j<k+3$. Second, instead of appealing to Condition (ii) of Definition 10.5.1 as
in [KM]’s proof of its Theorems 10.7.1, one makes use of its parametrized
generalization established in [KM] Theorems 11.6.1 (iii)-(iv).) Let
${\mathscr{P}}(Y;\delta)\subset{\mathscr{P}}(Y)$ denote the ball in
${\mathscr{P}}(Y)$ of radius $\delta$ centered at 0, and let
$N_{\mathscr{V}}(\delta)$ denote the tubular neighborhood of the zero-section
of ${\mathscr{V}}_{k+3}(Z_{\epsilon})$ whose intersection with the fiber of
${\mathscr{V}}_{k+3}(Z_{\epsilon})$ over any given
$(A,\Psi)\in{\mathscr{C}}_{k+3}(Z_{\epsilon})$, namely $L^{2}_{k+3,A}$, is a
ball of radius $\delta$. By the Sobolev embedding theorem, there is a constant
$\zeta>0$ such that
$\|\eta\|_{C^{k}_{A}(Z)}\leq\zeta\|\eta\|_{L^{2}_{k+3,A}(Z_{\epsilon})}$.
According to Fact 3.3 (2), there is an
$\textsl{z}^{\prime}(r)\in{\mathbb{R}}^{+}$ such that
${\mathscr{P}}(Y;\textsl{z}^{\prime})\times(\tilde{{\mathscr{M}}}_{r}\cup\tilde{{\mathscr{M}}}^{\prime}_{r})\subset{\mathfrak{Q}}^{-1}N_{\mathscr{V}}(\zeta^{-1}\zeta_{\mathfrak{p}})$.
We require that $2\|\hat{{\mathfrak{p}}}_{r}\|_{\mathscr{P}}$ is smaller than
every $i\in{\mathfrak{Y}}_{v}$’s version of $\textsl{z}^{\prime}(r)$.

(c) See Remark 3.13 and [KLT1]-[KLT5] for motivations of the condition Item
(2) in the preceding assumption.

######  Remarks 3.6.

(a) The notions ${\mathscr{C}}(M)$, ${\mathscr{B}}(M)$ are defined for
possibly disconnected 3- or 4-manifold $M$. The formula (3.2) for
$\operatorname{CSD}_{\mu}$ also applies when the 3-manifold $Y$ is
disconnected. Though in [KM] the Banach space of large tame perturbations
${\mathscr{P}}$ is defined for connected 3-manifolds, it generalizes readily
to the disconnected case: Suppose $M=\bigcup_{k}M_{k}$ is a closed
$\mathop{\mathrm{Spin}}\nolimits^{c}$ 3-manifold with connected components
$M_{k}$. Take ${\mathscr{P}}(M):=\prod_{k}{\mathscr{P}}(M_{k})$. Then each
${\mathfrak{q}}=({\mathfrak{q}}_{1},\ldots,{\mathfrak{q}}_{k},\ldots)\in\prod_{k}{\mathscr{P}}(M_{k})={\mathscr{P}}(M)$
defines a section of the tangent bundle
$T{\mathscr{C}}(M)=\boxtimes_{k}T{\mathscr{C}}(M_{k})$ over
${\mathscr{C}}(M)=\prod_{k}{\mathscr{C}}(M_{k})$: given
${\mathfrak{c}}=({\mathfrak{c}}_{1},\ldots,{\mathfrak{c}}_{k},\ldots)\in\prod_{k}{\mathscr{C}}(M_{k})={\mathscr{C}}(M)$,
${\mathfrak{q}}({\mathfrak{c}}):=({\mathfrak{q}}_{1}({\mathfrak{c}}_{1}),,\ldots,{\mathfrak{q}}_{k}({\mathfrak{c}}_{k}),\ldots)\in\boxtimes_{k}T{\mathscr{C}}(M_{k})=T{\mathscr{C}}(M)$.
As before, given a reference connection $B_{0}\in{\mathscr{C}}(M)$, the
function
$f_{\mathfrak{q}}\colon\thinspace{\mathscr{C}}(M)=\prod_{k}{\mathscr{C}}(M_{k})\to{\mathbb{R}}$
is defined to be the unique gauge-invariant function with formal gradient
${\mathfrak{q}}$ and with $f_{\mathfrak{q}}(B_{0},0)=0$. (So
$f_{\mathfrak{q}}({\mathfrak{c}})=\sum_{k}f_{{\mathfrak{q}}_{k}}({\mathfrak{c}}_{k})$.)
The properties (1)-(2) in Fact 3.3 remain valid for such generalized
${\mathfrak{q}}$ when $M$ is compact. In this way, the definitions of
${\mathfrak{F}}_{\mu,{\mathfrak{q}}}$,
$\operatorname{CSD}_{\mu,{\mathfrak{q}}}$ both generalize to the case of
disconnected manifolds.

(b) Let $X$ be a MCE with ending 3-manifolds $Y_{i}$, $i\in{\mathfrak{Y}}$ and
let $\mu$ be a closed 2-form on $X$ with the closed 2-form $\mu_{i}$ on $Y$ as
it $Y_{i}$-end limit for each $i$. Let $\hat{{\mathfrak{p}}}$ be a nonlocal
perturbation of the type given in Equation (24.2) of [KM]. For our
applications, the relevant disconnected 3-manifold $M=\bigcup_{k}M_{k}$
usually consists of slices of $X$ of the form $M_{k}=Y_{i_{k}:s_{k}}$. When
the 3-manifold $M$ under discussion is clear from the context, we often
abbreviate the restriction of $\mu$ to $M$ as $\mu\big{|}=\mu\big{|}_{M}$. For
any such $M$, the nonlocal perturbation $\hat{{\mathfrak{p}}}$ on $X$ also
induces an element in the Banach space of large tame perturbations, denoted
$\hat{{\mathfrak{p}}}\big{|}_{M}$ or simply
$\hat{{\mathfrak{p}}}\big{|}\in{\mathscr{P}}(M)$: In the notation of (3.6),
$\hat{{\mathfrak{p}}}|_{Y_{i:s}}=\chi_{i}(s)\hat{{\mathfrak{q}}}_{i}+\lambda_{i}(s)\hat{{\mathfrak{p}}}^{\prime}_{i}\in{\mathscr{P}}(Y_{i})$.
To make the notations less cumbersome, when $M$ is understood from the
context, $f_{\hat{{\mathfrak{p}}}|}$,
$\operatorname{CSD}_{\mu|,\hat{{\mathfrak{p}}}|}^{M}$ are also abbreviated
respectively as $f_{\hat{{\mathfrak{p}}}}$,
$\operatorname{CSD}_{\mu,\hat{{\mathfrak{p}}}}$.

###  3.2 $Y_{i}$-end limits of admissible Seiberg-Witten solutions

We begin with some general definitions and observations.

######  Definition 3.7.

Let $Y$ be a closed $\mathop{\mathrm{Spin}}\nolimits^{c}$ 3-manifold and
$B_{0}\in\mathop{\mathrm{Conn}}\nolimits(Y)$ be a reference connection as in
(3.2). A connection $B\in\mathop{\mathrm{Conn}}\nolimits(Y)$ is said to be in
a Coulomb gauge (with respect to $B_{0}$) if the following holds:

(3.9) $d^{*}(\delta B)=0,\quad\text{where $\delta B=B-B_{0}$.}$

An element $(B,\Phi)\in{\mathscr{C}}(Y)$ is said to be in a Coulomb gauge
(with respect to $B_{0}$) if $B$ is in a Coulomb gauge.

Fix an orthonormal basis $\\{[h_{k}]\\}_{k}$ of $H^{1}(Y;{\mathbb{Z}})$, where
the latter space is equipped with the inner product given by the Riemannian
metric on $Y$. Let $h_{k}\in\Omega^{1}(Y)$ denote the harmonic representative
of $[h_{k}]$. We say that a $B\in\mathop{\mathrm{Conn}}\nolimits(Y)$ or a
$(B,\Phi)\in{\mathscr{C}}(Y)$ is in the normalized Coulomb gauge (with respect
to $B_{0}$ and the basis $\\{[h_{k}]\\}_{k}$) if in addition to (3.9), it also
satisfies

(3.10) $\text{$0\leq\Big{\langle}h_{k},\frac{i}{4\pi}\delta B\Big{\rangle}<1$
$\forall k$}.$

For each gauge equivalence class
$[(B,\Phi)]\in\big{(}\mathop{\mathrm{Conn}}\nolimits({\mathbb{S}}_{Y})\times\Gamma({\mathbb{S}}_{Y})\big{)}/{\mathscr{G}}$,
there is a unique representative in the normalized Coulomb gauge. We usually
denote this particular representative as $[(B,\Phi)]_{c}\in{\mathscr{C}}(Y)$.

Working in the normalized Coulomb gauge has the following advantage: If $B$
(or $(B,\Phi)$ is in a normalized Coulomb gauge, then by a Gårding inequality,

(3.11) $\|\delta B\|_{L^{2}}\leq\zeta_{b}\big{(}\|(d+d^{*})\delta
B\|_{L^{2}}+1\big{)}=\zeta_{b}\big{(}\|F_{B}-F_{B_{0}})\|_{L^{2}}+1\big{)}$

for a positive constant $\zeta_{b}$ depending only on the metric on $Y$. We
shall need the following simple consequence of the preceding observations:

######  Lemma 3.8.

Let $Y$ be closed $\mathop{\mathrm{Spin}}\nolimits^{c}$ 3-manifold $Y$, and
fix a reference connection $B_{0}\in\mathop{\mathrm{Conn}}\nolimits(Y)$.
Recall that $f_{\mathfrak{q}}$ denotes the gauge-invariant function on
${\mathscr{C}}(Y)$ with formal gradient ${\mathfrak{q}}\in{\mathscr{P}}(Y)$
and with $f_{\mathfrak{q}}(B_{0},0)=0$. There is a constant $\zeta$ depending
only on the metric of $Y$ such that such that

$|f_{\mathfrak{q}}(B,\Phi)|\leq\zeta\|{\mathfrak{q}}\|_{\mathscr{P}}\,\big{(}\|F_{B}-F_{B_{0}}\|^{2}_{L^{2}}+\|\Phi\|^{2}_{L^{2}}+1\big{)}\quad\text{$\forall$
${\mathfrak{q}}\in{\mathscr{P}}(Y)$ and $(B,\Phi)\in{\mathscr{C}}(Y)$.}$

Proof. Since $f_{\mathfrak{q}}$ is gauge-invariant, it suffices to consider
the case when $(B,\Phi)$ is in the normalized Coulomb gauge. Write $\delta
B:=B-B_{0}$ as before and let $(B_{t},\Phi_{t}):=(B_{0}+t\delta B,t\Phi)$ for
$t\in[0,1]$. Then according to (3.11) and (3.4),

(3.12) $\begin{split}|f_{\mathfrak{q}}(B,\Phi)|&\leq\|(\delta
B,\Phi)\|_{L^{2}}\|{\mathfrak{q}}(B_{t},\Phi_{t})\|_{L^{2}}\quad\text{for a
certain $t\in[0,1]$}\\\ &\leq
m\|{\mathfrak{q}}\|_{\mathscr{P}}\big{(}\zeta_{b}\|F_{B}-F_{B_{0}}\|_{L^{2}}+\zeta_{b}+\|\Phi\|_{L^{2}}\big{)}(\|\Phi_{t}\|_{L^{2}}+1)\\\
&\leq\zeta\|{\mathfrak{q}}\|_{\mathscr{P}}\,\big{(}\|F_{B}-F_{B_{0}}\|^{2}_{L^{2}}+\|\Phi\|^{2}_{L^{2}}+1\big{)}.\end{split}$

$\Box$

Returning now to the setting of this article: Let $(X,\nu)$ be an admissible
pair and let $(Y_{i},\nu_{i})$ denote the limiting pair of $(X,\nu)$ of the
$Y_{i}$-end. Let $\mu_{r}$, $\hat{{\mathfrak{p}}}_{r}$ be as in the statement
of Theorem 1.7, and write $\mu_{i,r}=r\nu_{i}+w_{i,r}\in\Gamma(Y_{i})$ for the
$Y_{i}$-end limit of $\mu_{r}$. As previously mentioned, we frequently drop
the subscript $r$ when $Y_{i}$ is a vanishing end, since in this case
$\mu_{i,r}=w_{i}$ is independent of $r$. Let $(A,\Psi)=(A_{r},\Psi_{r})$ be an
admissible solution to the Seiberg-Witten equations
${\mathfrak{S}}_{\mu_{r},\hat{{\mathfrak{p}}}_{r}}(A,\Psi)=0$. The standard
compactness/properness results in Seiberg-Witten theory (see e.g. §10.7 in
[KM]) imply that the $Y_{i}$-end limit of $(A,\Psi)$ is a solution to the
3-dimensional Seiberg-Witten equations
${\mathfrak{F}}_{\mu_{i,r},{\mathfrak{q}}_{i}}(B_{i},\Phi_{i})=0$. Moreover,
$(A,\Psi)\big{|}_{Y_{i:L}}$ converges to $(B_{i},\Phi_{i})$ in $C^{k}(Y_{i})$
topology as $L\to\infty$. In the remainder of this subsection, we gather some
facts about such 3-dimensional Seiberg-Witten solutions that will be of use
later. In what follows, all statements concern only the pair of 3-manifold and
harmonic 2-form $(Y_{i},\nu_{i})$, the closed 2-form and the nonlocal
perturbation, $w_{i,r}$ and ${\mathfrak{q}}_{i,r}$, over $Y_{i}$; they are not
necessarily ending pairs and $Y_{i}$-end limits of a 4-dimensional $(X,\nu)$,
$w_{r}$, $\hat{{\mathfrak{p}}}_{r}$. The expression $i\in{\mathfrak{Y}}_{v}$
or $i\in{\mathfrak{Y}}_{m}$ only serves to indicate whether $\nu_{i}\equiv 0$
or otherwise. $w_{i,r}$ and ${\mathfrak{q}}_{i,r}$ are assumed to satisfy the
uniform bounds $\|w_{i,r}\|_{C^{k+3}}\leq\varsigma_{w}$ and
$\|{\mathfrak{q}}_{i,r}\|_{\mathscr{P}}\leq\textsl{z}_{\mathfrak{p}}$ for an
integer $k\geq 2$. For simplicity we also assume that the cohomology class
$[w_{i,r}]$ is independent of $r$, and that ${\mathfrak{q}}_{i,r}=0$ when
$i\in{\mathfrak{Y}}_{m}$, as in Assumption 3.4. (The preceding conditions hold
if $w_{i,r}$ and ${\mathfrak{q}}_{i,r}$ arise as the $Y_{i}$-end limits of
certain 4-dimensional $w_{r}$ and $\hat{{\mathfrak{p}}}_{r}$ that satisfy
Assumption 3.4.) The constants mentioned below will depend on $w_{i,r}$,
${\mathfrak{q}}_{i,r}$ only through $\varsigma_{w}$,
$\textsl{z}_{\mathfrak{p}}$, and $[w_{i,r}]$, not on the precise form of
$w_{i,r}$, ${\mathfrak{q}}_{i,r}$.

The next lemma is more or less standard, and is usually the first step in
typical proofs of compactness results in Seiberg-Witten theory. A short sketch
of proof is nevertheless provided, both in order to keep track of the
$r$-dependence of various coefficients, and because some formulas appearing
therein will be used again later.

######  Lemma 3.9.

Let $(Y_{i},\nu_{i})$, ${\mathfrak{q}}_{i}={\mathfrak{q}}_{i,r}$,
$i\in{\mathfrak{Y}}$ $\varsigma_{w}$, $\textsl{z}_{\mathfrak{p}}$ be as
described above, and define

${\rm a}_{i}:=\begin{cases}0&\text{when $i\in{\mathfrak{Y}}_{v}$;}\\\ 1&\text{
when $i\in{\mathfrak{Y}}_{m}$. }\end{cases}$

Let $\mu_{i,r}=r\nu_{i}+w_{i,r}$ and let
$(B_{i},\Phi_{i})=(B_{i,r},\Phi_{i,r})$ be a solution to the 3-dimensional
Seiberg-Witten equation
${\mathfrak{F}}_{\mu_{i,r},{\mathfrak{q}}_{i}}(B_{i},\Phi_{i})=0$. There exist
an $r$-independent constants $\zeta_{i}\in{\mathbb{R}}^{+}$ such that

(3.13) $\|\Phi_{i}\|_{L^{\infty}}\leq\zeta_{i}\,r^{{\rm a}_{i}/2}.$

Let $k\geq 2$ be as in Assumption 3.4. In the case when
$i\in{\mathfrak{Y}}_{v}$ and $(B_{i},\Phi_{i})$ is in the normalized Coulomb
gauge with respect to a reference connection $B_{0,i}$, $\eta:=(\delta
B_{i}=B_{i}-B_{0,i},\Phi_{i})$ satisfies

$\|\eta\|_{L^{2}_{k+4,B_{0,i}}}+\|\eta\|_{C^{k+2}_{B_{0,i}}}+\|\Phi_{i}\|_{L^{4}}\leq\zeta^{\prime}_{i}$

for some $r$-independent positive constant $\zeta^{\prime}_{i}$. The constant
$\zeta_{i}$ is determined by the metric on $Y_{i}$ and the norms of the
perturbations via $\|\nu_{i}\|_{C^{2}}$,
$\|w_{i,r}\|_{C^{2}}\leq\varsigma_{w}$, and
$\|{\mathfrak{q}}_{i}\|_{{\mathscr{P}}}\leq\textsl{z}_{\mathfrak{p}}$. The
constant $\zeta^{\prime}_{i}$ is determined by all the above, plus $B_{0,i}$.

Proof. The proof hinges on the 3-dimensional Weitzenböck formula

(3.14) $\mbox{\mbox{$\partial\mkern-7.5mu\mbox{/}\mkern
2.0mu$}}_{B_{i}}^{2}\Phi_{i}=\nabla_{B_{i}}^{*}\nabla_{B_{i}}\Phi_{i}+\frac{R_{g}}{4}\Phi_{i}+\frac{1}{2}\rho(F_{B_{i}})\Phi_{i},$

where $R_{g}$ denotes the scalar curvature. Applying this formula together
with (3.4) to the identity
$\|{\mathfrak{F}}_{\mu_{i,r}}(B_{i},\Phi_{i})\|_{L^{2}}^{2}=\|{\mathfrak{q}}_{i}(B_{i},\Phi_{i})\|_{L^{2}}^{2}$,
one has:

$\begin{split}&\frac{1}{4}\|F_{B_{i}}\|^{2}_{L^{2}}+\|\nabla_{B_{i}}\Phi_{i}\|^{2}_{L^{2}}+\big{\|}\frac{i}{4}\rho(\mu_{i,r})-(\Phi_{i}\Phi_{i}^{*})_{0}\big{\|}^{2}_{L^{2}}\\\
&\qquad+\int_{Y_{i}}\frac{R_{g}}{4}|\Phi_{i}|^{2}-\frac{i}{4}\int_{\hat{Y}_{[l,L]}}F_{B_{i}}\wedge*\mu_{i,r}\leq
4m^{2}\|{\mathfrak{q}}_{i}\|_{\mathscr{P}}^{2}\,\big{(}1+\|\Phi_{i}\|^{2}_{L^{2}}\big{)}.\\\
\end{split}$

Judicious applications of the triangle inequality then leads to

$\begin{split}\frac{1}{8}\|F_{B_{i}}\|^{2}_{L^{2}}+\|\nabla_{B_{i}}\Phi_{i}\|^{2}_{L^{2}}&+\frac{1}{2}\big{\|}\frac{i}{4}\rho(\mu_{i,r})-(\Phi_{i}\Phi_{i}^{*})_{0}\big{\|}^{2}_{L^{2}}\\\
&\leq\zeta\big{(}\|\mu_{i,r}\|_{C^{0}}^{2}+\|R_{g}\|_{C^{0}}^{2}+m^{4}\|{\mathfrak{q}}_{i}\|_{\mathscr{P}}^{4}\big{)}+\zeta^{\prime},\end{split}$

where $\zeta$, $\zeta^{\prime}$ are constants independent of anything. (For
more details, see (4.18) and subsequent discussions for 4-dimensional
analogs.)

According to Assumption 3.4, when $i\in{\mathfrak{Y}}_{v}$,
$\|\mu_{i,r}\|_{C^{0}}=\|w_{i,r}\|_{C^{0}}\leq\varsigma_{w}$ and
$\|{\mathfrak{q}}_{i}\|_{\mathscr{P}}\leq\textsl{z}_{\mathfrak{p}}$, and so it
follows that

$\|F_{B_{i}}\|^{2}_{L^{2}}+\|\nabla_{B_{i}}\Phi_{i}\|^{2}_{L^{2}}+\|\Phi_{i}\|_{L^{4}}^{4}\leq\zeta_{v},$

where $\zeta_{v}$ is an $r$-independent constant determined by the metric,
$\varsigma_{w}$ and $\textsl{z}_{\mathfrak{p}}$. With the preceding bound at
hand, a standard elliptic bootstrapping argument (see e.g. the proofs of
[KM]’s Lemma 5.1.2 and Theorem 10.7.1 for the 4-dimensional version) then
leads to

$\|\eta\|_{L^{2}_{k+4,B_{0,i}}}+\|\Phi_{i}\|_{C^{k+2}_{B_{0,i}}}\leq\zeta_{v}^{\prime},$

where $\zeta_{v}^{\prime}$ is a constant determined by the metric, $B_{0}$,
$\varsigma_{w}$ and $\textsl{z}_{\mathfrak{p}}$. Together with the preceding
bound on $\|\Phi_{i}\|_{L^{4}}$, this verified the assertions of the lemma in
the $i\in{\mathfrak{Y}}_{v}$ case.

In the case when $i\in{\mathfrak{Y}}_{m}$, ${\mathfrak{q}}_{i}=0$ and
$\|\mu_{i,r}\|_{C^{2}}\leq r\|\nu_{i}\|_{C^{2}}+\varsigma_{w}$. The preceding
argument provides an $r$-dependent bound on $\|\Phi_{i}\|_{L^{\infty}}$ that
grows faster with $r$ than what we need. Meanwhile, in this case the right
hand side of (3.14) vanishes by the second line of the Seiberg-Witten equation
${\mathfrak{F}}_{\mu_{i,r}}(B_{i},\Phi_{i})=0$. Combining this resulting
identity with the first line of the Seiberg-Witten equation, one has a
pointwise inequality of the form

$d^{*}d|\Phi_{i}|^{2}+|\nabla_{B_{i}}\Phi_{i}|^{2}+\frac{|\Phi_{i}|^{4}}{2}\leq\zeta
r|\Phi_{i}|^{2},$

where $\zeta$ is a constant depending only on the metric,
$\|\nu_{i}\|_{C^{2}}$, and $\varsigma_{w}$. This permits a straightforward
application of the strong maximum principle, which leads to the asserted bound
on $\|\Phi_{i}\|_{L^{\infty}}$. $\Box$

The next lemma is an ingredient for the proof of Theorem 1.2.

