u}_{i,-}$ and $\tilde{u}_{i,*_0}$ of $\bar{u}_i$ such that $\tilde{u}_{i,-}$ is close to $\bar{v}'_-$ after translation and passes
through $\overline{\mathfrak{m}}(\tau_i)$, $\tilde{u}_{i,*_0}$ is close to $\overline{v}''_{*0}$ after translation and nontrivially intersects $\sigma_{\infty}^{\tau_i}$, and the images of $\tilde{u}_{i,-}$ and $\tilde{u}_{i,*_0}$ are disjoint. This is a contradiction since we have an excess of $n^*$.

(3C) Suppose that $\overline{v}'_{-} \neq \emptyset$, $\overline{v}'_{*0} \neq \emptyset$, and $*_{0} = (1, j), j = 0, \dots, a$.

We claim that (a)–(d) from the proof of (1) hold, where we are only considering levels $\overline{v}_{*} < \overline{v}_{+}$. Indeed, if (a)–(c) do not hold, then the sum of $n^*$ of the ends is $\ge m - 2g$ by Lemmas 3.4.2 and 3.7.6, a contradiction. (The situation of $\overline{v}''_{-1,j}$ is not explicitly covered by Lemma 3.7.6, but the same proof works.) (d) is a consequence of (a)–(c). The claim implies that $(\beta_1)-(\beta_4)$ hold.

Next we claim that there must be a boundary point at $z_\infty$ on $\overline{v}_+$. Arguing by contradiction, if there is no boundary point at $z_\infty$ on $\overline{v}_+$, then a rescaling argument similar to (2B) gives $w_+ : \Sigma_+ \to \mathbf{CP}^1$ and $\pi_+ : \Sigma_+ \to \text{cl}(B_+)$ such that:

(i) $w_+(\partial \Sigma_+) \subset \{\phi = 0, \rho > 0\}$;

(ii) $w_+(z_0) = 0$ for some $z_0 \in \pi_+^{-1}(-\infty)$;

(iii) $w_+(\pi_+^{-1}(-\infty)) \subset \{\phi = 0, \rho \ge 0\}$;

(iv) $w_+(z_1) = \infty$ for some $z_1 \in \pi_+^{-1}(+\infty)$;

(v) $w_{+|\text{int}(\Sigma_+)}$ is a biholomorphism onto its image.

Here (iii) follows from observing that:

($\beta_2''$) There is no component of $\overline{v}''_{0,j}$ which is a cylinder from $\delta_0$ to $e$.

Indeed, $\mathfrak{r}$ contributes at least 2 towards I by Lemma 3.7.5, $I(\overline{v}''_{*0}) \ge 1$ if $*_{0} \neq +$, and a component of $\overline{v}''_{0,j}$ which is a cylinder from $\delta_0$ to $e$ contributes I = 2. Together with $(\beta_1)-(\beta_4)$, the total ECH index is at least 4, if there is a cylinder from $\delta_0$ to $e$, a contradiction.

By the Involution Lemmas, $\pi_+ \circ w_+^{-1}$, where defined, maps $\dot{\mathcal{R}}_0(0)$ to $\dot{\mathcal{L}}_{3/2}(-\infty)$, a contradiction. Hence there must be a boundary point at $z_\infty$ on $\overline{v}_+$. A similar argument implies that a boundary point at $z_\infty$ must lie on $\mathcal{L}_{3/2}$ and project to a large sector.

By Lemma 3.7.5, the boundary points at $z_\infty$ contribute at least 2 towards I. In addition, the constraint of lying in $\mathcal{L}_{3/2}$ contributes 1 towards I and the large sector condition contributes 1 towards I. These add up to at least I = 4, a contradiction.

(3D) Suppose that $\overline{v}'_{-} \neq \emptyset$, $\overline{v}'_{*0} \neq \emptyset$, and $*_{0} = (-1, j), j = 1, \dots, c+1$.

We would like to apply the argument from (3C) and (2E). The slight complication is that we may have ends of $\overline{v}''_{-1,j}$ that limit to $z_\infty$ but map to non-thin sectors. In particular, there may be components of $\overline{v}''_{-1,j}$ have at least two ends that limit to $z_\infty$ but has ind = 1 and it is not a priori clear that $(\beta_4)$ holds.

We claim that:

($\beta'_4$) Each positive end of $\overline{v}''_{-1,j}, j = 1, \dots, c$, that limits to $z_\infty$ contributes +1 towards I.
For ease of notation assume that $c = 1$ and $p_{0,0} = 2$ during the proof of the claim. Applying the rescaling argument without taking $m \to \infty$, we obtain $w_- : \Sigma_- \to \mathbf{CP}^1$ and $\pi_- : \Sigma_- \to \text{cl}(\mathbf{B}_-)$ such that:

(i) $w_-(z_0) = 0$ for some $z_0 \in \pi_-^{-1}(\bar{\mathfrak{m}}^b(+\infty))$;

(ii) $w_-$ maps the negative ends of $\Sigma_-$ to sectors $\mathcal{S}(\bar{b}_{k,l}, \bar{h}(\bar{b}_{k',l}'))$.

We now make crucial use of the fact that the sectors $\mathcal{S}(\bar{b}_{k,l}, \bar{h}(\bar{b}_{k',l}))$ corresponding to the negative ends of $\Sigma_-$ have different angles, provided the sectors are not both thin sectors; this follows from the definition of $\bar{a}$ from Section I.5.2.2. Since each thin sector corresponds to a thin strip of $\bar{v}_{-1,1}^\sharp$ which contributes $I = 1$, let us assume that there are no thin sectors in (ii). We use coordinates $(s, t) \in (-\infty, c] \times [0, 1]$ for the two negative ends $\mathcal{E}_{-,1}$, $\mathcal{E}_{-,2}$ of $\Sigma_-$ which agree with the coordinates $(s, t)$ on $\mathbf{B}_-$ and write $w_-^i := w_-|_{\mathcal{E}_{-,i}}$. Then $w_-^i(s, t) \approx c_i e^{\lambda_i(s+i\bar{t})}$, where $\lambda_i > \lambda_2 > 0$ without loss of generality; $c_1, c_2 \neq 0$; and the approximation gets better as $s \to -\infty$. On the other hand, suppose that $\text{ind}(\bar{v}_{-1,1}^\sharp) = 1$ and $\bar{v}_{-1,1}^\sharp$ is connected and has two positive ends $\mathcal{E}_{+,i}$, $i = 1, 2$, that limit to $z_\infty$. Here $\mathcal{E}_{+,i}$ is to be paired with $\mathcal{E}_{-,i}$. We use coordinates $(s', t') \in [c', \infty) \times [0, 1]$ for the positive ends $\mathcal{E}_{+,i}$ which agree with the coordinates on $\mathbf{B} = \mathbf{R} \times [0, 1]$. Then $\mathcal{E}_{+,1}$ approaches $z_\infty$ much faster than $\mathcal{E}_{+,2}$ does, as $s' \to +\infty$. This is inconsistent with our description of $w_-$, when we change coordinates $s' = s + \bar{c}$ for some $\bar{c}$. Hence each end $\mathcal{E}_{+,i}$ should be allowed to move independently and each positive end of $\bar{v}_{-1,1}^\sharp$ that limits to $z_\infty$ contributes +1 towards I.

($\beta'_4$) replaces $(\beta_4)$. We now take the limit $m \to \infty$ and argue as in (3C), which implies that the boundary point at $z_\infty$ must be on $\bar{v}_-$ and gives a total ECH contribution of at least 4. $\square$

**3.7.6.** *The case $\bar{v}'_- \neq \emptyset$.*

**Lemma 3.7.11.** — If $m \gg 0$, $\bar{u}_\infty \in \partial_{\{+\infty\}}\mathcal{M}$, and $\bar{v}'_- \neq \emptyset$, then no negative end of $\bar{v}_{0,j}^\sharp$, $1 \le j \le b$, is asymptotic to a multiple of $\delta_0$.

*Proof.* — Arguing by contradiction, suppose there exists some $\bar{v}_{0,j_0}^\sharp$, $1 \le j_0 \le b$, which has a negative end asymptotic to a multiple of $\delta_0$, say $\delta_0^{q_{0,j_0}}$. Since the negative end of $\bar{v}_{0,j_0}^\sharp$ contributes at least $m - q_{0,j_0}$ towards $n^*$, the following hold:

(i) at most one $\bar{v}_{0,j}^\sharp, j > 0$, has a negative end asymptotic to a multiple of $\delta_0$;

(ii) only one negative end of $\bar{v}_{0,j_0}^\sharp$ limits to a multiple of $\delta_0$;

(iii) $p_{0,0} \le p_{0,1} \le \cdots \le p_{0,j_0-1} \le p_{0,j_0} + q_{0,j_0}$ and $p_{0,j_0} \le p_{0,j_0+1} \le \cdots \le p_{0,b+1}$;

(iv) if $\mathcal{E}$ is a positive end of $\bar{v}_{0,j}^\sharp, j \ge 0$, that limits to $\delta_0^r$, then $n^*(\mathcal{E}) = r$; and

(v) $\cup_{j=1}^c \bar{v}_{-1,j}^\sharp$ is a union of $p_{0,0} > 0$ thin strips.
By Lemma 3.7.5, all the components besides $\bar{v}'_+$ have nonnegative ECH index. We have contributions $(\beta_1)$, $(\beta_3)$, $(\beta_4)$, $(\beta_6)$ as in Lemma 3.7.10, as well as the following variant of $(\beta_2)$:

($\tilde{\beta}'_2$) For $1 \le j < j_0$, $\bar{v}^\sharp_{0,j}$ is a union of cylinders from $\delta_0$ to $h$ or $e$ and

$$ \sum_{j=1}^{j_0-1} I(\bar{v}^\sharp_{0,j}) \ge p_{0,j_0} + q_{0,j_0} - p_{0,1}. $$

($\tilde{\beta}''_2$) For $j_0 + 1 \le j \le b$, $\bar{v}^\sharp_{0,j}$ is a union of cylinders from $\delta_0$ to $h$ or $e$ and

$$ \sum_{j=j_0+1}^{b} I(\bar{v}^\sharp_{0,j}) \ge p_{0,b+1} - p_{0,j_0+1}. $$

($\tilde{\beta}'''_2$) $I(\bar{v}^\sharp_{0,j_0}) \ge p_{0,j_0+1} - p_{0,j_0} + 1$.

($\tilde{\beta}'_2$) and ($\tilde{\beta}''_2$) are immediate from $n^*(\bar{v}^\sharp_{0,j_0}) \ge m - q_{0,j_0}$. ($\tilde{\beta}'''_2$) follows from the proof of Lemma 3.7.8, which implies that each positive end of $\bar{v}^\sharp_{0,j_0}$ that limits to $\delta_0$ additionally contributes one constraint.

Since $\bar{v}_{0,j_0}$ does not satisfy the partition condition at $\delta_0^{p_{0,j_0}+q_{0,j_0}}$, a straightforward calculation which takes into account the braiding near $\delta_0^{p_{0,j_0}+q_{0,j_0}}$ gives:

$$ (3.7.6) \qquad I(\bar{v}_{0,j_0}) \ge I(\bar{v}'_{0,j_0}) + I(\bar{v}''_{0,j_0}) + 2p_{0,j_0}. $$

Summing the contributions of $(\beta_1)$, $(\tilde{\beta}'_2)$–$(\tilde{\beta}''_2)$, $(\beta_3)$, $(\beta_4)$, and Equation (3.7.6), we obtain:

$$ (3.7.7) \qquad \sum_{\bar{v}_*} I(\bar{v}_*) \ge q_{0,j_0} + 2p_{0,j_0} + 1. $$

Here $q_{0,j_0} \ge 1$ by assumption. If $p_{0,j_0} > 0$ or $p_{0,j_0+1} > 0$, then $\sum_{\bar{v}_*} I(\bar{v}_*) \ge 4$, a contradiction. On the other hand, if $p_{0,j_0} = \dots = p_{0,b+1} = 0$, then $I(\bar{v}^\sharp_{0,j_0}) \ge 2$ by the usual rescaling argument (cf. Cases (3)–(6) of Theorem I.7.10.1). In this case the right-hand side of Equation (3.7.7) can be increased by one and $\sum_{\bar{v}_*} I(\bar{v}_*) \ge 3$, a contradiction. $\square$

**Lemma 3.7.12.** — If $m \gg 0$, $\bar{u}_\infty \in \partial_{\{+\infty\}} \mathcal{M}$, $\bar{v}'_- \ne \emptyset$, and $\bar{v}^\sharp_+$ has a negative end asymptotic to a multiple of $\delta_0$, then $\bar{u}_\infty \in A'_2$.

