$\rho_{k}=\tilde{\sigma}(x_{k})^{-1/2}\rho$, and the concentric balls
$B(x_{k},\rho_{k}/2)$ are mutually disjoint. Any such set $\Lambda^{v,\rho}$
has the following properties:

    * (i)

Given any compact $X_{\bullet}\subset X^{{}^{\prime}a}$, the set
$\Lambda_{X_{\bullet}}^{v,\rho}$ consists of no more than
$c_{v}\,\rho^{-2}(1+|X_{\bullet}|)$ elements.

    * (ii)

Given any ball ${\mathfrak{B}}_{R}=B\,(x,R)\subset X^{{}^{\prime}a}_{\delta}$
of radius $R$, with $(r\tilde{\sigma}(x))^{-1/2}\leq R<\tilde{\sigma}(x)/4$,
the set $\Lambda_{{\mathfrak{B}}_{R}}^{v,\rho}$ consists of no more than
$c_{v}^{\prime}\,\tilde{\sigma}(x)\,\rho^{-2}R^{2}$ elements.

In the above, the constants $\zeta$, $\zeta^{\prime}$ $\zeta_{0}$, and
$c_{v}$, $c^{\prime}_{v}$ (for each fixed $v$) depend only on $\nu$ and the
parameters listed in (4.15), and $r_{0}$ depends on e in addition to all the
above.

Proof. This is an analog of Lemmas I.3.6 and I.3.8 in [T] and follows from the
same argument with the use of Proposition I.3.1 by its counterpart in our
context, Proposition 6.11 above. $\Box$

###  6.4 (Further) refined curvature estimate

The next proposition is the counterpart of Proposition I.3.4 in [T] (and also
[Ts]’s Proposition 4.2) in our context. It improves the estimate for
$|F_{A}^{-}|$ given in Proposition 5.8 (and Lemma 6.8) over
$X^{{}^{\prime}a}_{\delta_{*}}$ by eliminating the coefficient
$\varepsilon^{\prime}$ therein, thereby making the first terms in the RHS of
(5.37) and (5.19) coincide.

The aforementioned improvement is made possible by the following consequence
of the monotonicity formula. This is in turn the counterpart of [T]’s Lemma
I.3.5 in our context.

######  Lemma 6.13.

Adopt the assumptions and notation of Lemma 4.6, and let $\delta_{*}$ be as in
Proposition 6.11. Then there exist positive constants $r_{0}$, $\zeta$,
$\zeta^{\prime}$ with the following significance: for $r\geq r_{0}$, there is
a smooth function ${\mathfrak{u}}$ on $X^{{}^{\prime}a}_{\delta_{*}}$, $0\leq
a\leq\frac{39}{8}$, satisfying the following properties:

  * •

$d^{*}d{\mathfrak{u}}\geq\zeta r\tilde{\sigma}$ where
$|\underline{\alpha}|\leq 2^{-1}$;

  * •

$\Big{|}d^{*}d{\mathfrak{u}}\Big{|}\leq\zeta^{\prime}r\tilde{\sigma}$;

  * •

$|{\mathfrak{u}}|\leq\zeta^{\prime}$.

In the above, the constants $\zeta$, $\zeta^{\prime}$ depend only on $\nu$ and
the parameters listed in (4.15), and $r_{0}$ depends on e in addition to all
the above.

Proof. Modify the proof of Lemma I.3.5 in [T] as follows.

  * •

Replace the parameter $r$ in [T] by $\tilde{r}=r\tilde{\sigma}$, and replace
balls of the form $B\,(x,\rho_{*})$ in [T] by their counterparts,
$B\,(x,\tilde{\sigma}^{-1/2}(x)\,\rho_{*})$, in our context.

  * •

Replace all appearances of $\alpha$ in [T]’s proof of its Lemma I.3.5 by
$\underline{\alpha}$. Replace the use of Lemmas I.3.6, I.3.8 in [T] by
Corollary 6.12 above, taking $\delta=\delta_{*}$.

  * •

In Step 2 of the proof of Lemma I.3.5 in [T], replace the set of balls
$\\{B_{i}\\}_{i}$ therein by the set of balls $\Lambda^{v,\rho}$ in the third
bullet of our Corollary 6.12; with $v=1/2$, and $\rho=r^{-1/2}$ as in [T].

  * •

In Step 3 of the proof of Lemma I.3.5 in [T], a function $u_{i}$ is introduced
for each element $B_{i}$ in $\\{B_{i}\\}_{i}$. As explained in the previous
bullet, The counterpart of [T]’s $B_{i}$ is a typical element,
$B\,(x_{k},\rho_{k})$, where $\rho_{k}=r^{-1/2}\tilde{\sigma}(x_{k})^{-1/2}$,
in $\Lambda^{1/2,r^{-1/2}}$. To this we associate a function
${\mathfrak{u}}_{k}$ on $X$, which plays the part of [T]’s $u_{i}$: In
parallel to the defining equation of [T]’s $u_{i}$ in (I.3.32) of [T],
${\mathfrak{u}}_{k}$ is defined to be the solution to:

$\begin{cases}\,\,\frac{1}{2}d^{*}d{\mathfrak{u}}_{k}=r\tilde{\sigma}(x_{k})\,(s_{k}-\kappa_{k}\,{\mathfrak{s}}_{k});&\\\
\,\,\text{$\forall i\in{\mathfrak{Y}}$, ${\mathfrak{u}}_{k}$ has $Y_{i}$-end
limit $0$.}&\end{cases}$

where $s_{i}$, ${\mathfrak{s}}_{i}$, $\kappa_{i}$ are defined as in [T],
modified according to the rules set forth by the first two bullets above. The
argument in [T] leading to its (I.3.36) shows that ${\mathfrak{u}}_{k}$ obeys:

(6.29)
$|{\mathfrak{u}}_{k}|+|d{\mathfrak{u}}_{k}|\leq\zeta_{4}\,(r\tilde{\sigma}(x_{k}))^{-3/2}\mathop{\mathrm{dist}\,}\nolimits(\cdot,x_{k})^{-3}$

where $\mathop{\mathrm{dist}\,}\nolimits(\cdot
x_{k})\geq\zeta^{\prime\prime}\rho_{k}$.

  * •

Take

${\mathfrak{u}}=\sum_{k\in\Lambda^{1/2,r^{-1/2}}}\chi\big{(}8\mathop{\mathrm{dist}\,}\nolimits(\cdot,x_{k})/\tilde{\sigma}(x_{k})\big{)}\,{\mathfrak{u}}_{k};\qquad
u:=\sum_{k\in\Lambda^{1/2,r^{-1/2}}}\,{\mathfrak{u}}_{k}.$

The arguments in Step 5 of [T]’s proof of its Lemma I.3.5 (again modified
according to the previously described rules) then confirm the bound on
${\mathfrak{u}}$ asserted in the last bullet of the lemma, and also that:

(6.30) $\begin{split}&\text{$d^{*}d\,u\geq\zeta r\tilde{\sigma}$ where
$|\underline{\alpha}|\leq
2^{-1}$;}\qquad\Big{|}d^{*}d\,u\Big{|}\leq\zeta^{\prime}r\tilde{\sigma}.\end{split}$

Note that given $x$ and $k$, the value of
$\Big{|}d^{*}d\Big{(}\chi\big{(}8\mathop{\mathrm{dist}\,}\nolimits(\cdot,x_{k})/\tilde{\sigma}(x_{k})\big{)}\,{\mathfrak{u}}_{k}-{\mathfrak{u}}_{k}\Big{)}\Big{|}$
at $x$ vanishes unless $k$ belongs to the subset

$\Lambda_{x}:=\\{k\,|\,x_{k}\in
B_{x}(x,3\tilde{\sigma}(x))-B(x,2\tilde{\sigma}(x)/5),\,k\in\Lambda^{1/2,r^{-1/2}}\\}.$

When $k\in\Lambda_{x}$, (6.29) may be used to bound this value by
$\zeta_{4}^{\prime}\,(r\tilde{\sigma}(x))^{-3/2}\tilde{\sigma}(x)^{-5}$.
Meanwhile, by Corollary 6.12 $\Lambda_{x}$ has at most
$\zeta_{5}\,r\tilde{\sigma}(x)^{3}$ elements. Together, these imply that
$\big{|}\,d^{*}d\,{\mathfrak{u}}-d^{*}d\,u\big{|}\,(x)\leq\zeta_{6}\,(r\delta_{*}^{3})^{-3/2}r\tilde{\sigma}(x)$.
Combined with (6.30), this confirms the first two bullets of the lemma. $\Box$

Here is the promised improved curvature estimate.

######  Proposition 6.14.

Re-introduce the notations $\varpi=|\nu|-|\alpha|^{2}$ and
$-u=|\nu|-|\psi|^{2}$. Adopt the assumptions and notation of Lemma 4.6, and
let $\delta_{*}$ be as in Proposition 6.11. Then there exist $r$-independent
constants $r_{0}$, $\zeta$, $\zeta^{\prime}$ such that for all $r\geq r_{0}$,

(6.31) $\begin{split}|F_{A}^{-}|&\leq
2^{-3/2}r\varpi+\zeta\tilde{\sigma}^{-2};\\\ |F_{A}^{-}|&\leq
2^{-3/2}r\,(-u)+\zeta^{\prime}\tilde{\sigma}^{-2}\quad\text{over
$X^{{}^{\prime}a}_{8\delta_{*}}$, $0\leq a\leq\frac{15}{4}$}.\end{split}$

In the above, the constants $\zeta$, $\zeta^{\prime}$ depend only on $\nu$ and
the parameters listed in (4.15), and $r_{0}$ depends on e in addition to all
the above.

Proof. Note that the two asserted inequalities are equivalent over
$X^{{}^{\prime}a}_{\delta_{0}}$ by way of Proposition 5.5; so it suffices to
establish either of them.

To begin, recall (5.22), and this time take
$\varepsilon=4\zeta_{3}^{-2}\tilde{\sigma}^{-2}r^{-1}|\nu|^{-1}\leq\zeta^{\prime\prime}r^{-1}\tilde{\sigma}^{-3}$.
Fix $\delta\geq\delta_{0}^{\prime}$, where $\delta_{0}^{\prime}$ is as in
Proposition 5.8, and let
$\varepsilon_{\delta}:=\zeta^{\prime\prime}r^{-1}\delta^{-3}$. Introduce the
following variant of the function $q_{0}$ in (5.25):

$q_{0}=q_{0,\delta}:=\textsc{s}+(2^{-3/2}+\varepsilon_{\delta})\,r\,u+\zeta_{2}r|\beta|^{2}.$

By Propositions 5.8, 5.4, 5.5, it satisfies the following variant of (5.39)
and (5.38):

$\begin{split}\big{(}\frac{d^{*}d}{2}+\frac{r|\psi|^{2}}{4}\big{)}q_{0,\delta}&\leq\zeta_{0}r\,\tilde{\sigma}^{-1};\\\
q_{0,\delta}&\leq\zeta(\ln
r)^{-1}r\,(-u)_{+}+\zeta^{\prime}\tilde{\sigma}^{-2}\quad\text{over
$X^{{}^{\prime}a}_{\delta}$}.\end{split}$

Combine this and Lemma 6.13 to find a constant $\zeta_{1}$ that is independent
of $r$, $\delta$, and $(A,\Psi)$, such that with
${\mathfrak{q}}_{0,\delta}:=q_{0,\delta}-\zeta_{1}\delta^{-2}{\mathfrak{u}}$,
one has for $\delta\geq\delta_{*}$:

$\begin{split}&\big{(}\frac{d^{*}d}{2}+\frac{r|\psi|^{2}}{4}\big{)}\,{\mathfrak{q}}_{0,\delta}\leq\zeta_{0}^{\prime}r\,\delta^{-2}|\psi|^{2};\\\
&\qquad{\mathfrak{q}}_{0,\delta}\leq\zeta\,(\ln
r)^{-1}r\,(-u)_{+}+\zeta^{\prime\prime}\delta^{-2}\quad\text{over
$X^{{}^{\prime}a}_{\delta}$, $0\leq a\leq\frac{39}{8}$. }\\\ \end{split}$

Together with (5.40) and the last bullet of Lemma 6.13, this in turn leads to
the existence of a constant $\zeta_{1}^{\prime}$ (again independent of $r$,
$\delta$, and $(A,\Psi)$), such that the function
${\mathfrak{q}}_{\delta}:={\mathfrak{q}}_{0,\delta}-\zeta_{1}^{\prime}\delta^{-2}$
on $X^{{}^{\prime}a}_{\delta}$ satisfies:

$\begin{split}&\big{(}\frac{d^{*}d}{2}+\frac{r|\psi|^{2}}{4}\big{)}\,{\mathfrak{q}}_{\delta}\leq
0;\\\ &{\mathfrak{q}}_{\delta}\leq\zeta\,(\ln
r)^{-1}r\,(-u)_{+}\quad\text{over $X^{{}^{\prime}a}_{\delta}$, $0\leq
a\leq\frac{39}{8}$. }\\\ &\text{$({\mathfrak{q}}_{\delta})_{+}$ is supported
on $U\subset V$,}\end{split}$

where $U$, $V$ are compact spaces defined as in the proof of 5.8. Now repeat
the remaining argument in the proof of Proposition 5.8, with
${\mathfrak{q}}_{\delta}$ playing the role of $q^{\prime}$ therein. To give
more details: We now have

$\gamma_{a,\delta}{\mathfrak{q}}_{\delta}=q_{1}+q_{2},\quad 0\leq
a\leq\frac{31}{8},$

where $\gamma_{a,\delta}$ is as in (5.42), and $q_{1}$, $q_{2}$ respectively
satisfy:

$\big{(}\frac{d^{*}d}{2}+\frac{r|\nu|}{4}\big{)}\,q_{1}=\xi^{\prime}_{\delta}\,\,\,\text{
over $V$};\qquad q_{1}|_{\partial V}=0,$

where

$\xi^{\prime}_{\delta}=\frac{d^{*}d}{2}(\gamma_{a,\delta}{\mathfrak{q}}_{\delta})-\gamma_{a,\delta}\frac{d^{*}d{\mathfrak{q}}_{\delta}}{2};$

and

(6.32)
$\begin{split}&\big{(}\frac{d^{*}d}{2}+\frac{r|\psi|^{2}}{4}\big{)}\,q_{2}=-\frac{ru}{4}q_{1}\\\
&\qquad q_{2}|_{\partial V}\leq 0.\\\ \end{split}$

The present version of $q_{1}$ is bounded by

$\begin{split}|q_{1}|&\leq\zeta_{2}^{\prime}(\ln
r)^{-1}r\,(-u)_{+}(r\delta)^{-1/2}e^{-\zeta_{g}(r\delta)^{1/2}\mathop{\mathrm{dist}\,}\nolimits(\cdot,\,X^{{}^{\prime}a+1}_{\delta}-X^{{}^{\prime}a}_{\delta})}\\\
&\qquad\qquad+\zeta_{2}(\ln
r)^{-1}r\,(-u)_{+}(r\delta^{3})^{-1/2}e^{-\zeta_{g}(r\delta)^{1/2}\mathop{\mathrm{dist}\,}\nolimits(\cdot,\,Z^{{}^{\prime}a}_{2\delta}-Z^{{}^{\prime}a}_{\delta})}.\\\
\end{split}$

Plugging this back in (6.32), we have:

$\begin{split}&\frac{d^{*}d}{2}\,(q_{2})_{+}\leq\zeta_{3}^{\prime}(\ln
r)^{-1}(r\tilde{\sigma})^{2}(r\delta)^{-1/2}e^{-\zeta_{g}(r\delta)^{1/2}\mathop{\mathrm{dist}\,}\nolimits(\cdot,\,X^{{}^{\prime}a+1}_{\delta}-X^{{}^{\prime}a}_{\delta})}\\\
&\qquad\qquad+\zeta_{3}^{\prime}(\ln
r)^{-1}(r\tilde{\sigma})^{2}(r\delta^{3})^{-1/2}e^{-\zeta_{g}(r\delta)^{1/2}\mathop{\mathrm{dist}\,}\nolimits(\cdot,\,Z^{{}^{\prime}a}_{2\delta}-Z^{{}^{\prime}a}_{\delta})}\\\
\\\ &\qquad(q_{2})_{+}|_{\partial V}=0,\qquad 0\leq a\leq\frac{31}{8}.\\\
\end{split}$

Noting again that the Green’s function $G$ for the differential operator
$\frac{d^{*}d}{2}$ with Dirichlet boundary condition on $V$ satisfies:

$|G(x,\cdot)|+\mathop{\mathrm{dist}\,}\nolimits(x,\cdot)\,|dG(x,\cdot)|\leq\zeta\mathop{\mathrm{dist}\,}\nolimits(x,\cdot)^{-2},$

this leads to the following bound for $q_{2}$ over
$X^{{}^{\prime}a}_{4\delta}\subset V$, $0\leq a\leq\frac{15}{4}$:

$q_{2}\leq\zeta_{4}\,\tilde{\sigma}^{-2}(\ln r)^{-1}r^{-1}\delta^{-4}.$

Combing this with the previously obtained bound for $q_{1}$ and the last
bullet of Lemma 6.13, we have:

$|F_{A}|\leq
2^{-3/2}\,r\,(-u)_{+}+\zeta_{4}^{\prime}\delta^{-2}\quad\text{$X^{{}^{\prime}a}_{4\delta}$,
$0\leq a\leq\frac{15}{4}$},$

when $\delta\geq\delta_{*}$, where $\zeta_{4}^{\prime}$ is independent of $r$,
$\delta$, and $(A,\Psi)$. Given a fixed $x\in X^{{}^{\prime}a}_{8\delta}$, we
take $\delta=\tilde{\sigma}(x)/2$ in the preceding formula. This leads to the
second line of (6.31), which in turn implies the first line. $\Box$

######  Remark 6.15.

The preceding improvement of Proposition 5.8 and and Lemma 6.8) is not
strictly necessary for the proof of our main theorems. (Proposition 5.8 and/or
Lemma 6.8 suffice for our purposes). Similarly, Proposition I.3.4 in [T]
improves the curvature estimate in Proposition I.2.4 therein, but the proof of
the main Theorem of part I of [T] uses only the latter. Nevertheless, the more
precise estimate in [T]’s Proposition I.3.4 plays a role in part IV of [T]. As
the present article is the counterpart of part I of [T], Proposition 6.14
should be useful for further studies on the relationship between the Seiberg-
Witten-Floer theory and ECH-type Floer theories.

###  6.5 Approximation by local models

Proposition I.4.2 of [T] (and correspondingly, Proposition 5.2 in [Ts]) has
the following counterpart in our context.

Let $\delta_{*}$ be as in be as in Proposition 6.11, and let $x\in
X^{{}^{\prime}a}_{2\delta_{*}}$. Re-introduce the notation
$\tilde{\sigma}_{x}=\tilde{\sigma}(x)$ and $\tilde{r}=r\tilde{\sigma}$;
$\tilde{r}_{x}=r\tilde{\sigma}_{x}$. Given an
$(A,\Psi)=(A_{r},\Psi_{r})=(A,(\alpha,\beta))$, let
$\underline{A}_{x},(\underline{\alpha}_{x},\underline{\beta}_{x}))$ be the
rescaled version of $(A,(\alpha,\beta))$ defined in [T]’s (I.4.7) (cf. also
Equation (5.1) of [Ts]), with the parameter $\lambda$ therein chosen to be
$(\tilde{r}_{x})^{1/2}$ instead.
($\underline{A}_{x},(\underline{\alpha}_{x},\underline{\beta}_{x}))$ was
called $(\underline{A},(\underline{\alpha},\underline{\beta}))$ in [T].

######  Lemma 6.16.

Adopt the assumptions and notation of Lemma 4.6. There is an $r_{0}>8$
depending on the same parameters listed in Proposition I.4.2 of [T] as well as
e, such that for all $r\geq r_{0}$, $x\in X^{{}^{\prime}a}_{2\delta_{*}}$,
$0\leq a\leq\frac{39}{8}$, and $(A,(\alpha,\beta))=(A,\Psi)=(A_{r},\Psi_{r})$,
there is a Seiberg-Witten solution on ${\mathbb{R}}^{4}$,
$(A_{0},(\alpha_{0},0))$, as described in Proposition I.4.1 of [T], which
approximates
$\underline{A}_{x},(\underline{\alpha}_{x},\underline{\beta}_{x}))$ in the
sense that it satisfies Items (1)–(4) of Proposition I.4.2 in [T].

Proof. The proof of Proposition I.4.2 in [T] can be copied, with the following
modifications:

  * •

As usual, the factors of $r$ in [T] are replaced by $\tilde{r}_{x}$.

  * •

Equation (I.4.8) of [T] has the following counterpart in our setting:

$\begin{split}&|\underline{\alpha}_{x}|+|F_{\underline{A}_{x}}|\leq\zeta_{1}\\\
&|\nabla_{\underline{A}_{x}}\underline{\alpha}_{x}|^{2}\leq\zeta\,(1-|\underline{\alpha}_{x}|^{2})_{+}+\zeta^{\prime}\tilde{\sigma}_{x}^{-2}\tilde{r}^{-1}\\\
&|\underline{\beta}_{x}|+|\nabla_{\underline{A}_{x}}\underline{\beta}_{x}|\leq\zeta_{1}^{\prime}\tilde{\sigma}_{x}^{-1}\tilde{r}^{-1/2},\end{split}$

where $\zeta$, $\zeta^{\prime}$, $\zeta_{1}$, $\zeta_{1}^{\prime}$ are
independent of $r$, $x$, and $(A,\Psi)$. Note that
$\tilde{\sigma}_{x}^{-1}\tilde{r}^{-1/2}\leq\zeta_{2}r^{-5/36}\to 0$ as
$r\to\infty$. These inequalities follow from Propositions 5.4, 5.5, 5.8, and
5.9.

  * •

Proposition 6.11 is used in place of [T]’s Proposition I.3.1. Proposition 5.8
(or its improved version, Proposition 6.14) is used in place of [T]’s
Proposition I.3.4.

  * •

The necessary integral bound on $r\,(|\nu|-|\alpha|^{2})$ (which, by
Proposition 5.4, 5.5, is equivalent to an integral bound on
$r\,(|\nu|-|\psi|^{2})$) is supplied by Lemma 6.2 and its amended versions in
Lemmas 6.5 and 6.9. The necessary integral bound on
$|F_{A}^{-}|^{2}-|F_{A}^{+}|^{2}$ is supplied by Proposition 4.12 and its
amendment in Proposition 6.4.

$\Box$

The preceding lemma leads to the following variant of Lemma I.4.5 in [T]:

######  Lemma 6.17.

Adopt the assumptions and notation of Lemma 4.6, and let $\delta_{*}$ be as in
be as in Proposition 6.11. Then the statement of Lemma I.4.5 in [T] holds for
$(A,\Psi)$ over $X^{{}^{\prime}a}_{2\delta_{*}}$, $0\leq a\leq\frac{39}{8}$,
with $\alpha$ and $r$ therein respectively replaced by $\underline{\alpha}$
and $\tilde{r}$. In other words, given $0<v<1$, there is a constant $z_{v}$
independent of $r$, $x$, and $(A,\Psi)$ (but depends on $v$), such that

$\mathop{\mathrm{dist}\,}\nolimits\,(\cdot,\alpha^{-1}(0))\leq
z_{v}\,\tilde{r}^{-1/2}\quad\text{over $X_{2\delta_{*},v}$},$

where $X_{\delta,v}$ is as in Corollary 6.12.