Suppose $i\in{\mathfrak{Y}}_{m}$. Let $\sigma(x)=\sigma_{Y}(x)$ denote the
distance from $x\in Y_{i}$ to $\nu_{i}^{-1}(0)$, and let $Y_{i,\delta}\subset
Y_{i}$ denote the set of all points with distance greater or equal to $\delta$
from $\nu_{i}^{-1}(0)$. Over $Y_{i}-\nu_{i}^{-1}(0)$, the Clifford action of
$\nu_{i}$ splits the spinor bundle ${\mathbb{S}}$ into a direct sum of eigen-
bundles, ${\mathbb{S}}=E\oplus E\otimes K^{-1}$, where $E$ to be the summand
corresponding to the eigenvalue $-i|\nu_{i}|$, and
$K^{-1}=\mathop{\mathrm{Ker}\,}\nolimits(*\nu_{i})\subset TY_{i}$ is a
subbundle of oriented 2-planes over $Y_{i}$. The Clifford action by
$\nu_{i}/|\nu_{i}|^{-1}$ endows $K^{-1}$ with a complex structure. Write

$\phi_{i}:=(2r)^{-1/2}\Phi_{i}=(\alpha,\beta)$

in terms of the preceding decomposition. Let $\nabla_{B_{i}}\alpha$,
$\nabla_{B_{i}}\beta$ respectively denote the covariant derivatives of
$\alpha\in\Gamma(E)$ and $\beta\in\Gamma(E\otimes K^{-1})$ with respect to the
connections induced fron $B_{i}$ and the Levi-Civita connection.

Let

(3.15) $\tilde{\sigma}:=\sigma\chi(\sigma)+1-\chi(\sigma).$

Note that our assumption that $\nu_{i}$ has nondegenerate zeros implies that
there exists a positive constant $\zeta_{i}$ satisfying

(3.16) $\zeta_{i}^{-1}\tilde{\sigma}\leq|\nu_{i}|\leq\zeta_{i}\tilde{\sigma}.$

######  Lemma 3.10.

Adopt the assumptions and notations in Lemma 3.9, and suppose that
$i\in{\mathfrak{Y}}_{m}$. There is a positive constant $r_{0}>1$ such that for
all $r\geq r_{0}$, the following holds: Let $(B_{i},\Phi_{i})$ be as in the
previous lemma, and $i\in{\mathfrak{Y}}_{m}$. Then:

  * (1)

There exists an $r$-independent positive constants $z^{\prime}$,
$z^{\prime\prime}$ such that over $Y_{i}$,

(3.17) $\begin{split}|\phi_{i}|^{2}&\leq|\nu_{i}|+zr^{-1/3};\\\
|\phi_{i}|^{2}&\leq|\nu_{i}|+z^{\prime}r^{-1}(\sigma^{-2}+1).\end{split}$

  * (2)

There exist positive constants $\textsc{o}>100$, $c$, $c^{\prime}$,
$\zeta^{\prime}$ and $\zeta$ that are independent of $r$ and $(B,\Phi)$, such
that for $\delta_{0}:=\textsc{o}r^{-1/3}$, the following pointwise estimates
hold on $Y_{i,\delta_{0}}$:

(3.18) $\displaystyle|\beta|^{2}\leq
c\tilde{\sigma}^{-3}r^{-1}(|\nu|-|\alpha|^{2})+\zeta\tilde{\sigma}^{-5}r^{-2};$
(3.19) $\displaystyle|\beta|^{2}\leq
c^{\prime}\tilde{\sigma}^{-3}r^{-1}(|\nu|-|\phi_{i}|^{2})+\zeta^{\prime}\tilde{\sigma}^{-5}r^{-2}.$

All the constants above ($\zeta$, $z$, $c$, o $r_{0}$ etc.) are determined by
$\|\nu_{i}\|_{C^{1}}$, $\varsigma_{w}$, and the metric on $Y_{i}$.

That $r$ is no less than the constant $r_{0}$ above will be a standing
assumption throughout the rest of this article.

Proof. The estimates (3.17) are the 3-dimensional analogs of Lemma 3.2 in [Ts]
and follow from the same proof. Similarly, a straightforward modification of
the proof of Proposition 3.1 in [Ts], with the dependence of constants therein
taken into account, yields the following:

There exist positive constants $r_{0}$, o, $c$, $c^{\prime}$, $\zeta^{\prime}$
and $\zeta$, which depend only on by $\|\nu_{i}\|_{C^{1}}$, $\varsigma_{w}$,
and the metric on $Y_{i}$, such that for any $r\geq r_{0}$ and
$\delta\geq\textsc{o}r^{-1/3}/2=\delta_{0}/2$, the following pointwise
estimate holds on $Y_{i,\delta}$:

$|\beta|^{2}\leq\frac{c}{8}\delta^{-3}r^{-1}(|\nu|-|\alpha|^{2})+\zeta\,(2\delta)^{-5}r^{-2}.$

Now, for a fixed $x\in Y_{i,\delta_{0}}$, we can apply the preceding formulae
for any choice of $\delta\geq\delta_{0}/2$, and thus obtaining infinitely many
bounds on the value $|\beta(x)|^{2}$. The inequality (3.18) is given by the
($x$-dependent) choice by setting $\delta=\tilde{\sigma}/2$. (Note that for
any $x\in Y_{i,\delta_{0}}$, $\tilde{\sigma}/2\geq\delta_{0}/2$ and so it is
an admissible choice of $\delta$).

The inequality (3.19) is a direct consequence of (3.18). $\Box$

######  Lemma 3.11.

Adopt the assumptions and notations in Lemma 3.9, and when
$i\in{\mathfrak{Y}}_{m}$, suppose in addition that $r\geq r_{0}$, where
$r_{0}>1$ is as in Lemma 3.10. There exist an $r$-independent constant
$\textsl{z}_{i}\in{\mathbb{R}}^{+}$ such that

$\|F_{B_{i}}\|_{L^{1}(Y_{i})}\leq\textsl{z}_{i}.$

In the case when $i\in{\mathfrak{Y}}_{v}$, the constant $\textsl{z}_{i}$ above
is determined by the metric on $Y_{i}$, $\varsigma_{w}$,and
$\|{\mathfrak{q}}_{i}\|_{{\mathscr{P}}}\leq\textsl{z}_{\mathfrak{p}}$. In the
case when $i\in{\mathfrak{Y}}_{m}$, the constant $\textsl{z}_{i}$ above is
determined by the metric on $Y_{i}$, $\|\nu_{i}\|_{C^{1}}$, $[w_{i,r}]$,
$\varsigma_{w}$, and the $\mathop{\mathrm{Spin}}\nolimits^{c}$ structure
${\mathfrak{s}}_{i}$ through the quantity

$\wp_{i}:=\frac{\pi}{2}c({\mathfrak{s}}_{i})\cdot[*\nu_{i}].$

Proof. When $i\in{\mathfrak{Y}}_{v}$, by the 3-dimensional Seiberg-Witten
equation ${\mathfrak{F}}_{\mu_{i,r,{\mathfrak{q}}_{i}}}(B_{i},\Phi_{i})=0$,
one may bound
$\int_{Y_{i}}|F_{B_{i}}|\leq\zeta_{1}(\|\Phi_{i}\|_{L^{\infty}}^{2}+\varsigma_{w})+\zeta_{2}\|{\mathfrak{q}}_{i}(B_{i},\Phi_{i})\|_{L^{2}}$,
where $\zeta_{1},\zeta_{2}$ depend only on the metric of $Y_{i}$. The asserted
$L^{1}$-bound on $F_{B_{i}}$ then follows from Lemma 3.9 and (3.4).

Suppose now that $i\in{\mathfrak{Y}}_{m}$. The proof in this case takes four
steps.

##### Step 1.

This step derives some pointwise bounds on $|F_{B_{i}}|$. By direct
computation:

(3.20) $\displaystyle|2F_{B_{i}}-iw_{i,r}|$ $\displaystyle=$ $\displaystyle
r\big{(}(|\nu_{i}|-|\alpha|^{2}+|\beta|^{2})^{2}+4|\alpha|^{2}\,|\beta|^{2}\big{)}^{1/2}$
$\displaystyle=$ $\displaystyle
r\big{(}(|\nu_{i}|-|\phi_{i}|^{2})^{2}+4|\nu_{i}|\,|\beta|^{2}\big{)}^{1/2}$
$\displaystyle\leq$ $\displaystyle
r\Big{|}|\nu_{i}|-|\phi_{i}|^{2}\Big{|}+2r|\nu_{i}|^{1/2}|\beta|$ (3.21)
$\displaystyle\leq$ $\displaystyle
r\Big{|}|\nu_{i}|-|\phi_{i}|^{2}\Big{|}+\zeta^{\prime}r^{2}\tilde{\sigma}^{3}|\beta|^{2}+\zeta^{\prime\prime}\tilde{\sigma}^{-2}\quad\text{
over $Y_{i}$,}$

where $\zeta^{\prime}$, $\zeta^{{}^{\prime\prime}}$ are constants depending
only on $\nu_{i}$. Thus, for $r\geq r_{0}$, one has:

(3.22)
$|F_{B_{i}}|\leq\begin{cases}\zeta_{f}\,r\tilde{\sigma}+\zeta^{\prime}_{f}\,\tilde{\sigma}^{-2}&\text{
over $Y_{i}$ by (\ref{17.1}) and (\ref{eq:Phi-ptws});}\\\
\zeta_{0}r\Big{|}|\nu_{i}|-|\phi_{i}|^{2}\Big{|}+\zeta^{\prime}_{0}\tilde{\sigma}^{-2}&\text{
over $Y_{i,\delta_{0}}$ by (\ref{17.2}) and
(\ref{ineq:3d-beta'}).}\end{cases}$

In the above, $r_{0}$ and $\delta_{0}$ are as in Lemma 3.10, and $\zeta_{f}$,
$\zeta_{f}^{\prime}$, $\zeta_{0}$, $\zeta_{0}^{\prime}$ are constants
depending only on $\nu_{i}$, $\varsigma_{w}$, and the metric.

##### Step 2.

Use the first line of Seiberg-Witten equation
${\mathfrak{F}}_{\mu_{i,r}}(B_{i},\Phi_{i})=0$ to write:

$\begin{split}\frac{r}{2}&\int_{Y_{i}}\big{(}|\nu_{i}|\,(|\nu_{i}|-|\alpha|^{2}+|\beta|^{2})\big{)}\\\
&\quad=\int_{Y_{i}}(iF_{B_{i}}-w_{i}/2)\wedge*\nu_{i}\\\ &\quad=\big{(}2\pi
c({\mathfrak{s}}_{i})-[w_{i}]/2\big{)}\cdot[*\nu_{i}]=4\wp_{i}-[w_{i}]\cdot[*\nu_{i}]/2.\end{split}$

Since
$r\int_{Y_{i}}\big{(}|\nu_{i}|(|\phi_{i}|^{2}-|\nu_{i}|)_{+}\big{)}\leq\zeta$
by Lemma 3.10 (1), the preceding formula implies that

(3.23)
$r\int_{Y_{i}}|\nu_{i}|\Big{|}|\nu_{i}|-|\phi_{i}|^{2}\Big{|}+2r\int_{Y_{i}}|\nu_{i}||\beta|^{2}\leq\zeta_{c},$

where $\zeta_{c}$ depends only on the metric, $\nu_{i}$, $\varsigma_{w}$,
$[w_{i}]$, and $\wp_{i}$.

##### Step 3.

While (3.23) is useful for bounding
$r\int_{Y_{i,\rho}}\Big{|}|\nu_{i}|-|\phi_{i}|^{2}\Big{|}$ for
$\rho\geq\delta_{0}$, the bound is hardly efficient when $\rho$ is small. This
step derives a better integral bound for
$r\Big{|}|\nu_{i}|-|\phi_{i}|^{2}\Big{|}$ near the zeros of $\nu_{0}$.

By assumption, the zero locus $\nu_{i}^{-1}(0)$ is nondegenerate; therefore it
consists of finitely many points $\nu_{i}^{-1}(0)=\bigcup_{p}p$, and there
exists a small positive $\rho_{0}<1$ and a function $f$ over
$Y_{i}-Y_{i,\rho_{0}}$, such that $Y_{i}-Y_{i,\rho_{0}}$ consists of mutually
disjoint balls $B_{p}(\rho_{0})$, and over
$Y_{i}-Y_{i,\rho_{0}}=\bigcup_{p}B_{p}(\rho_{0})$, $df=*\nu_{i}$ and
$|f|\leq\frac{|\nu|\sigma}{2}+O(\sigma^{3})$. Recall also from (3.16) that
$\zeta_{i}\sigma\geq|\nu_{i}|\geq\zeta_{i}^{-1}\sigma$ over
$Y_{i}-Y_{i,\rho_{0}}$.

Let $p\in\nu_{i}^{-1}(0)$, and let
${\mathscr{W}}_{p}(\rho):=\displaystyle\frac{r}{2}\displaystyle\int_{B_{p}(\rho)}\Big{(}|\nu_{i}|\Big{|}|\nu_{i}|-|\phi_{i}|^{2}\Big{|}\Big{)}$.
The first line of the Seiberg-Witten equation
${\mathfrak{F}}_{\mu_{r}}(B_{i},\Phi_{i})=0$, together with the second line of
(3.22), implies that for $\delta_{0}\leq\rho\leq\rho_{0}$ and $r\geq r_{0}$,

(3.24)
$\begin{split}{\mathscr{W}}_{p}(\rho)-\zeta_{1}\rho^{2}&\leq\frac{r}{2}\int_{B_{p}(\rho)}\big{(}|\nu_{i}|(|\nu_{i}|-|\phi_{i}|^{2}+2|\beta|^{2})\big{)}\\\
&=\int_{\partial B_{p}(\rho)}f\,(iF_{B_{i}}-w_{i,r}/2)\\\
&\leq\frac{\rho(1+\zeta^{\prime}\rho)}{2}\frac{d}{d\rho}{\mathscr{W}}_{p}(R)+\zeta_{2}\rho^{2},\\\
\end{split}$

where $\zeta_{1}$, $\zeta_{2}$, $\zeta^{\prime}$are constants depending only
on the metric, $\nu_{i}$ and $\varsigma_{w}$. Rewrite the preceding
differential inequality as:

$\frac{d}{d\rho}\big{(}e^{h}{\mathscr{W}}_{p}(R)\big{)}\geq-\zeta\rho,\quad\text{where
$h:=-2\ln\big{(}\frac{\rho}{1+\zeta^{\prime}\rho}\big{)}$,}$

and integrate from $\rho$ to $\rho_{0}$. One has:

$\rho^{-2}_{0}{\mathscr{W}}_{p}(\rho_{0})-\rho^{-2}{\mathscr{W}}_{p}(\rho)\geq-\zeta_{3}(\rho_{0}^{2}-\rho^{2}).$

Thus, from (3.23) one has

(3.25) ${\mathscr{W}}_{p}(\rho)\leq\zeta_{4}\rho^{2}.$

let $N\in{\mathbb{Z}}^{\geq 0}$ be the largest integer such that
$\delta_{0}^{\prime}:=2^{-N+1}\rho_{0}\geq\delta_{0}$. (Therefore
$2\delta_{0}\geq\delta_{0}^{\prime}\geq\delta_{0}$.) Write
$A_{n}:=A(2^{-n+1}\rho_{0},2^{-n}\rho_{0})$. Then it follows from (3.25) that
there is a constant $\zeta^{\prime\prime}$ depending only on the metric,
$\nu_{i}$ and $\varsigma_{w}$, such that for all $n,r,$ satisfying $0\leq
n\leq N$ and $r\geq r_{0}$,
$r\int_{A_{n}}\Big{|}|\nu_{i}|-|\phi_{i}|^{2}\Big{|}\leq\zeta^{\prime\prime}\,2^{-n}$.
Consequently

(3.26)
$r\int_{Y_{i,\delta_{0}^{\prime}}-Y_{i,\rho_{0}}}\Big{|}|\nu_{i}|-|\phi_{i}|^{2}\Big{|}\leq\zeta_{b}$

for a constant $\zeta_{b}$ depending only on the metric, $\nu_{i}$ and
$\varsigma_{w}$.

##### Step 4.

Divide $Y_{i}$ into the three regions: $Y_{i}-Y_{i,\delta_{0}^{\prime}}$,
$Y_{i,\delta_{0}^{\prime}}-Y_{i,\rho_{0}}$, and $Y_{i,\rho_{0}}$. Use the
first line of the pointwise bounds (3.22) to estimate the $L^{1}$ norm of
$F_{B_{i}}$ on the first region. Use the second line of the pointwise bounds
(3.22) for both the second and the third region; and combine with the integral
bounds (3.26) and (3.23) respectively for the second and the third region. We
have

$\begin{split}\int_{Y_{i}}|F_{B_{i}}|&=\int_{Y_{i}-Y_{i,\delta_{0}^{\prime}}}|F_{B_{i}}|+\int_{Y_{i,\delta_{0}^{\prime}}-Y_{i,\rho_{0}^{\prime}}}|F_{B_{i}}|+\int_{Y_{i,\delta_{0}^{\prime}}}|F_{B_{i}}|\leq\textsl{z}_{i},\end{split}$

where $\textsl{z}_{i}$ is a constant depending only on $\nu_{i}$,
$\varsigma_{w}$, and the metric. $\Box$

######  Lemma 3.12.

Adopt the assumptions and notations in Lemma 3.9. When
$i\in{\mathfrak{Y}}_{m}$, assume in addition that $r\geq r_{0}$, where $r_{0}$
is as in Lemma 3.11. There exist $r$-independent constants
$\zeta_{i},\zeta_{i}^{\prime},z^{\prime}_{i}\in{\mathbb{R}}^{+}$ with the
following significance:

Suppose that $(B_{i},\Phi_{i})=[(B_{i},\Phi_{i})]_{c}$ is in the normalized
Coulomb gauge with respect to $B_{0,i}$. Then:

  * (a)

One has the following bounds for $\delta B_{i}:=B_{i}-B_{i,0}$:

$\|\delta B_{i}\|_{L^{1}(Y_{i})}\leq\zeta_{i}$ and $\|\delta
B_{i}\|_{L^{\infty}(Y_{i})}\leq\zeta^{\prime}_{i}r^{\mathrm{a}_{i}/2}.$

  * (b)

The values of various versions of $\operatorname{CSD}$ functionals at
$(B_{i},\Phi_{i})$ are bounded as follows:

$\begin{split}|\operatorname{CSD}_{0}(B_{i},\Phi_{i})|&+|\operatorname{CSD}_{w_{i,r}}(B_{i},\Phi_{i})|+|\operatorname{CSD}_{w_{i,r},{\mathfrak{q}}_{i}}(B_{i},\Phi_{i})|\\\
&\quad+r^{-\mathrm{a}_{i}/2}|\operatorname{CSD}_{\mu_{i,r}}(B_{i},\Phi_{i})|+r^{-\mathrm{a}_{i}/2}|\operatorname{CSD}_{r\nu_{i}}(B_{i},\Phi_{i})|\leq
z_{i}^{\prime}r^{\mathrm{a}_{i}/2}.\end{split}$

In the case when $i\in{\mathfrak{Y}}_{v}$, the constants
$\zeta_{i},\zeta^{\prime}_{i},z^{\prime}_{i}$ above are determined by the
metric on $Y_{i}$, $\varsigma_{w}$,and
$\|{\mathfrak{q}}_{i}\|_{{\mathscr{P}}}\leq\textsl{z}_{\mathfrak{p}}$. In the
case when $i\in{\mathfrak{Y}}_{m}$, the constants
$\zeta_{i},\zeta^{\prime}_{i},z^{\prime}_{i}$ above are determined by the
metric on $Y_{i}$, $\|\nu_{i}\|_{C^{1}}$, $\varsigma_{w}$, $[w_{i}]$, and
$\mathop{\mathrm{Spin}}\nolimits^{c}$ structure ${\mathfrak{s}}_{i}$ through
the quantity $\wp_{i}$.

Proof. (a) When $i\in{\mathfrak{Y}}_{v}$, the asserted $L^{\infty}$-bound on
$\delta B_{i}$ follows from Lemma 3.9. The asserted $L^{1}$-bound on $\delta
B_{i}$ follows from Lemma 3.9 together with the fact that $\|\delta
B_{i}\|_{L^{1}}\leq\zeta\|\delta B_{i}\|_{L^{2}}$, where $\zeta$ is a constant
depending only on the metric of $Y_{i}$.

When $i\in{\mathfrak{Y}}_{m}$, combine the previous lemma and (3.10) to get:

$\begin{split}\|\delta
B_{i}\|_{L^{1}(Y_{i})}&\leq\zeta\|G\|_{L^{1}(Y_{i})}(\|F_{B_{i}}\|_{L^{1}(Y_{i})}+1)\\\
&\leq\zeta_{i},\end{split}$

where $G$ denotes the Green’s function for $d+d^{*}$ on $Y_{i}$. In the above,
$\zeta$ and $\zeta_{i}$ are $r$-independent constants: $\zeta$ depends only on
the metric of $Y_{i}$; $\zeta_{i}$ is determined by the metric on $Y_{i}$,
$\|\nu_{i}\|_{C^{1}}$, $[w_{i,r}]$, $\varsigma_{w}$, and $\wp_{i}$.

To get the asserted $L^{\infty}$ bound of $\delta B_{i}$, notice first that
Lemma 3.9 together with the Seiberg-Witten equation
${\mathfrak{F}}_{\mu_{i,r}}(B_{i},\Phi_{i})=0$ imply that
$\|F_{B}\|_{L^{\infty}}\leq\zeta r^{\mathrm{a}_{i}}$, where $\zeta$ is a
constant depending only on $\|\nu_{i}\|_{L^{\infty}}$ and $\varsigma_{w}$.
This in turn implies the asserted $L^{\infty}$ bound of $\delta B_{i}$ via a
standard elliptic regularity argument. (See e.g. [LT] Lemma 5.1 and references
therein.)

(b) When $i\in{\mathfrak{Y}}_{v}$, the asserted bounds on
$\operatorname{CSD}_{0}=\operatorname{CSD}_{r\nu_{i}}$ and
$\operatorname{CSD}_{w_{i}}=\operatorname{CSD}_{\mu_{i,r}}$ follows directly
from Lemma 3.9. The bound on $\operatorname{CSD}_{w_{i},{\mathfrak{q}}_{i}}$
follows from the preceding bound on $\operatorname{CSD}_{w_{i}}$, together
with a bound on $|f_{{\mathfrak{q}}_{i}}(B_{i},\Phi_{i})|$. This bound is
provided by combining Lemma 3.8 and Lemma 3.9.