Here $A'_2$ is as given in Step 4 of the proof of Theorem 3.3.1.

*Proof.* — This is similar to Lemma 3.7.11. The contributions of $(\beta_1)$, $(\tilde{\beta}'_2)$, $(\beta_3)$, $(\beta_4)$, and Equation (3.7.6) with $j_0 = b + 1$ add up to $q_+ + 2p_+$.
If $p_+ > 0$, then $\sum_{\bar{v}_*} I(\bar{v}_*) \ge q_+ + 2p_+ \ge 3$, a contradiction.

If $p_+ = 0$, then $q_+ + 2p_+ \ge 1$. By the usual rescaling argument, the negative end of $\bar{v}_+^\sharp$ that limits to $\delta_0$ additionally contributes +1 to I. Hence if $\sum_{\bar{v}_*} I(\bar{v}_*) = 2$, then $q_+ = 1$. This implies that $\bar{u}_\infty \in A'_2$. $\square$

**Lemma 3.7.13.** — If $m \gg 0$, $\bar{u}_\infty \in \partial_{(+\infty)}\mathcal{M}$, and $\bar{v}' \ne \emptyset$, then $\bar{u}_\infty \in A'_2$.

*Proof.* — Consider $\bar{u}_\infty \in \partial_{(+\infty)}\mathcal{M}$ such that $\bar{v}' \ne \emptyset$. By Lemma 3.7.10, there is no boundary point at $z_\infty$, no fiber component and no $\bar{v}_*''$ such that $\bar{v}_*' \cap \bar{v}_*'' \ne \emptyset$. By Lemmas 3.7.11 and 3.7.12, if $\bar{v}_{0,j_0}^\sharp$, $1 \le j_0 \le b+1$, has a negative end asymptotic to a multiple of $\delta_0$, then $\bar{u}_\infty \in A'_2$.

It remains to consider three possibilities, all of which satisfy $p_{0,0} \le \dots \le p_{0,b+1}$:

(1) Some $\bar{v}_{-1,j_0}^\sharp, j_0 = 1, \dots, c+1$, has a positive or negative end $\mathcal{E}$ at $z_\infty$ such that $n^*(\mathcal{E}) > m/2$.

(2) Some $\bar{v}_{1,j_0}^\sharp, j_0 = 0, \dots, a$, has a positive or negative end $\mathcal{E}$ at $z_\infty$ such that $n^*(\mathcal{E}) > m/2$.

(3) Some $\bar{v}_{0,j_0}^\sharp, j_0 = 0, \dots, b$, has a positive end $\mathcal{E}$ at a multiple of $\delta_0$ such that $n^*(\mathcal{E}) > m/2$.

It is clear that (1), (2), and (3) are mutually exclusive.

(1) is eliminated in a manner similar to (2E) of Lemma 3.7.10 and (2) in a manner similar to Case (6$_i$) of Lemma 3.4.28.

(3) In this case, (a)–(c) in the proof of Lemma 3.7.10 hold, with the exception of the level $\bar{v}_{0,j_0}$. Observe that $\bar{v}_{0,j_0}' \ne \emptyset$ and $\bar{v}_{0,j_0}^\sharp$ consists of:

- cylinders from $\delta_0$ to $e$ or $h$ which contribute $I=1$ or $2$ each; and

- a curve $\bar{v}_{0,j_0}^{\sharp 1}$ with one positive end $\mathcal{E}_1^\sharp$ which is asymptotic to $\delta_0^{c_1}$ and satisfies $n^*(\mathcal{E}_1^\sharp) = m + c_1$ and other positive ends which in total are asymptotic to $\delta_0^{c_2}$ and satisfy $n^* = c_2$.

We compute the contributions to the ECH index from the ends of $\bar{v}_{0,j_0}' \cup \bar{v}_{0,j_0}^\sharp$ that limit to multiples of $\delta_0$ at the positive end. Here is the list of such ends:

- the union $\mathcal{E}'$ of positive ends of $\bar{v}_{0,j_0}'$;

- $\mathcal{E}_1^\sharp$ satisfying $n^*(\mathcal{E}_1^\sharp) = m + c_1$;

- the union $\mathcal{E}_2^\sharp$ of positive ends of $\bar{v}_{0,j_0}^\sharp$ that correspond to punctures of $w_{0,j_0+1}$;

- the union $\mathcal{E}_3^\sharp$ of all other positive ends.

We use the formula:

$$ (3.7.8) \qquad \mathrm{ind}(\bar{u}) + (\tilde{\mu}_{\tau}(\bar{u}) - \mu_{\tau}(\bar{u}) - w_{\tau}(\bar{u})) \le \mathrm{I}(\bar{u}), $$
where the notation is as in Section I.3.4 and the equation follows from the adjunction inequality and [Hu2, Lemma 4.20]. The end $\mathcal{E}_1^\sharp$ has contributions $\tilde{\mu}_\tau = c_1$, $\mu_\tau = 1$, and $w_\tau = 1 - c_1$. Moreover, since the vanishing of the leading asymptotic eigenfunction corresponding to the end $\mathcal{E}_1^\sharp$ is a codimension two condition which contributes 2 to the Fredholm index, the extra contribution to I from $\mathcal{E}_1^\sharp$ is $c_1 - 1 - (1 - c_1) + 2 = 2c_1$. The contributions to I from $\mathcal{E}'$ and $\mathcal{E}_2^\sharp$, arising from a writhe computation, is $2(\deg \mathcal{E}' + \deg \mathcal{E}_2^\sharp) = 2(p_{0,j_0} + \deg \mathcal{E}_2^\sharp)$. Finally, the contributions to I from $\mathcal{E}_3^\sharp$ is $\deg \mathcal{E}_3^\sharp$ by the argument of Lemma 3.7.8.

The total ECH index of all the levels is $\ge 2c_1 + 2p_{0,j_0} \ge 4$, a contradiction. $\square$

**3.7.7.** *The case $\bar{v}'_\tau = \emptyset$.*

**Lemma 3.7.14.** — If $m \gg 0$, then there is no $\bar{u}_\infty \in \partial_{[+\infty]} \mathcal{M}$ such that $\bar{v}'_\infty = \emptyset$.

*Proof.* — Arguing by contradiction, suppose there exists $\bar{u}_\infty \in \partial_{[+\infty]} \mathcal{M}$ such that $\bar{v}'_\infty = \emptyset$. By Lemma 3.7.10, there is no boundary point at $z_\infty$, no fiber component, and no $\bar{v}''_\infty$ such that $\bar{v}'_\infty \cap \bar{v}''_\infty \neq \emptyset$.

The point constraint gives $n^*(\bar{v}^{\prime\prime}_\infty) \ge m$ and $I(\bar{v}^{\prime\prime}_\infty) \ge 2$. This immediately implies that (a)–(d) in the proof of Lemma 3.7.10 hold. In particular, $(\beta_2)$ from Lemma 3.7.10 holds.

If $p_{0,b+1} > 0$, then the proof of Lemma 3.7.8 gives us a contradiction. Hence $p_{0,b+1} = 0$. If a positive end of $\bar{v}''_\infty$ limits to $z_\infty$, then a component $\tilde{v}$ of $\bar{v}''_\infty$ must be a section of $\bar{W}_+$ from $z_\infty$ to $h$ or $e$. However, by the choice of $h$ from Section 3.2.3 and the proof of Lemma I.6.6.5, sections of $\bar{W}_+$ from $z_\infty$ to $h$ cannot exist, leaving us with the possibility that $\tilde{v}$ is a section from $z_\infty$ to $e$. Hence $I(\tilde{v}) \ge 1$ and the total ECH index is $\ge 3$, a contradiction. On the other hand, if $\bar{v}''_\infty = \bar{v}^b_\infty$, then $I(\bar{v}_{1,j}) \ge 1$ for some $j > 0$, which is again a contradiction. $\square$

*Proof of Lemma 3.3.10.* — Lemma 3.3.10 follows from Lemmas 3.7.13 and 3.7.14. $\square$

**3.8.** *Degeneration at $-\infty$, part II.* — Let $\bar{u}_\infty \in \partial_{[-\infty]} \mathcal{M}$ be the limit of $\bar{u}_i \in \mathcal{M}_{\tau_i}$, where $\tau_i \to -\infty$.

**Lemma 3.8.1.** — If $\bar{u}_\infty$ has no fiber components and $\bar{v}'_1 = \emptyset$, then the ECH index of each level is nonnegative.

*Proof.* — The proof is similar to that of Lemma 3.5.2 and uses the considerations of Lemma 3.7.5. $\square$

**3.8.1.** *The case $\bar{v}'_2 = \emptyset$.*
**Lemma 3.8.2.** — If $\bar{u}_{\infty} \in \partial_{[-\infty]} \mathcal{M}$ and $\bar{v}'_2 = \emptyset$, then:

(1) $\bar{u}_{\infty}$ is a 2-level building consisting of $\bar{v}_1$ with $I(\bar{v}_1) = 0$, $\bar{v}_2$ with $I(\bar{v}_2) = 2$, and $\bar{v}'_j = \emptyset$ for $j = 1, 2$;

(2) the left and right ends of $\bar{v}_1$ (= left and right ends of $\bar{v}_2$) do not limit to $z_{\infty}$;

(3) $\bar{u}_{\infty}$ has no fiber component;

(4) $\bar{u}_{\infty}$ has no boundary point at $z_{\infty}$.

*Proof.* — Suppose that $\bar{v}'_2 = \emptyset$. First observe that

$$ (3.8.1) \qquad n^*(\bar{v}_2(\mathcal{N}(q))) \ge m, $$

where $q \in \dot{\mathcal{F}}_2$ is the point such that $\bar{v}_2(q) = \bar{m}(-\infty)$.

We also have $I(\bar{v}_2) \ge 2$: If $\bar{u}_{\infty}$ has a fiber component $\tilde{v}$, then $\tilde{v}$ must pass through $\bar{m}(-\infty)$ and $I(\bar{v}_2) \ge 2g+2 \ge 4$ by an argument similar to that of Lemma I.7.5.5. On the other hand, if $\bar{u}_{\infty}$ does not have a fiber component, then $I(\bar{v}_2) \ge 2$, since passing through $\bar{m}(-\infty)$ is a generic codimension two condition.

(1) We claim that $\bar{v}'_1 = \emptyset$. Arguing by contradiction, if $\bar{v}'_1 \ne \emptyset$, then at least one of $\bar{v}^\sharp_{L,j}, j = 0, \dots, a$, has a right end $\mathcal{E}_R$ that limits to $z_\infty$. On the other hand, the end $\mathcal{E}_R$ satisfies

$$ (3.8.2) \qquad n^*(\mathcal{E}_R) \ge k_0 - 1 \gg 2g, $$

which contradicts Equation (3.8.1). This proves the claim.

The claim and Lemma 3.8.1 imply that each level of $\bar{u}_{\infty}$ has nonnegative ECH index. Hence $I(\bar{v}_2) = 2$, $I(\bar{v}_1) = 0$, and $a=b=c=d=0$ since a level $\bar{v}_{L,j}, j = 1, \dots, a$, is not a connector if and only if $I(\bar{v}_{L,j}) > 0$ (and the same holds for $\bar{v}_{R,j}, j = 1, \dots, b, \bar{v}_{B,j}, j = 1, \dots, c$, and $\bar{v}_{T,j}, j = 1, \dots, d$).

(2) If $\bar{v}_2$ has a right end $\mathcal{E}_R$ that limits to $z_\infty$, then $\mathcal{E}_R$ satisfies Equation (3.8.2), and we have a contradiction of Equation (3.8.1). If $\bar{v}_2$ has a left end that limits to $z_\infty$, then $\bar{v}_1$ has a right end $\mathcal{E}_R$ that limits to $z_\infty$ and satisfies Equation (3.8.2) (since $\bar{v}'_1 = \emptyset$), which is again a contradiction.

