Yi-Jen Lee

primarymsc200053C07 secondarymsc200052C15

# From Seiberg-Witten to Gromov:
MCE and Singular Symplectic Forms

Yi-Jen Lee  Institute of Mathematical Sciences,
the Chinese University of Hong Kong,

Shatin, N.T. Hong Kong<EMAIL_ADDRESS>
(October 28, 2015)

###### Abstract

Motivated by various possible generalizations of Taubes’s $SW=Gr$ theorem [T]
to Floer-theoretic setting, we prove certain variants of Taubes’s convergence
theorem in [T] (the first part of his proof of $SW=Gr$). In place of the
closed symplectic 4-manifold considered in [T], this article considers non-
compact manifolds with cylindrical ends, equipped with a self-dual harmonic
2-form with non-degenerate zeroes. This extends and simplifies some central
technical ingredients of the author’s prior work in [LT] and [KLT5]. Other
expected applications include: extending the $HM=PFH$ theorem in [LT] and the
$HM=HF$ theorem in [KLT1]-[KLT5] to TQFTs on both sides [L1]; definitions of
large-perturbation Seiberg-Witten analogs of Heegaard Floer theory’s link
Floer homologies and link cobordism invariants.

###### keywords:

Seiberg-Witten

###### keywords:

Floer theory

###### keywords:

pseudo-holomorphic curves

###### Contents

  1. 1 Introduction
    1. 1.1 Stating the main theorems
    2. 1.2 Motivations and applications
    3. 1.3 Relation to existing literature
    4. 1.4 Convention, notations, etc.
  2. 2 Preliminaries: the $Gr$ side
    1. 2.1 t-orbits and t-curves
    2. 2.2 Topology on the space of t-curves
    3. 2.3 $\mathop{\mathrm{Spin}}\nolimits^{c}$ structures
    4. 2.4 Energy of t-curves and chains of t-curves
    5. 2.5 Relative homology classes of (chains of) t-curves
    6. 2.6 Some basic facts about t-curves
  3. 3 Preliminaries: the $SW$ side
    1. 3.1 Setup and Assumptions
    2. 3.2 $Y_{i}$-end limits of admissible Seiberg-Witten solutions
    3. 3.3 t-convergence of large-$r$ Seiberg-Witten solutions
    4. 3.4 Strong t-convergence
    5. 3.5 Relative homotopy classes and relative homology classes
    6. 3.6 Taubes’s proof for $SW\Rightarrow Gr$: A synopsis
  4. 4 $L^{2}_{1,loc}$-bounds and integral estimates
    1. 4.1  ${\mathscr{E}}_{top}$ and ${\mathscr{E}}_{anal}$: the topological energy and the analytic energy
    2. 4.2  $L^{2}_{local}$ bounds on gauge invariant terms in terms of (modified) topological energies
    3. 4.3 Seiberg-Witten solutions as gradient flow lines of time-dependent CSD functionals
    4. 4.4 Lower bounds for ${\mathscr{E}}^{{}^{\prime}\mu_{r},\hat{{\mathfrak{p}}}}_{top}(\hat{Y}_{[l,L]})$
    5. 4.5 Lemma 4.6 and some of its variants
    6. 4.6 $L^{2}_{1,loc/A}$-bounds for $(A-A_{0},\Psi)$
    7. 4.7 Some integral estimates
  5. 5 A priori pointwise estimates
    1. 5.1 Estimates for $|\psi|^{2}$
    2. 5.2 Estimates for $|\beta|^{2}$
    3. 5.3 Estimating $|F_{A}|$
    4. 5.4 Estimating $|\nabla_{A}\alpha|$ and $|\nabla_{A}\beta|$
  6. 6 Monotonicity and its consequences
    1. 6.1 Bounding $\mathcal{W}_{Z_{\delta}}$
    2. 6.2 Integral bounds redux
    3. 6.3 The monotonicity formula
    4. 6.4 (Further) refined curvature estimate
    5. 6.5 Approximation by local models
    6. 6.6 Exponential decay
  7. 7 Proofs of the main theorems
    1. 7.1 Proving Theorem 1.7
    2. 7.2 SW and Gr versions of Floer-theoretic energies
    3. 7.3 Proving Theorem 1.9 and Proposition 1.12, the non-cylindrical case
    4. 7.4 Proving Theorem 1.9 and Proposition 1.12, the cylindrical case

##  1 Introduction

In a series of ground breaking papers [T], Taubes proved the equivalence of
the Seiberg-Witten invariant of closed symplectic 4-manifolds and a version of
Gromov invariant. The first article of the series, also technically the most
important of the four, $SW\Rightarrow Gr$, proves that by perturbing the
Seiberg-Witten equations by a large multiple of the symplectic form, the
Seiberg-Witten solutions converge in a technical sense to a union of connected
pseudo-holomorphic curves with weights. As the first step to the ultimate goal
of generalizing the equivalence theorem to smooth 4-manifolds with $b^{2+}>0$,
Taubes proved a generalization of this convergence theorem in [Ts]. In the
language adopted in this paper, this is rephrased as follows.

Let $X$ be a closed $\mathop{\mathrm{Spin}}\nolimits^{c}$ 4-manifold, and
${\mathbb{S}}_{X}={\mathbb{S}}={\mathbb{S}}^{+}\oplus{\mathbb{S}}^{-}$ be its
associated spinor bundle. The Seiberg-Witten equations on $X$ take the
following form:

(1.1)
$\begin{cases}\frac{1}{2}F_{A}^{+}-\rho^{-1}(\Psi\Psi^{*})_{0}+\frac{i}{4}\mu^{+}=0,&\\\
\mbox{\mbox{$\partial\mkern-7.5mu\mbox{/}\mkern
2.0mu$}}_{A}^{+}\Psi=0,&\end{cases}$

where $A$ is a connection on $\det{\mathbb{S}}^{+}$, $\Psi$ is a section on
${\mathbb{S}}^{+}$, and $\rho$,
$\mbox{\mbox{$\partial\mkern-7.5mu\mbox{/}\mkern
2.0mu$}}_{A}^{+}\colon\thinspace{\mathbb{S}}^{+}\to{\mathbb{S}}^{-}$ denote
respectively the Clifford multiplication and the Dirac operator. (We use the
definition given in Sections I.1.1-I.1.3 of [KM]). In the last term of the
first equation, $\mu$ denotes a 2-form, and $\mu^{+}$ its self-dual part. This
is often regarded as a perturbation.

######  Theorem 1.1.

[Ts] Let $X$ be a closed $\mathop{\mathrm{Spin}}\nolimits^{c}$ 4-manifold with
$b^{2+}>0$ as above, and let $\omega$ be a self-dual harmonic 2-form on $X$
vanishing transversely along an embedded 1-submanifold. Take the perturbation
two form $\mu^{+}$ in Equation (1.1) above to be

(1.2) $\mu^{+}_{r}=\frac{1}{2}r\omega+w^{+}_{r},$

where $r>1$, and $w^{+}_{r}$ is a smooth self-dual 2-form with
$\|w_{r}^{+}\|_{C^{2}}$ is bounded by an $r$-independent constant. Let
$\\{(A_{r},\Psi_{r})\\}_{r}$ be a sequence of corresponding solutions, with
$r\in\\{r_{1},r_{2},\ldots,r_{n},\ldots\\}$, $r_{n}\to\infty$ as $n\to\infty$.
Then there exists a t-curve $\mathbf{C}$ on $X-\omega^{-1}(0)$ with respect to
the almost complex structure determined by $\omega$ and the metric on $X$, and
a subsequence of the gauge equivalence classes $\\{[(A_{r},\Psi_{r})]\\}_{r}$
which t-converges to $\mathbf{C}$.