Proof. The proof identical to the proof of Lemma I.4.5 in [T], with the role
of [T]’s Proposition I.4.2 therein replaced by the preceding lemma. $\Box$

###  6.6 Exponential decay

The purpose of this subsection is to establish the counterpart of Lemma 6.1 in
[Ts] (which is a variant of Proposition I.4.4 in [T]) in our context, the
upcoming Proposition 6.19. The proof of this proposition makes use of a
corollary of the monotonicity formula, Lemma 6.18 below, which is in turn the
counterpart of Lemma I.4.6 in [T].

Before embarking on the aforementioned tasks, we introduce a notion of
“$Y_{i}$-end limits” for functions over (subdomains of)
$X^{{}^{\prime}a}_{\delta}$ (or more generally, sections of bundles over
$X^{{}^{\prime}a}_{\delta}$). The definition is parallel to the notion of
$Y_{i}$-end limits for functions/sections over (subdomains of) $X$, previously
introduced in Section 1.4 (6).

Given $\delta>0$ and $i\in{\mathfrak{Y}}_{m}$, let
$\sigma_{i}:=\mathop{\mathrm{dist}\,}\nolimits(\cdot,\nu_{i}^{-1}(0))$, and
$Y_{i,\delta}:=\\{y\in Y_{i}\,|\,\sigma_{i}\geq\delta\\}$. In what follows,
$Y_{i,\delta}$ is often identified with $Y_{i:\infty,\delta}:=Y_{i:\infty}\cap
X_{\delta}$. Observe that the admissibility of $\nu$ implies the following:
Let $g$ denote the metric on $X$ and recall
${\mathfrak{i}}\colon\thinspace\coprod_{i\in{\mathfrak{Y}}}[0,\infty)\times
Y_{i}\to X-X_{c}^{\circ}$, the isometry from Definition 1.3. Let ${\bf
Y}_{i,\delta,L}:={\mathfrak{i}}\big{(}[0,\infty)\times Y_{i,\delta}\big{)}$
when $i\in{\mathfrak{Y}}_{m}$. Given $i\in{\mathfrak{Y}}_{m}$ and a small
$\delta<1/8$, there is an $L=L_{\delta}>8$ and a diffeomorphism
${\mathfrak{i}}_{\delta}\colon\thinspace[L,\infty)\times
Y_{i,\delta}\to\hat{Y}_{i,L,\delta}:=X_{\delta}\,\cap\hat{Y}_{i,L}$, such
that:

$\begin{split}&\qquad{\bf
Y}_{i,2\delta,L}\subset\hat{Y}_{i,L,\delta}\subset{\bf Y}_{i,\delta/2,L};\\\
&\qquad{\mathfrak{i}}_{\delta}\big{|}_{{\bf
Y}_{i,2\delta,L}}={\mathfrak{i}}\big{|}_{{\bf
Y}_{i,2\delta,L}};\quad\text{{\small
s}}_{i}=\pi_{\mathbb{R}}\circ{\mathfrak{i}}_{\delta},\\\
&\|{\mathfrak{i}}_{\delta}^{*}\,g-{\mathfrak{i}}^{*}g\|_{C^{2}([l,\infty)\times
Y_{i,\delta})}+\|{\mathfrak{i}}_{\delta}^{*}\,\nu-\pi_{Y}^{*}\,\nu_{\infty}\|_{C^{2}([l,\infty)\times
Y_{i,\delta})}\leq\zeta e^{-\kappa_{i}\,l/2}\quad\forall l\geq L,\end{split}$

where $\kappa_{i}$ is the constants from (4.29), $\pi_{\mathbb{R}},\pi_{Y}$
respectively denote the projection to the first factor and the second factor
of the product ${\mathbb{R}}\times Y$, and $\zeta$ is a constant depending
only on the metric and $\nu$. Let ${\rm V}$ be a bundle over
$\overline{X_{\delta}^{\prime\prime}}$ (or its subdomains such as
$\overline{X^{{}^{\prime}a}_{\delta}}$ or
$\hat{Y}_{i,L}\cap\overline{X}_{\delta}$), and let ${\rm V}_{i}:={\rm
V}\big{|}_{Y_{i:\infty,\delta}}$. A section $q\in\Gamma({\rm
V}\big{|}_{X^{\prime\prime}_{\delta}})$ is said to have $q_{i}\in\Gamma({\rm
V}_{i})$ as a $Y_{i}$-end limit if $q$ extends to be defined over
$\overline{X_{\delta}^{\prime\prime}}$ with (the extended)
$q\big{|}_{Y_{i:\infty,\delta}}=q_{i}$; and

$\|{\mathfrak{i}}_{\delta}^{*}\,q-\pi_{Y}^{*}\,q_{i}\|_{C^{2}([l,\infty)\times
Y_{i,\delta})}+\|{\mathfrak{i}}_{\delta}^{*}\,q-\pi_{Y}^{*}q_{i}\|_{L^{2}_{1}([l,\infty)\times
Y_{i,\delta})}\to 0\quad\text{as $l\to\infty$.}$

Let ${\rm
d}_{\alpha}\,(\cdot):=\mathop{\mathrm{dist}\,}\nolimits(\cdot,\alpha^{-1}(0))$.
When $\alpha^{-1}(0)=\emptyset$, we set ${\rm
d}_{\alpha}\,(\cdot)=\mathop{\mathrm{dist}\,}\nolimits(\cdot,\alpha^{-1}(0))\equiv\infty$.
Recall the notation $\tilde{r}=r\tilde{\sigma}$. Let
$\tilde{r}_{x}:=\tilde{r}(x)$ and $\tilde{\sigma}_{x}=\tilde{\sigma}(x)$.

######  Lemma 6.18.

Adopt the assumptions and notation of Lemma 4.6, and let $\delta_{*}$ be as in
be as in Proposition 6.11. There exist positive constants $r_{0}>8$, $c$,
$c^{\prime}\geq 2^{5}\,c$, $\zeta_{0}$, $\zeta,\zeta^{\prime}$, that are
independent of $r$ and $(A,\Psi)$, such that for any $r\geq r_{0}$, there
exists a function $h_{r}$ over $X^{{}^{\prime}a}_{\delta_{*}}$, $0\leq
a\leq\frac{39}{8}$, satisfying

  * (1)

$d^{*}d\,h_{r}+\frac{r}{16}|\nu|h_{r}\geq-\zeta_{0}\exp\big{(}-(r\tilde{\sigma}^{3})^{1/2}/c^{\prime})$
where ${\rm d}_{\alpha}\geq c\tilde{r}^{-1/2}$.

  * (2)

$h_{r}\geq\zeta{\rm d}_{\alpha}^{-2}\exp\big{(}-\tilde{r}^{1/2}{\rm
d}_{\alpha}/c\big{)}$

  * (3)

$h_{r}\leq\zeta^{\prime}\tilde{r}\exp\big{(}-\tilde{r}^{1/2}{\rm
d}_{\alpha}/c\big{)}$ where ${\rm d}_{\alpha}\geq c\tilde{r}^{-1/2}$.

  * (4)

$h_{r}$ has a $Y_{i}$-end limit, denoted $h_{i,r}$, for every Morse end
$i\in{\mathfrak{Y}}_{m}$. This is a function over $Y_{i,\delta_{*}}$, which
satisfies 3-dimensional versions of (1)–(3) above.

In the above, the constants $c$, $c^{\prime}$, $\zeta_{0}$,
$\zeta,\zeta^{\prime}$ depend only on $\nu$ and the parameters listed in
(4.15), and $r_{0}$ depends on e in addition to all the above.

Proof. To construct $h_{r}$ and to verify that it satisfies Items (2) and (3)
above, repeat the arguments in the proof of (and the paragraph preceding)
Lemma I.4.6 in [T]. ($h_{r}$ is called $h$ in [T].) Modify the arguments of
[T] in like manner to what was done in the proof of Lemma 6.13:

  * •

Replace the parameter $r$ in [T] by $\tilde{r}=r\tilde{\sigma}$, and replace
balls of the form $B\,(x,\rho_{*})$ in [T] by their counterparts,
$B\,(x,\tilde{\sigma}^{-1/2}_{x}\,\rho_{*})$, in our context.

  * •

Use respectively Corollary 6.12 and Proposition 6.11 in place of Lemma I.3.6
and Proposition I.3.1 in [T].

  * •

Associate to each $y\in X^{\prime\prime}_{\delta_{*}/2}$ a function, $H_{y}$,
on $X$. This function $H_{y}$ is given by the formula in [T]’s (I.4.17), but
with $r$ replaced by $\tilde{r}_{y}$, and with the constant $c$ therein chosen
so that $H_{y}$ satisfies the following analog of [T]’s (I.4.18):

(6.33) $d^{*}d\,H_{y}+\frac{r}{32}|\nu(y)|\,H_{y}\geq 0\qquad\text{where ${\rm
d}_{\alpha}\geq c\tilde{r}_{y}^{-1/2}$.}$

(Recall the relation between $|\nu|$ and $\tilde{\sigma}$ from (5.1).) Choose
${\rm m}\geq 8$ such that for any given $y\in X^{\prime\prime}_{\delta_{*}}$,

(6.34) $\Big{|}|\nu|/|\nu(y)|-1\Big{|}<1/4$ over $B(y,2{\rm
m}^{-1}\tilde{\sigma}_{y})$.

Note that given $y\in X^{\prime\prime}_{\delta_{*}}$, the function
$\Big{|}d^{*}d\Big{(}\chi\big{(}{\rm
m}\mathop{\mathrm{dist}\,}\nolimits(\cdot,y)/\tilde{\sigma}_{y}\big{)}\,H_{y}-H_{y}\Big{)}\Big{|}$
is supported on $B(y,{\rm m}^{-1}\tilde{\sigma}_{y})-B(y,(2{\rm
m})^{-1}\tilde{\sigma}_{y})$, and by the defining formula for $H_{y}$, there
is constant $\zeta_{\rm m}$ depending on ${\rm m}$, but is independent of $r$,
$y$, $(A,\Psi)$, such that for all sufficiently large $r$ and $y\in
X^{\prime\prime}_{\delta_{*}}$,

(6.35) $\Big{|}d^{*}d\Big{(}\chi\big{(}{\rm
m}\mathop{\mathrm{dist}\,}\nolimits(\cdot,y)/\tilde{\sigma}_{y}\big{)}\,H_{y}-H_{y}\Big{)}\Big{|}\leq\zeta_{\rm
m}\,\tilde{r}_{y}^{1/2}\tilde{\sigma}_{y}^{-3}\exp\big{(}-\tilde{r}_{y}^{1/2}(2{\rm
m}c)^{-1}\tilde{\sigma}_{y}\big{)}.$

  * •

Let $\Lambda_{r}:=\Lambda^{-r^{1/2}}=\\{B\,(x_{k},\rho_{k})\\}_{k}$ denote the
version of $\Lambda^{\rho}$ from Corollary 6.12 with $\rho=r^{-1/2}$ and
$\delta=\delta_{*}$. Set

$h_{r}:=\sum_{k\in\Lambda_{r}}\chi\big{(}{\rm
m}\mathop{\mathrm{dist}\,}\nolimits(\cdot,x_{k})/\tilde{\sigma}(x_{k})\big{)}H_{x_{k}}.$

Items (2) and (3) asserted by the lemma follow from the arguments in the proof
of [T]’s Lemma I.4.6, modified as instructed in the first two bullets above.

To verify that $h_{r}$ satisfies Item (1) of the lemma, argue as in the last
bullet in the proof of Lemma 6.13: By (6.33), (6.34), (6.35), and Corollary
6.12,

$\begin{split}\big{(}d^{*}d\,h_{r}+\frac{r}{16}|\nu|h_{r}\big{)}(x)\,&\geq-\zeta^{\prime}_{\rm
m}\,r\tilde{\sigma}_{x}^{3}\sum_{k\in\Lambda_{r}\cap
B(x,3\tilde{\sigma}_{x})}\tilde{r}_{x_{k}}^{1/2}\tilde{\sigma}_{x_{k}}^{-3}\exp\big{(}-\tilde{r}_{x_{k}}^{1/2}(2{\rm
m}c)^{-1}\tilde{\sigma}_{x_{k}}\big{)},\\\ &\geq-\zeta_{\rm
m}^{\prime\prime}\,\exp\big{(}-(r\tilde{\sigma}_{x}^{3})^{1/2}(4{\rm
m}c)^{-1}\big{)}\qquad\text{where ${\rm d}_{\alpha}\geq
c\tilde{r}_{x}^{-1/2}$,}\end{split}$

when $x\in X^{\prime\prime}_{\delta_{*}}$ and $r$ is sufficiently large.

Lastly, with Items (1)-(3) now confirmed, Item (4) follows from the
admissibility assumption on $(A,\Psi)$ and the standard compactness/properties
results in Seiberg-Witten theory (cf. [KM]’s Section 10.7). $\Box$

######  Proposition 6.19.

Adopt the assumptions and notation of Lemma 4.6, and let $\delta_{*}$ be as in
be as in Proposition 6.11. There exist positive constants $r_{0}>8$,
$\zeta_{i}$, $\zeta_{i}^{\prime}$, $i=1,2,3$, and $c$ that are independent of
$r$, and $(A,\Psi)$, with the following significance: Suppose $r\geq r_{0}$,
Then over $X^{{}^{\prime}a}_{2^{5}\delta_{*}}$, $0\leq a\leq\frac{3}{2}$,

(6.36) $\displaystyle
r|\varpi|+|F_{A}|\leq\zeta_{2}r\tilde{\sigma}\exp\big{(}-(r\tilde{\sigma})^{1/2}{\rm
d}_{\alpha}/c\big{)}+\zeta_{2}^{\prime}\tilde{\sigma}^{-2};$ (6.37)
$\displaystyle|\nabla_{A}\underline{\alpha}|^{2}+r\tilde{\sigma}^{2}|\nabla_{A}\beta|^{2}\leq\zeta_{3}\,r\tilde{\sigma}\exp\big{(}-(r\tilde{\sigma})^{1/2}{\rm
d}_{\alpha}/c\big{)}+\zeta_{3}^{\prime}\,r^{-1}\tilde{\sigma}^{-5};$ (6.38)
$\displaystyle|\beta|^{2}\leq\zeta_{1}r^{-1}\tilde{\sigma}^{-2}\exp\,(-(r\tilde{\sigma})^{1/2}{\rm
d}_{\alpha}/c)+\zeta_{1}^{\prime}r^{-2}\tilde{\sigma}^{-5}.$

In the above, the constants $c$, $\zeta_{i},\zeta^{\prime}_{i}$ depend only on
$\nu$ and the parameters listed in (4.15), and $r_{0}$ depends on e in
addition to all the above.

Proof. Return to the beginning of the proof of Proposition 5.9.

Fix $\delta$ with $1\geq\delta\geq 2\delta_{0}^{\prime}$. Use (5.43) to find
constants $\zeta_{0}$, $\zeta_{1}^{\prime}$, $\zeta_{1}$ that are independent
of $r$, $\delta$ and $(A,\Psi)$, such that the function

$\mu=\mu_{\delta}:=\delta^{-2}\,|\nabla_{A}\underline{\alpha}|^{2}+\zeta_{0}\,r\,|\nabla_{A}\beta|^{2}$

satisfies

(6.39)
$\begin{split}\Big{(}d^{*}d+\frac{r}{8}|\nu|\Big{)}\,\mu_{\delta}&\leq\zeta_{1}r\varpi_{+}\mu_{\delta}\,+\zeta_{2}\delta^{-1}r^{-1}\tilde{\sigma}^{-7}\\\
&\quad+\zeta_{2}^{\prime}\tilde{\sigma}^{-6}+\zeta_{3}\delta^{-2}r\tilde{\sigma}^{-3}\varpi^{2}+\zeta_{3}^{\prime}r\tilde{\sigma}^{-4}\varpi_{+}\end{split}$

over $X^{{}^{\prime}a}_{\delta}$, $0\leq a\leq\frac{39}{8}$. In particular,
this implies that for all sufficiently large $r$ and $\delta\geq
2\delta^{\prime}_{0}$,

(6.40)
$\big{(}d^{*}d+\frac{r}{16}|\nu|\big{)}\,\mu_{\delta}\leq\zeta^{\prime}r\delta^{-3}\quad\quad\text{over
$X_{\delta}^{{}^{\prime}a}-X_{\delta,v}^{{}^{\prime}a}$, $0\leq
a\leq\frac{39}{8}$,}$

where $v=1-(9\zeta_{1})^{-1}$ and
$X^{{}^{\prime}a}_{\delta,v}:=\\{y\,|\,|\underline{\alpha}|(y)\leq v,\,y\in
X^{{}^{\prime}a}_{\delta}\\}$ as in Corollary 6.12.

The following observation will be useful. Given $z>0$, let

$U^{{}^{\prime}a}_{\delta,z}:=\\{y\,|\,\,{\rm d}_{\alpha}\,(y)\geq
z\,\tilde{r}^{-1/2},y\in X^{{}^{\prime}a}_{\delta}\\}.$

According to Lemma 6.17,
$X^{{}^{\prime}a}_{\delta}-X^{{}^{\prime}a}_{\delta,v}\supset
U^{{}^{\prime}a}_{\delta,z_{v}}$, where $z_{v}$ is the constant from Lemma
6.17. These noted, (6.40) implies:

(6.41)
$\big{(}d^{*}d+\frac{r}{16}|\nu|\big{)}\,\mu_{\delta}<\zeta^{\prime}r\delta^{-3}\qquad\text{over
$U^{{}^{\prime}a}_{\delta,z_{v}}$. }$

Note that by Propositions 5.9 and 5.5, there is a positive constant
$\zeta_{4}$ independent of $r$, $\delta$, $(A,\Psi)$, such that

(6.42) $\begin{split}\mu_{\delta}&\leq\zeta_{4}\,r\delta^{-1}\quad\text{over
$X_{\delta};\quad$ in particular, }\\\
\mu_{\delta}&\leq\zeta_{4}\,e^{z/c}\,r\delta^{-1}\exp\big{(}-\tilde{r}^{1/2}{\rm
d}_{\alpha}/c\big{)}\quad\text{over
$X^{{}^{\prime}a}_{\delta}-U^{{}^{\prime}a}_{\delta,z}$ for arbitrary
$c,z>0$.}\end{split}$

Now take $c$ to be the constant from Lemma 6.18, and set $z=\max\,(c,z_{v})$.
Then (6.41), (6.42), Lemma 6.18 can be used to find constants $\zeta_{5}$,
$\zeta_{5}^{\prime}$, which are independent of $r$, $\delta$, and $(A,\Psi)$,
so that:

(6.43)
$\begin{split}&\big{(}d^{*}d+\frac{r}{16}|\nu|\big{)}\,(\mu_{\delta}-\zeta_{5}\,\delta^{-2}h_{r}-\zeta_{5}^{\prime}\delta^{-4})\leq
0\quad\text{over $U^{{}^{\prime}a}_{\delta,z}$;}\\\
&\quad\mu_{\delta}-\zeta_{5}\,\delta^{-2}h_{r}\leq 0\quad\text{over
$X_{\delta}^{{}^{\prime}a}-U^{{}^{\prime}a}_{\delta,z}$}.\end{split}$

To be able to apply the maximum principle to (6.43), we still need another
comparison function, $q_{1}$ below, in order to offset the positive values of
$\mu_{\delta}\big{|}_{\partial U_{\delta,z}}$ along the $(\partial
X_{\delta})\,\cap U_{\delta,z}$ part of $\partial\,U_{\delta,z}$.

Recall the definition of ${\mathscr{X}}^{a}_{\delta}\subset
X_{\delta}^{{}^{\prime}a}$ and $\gamma_{a,\delta}$ from the beginning of
Section 5. Given any $i\in{\mathfrak{Y}}_{m}$, let $\mu_{\delta,i}$ denote the
$Y_{i}$-end limit of $\mu_{\delta}$ , and let $q_{1,i}$ be the function on
$Y_{i,\delta}$ solving the following Dirichlet boundary value problem:

(6.44)
$\begin{split}\big{(}d^{*}d+\frac{r}{16}|\nu_{i}|\big{)}\,q_{1,i}&=d^{*}d\big{(}(1-\chi(\sigma_{i}/\delta))\,\mu_{\delta,i}\big{)}-(1-\chi(\sigma_{i}/\delta))\,d^{*}d\,\mu_{\delta,i};\\\
q_{1,i}\big{|}_{\partial Y_{i,\delta}}&=0.\end{split}$

Then $q_{1}$ is defined to be the function on ${\mathscr{X}}^{a+1}_{\delta}$,
$0\leq a\leq\frac{31}{8}$, solving:

(6.45)
$\begin{split}&\big{(}d^{*}d+\frac{r}{16}|\nu|\big{)}\,q_{1}=d^{*}d\big{(}\gamma_{a,\delta}\,\mu_{\delta}\big{)}-\gamma_{a,\delta}\,d^{*}d\,\mu_{\delta};\\\
&\qquad\qquad q_{1}\big{|}_{\partial{\mathscr{X}}^{a+1}_{\delta}}=0;\\\
&\text{$\forall i\in{\mathfrak{Y}}_{m}$, $\,\,q_{1,i}$ is the $Y_{i}$-end
limit of $q_{1}$. }\end{split}$

Let

$\begin{split}{\rm
u}_{\delta}&:=\gamma_{a,\delta}\,\mu_{\delta}-q_{1}-\zeta_{5}\,\delta^{-2}h_{r}-\zeta_{5}^{\prime}\,\delta^{-4};\\\
{\rm
u}_{i,\delta}&:=(1-\chi(\sigma_{i}/\delta))\,\mu_{\delta,i}-q_{1,i}-\zeta_{5}\,\delta^{-2}h_{i,r}-\zeta_{5}^{\prime}\,\delta^{-4},\quad
i\in{\mathfrak{Y}}_{m}.\end{split}$

Then (6.45) and (6.43) together imply:

$\begin{split}&\big{(}d^{*}d+\frac{r}{16}|\nu|\big{)}\,{\rm u}_{\delta}\leq
0\quad\text{over ${\mathscr{X}}_{\delta}^{a+1}\cap U^{a+1}_{\delta,z}$ ;}\\\
&\qquad{\rm u}_{\delta}<0\qquad\text{over
$\,\partial\,({\mathscr{X}}_{\delta}^{a+1}\cap U^{a+1}_{\delta,z})$};\\\
&\text{$\forall i\in{\mathfrak{Y}}_{m}$, ${\rm u}_{i,\delta}$ is the
$Y_{i}$-end limit of ${\rm u}_{\delta}$,}\end{split}$

while ${\rm u}_{i,\delta}$ satisfies:

$\begin{split}&\big{(}d^{*}d+\frac{r}{16}|\nu_{i}|\big{)}\,{\rm
u}_{i,\delta}\leq 0\quad\text{over $U_{i,\delta,z}$;}\\\ &\quad{\rm
u}_{i,\delta}\big{|}_{\partial(U_{i,\delta,z})}<0\quad\forall
i\in{\mathfrak{Y}}_{m},\end{split}$

where $U_{i,\delta,z}\subset Y_{i,\delta}$ is the 3-dimensional analog of
$U^{a+1}_{\delta,z}$: $U_{i,\delta,z}:=\\{y\,|\,\,{\rm
d}_{\alpha_{i}}\,(y)\geq z\,(r\tilde{\sigma}_{i}(y))^{-1/2},y\in
Y_{i,\delta}\\}$, and $\overline{U^{a+1}_{\delta,z}}\cap
Y_{i:\infty,\delta}=U_{i,\delta,z}$. Thus, according to the maximum principle,