(3) Since $I(\bar{v}_2) = 2$, we cannot have a fiber component.

(4) A boundary point at $z_\infty$ contradicts Equation (3.8.1). $\square$

<figure><img src="image_26.png" /></figure>

**3.8.2.** *The case* $\bar{v}'_1 = \emptyset, \bar{v}'_2 \ne \emptyset$.

**Lemma 3.8.3.** — If $m \gg 0$, then there is no $\bar{u}_{\infty} \in \partial_{[-\infty]} \mathcal{M}$ such that $\bar{v}'_1 = \emptyset$ and $\bar{v}'_2 \ne \emptyset$.

*Proof.* — Arguing by contradiction, suppose there exists $\bar{u}_{\infty} \in \partial_{[-\infty]} \mathcal{M}$ such that $\bar{v}'_1 = \emptyset$ and $\bar{v}'_2 \ne \emptyset$. The analog of Lemma 3.5.1 holds, i.e.,

$$ \sum_{i=1}^{q} n^*(\mathcal{E}_i) + \sum_{i=1}^{r} n^*(\mathcal{E}'_i) = m, \qquad \sum_{i=1}^{q} n^{*,\text{alt}}(\mathcal{E}_i) + \sum_{i=1}^{r} n^{*,\text{alt}}(\mathcal{E}'_i) \ge m - 2g, $$
where $\mathcal{E}_i$, $i = 1, \dots, q$, are the ends that limit to $z_\infty$ and $\mathcal{E}'_i$, $i = 1, \dots, r$, are the neighborhoods of boundary points of type (P$_3$) of all the $\bar{v}^\sharp_{L,j}, j = 0, \dots, a$, and $\bar{v}^\sharp_{R,j}, j = 0, \dots, b$. The conclusion of Lemma 3.5.4 then holds by an almost identical argument. The proof of Lemma 3.5.12 then carries over without modification.
□

**3.8.3.** *The case $\bar{v}'_1 \neq \emptyset$, $\bar{v}'_2 \neq \emptyset$.*

**Lemma 3.8.4.** — Suppose $\bar{v}'_1 \neq \emptyset$ and $\bar{v}'_2 \neq \emptyset$. Let $\mathcal{E}_i$, $i = 1, \dots, q$, be the ends of all the $\bar{v}^\sharp_*$ that limit to $z_\infty$ and let $\mathcal{E}'_i$, $i = 1, \dots, r$, be the neighborhoods of the boundary points of type (P$_3$). Then one of the following holds:

(a) $\sum_{i=1}^q n^*(\mathcal{E}_i) + \sum_{i=1}^r n^*(\mathcal{E}'_i) \ge m - 2g$.

(b) Each end $\mathcal{E}_i$ is an end of $\bar{v}^\sharp_{B,j}, j = 1, \dots, c$, or $\bar{v}^\sharp_{T,j}, j = 0, \dots, d$, that projects to a thin sector of type $\mathfrak{S}(\bar{a}_{i',j'}, \bar{h}(\bar{a}_{i',j'}))$ or $\mathfrak{S}(\bar{b}_{i',j'}, \bar{h}(\bar{b}_{i',j'}))$. In particular, $\mathcal{E}_i$ is not a left or right end of $\bar{v}^\sharp_*$, where $*= (L,j), j=0, \dots, a$, or $(R,j), j=0, \dots, b$.

*Proof.* — We apply the continuation argument. If some $\mathcal{E}_i$, $i = 1, \dots, q$, does not project to a thin sector (of type $\mathfrak{S}(\bar{a}_{i',j'}, \bar{h}(\bar{a}_{i',j'}))$, $\mathfrak{S}(\bar{b}_{i',j'}, \bar{h}(\bar{b}_{i',j'}))$, or $\mathfrak{S}(\bar{b}_{i',j'}, \bar{a}_{i',j'})$), then the sectors $\pi_{D_{P_0}^2}(\mathcal{E}_i)$, $i = 1, \dots, q$, and $\pi_{D_{P_0}^2}(\mathcal{E}'_i)$, $i = 1, \dots, r$, will sweep out a neighborhood of $z_\infty$ with the exception of some thin sectors, implying (a).

On the other hand, suppose that all the ends $\mathcal{E}_i$, $i = 1, \dots, r$, project to thin sectors. We claim that there are no thin sectors of type $\mathfrak{S}(\bar{b}_{i',j'}, \bar{a}_{i',j'})$. Indeed, the number of left and right ends of all the $\bar{v}^\sharp_*$ that limit to $z_\infty$ must be equal and the right ends cannot map to thin sectors. This gives (b).
□

**Lemma 3.8.5.** — If $\bar{u}_\infty \in \partial_{\{\pm\infty\}} \mathcal{M}$, $\bar{v}'_1 \neq \emptyset$, and $\bar{v}'_2 \neq \emptyset$, then:

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

(2) if $\bar{v}''_*$ satisfies $\bar{v}'_* \cap \bar{v}''_* \neq \emptyset$ and $\bar{v}'_* \cap \bar{v}''_* \subset \text{int}(\bar{v}'_*)$, then $*= 1$ and $p_* = 1$;

(3) $\bar{u}_\infty$ has no boundary point of type (P$_1$) or (P$_2$).

*Proof.* — This is similar to the proof of Lemma 3.7.10 and uses Equations (3.7.1) and (3.7.2).

First observe that if $\bar{u}_\infty$ has a fiber component, a boundary point of type (P$_1$) or (P$_2$), or some $\bar{v}'_* \cap \bar{v}''_* \neq \emptyset$ with $\bar{v}''_* \cap \text{int}(\bar{v}^\sharp_*)$, then we are in the situation of Lemma 3.8.4(b). It is not hard to verify that $\text{I}(\bar{v}^\sharp_*) \ge 0$ for all $*$ with the exception of $\bar{v}_1$.

(1) Suppose $\bar{u}_\infty$ has a fiber component $\tilde{v}$. Then we have the following contributions towards I:

(β$_1$) $\text{I}(\bar{v}'_1) = -p_1 = -\text{deg}(\bar{v}'_1)$.

(β$_2$) $\sum_{j=1}^c \text{I}(\bar{v}^\sharp_{B,j}) = p_1$.

(β$_3$) If the fiber $\tilde{v}$ is a component of $\bar{v}_*$, then $\tilde{v}$ and the intersection $\tilde{v} \cap (\bar{v}_* - \tilde{v})$ contribute $2g + 2 \ge 4$ towards I.
The argument is similar to that of Lemma 3.7.10(1). This gives a total of $I > 2$, which is a contradiction.

(2) Suppose $\bar{v}'_{*0} \cap \bar{v}''_{*0} \neq \emptyset$ and $\bar{v}'_{*0} \cap \bar{v}''_{*0} \subset \text{int}(\bar{v}'_{*0})$. If $*0 \neq 1$, then $I(\bar{v}''_{*0}) \geq 1$ and the intersection points contribute at least $2p_{*0} \geq 2$. If $*0 = 1$ and $p_{*0} > 1$, then $I(\bar{v}''_{*0}) \geq 0$ and the intersection points contribute at least $2p_{*0} \geq 4$. Combined with the ECH contributions from $(\beta_1)$ and $(\beta_2)$, we have a contradiction.

(3) If $\bar{u}_{\infty}$ has a boundary point of type $(P_1)$, then the argument from (3B) of Lemma 3.7.10 implies an excess of $n^*$. If $\bar{u}_{\infty}$ has a boundary point of type $(P_2)$, then the analog of Lemma 3.7.5 implies that the boundary points at $z_{\infty}$ contribute $2p_{*0} \geq 2$; there is also a large sector which contributes an additional $+1$. Combined with $(\beta_1)$ and $(\beta_2)$ we obtain a total of $I > 2$, a contradiction. $\square$

**Lemma 3.8.6.** — If $m \gg 0$, $\bar{u}_{\infty} \in \partial_{(-\infty]}M$, $\bar{v}'_1 \neq \emptyset$, and $\bar{v}'_2 \neq \emptyset$, then there is no level $\bar{v}_*$ such that $\bar{v}'_* \cap \bar{v}''_* \neq \emptyset$ and $\bar{v}'_* \cap \bar{v}''_* \subset \text{int}(\bar{v}_*)$.

*Proof.* — Arguing by contradiction, suppose that $\bar{v}'_{*0} \cap \bar{v}''_{*0} \neq \emptyset$ and $\bar{v}'_{*0} \cap \bar{v}''_{*0} \subset \text{int}(\bar{v}'_{*0})$ for some $\bar{v}_{*0}$. Then $*0 = 1$ and $p_1 = 1$ by Lemma 3.8.5(2). By Lemma 3.8.4(b), we have $p_{L,j} = 1$ and $p_{R,j} = 1$ for all $j$. Hence we may assume that $a = b = 0$. Equation (3.7.2) then implies that $c = 1$ and $d = 0$.

By the rescaling argument with $m \to \infty$, we obtain a 2-level building $w_1 \cup w_2$, where

$$w_2 : \text{cl}(\mathbb{B}_{-\infty,2}) \to \mathbf{CP}^1, \quad w_1 : \text{cl}(\mathbb{B}_{-\infty,1}) \to \mathbf{CP}^1,$$

and (i)–(viii), given below, hold. Here $\text{cl}(\mathbb{B}_{-\infty,2})$ is obtained from $\mathbb{B}_{-\infty,2}$ by adding the left and right points at infinity, denoted $t = \pm\infty$; similarly $\text{cl}(\mathbb{B}_{-\infty,1})$ is obtained from $\mathbb{B}_{-\infty,1}$ by adding $s = \pm\infty$ and $t = \pm\infty$. The map $w_2$ satisfies the following:

(i) $w_2(\bar{m}^b(-\infty)) = 0$;

(ii) $w_2(\partial \text{cl}(\mathbb{B}_{-\infty,2})) \subset \{\phi = 0, \rho > 0\} \cup \{\infty\}$;

(iii) $w_2(t = +\infty) = \infty$ and $w_2(t = -\infty) \subset \{\phi = 0, \rho > 0\}$ (or vice versa);

(iv) $w_2|_{\text{int}(\mathbb{B}_{-\infty,2})}$ is a biholomorphism onto its image.

The map $w_1$ satisfies the following:

(v) $w_1(t = -\infty) = 0$ (or $w_1(t = +\infty) = 0$; for simplicity assume the former);

(vi) $w_1(\partial \text{cl}(\mathbb{B}_{-\infty,1})) \subset \{\phi = 0, \rho > 0\} \cup \{\infty\}$;

(vii) $w_1(\mathfrak{r}) = \infty$, where $\mathfrak{r} \in \bar{v}'_1 \cap \bar{v}''_1$;

(viii) $w_1|_{\text{int}(\mathbb{B}_{-\infty,1})}$ is a biholomorphism onto its image.

In view of the above, we consider the “tropical curves”

$$\Xi_i : \Gamma \to [-1, 1]/(-1 \sim 1) \times [0, d_i]$$
as in Section 3.5.6. The proof strategy of Lemmas 3.5.12 and 3.5.13 carry over to give us a contradiction; this is due to the disparity in the growth rates of the left and right ends of $\bar{u}_i$ when restricted to $[-2, 2] \times [1+L, r(\tau_i)-L]$.

The combination of Lemmas 3.8.5 and 3.8.6 give the following, which is the analog of Lemma 3.5.4(2),(3):

**Corollary 3.8.7.** — If $m \gg 0$, $\bar{u}_\infty \in \partial_{[-\infty]} \mathcal{M}$, $\bar{v}'_1 \neq \emptyset$, and $\bar{v}'_2 \neq \emptyset$, then there is no fiber component, no boundary point of type $(P_1)$ or $(P_2)$, and no intersection point $\mathbf{r} \in \bar{v}'_* \cap \bar{v}''_*$ such that $\mathbf{r} \in \bar{v}'_* \cap \bar{v}''_* \subset \text{int}(\bar{v}'_*)$.

Before embarking on the proof of Lemma 3.8.11, we prove Lemma 3.8.8, which is the analog of Lemma 3.5.4(4) and is a bit involved.