The terms ‘t-curves’ and ‘t-convergence’ above are respectively catch-all
phrases we coined for the kind of pseudo-holomorphic subvarieties and the
technical notion of convergence in Taubes’s result111Admittedly, this is a
poor choice of terminology. The “t-convergence” is not “real” convergence in
the usual sense: The sequence $\\{[(A_{r},\Psi_{r})]\\}_{r}$ does not converge
in any reasonable topological space.. They will be defined in §2.1 and §3.3.

##### Remark.

Only the $w_{r}^{+}=0$ case was discussed in [Ts]. However, the arguments in
[Ts] work to establish the slightly more general statement in the preceding
theorem.

In the special case when $X=S^{1}\times Y$, where $Y$ is a closed
$\mathop{\mathrm{Spin}}\nolimits^{c}$ 3-manifold with $b_{1}>0$, one may
choose a harmonic Morse-Novikov 1-form $\theta$ on $Y$, and take
$\omega=2(ds\wedge\theta)^{+}$ on $X$. Here, $s$ denotes the affine parameter
of $S^{1}={\mathbb{R}}/{\mathbb{Z}}$. With respect to the product metric,
$\omega$ is self-dual harmonic, and the $S^{1}$-invariant Seiberg-Witten
solutions on $X$ correspond to the Seiberg-Witten solutions on the 3-manifold
$Y$. Let $\check{\theta}$ denote the vector dual to $\theta$. The
$S^{1}$-invariant t-curves are products of $S^{1}$ with certain sets of
(weighted) trajectories of $\check{\theta}$ on $Y$. We call the latter
‘t-orbits’, with the precise definition given in §2.1. Thus, as a corollary to
the preceding theorem, an analogous statement for closed 3-manifolds also
holds. (A more precise statement is given in Proposition 3.15 below). For the
purpose of this article however, we need a slightly stronger notion of
convergence, which we call “strong t-convergence”. This will be defined in
§3.4.

The 3-dimensional Seiberg-Witten equation, arising from a dimensional
reduction of the four dimensional version, has the following general form:

(1.3)
${\mathfrak{F}}_{\mu}(B,\Phi):=\begin{cases}\frac{1}{2}*F_{B}+\rho^{-1}(\Phi\Phi^{*})_{0}+\frac{i}{4}*\mu=0,&\\\
\mbox{\mbox{$\partial\mkern-7.5mu\mbox{/}\mkern
2.0mu$}}_{B}\Phi=0,&\end{cases}$

where $B$ is a unitary connection on $\det\mathbb{S}$,
$\mathbb{S}={\mathbb{S}}_{Y}$ being the spinor-bundle associated to the given
$\mathop{\mathrm{Spin}}\nolimits^{c}$ structure, and $\Phi$ is a section of
$\mathbb{S}$. $\rho$ denotes the Clifford action, and
$\mbox{\mbox{$\partial\mkern-7.5mu\mbox{/}\mkern 2.0mu$}}_{B}$ denotes the
Dirac operator associated to $B$. $(\Phi\Phi^{*})_{0}$ denotes the traceless
part of $\Phi\Phi^{*}$. $\mu$ is a closed 2-form on $Y$. Two pairs $(B,\Phi)$,
$(B^{\prime},\Phi^{\prime})$ are said to be gauge equivalent if there exists a
$u\in C^{\infty}(Y,S^{1})$, such that
$(B^{\prime},\Phi^{\prime})=(B-2u^{-1}du,u\cdot\Phi)$. Recall also that a
closed 1-form $\nu$ on a 3-manifold $Y$ is said to be Morse-Novikov if $\nu$
has isolated, nondegenerate zeroes.

######  Theorem 1.2.

Let $(Y,{\mathfrak{s}})$ be a closed oriented
$\mathop{\mathrm{Spin}}\nolimits^{c}$ 3-manifold with $b_{1}>0$, and $\theta$
be a harmonic Morse-Novikov 1-form on $Y$. Let $\nu=*\theta$ and take
$\mu=\mu_{r}=r\nu+w_{r}$, where $w_{r}$ is a closed 2-form on $Y$, and $r>1$.
Suppose that $\|w_{r}\|_{C^{2}}$ is bounded by an $r$-independent constant.
Let $\\{(B_{r},\Phi_{r})\\}_{r}$ be a sequence of solutions to the
$\mu=\mu_{r}$’s version of (1.3), with
$r\in\\{r_{1},r_{2},\ldots,r_{n},\ldots\\}$, $r_{n}\to\infty$ as $n\to\infty$.
Then there exists a t-orbit $\boldsymbol{\gamma}$ on $Y-\theta^{-1}(0)$ with
respect to the flow of $\check{\theta}$, the vector field dual to $\theta$,
and a subsequence of the gauge equivalence classes
$\\{[(B_{r},\Phi_{r})]\\}_{r}$ which strongly t-converges to
$\boldsymbol{\gamma}$.

Furthermore,

(1.4)
$\int_{\tilde{\gamma}}\theta=\frac{1}{2}\big{(}c_{1}({\mathfrak{s}})\cdot[\theta]-\zeta_{\theta}\big{)}\geq
0,$

where $\zeta_{\theta}$ is a constant depending only on $\theta$ (and not the
spin structure ${\mathfrak{s}}$).

An explicit formula for the constant $\zeta_{\theta}$ in (1.4) is given in
Proposition 3.15.

The preceding theorem should be viewed as the first part of a (direct) proof
of a 3-dimensional variant of $SW=Gr$. In fact, the aforementioned variant has
a precise formulation as Conjecture 1.9 in [HL]. A slightly weaker version of
the latter conjecture is proved (indirectly) by combining the results in [HL,
MT]. See Section 3.4 for the proof of this theorem and terminologies in its
statement.

An explicit expression for the constant $\zeta_{\theta}$ in (1.4) will be
given in Section 3.3 below. Suffices to say for now that when
$\theta=d\tilde{f}$ for a $S^{1}$-valued Morse function $\tilde{f}$,
$\zeta_{\theta}$ is the maximal Euler characteristic of the regular fibers of
$\tilde{f}\colon\thinspace Y\to S^{1}$.

###  1.1 Stating the main theorems

The goal of this article is to prove an analog of Taubes’s theorem (Theorem
1.1 above) for 4-manifolds with cylindrical ends, equipped with a certain type
of harmonic self-dual 2-form $\omega$.

######  Definition 1.3.

A manifold with cylindrical ends (or “MCE” for short) is a connected oriented
manifold $X$ with a complete metric $g$, so that there is a compact manifold-
with-boundary $X_{c}\subset X$, and an isometry ${\mathfrak{i}}$ from the
closure of $X-X_{c}$ to $\coprod_{i\in{\mathfrak{Y}}}{\mathbb{R}}^{\geq
0}\times Y_{i}$, where $Y_{i}$ are closed connected manifolds and each half
cylinder ${\mathbb{R}}^{+}\times Y_{i}$ is endowed with the product metric.
The index set, ${\mathfrak{Y}}={\mathfrak{Y}}_{X}$, is a non-empty set of
finitely many elements. In the rest of this article, we shall sometimes
implicitly identify subspaces in the closure of $X-X_{c}$ and their image in
$\coprod_{i\in{\mathfrak{Y}}}{\mathbb{R}}^{\geq 0}\times Y_{i}$ under
${\mathfrak{i}}$.

We call the preimage (under ${\mathfrak{i}}$) of the half cylinder
${\mathbb{R}}^{\geq 0}\times Y_{i}$ above an end of $X$, or the $Y_{i}$-end
when we wish to be specific. It will be frequently denoted by $\hat{Y}_{i}$.
Let

$\text{{\small s}}_{i}\colon\thinspace{\mathbb{R}}^{\geq 0}\times
Y_{i}\simeq\hat{Y}_{i}\subset X\,\to{\mathbb{R}}^{\geq 0}$

denote the projection to the first factor. Denote $\text{{\small
s}}_{i}^{-1}[l,\infty)\subset\hat{Y}_{i}$ by $\hat{Y}_{i,l}$. Slightly abusing
terminology, we sometimes also call such $\hat{Y}_{i,l}$ an end of $X$. We
call each $Y_{i}$ an ending 3-manifold of $X$. Unless specified otherwise, the
ending 3-manifolds $Y_{i}$ are oriented such that ${\mathfrak{i}}$ is
orientation preserving.