$\begin{split}\sup_{U_{i,\delta,z}}{\rm u}_{i,\delta}&\leq 0;\\\
\sup_{{\mathscr{X}}_{\delta}^{a+1}\cap U^{a+1}_{\delta,z}}{\rm
u}_{\delta}&\leq\max_{i\in{\mathfrak{Y}}_{m}}\,(\sup_{U_{i,\delta,z}}{\rm
u}_{i,\delta})\leq 0,\end{split}$

Together with the second line of (6.43), this implies in particular that

(6.46)
$\mu_{\delta}\leq|q_{1}|+\zeta_{5}\,\delta^{-2}h_{r}+\zeta_{5}^{\prime}\,\delta^{-4}\quad\text{over
$X^{{}^{\prime}a}_{2\delta}\subset{\mathscr{X}}_{\delta}^{a+1}$.}$

We next give a pointwise bound for $|q_{1}|$, beginning by estimating its
$Y_{i}$-end limits, $|q_{1,i}|$. First, note that the (Dirichlet) Green’s
function for the elliptic system (6.44), denoted $G_{i,r}$, satisfies a bound
of the form:

$|G_{i,r}(x,\cdot)|+\mathop{\mathrm{dist}\,}\nolimits(x,\cdot)\,|dG_{i,r}(x,\cdot)|\leq{\rm
c}_{i}\mathop{\mathrm{dist}\,}\nolimits(x,\cdot)^{-1}\exp\,\big{(}-(r\delta)^{1/2}\mathop{\mathrm{dist}\,}\nolimits(x,\cdot)/{\rm
c}_{i}\big{)},$

where ${\rm c}_{i}$ is a constant depending only on the metric and $\nu_{i}$.
Multiply both sides of (6.44) by $G_{i,r}$ and integrate over $Y_{i,\delta}$.
The preceding bounds for $G_{i,r}$ and $dG_{i,r}$, together with the
($Y_{i}$-end limit of) the first line in (6.42) then leads to:

(6.47)
$\begin{split}|q_{1,i}|\leq\zeta^{\prime}_{i}r^{-1/2}\delta^{-5/2}\quad\text{over
$Y_{i,\delta}$},\\\ \end{split}$

where $\zeta^{\prime}_{i}$ is a constant independent of $r$, $\delta$, and
$(A,\Psi)$. Moreover, using (the $Y_{i}$-end limit of the bound from)
Proposition 5.9 and (3.23), one has for all sufficiently large $r$ and
$\delta\geq\delta_{*}$:

(6.48)
$\begin{split}|q_{1,i}|&\leq\zeta\delta^{-2}\sigma_{i}^{-1}\exp\,\big{(}-(2{\rm
c}_{i})^{-1}(r\delta)^{1/2}\sigma_{i}\big{)}\int_{A_{i,\delta}}\mu_{\delta,i}\\\
&\leq\zeta^{\prime}\delta^{-2}\sigma_{i}^{-1}\exp\,\big{(}-(2{\rm
c}_{i})^{-1}(r\delta)^{1/2}\sigma_{i}\big{)}\,\delta^{-3}\\\
&\leq\zeta^{\prime\prime}\,\exp\,\big{(}-(r\delta^{3})^{1/2}/{\rm
c}_{i}\big{)}\quad\text{over $Y_{i,4\delta}$},\end{split}$

where $A_{i,\delta}$ denotes the support of $d\chi(\sigma_{i}/\delta))$, and
$\zeta^{\prime\prime}$ is a constant independent of $r$, $\delta$, and
$(A,\Psi)$.

These $q_{1,i}$ being the $Y_{i}$-end limits of $q_{1}$, one may find an
$L>8$, such that

(6.49)
$\begin{split}&|q_{1}-({\mathfrak{i}}_{\delta}^{-1}\circ\pi_{Y})^{*}q_{1,i}|\leq
r^{-1}\quad\text{over $\hat{Y}_{i,L,\delta}$
$\forall\,i\in{\mathfrak{Y}}_{m}$, and hence by (\ref{bdd:q_1i}), }\\\
&|q_{1}|\leq\zeta_{7}r^{-1}\quad\text{over $\hat{Y}_{i,L,4\delta}$
$\forall\,i\in{\mathfrak{Y}}_{m}$.}\end{split}$

In the above, $\zeta_{7}$ is a constant independent of $r$, $\delta$, and
$(A,\Psi)$ (though $L$ may be dependent). Let
${\mathscr{X}}^{a+1}_{\delta,L}\subset{\mathscr{X}}^{a+1}_{\delta}$ be a
manifold with boundary with
${\mathscr{X}}^{a+1}_{\delta}-\bigcup_{i\in{\mathfrak{Y}}_{m}}\hat{Y}_{i,L+2,\delta}\subset{\mathscr{X}}^{a+1}_{\delta,L}\subset{\mathscr{X}}^{a+1}_{\delta}-\bigcup_{i\in{\mathfrak{Y}}_{m}}\hat{Y}_{i,L+3,\delta}$,
and let $\lambda_{L}$ be a smooth cutoff function supported on
${\mathscr{X}}^{a+1}_{\delta,L}$ which equals 1 over
${\mathscr{X}}^{a+1}_{\delta}-\bigcup_{i\in{\mathfrak{Y}}_{m}}\hat{Y}_{i,L+1,\delta}$.
By (6.45), the function $\lambda_{L}q_{1}$ now solves the Dirichlet boundary
valued problem:

$\begin{split}&\big{(}d^{*}d+\frac{r}{16}|\nu|\big{)}\,(\lambda_{L}q_{1})=d^{*}d\big{(}\gamma_{a,\delta}\,\mu_{\delta}\big{)}-\gamma_{a,\delta}\,d^{*}d\,\mu_{\delta}+d^{*}d\,(\lambda_{L}q_{1})-\lambda_{L}d^{*}dq_{1};\\\
&\qquad\qquad(\lambda_{L}q_{1})\big{|}_{\partial{\mathscr{X}}^{a+1}_{\delta,L}}=0.\\\
\end{split}$

The Green’s function for the preceding elliptic system satisfies a bound of
the form:

$|G_{r}(x,\cdot)|+\mathop{\mathrm{dist}\,}\nolimits(x,\cdot)\,|dG_{r}(x,\cdot)|\leq
c_{0}\mathop{\mathrm{dist}\,}\nolimits(x,\cdot)^{-2}\exp\,\big{(}-(r\delta)^{1/2}\mathop{\mathrm{dist}\,}\nolimits(x,\cdot)/c_{0}\big{)},$

where $c_{0}$ is a constant depending only on the metric and $\nu$. As usual,
multiply both sides of the first line in (6.45) by $G_{r}$ and integrate over
${\mathscr{X}}_{\delta}^{a+1}$. Integration by parts then gives:

$\begin{split}|q_{1}|&\leq\zeta_{1}\delta^{-2}\sigma^{-2}\exp\,\big{(}-(2c_{0})^{-1}(r\delta)^{1/2}\sigma\big{)}\int_{X^{{}^{\prime}a+1}_{\delta}-X^{{}^{\prime}a+1}_{2\delta}}\mu_{\delta}\\\
&\qquad+\zeta_{1}^{\prime}\operatorname{\text{{\small
s}}}_{*}^{-2}\exp\,\big{(}-(r\delta)^{1/2}\operatorname{\text{{\small
s}}}_{*}/c_{0}\big{)}\int_{X^{{}^{\prime}a+1}_{\delta}-X^{{}^{\prime}a}_{\delta}}\mu_{\delta}\\\
&\qquad+\zeta_{2}^{\prime}\operatorname{\text{{\small
s}}}_{L}^{-2}\exp\,\big{(}-(r\delta)^{1/2}\operatorname{\text{{\small
s}}}_{L}/c_{0}\big{)}\int_{\bigcup_{i\in{\mathfrak{Y}}_{m}}(\hat{Y}_{i,L+1,\delta}-\hat{Y}_{i,L+2,\delta})}|q_{1}|\quad\end{split}$

over
$X^{{}^{\prime}a-\frac{1}{8}}_{4\delta}-\bigcup_{i\in{\mathfrak{Y}}_{m}}\hat{Y}_{i,L,\delta}$
for all sufficiently large $r$ and $\delta\geq\delta_{*}$, where
$\operatorname{\text{{\small s}}}_{*}\geq 1$, $\operatorname{\text{{\small
s}}}_{L}\geq 1$ respectively denote distances to
$X^{{}^{\prime}a+1}_{\delta}-X^{{}^{\prime}a}_{\delta}$ and
$\bigcup_{i\in{\mathfrak{Y}}_{m}}(\hat{Y}_{i,L+1,\delta}-\hat{Y}_{i,L+2,\delta})$;
and $\zeta_{1}$, $\zeta_{1}^{\prime}$, $\zeta_{2}^{\prime}$ are constants
independent of $r$, $\delta$, $L$, and $(A,\Psi)$. Use Propositions 5.5, 5.9,
Lemma 6.2 and its amendments in Lemmas 6.1, 6.9 to estimate the first two
integrals in the preceding expression, and use (6.47), (6.49) to bound the
last integral. One has:

$|q_{1}|\leq\zeta_{7}^{\prime}\exp\,\big{(}-(r\delta^{3})^{1/2}/{\rm
c}_{0}\big{)}\leq\zeta_{8}r^{-1}\quad\text{over
$X^{{}^{\prime}a-\frac{1}{8}}_{4\delta}-\bigcup_{i\in{\mathfrak{Y}}_{m}}\hat{Y}_{i,L,4\delta}$,
$0\leq a\leq\frac{31}{8},$ }$

for all sufficiently large $r$ and $\delta\geq\delta_{*}$, where
$\zeta_{7}^{\prime}$, $\zeta_{8}$ are constants independent of $r$, $\delta$,
$L$, and $(A,\Psi)$. Combine the preceding bound with the second line of
(6.49), (6.46), the second line of (6.43), Item (3) of Lemma 6.18, and the
second line of (6.42) to see that

$\begin{split}&\delta^{-2}|\nabla_{A}\underline{\alpha}|^{2}+\zeta_{0}\,r\,|\nabla_{A}\beta|^{2}=\mu_{\delta}\\\
&\quad\leq\zeta_{5}\,\delta^{-2}h_{r}+\zeta_{6}^{\prime}\,\delta^{-4}\\\
&\quad\leq\zeta_{6}\,r\delta^{-1}\exp\big{(}-(r\delta)^{1/2}{\rm
d}_{\alpha}/c\big{)}+\zeta_{6}^{\prime}\,\delta^{-4}\quad\text{over
$X^{{}^{\prime}a}_{4\delta}$, $0\leq a\leq\frac{15}{4}$.}\end{split}$

Fix $x\in X^{{}^{\prime}a}_{8\delta_{*}}$ and take $\delta=\tilde{\sigma}(x)$
in the preceding expression. It follows from Item (3) of Lemma 6.18 that

(6.50)
$\begin{split}&|\nabla_{A}\underline{\alpha}|^{2}+\zeta_{0}\,r\tilde{\sigma}^{2}|\nabla_{A}\beta|^{2}/4\\\
&\qquad\leq\zeta_{5}\,h_{r}+\zeta_{6}^{\prime}\,\tilde{\sigma}^{-2}\quad\\\
&\qquad\leq\zeta_{6}\,\tilde{r}\exp\big{(}-\tilde{r}^{1/2}{\rm
d}_{\alpha}/c\big{)}+\zeta_{6}^{\prime}\,\tilde{\sigma}^{-2}\quad\text{over
$X^{{}^{\prime}a}_{8\delta_{*}}$, $0\leq a\leq\frac{15}{4}$}\end{split}$

where $\zeta_{6}$, $\zeta_{6}^{\prime}$ are constants independent of $r$,
$\delta$, $L$, and $(A,\Psi)$. Hold this for the moment before we return to
show how this leads to (6.37).

To prove the remaining assertions of the proposition, (6.36) and (6.38), it
suffices to verify the bound for $|\varpi|$; in fact, we shall show:

(6.51) $|\varpi|\leq\zeta\tilde{\sigma}\exp\big{(}-(r\tilde{\sigma})^{1/2}{\rm
d}_{\alpha}/c\big{)}+\zeta^{\prime}r^{-1}\tilde{\sigma}^{-2}\quad\text{over
$X^{{}^{\prime}a}_{16\,\delta_{*}}$, $0\leq a\leq\frac{21}{8}$.}$

Once (6.51) is established, the asserted bounds for $|F_{A}|$ and
$|\beta|^{2}$ would follow from Propositions 6.14 and 5.5.

To verify (6.51), note that the argument for (5.13) actually shows that the
absolute value of the left hand side of (5.13) is no greater than its right
hand side. Together with (4.55), Proposition 5.4, and (6.50), this implies:

(6.52)
$\begin{split}2^{-1}d^{*}d\,|\varpi|+4^{-1}r|\alpha|^{2}|\varpi|&\leq|\nu|\,|\nabla_{A}\underline{\alpha}|^{2}+\zeta_{1}^{\prime}|\nabla_{A}\beta|^{2}+\zeta_{2}^{\prime}\tilde{\sigma}^{-2}|\beta|^{2}+\zeta_{2}\tilde{\sigma}^{-1}\\\
&\leq\zeta\tilde{\sigma}h_{r}+\zeta^{\prime}\tilde{\sigma}^{-1}\quad\text{over
$X^{{}^{\prime}a}_{8\delta_{*}}$, $0\leq a\leq\frac{15}{4}$.}\end{split}$

Meanwhile, by Proposition 5.4,

(6.53) $\begin{split}|\varpi|&\leq|\nu|\quad\text{over
$X^{{}^{\prime}a}_{\delta_{*}}$;\quad in particular, }\\\
|\varpi|&\leq\zeta_{4}\,e^{z/c}\,\tilde{\sigma}\exp\big{(}-\tilde{r}^{1/2}{\rm
d}_{\alpha}/c\big{)}\quad\text{over
$X^{{}^{\prime}a}_{\delta_{*}}-U^{{}^{\prime}a}_{\delta_{*},z}$ for arbitrary
$c,z>0$.}\end{split}$

Recall that $X^{{}^{\prime}a}_{\delta}-X^{{}^{\prime}a}_{\delta,v}\supset
U^{{}^{\prime}a}_{\delta,z_{v}}$, where $z_{v}$ is the constant from Lemma
6.17. Let $z_{*}=\max\,(z,z_{1/4})$, where $z$ is as in the paragraph after
(6.42), and combine (6.52), (6.53) with Lemma 6.18 to find constants
$\zeta_{3}$, $\zeta_{3}^{\prime}$ that are independent of $r$, and $(A,\Psi)$,
so that:

(6.54)
$\begin{split}&\big{(}d^{*}d+\frac{r}{8}|\nu|\big{)}\,\big{(}|\varpi|-\zeta_{3}\,r^{-1}h_{r}-\zeta_{3}^{\prime}r^{-1}\tilde{\sigma}^{-2}\big{)}\leq
0\quad\text{over
$X^{{}^{\prime}a}_{8\delta_{*}}-X^{{}^{\prime}a}_{8\delta_{*},1/4}\supset
X^{{}^{\prime}a}_{8\delta_{*}}\cap U^{{}^{\prime}a}_{8\delta_{*},z_{*}}$;}\\\
&\quad|\varpi|-\zeta_{3}\,r^{-1}h_{r}\leq 0\quad\text{over
$X^{{}^{\prime}a}_{8\delta_{*}}-U^{{}^{\prime}a}_{8\delta_{*},z_{*}}$, $0\leq
a\leq\frac{15}{4}$}.\end{split}$

Noting that (6.54), (6.53) are respectively parallel to (6.43) and (6.42), a
repetition of the arguments following (6.43) leads to (6.51).

We now return to the verification of (6.37). Given (6.50), it suffices to
verify (6.37) over the region where $\tilde{r}\exp\big{(}-\tilde{r}^{1/2}{\rm
d}_{\alpha}/c\big{)}\leq\tilde{\sigma}^{-2}$. That is, where ${\rm
d}_{\alpha}\geq\frac{c}{\tilde{r}}\ln(\tilde{r}\tilde{\sigma}^{2})$. Denote
this region in $X^{{}^{\prime}a}_{16\delta_{*}}$, $0\leq a\leq\frac{21}{8}$ by
$V_{r}$. Reconsider the inequality (6.39), and the arguments that follow.
Armed with (6.52), (6.41) may be improved as:

$\begin{split}&\big{(}d^{*}d+\frac{r}{16}|\nu|\big{)}\,\mu_{\delta}<\zeta^{\prime}_{1}\tilde{\sigma}^{-6}\\\
&\quad+\zeta_{4}^{\prime}r\tilde{\sigma}^{-1}\delta^{-2}\exp\big{(}-2(r\tilde{\sigma})^{1/2}{\rm
d}_{\alpha}/c\big{)}+\zeta_{5}^{\prime}r\tilde{\sigma}^{-3}\exp\big{(}-(r\tilde{\sigma})^{1/2}{\rm
d}_{\alpha}/c\big{)}\quad\text{over $U^{{}^{\prime}a}_{\delta,z_{v}}$.
}\end{split}$

In particular,

$\big{(}d^{*}d+\frac{r}{16}|\nu|\big{)}\,\mu_{\delta}<\zeta^{\prime}_{2}\tilde{\sigma}^{-6}\quad\text{over
$V_{r}$. }$

Noting that over $\partial V_{r}$,
$\mu_{\delta}\leq\zeta\delta^{-2}\tilde{\sigma}^{-2}$ and appealing to Lemma
6.18 again, we have the following analog of (6.43):

$\begin{split}&\big{(}d^{*}d+\frac{r}{16}|\nu|\big{)}\,(\mu_{\delta}-\zeta_{6}\,\delta^{-2}h_{r}-\zeta_{6}^{\prime}r^{-1}\delta^{-7})\leq
0\quad\text{over $V_{r}$;}\\\
&\quad\mu_{\delta}-\zeta_{5}\,\delta^{-2}h_{r}\leq 0\quad\text{over $\partial
V_{r}$}.\end{split}$

Repeating the arguments following (6.43) with straightforward modifications
then leads to (6.37). $\Box$

##  7 Proofs of the main theorems

The section brings together the intermediate results obtained in Sections 4-6
above to prove the theorems claimed in Section 1. In this section, the
parameter $a$ in $X^{{}^{\prime}a}$ is set to satisfy $0\leq a\leq 3/2$.

###  7.1 Proving Theorem 1.7

Let $\text{{\small c}}_{r}:=(A_{r},\Psi_{r})$, $r\in\\{r_{n}\\}_{n}=:\Gamma$
be as in the statement of Theorem 1.7. As observed before, in this case by
Lemma 4.3, both inequalities in (4.13) in Lemma 4.6 hold with the constant e
determined by (4.8) via (4.11). In this section, we shall repeated invoke this
fact in combination with other results in previous sections, sometimes
implicitly.

##### Proof of Theorem 1.7.

(a): To find the desired t-curve $\mathbf{C}$ on $X^{{}^{\prime}a}$, associate
to each $(A_{r},\Psi_{r})$ a 2-current $\mathcal{F}_{r}$ on $X^{{}^{\prime}a}$
as follows. Given a compactly supported smooth 2-form $\mu$ on
$X^{{}^{\prime}a}$, let

$\mathcal{F}_{r}(\mu):=\frac{1}{2}\int_{X^{{}^{\prime}a}}\frac{i}{2\pi}(F_{A_{r}}-F_{A_{K}})\wedge\mu,$

where $F_{A_{K}}$ is the curvature form of the canonical connection on
$K^{-1}$. (As observed in [Ts], the 2-form $iF_{A_{K}}$ is singular at
$Z:=\nu^{-1}(0)$, but for any compact $X_{\bullet}\subset X^{\prime\prime}$,
the integral $\int_{X_{\bullet}}iF_{A_{K}}\wedge\mu$ is still finite because
$|F_{A_{K}}|\leq\zeta_{X_{\bullet}}\mathop{\mathrm{dist}\,}\nolimits(\cdot,Z)^{-2}$
near $Z\cap X_{\bullet}$). In fact, by our definition of MCE,

(7.1) $\|F_{A_{K}}\|_{L^{1}(X_{\bullet})}\leq\zeta_{K}|X_{\bullet}|.$

In the above, $\zeta_{K}$ is a positive constant that depends only on the
metric on $X$ and $\nu$. Together with (6.11) and Lemma 4.3, this implies that
there is a positive constant $\zeta_{E}$ that depends only on

(7.2) $\hss\left\\{\mkern-20.0mu\parbox{422.77661pt}{\begin{itemize}
\par\itemize@item@the metric and
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure on $X$, \par\itemize@item@the
relative homology class ${\rm h}(\text{{\scriptsize
h}})\in{\mathscr{H}}\big{(}(X,\nu),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$ (which
in turn depends implicitly on $\\{\tilde{\gamma}_{i}\\}_{i}$,
\par\itemize@item@$\nu$, $\varsigma_{w}$, $\textsl{z}_{\mathfrak{p}}$,
\par\end{itemize}}\right.$

such that

(7.3)
$\|F_{A_{r}^{E}}\|_{L^{1}(X_{\bullet})}\leq\zeta_{E}\,(|X_{\bullet}|+1).$

Thus, by Alaoglu’s theorem, there is a subsequence of
$\\{\mathcal{F}_{r}\\}_{r}$ which converges on $X_{\bullet}$. A
diagonalization argument then shows that a subsequence, $\Gamma_{0}$, of the
former converges to a current $\mathcal{F}$ on $X^{{}^{\prime}a}$. Abusing
notation, this subsequence is also denoted by $\\{\mathcal{F}_{r}\\}_{r}$.
With this done, the arguments in Sections 7(b)-(d) and 7(f) in [Ts] carry
through directly, with Lemma 4.1 therein substituted by our Corollary 6.12.
This gives the t-curve $\mathbf{C}$ and proves the first claim of the theorem
regarding t-convergence.

(b): By definition,

${\mathscr{F}}_{\omega}(\mathbf{C})\leq{\mathscr{F}}_{\nu}(\mathbf{C}|_{X^{\prime}})+\sup_{X_{*}\in{\mathfrak{X}}_{1}}{\mathscr{F}}_{\omega}(\mathbf{C}|_{X_{*}}),$

where ${\mathfrak{X}}_{1}=\\{X_{*}\,|\,X_{*}\subset
X^{{}^{\prime}a},|X_{*}|\leq 1\\}$. Consequently, it suffices to bound
${\mathscr{F}}_{\nu}(\mathbf{C}|_{X^{{}^{\prime}a}})$ and
${\mathscr{F}}_{\omega}(\mathbf{C}|_{X_{*}})$.