First we recall and modify some notation from Section I.5.7. We use the convention that $\star = (L,j), j = 1, \dots, a$, or $(R,j), j = 1, \dots, b+1$. The data $\vec{D}_{\star, \pm}$ at $z_\infty$ for $\bar{v}_\star$ is given by a $p = p_{\star, \pm}$-tuple of matchings

$$ (3.8.3) \qquad \{(i'_1, j'_1) \to (i_1, j_1), \dots, (i'_p, j'_p) \to (i_p, j_p)\}, $$

where $i_k, i'_k \in \{1, \dots, 2g\}, j_k, j'_k \in \{0, 1\}$ for $k=1, \dots, p$ and $i_k \neq i_l, i'_k \neq i'_l$ for $k \neq l$. Here the subscript $+$ (resp. $-$) in $\vec{D}_{\star, \pm}$ refers to the left (resp. right) end, $(i'_k, j'_k)$ corresponds to $\bar{b}'_{i'_k, j'_k}$, and $(i_k, j_k)$ corresponds to $\bar{a}_{i_k, j_k}$. We write

$$ \vec{D}_{\star, \pm} = \vec{D}'_{\star, \pm} \cup \vec{D}''_{\star, \pm}, $$

where $\vec{D}'_{\star, \pm}$ and $\vec{D}''_{\star, \pm}$ correspond to the ends of $\bar{v}'_\star, \bar{v}''_\star$, respectively.

**Lemma 3.8.8.** — If $\bar{u}_\infty \in \partial_{[-\infty]} \mathcal{M}$, $\bar{v}'_1 \neq \emptyset$ and $\bar{v}'_2 \neq \emptyset$, then every component of $\bar{v}''_\star$, $\star = (L,j), j = 1, \dots, a$, or $(R,j), j = 1, \dots, b+1$, is a thin strip.

*Proof.* — Arguing by contradiction, suppose there exists $\bar{v}''_{\star 0}$ which is not a union of thin strips; we take $\star_0 = (R, 1)$ without loss of generality. Then the following hold:

(a) $n^*(\bar{v}''_{\star 0}) = m$ and $n^{*, \text{alt}}(\bar{v}''_{\star 0}) \ge m - 2g$;

(b) each component $\tilde{v}$ of $\bar{v}''_{T,j}, j = 1, \dots, d$, or $\bar{v}''_{B,j}, j = 1, \dots, c$, is a thin strip with $z_\infty$ at the positive end and $I(\tilde{v}) = 1$;

(c) each component $\tilde{v}$ of $\bar{v}''_\star, \star \neq \star_0$, is a thin strip with $z_\infty$ at the left end and $I(\tilde{v}) = 1$;

(d) each component $\tilde{v}$ of $\bar{v}''_1$ limits to $z_\infty$ only at the left end and the top end, the left and top ends project to thin sectors, $\text{deg}(\tilde{v}) = 1$, and $I(\tilde{v}) = 1$.

(a) follows from the argument of Lemma 3.5.4(4)(a) and (b) and (c) follow immediately from (a). (d) The assertions about the ends of $\tilde{v}$ follow from (a). This then implies that $\tilde{v}$
projects to the domain bounded by $\bar{b}_i, \bar{h}(\bar{b}_i)$, and $\bar{h}(\bar{a}_i)$ in $\bar{S}$, in view of the positions of the
arcs $\bar{b}_{i,j}, \bar{a}_{i,j}, \bar{h}(\bar{b}_{i,j})$, and $\bar{h}(\bar{a}_{i,j})$ from Figure 6.

We now analyze the ends of $\bar{v}_{\star_0}^\sharp$ at $z_\infty$ in more detail. Let $l_+$ and $l_-$ be the number of left and right ends of $\bar{v}_{\star_0}^\sharp$ at $z_\infty$ and let $bp$ be the number of boundary points of type (P$_3$) on $\bar{v}_{\star_0}$. In view of (c) and (d), the total number of left ends of $\bar{v}_*$ that limit to $z_\infty$ besides those of $\bar{v}_{\star_0}^\sharp$ is $l_- - l_+$. By (a slight modification of) the continuation method, we obtain a cycle $\mathcal{Z}$ consisting of $2l_- + bp$ chords, where $l_-$ of the chords are of type $\mathfrak{S}(\bar{a}_{k,l}, \bar{b}_{k',l'})$, $l_+$ of the chords are of type $\mathfrak{S}(\bar{b}_{k,l}, \bar{a}_{k',l'})$, $bp$ of the chords are of type $\mathfrak{S}(\bar{a}_{k,l}, \bar{a}_{k',l'})$ or $\mathfrak{S}(\bar{b}_{k,l}, \bar{b}_{k',l'})$, and we are eliding the chords from $\bar{h}(\bar{a}_{k,l})$ to $\bar{a}_{k,l}$ and chords from $\bar{b}_{k,l}$ to $\bar{h}(\bar{b}_{k,l})$.

Let us write $p_* = \deg \bar{v}'_*$, $q_* = \deg \bar{v}_*^\sharp$, and $r_* = \deg \bar{v}_*^\flat$. We have $\mathrm{ind}(\bar{v}_{\star_0}^\sharp) \ge 2$ since there is an end that limits to $z_\infty$ or a boundary point of type (P$_3$) that projects to a large sector. Then $\mathrm{I}(\bar{v}_{\star_0}^\sharp) \ge 2$ by the ECH index inequality (cf. Lemma I.5.7.21) and considerations of $\mathcal{Z}$.

(1) Suppose that $\bar{v}'_{\star_0} \neq \emptyset$. We claim the following:

Claim 3.8.9. — $\mathrm{I}(\bar{v}_{\star_0}) \ge \mathrm{I}(\bar{v}'_{\star_0}) + \mathrm{I}(\bar{v}''_{\star_0}) + \delta_{\star_0}$, where $\delta_{\star_0} \ge 2$ if $bp > 0$ and $\delta_{\star_0} = 2p_{\star_0}$ if $bp = 0$.

*Proof of Claim 3.8.9.* — Each collection of boundary points of type (P$_3$) that map to the same point on the base contributes at least +2 towards I by the argument from Lemma 3.7.5. The inequality for $bp > 0$ then follows.

Next assume that $bp = 0$. Also assume that $l_+ = l_-$, since otherwise there exist thin strips of $\bar{v}_*$ from (c) or components from (d), all of which contribute > 0 to I, yielding a total of $I \ge 3$, a contradiction. Let $(s, t_2)$ be coordinates on $[-2, 2] \times [-1, 1]$. Let

$$ \check{C}'_{[-1,1]}, \check{C}''_{[-1,1]} \subset [-2, 2] \times [-1, 1] \times \bar{S} $$

be representatives of $\bar{v}'_{\star_0}$ and $\bar{v}''_{\star_0}$ and let $c'_{\pm 1}, c''_{\pm 1}$ be groomings on $A_{\varepsilon/2} = \partial D_{\varepsilon/2}^2 \times [-2, 2]$ and $A_{\varepsilon} = \partial D_{\varepsilon}^2 \times [-2, 2]$ corresponding to $\check{C}'_{[-1,1]}|_{l_2=\pm 1}$ and $\check{C}''_{[-1,1]}|_{l_2=\pm 1}$, such that the following hold:

$$ - c'_{+1} = c'_{-1} \text{ and } w(c'_{\pm 1}) = 0; $$

$$ - c''_{\pm 1} \text{ is obtained by intersecting } \pi_{[-2,2]\times\bar{S}}\text{-projections of the ends of } \bar{v}''_{\star_0} \text{ with } A_{\varepsilon}. $$

Here $\pi_{[-2,2]\times\bar{S}}$ is the projection $[-2, 2] \times \mathbf{R} \times \bar{S} \to [-2, 2] \times \bar{S}$. We also remark that, for sign consistency with the computations in Section I.5.7, we use the standard orientations on $A_{\varepsilon/2}$ and $A_{\varepsilon}$ but view $c'_{+1}$ and $c''_{+1}$ to be at the *negative end* and $c'_{-1}$ and $c''_{-1}$ to be at the *positive end* during the proof of this claim. By the above description of $\mathcal{Z}$:

$$ - w(c''_{+1}) = 0 \text{ or } -1 \text{ and } w(c''_{-1}) = 0 \text{ or } 1; $$

$$ - \text{the endpoints of } c''_{+1} \text{ and } c''_{-1} \text{ agree and are alternating.} $$
We now extend $\check{\mathcal{C}}'_{[-1,1]}$, $\check{\mathcal{C}}''_{[-1,1]}$ by concatenating with

$$ \check{\mathcal{C}}'_{[1,2]}, \check{\mathcal{C}}''_{[1,2]} \subset [-2, 2] \times [1, 2] \times \bar{S}, \\ \check{\mathcal{C}}'_{[-2,-1]}, \check{\mathcal{C}}''_{[-2,-1]} \subset [-2, 2] \times [-2, -1] \times \bar{S} $$

such that the ends of $\check{\mathcal{C}}'_{[-2,2]} \cup \check{\mathcal{C}}''_{[-2,2]}$ are groomed and have zero winding number. Writing $\mathfrak{w}$ for the writhe, Lemmas I.5.7.22 and I.5.7.23 imply that we have an additional contribution of at least

$$ 2p_{\star_0} = 2(\mathfrak{w}(\mathcal{c}'_1 \cup \mathcal{c}''_1) - \mathfrak{w}(\mathcal{c}'_{-1} \cup \mathcal{c}''_{-1})) $$

towards I.

<figure><img src="image_27.png" /></figure>

By Claim 3.8.9, $I(\bar{v}_{\star_0}) \ge 4$. Since the ECH indices of the other levels add up to at least 0 in view of (b), (c), and (d), we have a contradiction.

(2) Suppose that $\bar{v}'_{\star_0} = \emptyset$. Then there are no boundary points of type (P$_3$) by definition. Since $I(\bar{v}_{\star_0}) = I(\bar{v}''_{\star_0}) \ge 2$, the only nontrivial levels besides $\bar{v}_{\star_0}$ and $\bar{v}_1$ are $\bar{v}_{B,j}$, $j = 1, \dots, c$, which consist of thin strips and trivial strips. Again we have $l_+ = l_-$, since otherwise there exist thin strips of $\bar{v}_*$ from (c) or components from (d), all of which contribute > 0 to I, yielding a total of $I \ge 3$, a contradiction. Let us write $l_\star := l_+ = l_-$. We claim the following:

Claim 3.8.10. — $I(\bar{v}''_{\star_0}) \ge 2 + l_{\star_0}$.

*Proof of Claim 3.8.10.* — Let us first consider the case where:

(1) $\pi_S \circ \bar{v}''_{\star_0}|_{F''_{\star_0}}$ is a diffeomorphism onto its image, which contains $\bar{S} - \bar{a} - \bar{b}$;

(2) $\bar{v}''_{\star_0}$ limits to $z_\infty^{l_\star_0} \cup \mathbf{y}'$ to the left and to $z_\infty^{l_\star_0} \cup \mathbf{y}''$ to the right, where $\mathbf{y}' = \mathbf{y}''$.