Now suppose $i\in{\mathfrak{Y}}_{m}$. Use the 3-dimensional Seiberg-Witten
equation, ${\mathfrak{F}}_{\mu_{i,r}}(B_{i},\Phi_{i})=0$, Part (a) of the
lemma and Lemma 3.11 to get:

(3.27)
$\begin{split}|\operatorname{CSD}_{\mu}(B_{i},\Phi_{i})|&=\Big{|}-\frac{1}{8}\int_{Y}\delta
B_{i}\wedge\big{(}F_{B_{0,i}}+F_{B_{i}}+i\mu\Big{)}\Big{|}\\\
&\leq\frac{1}{8}\|\delta
B_{i}\|_{L^{\infty}(Y_{i})}\|F_{B_{0,i}}+F_{B_{i}}\|_{L^{1}(Y_{i})}+\frac{1}{8}\|\delta
B_{i}\|_{L^{1}(Y_{i})}\|\mu\|_{L^{\infty}(Y_{i})}\\\
&\leq\zeta_{1}r^{\mathrm{a}_{i}/2}+\zeta_{2}\|\mu\|_{L^{\infty}(Y_{i})},\end{split}$

where $\zeta_{1}$, $\zeta_{2}$ are constants determined by the constants
$\zeta_{i}$, $\zeta_{i}^{\prime}$ from Part (a), and the constant
$\textsl{z}_{i}$ in Lemma 3.11. Setting $\mu$ above to be $0$, $w_{i,r}$,
$\mu_{i,r}$, and $r\nu_{i}$, we arrive at the asserted bounds for the values
of $\operatorname{CSD}_{0}$,
$\operatorname{CSD}_{w_{i}}=\operatorname{CSD}_{w_{i},{\mathfrak{q}}_{i}}$,
$\operatorname{CSD}_{\mu_{i,r}}$ and $\operatorname{CSD}_{r\nu_{i}}$
respectively. $\Box$

######  Remark 3.13.

Recall that any $(B_{i},\Phi_{i})\in{\mathscr{C}}(Y_{i})$ may be written as
$(B_{i},\Phi_{i})=u_{i}\cdot[(B_{i},\Phi_{i})]_{c}$ for a gauge transformation
$u_{i}\in{\mathscr{G}}$. The values of various $\operatorname{CSD}$
functionals at $(B_{i},\Phi_{i})$ may be expressed in terms of those at
$[(B_{i},\Phi_{i})]_{c}$ and the cohomology class of $u_{i}$,
$[u_{i}]:=\frac{1}{2\pi}[-iu_{i}^{-1}du_{i}]\in H^{1}(Y_{i};{\mathbb{Z}})$,
via the following identity:

$\begin{split}&\operatorname{CSD}_{\mu,{\mathfrak{q}}_{i}}(B_{i},\Phi_{i})-\operatorname{CSD}_{\mu,{\mathfrak{q}}_{i}}([(B_{i},\Phi_{i})]_{c})\\\
&\qquad=\operatorname{CSD}_{\mu}(B_{i},\Phi_{i})-\operatorname{CSD}_{\mu}([(B_{i},\Phi_{i})]_{c})\\\
&\qquad=2\pi\big{(}\pi
c_{1}({\mathfrak{s}}_{i})-\frac{[\mu]}{4}\big{)}\cdot[u_{i}],\end{split}$

In particular, when $[w_{i,r}]=4\pi c_{1}({\mathfrak{s}}_{i})$, the bound for
$|\operatorname{CSD}_{w_{i,r}}(B_{i},\Phi_{i})|+|\operatorname{CSD}_{w_{i,r},{\mathfrak{q}}_{i}}(B_{i},\Phi_{i})|$
given in Part (b) of the preceding lemma holds for $(B_{i},\Phi_{i})$ in
arbitrary gauge.

###  3.3 t-convergence of large-$r$ Seiberg-Witten solutions

Let $({\mathscr{X}},\omega)$ be a symplectic 4-manifold with a riemannian
metric $g$ and an almost complex structure $J$ so that
$|\omega|\,g=\omega(\cdot,J\cdot)$. Let $\mu_{r}^{+}$ be defined in terms of
$\omega$ as in Equation (1.2). Fix a
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure on ${\mathscr{X}}$ and let
${\mathbb{S}}={\mathbb{S}}^{+}\oplus{\mathbb{S}}^{-}$ be the associated spinor
bundle. The Clifford action of $\omega$ splits ${\mathbb{S}}^{+}$ into a
direct sum of eigen-bundles, $E\oplus E\otimes K^{-1}$, where $E$ corresponds
to the eigenvalue $-|\omega|i$ and $K^{-1}$ is the anti-canonical bundle.
Given a $\mathop{\mathrm{Spin}}\nolimits^{c}$ connection $A$ on
${\mathbb{S}}^{+}$, let $A^{E}\in\mathop{\mathrm{Conn}}\nolimits(E)$ denote
the connection that it induces on $E$. Conversely, with the riemannian metric
on ${\mathscr{X}}$ fixed, an $A^{E}\in\mathop{\mathrm{Conn}}\nolimits(E)$
uniquely determines a $\mathop{\mathrm{Spin}}\nolimits^{c}$ connection on
${\mathbb{S}}^{+}$, and we have an isomorphism
$\mathop{\mathrm{Conn}}\nolimits({\mathbb{S}}^{+})\simeq\mathop{\mathrm{Conn}}\nolimits(E)$.

######  Definition 3.14.

Let $({\mathscr{X}},\omega)$, $J$ be as above. Let $\mathbf{C}:=[C,\tilde{C}]$
be a weighted $J$-holomorphic subvariety in ${\mathscr{X}}$. Let
${\mathscr{X}}_{1}\subset\cdots{\mathscr{X}}_{n}\subset{\mathscr{X}}_{n+1}\subset\cdots\subset{\mathscr{X}}$
be a countable exhaustionof ${\mathscr{X}}$ by open subsets with compact
closure, and let $\\{(A_{n},\Psi_{n})\\}_{n}$ be a corresponding sequence in
$\mathop{\mathrm{Conn}}\nolimits({\mathbb{S}}^{+})\times\Gamma({\mathbb{S}}^{+})$,
where $(A_{n},\Psi_{n})$ is defined over ${\mathscr{X}}_{n}$. This sequence is
said to t-converge to $\mathbf{C}$ if the following hold for any compact
subset $\mathcal{K}\subset X-\omega^{-1}(0)$,

  * (1)

Let $\Psi^{E}_{n}$ denote the $E$-component of $\Psi_{n}$. Then
$(\Psi^{E}_{n})^{-1}(0)$ are closed sets of Hausdorff dimension 2, and
$\lim_{n\to\infty}\mathop{\mathrm{dist}\,}\nolimits_{\mathcal{K}}\big{(}C\cap\mathcal{K},(\Psi^{E}_{n})^{-1}(0)\cap\mathcal{K}\big{)}=0$;

  * (2)

$\\{\frac{i}{2\pi}F_{A_{n}^{E}}\big{|}_{\mathcal{K}}\\}_{n}$ converges weakly
as currents to $\tilde{C}|_{\mathcal{K}}$.

Note that the definition of t-convergence above only depends on the gauge
equivalence class of $(A_{n},\Psi_{n})$. Therefore, we shall often refer to
the “t-convergence” of gauge equivalence classes of configurations.

The notion of t-convergence is similarly defined in the 3-dimensional case:
Let $M$ be a $\mathop{\mathrm{Spin}}\nolimits^{c}$ 3-manifold and $\theta$ be
a nowhere vanishing harmonic 1-form on $M$. Replace $\omega$ in the preceding
definition by $*\theta$, and replace $C_{k}$ by orbits of the dual vector to
$\theta$. Let $\Psi_{n}$ be sections of the spinor bundle ${\mathbb{S}}$
instead, and assume that $(\Psi^{E}_{n})^{-1}(0)$ now have Hausdorff dimension
1. Here, $E$ denotes the first summand in ${\mathbb{S}}=E\oplus E\otimes
K^{-1}$, where $K^{-1}$ is the 2-plane bundle
$\mathop{\mathrm{Ker}\,}\nolimits\theta\subset TM$, equipped with a complex
structure given by Clifford multiplication by $|\theta|^{-1}\theta$.

In either the 3-dimensional case $(M,\theta)$ or the 4-dimensional case
$({\mathscr{X}},\omega)$ described above, $A^{K}$ will denote the connection
on $K^{-1}$ induced from the Levi-Civita connection. For our applications,
$({\mathscr{X}},\omega)$ typically takes the form $((X_{\bullet}\cap
X_{0})^{\circ},2\nu^{+})$ for an admissible pair $(X,\nu)$. (Recall from
Section 1.4 the definition of $X_{0}:=X-\nu^{-1}(0)$ and $X_{\bullet}$.) In
the 3-dimensional case, typically $M=Y-\theta^{-1}(0)$ for a harmonic Morse
1-form on a closed $\mathop{\mathrm{Spin}}\nolimits^{c}$ 3-manifold $Y$. In
the former case, we say that a sequence of configurations on $X$ t-converges
to a t-curve $\mathbf{C}$ in $X_{\bullet}$ if it t-converges on the interior
of $X_{\bullet}\cap X_{0}$. In the latter case, a sequence of configurations
on $Y$ is said to t-converge to a t-orbit $\boldsymbol{\gamma}$ if it
t-converges on $Y-\theta^{-1}(0)$. In the 3-dimensional case, the assumption
that the zero locus of $\theta$ is nondegenerate implies the existence of a
constant $\zeta$ such that $|F_{A^{K}}|\leq\zeta\tilde{\sigma}^{-2}$, and
therefore $\|F_{A^{K}}\|_{L^{1}(Y)}\leq\zeta^{\prime}$ is finite. In the
4-dimensional case, the assumption that $\nu^{-1}(0)$ is non-degenerate,
together with the asymptotic condition of $\nu$, imply a similar pointwise
bound for $|F_{A^{K}}|$ and that
$\|F_{A^{K}}\|_{L^{1}(X_{\bullet})}\leq\zeta^{\prime}\,|X_{\bullet}|$ for a
constant $\zeta^{\prime}$ depending only on the metric and $\nu$. (Cf. Section
1.4 for the definition of $|X_{\bullet}|$.)

What follows is a restatement of Theorem 1.2 in weaker form.

######  Proposition 3.15.

Adopt the notations and assumption in Theorem 1.2. Then there exists a t-orbit
$\boldsymbol{\gamma}$ on $Y-\theta^{-1}(0)$ and a subsequence of the gauge
equivalence classes $\\{[(B_{r},\Phi_{r})]\\}_{r}$ which t-converges to
$\boldsymbol{\gamma}$.

Furthermore, $\int_{\tilde{\gamma}}\theta\geq 0$ and equals
$\frac{1}{2}(c_{1}({\mathfrak{s}})\cdot[\theta]-\zeta_{\theta})$, where

$\zeta_{\theta}=\int_{Y}\frac{i}{2\pi}F_{A^{K}}\wedge\theta,$

and $A^{K}$ is the connection on $K^{-1}$ induced by the Levi-Civita
connection. In particular, sequences $\\{(B_{r},\Phi_{r})\\}_{r}$ satisfying
the assumption of Theorem 1.2 exist only when
$c_{1}({\mathfrak{s}})\cdot[\theta]\geq\zeta_{\theta}$.

Proof. As mentioned in the introduction, the claim about t-convergence is a
consequence of [Ts]’s Theorem 1.1. Alternatively, it can be proved by going
through the proof in [Ts], which simplifies significantly in the 3-dimensional
setting. (See [LT] Section 3 for the case when $\theta^{-1}(0)=\emptyset$). As
the constituent flow lines of $\boldsymbol{\gamma}$ are flow lines of
$-\check{\theta}$, $\int_{\tilde{\gamma}}\theta\geq 0$. By the t-convergence
we also have

$\int_{\tilde{\gamma}}\theta=\lim_{r\to\infty}\int_{Y}\frac{i}{2\pi}F_{B^{E}_{r}}\wedge\theta=\frac{1}{2}(c_{1}({\mathfrak{s}})\cdot[\theta]-\zeta_{\theta}),$

where $B^{E}_{r}$ is the connection on $E$ induced from $B_{r}$. $\Box$

######  Remarks 3.16.

Note the cohomological nature of $\zeta_{\theta}$. Remove small radius
$\delta$ balls centered at points in $\theta^{-1}(0)$, and denote the
resulting 3-manifold with boundary $Y_{\delta}$. Then
$\frac{i}{2\pi}F_{A^{K}}$ defines a class $c_{1}(K^{-1})\in
H^{2}(Y_{\delta};{\mathbb{Z}})$, while $\theta$ defines a class $[\theta]$ in
$H^{1}(Y_{\delta},\partial Y_{\delta};{\mathbb{R}})\simeq
H_{2}(Y_{\delta};{\mathbb{R}})$. $\zeta_{\theta}$ is then the pairing
$\zeta_{\theta}=c_{1}(K^{-1})\cdot[\theta]$. In the case when $\theta=df$ for
a circle-valued Morse function $f\colon\thinspace
Y\to{\mathbb{R}}/{\mathbb{Z}}$, $\zeta_{\theta}$ is the maximal Euler
characteristic among the regular level surfaces of $f$. Note that the latter
is invariant under homotopy of $f$. Thus, $-\zeta_{\theta}\geq\|\theta\|_{T}$.
In fact, $-\zeta_{\theta}$ often coincides with $\|\theta\|_{T}$, such as in
the case when $\theta$ has no zeros. We believe that when $[\theta]$ is an
integral primitive class with its Poincaré dual represented by a connected
embedded minimal $\chi_{-}$ surface, $-\zeta_{\theta}=\|\theta\|_{T}$. In the
above, $\chi_{-}$ is as defined in [Th].

###  3.4 Strong t-convergence

Fix a closed connected $\mathop{\mathrm{Spin}}\nolimits^{c}$ 3-manifold
$(Y,{\mathfrak{s}})$ and let ${\mathbb{S}}$ denote its associated
$\mathop{\mathrm{Spin}}\nolimits^{c}$ bundle. Let
$\mathop{\mathrm{Conn}}\nolimits=\mathop{\mathrm{Conn}}\nolimits({\mathbb{S}})=:\mathop{\mathrm{Conn}}\nolimits(Y)$
denote the space of $\mathop{\mathrm{Spin}}\nolimits^{c}$-connections on
${\mathbb{S}}$, and let
${\mathscr{C}}:=\\{(A,\Psi)\\}=\mathop{\mathrm{Conn}}\nolimits(Y)\times\Gamma(Y,{\mathbb{S}})$.
Let ${\mathscr{G}}=C^{\infty}(Y,S^{1})$ be the gauge group. We use the short
hand
${\mathscr{B}}={\mathscr{B}}_{Y}={\mathscr{B}}(Y)={\mathscr{C}}/{\mathscr{G}}$.
Let $\Pi\colon\thinspace{\mathscr{C}}\to\mathop{\mathrm{Conn}}\nolimits$ be
the projection map taking $(B,\Phi)\in{\mathscr{C}}$ to
$B\in\mathop{\mathrm{Conn}}\nolimits$. It induces a map, also denoted $\Pi$,
from ${\mathscr{B}}$ to $\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}$.

######  Definition 3.17.

A sequence in ${\mathscr{B}}(Y)$ is said to be strongly t-convergent if it is
t-convergent and its image under $\Pi$ in
$\mathop{\mathrm{Conn}}\nolimits(Y)/{\mathscr{G}}(Y)$ is convergent in the
“current topology”, to be specified below.

The aforementioned topology is weaker than the customary $C^{\infty}$
topology, and requires knowledge of some basic structure of
$\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}$ to describe. To begin, note
that $\pi_{0}({\mathscr{G}})=H^{1}(Y;{\mathbb{Z}})$ is the fundamental group
of both ${\mathscr{B}}$ and $\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}$.
Let ${\mathscr{G}}_{0}\subset{\mathscr{G}}$ denote the identity component. The
universal cover of $\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}$,
$\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}_{0}$, is an affine space under
$\Omega^{1}(Y)/(d\Omega^{0}(Y))$. The latter has a bundle structure by way of
the following short exact sequence:

(3.28) $0\to
H^{1}(Y;{\mathbb{R}})\to\Omega^{1}(Y)/(d\Omega^{0}(Y))\stackrel{{\scriptstyle
d}}{{\to}}d\Omega^{1}(Y)\to 0.$

The deck transformations by $H^{1}(Y;{\mathbb{Z}}))$ on
$\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}_{0}$ preserves the fibers,
$H^{1}(Y;{\mathbb{R}})$, of this fibration; it acts freely on
$H^{1}(Y;{\mathbb{R}})$ by addition. (Recall that $H^{1}(Y;{\mathbb{Z}})$ is
torsion-free, cf. e.g. [D]’s Exercise 7.22, and we may and will identify it
with the integral lattice in $H^{1}(Y;{\mathbb{R}})$.) Consequently,
$\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}$ inherits the structure of a
torus bundle:

(3.29)
$\begin{CD}H^{1}(Y;{\mathbb{R}})/H^{1}(Y;{\mathbb{Z}})@>{}>{}>\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}\\\
@V{\operatorname{d}}V{}V\\\ \operatorname{B}^{2}({\mathfrak{s}}),\end{CD}$

where $\operatorname{B}^{2}({\mathfrak{s}})$ is the affine space under
$\operatorname{B}^{2}:=d\Omega^{1}(Y)$ consisting of all closed 2-forms with
cohomology class $c_{1}({\mathfrak{s}})/2\in H^{2}(Y;{\mathbb{Z}})$ modulo
torsions, and $\operatorname{d}$ is the map sending the gauge-equivalence
class $[B]\in\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}$ to the closed
2-form $\frac{i}{4\pi}F_{B}$. Note that by Poincaré duality,
$H^{1}(Y;{\mathbb{R}})/H^{1}(Y;{\mathbb{Z}})$ is canonically isomorphic to the
torus denoted as ${\mathbb{T}}_{Y}$ in Section 2.5; and we shall denote it by
the same notation. With the preliminaries out of the way, the current topology
on $\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}$ is that induced by the
current topology on the base space, $\operatorname{B}^{2}({\mathfrak{s}})$, of
(3.29), together with the standard topology on the fiber, ${\mathbb{T}}_{Y}$,
as a $b^{1}(Y)$-dimensional torus. Let $\operatorname{C}(Y)$,
$\tilde{\operatorname{C}}(Y)$ respectively denote the completion of
$\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}$ and
$\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}_{0}$ (with $C^{\infty}$
topology) with respect to the current topology, and let
${\mathscr{Z}}^{\mathbb{R}}({\mathfrak{s}})\supset\operatorname{B}^{2}({\mathfrak{s}})$
denote the space of closed 1-currents with cohomology class
$c_{1}({\mathfrak{s}})/2\in H^{2}(Y;{\mathbb{Z}})$ modulo torsions. This space
is an affine space under ${\mathscr{Z}}^{\mathbb{R}}(Y)$, the space of exact
1-currents. (Recall the notation convention from Section 2.5.) We have
fibrations:

(3.30) $\begin{CD}{\mathbb{T}}_{Y}@>{}>{}>\operatorname{C}(Y)\\\
@V{\operatorname{d}}V{}V\\\
{\mathscr{Z}}^{\mathbb{R}}({\mathfrak{s}}),\end{CD}\qquad\begin{CD}H^{1}(Y;{\mathbb{R}})@>{}>{}>\tilde{\operatorname{C}}(Y)\\\
@V{\operatorname{d}}V{}V\\\
{\mathscr{Z}}^{\mathbb{R}}({\mathfrak{s}}).\end{CD}$

When $(Y_{i},\nu_{i})$ is a Morse end, $\nu_{i}$ together with the metric on
$Y_{i}$ determines an isomorphism from
${\mathscr{Z}}^{\mathbb{R}}({\mathfrak{s}})$ to
${\mathscr{Z}}^{\mathbb{R}}_{(Y_{i},\nu_{i},{\mathfrak{s}}_{i})}$ as affine
spaces under ${\mathscr{Z}}^{\mathbb{R}}(Y)$, given by
$\frac{i}{4\pi}F_{B}\mapsto\frac{i}{2\pi}F_{B^{E}}$.

##### Proof of Theorem 1.2.

As explained in the paragraph prior to statement of the Theorem, by the
3-dimensional version of Theorem 1.1, there is a subsequence of
$\\{(B_{r},\Phi_{r})\\}_{r}$ that t-converges to a t-orbit
$\boldsymbol{\gamma}$. We show that this subsequence is in fact strongly
t-convergent.

Choose $b^{1}:=b^{1}(Y)$ smooth closed 2-forms
$\\{\xi_{1},\ldots,\xi_{b^{1}}\\}$ whose cohomology classes form a basis of
$H^{2}(Y;{\mathbb{R}})$, and denote by $\\{\xi_{k}^{*}\\}_{k=1}^{b^{1}}$ the
dual basis for $H^{1}(Y;{\mathbb{R}})$. Define a splitting
$\Omega^{1}(Y)/d\Omega^{0}(Y)\to H^{1}(Y;{\mathbb{R}})$ of the exact sequence
(3.28) given by

$a\mapsto\sum_{k}(\int_{Y}a\wedge\xi_{k})\,\xi_{k}^{*}.$

Since $(B_{r},\Phi_{r})$ are t-convergent, their curvature currents already
converge in $\operatorname{B}^{2}({\mathfrak{s}})$. Thus, by (3.28) it
suffices to check that there exists $u_{r}\in{\mathscr{G}}$ such that
$B_{r}^{\prime}=u_{r}\cdot B_{r}$ satisfies

(3.31) $i\int_{Y}(B_{r}^{\prime}-B_{0})\wedge\xi_{k}\quad\text{converges as
$r\to\infty$}$

for all $k$. By Stokes’ theorem and the convergence of $F_{B_{r}}$, for this
purpose the choice of these 1-forms may be arbitrary. By the 3-dimensional
version of Proposition 6.1 in [Ts], (3.31) can easily shown to be true if the
2-forms $\xi_{k}$ are chosen to be supported on small neighborhoods of
1-cycles in $Y$, which are disjoint from $\alpha^{-1}_{r}(0)$ and
$\nu^{-1}(0)$ for all sufficiently large $r$. $\Box$

###  3.5 Relative homotopy classes and relative homology classes

Let $X$ be $\mathop{\mathrm{Spin}}\nolimits^{c}$ MCE, and use
${\mathbb{S}}={\mathbb{S}}_{X}={\mathbb{S}}^{+}\oplus{\mathbb{S}}^{-}$ to
denote its associated spinor bundle. For each end $\hat{Y}_{i}$ of $X$, fix a
${\mathfrak{c}}_{i}\in{\mathscr{B}}_{Y_{i}}$. Denote by
${\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i}\\}_{i\in{\mathfrak{Y}}})=\\{(A,\Psi)\\}/\sim$
the quotient configuration space, where $(A,\Psi)$ is a pair consisting of a
$\mathop{\mathrm{Spin}}\nolimits^{c}$ connection on ${\mathbb{S}}^{+}$ and a
$\Psi\in\Gamma({\mathbb{S}}^{+})$, such that $(A,\Psi)$ is asymptotic to a
representative of ${\mathfrak{c}}_{i}$ on the $\hat{Y}_{i}$-end. Here,
$(A^{\prime},\Psi^{\prime})\sim(A,\Psi)$ if they are related by a gauge
transformation, i.e., $(A^{\prime},\Psi^{\prime})=u\cdot(A,\Psi)$ for some
$u\in C^{\infty}(X,S^{1})$. Given an element in
${\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i}\\}_{i})$, its relative homotopy class
refers to the element in
$\pi_{0}{\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i}\\}_{i})$ given by the path
component it is in. In the case when $X={\mathbb{R}}\times Y$ and $Y$ is a
$\mathop{\mathrm{Spin}}\nolimits^{c}$ closed 3-manifold, given
${\mathfrak{c}}_{-},{\mathfrak{c}}_{+}\in{\mathscr{B}}_{Y}$, we use
$\pi_{Y}({\mathfrak{c}}_{-},{\mathfrak{c}}_{+})$ to denote the set of relative
homotopy classes of quotient configurations with $-\infty$-limit
${\mathfrak{c}}_{-}$ and $+\infty$-limit ${\mathfrak{c}}_{+}$. This is a
torsor under the group $\pi_{1}({\mathscr{B}}_{Y})=H^{1}(Y;{\mathbb{Z}})$.
More generally, given any $X_{\bullet}\subset\overline{X}$, let

${\mathfrak{Y}}_{X_{\bullet}}:=\pi_{0}(\partial\overline{X_{\bullet}})\quad\text{and}\quad{\mathscr{B}}_{\partial\overline{X_{\bullet}}}:=\prod_{i\in{\mathfrak{Y}}_{X_{\bullet}}}{\mathscr{B}}_{Y_{i}}.$

Given
$\\{{\mathfrak{c}}_{i}\\}_{i\in{\mathfrak{Y}}_{X_{\bullet}}}\in{\mathscr{B}}_{\partial\overline{X_{\bullet}}}$,
let