The analog of the self-dual harmonic 2-form $\omega$ has to satisfy
appropriate asymptotic conditions. Correspondingly, the Seiberg-Witten
solutions $(A_{r},\Psi_{r})$ need also satisfy certain asymptotic conditions.

Regular Morse-Novikov 1-forms are generic in an appropriate sense.

######  Definition 1.4.

We call a Morse-Novikov closed 1-form $\theta$ on a closed oriented 3-manifold
regular if:

  * •

all finite-length flow lines $\gamma$ of the vector field dual to $\theta$ are
regular. Namely, either $\gamma$ is a flow line from an index 2 zero
$p\in\theta^{-1}(0)$ to an index 1 zero $q\in\theta^{-1}(0)$, where the
descending manifold from $p$ and ascending manifold of $q$ intersect
transversely at $\gamma$, or $\gamma$ is a closed orbit with nondegenerate
linearized Poincaré return map;

  * •

for any fixed $l>0$, there are finitely many such flow lines $\gamma$ with
$\int_{\gamma}\theta\leq l$.

######  Definition 1.5.

A smooth 2-form $\nu$ on a MCE $X$ is said to be admissible if it is a
nontrivial harmonic 2-form satisfying:

  * •

$\nu$ is asymptotic to a harmonic 2-form $v_{i}$ on $Y_{i}$ as $\text{{\small
s}}_{i}\to\infty$ on each end $\hat{Y_{i}}$. By this we mean the following:
$\tau_{-L}\nu$ converges to $\pi_{2}^{*}\nu_{i}$ on $[-1,1]\times Y_{i}$ in
the $C^{\infty}$-topology, where $\tau_{-L}$ denotes translation by
$-L\in{\mathbb{R}}$ in the ${\mathbb{R}}$-factor of
$\hat{Y}_{i}\simeq{\mathbb{R}}^{\geq 0}\times Y_{i}$, and $\pi_{2}$ denotes
projection to the $Y_{i}$-factor of $\hat{Y}_{i}\simeq{\mathbb{R}}^{\geq
0}\times Y_{i}$. The pair $(Y_{i},\nu_{i})$ is called the ending pair
associated to $\hat{Y}_{i}$. The harmonic form $\theta_{i}:=*_{3}\nu_{i}$ is
either Morse-Novikov and regular, in which case the corresponding end
$\hat{Y}_{i}$ is called a Morse end, or $\nu_{i}=0$, in which case the
corresponding end is called a vanishing end. Let
${\mathfrak{Y}}_{m}\subset{\mathfrak{Y}}$, and
${\mathfrak{Y}}_{v}:={\mathfrak{Y}}-{\mathfrak{Y}}_{m}\subset{\mathfrak{Y}}$
denote respectively the set of Morse ends and the set of vanishing ends.

  * •

There are functions ${\bf l}_{v},{\bf
l}_{v}^{+}\colon\thinspace{\mathfrak{Y}}_{v}\to{\mathbb{R}}^{+}$, ${\bf
l}_{v}(j)=:\textup{l}_{j}$, ${\bf
l}^{+}_{v}(j)=:\textup{l}^{+}_{j}>\textup{l}_{j}+10$, so that the restriction
of $\nu$ on

$X^{\prime\prime}:=X-\bigcup_{j\in{\mathfrak{Y}}_{v}}\hat{Y}_{j,\textup{l}^{+}_{j}}^{\circ}:\quad$

vanishes transversely along a 1-submanifold $\nu^{-1}(0)$ (possibly with
boundary). (In the above, $\hat{Y}_{j,l}^{\circ}=\text{{\small
s}}_{j}^{-1}(l,\infty)$ denotes the interior of $\hat{Y}_{j,l}$.) Let
$X^{\prime}:=X-\bigcup_{j\in{\mathfrak{Y}}_{v}}\hat{Y}_{j,\textup{l}_{j}}^{\circ}$.
Without loss of generality, we take $\textup{l}_{j}^{+}=\textup{l}_{j}+10$ for
the rest of this article. We shall also encounter intermediate domains lying
between $X^{\prime}$ and $X^{\prime\prime}$. These are defined as follows:
Given $a\in{\mathbb{R}}$, let

$X^{{}^{\prime}a}:=X-\bigcup_{j\in{\mathfrak{Y}}_{v}}\hat{Y}_{\textup{l}_{j}+a}^{\circ}.$

In particular, $X^{\prime}\subset X^{{}^{\prime}a}\subset X^{\prime\prime}$
when $0\leq a\leq 10$; $X^{\prime}=X^{{}^{\prime}0}$ and
$X^{\prime\prime}=X^{{}^{\prime}10}$. The value of $a$ in $X^{{}^{\prime}a}$
is left unspecified in some statements in this article. In such case, the
statement holds for all value of $a$ with $0\leq a\leq 10$, and the precise
value of $a$ does not matter.

  * •

$X$ has at least one Morse end.

A $\mathop{\mathrm{Spin}}\nolimits^{c}$ MCE $(X,{\mathfrak{s}})$ together with
an admissible form $\nu$ on it, denoted $(X,\nu)$, is called an admissible
pair.

In the case when $X=\mathbb{R}\times Y$ and $(X,\nu)$ is invariant under the
$\mathbb{R}$-action, the admissible pair is said to be cylindrical. In this
case $X$ has two ends, $[1,\infty)\times Y$ and $(-\infty,-1]\times
Y\simeq\mathbb{R}^{\geq 0}\times(-Y)$. We call the former the $+\infty$-end,
and the latter the $-\infty$-end.

All pairs of MCE’s and 2-forms $\nu$, $(X,\nu)$ are assumed to be admissible
in this article.

######  Remark 1.6.

In Definition 1.3’s terminology, the ending 3-manifolds of a cylindrical
$X={\mathbb{R}}\times Y$ are $Y$ and $-Y$.

Note that because the ending manifolds $Y_{i}$ are compact, $\nu$ decays
exponentially to $\nu_{i}$ in the $C^{k}$-norm. (Cf. e.g. (4.29) for a more
precise statement.) Meanwhile, the second bullet in the Definition 1.5 can be
met with a generic choice of $(X,\nu)$ and an appropriately chosen ${\bf
l}_{v}$. Fix such a choice. The condition also implies that over
$X^{\prime\prime}$ the restriction of $|\nabla\nu|$ to $\nu^{-1}(0)$ is
bounded below by a positive number.

The precise definition of admissible Seiberg-Witten solutions associated to an
admissible pair $(X,\nu)$ is given in §3.1 below. It suffices to say for now
that this is basically a finite energy condition that prescribes the
asymptotic behavior of the solutions. For a Morse end, the asymptotic
conditions are specified by certain strongly t-convergent Seiberg-Witten
solutions associated to the ending pair.

On “vanishing ends”, the asymptotic conditions are of the kind appearing in
typical Seiberg-Witten-Floer theories, as described in [KM].