We begin by bounding ${\mathscr{F}}_{\nu}(\mathbf{C}|_{X^{{}^{\prime}a}})$.
Fix a compact $X_{\bullet}\subset X^{{}^{\prime}a}$. Since
$\\{\mathcal{F}_{r}\\}_{r}$ converges to $\tilde{C}$ as currents,

(7.4)
${\mathscr{F}}_{\nu}(\mathbf{C}|_{X_{\bullet}})=\lim_{r\to\infty}\frac{1}{2}\int_{X_{\bullet}}\frac{i}{2\pi}(F_{A_{r}}-F_{A_{K}})\wedge\nu$

Note that by (7.1) and (4.29),

(7.5)
$\Big{|}\int_{X_{\bullet}}\frac{i}{2\pi}F_{A_{K}}\wedge\nu\Big{|}\leq\zeta^{\prime}_{K}$

for a positive constant $\zeta^{\prime}_{K}$ independent of $X_{\bullet}$.
Combine this with (7.4) and Lemmas 6.7, 4.3 to get

${\mathscr{F}}_{\nu}(\mathbf{C}|_{X_{\bullet}})\leq\zeta_{v},$

where $\zeta_{v}>0$ is independent of $X_{\bullet}$ and $r$, and is determined
by the items listed in (7.2). This implies that

${\mathscr{F}}_{\nu}(\mathbf{C}|_{X^{{}^{\prime}a}})\leq\zeta_{v}$

as well.

Fix $X_{*}\in{\mathfrak{X}}_{1}$. We now proceed to bound
${\mathscr{F}}_{\omega}(\mathbf{C}|_{X_{*}})$. By the convergence of
$\\{\mathcal{F}_{r}\\}_{r}$ again,

(7.6)
${\mathscr{F}}_{\omega}(\mathbf{C}|_{X_{*}})=\lim_{r\to\infty}\frac{1}{2}\int_{X_{*}}\frac{i}{2\pi}(F_{A_{r}}-F_{A_{K}})\wedge\omega$

Combine this with (7.1), Lemma 4.3, Lemma 4.16 and its amendment in
Proposition 6.4, Lemma 4.16, we have

${\mathscr{F}}_{\omega}(\mathbf{C}|_{X_{*}})\leq\zeta_{v}^{\prime}$

for any $X_{*}\in{\mathfrak{X}}_{1}$. Like $\zeta_{v}$, $\zeta_{v}^{\prime}$
only depends on the items listed in (7.2). Together with the bound
${\mathscr{F}}_{\nu}(\mathbf{C}|_{X^{{}^{\prime}a}})\leq\zeta_{v}$, this
implies Assertion (b) of Theorem 1.7. $\Box$

###  7.2 SW and Gr versions of Floer-theoretic energies

Compactness results in Floer theory typically hinge on bounds on certain
“topological energy”, whose value only depends on the relative homotopy class
of the solution. In Morse theory, this is just the change of the value of the
Morse function along the gradient flow. In the Seiberg-Witten setting, the
relevant notion of topological energy is ${\mathscr{E}}_{top}$ in [KM] (cf.
e.g. Equation (5.1) of [KM]). For $Gr$-variants of Floer theories (such as
ECH/PFH, SFT) that do not really have a Morse-theoretic model, local
convergence and global convergence are respectively controlled by
${\mathscr{F}}_{\omega}$ and ${\mathscr{F}}_{\nu}$. In Theorem 1.7, we
obtained a bound on ${\mathscr{F}}_{\omega}$ of the limiting t-curve
$\mathbf{C}$. For convenience of later discussions, rename this $\mathbf{C}$
as $\mathbf{C}_{0}$. Before moving on to the proof of Theorem 1.9, we need to
better understand the relation between ${\mathscr{F}}_{\nu}(\mathbf{C}_{0})$
and its counterpart on the SW side, ${\mathscr{E}}_{top}$. The two are related
via the quantity:

${\mathscr{V}}_{r}(X_{\bullet}):=\int_{X_{\bullet}}\frac{i}{2\pi}F_{A^{E}_{r}}\wedge\nu=\int_{X_{\bullet}}\frac{i}{4\pi}F_{A_{r}}\wedge\nu-\int_{X_{\bullet}}\frac{i}{4\pi}F_{A_{K}}\wedge\nu.$

Let

${\mathfrak{E}}_{r}(X_{\bullet}):=\frac{1}{\pi
r}{\mathscr{E}}_{top}^{\mu_{r},\hat{{\mathfrak{p}}}}(X_{\bullet})(A_{r},\Psi_{r})-\int_{X_{\bullet}}\frac{iF_{A_{K}}}{4\pi}\wedge\nu.$

It is easier to compare averaged versions of the three notions of energy,
${\mathfrak{E}}_{r}$, ${\mathscr{V}}_{r}$, ${\mathscr{F}}_{\nu}$, defined as
follows. Suppose $X_{\bullet}$ is of the following type:

(7.7) $X_{\bullet}=\begin{cases}\hat{Y}_{i,[l,L]},&\text{ with
$|\hat{Y}_{i,[l,L]}|\geq 2$, $l\geq 1/2$ or}\\\ X_{\bf l}\subset
X^{{}^{\prime}a}&\text{ ${\bf l}(i)\geq 1/2$ for every
$i\in{\mathfrak{Y}}_{m}$; ${\bf l}(i)=\textup{l}_{i}$ for
$i\in{\mathfrak{Y}}_{v}$}.\end{cases}$

Let

$\begin{split}\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,[l,L]})&:=\int_{-1/2}^{1/2}{\mathfrak{E}}_{r}(\hat{Y}_{i,[l+t,L+t]})\,dt.\\\
\bar{{\mathfrak{E}}}_{r}(X_{\bf
l})&:=\int_{-1/2}^{1/2}{\mathfrak{G}}_{r}(X_{{\bf l}+{\bf
t}})\,dt,\end{split}$

where ${\bf t}$ is the function given by ${\mathfrak{t}}(i)=t$ $\forall
i\in{\mathfrak{Y}}_{m}$; ${\bf t}(i)=0$ $\forall i\in{\mathfrak{Y}}_{v}$. $t$.
Let $\bar{{\mathscr{V}}}_{r}(X_{\bullet})$, $\bar{{\mathscr{F}}}_{\nu}({\bf
C}|_{X_{\bullet}})$ be similarly defined. Thus defined,
$\bar{{\mathfrak{E}}}_{r}$ (and also $\bar{{\mathscr{V}}}_{r}$,
$\bar{{\mathscr{F}}}_{\nu}$) are additive: Given $l<L<L^{\prime}$ with $L-l$,
$L^{\prime}-L$ no less than 2, for $i\in{\mathfrak{Y}}_{m}$

$\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,[l,L]})+\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,[L,L^{\prime}]})=\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,[l,L^{\prime}]});$

given functions ${\bf l}\colon\thinspace{\mathfrak{Y}}\to[0,\infty]$, ${\bf
L}\colon\thinspace{\mathfrak{Y}}\to[0,\infty]$ such that ${\bf L}(i)-{\bf
l}(i)\geq 2$ $\forall i\in{\mathfrak{Y}}_{m}$ and that $X_{\bf L}$, $X_{\bf
l}$ are of the type (7.7),

$\bar{{\mathfrak{E}}}_{r}(X_{\bf L})-\bar{{\mathfrak{E}}}_{r}(X_{\bf
l})=\sum_{i\in{\mathfrak{Y}}_{m}}\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,[{\bf
l}(i),{\bf L}(i)]}).$

From Theorem 1.7 we already know that

$\begin{split}&\lim_{r\to\infty}{\mathscr{V}}_{r}(X_{\bullet})={\mathscr{F}}_{\nu}(\mathbf{C}_{0}|_{X_{\bullet}})\quad\text{for
all compact $X_{\bullet}\subset X^{{}^{\prime}a}$. }\\\
&\lim_{r\to\infty}\bar{{\mathscr{V}}}_{r}(X_{\bullet})=\bar{{\mathscr{F}}}_{\nu}(\mathbf{C}_{0}|_{X_{\bullet}})\quad\text{for
all compact $X_{\bullet}\subset X^{{}^{\prime}a}$ of the type (\ref{ave-X}).
}\end{split}$

Meanwhile, by (6.14), Lemma 4.3, Lemma 4.15 and its amendment in Proposition
6.4, we know that for all $X_{\bullet}$, the difference between
${\mathfrak{E}}_{r}(X_{\bullet})-{\mathscr{V}}_{r}(X_{\bullet})$ is bounded by
a constant independent of $r$ and $X_{\bullet}$. Similarly for
$\bar{{\mathfrak{E}}}_{r}(X_{\bullet})-\bar{{\mathscr{V}}}_{r}(X_{\bullet})$.
We now show that the latter goes to 0 as $r\to\infty$.

######  Proposition 7.1.

Let $\\{(A_{r},\Psi_{r})\\}_{r\in\Gamma}$ be as in the statement of Theorem
1.7. Let $X_{\bullet}\subset X^{{}^{\prime}a}$ be of the type (7.7). Then
there exists positive constants $R_{S}>1$, $\zeta_{S}$ depending only on
(7.2), such that for all $r\geq R_{S}$,

$\Big{|}\bar{{\mathfrak{E}}}_{r}(X_{\bullet})-\bar{{\mathscr{V}}}_{r}(X_{\bullet})\Big{|}\leq\zeta_{S}\,r^{-3/7}.$

Proof. We verify the preceding inequality only for the case
$X_{\bullet}=X_{\bf l}$. The case when $X_{\bullet}=\hat{Y}_{i,[l,L]}$ is
similar. Write $\partial\overline{X_{\bf
l}}=\bigcup_{i\in{\mathfrak{Y}}}Y_{i:l_{i}}$, $l_{i}=\textup{l}_{i}$ when
$i\in{\mathfrak{Y}}_{v}$. Then

$\pi\big{(}\bar{{\mathfrak{E}}}_{r}(X_{\bullet})-\bar{{\mathscr{V}}}_{r}(X_{\bullet})\big{)}=r^{-1}\int_{-1/2}^{1/2}{\mathscr{E}}_{top}^{w_{r}}(X_{\bullet})\,dt.$

The right hand side of the preceding identity can be expressed as a sum of
three terms:

(7.8)
$\begin{split}&\pi\big{(}\bar{{\mathfrak{E}}}_{r}(X_{\bullet})-\bar{{\mathscr{V}}}_{r}(X_{\bullet})\big{)}\\\
&\quad=\frac{1}{4r}\int_{-1/2}^{1/2}\int_{X_{{\bf l}+{\bf
t}}}F_{A_{0}}\wedge(F_{A_{0}}+iw_{r})\,dt-\sum_{i\in{\mathfrak{Y}}}\int_{\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}\big{(}2\langle\psi,\mbox{\mbox{$\partial\mkern-7.5mu\mbox{/}\mkern
2.0mu$}}_{B}\psi\rangle+\partial_{s}|\nu|\big{)}\\\
&\qquad+\frac{1}{4r}\sum_{i\in{\mathfrak{Y}}}\int_{\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}(B-B_{0})\wedge(F_{B}+F_{B_{0}}+iw_{r})\wedge
ds.\\\ \end{split}$

These three terms are estimated as follows.

(i) For the first term on the RHS of (7.8),

$\Big{|}\frac{1}{4r}\int_{-1/2}^{1/2}\int_{X_{{\bf l}+{\bf
t}}}F_{A_{0}}\wedge(F_{A_{0}}+iw_{r})\,dt\Big{|}\leq\zeta_{1}r^{-1},$

where $\zeta_{1}$ depends only on the choice of $A_{0}$ and $w_{r}$. (In fact,
it equals 0 by the assumptions (3.3) and Assumption 3.4 (3).)

(ii) To estimate the second term on the RHS of (7.8), use the Seiberg-Witten
equation to rewrite:

(7.9)
$\begin{split}&-\int_{\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}\big{(}2\langle\psi,\mbox{\mbox{$\partial\mkern-7.5mu\mbox{/}\mkern
2.0mu$}}_{B}\psi\rangle+\partial_{s}|\nu|\big{)}=\int_{\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}\big{(}2\langle\psi,\partial_{s}\psi\rangle-\partial_{s}|\nu|\big{)}\\\
\\\
&\quad=\int_{\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}\partial_{s}(|\psi|^{2}-|\nu|).\end{split}$

Let $\delta\geq\textsc{o}r^{-1/3}$, where o is as in Proposition 5.5. Let
$v_{r}:=(\textsl{z}_{v}\delta)^{-1}|\nu|$, where $\textsl{z}_{v}$ is the
positive constant from (5.1). In particular, $\sigma\geq\delta$ where
$v_{r}\geq 1$. Write

$|\psi|^{2}-|\nu|=(1-\chi(v_{r}))\,(|\nu|\,(|\underline{\alpha}|^{2}-1)+|\beta|^{2})+\chi(v_{r})\,(|\psi|^{2}-|\nu|).$

We have:

(7.10)
$\begin{split}&\int_{\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}\partial_{s}(|\psi|^{2}-|\nu|)\\\
&\quad=\int_{\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}\partial_{s}\big{(}(1-\chi(v_{r}))\,(|\nu|\,(|\underline{\alpha}|^{2}-1)+|\beta|^{2})\big{)}\\\
&\qquad\qquad+\int_{\partial\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}\big{(}\chi(v_{r})\,(|\psi|^{2}-|\nu|)\big{)}\\\
&\quad=2\int_{\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}\big{(}(1-\chi(v_{r}))\,\big{(}|\nu|\,\langle\underline{\alpha},\partial_{s}\underline{\alpha}\rangle+\langle\beta,\partial_{s}\beta\rangle\big{)}\big{)}\\\
&\qquad\qquad+\int_{\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}\big{(}(1-\chi(v_{r}))\partial_{s}|\nu|(|\underline{\alpha}|^{2}-1)\big{)}\\\
&\qquad\qquad-(\textsl{z}_{v}\delta)^{-1}\int_{\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}\big{(}\chi^{\prime}(v_{r})\,\partial_{s}|\nu|(|\psi|^{2}-|\nu|)\big{)}\\\
&\qquad\qquad+\int_{\partial\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}\big{(}\chi(v_{r})(|\psi|^{2}-|\nu|)\big{)}.\end{split}$

By Proposition 5.4 and the assumptions on $\nu$, the last two terms above are
bounded together by

(7.11)
$\begin{split}&(\textsl{z}_{v}\delta)^{-1}\Big{|}\int_{\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}\big{(}\chi^{\prime}(v_{r})\,\partial_{s}|\nu|(|\psi|^{2}-|\nu|)\big{)}\Big{|}\\\
&\quad\quad\qquad+\Big{|}\int_{\partial\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}\big{(}\chi(v_{r})(|\psi|^{2}-|\nu|)\big{)}\Big{|}\\\
&\quad\leq\zeta_{0}\,\delta^{3},\end{split}$

where $\zeta_{0}$ depends only on the metric, $\nu$, $\varsigma_{w}$,
$\zeta_{\mathfrak{p}}$. The other two terms on the RHS of the second equality
of (7.10) are together bounded by

$\int_{\bar{Y}_{\delta}}\Big{(}2|\nu|\,|\underline{\alpha}|\,|\partial_{s}\underline{\alpha}|+2|\beta|\,|\partial_{s}\beta|+\Big{|}\partial_{s}|\nu|\Big{|}\,\Big{|}|\underline{\alpha}|^{2}-1\Big{|}\Big{)},$

where $\bar{Y}:=\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}$. Now take
$\delta=2^{4}\delta_{*}\geq\textsc{o}r^{-1/3}$, $\delta_{*}$ being as in
Propositions 6.11 and 6.19. The preceding integral may be bounded using
Proposition 6.19 by

(7.12)
$\begin{split}&\int_{\bar{Y}_{\delta}}\left(\zeta_{1}\sqrt{r\tilde{\sigma}^{3}}\exp\big{(}-(r\tilde{\sigma})^{1/2}\mathop{\mathrm{dist}\,}\nolimits(\cdot,\alpha^{-1}(0))/(2c)\big{)}+\zeta^{\prime}r^{-1/2}\tilde{\sigma}^{-3/2}+\zeta^{\prime\prime}r^{-1}\tilde{\sigma}^{-3}\right)\\\
&\quad\quad\leq\zeta_{1}\int_{\bar{Y}_{\delta}}\Big{(}\sqrt{r\tilde{\sigma}^{3}}\exp\big{(}-(r\tilde{\sigma})^{1/2}\mathop{\mathrm{dist}\,}\nolimits(\cdot,\alpha^{-1}(0))/(2c)\big{)}\Big{)}+\zeta^{\prime}_{1}r^{-1/2},\end{split}$

where $c$ is the constant from Proposition 6.19. To bound the first term on
the RHS of (7.12), first note that it vanishes when
$\alpha^{-1}(0)=\emptyset$. Assume consequently that
$\alpha^{-1}(0)\neq\emptyset$. Invoke the second bullet of Corollary 6.12 with
$\rho=r^{-1/2}$ to cover $\alpha^{-1}(0)\,\cap\,X_{\delta}$ by a set of balls,
$\Lambda^{\rho}=\\{B_{k}\\}_{k}$, where $B_{k}$ is centered at
$x_{k}\in\alpha^{-1}(0)\,\cap\,X_{\delta}$, and is of radius
$(\tilde{\sigma}(x_{k}))^{-1/2}\rho$. Let $B^{\prime}_{k}\supset B_{k}$ denote
the concentric ball with twice the radius, and let
$U=\bigcup_{k}B^{\prime}_{k}$. Define the function
$d_{k}(\cdot):=\mathop{\mathrm{dist}\,}\nolimits(\cdot,x_{k})$ on $X$. Given
$x\in X_{\delta}$, there exists $k_{x}\in\Lambda^{\rho}$ such that $B_{k_{x}}$
contains a point whose distance to $x$ equals
$\mathop{\mathrm{dist}\,}\nolimits(x,\alpha^{-1}(0))$. Then

$\begin{split}\mathop{\mathrm{dist}\,}\nolimits(x,\alpha^{-1}(0))\geq
d_{k_{x}}(x)-(\tilde{\sigma}(x_{k_{x}}))^{-1/2}\rho&\geq d_{k_{x}}(x)/2\\\
&\geq(\tilde{\sigma}(x_{k}))^{-1/2}\rho\quad\forall x\in
X_{\delta}-U.\end{split}$

Let $\bar{\Lambda}^{\rho}:=\Lambda^{\rho}_{\hat{Y}_{i,[l-1,l+1],\delta}}$. By
bullet 2 of by Corollary 6.12, $\bar{\Lambda}^{\rho}$ has at most $\zeta r$
elements. If $\bar{\Lambda}^{\rho}=\emptyset$, then
$\mathop{\mathrm{dist}\,}\nolimits(\cdot,\alpha^{-1}(0))\geq 1/2$ over
$\bar{Y}_{\delta}$, and in this case

$\begin{split}&\int_{\bar{Y}_{\delta}}\Big{(}\sqrt{r\tilde{\sigma}^{3}}\exp\big{(}-(r\tilde{\sigma})^{1/2}\mathop{\mathrm{dist}\,}\nolimits(\cdot,\alpha^{-1}(0))/(2c)\big{)}\Big{)}\leq\zeta_{2}^{\prime}(r\delta)^{-1/2}.\end{split}$

Otherwise, $\bar{\Lambda}^{\rho}\neq\emptyset$, and
$k_{x}\in\bar{\Lambda}^{\rho}$ for any $x\in\bar{Y}_{\delta}$. In this case

$\begin{split}&\int_{\bar{Y}_{\delta}}\Big{(}\sqrt{r\tilde{\sigma}^{3}}\exp\big{(}-(r\tilde{\sigma})^{1/2}\mathop{\mathrm{dist}\,}\nolimits(\cdot,\alpha^{-1}(0))/(2c)\big{)}\Big{)}\\\
&\qquad\leq\int_{U\cap\bar{Y}_{\delta}}\sqrt{r\tilde{\sigma}^{3}}+\int_{\bar{Y}_{\delta}-U}\Big{(}\sqrt{r\tilde{\sigma}^{3}}\exp\big{(}-(r\tilde{\sigma})^{1/2}\mathop{\mathrm{dist}\,}\nolimits(\cdot,\alpha^{-1}(0))/(2c)\big{)}\\\
&\qquad\leq\sum_{k\in\bar{\Lambda}^{\rho}}\int_{B_{k}^{\prime}}\sqrt{r\tilde{\sigma}^{3}}+\sum_{k\in\bar{\Lambda}^{\rho}}\int_{\bar{Y}_{\delta}-B_{k}^{\prime}}\sqrt{r\tilde{\sigma}^{3}}\exp\big{(}-(r\tilde{\sigma})^{1/2}d_{k}/(4c)\big{)}\\\
&\qquad\leq\zeta_{2}\,(r\delta)^{-1/2}.\end{split}$

Insert the preceding bounds for the first integral on the right hand side of
(7.12), and combine with Equations (7.11), (7.10) and (7.9), setting
$\delta=r^{-1/7}$, which is no less than $2^{4}\delta_{*}$ when $r$ is
sufficiently large and when $\epsilon\leq 1/7$ in the definition of
$\delta_{1,\epsilon}$. We then have:

$\text{the second term on the RHS of (\ref{E-V:sum})}\leq\zeta_{6}\,r^{-3/7}.$

(iii) To estimate the last term in (7.8), note that $F_{B}+F_{B_{0}}+iw_{r}$
is exact by Assumption 3.4 (2), and therefore

$\int_{\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}(B-B_{0})\wedge(F_{B}+F_{B_{0}}+iw_{r})\,ds=\int_{\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}(B^{\prime}-B_{i,0})\wedge(F_{B}+F_{B_{0}}+iw_{r})\,ds$

where $B^{\prime}(s)=A^{\prime}|_{Y_{i:s}}$, and $A^{\prime}=u\cdot A$ is in
the normalized Coulomb-Neumann gauge on $\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}$.
Arguing as in the proof of Lemma 3.12, combining Propositions 5.6 and 6.14
with a standard elliptic estimate yields

$\|A^{\prime}-A_{0}\|_{L^{\infty}(\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]})}\leq\zeta_{2}r^{1/2}.$

Together with the $L^{1}$-estimate for $F_{A}$ in (6.11), this gives:

$\frac{1}{4r}\Big{|}\int_{\hat{Y}_{i,[l_{i}-1/2,l_{i}+1/2]}}(B-B_{0})\wedge(F_{B}+F_{B_{0}}+iw_{r})\,ds\Big{|}\leq\zeta_{3}r^{-1/2}.$

Finally, the claimed inequality in the proposition follows by inserting the
bounds from (i), (ii), (iii) above into (7.8). $\Box$

In the upcoming proof of Theorem 1.9, we make use of a lower bound on
${\mathfrak{E}}_{r}$, which follows from an improved lower bound on
${\mathscr{E}}_{top}^{\mu_{r},\hat{{\mathfrak{p}}}}$.