${\mathscr{B}}_{X_{\bullet}}(\\{{\mathfrak{c}}_{i}\\}_{i\in{\mathfrak{Y}}_{X_{\bullet}}})=\\{(A,\Psi)\\}/\sim\quad\text{and}\quad\pi_{0}{\mathscr{B}}_{X_{\bullet}}(\\{{\mathfrak{c}}_{i}\\}_{i\in{\mathfrak{Y}}_{X_{\bullet}}})$

be respectively the direct generalization of
${\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i}\\})$ and
$\pi_{0}{\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i}\\})$, with $(A,\Psi)$ now
required to either be asymptotic to or restricts to a representative of
${\mathfrak{c}}_{i}$, depending on whether
$Y_{i}\subset\partial\overline{X_{\bullet}}$ is a ending 3-manifold of $X$ or
lies in the interior of $X$. The set
$\pi_{0}{\mathscr{B}}_{X_{\bullet}}(\\{{\mathfrak{c}}_{i}\\})$ is a torsor
under the group

$\pi_{X_{\bullet}}:=H^{1}(\partial\overline{X_{\bullet}};{\mathbb{Z}})/\mathop{\mathrm{Im}\,}\nolimits(i^{*}).$

Here, $i^{*}\colon\thinspace H^{1}(\overline{X_{\bullet}};{\mathbb{Z}})\to
H^{1}(\partial\overline{X_{\bullet}};{\mathbb{Z}})$ is the map which forms
part of the commutative diagram of long exact sequences below:

(3.32) $\begin{CD}\cdots
H_{3}(\overline{X_{\bullet}},\partial\overline{X_{\bullet}};{\mathbb{Z}})@>{}>{}>H_{2}(\partial\overline{X_{\bullet}};{\mathbb{Z}})@>{i}>{}>H_{2}(X_{\bullet};{\mathbb{Z}})@>{j}>{}>H_{2}(\overline{X_{\bullet}},\partial\overline{X_{\bullet}};{\mathbb{Z}})\cdots\\\
@V{\iota_{PD}}V{}V@V{\iota_{PD}}V{}V@V{\iota_{PD}}V{}V@V{\iota_{PD}}V{}V\\\
\cdots
H^{1}(X_{\bullet};{\mathbb{Z}})@>{i^{*}}>{}>H^{1}(\partial\overline{X_{\bullet}};{\mathbb{Z}})@>{d}>{}>H^{2}(\overline{X_{\bullet}},\partial\overline{X_{\bullet}};{\mathbb{Z}})@>{j^{*}}>{}>H^{2}(X_{\bullet};{\mathbb{Z}})\cdots.\end{CD}$

The rows above are parts of relative exact sequences of the pair
$(\overline{X_{\bullet}},\partial\overline{X_{\bullet}})$, and the vertical
maps $\iota_{PD}$ are the Poincaré duality maps. As a result of the above
commutative diagram, we have a chain of canonical isomorphisms:

$\begin{split}\pi_{X_{\bullet}}&=H^{1}(\partial\overline{X_{\bullet}};{\mathbb{Z}})/\mathop{\mathrm{Ker}\,}\nolimits
d\simeq\mathop{\mathrm{Im}\,}\nolimits(d)\\\
&\quad=\mathop{\mathrm{Ker}\,}\nolimits
j^{*}\simeq\mathop{\mathrm{Ker}\,}\nolimits j\subset
H_{2}(X_{\bullet};{\mathbb{Z}}),\end{split}$

where the first isomorphism
$H^{1}(\partial\overline{X_{\bullet}};{\mathbb{Z}})/\mathop{\mathrm{Ker}\,}\nolimits
d\simeq\mathop{\mathrm{Im}\,}\nolimits(d)$ is induced from $d$, and the second
isomorphism $\mathop{\mathrm{Ker}\,}\nolimits
j^{*}\simeq\mathop{\mathrm{Ker}\,}\nolimits j$ is the inverse of the Poincaré
duality map.

######  Remark 3.18.

Note that when $X_{\bullet}=X^{{}^{\prime}a}$ and (1.6) holds, the long exact
sequence in first row in (3.32) agrees with the
${\mathbb{K}}={\mathbb{Z}}$-version of the long exact sequence in the first
row in (2.1) (but with each showing a different portion of the long exact
sequence). Thus, recalling the defintion of ${\mathscr{H}}_{X}$ from Section
2.5, from (3.32) we have a canonical isomorphism

$\operatorname{h}\colon\thinspace\pi_{X}\stackrel{{\scriptstyle\sim}}{{\to}}{\mathscr{H}}_{X}\quad\text{
under the assumption (\ref{b_1=0}).}$

More generally, applying (3.32) to the case $X_{\bullet}=X^{{}^{\prime}a}$
yields a map, $\operatorname{h}$, from
$\pi_{X}\simeq\pi_{X^{{}^{\prime}a}}\simeq\mathop{\mathrm{Im}\,}\nolimits(d)\subset
H^{2}(\overline{X^{{}^{\prime}a}},\partial\overline{X^{{}^{\prime}a}};{\mathbb{Z}})$
to ${\mathscr{H}}_{X}\subset H_{2}(X^{{}^{\prime}a},\partial
X^{{}^{\prime}a};{\mathbb{Z}})$ via $j^{\prime}\circ\iota_{PD}^{-1}$, where
$j^{\prime}\colon\thinspace H_{2}(X^{{}^{\prime}a};{\mathbb{Z}})\to
H_{2}(X^{{}^{\prime}a},\partial X^{{}^{\prime}a};{\mathbb{Z}})$ is part of the
relative long exact sequence of the pair $(X^{{}^{\prime}a},\partial
X^{{}^{\prime}a})$.

Parallel to the relative homology classes discussed in Section 2.5,
concatenation defines maps among sets of relative homotopy classes in analogy
to the operations $*$, $\operatorname{c}_{\\{h_{i}\\}_{i}}$ in Section 2.5,
and we use the same notations for them: Assigned to every
$(c,c^{\prime})\in\pi_{Y}({\mathfrak{c}}_{-},{\mathfrak{c}})\times\pi_{Y}({\mathfrak{c}},{\mathfrak{c}}_{+})$
is an element $c^{\prime}*c\in\pi_{Y}({\mathfrak{c}}_{-},{\mathfrak{c}}_{+})$.
(This is nothing but the composition in the fundamental groupoid of
${\mathscr{B}}_{Y}$. In general, we use $\pi(M;a,b)$ to denote set of
morphisms from $a$ to $b$ in the fundamental groupoid of a topological space
$M$. Namely, it is the set of relative homotopy classes of paths in $M$ from
$a$ to $b$.) Fix $X_{\bullet}\subset\overline{X}$ and two arbitrary elements
${\mathfrak{c}}=\\{{\mathfrak{c}}_{i}\\}_{i\in{\mathfrak{Y}}_{X_{\bullet}}},{\mathfrak{c}}^{\prime}=\\{{\mathfrak{c}}_{i}^{\prime}\\}_{i\in{\mathfrak{Y}}_{X_{\bullet}}}$
in ${\mathscr{B}}_{\partial\overline{X_{\bullet}}}$. Then corresponding to
every
$c\in\pi_{\partial\overline{X_{\bullet}}}({\mathfrak{c}},{\mathfrak{c}}^{\prime}):=\prod_{i\in{\mathfrak{Y}}_{X_{\bullet}}}\pi_{Y_{i}}({\mathfrak{c}}_{i},{\mathfrak{c}}_{i}^{\prime})$,
there is an isomorphism (the concatenation map):

(3.33)
$\operatorname{c}_{c}\colon\thinspace\pi_{0}{\mathscr{B}}_{X_{\bullet}}({\mathfrak{c}})\to\pi_{0}{\mathscr{B}}_{X_{\bullet}}({\mathfrak{c}}^{\prime}).$

This is used to make sense of Condition (2) in the statement of Theorem 1.7.

######  Lemma 3.19.

(a) Given any $i\in{\mathfrak{Y}}_{m}$ and a strongly converging sequence
$\\{{\mathfrak{c}}_{i,r}\\}_{r}$, there is an $r_{0}>1$ with the following
significance: There is a distinguished element
$o_{Y_{i}}({\mathfrak{c}}_{i,r},{\mathfrak{c}}_{i,r^{\prime}})$ in
$\pi_{Y_{i}}({\mathfrak{c}}_{i,r},{\mathfrak{c}}_{i,r^{\prime}})$ for any
pairs of $r$, $r^{\prime}>r_{0}$. The number $r_{0}$ depends on the metric on
$Y_{i}$, but the distinguished element is independent of it.

(b) Let
$\\{\boldsymbol{{\mathfrak{c}}}_{r}|\,\boldsymbol{{\mathfrak{c}}}_{r}\in\Pi_{i\in{\mathfrak{Y}}}{\mathscr{B}}_{Y_{i}}\\}_{r}$
be a sequence satisfying condition (1) of Theorem 1.7. Namely,
$\boldsymbol{{\mathfrak{c}}}_{r}=({\mathfrak{c}}_{i,r})_{i}$ is such that
${\mathfrak{c}}_{i,r}={\mathfrak{c}}_{i}\in{\mathscr{B}}_{Y_{i}}$ is
independent of $r$ when $\hat{Y}_{i}$ is a vanishing end, and
$\\{{\mathfrak{c}}_{i,r}\\}_{r}\subset{\mathscr{B}}_{Y_{i}}$ is a strongly
t-converging sequence when $\hat{Y}_{i}$ is a Morse end. Let $r_{0}>1$ be such
that for any pair of $r,r^{\prime}>r_{0}$, and any $i\in{\mathfrak{Y}}_{m}$,
the distinguished element
$o_{Y_{i}}({\mathfrak{c}}_{i,r},{\mathfrak{c}}_{i,r^{\prime}})$ from item (a)
is defined. Set
$o_{Y_{i}}({\mathfrak{c}}_{i,r},{\mathfrak{c}}_{i,r^{\prime}}):=1\in\pi_{1}({\mathscr{B}}_{Y_{i}})=\pi_{Y_{i}}({\mathfrak{c}}_{i,r},{\mathfrak{c}}_{i,r^{\prime}})$
when $i\in{\mathfrak{Y}}_{v}$. Let
$c(r,r^{\prime}):=\big{(}o_{Y_{i}}({\mathfrak{c}}_{i,r},{\mathfrak{c}}_{i,r^{\prime}})\big{)}_{i}\in\prod_{i\in{\mathfrak{Y}}}\pi_{Y_{i}}({\mathfrak{c}}_{i,r},{\mathfrak{c}}_{i,r^{\prime}})$.
Then the concatenation map $\operatorname{c}_{c(r,r^{\prime})}$ defines a
canonical isomorphism from
$\pi_{0}{\mathscr{B}}_{X}(\boldsymbol{{\mathfrak{c}}}_{r})$ to
$\pi_{0}{\mathscr{B}}_{X}(\boldsymbol{{\mathfrak{c}}}_{r^{\prime}})$.

Proof. Item (b) in the statement of the lemma is a direct consequence of Item
(a). Item (a) follows from Item (a) of the next lemma, together with the
following observations:

(i) Recall the map
$\Pi\colon\thinspace{\mathscr{B}}_{Y}\to\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}\subset\operatorname{C}(Y)$
from the previous subsection. For every pair
${\mathfrak{c}},{\mathfrak{c}}^{\prime}\in{\mathscr{B}}_{Y}$, The map $\Pi$
induces an isomorphism

$\Pi_{*}\colon\thinspace\pi_{Y}({\mathfrak{c}},{\mathfrak{c}}^{\prime})\to\pi\,(\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}};\Pi{\mathfrak{c}},\Pi{\mathfrak{c}}^{\prime})=\pi\,(\operatorname{C}(Y);\Pi{\mathfrak{c}},\Pi{\mathfrak{c}}^{\prime}).$

Thus, to assign a distinguished element in
$\pi_{Y}({\mathfrak{c}},{\mathfrak{c}}^{\prime})$, it suffices to assign
distinguished elements in
$\pi\,(\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}};\Pi{\mathfrak{c}},\Pi{\mathfrak{c}}^{\prime})$
or $\pi(\operatorname{C}(Y);\Pi{\mathfrak{c}},\Pi{\mathfrak{c}}^{\prime})$.

(ii) By definition, $\\{{\mathfrak{c}}_{r}\\}_{r}\subset{\mathscr{B}}_{Y}$ is
strongly convergent iff
$\\{\Pi{\mathfrak{c}}_{r}\\}_{r}\subset\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}\subset\operatorname{C}(Y)$
converges in the current topology. $\Box$

The upcoming lemma is the $SW$ counterpart of Lemma 2.15 on the $Gr$ side. In
preparation for stating it, we introduce a counterpart of the map
$t_{\mathscr{H}}$ in Section 2.5.

Let
$\tilde{c}\colon\thinspace{\mathbb{R}}\to\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}$
be a path from $b$ to $b^{\prime}$ in
$\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}$. This corresponds to an
element $[A]$ in
$\mathop{\mathrm{Conn}}\nolimits\,(X;b,b^{\prime})/C^{\infty}(X,S^{1})$, where
$X={\mathbb{R}}\times Y$, and
$\mathop{\mathrm{Conn}}\nolimits\,(X;b,b^{\prime})\subset\mathop{\mathrm{Conn}}\nolimits\,(X)$
denotes the space of $\mathop{\mathrm{Spin}}\nolimits^{c}$ connections on
${\mathbb{S}}^{+}$ asymptotic respectively to representatives of
$b,b^{\prime}$ on its $-\infty$\- and $+\infty$-end. We extend the space
$\mathop{\mathrm{Conn}}\nolimits\,(X;b,b^{\prime})/C^{\infty}(X,S^{1})$ to a
space, denoted $\operatorname{C}(X;b,b^{\prime})$, that corresponds to the
space of paths in $\operatorname{C}(Y)$ from $b$ to $b^{\prime}$, where
$b,b^{\prime}\in\operatorname{C}(Y)\supset\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}$.
Let $A$ be a unitary connection on $\det{\mathbb{S}}^{+}$ representing $[A]$,
and let $h$ be a harmonic 2-form on $Y$ representing an element $[h]$ in
$H^{2}(Y;{\mathbb{R}})$. The integral
$\int_{X}\frac{iF_{A}}{4\pi}\wedge\pi_{2}^{*}h$ is finite and depends only on
the relative homotopy class of $\tilde{c}$, $[h]$, and the metric on $Y$. In
this way, the metric on $Y$ determines a map

$\jmath_{c}\colon\thinspace\pi\,(\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}};b,b^{\prime})\to
H^{1}(Y;{\mathbb{R}})=\mathop{\mathrm{Hom}\,}\nolimits(H^{2}(Y;{\mathbb{R}}),{\mathbb{R}}).$

In fact, $\jmath_{c}$ extends to a map from
$\pi\,(\operatorname{C}(Y);b,b^{\prime})$ to $H^{1}(Y;{\mathbb{R}})$, where
$b,b^{\prime}\in\operatorname{C}(Y)$. The latter map is also denoted by
$\jmath_{c}$, and may be regarded as a SW analog of the map $I_{\mathscr{H}}$
introduced in Lemma 2.14. A more precise description of the relation between
$\jmath_{c}$ and $I_{\mathscr{H}}$ will be given prior to Lemma 3.23. It
follows straightforwardly from construction that $\jmath_{c}$ intertwines with
the
$\pi_{1}(\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}})=\pi_{1}(\operatorname{C}(Y))\simeq
H^{1}(Y;{\mathbb{Z}})$-action on
$\pi\,(\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}};b,b^{\prime})$ (or more
generally $\pi_{1}(\operatorname{C}(Y),b,b^{\prime})$) and the
$H^{1}(Y;{\mathbb{Z}})\subset H^{1}(Y;{\mathbb{R}})$-action on
$H^{1}(Y;{\mathbb{R}})$. Thus, with the metric on $Y$ fixed, one has a
canonical isomorphism from
$\pi\,(\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}};b,b^{\prime})$ (or more
generally $\pi_{1}(\operatorname{C}(Y),b,b^{\prime})$) to an orbit of
$H^{1}(Y;{\mathbb{Z}})$ in $H^{1}(Y;{\mathbb{R}})$. Let
$t_{\operatorname{C}}(b,b^{\prime})$ denote the corresponding element in the
orbit space, $H^{1}(Y;{\mathbb{R}})/H^{1}(Y;{\mathbb{Z}})={\mathbb{T}}_{Y}$.
Summarizing, we have a map

$t_{\operatorname{C}}\colon\thinspace\operatorname{C}(Y)\times\operatorname{C}(Y)\to{\mathbb{T}}_{Y}.$

This map is an analog of the map $t_{\mathscr{H}}$ defined after Lemma 2.14 in
Section 2.5. It is continuous with respect to the current topology on
$\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}$ and $\operatorname{C}(Y)$.
Furthermore, recall from (3.29) and (3.30) the structure of
$\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}}$, $\operatorname{C}(Y)$ as
${\mathbb{T}}_{Y}$-bundles, $t_{\operatorname{C}}$ is
${\mathbb{T}}_{Y}$-equivariant both with respect to the
${\mathbb{T}}_{Y}$-action on the right factor of
$\operatorname{C}(Y)\times\operatorname{C}(Y)$, and the
${\mathbb{T}}_{Y}^{op}$-action on the left factor of
$\operatorname{C}(Y)\times\operatorname{C}(Y)$ (or
$(\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}})\times(\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}})$).
(${\mathbb{T}}_{Y}^{op}$ above refers to the inverse action of the torus
group.) It maps the diagonal to the identity element $0\in{\mathbb{T}}_{Y}$.

######  Lemma 3.20.

(a) Suppose $b,b^{\prime}\in\operatorname{C}(Y)$ are sufficiently close in the
sense that $t_{\operatorname{C}}(b,b^{\prime})$ falls in the ball
$B_{0}(1/2)\subset{\mathbb{T}}_{Y}$. Then there is a distinguished element
$\tilde{o}_{Y}(b,b^{\prime})$ in $\pi(\operatorname{C}(Y);b,b^{\prime})$. This
distinguished element is independent of the metric on $Y$, though the notion
of being “sufficiently close” does.

(b) Fix $X_{\bullet}\subset\overline{X}$. Suppose a pair
$\\{b_{i}\\}_{i},\\{b^{\prime}_{i}\\}_{i}\in\prod_{i\in{\mathfrak{Y}}_{X_{\bullet}}}\operatorname{C}(Y_{i})$
are sufficiently close in the sense described above. Given any
${\mathfrak{c}}_{i}\in\Pi^{-1}b_{i}$,
${\mathfrak{c}}_{i}^{\prime}\in\Pi^{-1}b_{i}^{\prime}$, let
$o_{Y_{i}}({\mathfrak{c}}_{i},{\mathfrak{c}}_{i}^{\prime}):=(\Pi_{*})^{-1}\tilde{o}_{Y_{i}}(b_{i},b^{\prime}_{i})\in\pi_{Y_{i}}({\mathfrak{c}}_{i},{\mathfrak{c}}_{i}^{\prime})$.
Then the concatenation map
$\operatorname{c}_{\\{o_{Y_{i}}({\mathfrak{c}}_{i},{\mathfrak{c}}_{i}^{\prime})\\}_{i}}$
defines a canonical isomorphism from
$\pi_{0}{\mathscr{B}}_{X_{\bullet}}(\\{{\mathfrak{c}}_{i}\\}_{i})$ to
$\pi_{0}{\mathscr{B}}_{X_{\bullet}}(\\{{\mathfrak{c}}_{i}^{\prime}\\}_{i})$ as
affine spaces under $\pi_{X_{\bullet}}$.

Proof. Item (a) follows directly from the discussion preceding to the
statement of the lemma. Item (b) generalizes Item (b) of the Lemma 3.19, and
follows from Item (a) above together with Part (i) in the proof of Lemma 3.19.
$\Box$

######  Remark 3.21.

The arguments in the proof the preceding lemma also establishes the following:
A choice of basis for $H^{1}(Y;{\mathbb{Z}})$ gives a way of (simultaneously)
identifying all sets of relative homotopy classes
$\pi_{Y}({\mathfrak{c}},{\mathfrak{c}}^{\prime})$,
${\mathfrak{c}},{\mathfrak{c}}^{\prime}\in{\mathscr{B}}_{Y}$, with
$H^{1}(Y;{\mathbb{Z}})$ (as affine spaces). (Recall that such a choice (cf.
(3.10)) is required to define the normalized Coulomb gauge, and has been fixed
implicitly in this article.) To see this, observe that as explained in Part
(i) in the proof of Lemma 3.19, it suffices to identify sets of relative
homotopy classes $\pi\,(\operatorname{C}(Y);b,b^{\prime})$, with
$H^{1}(Y;{\mathbb{Z}})$ (as affine spaces) for every pair
$b,b^{\prime}\in\operatorname{C}(Y)$. This is equivalent to (consistently)
choosing a base point in $\pi\,(\operatorname{C}(Y);b,b^{\prime})$ for each
pair $b,b^{\prime}\in\operatorname{C}(Y)$. To do so, note that a choice of
basis for $H^{1}(Y;{\mathbb{Z}})$ defines an isomorphism

$i_{h}\colon\thinspace{\mathbb{R}}^{b^{1}}\stackrel{{\scriptstyle\sim}}{{\to}}H^{1}(Y;{\mathbb{R}}).$

Recall also that $t_{\operatorname{C}}(b,b^{\prime})$ corresponds to an
isomorphism from $\pi\,(\operatorname{C}(Y);b,b^{\prime})$ to an orbit of
$H^{1}(Y;{\mathbb{Z}})$ in $H^{1}(Y;{\mathbb{R}})$. We define the base point
in $\pi\,(\operatorname{C}(Y);b,b^{\prime})$ to be the unique element whose
image under this isomorphism lies in $i_{h}\,([0,1)^{\times b^{1}})\subset
H^{1}(Y;{\mathbb{R}})$. Denote this base point by
$\tilde{o}_{Y}(b,b^{\prime})$, and given
${\mathfrak{c}},{\mathfrak{c}}^{\prime}\in{\mathscr{B}}_{Y}$, let

$o_{Y}({\mathfrak{c}},{\mathfrak{c}}^{\prime}):=(\Pi_{*})^{-1}\tilde{o}_{Y}(\Pi{\mathfrak{c}},\Pi{\mathfrak{c}}^{\prime})\in\pi_{Y}({\mathfrak{c}},{\mathfrak{c}}^{\prime})$

denote the corresponding base point in
$\pi_{Y}({\mathfrak{c}},{\mathfrak{c}}^{\prime})$. The argument for Item (b)
of Lemma 3.20 then implies that once a choice of basis is fixed for every
$H^{1}(Y_{i};{\mathbb{Z}})$, then for any given $X_{\bullet}$,
$\pi_{0}{\mathscr{B}}_{X_{\bullet}}(\\{{\mathfrak{c}}_{i}\\}_{i})$,
$\pi_{0}{\mathscr{B}}_{X_{\bullet}}(\\{{\mathfrak{c}}_{i}^{\prime}\\}_{i})$
are identified for every pair
$\\{{\mathfrak{c}}_{i}\\}_{i},\\{{\mathfrak{c}}_{i}^{\prime}\\}_{i}$ in
${\mathscr{B}}_{\partial\overline{X_{\bullet}}}$.