To ensure transversality, an additional nonlocal perturbation term,
$\hat{{\mathfrak{p}}}$, is added to the left hand side of the Seiberg-Witten
equation in [KM]. For this purpose, $\hat{{\mathfrak{p}}}$ can be set to be
zero on the Morse ends, and may be taken to be arbitrarily small on the
vanishing ends. Let
${\mathbb{S}}={\mathbb{S}}_{X}={\mathbb{S}}^{+}\oplus{\mathbb{S}}^{-}$ denote
the spinor bundle associated to the $\mathop{\mathrm{Spin}}\nolimits^{c}$
structure on $X$. Given a $\mathop{\mathrm{Spin}}\nolimits^{c}$-connection on
${\mathbb{S}}$, let $A$ denote the associated unitary connection on
$\det{\mathbb{S}}^{+}$, and let
$\mbox{\mbox{$\partial\mkern-7.5mu\mbox{/}\mkern
2.0mu$}}_{A}^{+}\colon\thinspace{\mathbb{S}}^{+}\to{\mathbb{S}}^{-}$ denote
the associated Dirac operator. Let $\Psi\in\Gamma({\mathbb{S}}^{+})$. The
general Seiberg-Witten equation on $X$ takes the form:

(1.5)
$\mathfrak{S}_{\mu,\hat{{\mathfrak{p}}}}\,(A,\Psi):=\Big{(}\frac{F_{A}^{+}}{2}-\rho^{-1}(\Psi\Psi^{*})_{0}+i\frac{\mu^{+}}{4},\mbox{\mbox{$\partial\mkern-7.5mu\mbox{/}\mkern
2.0mu$}}_{A}^{+}\Psi\Big{)}+\hat{{\mathfrak{p}}}\,(A,\Psi),$

where $\mu^{+}$ denotes the self-dual part of a 2-form $\mu$. (Cf. §1.4). Note
that $\mathfrak{S}_{\mu,\hat{{\mathfrak{p}}}}\,(A,\Psi)$ is invariant under
gauge transformations $(A,\Psi)\mapsto(A-2u^{-1}du,u\cdot\Psi)$, where $u$
maps $X$ to $S^{1}$. The gauge equivalence class of $(A,\Psi)$ represented by
$(A,\Psi)$ is denoted by $[(A,\Psi)]$.

######  Theorem 1.7 (local convergence).

Let $X$ be a 4-dimensional $\mathop{\mathrm{Spin}}\nolimits^{c}$ MCE and $\nu$
be an admissible 2-form on $X$. Given $r\geq 1$, let
$w_{r},\hat{{\mathfrak{p}}}_{r}$ be respectively an admissible closed 2-forms
and a nonlocal perturbation of the type described in [KM]. Suppose they
satisfy Assumption 3.4 below. Let $\mu_{r}:=r\nu+w_{r}$, and write

$\omega=2\nu^{+}$; $\mu_{r}^{+}=\frac{r}{2}\omega+w_{r}^{+}$.

Fix a sequence of positive real numbers $\Gamma:=\\{r_{n}\\}_{n}$,
$r_{n}\to\infty$ as $n\to\infty$, and a corresponding sequence of admissible
solutions $(A_{r},\Psi_{r})$ to the Seiberg-Witten equation
$\mathfrak{S}_{\mu_{r},\hat{{\mathfrak{p}}}_{r}}(A_{r},\Psi_{r})=0$ for
$r\in\\{r_{n}\\}_{n}$. Suppose that the sequence
$\\{(A_{r},\Psi_{r})\\}_{r\in\Gamma}$ satisfies:

  * (1)

If $\hat{Y}_{i}$ is a vanishing-end of $X$, then the $Y_{i}$-end limit of
$(A_{r},\Psi_{r})$, denoted $(B_{i},\Phi_{i})$, is independent of $r$. If
$\hat{Y}_{i}$ is a Morse-end of $X$, then the $Y_{i}$-end limit of
$(A_{r},\Psi_{r})$ strongly t-converges to a t-orbit $\boldsymbol{\gamma}_{i}$
as $r\to\infty$.

  * (2)

$\exists\,N\in{\mathbb{R}}^{+}$ such that $\forall r>N$, the relative homotopy
classes of $[(A_{r},\Psi_{r})]$ are identified via the canonical isomorphisms
in Lemma 3.19. Denote this class by h.

Then:

  * (a)

There is a t-curve $\mathbf{C}$ and an unbounded subsequence $\Gamma_{0}$ of
$\Gamma=\\{r_{n}\\}_{n}\subset{\mathbb{R}}^{+}$, such that
$\\{(A_{r},\Psi_{r})\\}_{r\in\Gamma_{0}}$ t-converges to $\mathbf{C}$ over
$X^{\prime}$.

  * (b)

Furthermore, $\mathbf{C}$ has finite $\omega$-energy, which is bounded above
by a positive constant determined by h (which depends on
$\boldsymbol{\gamma}_{i}$), the metric and the
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure of $X$, $\nu$ and the
constants $\varsigma_{w},\textsl{z}_{\mathfrak{p}}$ in Assumption 3.4.

######  Remark 1.8.

(a) See §3.5 for the definition of relative homotopy classes of Seiberg-Witten
solutions. The notion of $\omega$-energy of t-curves, as well as the
$\nu$-energy mentioned in the next theorem, are introduced in §2.4 below. They
are analogs of the notion of energies in symplectic field theory in §5.3 of
[BEHWZ].

(b) By Theorem 1.2, the sequence
$\\{(A_{r},\Psi_{r})\\}_{r\in\\{r_{n}\\}_{n}}$ in the statement of the
preceding theorem exists only when
$c_{1}({\mathfrak{s}}_{i})\cdot[*\nu_{i}]\geq-\zeta_{*\nu_{i}}$ for all Morse
ends $\hat{Y}_{i}$.

For Floer-theoretic purposes, a stronger version of convergence result is
often desired. In heuristic terms, a gauge equivalence class of Seiberg-Witten
solutions on a 4-dimensional cobordism is regarded as a (generalized) flow
line. This type of results say that a sequence of flow lines with the same end
points and relative homotopy class “weakly converge” to a “broken trajectory”
that is a concatenation of flow lines. (See Theorem 5.1.1 in [KM] for an
example.) The following theorem is intended to play a similar role. The
“broken trajectories” here in our setting are the so-called “chains of
t-curves”.

######  Theorem 1.9 (global convergence).

Adopt the assumptions and notations in Theorem 1.7; in particular,
$\\{\boldsymbol{\gamma}_{i}\\}_{i\in{\mathfrak{Y}}_{m}}$, h are as in
Conditions (1) and (2) in Theorem 1.7 respectively. Let
$\mathfrak{s}={\mathfrak{s}}_{X}$ denote the
$\mathop{\mathrm{Spin}}\nolimits^{c}$ structure of $X$. Then

  * (a)

There exists a chain of t-curves $\mathfrak{C}$ in $X^{\prime}$ with
$\mathop{\mathrm{Spin}}\nolimits^{c}$ structure $\mathfrak{s}$, $Y_{i}$-end
limits $\boldsymbol{\gamma}_{i}$ for $i\in{\mathfrak{Y}}_{m}$, together with a
subsequence $\Gamma^{\prime}=\\{r^{\prime}_{n}\\}_{n}$ of $\Gamma$,
$r^{\prime}_{n}\to\infty$ as $n\to\infty$, such that over $X^{\prime}$, the
corresponding subsequence of Seiberg-Witten solutions
$\\{(A_{r},\Psi_{r})\\}_{r\in\Gamma^{\prime}}$ weakly t-converges to
${\mathfrak{C}}$ in the sense of Definition 2.12.

  * (b)

${\mathfrak{C}}$ has finite $\omega$-energy, which has an upper bound
determined by the same factors listed in Item (b) in the statement of Theorem
1.7.

  * (c)

Suppose in addition that:

(1.6) $b^{1}(Y_{i})=0\quad\forall i\in{\mathfrak{Y}}_{v},$

and let ${\mathfrak{h}}$ be the isomorphism introduced in Lemma 3.23. (This is
a map from the set of relative homotopy classes of Seiberg-Witten gauge
equivalence classes to the set of relative homology classes of chains of
t-curves.) Then ${\mathfrak{C}}\subset X^{\prime}$ is of relative homology
class ${\mathfrak{h}}\,(\text{{\scriptsize h}})$.