######  Lemma 7.2.

There are positive constants $r_{0}$, $\zeta_{i}^{\prime}$,
$\zeta_{i}^{\prime\prime}$ depending only on (7.2) with the following
significance: For all $i\in{\mathfrak{Y}}$ and $l,L\geq 0$,

$\begin{split}{\mathscr{E}}_{top}^{\mu_{r},,\hat{{\mathfrak{p}}}}(\hat{Y}_{i,[l,L]})(A_{r},\Psi_{r})&\geq-r\zeta^{\prime}_{i}e^{-\kappa_{i}l},\\\
{\mathfrak{E}}_{r}(\hat{Y}_{i,[l,L]})(A_{r},\Psi_{r})&\geq-\zeta_{i}^{\prime\prime}e^{-\kappa_{i}l}\quad\text{if
$\hat{Y}_{i,[l,L]}\subset X^{{}^{\prime}a}$.}\end{split}$

Proof. By (4.38), (4.35), and (6.11),

${\mathscr{E}}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}(\hat{Y}_{i,[l,L]})(A_{r},\Psi_{r})\geq-r\zeta_{i}e^{-\kappa_{i}l}.$

By Lemma 6.6 and (4.29),

${\mathscr{E}}_{top}^{\mu_{r},\hat{{\mathfrak{p}}}}(\hat{Y}_{i,[l,L]})(A_{r},\Psi_{r})={\mathscr{E}}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}(\hat{Y}_{i,[l,L]})(A,\Psi)-\frac{ir}{4}\int_{\partial\hat{Y}_{i,[l,L]}}(B_{r}-B_{0})\wedge{\mathfrak{v}}\geq-r\zeta_{i}^{\prime}e^{-\kappa_{i}l}.$

So

${\mathfrak{E}}_{r}(\hat{Y}_{i,[l,L]})=\frac{1}{\pi
r}{\mathscr{E}}_{top}^{\mu_{r},\hat{{\mathfrak{p}}}}(\hat{Y}_{i,[l,L]})(A_{r},\Psi_{r})-\int_{\hat{Y}_{i,[l,L]}}\frac{iF_{A_{K}}}{4\pi}\wedge\nu\geq-\zeta_{i}^{\prime\prime}e^{-\kappa_{i}l}.$

$\Box$

######  Lemma 7.3.

Let $(A_{r},\Psi_{r})$ be as in Theorem 1.7 and let e, ${\mathbb{E}}$ be
respectively the constants from (4.8) and (4.11) determined (via Lemma 4.3) by
(4.7). There exists a positive constant $\zeta$ depending only on (7.2 such
that

${\mathfrak{E}}_{r}(X^{{}^{\prime}a})\leq\frac{\textsc{e}}{\pi}+\zeta;\quad\bar{{\mathfrak{E}}}_{r}(X^{{}^{\prime}a})\leq\frac{\textsc{e}}{\pi}+\zeta.$

If $X$ has no vanishing ends,
${\mathfrak{E}}_{r}(X)=\bar{{\mathfrak{E}}}_{r}(X^{{}^{\prime}a})$ and

$\lim_{r\to\infty}{\mathfrak{E}}_{r}(X)=\frac{{\mathbb{E}}}{\pi}-\int_{X}\frac{iF_{A_{K}}}{4\pi}\wedge\nu.$

Proof. Write

${\mathfrak{E}}_{r}(X^{{}^{\prime}a})=\frac{1}{\pi
r}{\mathscr{E}}_{top}^{\mu_{r},\hat{{\mathfrak{p}}}}(X)(A_{r},\Psi_{r})-\frac{1}{\pi
r}{\mathscr{E}}_{top}^{\mu_{r},,\hat{{\mathfrak{p}}}}(X-X^{{}^{\prime}a})(A_{r},\Psi_{r})-\int_{X^{{}^{\prime}a}}\frac{iF_{A_{K}}}{4\pi}\wedge\nu.$

The assertions of the lemma then follow directly from Lemma 7.2, (4.8), and
(4.11). $\Box$

###  7.3 Proving Theorem 1.9 and Proposition 1.12, the non-cylindrical case

We introduce some notations for convenience before embarking on the proofs.

Given two t-curves $\mathbf{C}=[C,\tilde{C}]$,
$\mathbf{C}^{\prime}=[C^{\prime},\tilde{C}^{\prime}]$ in $(X,\nu)$, and a
compact $X_{\bullet}\subset X^{{}^{\prime}a}$, let

$\text{{\small
d}}_{X_{\bullet}}(\mathbf{C}^{\prime},\mathbf{C}):=\mathop{\mathrm{dist}\,}\nolimits_{X_{\bullet}}(C^{\prime},C)+\|\tilde{C}^{\prime}-\tilde{C}\|_{op,X_{\bullet}},$

where $\|\cdot\|_{op,X_{\bullet}}$ denotes the norm as currents on
$X_{\bullet}$. The same expression is also used when either of the the entries
$\mathbf{C}$ or $\mathbf{C}^{\prime}$ on the left is replaced by a general
pseudo-holomorphic subvariety $C$ or $C^{\prime}$. In this case the
$\tilde{C}$ or $\tilde{C}^{\prime}$ on the right simply denotes the associated
current of the pseudo-holomorphic subvariety.

Given a t-orbit $\boldsymbol{\gamma}$ or a general multi-orbit $\gamma$ in
$(Y,\theta)$, let $\text{{\small d}}_{Y}(\boldsymbol{\gamma},\cdot)$,
$\text{{\small d}}_{Y}(\gamma,\cdot)$ be similarly defined.

Let $\text{{\small c}}_{r}:=(A_{r},\Psi_{r})$ be an admissible solution to
${\mathfrak{S}}_{\mu_{r},\hat{{\mathfrak{p}}}}(A_{r},\Psi_{r})=0$ on the
$\mathop{\mathrm{Spin}}\nolimits^{c}$ admissible pair $(X,\nu)$ as before, and
let $\mathbf{C}=[C,\tilde{C}]$ be a t-curve. Given a compact
$X_{\bullet}\subset X^{{}^{\prime}a}$, denote:

$\begin{split}\text{{\small d}}_{X_{\bullet}}(\text{{\small
c}}_{r},\mathbf{C})&:=\mathop{\mathrm{dist}\,}\nolimits_{X_{\bullet}}(\alpha_{r}^{-1}(0),C)+\Big{\|}\frac{i}{2\pi}F_{A^{E}_{r}}-\tilde{C}\Big{\|}_{op,X_{\bullet}}.\end{split}$

Recall from Section 2.3(a) the definition of the set of t-orbits,
${\mathbb{P}}(Y_{i},{\mathfrak{s}}_{i})$. Let

$\text{{\small
d}}_{i}:=\min_{\boldsymbol{\gamma},\boldsymbol{\gamma}^{\prime}\in{\mathbb{P}}({\mathfrak{s}}_{i}),\boldsymbol{\gamma}\neq\boldsymbol{\gamma}^{\prime}}\big{(}\mathop{\mathrm{dist}\,}\nolimits_{Y_{i}}(\gamma^{\prime},\gamma)+\|\tilde{\gamma}^{\prime}-\tilde{\gamma}\|_{op,Y_{i}}\big{)}>0.$

##### Proof of Theorem 1.9: the non-cylindrical case.

Assume that $(X,\nu)$ is non-cylindrical. The proof for the cylindrical case
requires only little modification and will be postponed to the next
subsection.

##### (a):

Let $\\{\operatorname{\text{{\small
c}}}_{r}:=(A_{r},\Psi_{r})\\}_{r\in\Gamma_{0}}$ be the sequence from the
conclusion of Theorem 1.7 (a). Rename the t-curve ${\bf C}$ in the statement
of Theorem 1.7 by ${\bf C}_{0}$. That is to say,
$\\{(A_{r},\Psi_{r})\\}_{r\in\Gamma_{0}}$ t-converges to ${\bf C}_{0}$ over
$X^{{}^{\prime}a}$. Given $i\in{\mathfrak{Y}}_{m}$, let
$\boldsymbol{\gamma}_{0,i}$ denote the $Y_{i}$-end limit of ${\bf C}_{0}$.
Write the sequence $\Gamma_{0}$ as $\\{r_{n}\\}_{n}$. Given $l\in[0,\infty]$,
let $\vec{l}\colon\thinspace{\mathfrak{Y}}_{X}\to[0,\infty]$ denote the
function with $\vec{l}_{i}=l$ when $i\in{\mathfrak{Y}}_{m}$,
$\vec{l}_{i}=\textup{l}_{i}$ when $i\in{\mathfrak{Y}}_{v}$.

Recalling the construction of ${\bf C}_{0}$ in the proof of Theorem 1.7 (a)
via diagonalization, we choose the diagonalization process such that the
subsequence $\Gamma_{0}=\\{r_{n}\\}_{n}\subset\Gamma$ so obtained
($\Gamma,\Gamma_{0}$ being as in the statement of Theorem 1.7) satisfies the
following: Let $\varepsilon_{n}:=e^{-n}$, $L_{n}:=e^{n}$, $r_{n}$ increases
with $n$, $r_{n}\to\infty$ as $n\to\infty$, and $\operatorname{\text{{\small
d}}}_{X_{\bullet}}(\operatorname{\text{{\small c}}}_{r_{n}},{\bf
C}_{0})\leq\varepsilon_{n}$ $\forall X_{\bullet}\subset X_{\vec{L}_{n}}$. We
find it convenient to restate the preceding statement in a re-parametrized
fashion: Let $n(r)\colon\thinspace\Gamma_{0}\to{\mathbb{N}}$ denote the
inverse function of the map $n\mapsto r_{n}$, and write
$\varepsilon_{T}(r):=\varepsilon_{n(r)}$,
$L_{T}(r):=L_{n(r)}=\varepsilon_{T}(r)^{-1}$. Then

(7.13) $\operatorname{\text{{\small
d}}}_{X_{\bullet}}(\operatorname{\text{{\small c}}}_{r},{\bf
C}_{0})\leq\varepsilon_{T}(r)\quad\forall X_{\bullet}\subset
X_{\vec{L}_{T}(r)},r\in\Gamma_{0}.$

Given $\varepsilon>0$, let $\ell_{i}(\varepsilon)\geq 0$ be the minimal $L$
such that

(7.14) $\operatorname{\text{{\small d}}}_{\hat{Y}_{i,[l,l+3]}}({\bf
C}_{0},{\mathbb{R}}\times\boldsymbol{\gamma}_{i,0})\leq\varepsilon$ $\forall
l\geq L$ and $\int_{\tilde{C}_{0}|_{\hat{Y}_{i,L}}}\nu\leq\varepsilon/3$.

Such $\ell_{i}$ exists because of Proposition 2.16, and because of (4.29).

Let $R(\varepsilon)>1$ denote the minimal $r$ such that all of the following
hold:

(7.15) $\begin{split}&\text{(i) $L_{T}(r)\geq\ell_{i}(\varepsilon)+3$ $\forall
i\in{\mathfrak{Y}}_{m}$;}\\\ &\text{(ii)
$\varepsilon_{T}(r)=L_{T}(r)^{-1}\leq(3\|\nu\|_{\infty})^{-1}\varepsilon$;}\\\
&\text{(iii) $\zeta_{S}r^{-3/7}\leq\varepsilon/3$ and $r\geq R_{S}$, where
$R_{S}$, $\zeta_{S}$ are as in Proposition \ref{cor:E-cp}.}\end{split}$

The preceding conditions for $R(\varepsilon)$, (7.13), the condition (7.14)
for $\ell(\varepsilon)$, and Proposition 7.1 together ensure that

(7.16) $\forall r\geq R(\varepsilon)$,
$\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,[L_{T}(r)-5/2,L_{T}(r)-1/2]})\leq\varepsilon$.

To proceed, we need the next two preliminary lemmas.

######  Lemma 7.4.

Fix $i\in{\mathfrak{Y}}_{m}$. Suppose that $I_{1}\subset I_{2}\subset\cdots$
is a sequence of intervals in ${\mathbb{R}}$ so that
$\bigcup_{n}I_{n}=:I_{\infty}$. Assign to each $n\in{\mathbb{Z}}^{+}$ a triple
$(r=r_{n},\text{{\small c}}_{r},m_{r})$, so that:

  * •

$\operatorname{\text{{\small c}}}_{r}$ is a solution to the $r$-th version of
Seiberg-Witten equation
${\mathfrak{S}}_{\mu_{r},\hat{{\mathfrak{p}}}}(\operatorname{\text{{\small
c}}}_{r})=0$;

  * •

$m_{r}\in{\mathbb{R}}^{\geq 0}$ is unbounded, and

  * •

$I_{n}$ is isomorphic by some translation $\tilde{\tau}_{n}$ to an interval
$I^{\prime}_{n}\subset[m_{r},\infty)$.

Let $\text{{\scriptsize b}}_{n}:=\tilde{\tau}_{n}^{*}(\text{{\small
c}}_{r_{n}}|_{\hat{Y}_{i,I^{\prime}_{n}}})$. Then there is a t-curve
$\mathbf{C}\subset I_{\infty}\times Y_{i}$ for the cylindrical admissible pair
$(I_{\infty}\times Y_{i},\pi_{2}^{*}\nu_{i})$, and a subsequence of
$\\{\text{{\scriptsize b}}_{n}\\}_{n}$, denoted by the same notation below,
which (locally) t-converges to $\mathbf{C}$ over $I_{\infty}\times Y_{i}$.
Moreover, there is an upper bound on ${\mathscr{F}}_{\omega}(\mathbf{C})$
depending only on h, ${\mathfrak{s}}$, and other data mentioned in the end of
Theorem 1.7.

Proof. Passing to a subsequence if necessary, we may assume that $m_{r}$
increases with $r$ and goes to infinity as $r\to\infty$. Meanwhile,
$\text{{\scriptsize b}}_{n}=:(\tilde{A}_{n},\tilde{\Psi}_{n})$ is a solution
to the Seiberg-Witten equation
${\mathfrak{S}}_{\tilde{\tau}_{n}^{*}\mu_{r},0}(\text{{\scriptsize b}}_{n})=0$
over $(\tilde{\tau}_{n})^{-1}\hat{Y}_{i,I^{\prime}_{n}}\simeq I_{n}\times
Y_{i}$. Over this,

(7.17)
$\|\tilde{\tau}_{n}^{*}(\nu|_{\hat{Y}_{i,I^{\prime}_{n}}})-\pi_{2}^{*}\nu_{i}\|_{C^{k}}\leq\zeta_{i}e^{-\kappa_{i}m_{r}}\to
0$

as $n\to\infty$, while
$\|\tilde{\tau}_{n}^{*}(w_{r}|_{\hat{Y}_{i,I^{\prime}_{n}}})-\pi_{2}^{*}w_{i,r}\|_{C^{k}}=0$
by assumption.

Because of (7.3) (and a diagonalization argument), there is a subsequence of
$\\{\text{{\scriptsize b}}_{n}\\}_{n}$ (again denoted by the same notation),
and a current $\tilde{C}$ on $I_{\infty}\times Y_{i}$, so that over any
compact subset $V\subset I_{k}\times Y_{i}$ the corresponding sequence of
currents $\\{\frac{i}{2\pi}F_{\tilde{A}^{E}_{n}}|_{V}\\}_{n\geq k}$ converge
to $\tilde{C}|_{V}$. Following Steps 5 and 6 of Section 3.6’s summary of
Taubes’s arguments, the support of $\tilde{C}$ is a $J_{i}$-holomorphic
subvariety $C$, where $J_{i}$ is the almost complex structure on
$I_{\infty}\times(Y_{i}-\nu_{i}^{-1}(0)$ defined by
$(\pi_{2}^{*}\nu_{i})^{+}$. Moreover, $\mathbf{C}=(C,\tilde{C})$ is a t-curve
for the cylindrical admissible pair $(I_{\infty}\times
Y_{i},\pi_{2}^{*}\nu_{i})$ and $\text{{\scriptsize b}}_{n}$ t-converges to
$\mathbf{C}$. The claim about the $\omega$-energy of $\mathbf{C}$ follows from
the same argument as in the proof of Theorem 1.7. $\Box$

######  Lemma 7.5.

Fix $i\in{\mathfrak{Y}}_{m}$ and a positive number $\epsilon$,
$\epsilon\leq\text{{\small d}}_{i}/8$. Let $\\{\operatorname{\text{{\small
c}}}_{r}\\}_{r\in\Gamma_{0}}$ be the sequence from the conclusion of Theorem
1.7, chosen in the aforementioned manner, and let $L_{T}(r)$ be as in (7.13).
Then there exist positive constants $h_{i}=h_{i}(\epsilon)\leq\text{{\small
d}}_{i}/8$, ${\mathscr{R}}_{i}={\mathscr{R}}_{i}(\epsilon)$ such that the
following holds: For any $r\geq{\mathscr{R}}_{i}$, $r\in\Gamma_{0}$, and
$l\geq L_{T}(r)$, if

$\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,[l+1/2,l+3/2]})\leq h_{i},$

then $\text{{\small d}}_{\hat{Y}_{i,[l,l+3]}}(\text{{\small
c}}_{r},{\mathbb{R}}\times\boldsymbol{\gamma})<\epsilon$ for some t-orbit
$\boldsymbol{\gamma}$ on $(Y_{i},\nu_{i})$.

Proof. Suppose the contrary. Then for any pair of positive numbers $\delta>0$
and ${\rm r}\geq 1$, there exist $r\geq{\rm r}$, $r\in\Gamma_{0}$, and $l\geq
L_{T}(r)$ such that

(7.18) $\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,[l+1/2,l+3/2]})\leq\delta$

and

(7.19) $\text{{\small d}}_{\hat{Y}_{i,[l,l+3]}}(\text{{\small
c}}_{r},{\mathbb{R}}\times\boldsymbol{\gamma})\geq\epsilon\quad\text{for all
$\boldsymbol{\gamma}\in{\mathbb{P}}({\mathfrak{s}}_{i})$.}$

Choose a sequence of $(\delta,{\rm r})$, denoted $\\{(\delta_{n},{\rm
r}_{n})\\}_{n}$, with $\lim_{n\to\infty}\delta_{n}\to 0$ and ${\rm
r}_{n}\to\infty$ as $n\to\infty$, and denote the corresponding $r,l$ by
$r^{\prime}_{n}$, $l_{n}$ respectively. Note that $l_{n}\to\infty$ as
$n\to\infty$, since $l_{n}\geq L_{T}(r^{\prime}_{n})$ and
$L_{T}(r^{\prime}_{n})\to\infty$ as $n\to\infty$. Apply Lemma 7.4 with
$I_{1}=I_{2}\cdots=I_{\infty}=[0,3]$, and with
$I_{n}^{\prime}=[l_{n},l_{n}+3]$. This gives a t-curve
$\mathbf{C}=[C,\tilde{C}]$ on the cylindrical admissible pair
$(\hat{Y}_{i,[0,3]},\pi_{2}^{*}\nu_{i})$ and a subsequence of
$\\{(\delta_{n},{\rm r}_{n})\\}_{n}$ (denoted by the same notation), such that
with $\operatorname{\text{{\small
c}}}_{n}^{\prime}:=\operatorname{\text{{\small c}}}_{r_{n}}$,

(7.20) $\text{{\small d}}_{\hat{Y}_{i,[0,3]}}(\tau^{*}_{l_{n}}(\text{{\small
c}}^{\prime}_{n}|_{\hat{Y}_{i,[l_{n},l_{n}+3]}}),\mathbf{C})=\text{{\small
d}}_{\hat{Y}_{i,[l_{n},l_{n}+3]}}(\text{{\small
c}}^{\prime}_{n},\tau_{-l_{n}}\mathbf{C})\to 0\quad\text{as $n\to\infty$. }$

Combining with Proposition 7.1, this implies that

$\begin{split}0\leq&\int_{-1/2}^{1/2}{\mathscr{F}}_{\nu}(\tilde{C}|_{\hat{Y}_{i,[1/2+t,3/2+t]}})\,dt\\\
&\quad=\lim_{n\to\infty}\bar{{\mathscr{V}}}_{r_{n}}(\hat{Y}_{i,[l_{n}+1/2,l_{n}+3/2]})\\\
&\quad=\lim_{n\to\infty}\bar{{\mathfrak{E}}}_{r_{n}}(\hat{Y}_{i,[l_{n}+1/2,l_{n}+3/2]})=0.\end{split}$

The $v$-energy ${\mathscr{F}}_{\nu}$ for t-curves being nonnegative, this
implies that ${\mathscr{F}}_{\nu}(\tilde{C}|_{\hat{Y}_{i,[1,2]}})=0$. But by
Lemma 2.9, this in turn implies that
$\mathbf{C}|_{\hat{Y}_{i,[1,2]}}=[1,2]\times\boldsymbol{\gamma}\subset[1,2]\times
Y_{i}\simeq\hat{Y}_{i,[1,2]}$ for some
$\boldsymbol{\gamma}\in{\mathbb{P}}({\mathfrak{s}}_{i})$, and hence
$\mathbf{C}=[0,3]\times\boldsymbol{\gamma}\subset\hat{Y}_{i,[0,3]}$. This,
together with (7.20), contradict the assumption (7.19). $\Box$

To continue with the proof of Theorem 1.9, fix $\epsilon\leq\text{{\small
d}}_{i}/8$, and let $\varepsilon_{0}:=h_{i}(\epsilon)$, where
$h_{i}(\epsilon)$ is as in Lemma 7.5. Divide $[L_{T}(r)-5/2,\infty)$ into
intervals $I^{*}$ of length 2. For each $i\in{\mathfrak{Y}}_{m}$, let
$\invbreve{{\mathscr{I}}}_{i,r}\subset[L_{T}(r)-5/2,\infty)\subset{\mathbb{R}}^{+}$
be the union of those $I^{*}$ satisfying

(7.21) $\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,I^{*}})\geq\varepsilon_{0}.$

Let $\breve{{\mathscr{I}}}_{i,r}$ be the closure of
$[L_{T}(r)-5/2,\infty)-\invbreve{{\mathscr{I}}}_{i,r}$. By Lemma 7.5, this
implies that when $r\geq{\mathscr{R}}_{i}(\epsilon)$, for each of the length-2
interval, $I^{*}=[l-5/2,l-1/2]\subset\breve{{\mathscr{I}}}_{i,r}$, there is a
t-orbit $\boldsymbol{\gamma}_{*}\in{\mathbb{P}}({\mathfrak{s}}_{i})$ so that

(7.22) $\operatorname{\text{{\small
d}}}_{\hat{Y}_{i,[l-3,l]}}(\operatorname{\text{{\small
c}}}_{r},{\mathbb{R}}\times\boldsymbol{\gamma}_{*})\leq\epsilon.$

Since ${\mathbb{P}}({\mathfrak{s}}_{i})$ is discrete, this t-orbit
$\boldsymbol{\gamma}_{*}$ only depends on the connected component of
$\breve{{\mathscr{I}}}_{i,r}$ that $I^{*}$ lies in. Let $\Lambda_{i,r}+1$ be
the number of connected components of $\breve{{\mathscr{I}}}_{i,r}$. The set
of connected components of $\breve{{\mathscr{I}}}_{i,r}$ is naturally ordered,
and we denote it by
$\\{\breve{{\mathscr{I}}}_{i,r}^{(k)}\\}_{k=0}^{\Lambda_{i,r}}$: for any pair
$k<k^{\prime}$ and any $l\in\breve{{\mathscr{I}}}_{i,r}^{(k)}$ and
$l^{\prime}\in\breve{{\mathscr{I}}}_{i,r}^{(k^{\prime})}$, $l<l^{\prime}$.
Denote the t-orbit associated to the component
$\breve{{\mathscr{I}}}^{(k)}_{i,r}$ by $\boldsymbol{\gamma}_{i,r}^{(k)}$. By
(7.16), the interval
$[L_{T}(r)-5/2,L_{T}(r)-1/2]\subset\breve{{\mathscr{I}}}_{i,r}^{(0)}$, and it
follows from the definition of $R(\varepsilon)$ that
$\boldsymbol{\gamma}_{i,r}^{(0)}=\boldsymbol{\gamma}_{0,i}$ $\forall r\geq
R(\varepsilon_{0})$. Let $r_{0}$ denote the maximum among $R(\varepsilon_{0})$
and ${\mathscr{R}}_{i}(\epsilon)$ $\forall i\in{\mathfrak{Y}}_{m}$, and assume
that $r\geq r_{0}$ from now on.