######  Remark 3.22.

Fix a reference connections $A_{0}$ on ${\mathbb{S}}^{+}$ as in (3.3). Then
for every given $X_{\bullet}$, $(A_{0},0)|_{X_{\bullet}}$ determines a base
point in
$\pi_{0}({\mathscr{B}}_{X_{\bullet}}(\\{{\mathfrak{c}}_{0,i}\\}_{i}))$, where
$i\in{\mathfrak{Y}}_{X_{\bullet}}$, and
${\mathfrak{c}}_{0,i}\in{\mathscr{B}}_{Y_{i}}$ denotes the restriction of
$(A_{0},0)$ to the $i$-th connected component of
$\partial\overline{X_{\bullet}}$. This in turn defines an isomorphism

$\imath_{A_{0}}\colon\thinspace\pi_{0}({\mathscr{B}}_{X_{\bullet}}(\\{{\mathfrak{c}}_{0,i}\\}_{i}))\stackrel{{\scriptstyle\sim}}{{\to}}\pi_{X_{\bullet}}$

as affine spaces.

Thus, a choice of $A_{0}$ determines, for any given
$X_{\bullet}\subset\overline{X}$, a way of simultaneously identifying
$\pi_{0}({\mathscr{B}}_{X_{\bullet}}(\\{{\mathfrak{c}}_{i}\\}_{i}))\simeq\pi_{X_{\bullet}}$
for all
$\\{{\mathfrak{c}}_{i}\\}_{i}\in{\mathscr{B}}_{\partial\overline{X_{\bullet}}}$:
Given any
$\\{{\mathfrak{c}}_{i}\\}_{i}\in\prod_{i\in{\mathfrak{Y}}_{X_{\bullet}}}{\mathscr{B}}_{Y_{i}}$,
Let $h_{A_{0}}$ denote the isomorphism

$h_{A_{0}}:=\imath_{A_{0}}\circ\operatorname{c}_{\\{o_{Y_{i}}({\mathfrak{c}}_{i},{\mathfrak{c}}_{0,i})\\}_{i\in{\mathfrak{Y}}_{X_{\bullet}}}}\colon\thinspace\pi_{0}({\mathscr{B}}_{X_{\bullet}}(\\{{\mathfrak{c}}_{i}\\}_{i\in{\mathfrak{Y}}_{X_{\bullet}}}))\stackrel{{\scriptstyle\sim}}{{\to}}\pi_{X_{\bullet}}.$

When $X_{\bullet}$ is of the form $\hat{Y}_{i,I}$, this isomorphism agrees
with the isomorphism
$\pi_{Y_{i}}({\mathfrak{c}},{\mathfrak{c}}^{\prime})\stackrel{{\scriptstyle\sim}}{{\to}}H^{1}(Y_{i};{\mathbb{Z}})$
introduced in Remark 3.21 under our assumptions on $A_{0}$.

We next compare the relative homology classes of t-curves and the relative
homotopy classes of Seiberg-Witten quotient configurations. Recall from
Section 2.5 the definition of
${\mathscr{H}}^{\mathbb{R}}(X^{{}^{\prime}a},\nu,{\mathfrak{s}},\\{\tilde{\gamma}_{i}\\})$
and various other notions. Let
$c=[(A,\Psi)]\in{\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i}\\}_{i})$, with
${\mathfrak{c}}_{i}=[(B_{i},\Phi_{i})]\in{\mathscr{B}}_{Y_{i}}$. Assign to $c$
the 2-current $\tilde{c}=\frac{i}{2\pi}F_{A^{E}}$ on $X^{{}^{\prime}a}$. This
current determines a class in
${\mathscr{H}}^{\mathbb{R}}(X^{{}^{\prime}a},\nu,{\mathfrak{s}},\\{\tilde{{\mathfrak{c}}}_{i}\\}_{i\in{\mathfrak{Y}}_{m}})$,
where
$\tilde{{\mathfrak{c}}}_{i}=\frac{i}{2\pi}F_{B_{i}^{E}}\in{\mathscr{Z}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i})\subset{\mathscr{Z}}^{\mathbb{R}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i})$.
This class only depends on the relative homotopy class of $c$, and so in this
way we have a map

${\mathfrak{h}}^{\prime}\colon\thinspace\pi_{0}{\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i}\\}_{i\in{\mathfrak{Y}}})\to{\mathscr{H}}^{\mathbb{R}}((X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{{\mathfrak{c}}}_{i}\\}_{i\in{\mathfrak{Y}}_{m}}).$

This map intertwines with the $\pi_{X}$-action on
$\pi_{0}{\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i}\\}_{i\in{\mathfrak{Y}}})$, and
the $\operatorname{h}(\pi_{X})\subset{\mathscr{H}}^{\mathbb{R}}_{X}$-action on
${\mathscr{H}}^{\mathbb{R}}((X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{{\mathfrak{c}}}_{i}\\}_{i\in{\mathfrak{Y}}_{m}})$.
(The map $\operatorname{h}\colon\thinspace\pi_{X}\to{\mathscr{H}}_{X}$ is as
defined in Remark 3.18.) It is also natural with respect to the concatenation
maps on both sides. In the special case when $(X,\nu)=({\mathbb{R}}\times
Y_{i},\pi_{2}^{*}\nu_{i})$, $i\in{\mathfrak{Y}}_{m}$, is cylindrical, the map
${\mathfrak{h}}^{\prime}\colon\thinspace\pi_{Y_{i}}({\mathfrak{c}},{\mathfrak{c}}^{\prime})\to{\mathscr{H}}^{\mathbb{R}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{{\mathfrak{c}}},\tilde{{\mathfrak{c}}}^{\prime})$
by construction factors through a map
$\underline{{\mathfrak{h}}}\colon\thinspace\pi\,(\operatorname{C}(Y_{i}),\Pi{\mathfrak{c}},\Pi{\mathfrak{c}}^{\prime})\to{\mathscr{H}}^{\mathbb{R}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{{\mathfrak{c}}},\tilde{{\mathfrak{c}}}^{\prime})$:
${\mathfrak{h}}^{\prime}=\underline{{\mathfrak{h}}}\circ\Pi_{*}$. It follows
from construction that in this case,

$\jmath_{c}=\iota_{PD}\circ I_{{\mathscr{H}}}\circ\underline{{\mathfrak{h}}},$

where $\iota_{PD}$ again denotes the Poincaré map.

Next, consider the case when (1.6) holds. Recall from Remark 3.18 that
$\pi_{X}\stackrel{{\scriptstyle{\rm h}}}{{\simeq}}{\mathscr{H}}_{X}$ in this
case.

######  Lemma 3.23.

Assume (1.6); namely, all vanishing ends of $(X,\nu)$ has zero first betti
numbers. For each $i\in{\mathfrak{Y}}_{m}$, let
$\\{{\mathfrak{c}}_{i,r}\\}_{r}\in{\mathscr{B}}_{Y_{i}}$ be a sequence that
strongly t-converges to a t-orbit $\boldsymbol{\gamma}_{i}$, and for each
$i\in{\mathfrak{Y}}_{v}$, let
${\mathfrak{c}}_{i,r}={\mathfrak{c}}_{i}\in{\mathscr{B}}_{Y_{i}}$ be
independent of $r$. Let $r_{0}>1$ be as in Lemma 3.19. Then:

(a) For all $r>r_{0}$, there is a canonical isomorphism ${\mathfrak{h}}$ from
$\pi_{0}{\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i,r}\\}_{i\in{\mathfrak{Y}}})$ to
an ${\mathscr{H}}_{X}$-orbit in
${\mathscr{H}}^{\mathbb{R}}\big{(}(X^{{}^{\prime}a},\nu),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$
as affine spaces under $\pi_{X}\simeq{\mathscr{H}}_{X}$.

(b) If moreover, the sequences $\\{{\mathfrak{c}}_{i,r}\\}_{r}$,
$i\in{\mathfrak{Y}}_{m}$ arise as the $Y_{i}$-end limits of a sequence of
admissible solutions $\\{(A_{r},\Psi_{r})\\}_{r}$ to
${\mathfrak{S}}_{\mu_{r},\hat{{\mathfrak{p}}}}(A_{r},\Psi_{r})=0$ as in the
assumptions of Theorem 1.7, and the assumptions of Theorem 1.7 hold. Then the
image of the map ${\mathfrak{h}}$ is
${\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}\subset{\mathscr{H}}^{\mathbb{R}}\big{(}(X^{{}^{\prime}a},\nu),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$.
Namely, in this case ${\mathfrak{h}}$ is a canonical isomorphism from
$\pi_{0}{\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i,r}\\}_{i})$ to
${\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$
as affine spaces under $\pi_{X}\simeq{\mathscr{H}}_{X}$.

Proof. By construction, we have the following commutative diagram under the
assumptions of the lemma: For any $r,r^{\prime}>r_{0}$

(3.34)

where all the maps are morphisms of affine spaces under
$\pi_{X}\simeq{\mathscr{H}}_{X}\subset{\mathscr{H}}_{X}^{\mathbb{R}}$; the
horizontal map in the top row is the canonical isomorphism from Lemma 3.19;
the horizontal maps in the bottom row are the canonical isomorphisms from
Lemma 2.14. The composition $i_{\infty}\circ{\mathfrak{h}}^{\prime}$ maps
$\pi_{0}{\mathscr{B}}_{X}(\\{{\mathfrak{c}}_{i,r}\\}_{i\in{\mathfrak{Y}}})$ to
another orbit of the ${\mathscr{H}}_{X}$-action in
${\mathscr{H}}^{\mathbb{R}}\big{(}(X^{{}^{\prime}a},\nu),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$.
The canonical isomorphism claimed in Item (a) of the statement of the present
lemma is defined to be this composition map:
${\mathfrak{h}}:=i_{\infty}\circ{\mathfrak{h}}^{\prime}$.

The proof of Item (b) will follows as a by-product of the proof of Theorem 1.9
(c), and will be deferred to Section 7.3 $\Box$

###  3.6 Taubes’s proof for $SW\Rightarrow Gr$: A synopsis

The proof of Theorem 1.7 follows the outline of Taube’s arguments in [T]. To
serve as a roadmap for the remainder of this article, a brief summary of the
ingredients in [T] is provided here for reader’s convenience. For each step
listed below, we indicate where it takes place in [T], [Ts], and the present
article, under their respectively contexts.

Let $(X,\nu)$ be an admissible pair and let $(A,\Psi)=(A_{r},\Psi_{r})$ be as
in the statement of Theorem 1.7. Write

$\Psi=(r/2)^{1/2}\psi,\quad\text{and $\psi=(\alpha,\beta)$}$

according to the decomposition ${\mathbb{S}}^{+}=E\oplus E\otimes K^{-1}$ on
$X^{{}^{\prime}a}$. The $\mathop{\mathrm{Spin}}\nolimits^{c}$-connection
corresponding to $A$ induces a connection on $E$, and together with the Levi-
Civita connection, also a connection on $E\otimes K^{-1}$. We use
$\nabla_{A}\alpha$, $\nabla_{A}\beta$ to denote the covariant derivative with
respect to the aforementioned induced connections.

Taubes’s proof proceeds with the steps listed below:

  * (1)

Obtaining pointwise estimates of
$|\beta|,|F_{A}|,|\nabla_{A}\alpha|,|\nabla_{A}\beta|$ in terms of the “energy
density” $r|\varpi|:=r||\nu|-|\alpha|^{2}|$. The estimates show that these
quantities are interesting where $\alpha$ is small. Cf. [Ts] Sections 3 b)-e),
[T] Section I.2. Section 5 in this article contains the corresponding
estimates on a MCE.

  * (2)

Obtaining an integral bound for $r|\varpi|$ in terms of constants depending
only on the $\mathop{\mathrm{Spin}}\nolimits^{c}$ structure and the metric.
This is basically equivalent to an integral bound on
$\frac{iF_{A}}{2\pi}\wedge\nu$. Cf. [Ts] Section 3 a) and references therein.
The corresponding results are establishes in §4.7 below.

  * (3)

Establishing a monotonicity formula for $\mathcal{W}_{B}$, which is a certain
notion of “energy” over a ball $B$ in $X$. Cf. [Ts] Section 4 and [T] Section
I.3 and the beginning of Section 6 below. Roughly speaking,
${\mathscr{W}}_{B}$ is the integral of the “energy density” $r|\varpi|$ over
$B$. Note that this notion of energy is however of a different nature from
other, more typical notions of energy for Seiberg-Witten theory on MCE, such
as the Chern-Simons-Dirac functional, or more generally, the topological
energy ${\mathscr{E}}_{top}$ introduced in [KM]. This ${\mathscr{W}}_{B}$
should be regarded as an analogue of the notion of energy on the Gromov side,
namely the area of the holomorphic curves. As a result of the monotonicity
formula, one obtains an $r$-independent bound on the 2-dimensional Hausdorff
measure of $\alpha^{-1}(0)$, and hence also on that of the t-curve that
$\alpha^{-1}(0)$ geometrically converges to as $r\to\infty$. The relevant
monotonicity formula in our context is given in Section 6 below.

  * (4)

Rescaling that $(A,\psi)$ over a ball of radius $O(r^{-1/2})$ yields an
approximate solution to the version of the Seiberg-Witten equation (1.1) on
the Euclidean space $X=\R^{4}=\\{(x_{1},\cdots,x_{4})\\}$ and
$\mu^{+}=\frac{1}{2}(dx_{1}\wedge dx_{2}+dx_{3}\wedge dx_{4})$. The behavior
of Seiberg-Witten solutions for such $(X,\mu^{+})$ are well-understood in
terms of vortex solutions on ${\mathbb{R}}^{2}$, and these give the local
models for $(A,\psi)$. In particular, combining with Items (1)–(3) above, this
implies the exponential decay of
$|\beta|,|F_{A^{E}}|,|\nabla_{A}\alpha|,|\nabla_{A}\beta|$ away from
$\alpha^{-1}(0)$. This step appeared in Section 6 of [Ts] and Section I.4 of
[T]. This adapts easily to our context; see Sections 6.6 and 6.5 below.

  * (5)

Regarding $\frac{i}{2\pi}F_{A^{E}}$ as a 2-current , Items (1) and (2) above
imply that this current is bounded independently of $r$. Thus, Alaoglu’s
theorem (cf. e.g. [Rd] Theorem 10.6.17) implies that these currents converge
to a current $\mathcal{F}$ in weak∗ topology. Furthermore, via (3) and (4) it
is shown that the support of $\mathcal{F}$, denoted temporarily by $C$, is a
closed space of 2-dimensional measure, and is the $r\to\infty$ limit of
$\alpha^{-1}(0)$ in the sense of ((1)). Cf. the first half of Section I.5 of
[T]; Sections 7(a) and (b) of [Ts], and Section 7.1 in this article.

  * (6)

Using Item (4), it is shown that the current $\mathcal{F}$ defines a “positive
cohomology assignment”, namely, the map from the set of a certain kind of
generic disks in $X$ (the “admissible disks”) to $\Z$, whose value is positive
for admissible pseudo-holomorphic disks which intersect $C$. Taubes showed
that this fact guarantees that $C$ is a pseudo-holomorphic curve. Cf. Section
6 of [T] and Sections 7(c)-(e) of [Ts]. This part of Taubes’ argument applies
directly in our setting with no need for modification.

##  4 $L^{2}_{1,loc}$-bounds and integral estimates

The main results of this section come in two groups. The first group of main
results, Propositions 4.12 and 4.13, provide $L^{2}_{1}$-bounds on solutions
$(A_{r},\Psi_{r})$ to the Seiberg-Witten equation
${\mathfrak{S}}_{\mu_{r},\hat{{\mathfrak{p}}}}(A_{r},\Psi_{r})=0$, with
careful control on the growth rate of these bounds on $r$. While (for each
fixed $r$) such $L^{2}_{1}$ bounds usually serve as the starting point of
typical proofs compactness results in Seiberg-Witten theory (cf. e.g. [KM]’s
Corollary 10.6.2 and Theorem 5.2.1 (ii)), the usual arguments to derive them
(e.g. those in [KM]), do not provide sufficient bounds on the $r$-growth rate
needed for t-convergence. The second group of main results, Lemmas 4.16 and
4.17, supply the type of “energy bounds” required to begin the proof of
$t$-convergence results. This corresponds to Step (2) of the list in Section
3.6.

###  4.1  ${\mathscr{E}}_{top}$ and ${\mathscr{E}}_{anal}$: the topological
energy and the analytic energy

We begin with some general definitions and observations. Similar to what was
done in [KM] (Definition 4.5.4 therein), we introduce two Floer-theoretic
notion of “energy”, ${\mathscr{E}}_{anal}(A,\Psi)$ and
${\mathscr{E}}_{top}(A,\Psi)$ below. They are related when $(A,\Psi)$
satisfies a Seiberg-Witten equation. (The subscripts “anal” and “top” signify
“analytical” and “topological” respectively.) As is typical in Floer theories,
${\mathscr{E}}_{anal}$ is useful for estimating $L^{2}_{1}$-norms, while
${\mathscr{E}}_{top}$ depends only on the end points of the Floer trajectory
and its relative homotopy class. In the cylindrical case,
${\mathscr{E}}_{top}$ is the difference in the values of the CSD functional at
end points (cf. §3.1 for the definition of CSD functionals).

Let $X$ be a $\mathop{\mathrm{Spin}}\nolimits^{c}$ MCE with ending 3-manifolds
$Y_{i}$, and let $\mu$ be a closed 2-form on $X$ that has a closed 2-form
$\mu_{i}$ as the $Y_{i}$-end limit for each $i\in{\mathfrak{Y}}$, in the sense
defined in Section 1.4. Recall also from Section 1.4 the definition of
$X_{\bullet}$. (Note in particular that the boundary components of
$X_{\bullet}$ consist of slices of the form $Y_{i:l}$).

######  Definition 4.1.

Fix a compact $X_{\bullet}\subset X$. Given
$(A,\Psi)\in{\mathscr{C}}(X_{\bullet})$, let

(4.1)
$\begin{split}{\mathscr{E}}_{anal}^{\mu}(X_{\bullet})(A,\Psi)=&\frac{1}{4}\int_{X_{\bullet}}|F_{A}|^{2}+\int_{X_{\bullet}}|\nabla_{A}\Psi|^{2}+\int_{X_{\bullet}}\Big{|}\frac{i}{4}\rho(\mu^{+})-(\Psi\Psi^{*})_{0}\Big{|}^{2}\\\
&\qquad+\int_{X_{\bullet}}\frac{R_{g}}{4}|\Psi|^{2}-\frac{i}{4}\int_{X_{\bullet}}F_{A}\wedge*_{4}\mu,\\\
\end{split}$

where $R_{g}$ denotes the scalar curvature.

Over a slice $Y_{i:s}:=\text{{\small s}}_{i}^{-1}(s)\simeq Y_{i}$ of
$\bar{X}$, let $(B,\Phi)=(B(s),\Phi(s))=(A,\Psi)\big{|}_{Y_{i:s}}$ denote the
restriction of $(A,\Psi)$ to $\text{{\small s}}_{i}^{-1}(s)\subset X$. We
define

(4.2)
${\mathscr{E}}_{top}^{\mu}(X_{\bullet})(A,\Psi)=\frac{1}{4}\int_{X_{\bullet}}F_{A}\wedge
F_{A}-\int_{\partial\overline{X_{\bullet}}}\langle\Phi,\mbox{\mbox{$\partial\mkern-7.5mu\mbox{/}\mkern
2.0mu$}}_{B}\Phi\rangle+\frac{i}{4}\int_{X_{\bullet}}F_{A}\wedge\mu.$

We then define

(4.3)
${\mathscr{E}}_{top}^{\mu,\hat{{\mathfrak{p}}}}(X_{\bullet})(A,\Psi)={\mathscr{E}}_{top}^{\mu}(X_{\bullet})(A,\Psi)-2\,f_{\hat{{\mathfrak{p}}}}((A,\Psi)|_{\partial\overline{X_{\bullet}}}).$

The preceding definition of ${\mathscr{E}}_{top}^{\mu}$ and
${\mathscr{E}}^{\mu}_{anal}$ is motivated by the following identity:

(4.4)
$\|\mathfrak{S}_{\mu_{r}}(A,\Psi)\|_{L^{2}(X_{\bullet})}^{2}=\mathcal{E}_{anal}^{\mu_{r}}(X_{\bullet})(A,\Psi)-{\mathcal{E}}_{top}^{\mu_{r}}(X_{\bullet})(A,\Psi).\quad$

In particular, when $(A,\Psi)\in{\mathscr{C}}(X_{\bullet})$ is a solution to
the Seiberg-Witten equation
${\mathfrak{S}}_{\mu,\hat{{\mathfrak{p}}}}(A,\Psi)=0$, one has:

${\mathscr{E}}_{anal}^{\mu}(X_{\bullet})(A,\Psi)={\mathscr{E}}_{top}^{\mu,\hat{{\mathfrak{p}}}}(X_{\bullet})(A,\Psi)+\|\hat{{\mathfrak{p}}}(A,\Psi)\|_{L^{2}(X_{\bullet})}^{2}.$

This identity is used in Part 1 of next subsection to bound the square terms
in the first line of (4.1) in terms of ${\mathscr{E}}_{top}$. What follows are
some observations demonstrating the topological nature of
${\mathscr{E}}_{top}$.