  * (d)

Assuming that $X$ has no vanishing ends. Then the $\nu$-energy of
${\mathfrak{C}}$ is determined by $\nu$, the
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure,
$\\{\boldsymbol{\gamma}_{i}\\}_{i}$ and $\operatorname{\text{{\scriptsize
h}}}$ via the explicit formula (7.33).

See Definition 2.11 below for the definitions of chains of t-curves, as well
as their $\mathop{\mathrm{Spin}}\nolimits^{c}$ structures and relative
homology classes. The term “cylindrical” in Case (i) above refers to the case
when $X$ is a 4-dimensional cylinder $\mathbb{R}\times Y$, and $\mu_{r}$,
$\hat{{\mathfrak{p}}}_{r}$ are invariant under translation in the $\mathbb{R}$
factor. Meanwhile, $\tau_{L}$ denotes translation by $L$ in the
$\mathbb{R}$-factor of ${\mathbb{R}}\times Y$. (Cf. §1.4).

######  Remark 1.10.

As with typical Floer theories, constraints on the relative homotopy class
(Condition (2) of Theorem 1.7 in our setting; also assumed in Theorem 1.9) may
be replaced by the more general condition of a uniform upper bound on a
certain “topological energy”. Compare e.g. [KM]’s Theorem 24.6.2 and
Proposition 24.6.4. The corresponding “energy-bound” condition in our setting
takes the form of:

(1.7)
$r^{-1}{\mathscr{E}}_{top}^{\mu_{r},\hat{{\mathfrak{p}}}_{r}}(X)(A_{r},\Psi_{r})\leq\textsc{e}\quad\text{for
an $r$-independent constant $\textsc{e}$,}$

where ${\mathscr{E}}_{top}^{\mu,\hat{{\mathfrak{p}}}}$ is a natural
generalization of [KM]’s “perturbed topological energy”
${\mathscr{E}}_{top}^{\hat{{\mathfrak{p}}}}$. (Cf. (4.13) below.) Fixing the
relative homotopy class h as in Condition (2) of Theorem 1.7 leads to a bound
of the preceding form (with $\textsc{e}_{p}$ depending on h); see Lemma 4.3
below. As explained in §7.2, such a bound on the (Seiberg-Witten) topological
energy is virtually equivalent to a bound on the corresponding symplectic
version of energy, which is required in Gromov-type compactness theorems. A
special case of such a generalization is used in the proof of $HM=PFH$ in
[LT].

###  1.2 Motivations and applications

This article is intended to provide some essential analytic foundations of
various speculated relations between Seiberg-Witten-Floer homologies and
$Gr$-type Floer homologies (such as Hutchings-Taubes’s $ECH$/$PFH$ [HT1]).
More specifically, it began as a part of the author’s program of proving the
equivalence of Seiberg-Witten-Floer homology ($\mathring{HM}$) and Heegaard
Floer homology ($HF^{\circ}$) [L]; and the earliest incarnation of the present
paper dates back to 2006. Since then, several major progress has been made on
the relation between the two types of Floer theories. In particular, due to
certain technical difficulties, the program in [L] has been since been
modified to take advantage of Taubes’s monumental work on the equivalence of
$HM$ and $ECH$ in 2008 [Te]. See [KLT1], [KLT2], [KLT3], [KLT4], [KLT5]. This
renders the results stated here not immediately relevant to our original goal
(which partially accounts for its delay in appearance). However, the
aforementioned relation comes in many forms, (some even lacking precise
formulation), and is still very far from being completely understood. This
article is expected to serve as technical basis for possible results in this
general direction. Here we briefly describe some of them, as well as their
background.

##### (a) $PFH$ and TQFT.

The definition of purported $Gr$ analogs of $HM$ is itself a difficult
subject. Certain variants, called Periodic Floer homology ($PFH$) and Embedded
Floer homology ($ECH$), are proposed by Hutchings, and shown to be well-
defined by Hutchings and Taubes [HT1]. The latter ($ECH$) is associated to
$\mathop{\mathrm{Spin}}\nolimits^{c}$ 3-manifolds equipped with a contact
form. As mentioned before, it is shown to be equivalent to the (unperturbed
version) of $HM$ by Taubes. In fact, some of the expected properties of $ECH$
are difficult to establish directly, and this relation with the better-
developed $HM$ are used to justify them, resulting in important consequences.
For example, the independence of $ECH$ on the contact form and the definition
of cobordism maps in $ECH$ both hinge on their counterparts in $HM$, and these
are in turn the basis for the proofs of 3-dimensional Weinstein conjecture
[Tw], Arnolds’s conjecture [HT2], and Hutchings’s $ECH$ capacities [H].

The closely related $PFH$ pertains to a special case of the setting studied in
this paper. These are defined for 3-manifolds that are mapping tori, and they
are shown to be equivalent to (a perturbed version) of $HM$ in the author’s
joint work with Taubes [LT]. Given Taubes’s prior work on $HM=ECH$, the only
new ingredient of this proof consists of a special case of this article’s
Theorem 1.7, namely the case when $(X,\nu)=({\mathbb{R}}\times
Y,\pi_{2}^{*}*_{3}\theta)$ is cylindrical, and the harmonic 1-form $\theta$ is
nowhere vanishing. One naturally expects that this equivalence extends to an
equivalence of the TQFT on both sides, and the theorems of this article in
that case when $X$ is a symplectic cobordism would form the main part of the
proof. Unfortunately, like $ECH$, cobordism maps for $PFH$ is not at present
(directly) defined. In $ECH$, it is defined indirectly through its Seiberg-
Witten counterparts by Taubes’s $HM=ECH$). A similar indirect definition of
PFH cobordism maps via the main theorems of this article is very likely
possible. Alternatively, for heuristic reasons one expects the purported TQFT
for $PFH$ to be equivalent to Usher’s TQFT for what he calls the “FCOB
category” [U2]. based on Donaldson’s result on the equivalence of (closed)
symplectic 4-manifolds and Lefschetz pencils, Usher introduces in [U2] a
version of TQFT on what is called the “FCOB category” in [U2]. The latter
$TQFT$ is rigorously defined; and therefore one has a precisely formulated
conjecture:

######  Conjecture 1.11.

[U2] Usher’s TQFT is equivalent to the restriction of (perturbed versions of)
Seiberg-Witten $TQFT$, restricted to the subcategory of $FCOB$.

The main results of this article are expected to play a key role in a proof of
this conjecture.

The aforementioned heuristic reasoning also leads to other variants of this
conjecture: In the context of closed symplectic manifolds, The TQFTs of
Usher’s and of $PFH$’s are respectively the counterparts of Donaldson-Smith’s
invariant [DS] for symplectic Lefschetz pencils, and Taubes’s Gromov invariant
for symplectic 4-manifolds. The equivalence of the two has been established by
Usher [U1]. Hence by Taubes’s $SW_{4}=Gr$ [T], the Donaldson-Smith invariant
is equivalent to the Seiberg-Witten invariant for closed 4-manifolds. This is
the counterpart of Conjecture 1.11 above.

As $PFH$ concerns the setting of Theorem 1.2 in the special case when the
harmonic $1$-form has no zeroes, in full generality Theorems 1.2 and 1.7, 1.9
suggest that there should be a generalized version of the $PFH$ TQFT, which is
defined for a larger category containing $FCOB$ as a subcategory, whose
objects consist of closed 3-manifolds with $b_{1}>0$ (equipped with a
nondegenerate harmonic 1-form $\theta$), and whose morphisms are 4-dimensional
cobordisms with $b^{+}>0$, equipped with nondegenerate harmonic self-dual
2-forms that appear as $\omega$ in the setup of Theorems 1.7, 1.9. The closed
4-manifold counterpart of this larger category consists of pairs $(X,\omega)$
described in the statement of Taubes’s Theorem 1.1. Such manifolds are called
“near symplectic manifolds” by Donaldson and some other authors. Donaldson’s
description of symplectic manifolds as Lefschetz pencils has been generalized
to near-symplectic 4-manifolds [AKM]. This endows every near-symplectic
manifold with a “singular Lefschetz pencil”. Correspondingly, Perutz defined a
“Lagrangian-matching invariant” for singular Lefschetz fibrations [P], which
can be regarded as a generalization of Donaldson-Smith’s invariant for
symplectic Lefschetz invariants. A TQFT version of this “Lagrangian-matching
invariant” is briefly outlined in Section 1 of [P], and currently being
developed by Lekili and Perutz [LP]. This is supposed to be equivalent to the
putative “generalized $PFH$ TQFT mentioned above, and Conjecture 1.11 has a
corresponding generalization in terms of these invariants.