Correspondingly, label the connected components of
$\invbreve{{\mathscr{I}}}_{i,r}$ by $\invbreve{{\mathscr{I}}}_{i,r}^{(k)}$,
$k\in{\mathbb{Z}}^{+}$, so that $\invbreve{{\mathscr{I}}}_{i,r}^{(k)}$ lies
between $\breve{{\mathscr{I}}}_{i,r}^{(k-1)}$ and
$\breve{{\mathscr{I}}}_{i,r}^{(k)}$. We claim that each
$\invbreve{{\mathscr{I}}}_{i,r}^{(k)}$ has finite length, and the number of
connected components of $\invbreve{{\mathscr{I}}}_{i,r}$, namely
$\Lambda_{i,r}$, is bounded independently of $r$. To see this, write:

(7.23) $\bar{{\mathfrak{E}}}_{r}(X^{{}^{\prime}a})(\text{{\small
c}}_{r})=\bar{{\mathfrak{E}}}_{r}(X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{\frac{5}{2}}})(\text{{\small
c}}_{r})+\sum_{i\in{\mathfrak{Y}}_{m}}\Big{(}\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,\breve{{\mathscr{I}}}_{i,r}})(\text{{\small
c}}_{r})+\sum_{k=1}^{\Lambda_{i,r}}\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,\invbreve{{\mathscr{I}}}_{i,r}^{(k)}})(\text{{\small
c}}_{r})\Big{)}$

By Proposition 7.1 and (7.13), $\forall r\geq r_{0}$

$\begin{split}\bar{{\mathfrak{E}}}_{r}(X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{\frac{5}{2}}})(\text{{\small
c}}_{r})&\geq\bar{{\mathscr{V}}}_{r}(X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{\frac{5}{2}}})(\text{{\small
c}}_{r})-\zeta_{S}\,r^{-3/7}\\\
&\geq\int_{-1/2}^{1/2}{\mathscr{F}}_{\nu}(\tilde{C}_{0}|_{X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{\frac{5}{2}}+\vec{t}}})\,dt-\|\nu\|_{\infty}\varepsilon_{T}(r)-\zeta_{S}\,r^{-3/7}\\\
&\geq-\|\nu\|_{\infty}\,\varepsilon_{T}(r)-\zeta_{S}\,r^{-3/7}.\end{split}$

By definition, for each pair of $i,k$,
$\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,\invbreve{{\mathscr{I}}}_{i,r}^{(k)}})(\text{{\small
c}}_{r})\geq|\invbreve{{\mathscr{I}}}_{i,r}^{(k)}|\varepsilon_{0}/2$. By Lemma
7.2, for each $i\in{\mathfrak{Y}}_{m}$

$\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,\breve{{\mathscr{I}}}_{i,r}})(\text{{\small
c}}_{r})\geq-\zeta_{i}^{\prime\prime}e^{-\kappa_{i}L_{T}(r)}$

Combine the above bounds with (7.23) to get:

(7.24)
$\begin{split}\sum_{i\in{\mathfrak{Y}}_{m}}\sum_{k=1}^{\Lambda_{i,r}}\,|\invbreve{{\mathscr{I}}}_{i,r}^{(k)}|\,\varepsilon_{0}/2&\leq\sum_{i\in{\mathfrak{Y}}_{m}}\sum_{k=1}^{\Lambda_{i,r}}\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,\invbreve{{\mathscr{I}}}_{i,r}^{(k)}})(\text{{\small
c}}_{r})\\\ &\leq\bar{{\mathfrak{E}}}_{r}(X^{{}^{\prime}a})(\text{{\small
c}}_{r})+\|\nu\|_{\infty}\varepsilon_{T}(r)+\zeta_{S}r^{-3/7}+\sum_{i\in{\mathfrak{Y}}_{m}}\zeta_{i}^{\prime\prime}e^{-\kappa_{i}L_{T}(r)}.\end{split}$

By Lemma 7.3, the right hand side of the preceding inequality has an
$r$-independent upper bound. This in turn gives an $r$-independent upper bound
on the left hand side for all $r\geq r_{0}$. Recalling that every
$|\invbreve{{\mathscr{I}}}_{i,r}^{(k)}|$ is a positive even integer, this
gives an upper bound for each $\Lambda_{i,r}$ and
$|\invbreve{{\mathscr{I}}}_{i,r}^{(k)}|$.

The preceding uniform bounds on $\Lambda_{i,r}$ enables us to pass to a
subsequence $\Gamma_{1}$ of $\Gamma_{0}$, such that
$\Lambda_{i,r}=\Lambda_{i}$ $\forall r\in\Gamma_{1}$ for some
$\Lambda_{i}\in{\mathbb{Z}}^{\geq 0}$. Given $k,k^{\prime}$, $1\leq
k<k^{\prime}\leq\Lambda_{i}$, Let $\operatorname{\text{{\small
m}}}_{i,r}(k,k^{\prime})$ denote the distance between the centers of intervals
$\invbreve{{\mathscr{I}}}_{i,r}^{(k)}$ and
$\invbreve{{\mathscr{I}}}_{i,r}^{(k^{\prime})}$. Let
$\operatorname{\text{{\small m}}}_{i,r}(0,k)$ denote the distance between the
center of $\invbreve{{\mathscr{I}}}_{i,r}^{(k)}$ and $L_{T}(r)$. We choose the
subsequence $\Gamma_{1}$ so that:

(7.25) $\hss\left\\{\mkern-20.0mu\parbox{422.77661pt}{\begin{itemize}
\par\itemize@item@For every $i$ and $k\in\Lambda_{i}$, the t-orbit
$\boldsymbol{\gamma}_{i,r}^{(k)}$ is independent of $r$. This is possible
because ${\mathbb{P}}({\mathfrak{s}}_{i})$ consists of finitely many elements.
\par\itemize@item@Given $i\in{\mathfrak{Y}}_{m}$, and any pair $k,k^{\prime}$,
$0\leq k<k^{\prime}\leq\Lambda_{i}$, the distance $\operatorname{\text{{\small
m}}}_{i,r}(k,k^{\prime})$ is non-decreasing with respect to $r$.
\par\end{itemize}}\right.$

Fix $i\in{\mathfrak{Y}}_{m}$. Let $k_{1}$ be the smallest $k$ such that
$\operatorname{\text{{\small m}}}_{i,r}(0,k)$ is unbounded with respect to
$r$. Let ${\mathfrak{I}}_{1}$ denote the finite interval
${\mathfrak{I}}_{1}\supset\invbreve{{\mathscr{I}}}_{i,r}^{(1)}\cup\invbreve{{\mathscr{I}}}_{i,r}^{(k_{1})}$
with
$\partial{\mathfrak{I}}_{1}\subset\partial\invbreve{{\mathscr{I}}}_{i,r}^{(1)}\cup\partial\invbreve{{\mathscr{I}}}_{i,r}^{(k_{1})}$
if $k_{1}$ exists; otherwise set ${\mathfrak{J}}_{1}=\emptyset$. Given any
pair $k$, $k^{\prime}$ such that $k<k^{\prime}\leq\Lambda_{i}$ and
$\operatorname{\text{{\small m}}}_{i,r}(k,k^{\prime})$ is bounded with respect
to $r$, we form a finite interval
${\mathfrak{I}}_{*}\supset\invbreve{{\mathscr{I}}}_{i,r}^{(k)}\cup\invbreve{{\mathscr{I}}}_{i,r}^{(k^{\prime})}$
with
$\partial{\mathfrak{I}}_{*}\subset\partial\invbreve{{\mathscr{I}}}_{i,r}^{(k)}\cup\partial\invbreve{{\mathscr{I}}}_{i,r}^{(k^{\prime})}$.
Given $i$ and $r$, let ${\mathfrak{J}}_{i,r}\subset[L_{T}(r)-1/2,\infty)$
denote
$\bigcup_{*}{\mathfrak{J}}_{*}\cup{\mathfrak{J}}_{1}\cup\bigcup_{k}\invbreve{{\mathscr{I}}}_{i,r}^{(k)}$,
where the index $k$ runs over all $k$ such that
$\invbreve{{\mathscr{I}}}_{i,r}^{(k)}\subset[L_{T}(r)-1/2,\infty)-\bigcup_{*}{\mathfrak{J}}_{*}-{\mathfrak{J}}_{1}$.
Let ${\mathfrak{K}}_{i}$ denote the number of connected components of
${\mathfrak{J}}_{i,r}$, and when ${\mathfrak{K}}_{i}>0$, let
$\\{{\mathfrak{I}}_{i,r}^{k}\\}_{k=1}^{{\mathfrak{K}}_{i}}$ denote the set of
connected components of ${\mathfrak{J}}_{i,r}$. These connected components are
ordered such that ${\mathfrak{I}}_{i,r}^{k^{\prime}}$ lies to the right of
${\mathfrak{I}}_{i,r}^{k}\subset[L_{T}(r)-1/2,\infty)$ for any $k,k^{\prime}$
such that $1\leq k<k^{\prime}\leq{\mathfrak{K}}_{i}$. For any $1\leq
k<{\mathfrak{K}}_{i}$, the intervals ${\mathfrak{I}}_{i,r}^{k}$ and
${\mathfrak{I}}_{i,r}^{k+1}$ are separated by one of the connected components
of $\breve{{\mathscr{I}}}_{i,r}$, say
$\breve{{\mathscr{I}}}_{i,r}^{(k^{\prime})}$. We rename the t-orbit associated
to this connected component, $\boldsymbol{\gamma}_{i,r}^{(k^{\prime})}$, by
$\boldsymbol{\gamma}_{i}^{k}$. (Recall the second bullet of (7.25).)
Meanwhile, ${\mathfrak{J}}_{i,r}^{1}$ is adjacent to the first connected
component of $\breve{{\mathscr{I}}}_{i,r}$, i.e.
$\breve{{\mathscr{I}}}_{i,r}^{(0)}$, on the left; and
${\mathfrak{J}}_{i,r}^{{\mathfrak{K}}_{i}}$ is adjacent to the last connected
component of $\breve{{\mathscr{I}}}_{i,r}$, i.e.
$\breve{{\mathscr{I}}}_{i,r}^{(\Lambda_{i})}$, which is a half-infinite
interval of the form $[L_{+},\infty)$. The t-orbits associated to
$\breve{{\mathscr{I}}}_{i,r}^{(0)}$ and
$\breve{{\mathscr{I}}}_{i,r}^{(\Lambda_{i})}$ are respectively
$\boldsymbol{\gamma}_{0,i}$ and $\boldsymbol{\gamma}_{i}$.
($\boldsymbol{\gamma}_{i}$ is as in Condition (1) in the statement of Theorem
1.7.) Let $\boldsymbol{\gamma}_{i}^{0}:=\boldsymbol{\gamma}_{0,i}$ and
$\boldsymbol{\gamma}_{i}^{{\mathfrak{K}}_{i}}:=\boldsymbol{\gamma}_{i}$. Note
that when ${\mathfrak{K}}_{i}=0$, ${\mathfrak{J}}_{i,r}=\emptyset$ and
$\boldsymbol{\gamma}_{0,i}=\boldsymbol{\gamma}_{i}$.

Recall also that by (7.24), the length
$|\invbreve{{\mathscr{I}}}_{i,r}^{(k)}|$ has an upper bound independent of
$r$, $i$, and $k$. This means that there is a finite number
$\operatorname{\text{{\small m}}}\geq 1$ so that
$|{\mathfrak{I}}_{i,r}^{k}|\leq 2\operatorname{\text{{\small m}}}$ for all
$i,k$ and $r$. Let $L^{i}_{k,r}\in[L_{T}(r)-1/2,\infty)$ denote the mid point
of the interval ${\mathfrak{I}}_{i,r}^{k}$, and let
${\mathfrak{Y}}^{\prime}_{m}=\\{i\,|\,i\in{\mathfrak{Y}}_{m},{\mathfrak{K}}_{i}>0\\}$.
By construction, we have:

(7.26) $\hss\left\\{\mkern-20.0mu\parbox{422.77661pt}{\begin{itemize}
\par\itemize@item@Given $i\in{\mathfrak{Y}}^{\prime}_{m}$, for every $k$ with
$1\leq k\leq{\mathfrak{K}}_{i}-1$,
$\Delta^{i}_{k,r}:=L^{i}_{k+1,r}-L^{i}_{k,r}$ is non-decreasing with respect
to $r$. Also, $\Delta^{i}_{0,r}:=L^{i}_{1,r}-L_{T}(r)$ is non-decreasing with
respect to $r$. (cf. the second bullet in (\ref{def:Gamma1}).)
\par\itemize@item@Given $i\in{\mathfrak{Y}}^{\prime}_{m}$, for every $k$ with
$0\leq k\leq{\mathfrak{K}}_{i}-1$,
$\lim_{r\to\infty}\Delta^{i}_{k,r}\to\infty$ as $r\to\infty$.
\par\end{itemize}}\right.$

In particular, there exists an $R_{1}\geq R_{0}$ such that
$\Delta^{i}_{k,r}>2\operatorname{\text{{\small m}}}+4$ for all
$i\in{\mathfrak{Y}}^{\prime}_{m}$, $k\in\\{0,\ldots,{\mathfrak{K}}_{i}-1\\}$,
$\forall r\geq R_{1}$. Assume that $r\geq R_{1}$ from now on.

Fix $i\in{\mathfrak{Y}}^{\prime}_{m}$. Given $r$ and $k$, $1\leq
k\leq{\mathfrak{K}}_{i}$, let
$I^{{}^{\prime}i}_{k,r}\supset{\mathfrak{J}}^{k}_{i,r}$ be defined by

$I^{{}^{\prime}i}_{k,r}:=\begin{cases}[L^{i}_{k-1,r}+\operatorname{\text{{\small
m}}},L^{i}_{k+1,r}-\operatorname{\text{{\small m}}}]&\text{when $2\leq
k\leq{\mathfrak{K}}_{i}-1$};\\\
[L_{T}(r)-3,L^{i}_{2,r}-\operatorname{\text{{\small m}}}]&\text{when
$k=1$};\\\ [L^{i}_{{\mathfrak{K}}_{i}-1,r}+\operatorname{\text{{\small
m}}},\infty)&\text{when $k={\mathfrak{K}}_{i}$}.\end{cases}$

Let $I_{k,r}^{i}:=\tau_{-L^{i}_{k,r}}I^{{}^{\prime}i}_{k,r}$. That is,

$I^{i}_{k,r}:=\begin{cases}[-\Delta^{i}_{k-1,r}+\operatorname{\text{{\small
m}}},\Delta^{i}_{k,r}-\operatorname{\text{{\small m}}}]&\text{when $2\leq
k\leq{\mathfrak{K}}_{i}-1$};\\\
[-\Delta^{i}_{0,r}-3,\Delta^{i}_{1,r}-\operatorname{\text{{\small
m}}}]&\text{when $k=1$};\\\
[-\Delta{}^{i}_{{\mathfrak{K}}_{i}-1,r}+\operatorname{\text{{\small
m}}},\infty)&\text{when $k={\mathfrak{K}}_{i}$}.\end{cases}$

Then by (7.26), for every fixed $1\leq k\leq{\mathfrak{K}}_{i}$,
$I^{i}_{k,r}\subset I^{i}_{k,r^{\prime}}$ when $r^{\prime}>r$, and together
they form a nested sequence of intervals exhausting
${\mathbb{R}}=I^{i}_{k,\infty}$. Let

$\breve{{\mathfrak{I}}}^{i}_{k,r}:=\begin{cases}I^{{}^{\prime}i}_{k,r}\cap
I^{{}^{\prime}i}_{k+1,r}=[L_{k,r}^{i}+\operatorname{\text{{\small
m}}},L_{k+1,r}^{i}-\operatorname{\text{{\small
m}}}]\subset\breve{{\mathscr{I}}}_{i,r}&\text{when $1\leq
k\leq{\mathfrak{K}}_{i}-1$;}\\\
[L_{T}(r)-5/2,L_{1,r}^{i}-\operatorname{\text{{\small
m}}}]\subset\breve{{\mathscr{I}}}_{i,r}^{(0)}&\text{when $k=0$;}\\\
[L_{{\mathfrak{K}}_{i},r}^{i}+\operatorname{\text{{\small
m}}},\infty)\subset\breve{{\mathscr{I}}}_{i,r}^{(\Lambda_{i})}&\text{when
$k={\mathfrak{K}}_{i}$.}\\\ \end{cases}$

Then by construction (recall (7.22)), for $r\geq R_{1}$,

(7.27) $\operatorname{\text{{\small
d}}}_{\hat{Y}_{i,I^{*}}}(\operatorname{\text{{\small
c}}}_{r},{\mathbb{R}}\times\boldsymbol{\gamma}_{i}^{k})\leq\epsilon\quad\forall\,I^{*}\subset\breve{{\mathfrak{I}}}_{k,r}^{i},\,|I^{*}|=2.$

Consider now the sequence $\\{\text{{\scriptsize
b}}_{r}:=\tau_{-L^{i}_{k,r}}\operatorname{\text{{\small
c}}}_{r}|_{\hat{Y}_{i,I_{k,r}^{\prime}}}\\}_{r\in\Gamma_{1}}$.
$\text{{\scriptsize b}}_{r}$ is defined on $I_{k,r}^{i}\times Y_{i}$. Apply
Lemma 7.4 to $\\{\text{{\scriptsize b}}_{r}\\}_{r\in\Gamma_{1}}$ to get a
t-curve ${\bf C}_{i,k}$ on $({\mathbb{R}}\times Y_{i},\pi_{2}^{*}\nu_{i})$ and
a subsequence $\Gamma^{i}_{k}$ of $\Gamma_{1}$ so that $\\{\text{{\scriptsize
b}}_{r}\\}_{r\in\Gamma_{k}^{i}}$ t-converges to ${\bf C}_{i,k}$. More
specifically, the subsequence $\Gamma^{i}_{k}$ is chosen via diagonalization
so that

(7.28) $\operatorname{\text{{\small d}}}_{I^{*}\times
Y_{i}}(\text{{\scriptsize b}}_{r_{n}},{\bf C}_{i,k})\leq e^{-n}\quad\forall
I^{*}\subset I^{i}_{k,r_{n}},\forall n,$

where $r_{n}$ now denotes the $n$-th element in the sequence
$\Gamma^{i}_{k}=\\{r_{n}\\}_{n\in{\mathbb{Z}}^{+}}$. Comparing the preceding
inequality with (7.27), we see that the $-\infty$-limit of ${\bf C}_{i,k}$ is
$\boldsymbol{\gamma}_{i}^{k-1}$, and its $+\infty$-limit is
$\boldsymbol{\gamma}_{i}^{k}$. Thus, when ${\mathfrak{K}}_{i}>0$, $\\{{\bf
C}_{i,k}\\}_{k=1}^{{\mathfrak{K}}_{i}}$ forms a chain of t-curves on
$({\mathbb{R}}\times Y_{i},\pi_{2}^{*}\nu_{i})$, ${\mathfrak{C}}_{i}$, with
$-\infty$-limit $\boldsymbol{\gamma}_{i}^{0}=\boldsymbol{\gamma}_{i,0}$, and
with $+\infty$-limit
$\boldsymbol{\gamma}_{i}^{{\mathfrak{K}}_{i}}=\boldsymbol{\gamma}_{i}$. When
${\mathfrak{K}}_{i}=0$, $\breve{{\mathscr{I}}}_{i,r}=[L_{T}(r)-5/2,\infty)$ is
connected; by (7.22), $\boldsymbol{\gamma}_{0,i}=\boldsymbol{\gamma}_{i},$ and

$\operatorname{\text{{\small
d}}}_{\hat{Y}_{i,I^{*}}}(\operatorname{\text{{\small
c}}}_{r},{\mathbb{R}}\times\boldsymbol{\gamma}_{i})\leq\epsilon\quad\forall\,I^{*}\subset[L_{T}(r)-5/2,\infty),\,|I^{*}|=2.$

In this case, let ${\mathfrak{C}}_{i}$ be the 0-component chain of t-curves
with a single rest orbit $\boldsymbol{\gamma}_{0,i}=\boldsymbol{\gamma}_{i}$.
Now ${\mathfrak{C}}:=\\{{\bf
C}_{0},\\{{\mathfrak{C}}_{i}\\}_{i\in{\mathfrak{Y}}_{m}}\\}$ forms a chain of
t-curves on $(X^{{}^{\prime}a},\nu)$, with $Y_{i}$-end limit
$\boldsymbol{\gamma}_{i}$ for each $i\in{\mathfrak{Y}}_{m}$. Finally, choose
an order of the finite set

$\\{(i,k)\,|\,\text{$i\in{\mathfrak{Y}}_{m}$ with
${\mathfrak{K}}_{i}>0$},\,1\leq k\leq{\mathfrak{K}}_{i}\\},$

and choose consecutive subsequences $\Gamma^{i}_{k}\subset\Gamma_{1}$ in this
order: $\Gamma^{i}_{k}\subset\Gamma^{i^{\prime}}_{k^{\prime}}$ whenever
$(i,k)>(i^{\prime},k^{\prime})$ to get a subsequence
$\Gamma^{\prime}=\bigcap_{\\{(i,k)\\}}\Gamma^{i}_{k}\subset\Gamma_{1}$. Then
$\Gamma^{\prime}$ and ${\mathfrak{C}}$ together satisfies Item (a) asserted by
Theorem 1.9.

##### (b) and (d):

Recall that
${\mathscr{F}}_{\omega}({\mathfrak{C}})={\mathscr{F}}_{\omega}({\bf
C}_{0})+\sum_{i\in{\mathfrak{Y}}^{\prime}_{m},{\mathfrak{K}}_{i}>0}\sum_{k=1}^{{\mathfrak{K}}_{i}}{\mathscr{F}}_{\omega}({\bf
C}_{i,k})$. We already provided a bound on ${\mathscr{F}}_{\omega}({\bf
C}_{0})$ in the proof of Theorem 1.7 (b). Thus, to verify Item (b) in the
statement of the theorem, it suffices to bound each
${\mathscr{F}}_{\omega}({\bf C}_{i,k})$. As in the proof of Theorem 1.7 (b),
this reduces to the following two tasks: (i) obtaining a $l$-independent bound
on ${\mathscr{F}}_{\omega}({\bf C}_{i,k}|_{[l,l+1]})$; (ii) bounding
${\mathscr{F}}_{\nu}({\bf C}_{i,k})$.