Let $A_{0}$, $B_{0,i}$ be the reference connections from (3.3). Recall the
definitions of CSD functionals from (3.1), (3.2), and Remark 3.6. It is often
convenient to re-express the topological energies ${\mathscr{E}}_{top}^{\mu}$,
${\mathscr{E}}_{top}^{\mu,\hat{{\mathfrak{p}}}}$ in terms of the CSD
functionals: By the Stokes’ theorem,

(4.5)
$\begin{split}{\mathscr{E}}_{top}^{\mu}(X_{\bullet})(A,\Psi)&=\frac{1}{4}\int_{X_{\bullet}}F_{A_{0}}\wedge
F_{A_{0}}+\frac{i}{4}\int_{X_{\bullet}}F_{A_{0}}\wedge\mu-2\operatorname{CSD}_{\mu}^{\partial\overline{X_{\bullet}}}(B,\Phi);\\\
{\mathscr{E}}_{top}^{\mu,\hat{{\mathfrak{p}}}}(X_{\bullet})(A,\Psi)&=\frac{1}{4}\int_{X_{\bullet}}F_{A_{0}}\wedge
F_{A_{0}}+\frac{i}{4}\int_{X_{\bullet}}F_{A_{0}}\wedge\mu_{r}-2\operatorname{CSD}_{\mu,\hat{{\mathfrak{p}}}}^{\partial\overline{X_{\bullet}}}(B,\Phi).\\\
\end{split}$

Given the formulas (4.2), (4.3), ${\mathscr{E}}_{top}^{\mu}$ and
${\mathscr{E}}_{top}^{\mu,\hat{{\mathfrak{p}}}}$ are by definition gauge-
invariant. Moreover, ${\mathscr{E}}_{top}^{\mu}(X_{\bullet})(A,\Psi)$,
${\mathscr{E}}_{top}^{\mu,\hat{{\mathfrak{p}}}}(X_{\bullet})(A,\Psi)$ are can
be computed from the relative homotopy class of $[(A,\Psi)]$ via the following
explicit formulas: Recalling that $[(B,\Phi)]_{c}\in{\mathscr{C}}(Y)$ denotes
the representative of $[(B,\Phi)]\in{\mathscr{B}}(Y)$ in the normalized
Coulomb gauge, one may re-express (4.5) as

(4.6)
$\begin{split}&{\mathscr{E}}_{top}^{\mu}(X_{\bullet})(A,\Psi)=\frac{1}{4}\int_{X_{\bullet}}F_{A_{0}}\wedge
F_{A_{0}}+\frac{i}{4}\int_{X_{\bullet}}F_{A_{0}}\wedge\mu\\\
&\qquad\quad-2\operatorname{CSD}_{\mu}^{\partial\overline{X_{\bullet}}}([(B,\Phi)]_{c})-2i^{*}\big{(}(\pi
c_{1}({\mathfrak{s}}_{X}|_{X_{\bullet}})-\frac{\mu|_{X_{\bullet}}}{4}\big{)}\cdot
h\,(A,\Psi);\\\
&{\mathscr{E}}_{top}^{\mu,\hat{{\mathfrak{p}}}}(X_{\bullet})(A,\Psi)=\frac{1}{4}\int_{X_{\bullet}}F_{A_{0}}\wedge
F_{A_{0}}+\frac{i}{4}\int_{X_{\bullet}}F_{A_{0}}\wedge\mu\\\
&\qquad\quad-2\operatorname{CSD}_{\mu,\hat{{\mathfrak{p}}}}^{\partial\overline{X_{\bullet}}}([(B,\Phi)]_{c})-2i^{*}\big{(}(\pi
c_{1}({\mathfrak{s}}_{X}|_{X_{\bullet}})-\frac{\mu|_{X_{\bullet}}}{4}\big{)}\cdot
h\,(A,\Psi),\end{split}$

where $h\,(A,\Psi)\in
H^{1}(\partial\overline{X_{\bullet}};{\mathbb{Z}})/\mathop{\mathrm{Im}\,}\nolimits
i^{*}\simeq\pi_{X_{\bullet}}$ denotes the image of the relative homology class
of $[(A,\Psi)]$ under the map $h_{A_{0}}$ from Remark 3.22, and
$i^{*}\colon\thinspace H^{*}(\overline{X_{\bullet}};{\mathbb{Z}})\to
H^{*}(\partial\overline{X_{\bullet}};{\mathbb{Z}})$ is part of the relative
long exact sequence of the pair
$(\overline{X_{\bullet}},\partial\overline{X_{\bullet}})$. Note that the
pairing between
$H^{1}(\partial\overline{X_{\bullet}};{\mathbb{Z}})/\mathop{\mathrm{Im}\,}\nolimits
i^{*}\simeq
H_{2}(\partial\overline{X_{\bullet}};{\mathbb{Z}})/\mathop{\mathrm{Ker}\,}\nolimits
i$ and $\mathop{\mathrm{Im}\,}\nolimits(i^{*}\colon\thinspace
H^{2}(\overline{X_{\bullet}};{\mathbb{Z}})\to
H^{2}(\partial\overline{X_{\bullet}};{\mathbb{Z}}))$ is well-defined because
the subspace $\mathop{\mathrm{Im}\,}\nolimits(i^{*}\colon\thinspace
H^{2}(\overline{X_{\bullet}};{\mathbb{Z}})\to
H^{2}(\partial\overline{X_{\bullet}};{\mathbb{Z}}))$ pairs trivially with
$\mathop{\mathrm{Ker}\,}\nolimits(i\colon\thinspace
H_{2}(\partial\overline{X_{\bullet}};{\mathbb{Z}})\to
H_{2}(\overline{X_{\bullet}};{\mathbb{Z}}))$.

To summarize:

######  Lemma 4.2.

Let $X$, $\mu$, $\hat{{\mathfrak{p}}}$ be as before. Given arbitrary (possibly
non-compact) $X_{\bullet}\subset X$ and an admissible
$(A,\Psi)\in{\mathscr{C}}(X_{\bullet})$, the topological energy
${\mathscr{E}}_{top}^{\mu}(X_{\bullet})(A,\Psi)$ is well-defined. Moreover,
their values depend only on the
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure of $X$,
$[(A,\Psi)|_{\partial\overline{X_{\bullet}}}]\in{\mathscr{B}}(\partial\overline{X_{\bullet}})$,
and the relative homotopy class of $[(A,\Psi)]$.

Moreover, combining (4.6) and Lemmas 3.12, 3.23, one has:

######  Lemma 4.3.

Adopt the assumptions and notations in the statement of Theorem 1.7. (In
particular, recall that $(A_{r},\Psi_{r})$ are Seiberg-Witten solutions
${\mathfrak{S}}_{\mu_{r},\hat{{\mathfrak{p}}}}(A_{r},\Psi_{r})=0$ with
$Y_{i}$-end limits proscribed by
$\\{\boldsymbol{\gamma}_{i}\\}_{i\in{\mathfrak{Y}}_{m}}$,
$\\{(B_{i},\Phi_{i})\\}_{i\in{\mathfrak{Y}}_{v}}$, and with fixed relative
homotopy class h.) Then there exist a constant
$\textsc{e},\in{\mathbb{R}}^{+}$ depending only on $\|\nu\|_{C^{1}}$,
$\varsigma_{w}$, and:

(4.7) $\hss\left\\{\mkern-20.0mu\parbox{422.77661pt}{\begin{itemize}
\par\itemize@item@the $\mathop{\mathrm{Spin}}\nolimits^{c}$-structure on $X$,
\par\itemize@item@the relative homotopy class $\text{{\scriptsize h}}$, or
equivalently ${\mathfrak{h}}(\text{{\scriptsize h}})$ (and hence also
implicitly on $\\{\boldsymbol{\gamma}_{i}\\}_{i\in{\mathfrak{Y}}_{m}}$),
\par\itemize@item@the cohomology class $[\nu]$, \par\end{itemize}}\right.$

such that for all $r\geq r_{0}$ ($r_{0}$ being larger or equal to that in
Lemma 3.12),

(4.8)
$|{\mathscr{E}}_{top}^{\mu_{r}}(X)(A_{r},\Psi_{r})|\leq\textsc{e}r,\quad\text{and}\quad|{\mathscr{E}}_{top}^{\mu_{r},\hat{{\mathfrak{p}}}}(X)(A_{r},\Psi_{r})|\leq\textsc{e}\,r.$

The next lemma says more about the coefficient e above.

######  Lemma 4.4.

(a) Let $Y_{i}$ be a Morse end. Suppose $\\{(B_{r},\Phi_{r})\\}_{r}$ is a
sequence of Seiberg-Witten solutions
${\mathfrak{F}}_{\mu_{i,r}}(B_{r},\Phi_{r})=0$ that strongly t-converges to
$\boldsymbol{\gamma}$. Then the limit

$\lim_{r\to\infty}(r^{-1}\operatorname{CSD}_{\mu_{i,r}}([B_{r},\Phi_{r}]_{c}))=\lim_{r\to\infty}(r^{-1}\operatorname{CSD}_{\mu_{i,r},{\mathfrak{q}}_{i}}([B_{r},\Phi_{r}]_{c}))$

exists, and

(4.9)
$\Big{|}\lim_{r\to\infty}(r^{-1}\operatorname{CSD}_{\mu_{i,r}}([B_{r},\Phi_{r})]_{c})\Big{|}\leq\frac{\pi}{2}\sqrt{b^{1}(Y_{i})}\,|[\nu_{i}]|.$

(b) Adopt the notations and assumptions of Theorem 1.7. Then the limit

${\mathbb{E}}:=\lim_{r\to\infty}(r^{-1}{\mathscr{E}}_{top}^{\mu_{r},\hat{{\mathfrak{p}}}}(A_{r},\Psi_{r}))$

exists, and equals
$\lim_{r\to\infty}(r^{-1}{\mathscr{E}}_{top}^{\mu_{r}}(A_{r},\Psi_{r}))$. It
is determined by the items listed in (4.7) via the formula (4.11) below.

Proof. (a): Let ${\mathfrak{c}}_{r}$ denote the gauge equivalence class of
$(B_{r},\Phi_{r})$. Recall the map
$\jmath_{c}\colon\thinspace\pi\,(\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}};[B_{0}],\Pi{\mathfrak{c}}_{r})\to
H^{1}(Y_{i};{\mathbb{R}})\simeq H_{2}(Y_{i};{\mathbb{R}})$ from Section 3.5
and let
$\tilde{c}_{0}({\mathfrak{c}}_{r})\in\pi\,(\mathop{\mathrm{Conn}}\nolimits/{\mathscr{G}};[B_{0}],\Pi{\mathfrak{c}}_{r})$
be the element represented by the path $s\mapsto
B_{0}+(1-\chi(s))(B_{r}-B_{0})$ on $\mathop{\mathrm{Conn}}\nolimits(Y_{i})$.
Then

(4.10)
$\begin{split}r^{-1}\operatorname{CSD}_{\mu_{i,r}}([B_{r},\Phi_{r}]_{c})&=r^{-1}\operatorname{CSD}_{\mu_{i,r},{\mathfrak{q}}_{i}}([B_{r},\Phi_{r}]_{c})\\\
&=r^{-1}\operatorname{CSD}_{w_{i,r}}({\mathfrak{c}}_{r})-\frac{i}{8}\int_{Y_{i}}([B_{r}]_{c}-B_{0})\wedge\nu_{i}\\\
&=r^{-1}\operatorname{CSD}_{w_{i,r}}({\mathfrak{c}}_{r})-\frac{\pi}{2}[\nu_{i}]\cdot\jmath_{c}(\tilde{c}_{0}({\mathfrak{c}}_{r})).\\\
\end{split}$

The assumption that $(B_{r},\Phi_{r})$ strongly converges implies that the
limit $\lim_{r\to\infty}\jmath_{c}(\tilde{c}_{0}({\mathfrak{c}}_{r}))$ exists,
and its norm is bounded by $\sqrt{b^{1}(Y_{i})}$. Denote this limit by
$\jmath_{h}(\boldsymbol{\gamma})\in H_{2}(Y_{i};{\mathbb{R}})$. Together with
Lemma 3.12, this implies that for $i\in{\mathfrak{Y}}_{m}$,

$\lim_{r\to\infty}(r^{-1}\operatorname{CSD}_{\mu_{i,r}}([B_{r},\Phi_{r}]_{c}))=-\frac{\pi}{2}[\nu_{i}]\cdot\jmath_{h}(\boldsymbol{\gamma}),$

and hence Assertion (a) of the lemma.

(b): By (4.6),

$\begin{split}r^{-1}{\mathscr{E}}_{top}^{\mu_{r}}(A_{r},\Psi_{r})&=\frac{1}{4r}\int_{X}F_{A_{0}}\wedge
F_{A_{0}}+\frac{i}{4r}\int_{X}F_{A_{0}}\wedge\mu_{r}\\\
&\qquad-2r^{-1}\sum_{i\in{\mathfrak{Y}}}\operatorname{CSD}_{\mu_{i,r}}^{Y_{i}}([(B_{i,r},\Phi_{i,r})]_{c})+i^{*}[\nu]\cdot
h_{A_{0}}(\operatorname{\text{{\scriptsize h}}})/2;\\\
r^{-1}{\mathscr{E}}_{top}^{\mu_{r},\hat{{\mathfrak{p}}}}(A_{r},\Psi_{r})&=\frac{1}{4r}\int_{X}F_{A_{0}}\wedge
F_{A_{0}}+\frac{i}{4r}\int_{X}F_{A_{0}}\wedge\mu_{r}\\\
&\qquad-2r^{-1}\sum_{i\in{\mathfrak{Y}}}\operatorname{CSD}_{\mu_{i,r},{\mathfrak{q}}_{i}}^{Y_{i}}([(B_{i,r},\Phi_{i,r})]_{c})+i^{*}[\nu]\cdot
h_{A_{0}}(\operatorname{\text{{\scriptsize h}}})/2,\end{split}$

where $h_{A_{0}}$ is the map defined in Remark 3.22.

Combined with Assertion (a) of the lemma and Lemma 3.12, this gives

(4.11)
$\begin{split}{\mathbb{E}}&:=\lim_{r\to\infty}\big{(}r^{-1}{\mathscr{E}}_{top}^{\mu_{r},\hat{{\mathfrak{p}}}}(A_{r},\Psi_{r})\big{)}=\lim_{r\to\infty}\big{(}r^{-1}{\mathscr{E}}_{top}^{\mu_{r}}(A_{r},\Psi_{r})\big{)}\\\
&=\frac{i}{4}\int_{X}F_{A_{0}}\wedge\nu+\pi\sum_{i\in{\mathfrak{Y}}_{m}}[\nu_{i}]\cdot\jmath_{h}(\boldsymbol{\gamma}_{i})+i^{*}[\nu]\cdot
h_{A_{0}}(\operatorname{\text{{\scriptsize h}}})/2.\end{split}$

$\Box$

######  Remarks 4.5.

(a) By Theorem 1.2 and the fact that there are finitely many t-orbits with a
fixed $\mathop{\mathrm{Spin}}\nolimits^{c}$ structure, the bound (4.9) holds
for any sequence of solutions to the 3-dimensional Seiberg-Witten equations
${\mathfrak{F}}_{\mu_{i,r}}(B_{r},\Phi_{r})=0$.

(b) By Item (b) of the previous lemma, the constant e in Lemma 4.3 may be
chosen to depend only on (4.7) (though $r_{0}$ may still depend on
$\|\nu\|_{C^{1}}$ and $\varsigma_{w}$).

To obtain the first group of results mentioned in the beginning of the present
section, we need a generalization of Lemma 4.3 to general $X_{\bullet}\subset
X$, with counterparts of the coefficient e in (4.8) independent of both $r$
and $X_{\bullet}$. This is much more difficult to achieve, mainly due to the
fact that the perturbation form $\nu$ is not translation-invariant on the ends
of $X$; cf. the second paragraph of Section 1.3. In fact, instead of bounding
${\mathscr{E}}_{top}^{\mu_{r}}$ and
${\mathscr{E}}_{top}^{\mu_{r},\hat{{\mathfrak{p}}}}$, we find it more
convenient to work with a modified version of them, which agree with them in
the case when $\nu$ is translation-invariant on the ends.

First, on each $Y_{i:s}\subset X-X_{c}$, let $\nu_{+}=\nu_{+}(s)$,
$\mu_{+}=\mu_{+}(s)$ respectively denote the $s$-dependent closed 2-forms on
$Y_{i}$:

$\nu_{+}(s):=2\nu^{+}|_{Y_{i:s}};\quad\mu_{+}(s)=2(\mu_{r}^{+})|_{Y_{i:s}}=r\nu_{+}+w_{i,r}$

Modifying (4.5), we set

(4.12)
$\begin{split}{\mathscr{E}}_{top}^{{}^{\prime}\mu_{r}}(X_{\bullet})(A,\Psi)&:={\mathscr{E}}_{top}^{\mu_{r}}(X_{\bullet})(A,\Psi)+\frac{i}{4}\int_{\partial\overline{X_{\bullet}}}(A-A_{0})\wedge(*_{4}\mu_{r})\\\
&=\frac{1}{4}\int_{X_{\bullet}}F_{A_{0}}\wedge
F_{A_{0}}+\frac{i}{4}\int_{X_{\bullet}}F_{A_{0}}\wedge\mu_{r}-2\operatorname{CSD}_{\mu_{+}}^{\partial\overline{X_{\bullet}}}(B,\Phi).\\\
\end{split}$

Define
${\mathscr{E}}_{top}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}(X_{\bullet})$
similarly by replacing the term ${\mathscr{E}}_{top}^{\mu_{r}}(X_{\bullet})$
above with ${\mathscr{E}}_{top}^{\mu_{r},\hat{{\mathfrak{p}}}}(X_{\bullet})$.
Note that

${\mathscr{E}}_{top}^{{}^{\prime}\mu_{r}}(X)={\mathscr{E}}_{top}^{\mu_{r}}(X);\qquad{\mathscr{E}}_{top}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}(X)={\mathscr{E}}_{top}^{\mu_{r},\hat{{\mathfrak{p}}}}(X).$

Therefore the bounds in (4.8) hold for
${\mathscr{E}}_{top}^{{}^{\prime}\mu_{r}}(X)$,
${\mathscr{E}}_{top}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}(X)$ as well.

A preliminary version of the aforementioned generalization of Lemma 4.3 is
stated in terms of ${\mathscr{E}}_{top}^{{}^{\prime}\mu_{r}}$ and
${\mathscr{E}}_{top}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}$ as follows:

######  Lemma 4.6.

Let $(A,\Psi)=(A_{r},\Psi_{r})$ be an admissible solution to the Seiberg-
Witten equation
${\mathfrak{S}}_{\mu_{r},\hat{{\mathfrak{p}}}}(A_{r},\Psi_{r})=0$ on $X$ that
satisfies in addition

(4.13) $\text{either
(a)}\,\,{\mathscr{E}}_{top}^{\mu_{r}}(X)(A_{r},\Psi_{r})\leq\textsc{e}\,r\quad\text{
or
(b)}\,\,{\mathscr{E}}_{top}^{\mu_{r},\hat{{\mathfrak{p}}}}(X)(A_{r},\Psi_{r})\leq\textsc{e}\,r$

for a positive constant e independent of $r$ and $(A,\Psi)$. (In particular,
according to Lemma 4.3 and Remarks 4.5, this holds for those
$(A_{r},\Psi_{r})$ from the statement of Theorem 1.7, with e determined by
(4.7) via (4.11).) Then there exist constants $\zeta>0$ $\zeta^{\prime}>0$, a
function
$\boldsymbol{\hat{\mathop{\mathrm{l}}}}\colon\thinspace{\mathfrak{Y}}\to{\mathbb{R}}^{+}$
depending only on $\nu$, and an $r_{0}>8$ depending only on the constants e,
${\mathfrak{l}}^{\prime}_{i}$, $i\in{\mathfrak{Y}}_{v}$, and $\nu$, such that
for all $r\geq r_{0}$ and ${\bf l}_{r}:=(\ln
r)\,\boldsymbol{\hat{\mathop{\mathrm{l}}}}$,

(4.14) $\begin{split}\text{(i) }&\begin{cases}-\zeta^{\prime}r\ln
r\leq{\mathscr{E}}_{top}^{{}^{\prime}\mu_{r}}(X_{\bullet})(A_{r},\Psi_{r})\leq
r(\textsc{e}+\zeta)&\text{assuming (\ref{assume:EtopX-ubdd}) (a)}\\\
-\zeta^{\prime}_{p}r\ln
r\leq{\mathscr{E}}_{top}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}(X_{\bullet})(A_{r},\Psi_{r})\leq
r(\textsc{e}+\zeta_{p})&\text{assuming (\ref{assume:EtopX-ubdd})
(b)}\end{cases}\quad\text{$\forall X_{\bullet}\supset X_{{\bf l}_{r}}$};\\\
\text{(ii)
}&-\zeta^{\prime}_{5}r\leq{\mathscr{E}}_{top}^{{}^{\prime}\mu_{r}}(X_{\bullet})(A_{r},\Psi_{r})={\mathscr{E}}_{top}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}(X_{\bullet})(A_{r},\Psi_{r})\leq
r(\textsc{e}+\zeta_{e}\ln r)\quad\text{$\forall X_{\bullet}\subset
X-\mathring{X}_{{\bf l}_{r}}$}.\end{split}$

The positive constants $\zeta$, $\zeta^{\prime}$, $\zeta^{\prime}_{5}$,
$\zeta_{e}$ above depend only on the metric, the
$\mathop{\mathrm{Spin}}\nolimits^{c}$ structure, $\nu$, $\varsigma_{w}$,
$B_{0}$, and the constants $\textsl{z}_{i}$ in Lemma 3.11. $\zeta_{p}$,
$\zeta^{\prime}_{p}$ are positive constants that depend only on the preceding
list of parameters, together with the constant $\textsl{z}_{p}$ in Assumption
3.4. In particular, these constants as well as
$\boldsymbol{\hat{\mathop{\mathrm{l}}}}$ and $r_{0}$ above are independent of
$X_{\bullet}$ and $r$.

A proof of the preceding lemma will be given in Section 4.5. The undesirable
factors of $\ln r$ in (4.14) appear due to the previously-mentioned trouble
with non-translation invariant $\nu$ at the ends (cf. (4.38) and remarks that
follow). They will eventually be removed in Section 6.2. (See Proposition
6.4.)

Looking ahead, the definitions and lemmas above are relevant to the first
group of results mentioned in the beginning of this section in the following
manner: The typical first step towards such results is to use the relation
between analytical and topological energies to bound
$L^{2}(X_{\bullet})$-norms of gauge invariant terms such as $F_{A}$ and
$\nabla_{A}\Psi$ in terms of the topological energy over $X_{\bullet}$. (In
our context, this appears as Lemmas 4.7 and 4.8 below.) Upper bounds on the
topology energy over $X_{\bullet}$, such as those given in Lemma 4.6 and
Proposition 6.4 in our context, then give rise to upper bounds on the
aforementioned squares of $L^{2}(X_{\bullet})$-norms of gauge invariant terms.
The latter are used to obtain $L^{2}_{1,A}(X_{\bullet})$ bounds on Seiberg-
Witten solutions. (This is done in Section 4.6 in our context.)

In existing literature,$(X,\nu)$ is cylindrical on the ends, and upper bounds
on the topological energy over $X_{\bullet}$ follows directly from a
corresponding bound over $X$. (The analog of Lemma 4.3 in our context.) This
is due to the fact that the $\operatorname{CSD}$ functional is decreasing on a
Floer trajectory, which can be interpreted as the gradient flow line of the
$\operatorname{CSD}$ functional. In the more general setting of ours, local
upper bounds for ${\mathscr{E}}_{top}^{\mu}$ likewise follows from a lower
bound on ${\mathscr{E}}_{top}^{\mu}$ for $X_{\bullet}$ that are contained in
the cylindrical ends of $X$; see Section 4.4. This lower bound makes use of an
interpretation of the Floer trajectory as a gradient flow line of a time-
dependent $\operatorname{CSD}$-functional; see Section 4.3 below.