If we choose a multivalued trivialization $\tau$ to be compatible with $\vec{D}''_{\star_0, \pm}$ (cf. Section I.5.7.3), then

$$ \begin{aligned} \mathrm{ind}(\bar{v}_{\star_0}^{\sharp}) &= -\chi(F_{\star_0}^{\sharp}) + \deg \bar{v}_{\star_0}^{\sharp} + \mu_{\tau}(\bar{v}_{\star_0}^{\sharp}) + 2c_1((\bar{v}_{\star_0}^{\sharp})^*T\bar{S}, \tau) \\ &= -(1 - 2g - (\deg \bar{v}_{\star_0}^{\sharp} - l_{\star_0})) + \deg \bar{v}_{\star_0}^{\sharp} + (l_{\star_0} + 1) + 2(1 - 2g) \\ &= 2 - 2g + 2 \deg \bar{v}_{\star_0}^{\sharp}. \end{aligned} $$

Hence $I(\bar{v}_{\star_0}^{\sharp}) \ge 2 - 2g + 2 \deg \bar{v}_{\star_0}^{\sharp}$. We also have $I(\bar{v}_{\star_0}^{\flat}) \ge 0$ and $\langle \bar{v}_{\star_0}^{\sharp}, \bar{v}_{\star_0}^{\flat} \rangle = \deg \bar{v}_{\star_0}^{\flat}$, where $\langle, \rangle$ is the algebraic intersection number. Summing the contributions, we obtain:

$$ \begin{aligned} \mathbf{(3.8.4)} \qquad & I(\bar{v}_{\star_0}) = I(\bar{v}_{\star_0}^{\sharp}) + I(\bar{v}_{\star_0}^{\flat}) + 2\langle \bar{v}_{\star_0}^{\sharp}, \bar{v}_{\star_0}^{\flat} \rangle \\ & \ge 2 - 2g + 2 \deg \bar{v}_{\star_0}^{\sharp} + 2 \deg \bar{v}_{\star_0}^{\flat} \ge 2 + \deg \bar{v}_{\star_0}^{\sharp}. \end{aligned} $$
In general, the cases with smallest ECH indices occur when elements of $\mathbf{y}'$ are of type $x_{i_1}^{\#}$ or $x_{i_3}^{\#}$ and elements of $\mathbf{y}''$ are of type $x_{i_2}^{\#}$. This has the effect of decreasing the lower bound in Equation (3.8.4) by the cardinality of $\mathbf{y}'$. Hence $I(\bar{v}_{\star_0}) \ge 2 + l_{\star_0}$. $\square$

Since $l_{\star_0} \ge 1$, we have $I(\bar{v}_{\star_0}) \ge 3$, which is a contradiction. $\square$

**Lemma 3.8.11.** — If $m \gg 0$, then there is no $\bar{u}_{\infty} \in \partial_{(-\infty, \infty)} \mathcal{M}$ such that $\bar{v}'_1 \ne \emptyset$ and $\bar{v}'_2 \ne \emptyset$.

*Proof.* — Corollary 3.8.7 and Lemma 3.8.8 imply the analog of Lemma 3.5.4. The proof of Lemma 3.5.13 then carries over with no change. $\square$

*Proof of Lemma 3.3.11.* — Suppose $\bar{u}_{\infty} \in \partial_{(-\infty, \infty)} \mathcal{M}$. If $\bar{v}'_2 = \emptyset$, then we are in the situation of Lemma 3.3.11 by Lemma 3.8.2. On the other hand, Lemmas 3.8.3 and 3.8.11 imply that $\bar{v}'_2 = \emptyset$. $\square$

**3.9.** *Breaking in the middle, part II.* — Let $\bar{u}_{\infty} \in \partial_{(-\infty, +\infty)} \mathcal{M}$ be the limit of $\bar{u}_i \in \mathcal{M}_{\tau_i}$, where $\tau_i \to T'$.

**Lemma 3.9.1.** — If $\bar{u}_{\infty}$ has no fiber components, then the ECH index of each level $\bar{v}_* \ne \bar{v}_0$ is nonnegative and the only components of $\bar{u}_{\infty}$ which have negative ECH index are the following:

- branched covers of $\sigma_{\infty}^{T'}$; and

- at most one component $\tilde{v}$ of $\bar{v}_0''$ with $I(\tilde{v}) = -1$.

*Proof.* — Similar to the proofs of Lemmas 3.7.4 and 3.7.5. $\square$

The following lemma is analogous to Lemma 3.7.10.

**Lemma 3.9.2.** — If $m \gg 0$ and $\bar{u}_{\infty} \in \partial_{(-\infty, +\infty)} \mathcal{M}$, then the following hold:

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

(2) there is no level $\bar{v}_*$ such that $\bar{v}'_* \cap \bar{v}''_* \ne \emptyset$ and $\bar{v}'_* \cap \bar{v}''_* \subset \text{int}(\bar{v}^*)$; and

(3) $\bar{u}_{\infty}$ has no boundary point at $z_{\infty}$.

*Proof.* — (1) Arguing by contradiction, suppose that $\bar{u}_{\infty}$ has a fiber component $\tilde{v}$. Then $n^*(\tilde{v}) \ge m$ and $\tilde{v}$ can be eliminated by arguing as in Lemma 3.7.10(1), in view of the following contributions towards I:

(γ1) $I(\bar{v}'_0) = -p_0$.

(γ2) $\cup_{j=1}^c \bar{v}_{-1,j}^{\#}$ is a union of $p_0$ trivial strips and $\sum_{j=1}^c I(\bar{v}_{-1,j}^{\#}) = p_0$.

(γ3) If $\tilde{v}$ is a component of $\bar{v}_*$, then $\tilde{v}$ and the intersection $\tilde{v} \cap (\bar{v}_* - \tilde{v})$ contribute $2g + 2 \ge 4$ towards I.
(γ_4) After removing any fiber components $\tilde{v}$, all the levels $\neq \bar{v}_0$ have nonnegative ECH index and $I(\bar{v}_0) \geq -1$ by Lemma 3.9.1.

The sum of the above ECH indices is at least 3, a contradiction.

(2), (3) Similar to (2) and (3) of Lemma 3.7.10. Observe that the sum of ECH indices of the levels was at least 4 in all the cases of Lemma 3.7.10 that were not ruled out by other means; in the present case the sum will be at least 3. (Note that the ECH indices added up to only 3 in (2C) of Lemma 3.7.10, but we do not have levels $\bar{v}_{0,j}, j = 1, \dots, b$, in the current situation.) $\square$

**Lemma 3.9.3.** — If $\bar{u}_\infty \in \partial_{(-\infty, +\infty)} \mathcal{M}$ and $\bar{v}'_0 = \emptyset$, then $\bar{u}_\infty$ is one of the following 2-level buildings:

(1) $\bar{v}_{1,1}$ with $I=1$ from $\mathbf{z}$ to some $\mathbf{z}'$ consisting of

(i) one thin strip and trivial strips or

(ii) one nontrivial component of $\bar{v}_{1,1}''$ with image in W and trivial strips; and $\bar{v}_0 = \bar{v}_0''$ with $I=1$ and $n^* \le m + |\mathcal{I}|$ from $\mathbf{z}'$ to $\mathbf{y}'$.

(2) $\bar{v}_0 = \bar{v}_0''$ with $I=1$ and $n^* \le m + |\mathcal{I}|$ from $\mathbf{z}$ to some $\mathbf{y}''$; and $\bar{v}_{-1,1}$ with $I=1$ from $\mathbf{y}''$ to $\mathbf{y}'$.

*Proof*. — Suppose that $\bar{v}'_0 = \emptyset$. Since passing through $\bar{m}(T')$ is a codimension two condition, we have $I(\bar{v}_0) \ge 1$. Hence there can be only one other nontrivial level — either $\bar{v}_{1,1}$ or $\bar{v}_{-1,1}$ — and $I(\bar{v}_{\pm 1,1}) \ge 1$. By Lemma 3.9.1, $I(\bar{v}_0) = 1$ and $I(\bar{v}_{\pm 1,1}) = 1$. Moreover, $n^*(\bar{v}_{\pm 1,1}) \le |\mathcal{I}|$ since $n^*(\bar{v}_0) \ge m$. This means that there are no components $\bar{v}_*^\pm$ that limit to $z_\infty$ at the negative end and all the positive ends $\mathcal{E}_+$ that limit to $z_\infty$ satisfy $n^*(\mathcal{E}_+) = 1$. The lemma follows. $\square$

**Lemma 3.9.4.** — For each interval $[-T, T]$, there exists $m \gg 0$ such that there is no sequence of curves $\bar{u}_i \in \mathcal{M}_{\tau_i}$, $\tau_i \to T' \in [-T, T]$, that limits to $\bar{u}_\infty$ for which $\bar{v}'_0 \neq \emptyset$.

*Proof*. — This is similar to Lemma 3.6.5 and will be omitted. $\square$

*Proof of Lemma 3.3.12.* — This is argued in the same way as Lemma 3.3.8. Suppose $\bar{u}_\infty \in \partial_{(-\infty, +\infty)} \mathcal{M}$. If $\bar{v}'_0 = \emptyset$, then Lemma 3.9.3 implies that $\bar{u}_\infty$ is as described in Lemma 3.3.12. On the other hand, by Lemma 3.9.4, for any $T > 0$ there exists $m \gg 0$ such that $\bar{v}'_0 = \emptyset$ in $\bar{u}_\infty \in \partial_{[-T, +T]} \mathcal{M}$. The possibilities where $m_i \to \infty$, $T_i \to \infty$, $\bar{u}_{ij} \to \bar{u}_{i\infty} \in \partial_{[\pm T_i]} \mathcal{M}$ are treated in the same way as in Lemmas 3.7.14, 3.8.3, and 3.8.11. $\square$

# 4. Homotopy of cobordisms II

In this section we consider the homotopy of cobordisms corresponding to $\Phi \circ \Psi$. Many of the constructions for $\overline{W}_\tau^\pm$ are similar to those of $\overline{W}_\tau^\pm$ with minor modifications.
We first give a brief description of $\overline{W}_{\tau} = \overline{W}_{\tau}^{\mp}$. (If $\mp$ is understood, as it will be in the rest of this section, then it will be omitted.) The base $B_{\tau}$ of $\overline{W}_{\tau}$ is biholomorphic to an infinite cylinder with a disk removed. As $\tau \to +\infty$, $\overline{W}_{\tau}$ degenerates to the stacking of $\overline{W}_{-}$ “on top of” $\overline{W}_{+}$. On the other hand, as $\tau \to -\infty$, $\overline{W}_{\tau}$ degenerates to $\overline{W}_{-\infty}$, whose base $B_{-\infty}$ is given by:

$$ (4.0.1) \qquad B_{-\infty} = ((\mathbf{R} \times (\mathbf{R}/2\mathbf{Z})) \sqcup \mathbf{E}) / \sim, $$

where $E = \{|z| \le 1\} \subset \mathbf{C}$ and $\sim$ identifies $(0, \frac{3}{2}) \in \mathbf{R} \times (\mathbf{R}/2\mathbf{Z})$ with $0 \in E$.

The degeneration for $\tau \to -\infty$ can be described in an equivalent way as a neck-stretching along a stable Hamiltonian hypersurface which is the preimage of a boundary-parallel loop in the base.

**4.1. Construction of the homotopy of cobordisms for $\Phi \circ \Psi$**

**4.1.1. Definition of the family $\overline{W}_{\tau}$.** — Let $l \in (0, \infty)$ and $r \in (0, 1]$. Consider the fibration

$$ \pi : \mathbf{R} \times \overline{\mathbf{N}} \to \mathbf{R} \times (\mathbf{R}/2\mathbf{Z}), $$

where $\overline{\mathbf{N}}$ is viewed as $(\overline{\mathbf{S}} \times [0, 2])/(x, 2) \sim (\overline{\mathfrak{h}}(x), 0)$ and $(s, t)$ are coordinates on $\mathbf{R} \times (\mathbf{R}/2\mathbf{Z})$. We define $\overline{W}_{l,r} = \pi^{-1}(B_{l,r})$, where the base $B_{l,r}$ is obtained by smoothing the corners of

$$ (\mathbf{R} \times (\mathbf{R}/2\mathbf{Z})) - ((-l/2, l/2) \times ((3-r)/2, (3+r)/2)). $$

Next choose a function

$$ (4.1.1) \qquad \eta = (l, r) : \mathbf{R} \to (0, \infty) \times (0, 1], $$

which is obtained by smoothing

$$ \eta_0(\tau) = \begin{cases} (\tau + 1, 1), & \text{for } \tau \ge 0; \\ (e^{\tau}, e^{\tau}), & \text{for } \tau \le 0; \end{cases} $$

near $\tau = 0$.

We then define $\overline{W}_{\tau} = \overline{W}_{\eta(\tau)}$ and $B_{\tau} = B_{\eta(\tau)}$. Let $\pi_{B_{\tau}} : \overline{W}_{\tau} \to B_{\tau}$ be the projection along $\{(s, t)\} \times \overline{\mathbf{S}}$.