A more modest project is to understand the relation between $HM$ and $PFH$
cobordism maps in special cases. Hutchings communicated to us that $PFH$
cobordism maps can be well-defined for some simple cobordisms equipped with
Lefschetz fibration structure. G.H. Chen is working on generalizing these and
establishing the equivalence of $HM$ and $PFH$ cobordism maps in these special
cases. Also according Hutchings, one complication of directly define
$ECH$/$PFH$-cobordism maps is that, unlike the case of Floer homologies, such
maps would inevitably involve counting (nontrivially-) broken trajectories
(analogs of our “chains of t-curves” consisting of more than one t-curves),
even after generic perturbation. We hope that by exploring some explicit
examples of Theorem 1.9 one may gain some insight on how to correctly count
holomorphic curves in cobordisms for the definition of $ECH$/$PFH$-cobordism
maps.

##### (b) Seiberg-Witten invariants and Heegaard Floer invariants.

One common essential ingredient of both the proof of $HM=HF$ in [KLT1]-[KLT5]
and the original program [L] is a positivity result, exemplified by
Propositions 3.4 and 3.7 in [KLT5]. (A special case of this result is also a
major ingredient of [KLT4]). Their counterparts in [L] are Claim (1) of its
Section 6.4, and the last part of its sketch of proof for Theorem 8.2. These
type of positive results serve the purpose of showing that the differential
and cobordism maps between Floer complexes preserve the filtration, ensuring
that the “filtered” Floer homologies and cobordism maps among them are well-
defined.

The following proposition is a by-product of the proof of Theorem 1.9, given
in Sections 7.3, 7.4 below. The positivity results needed in [L] are special
cases of this proposition.

######  Proposition 1.12.

Let $\\{(A_{r},\Psi_{r})\\}_{r\in\Gamma^{\prime}}$ and ${\mathfrak{C}}$ be
respectively the sequence of Seiberg-Witten solutions and the chain of
t-curves from the conclusion of Theorem 1.9. Suppose that
$P\subset\mathring{X}^{\prime}$ is a pseudo-holomorphic submanifold with the
following properties:

(1.8) $\hss\left\\{\mkern-20.0mu\parbox{422.77661pt}{\begin{itemize}
\par\itemize@item@$P$ is disjoint from $\nu^{-1}(0)$, and its distance to the
latter is bounded below by a positive number. \par\itemize@item@On each Morse
end $\hat{Y}_{i}$, $P$ is asymptotic to a union of flow lines of the vector
field dual to $*\nu_{i}$, denoted $\textsl{p}_{i}$, in this sense of
Definition \ref{def:asymp-cur}. ($\textsl{p}_{i}$ may be empty.) For every
$i\in{\mathfrak{Y}}_{m}$, $\textsl{p}_{i}$ is disjoint from all the rest
orbits of ${\mathfrak{C}}$. \par\itemize@item@When $(X,\nu)$ is non-
cylindrical: Write ${\mathfrak{C}}=\\{{\bf
C}_{0},\\{{\mathfrak{C}}_{i}\\}_{i\in{\mathfrak{Y}}_{m}}\\}$. Then $P$
intersects $C_{0}$ transversely in a discrete set in the interior of $P$, and
for each ${\mathfrak{C}}_{i}$ with at least one component,
${\mathbb{R}}\times\textsl{p}_{i}$ intersects each component of
${\mathfrak{C}}_{i}$ transversely in a discrete subset in the interior of
${\mathbb{R}}\times\textsl{p}_{i}$. When $(X,\nu)$ is cylindrical: Let
$\textsl{p}_{\pm}$ denote the limit of $P$ in the $\pm\infty$-end. Then every
component of ${\mathfrak{C}}$ intersects all three of $P$,
${\mathbb{R}}\times\textsl{p}_{\pm}$ transversely in discrete points in the
interior; and for every rest orbit $\boldsymbol{\gamma}$ of ${\mathfrak{C}}$,
$P$ intersects ${\mathbb{R}}\times\boldsymbol{\gamma}$ transversely in
discrete points in the interior. \par\end{itemize}}\right.$

Then $\lim_{r\to\infty}\int_{P}\frac{i}{2\pi}F_{A_{r}^{E}}$ exists and is a
non-negative integer. (See §3.14 for the definition of $A^{E}$). Moreover, if
${\mathfrak{Y}}_{v}=\emptyset$, this limit is determined by the relative
homotopy class h (which depends implicitly on $\nu$ and
$\\{\boldsymbol{\gamma}_{i}\\}_{i\in{\mathfrak{Y}}}$) and the (relative)
homology class of $P$ relative to $\\{\textsl{p}_{i}\\}_{i\in{\mathfrak{Y}}}$.
In the above, two pseudo-holomorphic subvarieties $P,P^{\prime}$ are said to
be homologous relative to $\\{\textsl{p}_{i}\\}_{i\in{\mathfrak{Y}}}$ if they
both satisfy (1.8) with the same $\\{\textsl{p}_{i}\\}_{i\in{\mathfrak{Y}}}$,
and $P-P^{\prime}$ bounds a chain with closed support (cf. the footnote in
Section 2.3).

See Definition 2.11 for terminology and notation (e.g. rest orbits,
components) in the preceding statement.

In fact, with this proposition in place, [KLT5] can be significantly
shortened. This will be explained in more detail in the next subsection. Its
generality also renders the definition of filtered Seiberg-Witten counterparts
of some of the extensions of $HF^{\circ}$ immediate. For example, the knot
Floer homology $HFK^{\circ}(Y,K)$ in [OS] has a $HMT^{\circ}$ counterpart
$HMT^{\circ}(Y,K)$ by a straightforward translation of Ozsvath-Szabo’s
construction to the large-perturbation Seiberg-Witten theory according to the
prescription of Section 6 of [L], by introduction to a second filtration to
$CMT^{\circ}$ associated to $\underline{\gamma}_{w}$, the 1-cycle in
$\underline{Y}$ corresponding to the second base point $w$ in [OS]. Modifying
$HMT^{\circ}(Y,K)$ in the same way as [KLT1]-[KLT5], one also has an extension
of the latter’s $H^{\circ}$ to knots. Theorem 1.1 of [KLT5] states that
$H^{\circ}$ is isomorphic to $HMT^{\circ}$ tensored with copies of
$H_{*}(S^{1};{\mathbb{Z}})$. The aforementioned extension of $H^{\circ}$ is
related to $HMT^{\circ}(Y,K)$ in the same manner.