(i): Fix $i$ with ${\mathfrak{K}}_{i}>0$ and $k$. Given $l\in{\mathbb{R}}$,
there exists ${\mathfrak{r}}_{l}\geq R_{1}$ such that $[l,l+1]\subset
I^{i}_{k,r}$ $\forall r\geq{\mathfrak{r}}_{l}$. Let $r_{n}$ denote the $n$-th
element of $\Gamma_{k}^{i}$, and consider an $r_{n}\geq{\mathfrak{r}}_{l}$.
Then according to (7.28) and (6.11),

$\begin{split}{\mathscr{F}}_{\omega}({\bf
C}_{i,k}|_{[l,l+1]})-\frac{1}{2}c_{1}({\mathfrak{s}}_{i})\cdot[\theta]+\zeta_{\theta}&\leq\int_{\hat{Y}_{i,[L^{i}_{k,r_{n}}+l,L^{i}_{k,r_{n}}+l+1]}}\frac{iF_{A_{r_{n}}^{E}}}{2\pi}\wedge\nu_{i}+e^{-n}\|\nu_{i}\|_{\infty}\\\
&\leq\zeta^{\prime}+e^{-n}\|\nu_{i}\|_{\infty},\end{split}$

where $\theta=*\nu_{i}$ and $\zeta_{\theta}$ is as in Proposition 3.15, and
$\zeta^{\prime}>0$ is independent of $n$ (equivalently, $r=r_{n}$), $k$, and
$l$. This gives us an $l$-independent bound

(7.29) ${\mathscr{F}}_{\omega}({\bf
C}_{i,k}|_{[l,l+1]})\leq\frac{1}{2}c_{1}({\mathfrak{s}}_{i})\cdot[\theta]+\zeta^{\prime}-\zeta_{\theta}.$

(ii): When ${\mathfrak{K}}_{i}>1$, let
$l^{i}_{k,r}:=(L^{i}_{k,r}+L^{i}_{k+1,r})/2$ denote the mid point of the
interval $\breve{{\mathfrak{I}}}^{i}_{k,r}$ for $1\leq
k\leq{\mathfrak{K}}_{i}-1$. Define $J^{{}^{\prime}i}_{k,r}$,
$I^{{}^{\prime}i}_{k,r}\supset
J^{{}^{\prime}i}_{k,r}\supset{\mathfrak{I}}^{k}_{i,r}$ by

$J^{{}^{\prime}i}_{k,r}:=\begin{cases}[l^{i}_{k-1,r},l^{i}_{k,r}]&\text{when
${\mathfrak{K}}_{i}>1$ and $2\leq k\leq{\mathfrak{K}}_{i}-1$;}\\\
[L_{T}(r)-\frac{5}{2},l^{i}_{1,r}]&\text{when ${\mathfrak{K}}_{i}>1$ and
$k=1$;}\\\ [l^{i}_{{\mathfrak{K}}_{i}-1,r},\infty\big{)}&\text{when
${\mathfrak{K}}_{i}>1$ and $k={\mathfrak{K}}_{i}$;}\\\
[L_{T}(r)-\frac{5}{2},\infty)&\text{when ${\mathfrak{K}}_{i}=1$ and
$k=1$.}\end{cases}$

Let $J^{i}_{k,r}=\tau_{-L^{i}_{k,r}}J^{{}^{\prime}i}_{k,r}$. In other words,

$J^{i}_{k,r}:=\begin{cases}\big{[}-\frac{\Delta^{i}_{k-1,r}}{2},\frac{\Delta^{i}_{k,r}}{2}\big{]}&\text{when
${\mathfrak{K}}_{i}>1$ and $2\leq k\leq{\mathfrak{K}}_{i}-1$;}\\\
\big{[}-\Delta^{i}_{0,r}-\frac{5}{2},\frac{\Delta^{i}_{1,r}}{2}\big{]}&\text{when
when ${\mathfrak{K}}_{i}>1$ and $k=1$;}\\\
\big{[}-\frac{\Delta^{i}_{{\mathfrak{K}}_{i}-1,r}}{2},\infty\big{)}&\text{when
${\mathfrak{K}}_{i}>1$ and $k={\mathfrak{K}}_{i}$.}\\\
\big{[}-\Delta^{i}_{0,r}-\frac{5}{2},\infty)&\text{when when
${\mathfrak{K}}_{i}=1$ and $k=1$;}\\\ \end{cases}$

Then $J^{i}_{k,r}\subset I^{i}_{k,r}$, yet according to (7.26),
$\\{J^{i}_{k,r}\\}_{r\in\Gamma^{\prime}}$ still forms a nested sequence of
intervals exhausting ${\mathbb{R}}$. Meanwhile, for every fixed
$i\in{\mathfrak{Y}}^{\prime}_{m}$ and $r$,
$\bigcup_{k=1}^{{\mathfrak{K}}_{i}}J^{{}^{\prime}i}_{k,r}=[L_{T}(r)-5/2,\infty)$,
and the interiors of $J^{{}^{\prime}i}_{k,r}$ are mutually disjoint. Now

(7.30)
$\begin{split}\bar{{\mathfrak{E}}}_{r}(X^{{}^{\prime}a})&=\bar{{\mathfrak{E}}}_{r}(X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{\frac{5}{2}}})+\sum_{i\in{\mathfrak{Y}}^{\prime}_{m}}\sum_{k=1}^{{\mathfrak{K}}_{i}}\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,J^{{}^{\prime}i}_{k,r}})+\sum_{i\in{\mathfrak{Y}}_{m}-{\mathfrak{Y}}^{\prime}_{m}}\bar{{\mathfrak{E}}}_{r}(\hat{Y}_{i,L_{T}(r)-\frac{5}{2}})\\\
&=\bar{{\mathscr{V}}}_{r}(X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{\frac{5}{2}}})+\sum_{i\in{\mathfrak{Y}}^{\prime}_{m}}\sum_{k=1}^{{\mathfrak{K}}_{i}}\bar{{\mathscr{V}}}_{r}(\hat{Y}_{i,J^{{}^{\prime}i}_{k,r}})+\sum_{i\in{\mathfrak{Y}}_{m}-{\mathfrak{Y}}^{\prime}_{m}}\bar{{\mathscr{V}}}_{r}(\hat{Y}_{i,L_{T}(r)-\frac{5}{2}})+{\mathscr{O}}(r^{-3/7})\\\
&=\bar{{\mathscr{F}}}_{\nu}({\bf
C}_{0}|_{X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{\frac{5}{2}}}})+\sum_{i\in{\mathfrak{Y}}^{\prime}_{m}}\sum_{k=1}^{{\mathfrak{K}}_{i}}\bar{{\mathscr{F}}}_{\nu}({\bf
C}_{i,k}|_{J^{i}_{k,r}\times
Y_{i}})+{\mathscr{O}}(e^{-n_{r}})+{\mathscr{O}}(r^{-3/7})\\\
&\qquad+\sum_{i\in{\mathfrak{Y}}^{\prime}_{m}}\sum_{k=1}^{{\mathfrak{K}}_{i}}\int_{-1/2}^{1/2}\int_{\tau_{t}\hat{Y}_{i,J^{{}^{\prime}i}_{k,r}}}\frac{i}{2\pi}F_{A^{E}}\wedge(\nu-\pi_{2}^{*}\nu_{i})\,dt\\\
&\qquad+\sum_{i\in{\mathfrak{Y}}_{m}-{\mathfrak{Y}}^{\prime}_{m}}\int_{-1/2}^{1/2}\int_{\hat{Y}_{i,L_{T}(r)-\frac{5}{2}+t}}\frac{i}{2\pi}F_{A^{E}}\wedge(\nu-\pi_{2}^{*}\nu_{i})\,dt,\\\
\end{split}$

where $r\in\Gamma^{\prime}$ and $n_{r}\in{\mathbb{Z}}^{+}$ are related as
follows: $r$ is the $n_{r}$-th element in $\Gamma^{\prime}$. In the above, the
second equality follows from Proposition 7.1, and the third equality is a
consequence of the construction of $\Gamma^{\prime}$ and ${\bf C}_{i,k}$; see
in particular (7.13) and (7.28). Using (6.11) and (4.29) to estimate the last
two terms above, we have:

(7.31)
$\begin{split}\bar{{\mathfrak{E}}}_{r}(X^{{}^{\prime}a})&=\bar{{\mathscr{F}}}_{\nu}({\bf
C}_{0}|_{X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{\frac{5}{2}}}})+\sum_{i\in{\mathfrak{Y}}_{m}^{\prime}}\sum_{k=1}^{{\mathfrak{K}}_{i}}\bar{{\mathscr{F}}}_{\nu}({\bf
C}_{i,k}|_{J^{i}_{k,r}\times Y_{i}})\\\
&\qquad\quad+{\mathscr{O}}(e^{-n_{r}})+{\mathscr{O}}(r^{-3/7})+\sum_{i\in{\mathfrak{Y}}_{m}}{\mathscr{O}}(e^{-\kappa_{i}L_{T}(r)})\end{split}$

Together with Lemma 7.3, the preceding identity then implies that

$\bar{{\mathscr{F}}}_{\nu}({\bf
C}_{0}|_{X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{\frac{5}{2}}}})+\sum_{i\in{\mathfrak{Y}}^{\prime}_{m}}\sum_{k=1}^{{\mathfrak{K}}_{i}}\bar{{\mathscr{F}}}_{\nu}({\bf
C}_{i,k}|_{J^{i}_{k,r}\times Y_{i}})\leq\frac{\textsc{e}}{\pi}+\zeta$

for all sufficiently large $r$. Note that as functions of $r$,
$\bar{{\mathscr{F}}}_{\nu}({\bf
C}_{0}|_{X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{\frac{5}{2}}}})$,
$\bar{{\mathscr{F}}}_{\nu}({\bf C}_{i,k}|_{J^{i}_{k,r}\times Y_{i}})$ are all
nondecreasing, nonnegative functions, and recall that $\\{J^{i}_{k,r}\\}_{r}$
exhausts ${\mathbb{R}}$ when $i\in{\mathfrak{Y}}_{m}^{\prime}$. Consequently,
$\lim_{r\to\infty}\bar{{\mathscr{F}}}_{\nu}({\bf
C}_{0}|_{X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{\frac{5}{2}}}})=\bar{{\mathscr{F}}}_{\nu}({\bf
C}_{0})$, and for all $i\in{\mathfrak{Y}}^{\prime}_{m},k$, the limit
$\lim_{r\to\infty}\bar{{\mathscr{F}}}_{\nu}({\bf C}_{i,k}|_{J^{i}_{k,r}\times
Y_{i}})$ exists and equals ${\mathscr{F}}_{\nu}({\bf C}_{i,k})$. Take the
$r\to\infty$ limit on both sides of the identity (7.31) and apply Lemma 7.3.
We get:

(7.32)
$\bar{{\mathscr{F}}}_{\nu}({\mathfrak{C}}):=\bar{{\mathscr{F}}}_{\nu}(\mathbf{C}_{0})+\sum_{i\in{\mathfrak{Y}}^{\prime}_{m}}\sum_{k=1}^{{\mathfrak{K}}_{i}}{\mathscr{F}}_{\nu}({\bf
C}_{i,k})\leq\textsc{e}/\pi+\zeta,$

where $\zeta$ is as in Lemma 7.3. Also, when $X$ has no vanishing ends, by
(4.11)

(7.33)
$\begin{split}{\mathscr{F}}_{\nu}({\mathfrak{C}})&=\bar{{\mathscr{F}}}_{\nu}({\mathfrak{C}})\\\
&=\lim_{r\to\infty}\big{(}(\pi
r)^{-1}{\mathscr{E}}_{top}^{\mu_{r}}(A_{r},\Psi_{r})\big{)}-\int_{X}\frac{iF_{A_{K}}}{4\pi}\wedge\nu\\\
&=\frac{i}{2\pi}\int_{X}F_{A_{0}^{E}}\wedge\nu+\frac{1}{\pi}\sum_{i\in{\mathfrak{Y}}_{m}}[\nu_{i}]\cdot\jmath_{h}(\boldsymbol{\gamma}_{i})+\frac{1}{2\pi}i^{*}[\nu]\cdot
h_{A_{0}}(\operatorname{\text{{\scriptsize h}}}),\end{split}$

as claimed in Item (d) in the statement of the theorem. Meanwhile, Item (b) in
the statement of the theorem follows from a combination of (7.32), (7.29), and
Theorem 1.7 (b).

##### (c):

Let $l^{i}_{0,r}:=L_{T}(r)-3/2$. Let $u_{r}\in{\mathscr{G}}$ be such that
$(A_{r}^{\prime},\Psi_{r}^{\prime}):=u_{r}\cdot(A_{r},\Psi_{r})$ is in a
temporal gauge on $\hat{Y}_{i,\breve{{\mathfrak{I}}}^{i}_{k,r}}$ $\forall
i\in{\mathfrak{Y}}_{m}$ and $k\in\\{0,1,\ldots,{\mathfrak{K}}_{i}\\}$. Given
$i\in{\mathfrak{Y}}^{\prime}_{m}$ and
$k\in\\{0,1,\ldots,{\mathfrak{K}}_{i}-1\\}$, let
$B^{i}_{k,r}\in\mathop{\mathrm{Conn}}\nolimits(Y_{i})$,
$\Phi^{i}_{k,r}\in\Gamma({\mathbb{S}}_{i})$ be

$B^{i}_{k,r}:=B_{0}+\int_{-1/2}^{1/2}(B_{r}(l^{i}_{k,r}+t)-B_{0})\,dt,\quad\Phi^{i}_{k,r}:=\int_{-1/2}^{1/2}\Phi_{r}(l^{i}_{k,r}+t)\,dt,$

where $(B_{r}(s),\Phi_{r}(s))=(A_{r}^{\prime},\Psi^{\prime}_{r})|_{Y_{i:s}}$.
When $i\in{\mathfrak{Y}}_{v}$, set ${\mathfrak{K}}_{i}=0$, and for all
$i\in{\mathfrak{Y}}$, let
${\mathfrak{c}}^{i}_{{\mathfrak{K}}_{i},r}:={\mathfrak{c}}_{i,r}$. When
$i\in{\mathfrak{Y}}^{\prime}_{m}$, let
${\mathfrak{c}}^{i}_{k,r}\in{\mathscr{B}}_{Y_{i}}$ denote the gauge-
equivalence class of $(B^{i}_{k,r},\Phi^{i}_{k,r})$ when
$k\in\\{0,1,\ldots,{\mathfrak{K}}_{i}-1\\}$. Recall that given
${\mathfrak{c}}=[(B,\Phi)]\in{\mathscr{B}}_{Y_{i}}$, $\tilde{{\mathfrak{c}}}$
denotes the 1-current

$\tilde{{\mathfrak{c}}}=\frac{iF_{B^{E}}}{2\pi}=\frac{iF_{B}}{4\pi}-\frac{iF_{B^{K_{i}}}}{4\pi},$

where $K_{i}^{-1}$ denotes the anti-canonical bundle over $Y_{i}$ defined by
$\nu_{i}$, and $B^{K_{i}}$ denotes the connection on $K_{i}^{-1}$ determined
by the Levi-Civita connection. Let $[x]_{a}\subset{\mathbb{R}}$ denote the
length $a$ interval centered at $x$. Given a 1-form $\mu$ on $Y_{i}$, the
evaluation of the exact 1-current
$\tilde{{\mathfrak{c}}}_{k,r}^{i}-\tilde{\gamma}_{i}^{k}$ on $\mu$ is equal to
the evaluation of the 2-current
$\frac{iF_{A_{r}}}{4\pi}-\frac{iF_{B^{K_{i}}}}{4\pi}-\pi_{2}^{*}\tilde{\gamma}_{i}^{k}$
on the 2-form $\pi_{2}^{*}\mu\wedge ds$ over $\hat{Y}_{i,[l^{i}_{k,r}]_{1}}$.
The absolute value of this is bounded by

$\|\mu\|_{\infty}\Big{(}\operatorname{\text{{\small
d}}}_{\hat{Y}_{i,[l^{i}_{k,r}]_{1}}}(\operatorname{\text{{\small
c}}}_{r},{\mathbb{R}}\times\boldsymbol{\gamma}^{k}_{i})+\Big{\|}\frac{iF_{A_{K}}}{4\pi}-\frac{iF_{B^{K_{i}}}}{4\pi}\Big{\|}_{L^{1}(\hat{Y}_{i,[l^{i}_{k,r}]_{1}})}\Big{)},$

and therefore

$\begin{split}&\|\tilde{{\mathfrak{c}}}_{k,r}^{i}-\tilde{\gamma}_{i}^{k}\|_{op}\leq\operatorname{\text{{\small
d}}}_{\hat{Y}_{i,[l^{i}_{k,r}]_{1}}}(\operatorname{\text{{\small
c}}}_{r},{\mathbb{R}}\times\boldsymbol{\gamma}^{k}_{i})+\Big{\|}\frac{iF_{A^{K}}}{4\pi}-\frac{iF_{B^{K_{i}}}}{4\pi}\Big{\|}_{L^{1}(\hat{Y}_{i,[l^{i}_{k,r}]_{1}})}\\\
&\qquad\leq\operatorname{\text{{\small
d}}}_{\hat{Y}_{i,[-s^{i}_{k,r}]_{1}}}(\tau_{-L^{i}_{k+1,r}}\operatorname{\text{{\small
c}}}_{r},{\bf C}_{i,k+1})+\operatorname{\text{{\small
d}}}_{\hat{Y}_{i,[-s^{i}_{k,r}]_{1}}}({\bf
C}_{i,k+1},{\mathbb{R}}\times\boldsymbol{\gamma}^{k}_{i})+\zeta_{i}e^{-\kappa_{i}l^{i}_{k,r}}\\\
&\qquad\to 0\quad\text{as $r\to\infty$},\end{split}$

where $s^{i}_{k,r}=\Delta^{i}_{k,r}/2$ when
$k\in\\{1,\ldots,{\mathfrak{K}}_{i}-1\\}$, and
$s^{i}_{0,r}:=\Delta^{i}_{0,r}+3/2$. In the above, we used (4.29), (7.28) and
the facts that $s^{i}_{k,r}$, $s^{i}_{k,r}$ both go to $\infty$ as
$r\to\infty$, and that the $-\infty$-limit of ${\bf C}_{i,k+1}$ is
$\boldsymbol{\gamma}^{k}_{i}$. Given that
$\tilde{{\mathfrak{c}}}_{k,r}^{i}=\frac{i}{2\pi}F_{B^{i}_{k,r}}$ converges to
$\tilde{\gamma}_{i}^{k}$, The arguments in the proof of Theorem 1.2 in Section
3.4 may be combined with (7.28) to show that
$[B^{i}_{k,r}]\in\mathop{\mathrm{Conn}}\nolimits(Y_{i})/{\mathscr{G}}\subset\operatorname{C}(Y_{i})$
converges in the current topology. We denote this limit by
$[B^{i}_{k,\infty}]$. Thus, the arguments in Lemma 3.19 shows that there
exists an $R_{2}\geq R_{1}$ such that for all $r,r^{\prime}$ satisfying
$r>r^{\prime}\geq R_{2}$, we have canonical isomorphisms of affine spaces

$\begin{split}&\pi_{Y_{i}}({\mathfrak{c}}^{i}_{k-1,r^{\prime}},{\mathfrak{c}}^{i}_{k,r^{\prime}})\stackrel{{\scriptstyle\sim}}{{\to}}\pi_{Y_{i}}({\mathfrak{c}}^{i}_{k-1,r},{\mathfrak{c}}^{i}_{k,r})\quad\forall
i\in{\mathfrak{Y}}^{\prime}_{m},\,k\in\\{1,\ldots,{\mathfrak{K}}_{i}\\};\\\
&\pi_{0}({\mathscr{B}}_{X}(\\{{\mathfrak{c}}^{i}_{0,r^{\prime}}\\}_{i\in{\mathfrak{Y}}})\stackrel{{\scriptstyle\sim}}{{\to}}\pi_{0}({\mathscr{B}}_{X}(\\{{\mathfrak{c}}^{i}_{0,r}\\}_{i\in{\mathfrak{Y}}})\end{split}$

via concatenations with the distinguished elements
$o_{Y_{i}}({\mathfrak{c}}^{i}_{k,r^{\prime}},{\mathfrak{c}}^{i}_{k,r})\in\pi_{Y_{i}}({\mathfrak{c}}^{i}_{k,r^{\prime}},{\mathfrak{c}}^{i}_{k,r})$
for $0\leq k\leq{\mathfrak{K}}_{i}$, $i\in{\mathfrak{Y}}_{m}$. (When
$i\in{\mathfrak{Y}}_{v}$,
${\mathfrak{c}}^{i}_{0,r}={\mathfrak{c}}^{i}_{0,r^{\prime}}:=[(B_{i},\Phi_{i})]$
is the $Y_{i}$-end limit of $[(A_{r},\Psi_{r})]$, and
$o_{Y_{i}}({\mathfrak{c}}^{i}_{0,r^{\prime}},{\mathfrak{c}}^{i}_{0,r})$ is the
identity in this case.) Recall from Lemma 3.23 the definition of
${\mathfrak{h}}^{\prime}$; and that
${\mathfrak{h}}=i_{\infty}\circ{\mathfrak{h}}^{\prime}$. Each of the preceding
maps fits into its respective version of the diagram (3.34):

and for $i\in{\mathfrak{Y}}^{\prime}_{m}$,
$k\in\\{1,\ldots,{\mathfrak{K}}_{i}\\}$,

$\textstyle{\pi_{Y_{i}}({\mathfrak{c}}^{i}_{k-1,r^{\prime}},{\mathfrak{c}}^{i}_{k,r^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{*}$$\scriptstyle{{\mathfrak{h}}^{\prime}}$$\textstyle{\pi_{Y_{i}}({\mathfrak{c}}^{i}_{k-1,r},{\mathfrak{c}}^{i}_{k,r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathfrak{h}}^{\prime}}$$\scriptstyle{{\mathfrak{h}}}$$\textstyle{{\mathscr{H}}^{\mathbb{R}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{{\mathfrak{c}}}^{i}_{k-1,r^{\prime}},\tilde{{\mathfrak{c}}}^{i}_{k,r^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{r^{\prime},r}}$$\textstyle{{\mathscr{H}}^{\mathbb{R}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{{\mathfrak{c}}}^{i}_{k-1,r},\tilde{{\mathfrak{c}}}^{i}_{k,r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\infty}}$$\textstyle{{\mathscr{H}}^{\mathbb{R}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i}^{k-1},\tilde{\gamma}_{i}^{k}),}$

where the horizonal map in the top row labeled $*$ is given by $c\mapsto
o_{Y_{i}}({\mathfrak{c}}^{i}_{k-1,r},{\mathfrak{c}}^{i}_{k-1,r^{\prime}})*c*o_{Y_{i}}({\mathfrak{c}}^{i}_{k,r^{\prime}},{\mathfrak{c}}^{i}_{k,r})$.
Recalling that in the cylindrical case, ${\mathfrak{h}}^{\prime}$ factors as
$\underline{{\mathfrak{h}}}\circ\Pi_{*}$ and combing this with the fact that
${\mathfrak{h}}^{\prime}$, ${\mathfrak{h}}$, $\underline{{\mathfrak{h}}}$
preserve concatenations, we see that the horizonal maps $i_{r^{\prime},r}$,
$i_{\infty}$ are given by

(7.34)
$i_{r^{\prime},r}=\begin{cases}\operatorname{c}_{\\{\underline{{\mathfrak{h}}}\,(\tilde{o}_{Y_{i}}([B^{i}_{0,r^{\prime}}],[B^{i}_{0,r}]))\\}_{i}}&\text{in
1st CD;}\\\ \text{the map
$h\mapsto\underline{{\mathfrak{h}}}\,(\tilde{o}_{Y_{i}}([B^{i}_{k-1,r}],[B^{i}_{k-1,r^{\prime}}])*h*\underline{{\mathfrak{h}}}\,(\tilde{o}_{Y_{i}}([B^{i}_{k,r^{\prime}}],[B^{i}_{k,r}]))$}&\text{in
2nd CD;}\end{cases}$ (7.35)
$i_{\infty}=\begin{cases}\operatorname{c}_{\\{\underline{{\mathfrak{h}}}\,(\tilde{o}_{Y_{i}}([B^{i}_{0,r}],[B^{i}_{0,\infty}]))\\}_{i}}&\text{in
1st CD;}\\\ \text{the map
$h\mapsto\underline{{\mathfrak{h}}}\,(\tilde{o}_{Y_{i}}([B^{i}_{k-1,\infty}],[B^{i}_{k-1,r}])*h*\underline{{\mathfrak{h}}}\,(\tilde{o}_{Y_{i}}([B^{i}_{k,r}],[B^{i}_{k,\infty}]))$}&\text{in
2nd CD.}\end{cases}$

(In the above, “CD” stands for “commutative diagram”.)