As $\tau \to +\infty$, the cobordism $\overline{W}_{\tau}$ approaches the concatenation of $\overline{W}_{-}$ and $\overline{W}_{+}$. See Figure 17. On the other hand, as $\tau \to -\infty$, the cobordism $\overline{W}_{\tau}$ can be viewed as degenerating to $\overline{W}_{-\infty} = (\overline{W}_{-\infty,1} \sqcup \overline{W}_{-\infty,2}) / \sim$, which we describe now: Consider the base $B_{-\infty} = (B_{-\infty,1} \sqcup B_{-\infty,2}) / \sim$, where $B_{-\infty,1} = \mathbf{R} \times (\mathbf{R}/2\mathbf{Z})$, $B_{-\infty,2} = E = \{|z| \le 1\}$, and $\sim$ identifies $(0, \frac{3}{2}) \in B_{-\infty,1}$ with $0 \in B_{-\infty,2}$. We also identify the asymptotic markers $\partial_s$ at $(0, \frac{3}{2})$ and $\{x = 0, y > 0\}$ at 0. We then set $\overline{W}_{-\infty,1} = \pi^{-1}(B_{-\infty,1})$ and $\overline{W}_{-\infty,2} = B_{-\infty,2} \times \overline{\mathbf{S}}$,
<figure><img src="image_28.png" /><figcaption>FIG. 17. — The bases of the family $\overline{W}_{\tau}^{\pm}$. The leftmost diagram represents $B_{-\infty}$, i.e., $\tau = -\infty$, and the parameter $\tau$ increases as we go to the right. The location of $\overline{m}^b(\tau)$ is indicated by $\times$</figcaption></figure>

and identify $(0, \frac{3}{2}, x) \sim (0, x)$, where $(0, \frac{3}{2}, x) \in \overline{W}_{-\infty, 1}$ and $(0, x) \in \overline{W}_{-\infty, 2}$. We write $\pi_{B_{-\infty, i}} : \overline{W}_{-\infty, i} \to B_{-\infty, i}$ for the projection along $\overline{S}$.

The alternate description of the degeneration as $\tau \to -\infty$ mentioned at the beginning of the section relies on the conformal equivalence between annuli $\{e^\tau \le |z| \le 1\} \subset \mathbf{C}$ and finite cylinders $[\tau, 0] \times \mathbb{S}^1$ for $\tau < 0$. The details are left to the reader.

**4.1.2. Marked points.** — We choose a 1-parameter family of marked points

$$ \overline{m}(\tau) = (\overline{m}^b(\tau), \overline{m}^f(\tau)) = (\overline{m}^b(\tau), z_\infty) \in \overline{W}_\tau $$

for $\tau \in \mathbf{R}$, such that the following hold:

(i) $\overline{m}^b(\tau)$ is on the segment $\{s > \frac{l(\tau)}{2}, t = \frac{3}{2}\}$;

(ii) as $\tau \to +\infty$, $\overline{m}(\tau)$ limits to $\overline{m}(+\infty) = (\overline{m}^b(+\infty), \overline{m}^f(+\infty))$, where $\overline{m}^b(+\infty) = (0, \frac{3}{2}) \in B_-$ and $\overline{m}^f(+\infty) = z_\infty$;

(iii) as $\tau \to -\infty$, $\overline{m}(\tau)$ limits to $\overline{m}(-\infty) = (\overline{m}^b(-\infty), \overline{m}^f(-\infty))$, where $\overline{m}^b(-\infty) = \frac{i}{2} \in B_{-\infty, 2}$ and $\overline{m}^f(-\infty) = z_\infty$.

**Remark 4.1.1.** — Here is a slightly more detailed description of the convergence of the marked points: For $\tau \gg 0$ we take $\overline{m}^b(\tau) = (\frac{\tau+5}{2}, 0)$. For $\tau \ll 0$ we define sets $Q_\tau = B_\tau - B_{-\sqrt{|\tau|}}$ and maps

$$ \mathbf{r}_\tau : Q_\tau \to \mathbf{R}^2, \quad \mathbf{r}_\tau(s, t) = 2e^{-\tau} (s, t - \frac{3}{2}). $$

The images $\mathbf{r}_\tau(Q_\tau)$ form an exhaustion of a set $Q_{-\infty}$ obtained by smoothing the boundary of $\mathbf{R}^2 - ((-1, 1) \times (-1, 1))$. Let $\mathfrak{u} : \mathbb{D}^2 - \{0\} \to Q_{-\infty}$ be a diffeomorphism which is holomorphic in the interior. By smoothing the boundary of $\mathbf{R}^2 - ((-1, 1) \times (-1, 1))$ symmetrically, we choose $\mathfrak{u}$ so that the imaginary axis is mapped to the axis $\{t = 0\}$. Let $\mathfrak{u}(\frac{i}{2}) = (s_{-\infty}, 0)$. Then for $\tau \ll 0$ we take $\overline{m}^b(\tau) = \mathbf{r}_\tau^{-1}(s_{-\infty}, 0) = (\frac{e^\tau}{2} s_{-\infty}, \frac{3}{2})$.
**4.1.3.** *Stable Hamiltonian structures and symplectic forms.* — We first consider $\overline{W}_{\tau}$. The stable Hamiltonian structure on $\overline{N} = (\overline{S} \times [0, 2]) / \sim$ is obtained from $(dt, \overline{\omega})$ on $\overline{S} \times [0, 2]$ by passing to the quotient. The 2-plane field is $\xi_{\tau} = T\overline{S}$ and the Hamiltonian vector field is $R_{\tau} = \partial_{t}$. The symplectic form $\overline{\Omega}_{\tau}$ is the restriction to $\overline{W}_{\tau}$ of $ds \wedge dt + \overline{\omega}$, defined on $\mathbf{R} \times \overline{S} \times [0, 2]$ and sent to the quotient $\mathbf{R} \times \overline{N}$.

Next we consider $\overline{W}_{-\infty}$. For $\overline{W}_{-\infty,1} = \overline{N}$ the Hamiltonian structure is $(dt, \overline{\omega})$ and the symplectic form is $\overline{\Omega}_{-\infty,1} = ds \wedge dt + \overline{\omega}$. The symplectic form on $\overline{W}_{-\infty,2} = B_{-\infty,2} \times \overline{S}$ is a product symplectic form $\overline{\Omega}_{-\infty,2} = \omega_{-\infty,2} \oplus \overline{\omega}$, where $\omega_{-\infty,2}$ is the standard area form on the unit disk.

**4.2. Holomorphic curves and moduli spaces.**

**4.2.1.** *Lagrangian boundary conditions.* — Consider the cobordism $(\overline{W}_{\tau}, \overline{\Omega}_{\tau})$. We place a copy of $\overline{a}$ on $\pi_{B_{\tau}}^{-1}(\frac{l(\tau)}{2}, \frac{3}{2})$ and parallel transport along the vertical boundary $\partial_{\overline{v}} \overline{W}_{\tau} := \pi^{-1}(\partial B_{\tau})$ using the symplectic connection $\overline{\Omega}_{\tau}$ to obtain $L_{\overline{a}}^{\tau}$. Similarly, we place a copy of $\overline{a}$ over $1 \in B_{-\infty,2}$ and parallel transport along $\partial B_{-\infty,2}$ to obtain $L_{\overline{a}}^{-\infty,2} = \partial B_{-\infty,2} \times \overline{a}$. The Lagrangian submanifolds $L_{\overline{a}}^{\tau}$, $L_{\overline{a}_i}^{\tau}$, $L_{\overline{a}_i}^{-\infty}$ are defined similarly. Note that the Lagrangians close up because the monodromy of the fibration $\pi_{B_{\tau}} : \overline{W}_{\tau} \to B_{\tau}$ is trivial along $\partial B_{\tau}$.

**4.2.2. Almost complex structures.**

*Definition 4.2.1.*

(1) An almost complex structure $\overline{J}_{\tau}$ on $\overline{W}_{\tau}$ is admissible if $\overline{J}_{\tau}$ is the restriction of some $\overline{J}' \in \mathcal{J}_{\overline{W}}$. In this case $\overline{J}_{\tau}$ is said to be compatible with $\overline{J}'$.

(2) An almost complex structure $\overline{J}_{-\infty,1}$ on $\mathbf{R} \times \overline{N}$ is admissible if $\overline{J}_{-\infty,1} \in \mathcal{J}_{\overline{W}}$.

(3) An almost complex structure $\overline{J}_{-\infty,2}$ on $B_{-\infty,2} \times \overline{S}$ is admissible if it is a product complex structure.

The space of admissible $\overline{J}_{\tau}, \overline{J}_{-\infty,1}$ and $\overline{J}_{-\infty,2}$ will be denoted by $\mathcal{J}_{\overline{W}_{\tau}}, \mathcal{J}_{\overline{W}_{-\infty,1}}$ and $\mathcal{J}_{\overline{W}_{-\infty,2}}$, respectively.

*Definition 4.2.2.* — A family $\{\overline{J}_{\tau} \in \overline{J}_{\tau}\}_{\tau \in \mathbf{R}}$ of almost complex structures is admissible if there exist $\overline{J} \in \mathcal{J}_{\overline{W}}$, $\overline{J} \in \mathcal{J}_{\overline{W}}$, $\overline{J}_{+} \in \mathcal{J}_{\overline{W}_{+}}$, $\overline{J}_{-} \in \mathcal{J}_{\overline{W}_{-}}$, $\overline{J}_{-\infty,1} \in \mathcal{J}_{\overline{W}_{-\infty,1}}$ and $\overline{J}_{-\infty,2} \in \mathcal{J}_{\overline{W}_{-\infty,2}}$ such that the following hold:

(1) $\overline{J}_{\tau}$ converges to $(\overline{J}_{-\infty,1}, \overline{J}_{-\infty,2})$ as $\tau \to -\infty$;

(2) $\overline{J}_{\tau}$ converges to $(\overline{J}_{+}, \overline{J}_{-})$ as $\tau \to +\infty$;

(3) $\overline{J}_{+}$ and $\overline{J}_{-}$ are compatible with $\overline{J}'$ and $\overline{J}$; and

(4) $\overline{J}_{\tau}$ is compatible with $\overline{J}' = \overline{J}_{-\infty,1}$.

For $\tau \to +\infty$ the convergence of almost complex structures is to be understood in the sense of neck-stretching as in [BEHWZ, Section 3.4]. For $\tau \to -\infty$ we gave two equivalent descriptions of the
limit: one as nodal degeneration and one as a neck-stretching. If the limit is viewed as a neck-stretching, then the convergence of the almost complex structures is again as in [BEHWZ, Section 3.4]. If the limit is viewed as a nodal degeneration, then $\bar{J}_{-\infty,1}$ is the $C_{\text{loc}}^{\infty}$ limit of $\bar{J}_{\tau}$ and $\bar{J}_{-\infty,1}$ is the pullback by the uniformization map $\mathfrak{u}$ of the $C_{\text{loc}}^{\infty}$ limit of the restriction of $\bar{J}_{\tau}$ to $\mathcal{Q}_{\tau}$ after applying the rescaling map $\tau_{\tau}$; see Remark 4.1.1 for the definitions of $\mathfrak{u}$ and $\tau_{\tau}$. Strictly speaking, the limit produces almost complex structures on $\bar{W}_{-\infty,1} - \pi^{-1}((0, \frac{3}{2}))$ and $\bar{W}_{-\infty,2} - \{(0) \times \bar{S}\}$, but they can be extended over the missing points. The space of all admissible $\{\bar{J}_{\tau} \in \bar{\mathcal{J}}_{\tau}\}_{\tau \in \mathbb{R}}$ will be denoted by $\bar{\mathcal{I}}$.

**4.2.3. Notation and conventions.** — We will be using the notation and conventions from Section 3.2.3, with the exception of intersection numbers $n^*(\bar{u})$.

*Intersection numbers.* Let $\delta_{\rho_0, \phi_0}$ be a closed orbit of the Hamiltonian vector field $\partial_t$ which lies on the torus $\{\rho = \rho_0\} \subset \bar{S} \times [0, 2]/\sim$ for $\rho_0 > 0$ sufficiently small and which passes through the point $(t, \rho, \phi) = (0, \rho_0, \phi_0)$. We assume additionally that $\delta_{\rho_0, \phi_0}$ does not intersect the projections of the Lagrangians of $\bar{W}_{\tau}, \bar{W}_{+}$ and $\bar{W}_{-}$ to $\bar{N}$.