###  4.2  $L^{2}_{local}$ bounds on gauge invariant terms in terms of
(modified) topological energies

Let $(A_{r},\Psi_{r})=(A,\Psi)$ be an admissible solution to the Seiberg-
Witten equation
${\mathfrak{S}}_{\mu_{t},\hat{{\mathfrak{p}}}}(A_{r},\Psi_{r})=0$ with
$Y_{i}$-end limit $(B_{i},\Phi_{i})$, and let $A_{0}$ $\\{B_{0,i}\\}_{i}$ be
the reference connections fixed previously. Recall the definition of
$|X_{\bullet}|$ from Section 1.4 and let

$|X_{\bullet}|_{1}:=\min\\{1,|X_{\bullet}|\\}.$

Recall also the definitions of $X^{{}^{\prime}a}$, $X^{\prime\prime}$ from
Definition 1.5, and note that $\hat{{\mathfrak{p}}}(A,\Psi)$ is supported on
the vanishing ends; in fact, on $X-X^{\prime\prime}\supset
X-X^{{}^{\prime}a}$. Let

$X_{\bullet,v}:=X_{\bullet}\cap(X-X^{\prime\prime})\qquad\text{and}\qquad
X_{\bullet,m}:=X_{\bullet}\cap X^{\prime\prime}.$

In this subsection, we show that:

######  Lemma 4.7.

Let $X_{\bullet}\subset X$ be compact and let $r\geq 1$. Then there exist
positive constants $\zeta_{0}$, $\zeta^{\prime}$, $\zeta^{\prime}_{p}$,
$\zeta^{\prime\prime}$, $\zeta^{\prime\prime}_{p}$ that depend only on

(4.15) $\hss\left\\{\mkern-20.0mu\parbox{422.77661pt}{\begin{itemize}
\par\itemize@item@the metric, \par\itemize@item@the
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure on $X$, \par\itemize@item@the
cohomology class of $\nu$, \par\itemize@item@the constant $\varsigma_{w}$ in
Assumption \ref{assume}, \par\itemize@item@the constant
$\textsl{z}_{\mathfrak{p}}$ in Assumption \ref{assume},
\par\end{itemize}}\right.$

such that the following hold:

(4.16)
$\begin{split}\text{(a)}&\quad\frac{1}{8}\int_{X_{\bullet}}|F_{A}|^{2}+\int_{X_{\bullet}}|\nabla_{A}\Psi|^{2}+\frac{1}{2}\int_{X_{\bullet}}\big{|}\frac{i}{4}\rho(\mu_{r}^{+})-(\Psi\Psi^{*})_{0}\big{|}^{2}\\\
&\qquad\quad\leq{\mathscr{E}}_{top}^{{}^{\prime}\mu_{r}}(X_{\bullet})(A,\Psi)+r\,(\zeta_{0}|X^{\prime}_{\bullet,m}|+\zeta^{\prime}|X_{\bullet}|_{1})+\zeta^{\prime\prime}|X_{\bullet}|;\\\
\text{(b)}&\quad\frac{1}{16}\int_{X_{\bullet}}|F_{A}|^{2}+\int_{X_{\bullet}}|\nabla_{A}\Psi|^{2}+\frac{1}{4}\int_{X_{\bullet}}\big{|}\frac{i}{4}\rho(\mu_{r}^{+})-(\Psi\Psi^{*})_{0}\big{|}^{2}\\\
&\qquad\quad\leq{\mathscr{E}}_{top}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}(X_{\bullet})(A,\Psi)+r\,(\zeta_{0}|X^{\prime}_{\bullet,m}|+\zeta^{\prime}_{p}|X_{\bullet}|_{1})+\zeta^{\prime\prime}_{p}|X_{\bullet}|.\end{split}$

Proof. To begin, combine (4.4) with the Seiberg-Witten equation
${\mathfrak{G}}_{\mu_{r},\hat{{\mathfrak{p}}}}(A_{r},\Psi_{r})=0$ to get:

(4.17)
$\begin{split}&\frac{1}{4}\int_{X_{\bullet}}|F_{A}|^{2}+\int_{X_{\bullet}}|\nabla_{A}\Psi|^{2}+\int_{X_{\bullet}}\big{|}\frac{i}{4}\rho(\mu_{r}^{+})-(\Psi\Psi^{*})_{0}\big{|}^{2}\\\
&\quad=-\int_{X_{\bullet}}\frac{R_{g}}{4}|\Psi|^{2}+\frac{i}{4}\int_{X_{\bullet}}F_{A}\wedge*_{4}\mu_{r}+{\mathscr{E}}_{top}^{\mu_{r}}(X_{\bullet})(A,\Psi)+\|\hat{{\mathfrak{p}}}(A,\Psi)\|_{L^{2}(X_{\bullet})}^{2}\\\
&\quad=-\int_{X_{\bullet}}\frac{R_{g}}{4}|\Psi|^{2}+\|\hat{{\mathfrak{p}}}(A,\Psi)\|_{L^{2}(X_{\bullet})}^{2}+\frac{i}{4}\int_{X_{\bullet}}F_{A}\wedge*_{4}w_{r}\\\
&\qquad\quad+\frac{ir}{4}\int_{X_{\bullet}}F_{A_{0}}\wedge*_{4}\nu+{\mathscr{E}}_{top}^{{}^{\prime}\mu_{r}}(X_{\bullet})(A,\Psi).\end{split}$

The terms in the third and fourth line above are bounded in Steps (1)-(3)
below.

(1) The terms $-\int_{X_{\bullet}}\frac{R_{g}}{4}|\Psi|^{2}$,
$\|\hat{{\mathfrak{p}}}(A,\Psi)\|_{L^{2}(X_{\bullet})}^{2}$ are bounded by the
same general trick. Observe that for any real valued function
${\mathfrak{f}}$,

$\begin{split}0&\leq\Big{|}\frac{i}{4}\rho(\mu^{+}_{r})-\big{(}1-\frac{{\mathfrak{f}}}{2|\Psi|^{2}}\big{)}(\Psi\Psi^{*})_{0}\Big{|}^{2}\\\
&\leq\Big{|}\frac{i}{4}\rho(\mu^{+}_{r})-(\Psi\Psi^{*})_{0}\Big{|}^{2}+\frac{{\mathfrak{f}}^{2}}{4}-{\mathfrak{f}}|\Psi|^{2}+\frac{|{\mathfrak{f}}|}{8}|\mu_{r}|\end{split}$

where $\Psi\neq 0$. So, for general $\Psi$,

(4.18)
${\mathfrak{f}}|\Psi|^{2}\leq\Big{|}\frac{i}{4}\rho(\mu^{+}_{r})-(\Psi\Psi^{*})_{0}\Big{|}^{2}+\frac{{\mathfrak{f}}^{2}}{4}+\frac{|{\mathfrak{f}}|}{8}|\mu_{r}|.$

Taking ${\mathfrak{f}}=\pm R_{g}$ in (4.18) then gives us

(4.19)
$\begin{split}\Big{|}-\int_{X_{\bullet}}\frac{R_{g}}{4}|\Psi|^{2}\Big{|}&\leq\frac{1}{4}\Big{\|}\frac{i}{4}\rho(\mu^{+}_{r})-(\Psi\Psi^{*})_{0}\Big{\|}^{2}_{L^{2}(X_{\bullet})}+\int_{X_{\bullet}}\frac{R_{g}^{2}}{16}+\int_{X_{\bullet}}\frac{|R_{g}|}{32}|\mu_{r}|\\\
&\leq\frac{1}{4}\Big{\|}\frac{i}{4}\rho(\mu^{+}_{r})-(\Psi\Psi^{*})_{0}\Big{\|}^{2}_{L^{2}(X_{\bullet})}+\zeta_{g}|X_{\bullet}|+\zeta_{h}\,r(|X_{\bullet,m}|+|X_{\bullet}|_{1}).\end{split}$

In the above, $\zeta_{g}$, $\zeta_{h}$ are positive constants depending only
on the metric, $\varsigma_{w}$, and $\nu$. To go from the first line to the
second line, we also made use of the well-known fact that $\nu$ exponentially
decays to $\nu_{i}$ on the $Y_{i}$-end for each $i\in{\mathfrak{Y}}$ (cf.
(4.29)).

Meanwhile by (3.4),

$\begin{split}&\|\hat{{\mathfrak{p}}}(A,\Psi)\|_{L^{2}(X_{\bullet})}^{2}\leq
z\big{(}\|\Psi\|_{L^{2}(X_{\bullet,v})}^{2}+|X_{\bullet,v}|\big{)},\end{split}$

where $z$ is a positive constant depending only on the constant
$\textsl{z}_{\mathfrak{p}}$ and the metric on the vanishing ends. Taking
${\mathfrak{f}}=4z$ in (4.18) then gives us

(4.20)
$\begin{split}\|\hat{{\mathfrak{p}}}(A,\Psi)\|_{L^{2}(X_{\bullet})}^{2}&\leq\frac{1}{4}\Big{\|}\frac{i}{4}\rho(\mu^{+}_{r})-(\Psi\Psi^{*})_{0}\Big{\|}^{2}_{L^{2}(X_{\bullet,v})}+z_{1}|X_{\bullet,v}|_{1}+z_{2}r\int_{X_{\bullet,v}}|\nu|\\\
&\leq\frac{1}{4}\Big{\|}\frac{i}{4}\rho(\mu^{+}_{r})-(\Psi\Psi^{*})_{0}\Big{\|}^{2}_{L^{2}(X_{\bullet,v})}+z_{1}|X_{\bullet,v}|_{1}+z_{3}r|X_{\bullet,v}|_{1},\end{split}$

noting that $|\nu|$ decays exponentially to 0 on vanishing ends. (For a more
precise statement, see (4.29).) In the above, $z_{i}$ are positive constants
that depend only on $\nu$, the metrics, and the constants $\varsigma_{w}$,
$\textsl{z}_{\mathfrak{p}}$.

(2) The term $\frac{i}{4}\int_{X_{\bullet}}F_{A}\wedge*_{4}w_{r}$ in (4.17) is
bounded as follows.

(4.21)
$\frac{i}{4}\int_{X_{\bullet}}F_{A}\wedge*_{4}w_{r}\leq\frac{1}{16}\|F_{A}\|_{L^{2}(X_{\bullet})}^{2}+\|w_{r}\|_{L^{2}(X_{\bullet})}^{2}\leq\frac{1}{16}\|F_{A}\|_{L^{2}(X_{\bullet})}^{2}+z_{4}|X_{\bullet}|,$

where $z_{4}$ depends only on the constant $\varsigma_{w}$.

(3) The term $\frac{ir}{4}\int_{X_{\bullet}}F_{A_{0}}\wedge*_{4}\nu$ in (4.17)
is bounded as follows. Recall that by assumption, $F_{A_{0}}$ is the pull-back
of $F_{B_{0,i}}$ on the ends $\hat{Y}_{i}$, and that $\wp_{i}=0$ for vanishing
ends. Thus,

(4.22) $\begin{split}&\frac{ir}{4}\int_{X_{\bullet}}F_{A_{0}}\wedge*_{4}\nu\\\
&\qquad=\frac{ir}{4}\int_{X_{\bullet}\cap
X_{c}}F_{A_{0}}\wedge*_{4}\nu+\sum_{i\in{\mathfrak{Y}}}r\wp_{i}|X_{\bullet}\cap\hat{Y}_{i}|+r\sum_{i\in{\mathfrak{Y}}}\int_{X_{\bullet}\cap\hat{Y}_{i}}F_{A_{0}}\wedge*_{4}(\nu-\nu_{i})\\\
&\qquad=\frac{ir}{4}\int_{X_{\bullet}\cap
X_{c}}F_{A_{0}}\wedge*_{4}\nu+\sum_{i\in{\mathfrak{Y}}_{m}}r\wp_{i}|X_{\bullet}\cap\hat{Y}_{i}|+r\,\zeta_{5}\sum_{i\in{\mathfrak{Y}}}|X_{\bullet}\cap\hat{Y}_{i}|_{1}\\\
&\qquad\leq r\,(z_{5}|X_{\bullet,m}|+\zeta^{\prime}_{5}).\end{split}$

Here, $\zeta_{5}$ and $\zeta_{5}^{\prime}$ are positive constants depending
only on the choice of $A_{0}$ and $\nu$, while $z_{5}$ is a positive constant
depending only on the choice of $A_{0}$, the
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure, and $\nu$. In estimating the
last term in the second line above, we also used the exponential decay of
$\nu$ on the ends of $X$. (Cf. (4.29) for a precise statement.)

Combining the bounds (4.19), (4.20), (4.21), (4.22) with (4.17) and re-
arranging, we arrive at item (a) of (4.16). The inequality in item (b) follows
from item (a) when $X_{\bullet}\subset X^{{}^{\prime}a}$. By the additivity of
${\mathscr{E}}_{top}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}$, it remains to
verify the inequality on the vanishing ends. The proof in this case is similar
and is deferred to the next subsection. $\Box$

Recall the definition of $\wp_{i}$ from Lemma 3.11. For each
$i\in{\mathfrak{Y}}$, set

$\wp_{i}^{+}:=\begin{cases}\wp_{i}+\frac{\pi}{2}\|*\nu_{i}\|_{T}&\text{when
$\hat{Y}_{i}$ is a Morse end};\\\ 0&\text{when $\hat{Y}_{i}$ is a vanishing
end.}\end{cases}$

Note that according to Proposition 3.15 and Remarks 3.16,
$\wp_{i}^{+}\geq\frac{\pi}{2}(\|*\nu_{i}\|_{T}+\zeta_{*\nu_{i}})$ in our
context, with the right hand side,
$\frac{\pi}{2}(\|*\nu_{i}\|_{T}+\zeta_{*\nu_{i}})$ vanishing in many cases,
such as in the context of [LT].

When $X_{\bullet}=\hat{Y}_{i,[l,L]}$, the bound (4.16) (a) can be refined as
follows.

######  Lemma 4.8.

Let $(A,\Psi)=(A_{r},\Psi_{r})$ be as in Lemma 4.7. When
$X_{\bullet}=\hat{Y}_{i,[l,L]}$, we have:

(4.23)
$\begin{split}&\frac{1}{8}\int_{X_{\bullet}}|F_{A}|^{2}+\int_{X_{\bullet}}|\nabla_{A}\Psi|^{2}+\frac{1}{2}\int_{X_{\bullet}}\Big{|}\frac{i}{4}\rho(\mu_{r}^{+})-(\Psi\Psi^{*})_{0}\Big{|}^{2}\\\
&\qquad\quad\leq{\mathscr{E}}_{top}^{{}^{\prime}\mu_{r}}(X_{\bullet})(A,\Psi)+r\,(\wp_{i}^{+}+z_{i})|X_{\bullet}|+rz^{\prime}_{i}|X_{\bullet}|_{1}+\zeta^{\prime\prime}|X_{\bullet}|,\end{split}$

where $z_{i}$ is a non-negative constant depending only on the metric on
$Y_{i}$, and $z^{\prime}_{i}$ is a positive constant that only depends on the
metric on $X$, the cohomology class of $\nu$, and the constant $\varsigma_{w}$
in Assumption 3.4. (In particular, $z_{i}$ and $z^{\prime}_{i}$ are
independent of $r$ and the $\mathop{\mathrm{Spin}}\nolimits^{c}$ structure.)
Moreover, the constant $z_{i}$ has the following properties:
$z_{i}=0=\wp^{+}+z_{i}$ when $\hat{Y}_{i}$ is a vanishing end. When
$\hat{Y}_{i}$ is a Morse end and $Y_{i}$ is irreducible and atoroidal, then
for any $\epsilon>0$, there exists a metric on $Y_{i}$ so that its
corresponding $z_{i}$ is less than $\epsilon$. Meanwhile,
$\zeta^{\prime\prime}$ is a positive constant that depends only on the metric
and $\varsigma_{w}$.

Proof. To begin, recall [KM1]’s Theorem 2. Reformulated as [BD]’s Theorem 5.4,
it asserts that when $Y_{i}$ is irreducible, and atoroidal,

(4.24)
$\|*\nu_{i}\|_{T}=\frac{1}{4\pi}\inf_{h}\\{\|R_{h}\|_{h}\,\|\theta_{h}\|_{h}\\},$

where $h$ ranges through all Riemannian metrics on $Y_{i}$. In the above,
$\|\cdot\|_{h}$ denotes $L^{2}$-norm with respect to $h$, and $\theta_{h}$
denotes the harmonic representative of the cohomology class $[*\nu_{i}]$ with
respect to the metric $h$. $R_{h}$ denotes the scalar curvature on $Y_{i}$
corresponding to $h$. Recall also that $\|*\nu_{i}\|_{T}=\|[*\nu_{i}]\|_{T}$
is the Thurston semi-norm of $[*\nu_{i}]\in H^{1}(Y_{i};{\mathbb{R}})$.

By assumption, over $\hat{Y}_{i}\simeq[0,\infty)\times Y_{i}$, $g$ is the
product metric of the standard affine metric on ${\mathbb{R}}$ and a metric
$g_{i}$ on $Y_{i}$. Thus, $R_{g}|_{Y_{i:s}}=R_{g_{i}}$ for any $s\geq 0$.
Combined with the asymptotic behavior of $\nu$ (described explicitly in (4.27)
below), (4.24) implies that for any $s\geq 0$,

$\|R_{g}\|_{L^{2}(Y_{i:s})}\|\nu|_{Y_{i:s}}\|_{L^{2}(Y_{i:s})}\leq
4\pi\|*\nu_{i}\|_{T}+z_{i}+{\mathfrak{z}}_{i}(s),$

where
$z_{i}=z_{Y_{i},\nu_{i}}:=\|R_{g_{i}}\|_{L^{2}(Y_{i})}\|\nu_{i}\|_{L^{2}(Y_{i})}-4\pi\|*\nu_{i}\|_{T}$.
According to (4.24), $z_{i}$ has the properties asserted in the statement of
the lemma. ${\mathfrak{z}}_{i}(s)$ is a positive function on $[0,\infty)$ that
depends only on $\nu$ and exponentially decays to $0$ as $s\to\infty$.
(Explicitly, it is a multiple of the right hand side of (4.29).) Use the
preceding inequality to replace (4.19) in this context by:

(4.25)
$\begin{split}&\Big{|}-\int_{X_{\bullet}}\frac{R_{g}}{4}|\Psi|^{2}\Big{|}\\\
&\quad\leq\frac{1}{4}\Big{\|}\frac{i}{4}\rho(\mu^{+}_{r})-(\Psi\Psi^{*})_{0}\Big{\|}^{2}_{L^{2}(X_{\bullet})}+\int_{X_{\bullet}}\frac{R_{g}^{2}}{16}+\int_{X_{\bullet}}\frac{|R_{g}|}{32}|\mu_{r}|\\\
&\quad\leq\frac{1}{4}\Big{\|}\frac{i}{4}\rho(\mu^{+}_{r})-(\Psi\Psi^{*})_{0}\Big{\|}^{2}_{L^{2}(X_{\bullet})}+\zeta_{1}|X_{\bullet}|+\frac{r}{32}\big{(}\int_{l}^{L}ds\|R_{g}\|_{L^{2}(Y_{i:s})}\|\nu|_{Y_{i:s}}\|_{L^{2}(Y_{i:s})}\big{)}\\\
&\quad\leq\frac{1}{4}\Big{\|}\frac{i}{4}\rho(\mu^{+}_{r})-(\Psi\Psi^{*})_{0}\Big{\|}^{2}_{L^{2}(X_{\bullet})}+\zeta_{1}|X_{\bullet}|+r\,\big{(}\frac{\pi}{2}\|*\nu_{i}\|_{T}+z_{i}\big{)}|X_{\bullet}|+r\zeta_{2}|X_{\bullet}|_{1},\end{split}$

where $\zeta_{1}$, $\zeta_{2}$ are positive constants depending only on
$R_{g}$ and $\varsigma_{w}$. (In particular, they are independent of $r$ and
${\mathfrak{s}}_{i}$.)

Meanwhile, replace (4.22) in this context by

(4.26) $\frac{ir}{4}\int_{X_{\bullet}}F_{A_{0}}\wedge*_{4}\nu\leq
r\wp_{i}|X_{\bullet}|+r\zeta_{5}|X_{\bullet}|_{1},$

$\zeta_{5}$ being a positive constant depending only on $\nu$. Combine the
bounds (4.25), (4.21), (4.26) with (4.17) and re-arranging, we arrive at
(4.23). $\Box$

######  Remarks 4.9.

Since both $r$ and $|X_{\bullet}|$ can be arbitrarily large, applying (4.23)
to the case when $X={\mathbb{R}}\times Y$ and
$(A_{r},\Psi_{r})=(\hat{B}_{r},\hat{\Phi}_{r})$ where $(B_{r},\Phi_{r})$ is a
sequence of solutions to the 3-dimensional Seiberg-Witten equation
${\mathfrak{F}}_{\mu_{r}}(B_{r},\Phi_{r})=0$ from Theorem 1.2 implies that
$c_{1}({\mathfrak{s}})\cdot[*\nu]+\|*\nu\|_{T}\geq-\frac{2z_{Y,\nu}}{\pi}$
when the sequence of $(B_{r},\Phi_{r})$ in Theorem 1.2 exists. This bound is
comparable to [KM]’s Propositions 40.1.1 and 40.1.3, where certain non-
existence results of Seiberg-Witten solutions are obtained under similar
constraints on $c_{1}({\mathfrak{s}})$. In particular, in the case when
$b_{1}(Y)>0$, the aforementioned propositions in [KM] were used to prove that
for irreducible $Y$, the dual Thurston polytope is the convex hull of the
“Seiberg-Witten basic classes” (cf. [KM1] Theorem 1 and [KM] Theorem 41.5.2).

###  4.3 Seiberg-Witten solutions as gradient flow lines of time-dependent CSD
functionals

We begin with some preliminary observations on the behavior of the form $\nu$
on the ends, and introduce some notations along the way. In particular, the
previously imentioned exponential decay of $\nu$ on the ends of $X$ is made
precise.

Consider an end $\hat{Y}_{i}$ of $X$. Write the harmonic 2-form $\nu$ as

$\nu=\underline{\nu}+ds\wedge*_{Y}{\mathfrak{v}}\ \quad\text{on
$\hat{Y}_{i}$,}$

where $\underline{\nu}$, ${\mathfrak{v}}$ are $s$-dependent 2-forms on
$Y:=Y_{i}$, $*_{Y}$ denotes the 3-dimensional Hodge dual on $Y_{i}$, and
$s=\text{{\small s}}_{i}$. The harmonicity of $\nu$ implies that both
$\underline{\nu}$ and ${\mathfrak{v}}$ are closed 2-forms. Moreover,
${\mathfrak{v}}$ and $\underline{\nu}-\nu_{i}$ are $s$-dependent exact 2-forms
on $Y_{i:s}=\text{{\small s}}_{i}^{-1}(s)$, and satisfy

(4.27)
$\partial_{s}(\underline{\nu}-\nu_{i})=-d_{Y}*_{Y}{\mathfrak{v}}\quad\text{and}\quad\partial_{s}{\mathfrak{v}}=-d_{Y}*_{Y}(\underline{\nu}-\nu_{i}).$

Here, $d_{Y}$ denotes the exterior derivative on $Y_{i}$.

Let

$\xi_{\nu}:=\underline{\nu}-\nu_{i}+{\mathfrak{v}};\quad\xi_{\nu}^{\prime}:=\underline{\nu}-\nu_{i}-{\mathfrak{v}},$

so that in this notation, $\nu_{+}=\nu_{i}+\xi_{\nu}$ on $\hat{Y}_{i}$.
$\xi_{\nu}$ and $\xi_{\nu}^{\prime}$ are exact 2-forms satisfying

(4.28)
$\partial_{s}\xi_{\nu}=-d_{Y}*_{Y}\xi_{\nu};\quad\partial_{s}\xi_{\nu}^{\prime}=d_{Y}*_{Y}\xi_{\nu}^{\prime}.$

As $d_{Y}*_{Y}$ is a self-dual operator on the space of $L^{2}$ exact 2-forms
and $Y_{i}$ is closed, it has discrete real eigenvalues
$\\{\kappa_{j}^{(i)}\\}_{j}$ with $\min_{j}|\kappa_{j}^{(i)}|=:\kappa_{i}>0$.
Thus, there exists a constant $\zeta_{i}>0$ such that

(4.29)
$\begin{split}&\|\xi_{\nu}\|_{C^{k}(Y_{i:s})}+\|\underline{\nu}-\nu_{i}\|_{C^{k}(Y_{i:s})}+\|{\mathfrak{v}}\|_{C^{k}(Y_{i:s})}\leq\zeta_{i}e^{-\kappa_{i}s};\quad\text{and
via (\ref{DE:xi}), }\\\
&\|\nu-\pi_{2}^{*}\nu_{i}\|_{C^{k}(Y_{i,s})}\leq\,\zeta_{i}e^{-\kappa_{i}s}.\end{split}$

The ($s$-dependent) form $\nu_{+}$ on $Y_{i}$, introduced previously in
Section 4.1, is expressed in this part’s notation as:

$\begin{split}\nu_{+}&=\underline{\nu}+{\mathfrak{v}}.\\\
&=\nu_{i}+\xi_{\nu}.\end{split}$

Returning now to the task of interpreting of Seiberg-Witten solutions on the
ends of $X$, we choose to work in a gauge such that $(A,\Psi)|_{\hat{Y}_{i}}$
is in the temporal gauge over each end $\hat{Y}_{i}$ of $X$. Namely, over the
ends $X-X_{c}$ we can write

(4.30) $(A,\Psi)=(\partial_{s}+B,\Phi),$

where $(B(s),\Phi(s))$, $s\geq 0$, is a path in
$\mathop{\mathrm{Conn}}\nolimits(Y_{i})\times\Gamma({\mathbb{S}}_{Y_{i}})$.
Recall that the reference connection $A_{0}$ is chosen such that over the ends
of $X$, $A_{0}=\partial_{s}+B_{0}$, where $B_{0}=B_{0,i}$ on the $Y_{i}$-end.