###  1.3 Relation to existing literature

The main theorems of this article may also be viewed as an extension of a
counterpart of [HT3]’s Proposition 5.2, which partially generalizes the
$SW\Rightarrow Gr$ part of Taubes’s $HM=ECH$ proof in [Te]. In the latter,
${\mathbb{R}}$-invariant large-perturbation Seiberg-Witten equations on
${\mathbb{R}}\times Y$ are considered, with the role of $*_{3}\nu$ here played
by a contact form. This is extended to the case of 4-dimensional exact
symplectic cobordisms in [HT3]. The generalization is straightforward, as it
does not require new techniques. Taubes’s equivalence theorem has a sister
version in [LT], which asserts that (a different variant of) $HM$ is
isomorphic to $PFH$. Much of the proof for this theorem is similar to that in
[Te], except for the $SW\Rightarrow Gr$ part. In both settings, a crucial
“energy bound” is required. In [Te], this hinges on certain spectral flow
estimates, while in [LT], a more topological argument is used, basing on
harmonicity of $\nu$. The latter is also the strategy of this article; see
Section 4. In fact, what was proved in [LT] is the special case of our Theorem
1.7 when $X$ is cylindrical and $\nu$ is nowhere-vanishing. Thus, one may
expect that, like what happened in the $ECH$ case (Proposition 5.2 of [HT3]),
the results of this article can also be obtained by simple modifications of
the work in [LT]. This unfortunately turns out not to be the case.

In the case of [HT3], the contact condition makes it possible to choose the
perturbation form on $X$ so that it is identical with that in the cylindrical
case of [Te]. In the setting of this article, the condition that $\nu$ be
harmonic is rather rigid, and the behavior of the perturbation form on the
ends of $X$ is determined by that on its compact piece, $X_{c}$. In general,
it does not agree with the cylindrical case, but only asymptotic to it.
Surprisingly, this small difference turns out to be a major obstacle, making
the desired results substantially more elusive. See remarks near the end of
Section 4.1 and in the beginning of Section 4.4.

Another complication stems from the existence of zero locus of $\nu$. In [Ts],
the harmonic form $\omega$ is also allowed to vanish along a 1-dimensional
submanifold of $X$. While part of the ingredients in our proof can be obtained
from straightforward adaptation of their counterparts in [Ts], the same
existence of zero locus causes additional problems in our setting. In [Ts],
much of what was done in [T] easily carries over if one restricts his
attention to complement of small tubular neighborhoods of $\omega^{-1}(0)$.
This is no longer possible in our setting, since several important steps here
rely on global estimates where contribution from tubular neighborhood of
$\nu^{-1}(0)$ can not be ignored. Lack of knowledge on the behavior of
Seiberg-Witten solutions on this region thus becomes an unavoidable problem.

The purpose of [KLT1]-[KLT5]’s modification to the original program in [L] is
precisely to sidestep the aforementioned issues. In the particular setting of
[L]’s program, the zero locus in the cylindrical ${\mathbb{R}}\times M$ is
explicitly known. Thus, one may perform certain surgery operation along the
zero locus of $*_{3}\nu$, and “perturb” $\nu$ near the surgered part so that
$*_{3}\nu$ approximates certain standard contact 3-form. This makes $\nu$
nowhere-vanishing, and also allows one to take advantage Taubes’s proof of
$HM=ECH$, which was not available when [L] was written.

The major part of [KLT5] is dedicated to proving a variant of Proposition 1.12
for the connecting sum cobordisms. (Cf. Propositions 3.9-3.14 in [KLT5]). Here
one also encounter the first issue above, arising from the asymptotic behavior
of the perturbation form. Because of this problem, the results there only
applies to very special types of manifolds equipped with rather stringent
geometric conditions, essentially only the specific cobordisms used in the
article. For example, the perturbation form there is required to be
${\mathbb{R}}$-invariant on the ends (cf. e.g. (2.12) of [KLT5]), and the
pseudo-holomorphic submanifold $P$ (in Proposition 1.12’s notation) is a
cylinder, near which the metric is “standard”. As mentioned before, this does
not hold in most cases. Therefore the entire Section 9 therein (about 60
pages) is dedicated to constructing such metrics and forms case by case for
the specific cobordisms needed in [KLT5]. Except for its Proposition 3.13,
where $X={\mathbb{R}}\times Y$ and the perturbation form is one that
interpolates the harmonic version of [L] and the modified version of
[KLT1]-[KLT5], all other positivity results in [KLT5] are special cases of
Proposition 1.12 above, and its Proposition 3.13 says that as far as [KLT5]
part of the program is concerned, it makes no difference to either work along
the original plan in [L] and or its modified form in [KLT1]-[KLT5].

Alternatively, an avatar (2006-2008) of the present article also assumed that
the perturbation form is constant on the ends, and this is made permissible by
only requiring the form $\nu$ to be Lipshitz, not necessarily smooth. (The
assumptions of [KM] is sufficiently general to apply in this case). In this
way, the proof therein was no more difficult than that in [LT]. The trade off
is that one would have to deal with Lipshitz almost complex structures and
Lipshitz holomorphic curves.

One should also note that in [KM], the perturbations used to define the
differentials in $HM$ and the cobordism maps are also assumed to be constant
on the ends. Though this is in general not the case when applying our theorem,
we believe that the results in [KM] may be easily generalized to our setting,
as our perturbations decay to constant ones on the ends. One should be able to
do so by either going through the arguments in [KM], or use a direct limit
argument.

###  1.4 Convention, notations, etc.

Here is a list of notations and conventions frequently used throughout this
article.

  1. 1.

$W^{\circ}$ or $\mathring{W}$ denotes the interior of a topological space $W$.

  2. 2.

$p.d.$ stands for “Poincaré dual”.

  3. 3.

Given an interval $I\subset{\mathbb{R}}^{\geq 0}$, we use $\hat{Y}_{i,I}$ to
denote the subset $\text{{\small s}}_{i}^{-1}I$ in the end $\hat{Y}_{i}$, and
$\hat{Y}_{i,L}:=\hat{Y}_{i,[L,\infty)}$. The latter should be contrasted with

$Y_{i:L}:=\text{{\small s}}_{i}^{-1}(L).$

We shall often omit the index $i$ when the statement refers to any of the ends
of $X$. The interval $I$ above is allowed to be either open or closed, or of
the mixed type $[L,L^{\prime})$ or $(L,L^{\prime}]$. We use
$\hat{Y}_{i,[L,\infty]}\simeq[L,\infty]\times Y_{i}\supset[L,\infty)\times
Y_{i}\simeq\hat{Y}_{i,L}$ to denote the natural compactification of
$\hat{Y}_{i,L}$ by adding a $Y_{i:\infty}\simeq Y_{i}$ at infinity.
Correspondingly,
$\overline{X}=X_{c}\cup\bigcup_{i\in{\mathfrak{Y}}}\hat{Y}_{i,[L,\infty]}\supset
X$ denotes the compactification of $X$ by adding
$\bigcup_{i\in{\mathfrak{Y}}}Y_{i:\infty}$ at infinity. Given a subspace
$M\subset\overline{X}$, $\overline{M}$ denotes its closure in $\overline{X}$.

  4. 4.

$\pi_{2}\colon\thinspace I\times Y\to Y$ denotes the projection to the second
factor of the product.

  5. 5.

Given a bundle ${\rm V}$ over $M$, $C^{k}_{A}(M;{\rm V})$, $L^{p}_{k,A}(M;{\rm
V})$ respectively denote the spaces of $C^{k}$\- and $L^{p}_{k}$-sections
associated to a connection $A$ on ${\rm V}$. Occasionally $M$, $V$, or the
connection $A$ is omitted from the notation when it is clear from the context.
The bundles appearing in this article are typically constructed out of
$T^{*}M$ and/or the spinor bundle over $M$. When the connection $A$ is omitted
from the notation, e.g. $L^{p}_{k}(M)$, the implicit connection $A$ that
induced from the Levi-Civita connection on $T^{*}M$, and/or the (previously
specified) reference connection $A_{0}$ on the spinor bundle.

  6. 6.

Given a bundle ${\rm V}$ over $\overline{X}$ with ${\rm V}_{i}:={\rm
V}\big{|}_{Y_{i:\infty}}$, a section $q\in\Gamma({\rm V}\big{|}_{X})$ is said
to have $q_{i}\in\Gamma({\rm V}_{i})$ as a $Y_{i}$-end limit if

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

  7. 7.