Recall that the relative homology class of ${\mathfrak{C}}$ is defined to be
the composition

$[{\mathfrak{C}}]=\operatorname{c}_{\\{[{\mathfrak{C}}_{i}]\\}_{i}\in{\mathfrak{Y}}_{m}}([{\bf
C}_{0}]),$

and

$[{\mathfrak{C}}_{i}]:=\begin{cases}[\tilde{C}_{i,{\mathfrak{K}}_{i}}]*\cdots*[\tilde{C}_{i,2}]*[\tilde{C}_{i,1}]&\text{when
$i\in{\mathfrak{Y}}^{\prime}_{m}$;}\\\ 0\in
H_{2}(Y_{i};{\mathbb{Z}})&\text{when ${\mathfrak{K}}_{i}=0$.}\end{cases}$

Denote by $\text{{\scriptsize h}}_{r}$ the relative homotopy class of
$(A_{r},\Psi_{r})$ and recall that by assumption, versions of
$\text{{\scriptsize h}}_{r}$ corresponding to different, sufficiently large
$r$ are identified via the canonical isomorphisms in Lemma 3.19 and
collectively denoted as h. To compare
${\mathfrak{h}}(\operatorname{\text{{\scriptsize h}}})$ with
$[{\mathfrak{C}}]$, first fix $r>R_{2}$ and identify
$\operatorname{\text{{\scriptsize h}}}=\operatorname{\text{{\scriptsize
h}}}_{r}$ with the relative homotopy class of $(A_{r},\Psi_{r})$ in
$\pi_{0}{\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i,r}\\}_{i\in{\mathfrak{Y}}})$.
We shall express $\operatorname{\text{{\scriptsize h}}}_{r}$ as a composition,

(7.36) $\operatorname{\text{{\scriptsize
h}}}_{r}=\operatorname{c}_{\\{\operatorname{\text{{\scriptsize
h}}}_{i,r}\\}_{i}}(\operatorname{\text{{\scriptsize h}}}_{0,r}),\quad$

where $\operatorname{\text{{\scriptsize
h}}}_{0,r}\in\pi_{0}{\mathscr{B}}_{X}(\\{{\mathfrak{c}}^{i}_{0,r}\\}_{i\in{\mathfrak{Y}}})$,
and

(7.37) $\operatorname{\text{{\scriptsize
h}}}_{i,r}=\begin{cases}\operatorname{\text{{\scriptsize
h}}}^{r}_{i,{\mathfrak{K}}_{i}},*\cdots*\operatorname{\text{{\scriptsize
h}}}^{r}_{i,2}*\operatorname{\text{{\scriptsize
h}}}^{r}_{i,1},\quad\operatorname{\text{{\scriptsize
h}}}^{r}_{i,k}\in\pi_{Y_{i}}({\mathfrak{c}}^{i}_{k-1,r},{\mathfrak{c}}^{i}_{k,r})&\text{when
$i\in{\mathfrak{Y}}^{\prime}_{m}$};\\\ 0\in
H_{2}(Y;{\mathbb{Z}})\simeq\pi_{Y_{i}}({\mathfrak{c}}_{i,r},{\mathfrak{c}}_{i,r})&\text{when
$i\in{\mathfrak{Y}}-{\mathfrak{Y}}^{\prime}_{m}$},\\\ \end{cases}$

such that

(7.38) ${\mathfrak{h}}(\operatorname{\text{{\scriptsize h}}}_{0,r})=[{\bf
C}_{0}]$ and ${\mathfrak{h}}(\operatorname{\text{{\scriptsize
h}}}^{r}_{i,k})=[{\bf C}_{i,k}]$ for all $i\in{\mathfrak{Y}}^{\prime}_{m}$ and
$k\in\\{1,\ldots,{\mathfrak{K}}_{i}\\}$.

Once (7.36), (7.37), (7.38) are established, modulo Lemma 3.23(b), the proof
of Item (c) in the statement of the theorem follows directly from the
naturality of ${\mathfrak{h}}$ under concatenation/composition maps.

To define $\operatorname{\text{{\scriptsize h}}}_{0,r}$,
$\operatorname{\text{{\scriptsize h}}}^{r}_{i,k}$, we begin with a
modification of
$[(A_{r},\Psi_{r})]\in{\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i,r}\\}_{i\in{\mathfrak{Y}}})$.
Introduce the cutoff function $\lambda(s):=\chi(2s)\chi(-2s)$ and let
$(\hat{B}^{i}_{k,r},\hat{\Phi}^{i}_{k,r})$ denotes the pull back of
$(B^{i}_{k,r},\Phi^{i}_{k,r})$ to $\hat{Y}_{i}$. Set

$\begin{split}\tilde{A}_{r}:&=A^{\prime}_{r}+\sum_{i\in{\mathfrak{Y}}^{\prime}_{m}}\sum_{k=0}^{{\mathfrak{K}}_{i}-1}\big{(}\lambda(s-l^{i}_{k,r})(\hat{B}^{i}_{k,r}-A^{\prime}_{r})\big{)},\\\
\tilde{\Psi}_{r}:&=\Psi^{\prime}_{r}+\sum_{i\in{\mathfrak{Y}}^{\prime}_{m}}\sum_{k=0}^{{\mathfrak{K}}_{i}-1}\big{(}\lambda(s-l^{i}_{k,r})(\hat{\Phi}^{i}_{k,r}-\Psi^{\prime}_{r})\big{)}.\end{split}$

Then $[(\tilde{A}_{r},\tilde{\Psi}_{r})]$ represents the same relative
homotopy class as $[(A_{r},\Psi_{r})]$, namely
$\operatorname{\text{{\scriptsize h}}}_{r}$. Now, we define
$\operatorname{\text{{\scriptsize h}}}_{0,r}$ to be the relative homotopy
class of $[(A_{0,r},\Psi_{0,r})]$, which is in turned given as follows: Define
$(A_{0,r},\Psi_{0,r})\in\mathop{\mathrm{Conn}}\nolimits(X)\times\Gamma({\mathbb{S}}_{X}^{+})$
by setting $(A_{0,r},\Psi_{0,r})=(\tilde{A}_{r},\tilde{\Psi}_{r})$ over
$X^{+}_{\vec{L}_{T}(r)-\vec{3/2}}:=X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{3/2}}\cup(X-X^{{}^{\prime}a})$,
and then extending over $X$ by the $s$-independent configurations
$(\hat{B}^{i}_{0,r},\hat{\Phi}^{i}_{0,r})$, $i\in{\mathfrak{Y}}_{m}^{\prime}$,
over the complement of $X^{+}_{\vec{L}_{T}(r)-\vec{3/2}}$.

Similarly, when $i\in{\mathfrak{Y}}^{\prime}_{m}$,
$\operatorname{\text{{\scriptsize h}}}^{r}_{i,k}$ is defined to be the
relative homotopy class of $[(A_{k,r}^{i},\Psi_{k,r}^{i})]$, where
$(A_{k,r}^{i},\Psi_{k,r}^{i})\in\in\mathop{\mathrm{Conn}}\nolimits({\mathbb{R}}\times
Y_{i})\times\Gamma({\mathbb{S}}_{{\mathbb{R}}\times Y_{i}}^{+})$ is given by
setting
$(A_{k,r}^{i},\Psi_{k,r}^{i})=\tau_{-L^{i}_{k,r}}(\tilde{A}_{r},\tilde{\Psi}_{r})$
over $J^{i}_{k,r}\times Y_{i}$, and then extending over ${\mathbb{R}}\times
Y_{i}$ by the $s$-independent configurations over the complement, namely
$\hat{B}^{i}_{k-1,r}$ and $\hat{B}^{i}_{k,r}$ respectively over the
$-\infty$-end and the $+\infty$-end. With $\operatorname{\text{{\scriptsize
h}}}^{r}_{i,k}$, $\operatorname{\text{{\scriptsize h}}}_{0,r}$ so defined, the
identities in (7.36), (7.37) hold by construction.

Let $\\{\mu_{j}\\}_{j}$ be a set of closed 2-forms on $X$ that represents a
basis of $H^{2}(X;{\mathbb{R}})$. We choose $\mu_{j}$ so that for every $j$,

  * •

$\mu_{j}|_{\hat{Y}_{i}}=\pi_{2}^{*}\mu_{i,j}$ for a closed 2-forn $\mu_{i,j}$
on $Y_{i}$ for all $i\in{\mathfrak{Y}}$.

  * •

When $i\in{\mathfrak{Y}}_{v}$, $\mu_{i,j}=0$. This is possible because of the
condition (1.6).

  * •

When $i\in{\mathfrak{Y}}_{m}$, $\mu_{i,j}$ is supported on the complement of
$\bigcup_{\boldsymbol{\gamma}\in{\mathbb{P}}({\mathfrak{s}}_{i})}\gamma\subset
Y_{i}$. This is possible because ${\mathbb{P}}({\mathfrak{s}}_{i})$ consists
of finitely many elements, and the closure of each $\boldsymbol{\gamma}$ is a
compact 1-manifold with (possibly empty) boundary.

To verify the first identity in (7.38), it suffices to verify that for every
$j$, the pairing

(7.39)
$\big{\langle}[\mu_{j}],{\mathfrak{h}}^{\prime}(\operatorname{\text{{\scriptsize
h}}}_{0,r})-(i_{\infty})^{-1}[{\bf C}_{0}]\big{\rangle}=0,\quad r>R_{2}.$

Given (7.35), the left hand side of the preceding expression equals

$\begin{split}&\int_{-1/2}^{1/2}\Big{(}\int_{X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{3/2}+\vec{t}}}\frac{iF_{A_{0,r}^{E}}}{2\pi}\wedge\mu_{j}-\int_{\tilde{C}_{0}|_{X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{3/2}+\vec{t}}}}\mu_{j}\\\
&\qquad-\sum_{i\in{\mathfrak{Y}}^{\prime}_{m}}\Big{(}\int_{\tilde{C}_{0}|_{\hat{Y}_{i,L_{T}(r)-3/2+t}}}\pi_{2}^{*}\mu_{i,j}-\lim_{r^{\prime}\to\infty}\int_{Y_{i}}\big{(}\frac{i}{2\pi}(B_{0,r^{\prime}}^{E}-B_{0,r}^{E})\big{)}\wedge\mu_{i,j}\Big{)}\Big{)}dt\\\
&=\int_{-1/2}^{1/2}\Big{(}\int_{X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{3/2}+\vec{t}}}\frac{iF_{A_{r}^{E}}}{2\pi}\wedge\mu_{j}-\int_{\tilde{C}_{0}|_{X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{3/2}+\vec{t}}}}\mu_{j}\\\
&\qquad\quad-\sum_{i\in{\mathfrak{Y}}^{\prime}_{m}}\Big{(}\int_{\tilde{C}_{0}|_{\hat{Y}_{i,L_{T}(r)-3/2+t}}}\pi_{2}^{*}\mu_{i,j}-\lim_{r^{\prime}\to\infty}\int_{Y_{i}}\big{(}\frac{i}{2\pi}(B_{0,r^{\prime}}^{E}-B_{0,r}^{E})\big{)}\wedge\mu_{i,j}\Big{)}\Big{)}dt,\end{split}$

and is independent of $r$. However, according to (7.13), the fact that ${\bf
C}_{0}$ is asymptotic to a t-orbit, and the convergence of $[B_{0,r}]$ in
current topology, the preceding expression converges to 0 as $r\to\infty$. We
have thus confirmed the first identity in (7.38).

The second identity in (7.38) for each given $i,k$ is verified similarly, with
the roles of $A_{0,r}$, ${\bf C}_{0}$,
$X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{3/2}}$ above respectively replaced by
$A_{k,r}^{i}$, ${\bf C}_{i,k}$, and $J^{i}_{k,r}\times Y_{i}$, and with the
role of (7.13) in the preceding argument now played by (7.28). $\Box$

Finally, we prove Lemma 3.23(b).

##### Proof of Lemma 3.23(b).

Recall that the currents $\\{\tilde{\gamma}_{i}\\}_{i\in{\mathfrak{Y}}_{m}}$
are integral, and
${\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$
embeds in
${\mathscr{H}}^{\mathbb{R}}\big{(}(X^{{}^{\prime}a},\nu),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$
as an orbit of the ${\mathscr{H}}_{X}$-action. The map
${\mathfrak{h}}=i_{\infty}\circ{\mathfrak{h}}^{\prime}$ maps
$\pi_{0}{\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i,r}\\}_{i\in{\mathfrak{Y}}})\simeq\pi_{0}{\mathscr{B}}_{X^{{}^{\prime}a}}(\\{{\mathfrak{c}}_{i,r}\\}_{i\in{\mathfrak{Y}}_{m}})$
to another orbit of the ${\mathscr{H}}_{X}$-action in
${\mathscr{H}}^{\mathbb{R}}\big{(}(X^{{}^{\prime}a},\nu),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$.
We claim that these two orbits are identical. To see this, note that by
Theorem 1.9 (a), there is a subsequence, also denoted
$\\{(A_{r},\Psi_{r})\\}_{r}$, which weakly converges to a chain of t-curves,
${\mathfrak{C}}$. The associated current $\tilde{{\mathfrak{C}}}$ is integral,
and has relative homology class in
${\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}\subset{\mathscr{H}}^{\mathbb{R}}\big{(}(X^{{}^{\prime}a},\nu),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$.
Meanwhile, $[{\mathfrak{C}}]={\mathfrak{h}}(\operatorname{\text{{\scriptsize
h}}})$ is in the image of ${\mathfrak{h}}$. This implies that the
${\mathscr{H}}_{X}$-orbits
${\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$
and
${\mathfrak{h}}\,(\pi_{0}{\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i,r}\\}_{i\in{\mathfrak{Y}}}))$
in
${\mathscr{H}}^{\mathbb{R}}\big{(}(X^{{}^{\prime}a},\nu),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$
are identical. $\Box$

##### Proof of Proposition 1.12, the non-cylindrical case.

Adopt the notations from the proof of Theorem 1.9 above. By the assumption
(1.8), $C_{0}\cap P$, $C_{i,k}\cap({\mathbb{R}}\times\textsl{p}_{i})$ $\forall
i,k$ all consist of finitely many points. Thus, there exists an $R_{3}\geq
R_{2}$ such that for all $r\geq R_{3}$:

  * •

$C_{0}\cap P$ lies in the interior of
$X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{3}}$;

  * •

$C_{i,k}\cap({\mathbb{R}}\times\textsl{p}_{i})$ lies in the interior of
$J^{i}_{k,r}\times Y_{i}$ for all $i$, $k$.

  * •

For all $i\in{\mathfrak{Y}}_{m}$, $\operatorname{\text{{\small
d}}}_{\hat{Y}_{i,I_{*}}}(P,{\mathbb{R}}\times\textsl{p}_{i})<\epsilon/4$ for
all $I_{*}\subset[L_{T}(r)-5/2,\infty)$ with length $|I_{*}|=2$.

Let $\tilde{P}\subset X^{{}^{\prime}a}$ be a subvariety such that:

  * •

$\tilde{P}$ agrees with $P$ over $X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{3}}$;

  * •

it agrees with ${\mathbb{R}}\times\textsl{p}_{i}$ over
$\hat{Y}_{i,L_{T}(r)-5/2}$ $\forall i\in{\mathfrak{Y}}_{m}$, and

  * •

$\operatorname{\text{{\small
d}}}_{\hat{Y}_{Y_{i,[L_{T}(r)-3,L_{T}(r)-5/2]}}}(P,\tilde{P})+\operatorname{\text{{\small
d}}}_{\hat{Y}_{Y_{i,[L_{T}(r)-3,L_{T}(r)-5/2]}}}(\tilde{P},{\mathbb{R}}\times\textsl{p}_{i})\leq\epsilon$.

  * •

$P-\tilde{P}$ is a boundary.

Write

$\begin{split}\int_{P}\frac{i}{2\pi}F_{A_{r}^{E}}&=\int_{\tilde{P}\cap
X^{{}^{\prime}a}_{\vec{L}_{T}(r)-\vec{\frac{5}{2}}}}\frac{i}{2\pi}F_{A_{r}^{E}}+\sum_{i\in{\mathfrak{Y}}^{\prime}_{m}}\sum_{k=1}^{{\mathfrak{K}}_{i}}\int_{\tilde{P}\cap\hat{Y}_{i,J^{{}^{\prime}i}_{k,r}}}\frac{i}{2\pi}F_{A_{r}^{E}}\\\
\end{split}$

Taking the $r\to\infty$ limit, by (7.13) and (7.28) we then have

$\lim_{r\to\infty}\int_{P}\frac{i}{2\pi}F_{A_{r}^{E}}=\\#\,(P\cap{\bf
C}_{0})+\sum_{i\in{\mathfrak{Y}}^{\prime}_{m}}\sum_{k=1}^{{\mathfrak{K}}_{i}}\\#\,\big{(}({\mathbb{R}}\times\textsl{p}_{i})\cap{\bf
C}_{i,k}\big{)}.$

Now observe that each term on the right hand side is a non-negative integer,
since $P$, ${\mathbb{R}}\times\textsl{p}_{i}$, $C_{0}$, $C_{i,k}$ are all
pseudo-holomorphic. $\Box$

###  7.4 Proving Theorem 1.9 and Proposition 1.12, the cylindrical case

The proofs require only minor modifications of the proofs in the non-
cylindrical case.

##### Proof of Theorem 1.9, the cylindrical case.

Begin as with the proof of the noncylindrical case in the previous subsection:
Rename the t-curve ${\bf C}$ in the statement of Theorem 1.7 as ${\bf C}_{0}$.
That is, $\\{(A_{r},\Psi_{r})\\}_{r\in\Gamma_{0}}$ t-converges to ${\bf
C}_{0}$ over $X={\mathbb{R}}\times Y$. As noted in Remark 1.6, in this case we
may regard $X$ as an MCE with two ends:
${\mathfrak{Y}}={\mathfrak{Y}}_{m}=\\{\pm\\}$ consists two elements, with the
elements $+/-$ labelling respectively the $+\infty$/$-\infty$-ends; their
corresponding ending 3-manifolds are $Y_{\pm}=\pm Y$. Rerunning the arguments
in the previous subsection, we get a subsequence $\Gamma^{\prime}$ and two
chains of t-curves ${\mathfrak{C}}_{\pm}$ on ${\mathbb{R}}\times(\pm Y)$,
where ${\mathfrak{C}}_{\pm}$ has $-\infty$-limit
$\pm\boldsymbol{\gamma}_{0,\pm}$ and $+\infty$-limit
$\pm\boldsymbol{\gamma}_{\pm}$. Let
$\iota\colon\thinspace{\mathbb{R}}\times(-Y)\to{\mathbb{R}}\times Y$ denote
the orientation preserving map $(s,x)\mapsto(-s,x)$. Write

${\mathfrak{C}}=\begin{cases}\\{\iota({\bf
C}_{-,{\mathfrak{K}}_{-}}),\iota({\bf
C}_{-,{\mathfrak{K}}_{-}-1}),\ldots,\iota({\bf C}_{-,1}),{\bf C}_{0},{\bf
C}_{+,1},\ldots,{\bf C}_{+,{\mathfrak{K}}_{+}}\\}&\text{when
${\mathfrak{K}}_{-}>0$; ${\mathfrak{K}}_{+}>0$};\\\ \\{\iota({\bf
C}_{-,{\mathfrak{K}}_{-}}),\iota({\bf
C}_{-,{\mathfrak{K}}_{-}-1}),\ldots,\iota({\bf C}_{-,1}),{\bf
C}_{0}\\}&\text{when ${\mathfrak{K}}_{-}>0$; ${\mathfrak{K}}_{+}=0$};\\\
\\{{\bf C}_{0},{\bf C}_{+,1},\ldots,{\bf
C}_{+,{\mathfrak{K}}_{+}}\\}&\text{when ${\mathfrak{K}}_{-}=0$;
${\mathfrak{K}}_{+}>0$};\\\ \\{{\bf C}_{0}\\}&\text{when
${\mathfrak{K}}_{-}=0$; ${\mathfrak{K}}_{+}=0$}\end{cases}$

if ${\bf C}_{0}$ is non-constant; and when ${\bf C}_{0}$ is constant, let

${\mathfrak{C}}=\begin{cases}\\{\iota({\bf
C}_{-,{\mathfrak{K}}_{-}}),\iota({\bf
C}_{-,{\mathfrak{K}}_{-}-1}),\ldots,\iota({\bf C}_{-,1}){\bf
C}_{+,1},\ldots,{\bf C}_{+,{\mathfrak{K}}_{+}}\\}&\text{when
${\mathfrak{K}}_{-}>0$; ${\mathfrak{K}}_{+}>0$};\\\ \\{\iota({\bf
C}_{-,{\mathfrak{K}}_{-}}),\iota({\bf
C}_{-,{\mathfrak{K}}_{-}-1}),\ldots,\iota({\bf C}_{-,1})\\}&\text{when
${\mathfrak{K}}_{-}>0$; ${\mathfrak{K}}_{+}=0$};\\\ \\{{\bf
C}_{+,1},\ldots,{\bf C}_{+,{\mathfrak{K}}_{+}}\\}&\text{when
${\mathfrak{K}}_{-}=0$; ${\mathfrak{K}}_{+}>0$}.\\\ \end{cases}$

Lastly, when ${\bf C}_{0}$ is constant and ${\mathfrak{K}}_{\pm}=0$, let
${\mathfrak{C}}$ be the 0-component chain of t-curves with a single rest point
$\boldsymbol{\gamma}_{+}$. The arguments from the previous subsections then
shows that $\\{(A_{r},\Psi_{r})\\}_{r\in\Gamma^{\prime}}$ weakly t-converges
to ${\mathfrak{C}}$, which have the properties described in Items (b)-(d) in
the statement of the theorem. (Note that when $(Y,\nu)$ is cylindrical, the
conditions in Items (c) are (d) are both met.) The case when ${\bf C}_{0}$ is
constant and ${\mathfrak{K}}_{\pm}=0$ occurs only when
$\\{(A_{r},\Psi_{r})\\}_{r\in\Gamma_{0}}$ is such that there exists $r_{0}>1$,
such that $\forall r\geq r_{0}$, $(A_{r},\Psi_{r})$ is constant. This means
that $(A_{r},\Psi_{r})=(\hat{B}_{r},\hat{\Phi}_{r})$, where $(B_{r},\Phi_{r})$
strongly t-converges to $\boldsymbol{\gamma}_{-}=\boldsymbol{\gamma}_{+}$ by
assumption. In this case, ${\mathfrak{C}}$ is the 0-component chain of
t-curves with a single rest point
$\boldsymbol{\gamma}_{-}=\boldsymbol{\gamma}_{+}$. $\Box$

##### Proof of Proposition 1.12, the cylindrical case.

Let ${\bf C}_{0}$, ${\mathfrak{C}}$ be as in the proof of the cylindrical case
of Theorem 1.9 above. Then simply repeat the arguments in the proof for the
noncylindrical case of Proposition 1.12 in the previous subsection. $\Box$

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