We then write $(\sigma_{\infty}^{*})^{\dagger}$ for the restriction of $\mathbf{R} \times \delta_{\rho_0, \phi_0}$ to $\bar{W}_{*}$, where $* = \emptyset, '$, $\tau$, $+$, $-$ or $(-\infty, 1)$. For $\bar{W}_{-\infty, 2}$ we write

$$ (\sigma_{\infty}^{-\infty, 2})^{\dagger} = \mathbf{B}_{-\infty, 2} \times \{\rho = \rho_0, \phi = \phi_0 + 3\pi/2m + 2\pi k/m, k \in \mathbb{Z}\}. $$

Finally we define $n^*(\bar{u}) = \langle \bar{u}, (\sigma_{\infty}^{*})^{\dagger} \rangle$, where $* = \emptyset, '$, $\tau$, $+$, $-$ or $(-\infty, 1)$.

**4.2.4. Holomorphic maps to $\bar{W}_{\tau}$** — Let $(F, j)$ be a compact Riemann surface, possibly disconnected, with $2g$ boundary components and two sets of interior punctures $\mathbf{p}^{+} = \{p_{1}^{+}, \dots, p_{k_{+}}^{+}\}$ and $\mathbf{p}^{-} = \{p_{1}^{-}, \dots, p_{k_{-}}^{-}\}$ such that each component of $F$ has nonempty boundary, at least one puncture from $\mathbf{p}^{+}$, and at least one puncture from $\mathbf{p}^{-}$. We write $\dot{\mathbf{F}} = \mathbf{F} - \mathbf{p}^{+} - \mathbf{p}^{-}$.<fnref n="14" />

*Definition 4.2.3.* — Let $\bar{J}_{\tau} \in \mathcal{J}_{\bar{W}_{\tau}}$ and let $\gamma = \prod \gamma_{i}^{m_{i}}$, $\gamma' = \prod (\gamma_{i}')^{m'_{i}} \in \mathcal{O}_{k}$.

A degree $k$ multisection of $(\bar{W}_{\tau}, \bar{J}_{\tau})$ from $\gamma$ to $\gamma'$ is a holomorphic map

$$ \bar{u} : (\dot{\mathbf{F}}, j) \to (\bar{W}_{\tau}, \bar{J}_{\tau}) $$

which is a degree $k$ multisection of $\pi_{B_{\tau}} : \bar{W}_{\tau} \to \mathbf{B}_{\tau}$ and which additionally satisfies the following:

(1) $\bar{u}(\partial \mathbf{F}) \subset \mathbf{L}_{\mathbf{a}}^{\tau}$;

(2) $\bar{u}$ maps each connected component of $\partial \mathbf{F}$ to a different $\mathbf{L}_{\mathbf{a}_{i}}^{\tau}$;

(3) $\lim_{w \to p_{i}^{+}} \pi_{\mathbf{R}} \circ \bar{u}(w) = +\infty$ and $\lim_{w \to p_{i}^{-}} \pi_{\mathbf{R}} \circ \bar{u}(w) = -\infty$;

(4) $\bar{u}$ converges to a cylinder over a multiple of some $\gamma_j$ near each puncture $p_i^+$ so that the total multiplicity of $\gamma_j$ over all the $p_i^+$'s is $m_j$ (and similarly for $p_i^-$).

Here $\pi_{\mathbf{R}} : \bar{W}_{\tau} \to \mathbf{R}$ is the projection to the s-coordinate.

<footnote n="14">¹⁴ Observe that $\dot{\mathbf{F}}$ has no boundary punctures.</footnote>
$A(\overline{W}_{\tau}, \overline{J}_{\tau})$-curve from $\boldsymbol{\gamma}$ to $\boldsymbol{\gamma}'$ is a degree 2g multisection of $(\overline{W}_{\tau}, \overline{J}_{\tau})$ satisfying $n^*(\bar{u}) = m$.

Let $\mathcal{M}_{\overline{J}_{\tau}}(\boldsymbol{\gamma}, \boldsymbol{\gamma}')$ be the moduli space of degree $k$ multisections of $(\overline{W}_{\tau}, \overline{J}_{\tau})$ from $\boldsymbol{\gamma}$ to $\boldsymbol{\gamma}'$ and let $\mathcal{M}_{\overline{J}_{\tau}}(\boldsymbol{\gamma}, \boldsymbol{\gamma}'; \overline{\boldsymbol{m}}(\tau))$ be the moduli space of curves with a point constraint $\overline{\boldsymbol{m}}(\tau)$; as usual, they are called moduli spaces of curves passing through $\overline{\boldsymbol{m}}(\tau)$. Topological considerations based on the intersection numbers $n^*$ imply that the forgetful map

$$ \mathcal{M}_{\overline{J}_{\tau}}^{n^*=m}(\boldsymbol{\gamma}, \boldsymbol{\gamma}'; \overline{\boldsymbol{m}}(\tau)) \to \mathcal{M}_{\overline{J}_{\tau}}^{n^*=m}(\boldsymbol{\gamma}, \boldsymbol{\gamma}') $$

is injective.

We write

$$ \mathcal{M}_{\{\overline{J}_{\tau}\}}(\boldsymbol{\gamma}, \boldsymbol{\gamma}') = \{(\tau, u) \mid \tau \in \mathbf{R}, u \in \mathcal{M}_{\overline{J}_{\tau}}(\boldsymbol{\gamma}, \boldsymbol{\gamma}')\}, $$

$$ \mathcal{M}_{\{\overline{J}_{\tau}\}}(\boldsymbol{\gamma}, \boldsymbol{\gamma}'; \overline{\boldsymbol{m}}) = \{(\tau, u) \mid \tau \in \mathbf{R}, u \in \mathcal{M}_{\overline{J}_{\tau}}(\boldsymbol{\gamma}, \boldsymbol{\gamma}'; \overline{\boldsymbol{m}}(\tau))\}. $$

**4.2.5. Holomorphic maps to $\overline{W}_{-\infty}$.** — We now describe the curves in $\overline{W}_{-\infty,i}$.

**Definition 4.2.4.** — Let $\overline{J}_{-\infty,1} \in \mathcal{J}_{\overline{W}_{-\infty,1}}$ and let $\boldsymbol{\gamma} = \prod \gamma_i^{m_i}, \boldsymbol{\gamma}' = \prod (\gamma_i')^{m_i'} \in \hat{\mathcal{O}}_k$. A degree $k$ multisection of $(\overline{W}_{-\infty,1}, \overline{J}_{-\infty,1})$ from $\boldsymbol{\gamma}$ to $\boldsymbol{\gamma}'$ is a holomorphic map

$$ \bar{u} : (\dot{\mathcal{F}}, j) \to (\overline{W}_{-\infty,1}, \overline{J}_{-\infty,1}) $$

which is a degree $k$ multisection of $\pi_{B_{-\infty,1}} : \overline{W}_{-\infty,1} \to B_{-\infty,1}$ and which is asymptotic to $\boldsymbol{\gamma}$ and $\boldsymbol{\gamma}'$ at the positive and negative ends. Here $(\dot{\mathcal{F}}, j)$ is a punctured Riemann surface. A $(\overline{W}_{-\infty,1}, \overline{J}_{-\infty,1})$-curve from $\boldsymbol{\gamma}$ to $\boldsymbol{\gamma}'$ is a degree 2g multisection of $(\overline{W}_{-\infty,1}, \overline{J}_{-\infty,1})$ satisfying $n^*(\bar{u}) = 0$.

**Definition 4.2.5.** — Let $\overline{J}_{-\infty,2} \in \mathcal{J}_{\overline{W}_{-\infty,2}}$. A degree $k$ multisection of $(\overline{W}_{-\infty,2}, \overline{J}_{-\infty,2})$ is holomorphic map

$$ \bar{u} : (\mathcal{F}, j) \to (\overline{W}_{-\infty,2}, \overline{J}_{-\infty,2}) $$

which is a degree $k$ multisection of $\pi_{B_{-\infty,2}} : \overline{W}_{-\infty,2} \to B_{-\infty,2}$ and which additionally satisfies the following:

(1) $(\mathcal{F}, j)$ is a compact Riemann surface with $k$ boundary components;

(2) $\bar{u}$ maps each component of $\partial \mathcal{F}$ to a different $L_{\hat{a}_i}^{-\infty,2}$.

$A(\overline{W}_{-\infty,2}, \overline{J}_{-\infty,2})$-curve is a degree 2g multisection of $(\overline{W}_{-\infty,2}, \overline{J}_{-\infty,2})$ satisfying $n^*(\bar{u}) = m$. A degenerate $(\overline{W}_{-\infty,2}, \overline{J}_{-\infty,2})$-curve consists of 2g copies of $B_{-\infty,2} \times \{pt\}$ and a singular fiber $\{0\} \times \overline{S}$.

Let $\mathcal{M}_{\overline{J}_{-\infty,1}}(\boldsymbol{\gamma}, \boldsymbol{\gamma}')$ be the moduli space of degree 2g multisections of $(\overline{W}_{-\infty,1}, \overline{J}_{-\infty,1})$ from $\boldsymbol{\gamma}$ to $\boldsymbol{\gamma}'$, let $\mathcal{M}_{\overline{J}_{-\infty,2}}$ be the moduli space of degree 2g multisections of $(\overline{W}_{-\infty,2}, \overline{J}_{-\infty,2})$,
and let $\mathcal{M}_{\bar{\mathcal{J}}_{-\infty,2}}(\bar{\mathfrak{m}}(-\infty)) \subset \mathcal{M}_{\bar{\mathcal{J}}_{-\infty,2}}$ be the subset consisting of multisections satisfying a point constraint at $\bar{\mathfrak{m}}(-\infty)$.

We define the evaluation maps

$$ev_{\bar{S},1}: \mathcal{M}_{\bar{\mathcal{J}}_{-\infty,1}}(\boldsymbol{\gamma}, \boldsymbol{\gamma}') \to \mathrm{Sym}^{2g}(\bar{S}), \quad ev_{\bar{S},2}: \mathcal{M}_{\bar{\mathcal{J}}_{-\infty,2}} \to \mathrm{Sym}^{2g}(\bar{S})$$

as in Definition 2.1.1, by intersecting the holomorphic curves in $\mathcal{M}_{\bar{\mathcal{J}}_{-\infty,1}}(\boldsymbol{\gamma}, \boldsymbol{\gamma}')$ (resp. in $\mathcal{M}_{\bar{\mathcal{J}}_{-\infty,2}}$) with the fiber of the fibration $\bar{W}_{-\infty,1} \to \mathbf{R} \times \mathbf{R}/2\mathbf{Z}$ over $(0, \frac{3}{2})$ which we identify with $\bar{S}$ (resp. with the fiber $\{0\} \times \bar{S}$). Given $\boldsymbol{\zeta} \in \mathrm{Sym}^{2g}(\bar{S})$, we define

$$\mathcal{M}_{\bar{\mathcal{J}}_{-\infty,1}}(\boldsymbol{\gamma}, \boldsymbol{\gamma}', \boldsymbol{\zeta}) = ev_{\bar{S},1}^{-1}(\boldsymbol{\zeta}) \subset \mathcal{M}_{\bar{\mathcal{J}}_{-\infty,1}}(\boldsymbol{\gamma}, \boldsymbol{\gamma}')$$

$$\mathcal{M}_{\bar{\mathcal{J}}_{-\infty,2}}(\boldsymbol{\zeta}) = ev_{\bar{S},2}^{-1}(\boldsymbol{\zeta}) \subset \mathcal{M}_{\bar{\mathcal{J}}_{-\infty,2}},$$

$$\mathcal{M}_{\bar{\mathcal{J}}_{-\infty,2}}(\boldsymbol{\zeta}, \bar{\mathfrak{m}}(-\infty)) = \mathcal{M}_{\bar{\mathcal{J}}_{-\infty,2}}(\bar{\mathfrak{m}}(-\infty)) \cap \mathcal{M}_{\bar{\mathcal{J}}_{-\infty,2}}(\boldsymbol{\zeta}).$$

In order to highlight the special role of the point $z_\infty$ we will write

$$\boldsymbol{\zeta} = \zeta_0^{r_0} \zeta_1^{r_1} \dots \zeta_l^{r_l},$$

where $r_0 \in \mathbf{Z}^{\ge 0}$, $r_1, \dots, r_l \in \mathbf{Z}^{> 0}$, $r_0 + \dots + r_l = 2g$, $\zeta_0 = z_\infty$, and $\zeta_1, \dots, \zeta_l \in \bar{S} - \{z_\infty\}$. Also, for sake of brevity, we will denote $\mathfrak{Z} = \mathrm{Sym}^{2g}(\bar{S})$.