Restricting to an end $\hat{Y}_{i}$ of $X$, the 4-dimensional Seiberg-Witten
equation ${\mathfrak{S}}_{\mu_{r},\hat{{\mathfrak{p}}}}(A,\Psi)=0$ may be re-
expressed in terms of $(B(s),\Phi(s))$ over $\hat{Y}_{i}$ as

(4.31)
$\big{(}\frac{1}{2}\partial_{s}B,\partial_{s}\Phi\big{)}+\mathfrak{F}_{\mu_{+},\hat{{\mathfrak{p}}}}(B,\Phi)=0,$

where
$\hat{{\mathfrak{p}}}=\hat{{\mathfrak{p}}}(s)=\chi_{i}(s){\mathfrak{q}}_{i}+\lambda_{i}(s){\mathfrak{p}}^{\prime}_{i}$
is regarded as a path of tame perturbations as in (3.6).

Square both sides of the previous equation and use integration by parts plus
(4.27) to get:

(4.32)
$\begin{split}&\|\partial_{s}B\|_{L^{2}(\hat{Y}_{[l,L]})}^{2}+4\|\partial_{s}\Phi\|^{2}_{L^{2}(\hat{Y}_{[l,L]})}+4\|\mathfrak{F}_{\mu_{+},\hat{{\mathfrak{p}}}}(B,\Phi)\|_{L^{2}(\hat{Y}_{[l,L]})}^{2}\\\
&\quad\quad=8\operatorname{CSD}_{\mu_{+}(l),\hat{{\mathfrak{p}}}(l)}^{Y}(B,\Phi)|_{Y_{l}}-8\operatorname{CSD}_{\mu_{+}(L),\hat{{\mathfrak{p}}}(L)}^{Y}(B,\Phi)|_{Y_{L}}\\\
&\qquad\quad-
ir\int_{\hat{Y}_{[l,L]}}ds\,(*_{Y}\xi_{\nu})\wedge(F_{B}-F_{B_{0}})-8\int_{\hat{Y}_{[l,L]}}(\partial_{s}f_{\hat{{\mathfrak{p}}}(s)})(B,\Phi)\,ds\end{split}$

for any interval $[l,L]\subset[0,\infty]$. (Here $Y$ stands for any of the
ending 3-manifolds $Y_{i}$.)

In what follows we use the preceding formula in two ways: (i) In the remainder
of this subsection, we use the upper bound on the right hand side of (4.32) to
get an upper bound on the square terms on the left hand side; in particular,
this leads to the completion of the proof of (4.16) (b). (ii) In the
subsequent subsection, we use the non-negativity of the left hand side to
obtain a lower bound of the first two terms on the right hand side, and hence
a lower bound on
${\mathscr{E}}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}_{top}(\hat{Y}_{[l,L]})(A,\Psi)$,
generalizing the positivity results in the cylindrical case mentioned towards
the end of Section 4.1.

##### Verifying the remaining case of (4.16) (b).

As observed in the previous subsection, it remains to verify the claimed
inequality for the case when $X_{\bullet}$ lies in vanishing ends.
Nevertheless, assume for the moment that $X_{\bullet}=\hat{Y}_{[l,L]}$,
$\hat{Y}=\hat{Y}_{i}$ is a general end of $X_{\bullet}$. (Either vanishing or
Morse.)

To begin, recall from Assumption 3.4 (3) and use the 3-dimensional Weitzenböck
formula (3.14) to get

$\begin{split}&\frac{1}{4}\|\partial_{s}B\|_{L^{2}(\hat{Y}_{[l,L]})}^{2}+\|\partial_{s}\Phi\|^{2}_{L^{2}(\hat{Y}_{[l,L]})}+\|\mathfrak{F}_{\mu_{+}}(B,\Phi)\|_{L^{2}(\hat{Y}_{[l,L]})}^{2}\\\
&\quad=\frac{1}{4}\int_{\hat{Y}_{[l,L]}}|F_{A}|^{2}+\int_{\hat{Y}_{[l,L]}}|\nabla_{A}\Psi|^{2}+\int_{\hat{Y}_{[l,L]}}\big{|}\frac{i}{4}\rho(\mu_{r}^{+})-(\Psi\Psi^{*})_{0}\big{|}^{2}\\\
&\qquad+\int_{\hat{Y}_{[l,L]}}\frac{R_{g}}{4}|\Psi|^{2}-\frac{i}{4}\int_{\hat{Y}_{[l,L]}}F_{A}\wedge*_{4}w_{r}-\frac{ir}{4}\int_{\hat{Y}_{[l,L]}}ds\,F_{B}\wedge(*_{Y}\nu_{+}).\\\
\end{split}$

Inserting both the preceding formula and (4.32) into the following inequality

$\begin{split}&\frac{1}{4}\|\partial_{s}B\|_{L^{2}(\hat{Y}_{[l,L]})}^{2}+\|\partial_{s}\Phi\|^{2}_{L^{2}(\hat{Y}_{[l,L]})}+\|\mathfrak{F}_{\mu_{+}}(B,\Phi)\|_{L^{2}(\hat{Y}_{[l,L]})}^{2}\\\
&\qquad\leq\frac{1}{4}\|\partial_{s}B\|_{L^{2}(\hat{Y}_{[l,L]})}^{2}+\|\partial_{s}\Phi\|^{2}_{L^{2}(\hat{Y}_{[l,L]})}+\|\mathfrak{F}_{\mu_{+},\hat{{\mathfrak{p}}}}(B,\Phi)\|_{L^{2}(\hat{Y}_{[l,L]})}^{2}+\|\hat{{\mathfrak{p}}}(A,\Psi)\|_{L^{2}(\hat{Y}_{[l,L]})}^{2},\end{split}$

we have

(4.33)
$\begin{split}&\frac{1}{4}\int_{\hat{Y}_{[l,L]}}|F_{A}|^{2}+\int_{\hat{Y}_{[l,L]}}|\nabla_{A}\Psi|^{2}+\int_{\hat{Y}_{[l,L]}}\big{|}\frac{i}{4}\rho(\mu_{r}^{+})-(\Psi\Psi^{*})_{0}\big{|}^{2}\\\
&\quad\leq-2\operatorname{CSD}_{\mu_{+},\hat{{\mathfrak{p}}}}^{\partial\hat{Y}_{[l,L]}}(B,\Phi)+r\wp_{i}(L-l)-2\int_{\hat{Y}_{[l,L]}}(\partial_{s}\,f_{\hat{{\mathfrak{p}}}(s)})(B,\Phi)+\|\hat{{\mathfrak{p}}}(A,\Psi)\|_{L^{2}(\hat{Y}_{[l,L]})}^{2}\\\
&\qquad-\int_{\hat{Y}_{[l,L]}}\frac{R_{g}}{4}|\Psi|^{2}+\frac{i}{4}\int_{\hat{Y}_{[l,L]}}F_{A}\wedge*_{4}w_{r}+\frac{ir}{4}\int_{\hat{Y}_{[l,L]}}ds\,(*_{Y}\xi_{\nu})\wedge
F_{B_{0}}.\end{split}$

Argue as in (4.25), (4.20), (4.21), (4.22) to bound the fourth to the last
terms on the right hand side of the preceding formula and rearranging, we get:

(4.34)
$\begin{split}&\frac{1}{8}\int_{\hat{Y}_{[l,L]}}|F_{A}|^{2}+\int_{\hat{Y}_{[l,L]}}|\nabla_{A}\Psi|^{2}+\frac{1}{2}\int_{\hat{Y}_{[l,L]}}\big{|}\frac{i}{4}\rho(\mu_{r}^{+})-(\Psi\Psi^{*})_{0}\big{|}^{2}\\\
&\quad\leq-2\operatorname{CSD}_{\mu_{+},\hat{{\mathfrak{p}}}}^{\partial\hat{Y}_{[l,L]}}(B,\Phi)+r(\wp_{i}^{+}+z_{i})|\hat{Y}_{[l,L]}|+\zeta
r|\hat{Y}_{[l,L]}|_{1}+\zeta^{\prime}|\hat{Y}_{[l,L]}|\\\
&\quad\quad-2\int_{\hat{Y}_{[l,L]}}(\partial_{s}\,f_{\hat{{\mathfrak{p}}}(s)})(B,\Phi),\\\
\end{split}$

where $\zeta,\zeta^{\prime}$ are positive constants that depend only on the
choice of reference connection $A_{0}$, the metric, $\nu$, and the constants
$\varsigma_{w}$, $\textsl{z}_{\mathfrak{p}}$. Note that over
$\hat{Y}_{[l,L]}$,

(4.35)
$\begin{split}&-2\operatorname{CSD}_{\nu_{+},\hat{{\mathfrak{p}}}}^{\partial\hat{Y}_{[l,L]}}(B,\Phi)-{\mathscr{E}}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}(\hat{Y}_{[l,L]})(A,\Psi)\\\
&\quad=-\frac{i}{4}\int_{\hat{Y}_{[l,L]}}F_{B_{0}}\wedge\mu_{r}\leq\zeta_{e}r\,e^{-\kappa_{i}l}|\hat{Y}_{[l,L]}|_{1},\end{split}$

where $\zeta_{e}$ is a positive constant depending only on $B_{0}$ and $\nu$.
Thus, the bound (4.34) is equivalent to

(4.36)
$\begin{split}&\frac{1}{8}\int_{\hat{Y}_{[l,L]}}|F_{A}|^{2}+\int_{\hat{Y}_{[l,L]}}|\nabla_{A}\Psi|^{2}+\frac{1}{2}\int_{\hat{Y}_{[l,L]}}\big{|}\frac{i}{4}\rho(\mu_{r}^{+})-(\Psi\Psi^{*})_{0}\big{|}^{2}\\\
&\quad\leq{\mathscr{E}}^{{}^{\prime}\mu_{r}\hat{{\mathfrak{p}}}}(\hat{Y}_{[l,L]})(A,\Psi)+r(\wp_{i}^{+}+z_{i})|\hat{Y}_{[l,L]}|+\zeta_{o}r|\hat{Y}_{[l,L]}|_{1}+\zeta^{\prime}_{o}|\hat{Y}_{[l,L]}|\\\
&\quad\quad-2\int_{\hat{Y}_{[l,L]}}(\partial_{s}f_{\hat{{\mathfrak{p}}}})(B,\Phi),\\\
\end{split}$

where $\zeta_{o},\zeta^{\prime}_{o}$ are likewise positive constants that
depend only on the choice of reference connection $A_{0}$, the metric, $\nu$,
and the constants $\varsigma_{w}$, $\textsl{z}_{\mathfrak{p}}$.

We now estimate the last term above,
$-2\int_{\hat{Y}_{[l,L]}}(\partial_{s}\,f_{\hat{{\mathfrak{p}}}(s)})(B,\Phi)$.
This vanishes unless $\hat{Y}_{i}$ is a vanishing end; so we assume from now
on that $\hat{Y}=\hat{Y}_{i}$ is a vanishing end. According to (3.6) (and
recalling the definitions and assumptions on $\chi_{i}$, $\lambda_{i}$ there),

$(\partial_{s}f_{\hat{{\mathfrak{p}}}})(B,\Phi)=\chi^{\prime}_{i}(s)\,f_{{\mathfrak{q}}_{i}}(B,\Phi)+\lambda^{\prime}_{i}(s)\,f_{{\mathfrak{p}}^{\prime}}(B,\Phi).$

This is supported on
$\hat{Y}_{[{\mathfrak{l}}_{i},{\mathfrak{l}}_{i}^{\prime}]}$, and by our
assumption on the cutoff functions $\chi_{i}$ and $\lambda_{i}$, its absolute
value is bounded by
$|f_{{\mathfrak{q}}_{i}}(B(s),\Phi(s))|+|f_{{\mathfrak{p}}^{\prime}}(B(s),\Phi(s))|$.
At each $s\in[{\mathfrak{l}}_{i},{\mathfrak{l}}_{i}^{\prime}]$,
$|f_{{\mathfrak{q}}_{i}}(B(s),\Phi(s))|$ and
$|f_{{\mathfrak{p}}^{\prime}}(B(s),\Phi(s))|$ can be bounded via a variant of
Lemma 3.8: Applying the triangle inequality differently in the last step of
(3.12) and recalling Assumption 3.4 4d), one may arrange so that:

$\begin{split}|f_{{\mathfrak{q}}_{i}}(B(s),\Phi(s))|+|f_{{\mathfrak{p}}^{\prime}}(B(s),\Phi(s))|&\leq\frac{1}{32}\|F_{B}-F_{B_{0}}\|_{L^{2}(Y_{i:s})}^{2}+\zeta_{2}^{\prime}\big{(}\|\Phi\|_{L^{2}(Y_{i:s})}^{2}+1\big{)},\end{split}$

where $\zeta^{\prime}_{2}$ is a positive constants depending only on
$\textsl{z}_{\mathfrak{p}}$ and the metric on $Y_{i}$. Integrating the
preceding inequality over
$I:=[l,L]\cap[{\mathfrak{l}}_{i},{\mathfrak{l}}_{i}^{\prime}]\subset{\mathbb{R}}$,
and appealing again to (4.18), we get

(4.37)
$\begin{split}&\Big{|}-2\int_{\hat{Y}_{[l,L]}}(\partial_{s}f_{\hat{{\mathfrak{p}}}})(B,\Phi)\Big{|}\\\
&\quad\leq\frac{1}{16}\|F_{B}-F_{B_{0}}\|_{L^{2}(\hat{Y}_{I})}^{2}+\zeta^{\prime}_{3}\big{(}\|\Psi\|^{2}_{L^{2}(\hat{Y}_{I})}+|\hat{Y}_{I}|\big{)}\\\
&\quad\leq\frac{1}{16}\|F_{B}-F_{B_{0}}\|_{L^{2}(\hat{Y}_{I})}^{2}+\frac{1}{4}\Big{\|}\frac{i}{4}\rho(\mu^{+}_{r})-(\Psi\Psi^{*})_{0}\Big{\|}^{2}_{L^{2}(\hat{Y}_{I})}+\zeta^{\prime}_{4}r|\hat{Y}_{I}|,\end{split}$

where $\zeta_{3}^{\prime}$ is a positive constant depending only on
$\textsl{z}_{\mathfrak{p}}$ and the metric on $Y_{i}$, and
$\zeta_{4}^{\prime}$ is a positive constant depending only on $\nu$, and
$\varsigma_{w}$, $\textsl{z}_{\mathfrak{p}}$, and the metric. Insert the
preceding estimate into (4.36), we arrive at $(\ref{eq:E-top-bdd0})$ (b).
$\Box$

###  4.4 Lower bounds for
${\mathscr{E}}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}_{top}(\hat{Y}_{[l,L]})$

We need a lower bound that is independent of $\hat{Y}_{[l,L]}$,
$[l,L]\subset[0,\infty]$ being arbitrary. When $\hat{Y}=\hat{Y}_{i}$ is a
vanishing end, suppose in addition that $l\geq{\mathfrak{l}}_{i}^{\prime}$, so
that the last term of (4.32) vanishes on such $\hat{Y}_{[l,L]}$. Then, for an
admissible solution $(A,\Psi)=(A_{r},\Psi_{r})$ to
${\mathfrak{S}}_{\mu_{r},\hat{{\mathfrak{p}}}}(A_{r},\Psi_{r})=0$, we have:

(4.38)
$\begin{split}&-2\operatorname{CSD}_{\mu_{+},\hat{{\mathfrak{p}}}}^{\partial\hat{Y}_{[l,L]}}(B,\Phi)\\\
&\quad\quad=\frac{1}{2}\|\partial_{s}B\|_{L^{2}(\hat{Y}_{i,l})}^{2}+2\|\partial_{s}\Phi\|^{2}_{L^{2}(\hat{Y}_{i,l})}-\frac{ir}{4}\int_{\hat{Y}_{[l,L]}}ds\,(*_{Y}\xi_{\nu})\wedge(F_{B}-F_{B_{0}})\\\
&\quad\quad\geq-\frac{ir}{4}\int_{\hat{Y}_{[l,L]}}ds\,(*_{Y}\xi_{\nu})\wedge(F_{B}-F_{B_{0}}).\end{split}$

Ideally, $1/r$ times the last term above should be bounded independently of
both $r$ and $\hat{Y}_{[l,L]}$. Unfortunately, this is far from
straightforward. Note that similar terms do not appear in the setting of [LT]
or [HT3]. What follows is a first attempt to tackle this trouble-making term.

To proceed, use (4.32), (4.28), (4.29), Lemma 3.11 to get that for $r\geq
r_{0}$,

$\begin{split}&\frac{1}{2}\|\partial_{s}B\|_{L^{2}(\hat{Y}_{i,l})}^{2}+2\|\partial_{s}\Phi\|^{2}_{L^{2}(\hat{Y}_{i,l})}\\\
&\quad=-2\operatorname{CSD}_{\mu_{+},\hat{{\mathfrak{p}}}}^{\partial\hat{Y}_{i,[l,\infty]}}(B,\Phi)+\frac{ir}{4}\int_{\hat{Y}_{i,l}}ds\,(*_{Y}\xi_{\nu})\wedge(F_{B_{i}}-F_{B_{0}})\\\
&\qquad\quad-\frac{ir}{4}\int_{l}^{\infty}\,ds\int_{Y_{i:s}}(\partial_{s}\xi_{\nu})\wedge\big{(}\int_{s}^{\infty}\partial_{t}B(t)dt\big{)}\\\
&\quad\leq-2\operatorname{CSD}_{\mu_{+},\hat{{\mathfrak{p}}}}^{\partial\hat{Y}_{i,[l,\infty]}}(B,\Phi)+\zeta_{i}r\,e^{-\kappa_{i}l}+\frac{1}{4}\|\partial_{s}B\|_{L^{2}(\hat{Y}_{i,l})}^{2}+\zeta^{\prime}_{i}r^{2}e^{-2\kappa_{i}l},\end{split}$

where $\zeta_{i}$, $\zeta_{i}^{\prime}$ are positive constants; $\zeta_{i}$
depends only on $\nu$, $B_{0}$, the constant $\textsl{z}_{i}$ in Lemma 3.11;
$\zeta_{i}^{\prime}$ depends only on $\nu$. Rearranging, we get

$-2\operatorname{CSD}_{\mu_{+},\hat{{\mathfrak{p}}}}^{\partial\hat{Y}_{i,[l,\infty]}}(B,\Phi)\geq-\zeta_{i}^{\prime\prime}(r^{2}e^{-2\kappa_{i}l}+1),$

and by (4.35),
${\mathscr{E}}^{{}^{\prime}\mu_{r}}_{top}(\hat{Y}_{i,l})(A,\Psi)={\mathscr{E}}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}_{top}(\hat{Y}_{i,l})(A,\Psi)$
satisfies a similar inequality:

${\mathscr{E}}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}_{top}(\hat{Y}_{i,l})(A,\Psi)\geq-\underline{\zeta}_{i}(r^{2}e^{-2\kappa_{i}l}+1).$

In particular, setting $\hat{\mathop{\mathrm{l}}}_{i}:=\kappa_{i}^{-1}$, and
choosing $r_{0}$ to be greater than
$\max_{i\in{\mathfrak{Y}}_{v}}\exp\,(4{\mathfrak{l}}_{i}^{\prime}/\hat{\mathop{\mathrm{l}}}_{i})$,
we have that for $r\geq r_{0}$,

(4.39)
$\begin{split}-2\operatorname{CSD}_{\mu_{+},\hat{{\mathfrak{p}}}}^{\partial
Y_{i,[l,\infty]}}(B,\Phi)&\geq-2\,\zeta_{i}^{\prime\prime}r;\\\
{\mathscr{E}}^{{}^{\prime}\mu_{r}}_{top}(\hat{Y}_{i,l})(A,\Psi)={\mathscr{E}}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}_{top}(\hat{Y}_{i,l})(A,\Psi)&\geq-2\underline{\zeta}_{i}r\qquad\forall
l\geq\big{(}\frac{\ln r}{2}\big{)}\hat{\mathop{\mathrm{l}}}_{i}.\end{split}$

In the above, the positive constants
$\zeta_{i}^{\prime\prime},\underline{\zeta}_{i}>0$ depend only on $\nu$,
$B_{0}$, and the constant $\textsl{z}_{i}$ in Lemma 3.11.

Let
$\boldsymbol{\hat{\mathop{\mathrm{l}}}}\colon\thinspace{\mathfrak{Y}}\to{\mathbb{R}}^{+}$
be the function given by
$\boldsymbol{\hat{\mathop{\mathrm{l}}}}(i)=(\hat{\mathop{\mathrm{l}}}_{i})_{i}$,
and write ${\bf l}_{r}:=(\ln r)\,\boldsymbol{\hat{\mathop{\mathrm{l}}}}$.
Under the assumption (4.13) (b), the preceding pair of inequalities then
implies that for $r\geq r_{0}$,

(4.40)
$\begin{split}{\mathscr{E}}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}_{top}(X_{\bf
l})(A,\Psi)&={\mathscr{E}}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}_{top}(X)(A,\Psi)-\sum_{i\in{\mathfrak{Y}}}{\mathscr{E}}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}_{top}(\hat{Y}_{i,l})(A,\Psi)\\\
&\leq
r\,(\textsc{e}+2\sum_{i\in{\mathfrak{Y}}}\underline{\zeta}_{i})\quad\forall\,{\bf
l}\geq\frac{{\bf l}_{r}}{2}\quad\text{assuming (\ref{assume:EtopX-ubdd}) (b);
similarly, }\\\ {\mathscr{E}}^{{}^{\prime}\mu_{r}}_{top}(X_{\bf
l})(A,\Psi)&\leq
r\,(\textsc{e}+2\sum_{i\in{\mathfrak{Y}}}\underline{\zeta}_{i})\quad\forall\,{\bf
l}\geq\frac{{\bf l}_{r}}{2}\quad\text{assuming (\ref{assume:EtopX-ubdd})
(a).}\end{split}$

Combining the preceding upper bounds for
${\mathscr{E}}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}_{top}(X_{\bf l})$,
${\mathscr{E}}^{{}^{\prime}\mu_{r}}_{top}(X_{\bf l})$ with (4.16), we have for
any $r\geq r_{0}$ and $\forall{\bf
l}\geq\boldsymbol{\hat{\mathop{\mathrm{l}}}}_{r}/2$ with ${\bf l}(i)=:l_{i}$,

(4.41) $\begin{split}&\frac{1}{8}\int_{X_{{\bf l}}}|F_{A}|^{2}+\int_{X_{{\bf
l}}}|\nabla_{A}\Psi|^{2}+\frac{1}{2}\int_{X_{{\bf
l}}}\big{|}\frac{i}{4}\rho(\mu_{r}^{+})-(\Psi\Psi^{*})_{0}\big{|}^{2}\\\
&\qquad\quad\leq
r\,(\textsc{e}+\zeta)+\zeta^{\prime}\sum_{i\in{\mathfrak{Y}}}r^{\mathrm{a}_{i}}l_{i}\quad\text{assuming
(\ref{assume:EtopX-ubdd}) (a)};\\\ &\frac{1}{16}\int_{X_{{\bf
l}}}|F_{A}|^{2}+\int_{X_{{\bf