**4.2.6. Indices.** — We now briefly discuss the Fredholm index $\mathrm{ind}(\bar{u})$ and the ECH index $\mathrm{I}(\bar{u})$ of a $\bar{W}_\tau$-curve $\bar{u}: \dot{\mathcal{F}} \to \bar{W}_\tau$ from $\boldsymbol{\gamma}$ to $\boldsymbol{\gamma}'$. This is similar to Section 3.2.6.

Let $\tilde{\bar{W}}_\tau = \bar{W}_\tau - \{s > \frac{l(\tau)}{2} + 1\} - \{s < -\frac{l(\tau)}{2} - 1\}$, where $l(\tau)$ is given in Equation (4.1.1). Let $\tilde{\bar{u}}: \dot{\mathcal{F}} \to \tilde{\bar{W}}_\tau$ be the compactification of $\bar{u}$, where $\tilde{\mathcal{F}}$ is obtained by performing a real blow-up of $\mathcal{F}$ at its boundary punctures.

The trivialization $\tau^*$ of $\bar{T}\bar{S}$ along $L_\mathbf{a}^\tau$ is defined as in Section I.4.4.2: We pick a point $p \in \partial B_\tau$, define $\tau^*$ of $\bar{T}\bar{S}|_{\pi_{B_\tau}^{-1}(\rho)}$ along $\hat{\mathbf{a}}$ by choosing a nonsingular tangent vector field along $\hat{\mathbf{a}}$, and then parallel transport $\tau^*$ along $\partial \bar{W}_\tau$. We also choose $\tau^*$ along $\boldsymbol{\gamma}$ and $\boldsymbol{\gamma}'$.

Let $\mathrm{Q}_\tau^*(\tilde{\bar{u}})$ be the relative intersection form given by normalizing $\tilde{\bar{u}}$ near $s = \pm(\frac{l(\tau)}{2} + 1)$ as in [Hul, Definition 2.4] and intersecting $\tilde{\bar{u}}$ and a pushoff of $\tilde{\bar{u}}$. Here the pushoff along $\partial \tilde{\bar{W}}_\tau$ is in the direction of $\mathcal{J}\tau^*$. Then

$$(4.2.1) \qquad \mathrm{I}(\bar{u}) = c_1(\tilde{\bar{u}}^* \bar{T}\bar{S}, \tau^*) + \mathrm{Q}_\tau^*(\tilde{\bar{u}}) + \tilde{\mu}_\tau^*(\boldsymbol{\gamma}) - \tilde{\mu}_{\tau^*}(\boldsymbol{\gamma}') - 2g,$$

$$(4.2.2) \qquad \mathrm{ind}(\bar{u}) = -\chi(\dot{\mathcal{F}}) + 2c_1(\tilde{\bar{u}}^* \bar{T}\bar{S}, \tau^*) + \mu_{\tau^*}(\boldsymbol{\gamma}) - \mu_{\tau^*}(\boldsymbol{\gamma}') - 4g,$$

These index formulas are obtained by adding the index formulas for holomorphic curves in $\bar{W}_\pm$ of [I]. In deriving Equation (4.2.2) from Propositions I.5.5.2 and I.5.5.5 we should observe that every term in the Fredholm index formula is additive except for the Euler
characteristic: if a surface $\dot{F}$ is obtained by gluing surfaces $\dot{F}_+$ and $\dot{F}_-$ along $2g$ strip-like ends, then $\chi(\dot{F}_+) + \chi(\dot{F}_-) = \chi(\dot{F}) + 2g$. The index inequality holds as usual:

$$ (4.2.3) \qquad \mathrm{ind}(\bar{u}) + 2\delta(\bar{u}) \le \mathrm{I}(\bar{u}), $$

where $\delta(\bar{u}) \ge 0$ and equals zero if and only if $\bar{u}$ is an embedding. We also have

$$ (4.2.4) \qquad \mathrm{I}(\sigma_{\infty}^{\tau}) = \mathrm{ind}(\sigma_{\infty}^{\tau}) = -1. $$

The Fredholm and ECH indices for $\overline{W}_{-\infty,i}$-curves can be defined and computed similarly.

**Remark 4.2.6.** — The Fredholm and ECH indices for $\overline{W}_{\tau}$ and $\overline{W}_{-\infty,i}$ do not take into account the point constraint $\overline{m}(\tau)$ and the condition “passing through $\overline{m}(\tau)$” is a codimension 2 condition. Moreover, the indices for $\overline{W}_{-\infty,i}$, $i = 1, 2$, do not take into account the constraints 3.

We now have the following:

**Lemma 4.2.7.** — Let $\bar{u}$ be a $\overline{W}_{-\infty,2}$-curve. Then

$$ [\bar{u}] = 2g[B_{-\infty,2} \times \{pt\}] + [\{pt\} \times \overline{S}] \in \mathrm{H}_2(\overline{W}_{-\infty,2}, \partial B_{-\infty,2} \times \overline{S}), $$

$\bar{u}$ consists of $0 \le k < 2g$ copies of $B_{-\infty,2} \times \{pt\}$ together with an irreducible component in the class $(2g-k)[B_{-\infty,2} \times \{pt\}] + [\{pt\} \times \overline{S}]$, and $\mathrm{I}(\bar{u}) = 4g + 2$.

**Proof.** — Let $\pi_{\overline{S}}: \overline{W}_{-\infty,2} \to \overline{S}$ be the projection along $B_{-\infty,2}$. Since $\overline{J}_{-\infty,2}$ is a split complex structure, $\pi_{\overline{S}}$ is a holomorphic map. Hence if $\bar{u}$ is a $\overline{W}_{-\infty,2}$-curve, then $\pi_{\overline{S}} \circ \bar{u}$ either maps to a point on some $\hat{a}_i$ or to all of $\overline{S}$. The lemma follows by listing all the possibilities. $\square$

We also have the following, which is stated without proof.

**Lemma 4.2.8.** — Let $\bar{u}$ be a degree $2g$ multisection of $(\overline{W}_{-\infty,2}, \overline{J}_{-\infty,2})$ satisfying $n^*(\bar{u}) = 0$. Then $\bar{u} = \bar{u}' \cup \bar{u}''$ consists of $\bar{u}'$ which is a degree $k$ cover of $\sigma_{\infty}^{-\infty,2}$ and $\bar{u}''$ which is the union of $2g-k$ copies of $B_{-\infty,2} \times \{pt\}$, and $\mathrm{I}(\bar{u}) = 2g$.

**4.2.7. Regularity.**

**Definition 4.2.9.** — The family $\{\overline{J}_{\tau}\}_{\tau \in \mathbb{R}} \in \overline{\mathcal{I}}$ is regular if:

(1) $\mathcal{M}_{\overline{J}_{\tau}}^{\dagger}(\delta_0^{\rho}\boldsymbol{\gamma}, \delta_0^{\rho}\boldsymbol{\gamma}')$ is transversely cut out for all $\delta_0^{\rho}\boldsymbol{\gamma}, \delta_0^{\rho}\boldsymbol{\gamma}' \in \mathcal{O}_k, k \le 2g$;

(2) the restriction $\overline{J}$ of $\overline{J}_{\tau}$ to the positive and negative ends (the restriction is independent of $\tau$) is regular; and
(3) $\bar{J}_-$ and $\bar{J}_+$ in the limit $\tau \to +\infty$ are regular.

Let $\overline{\mathcal{I}}^{\text{reg}}$ be the space of regular $\{\bar{J}_{\tau}\} \in \overline{\mathcal{I}}$. As usual we have:

**Lemma 4.2.10.** — The generic $\{\bar{J}_{\tau}\} \in \overline{\mathcal{I}}$ is regular.

We also introduce the perturbations of $\{\bar{J}_{\tau}\}$ to ensure that passing through $\overline{\mathfrak{m}}(\tau)$ is a generic condition. Let $\mathfrak{p}(\tau) \in \mathcal{B}_{\tau}$ be a family of points such that:

$$ \lim_{\tau \to +\infty} \mathfrak{p}(\tau) = \mathfrak{p}(+\infty) \in \mathcal{B}_{-}, \quad \lim_{\tau \to -\infty} \mathfrak{p}(\tau) = \mathfrak{p}(-\infty) \in \mathcal{B}_{-\infty,1} $$

and $\mathfrak{p}(\tau) \neq \overline{\mathfrak{m}}^{\flat}(\tau)$ for all $\tau \in [-\infty, \infty]$. We then define the families $\{U_{\tau} = U_{\varepsilon, \delta, \mathfrak{p}(\tau)}\}$, $\{K_{\tau} = K_{\mathfrak{p}(\tau), \delta}\}$, and $\{\bar{J}_{\tau}^{\flat} = \bar{J}_{\tau}^{\flat}(\varepsilon, \delta, \mathfrak{p}(\tau))\}$ as in Section 3.2.7. Also, the modifier $\{K_{\tau}\}$ means “passing through $K_{\tau}$ for an appropriate $\tau$.”

**Definition 4.2.11.** — The family $\{\bar{J}_{\tau}^{\flat}\}$ is $\{K_{\tau}\}$-regular with respect to $\overline{\mathfrak{m}}$ if the moduli spaces $\mathcal{M}_{\{\bar{J}_{\tau}^{\flat}\}}^{\flat, \{K_{\tau}\}}(\delta_{0}^{\flat}\mathbf{y}, \delta_{0}^{\flat}\mathbf{y}'; \overline{\mathfrak{m}})$ are transversely cut out.

The following lemmas are analogous to Lemmas 3.2.21 and 3.2.22:

**Lemma 4.2.12.** — A generic $\{\bar{J}_{\tau}^{\flat}\}_{\tau \in \mathcal{R}}$ is $\{K_{\tau}\}$-regular with respect to $\overline{\mathfrak{m}}$.

**Lemma 4.2.13.** — If $\{\bar{J}_{\tau}\}$ is a generic family, then for $\varepsilon, \delta > 0$ sufficiently small, there exist a generic family $\{\bar{J}_{\tau}^{\flat}(\varepsilon, \delta, \mathfrak{p}(\tau))\}$ which is $\{K_{\mathfrak{p}(\tau), \delta}\}$-regular with respect to $\overline{\mathfrak{m}}$ and disjoint finite subsets $\mathcal{T}_1, \mathcal{T}_2 \subset \mathcal{R}$ with the following properties:

(1) $\tau \in \mathcal{T}_1$ if and only if there exists $\bar{v}_{\tau} \in \mathcal{M}_{\bar{J}_{\tau}^{\flat}}^{\flat, s, \text{irr}, \text{ind}=-1}(\delta_{0}^{\flat}\mathbf{y}, \delta_{0}^{\flat}\mathbf{y}')$ for some $\delta_{0}^{\flat}\mathbf{y}$ and $\delta^{\flat}\mathbf{y}'$.

(2) $\tau \in \mathcal{T}_2$ if and only if there exists $\bar{v}_{\tau} \in \mathcal{M}_{\bar{J}_{\tau}^{\flat}}^{\flat, s, \text{irr}, \{K_{\tau}\}, \text{ind}=1}(\delta_{0}^{\flat}\mathbf{y}, \delta_{0}^{\flat}\mathbf{y}'; \overline{\mathfrak{m}})$ for some $\delta_{0}^{\flat}\mathbf{y}$ and $\delta^{\flat}\mathbf{y}'$.

Moreover, for each $\tau \in \mathcal{T}_i$ there is a unique such irreducible curve $\bar{v}_{\tau}$.

**4.3.** *Proof of the other half of Theorem 1.0.1.* — In this subsection we prove the “other half” of Theorem 1.0.1, i.e., $\Phi \circ \Psi = \text{id}$ on the level of homology, assuming the results of Sections 4.4–4.5.

We make the following simplifying assumption, which is possible by Theorem I.2.5.2.

(††)$_{2g}$ All the elliptic orbits of the stable Hamiltonian vector field corresponding to $h: S \to S$ that intersect $S \times \{0\}$ at most $2g$ times have linearized first return maps which are $\varepsilon$-rotations with $0 < |\varepsilon| < \frac{\pi}{g}$ with respect to some trivialization.