Given a function $\mathbf{L}\colon\thinspace{\mathfrak{Y}}\to[0,\infty]$ with
${\bf L}(i)=L_{i}$, let $X_{\mathbf{L}}\subset\overline{X}$ denote the subset
$\\{x\,|\,\text{{\small s}}_{i}(x)\leq L_{i}\,\forall i,\,\text{wherever
$\text{{\small s}}_{i}$ is defined}\\}$. Given two functions ${\bf
l},\mathbf{L}\colon\thinspace{\mathfrak{Y}}\to[0,\infty]$ We write
$\mathbf{l}<\mathbf{L}$ if $X_{\mathbf{l}}\subset X_{\mathbf{L}}^{\circ}$, and
we write $\mathbf{l}\leq\mathbf{L}$ if $X_{\mathbf{l}}\subset X_{\mathbf{L}}$.
Also, let $\hat{Y}_{[{\bf l},{\bf
L}]}:=\bigcup_{i\in{\mathfrak{Y}}}\hat{Y}_{i,[l_{i},L_{i}]}$.

  8. 8.

The notation $X_{\bullet}$ will be used to denote any of the $X_{\bf L}$,
$\hat{Y}_{i,I}$ above, or their interiors. Given $X_{\bullet}\subset\bar{X}$,
$\overline{X_{\bullet}}$ denotes its closure in $\bar{X}$. The length of
$X_{\bullet}$, denoted by $|X_{\bullet}|$, is given by:
$|X_{\bullet}|:=|\overline{X_{\bullet}}|\in[0,\infty]$ and

$|X_{\bullet}|:=\begin{cases}|I|=L-l&\text{when $X_{\bullet}=\hat{Y}_{I}$, and
$I=[l,L]$.}\\\ 1+\sum_{i}L_{i}&\text{when $X_{\bullet}=X_{\bf L}$, ${\bf
L}(i)=:L_{i}$.}\end{cases}$

  9. 9.

Given an admissible pair $(X,\nu)$, let $X_{\delta}\subset X$ denote the
subspace of $X$ consisting of points whose distance to $\nu^{-1}(0)$ is
greater than $\delta$, and let $Z_{\delta}:=X-X_{\delta}\subset X$. Given
$X_{\bullet}\subset\overline{X}$, $X_{\bullet,\delta}:=X_{\delta}\cap
X_{\bullet}$ and $Z_{\bullet,\delta}:=Z_{\delta}\cap X_{\bullet}$. For a
closed 3-manifold $Y$ and a 1-form $\theta$ over $Y$, $Y_{\delta}\subset Y$ is
similarly defined.

  10. 10.

$\tau_{L}\colon\thinspace[l,\infty)\times Y\to[l+L,\infty)\times Y$ denotes
translation in the first factor. Slightly abusing the notation, we also
sometimes write $\tau_{-L}^{*}$ as $\tau_{L}$. I.e., given a function or
section $\xi(s,p)$ on a half cylinder $\hat{Y}\simeq{\mathbb{R}}^{\geq
0}\times Y$ and $L\in{\mathbb{R}}$, $(\tau_{L}\xi)(s,p):=\xi(s-L,p)$ on
$[L,\infty)\times Y$.

  11. 11.

Throughout this article, a “$k$-dimensional current” $\tilde{C}$ on an
$n$-manifold $X$ refers to a linear functional on the space of compactly-
supported smooth $(n-k)$-forms on $X$, where the latter space is endowed with
the topology defined by the sup-norm. The convergence of currents refers to
the convergence in weak∗ topology. Given a smooth $(n-k)$-form $\nu$, we
sometimes use $\int_{\tilde{C}}\nu$ or $\langle\tilde{C},\nu\rangle$ to denote
the value of $\tilde{C}$ on $\nu$.

  12. 12.

A ($k$-dimensional) subvariety $C$ of a manifold $X$ is the image of an
immersion $u_{0}\colon\thinspace C_{0}\to X$, where $C_{0}$ is a
($k$-dimensional) manifold (possibly disconnected or empty) containing a
countable set $\Lambda\subset C_{0}$ such that $u_{0}\Big{|}_{C_{0}-\Lambda}$
is an embedding. The immersion $u_{0}$ is said to be a representing map of
$C$. Given a $k$-form $\vartheta$ on $X$,
$\int_{C}\vartheta:=\int_{C_{0}}u_{0}^{*}\vartheta$. The current associated to
$C$, often denoted $\tilde{C}$, refers to that given by
$\vartheta\mapsto\int_{\tilde{C}}\vartheta:=\int_{C}\vartheta$. An irreducible
component of $C$ is the closure of a component of $C-u_{0}(\Lambda)$. $C$ is
irreducible if it has only one irreducible component.

  13. 13.

A weighted subvariety ${\bf C}$ in a manifold $X$ with underlying subvariety
$C$ is a set of pairs $\\{(C_{a},m_{a})\\}_{a}$, where $C_{a}$ are the
irreducible components of $C$, and $m_{a}\in{\mathbb{Z}}^{+}$ is the weight
associated to the irreducible component $C_{a}$. The current associated to
${\bf C}$, denoted $\tilde{C}$, is $\sum_{a}m_{a}\tilde{C}_{a}$. Since the
underlying subvariety $C$ and its associated current $\tilde{C}$ together
determine the weighted subvariety ${\bf C}$, it is alternatively denoted as a
pair ${\bf C}=[C,\tilde{C}]$.

  14. 14.

Let ${\bf C}=\\{(C_{a},m_{a})\\}_{a}$ be an oriented weighted subvariety, and
$P$ be an oriented submanifold in the oriented manifold $X$. Let
$u_{0}\colon\thinspace C_{0}\to X$ be the immersion representing the
underlying subvariety $C=\bigcup_{a}C_{a}$ of ${\bf C}$. Write
$C_{0}=\coprod_{a}C^{\prime}_{a}$, where $C^{\prime}_{a}$ is the connected
component which is the preimage of $C_{a}$ under $u_{0}$. Then $P$ intersects
${\bf C}$ transversely if $u_{0}$ intersects $P\subset X$ transversely. The
intersection number $\\#\,(P\cap{\bf
C}):=\sum_{a}m_{a}\,\chi\big{(}(u_{0}|_{C^{\prime}_{a}})^{-1}P\big{)}$, where
$\chi$ denotes the Euler characteristic.

  15. 15.

The letters $\zeta$, $z$, o, or any of their decorated forms (e.g.
$\zeta_{i}$, $\zeta^{\prime}$) customarily denote positive constants
independent of $r$, $\delta$, and $(A,\Psi)$ or $(B,\Phi)$, and whose precise
values are immaterial. Unless otherwise specified, they can be taken so as to
increase with each appearance. Likewise, $r_{0}\geq 1$ is a constant whose
precise value may change with appearances. It is taken so as to increase with
each appearance.

  16. 16.

Unless otherwise specified, $\chi\colon\thinspace{\mathbb{R}}\to{\mathbb{R}}$
denotes a smooth cutoff function supported on $(-\infty,2]$, and equal 1 on
$(-\infty,1]$.

  17. 17.

$B(x,R)$ or $B_{x}(R)$ denotes a geodesic ball centered at $x$ with radius
$R$.

  18. 18.

$*_{3}$ and $*_{4}$ respectively denote Hodge dual on 3-manifolds and
4-manifolds. Sometimes we also use $*_{Y}$ to denote the Hodge dual on the
specific manifold $Y$.

  19. 19.

Given a $\mathop{\mathrm{Spin}}\nolimits^{c}$ 4-manifold $X$ with spinor
bundle ${\mathbb{S}}^{+}\oplus{\mathbb{S}}^{-}$, we use
$\mathop{\mathrm{Conn}}\nolimits({\mathbb{S}}^{+})$,
$\mathop{\mathrm{Conn}}\nolimits(\det{\mathbb{S}})$ to denote respectively the
space of $\mathop{\mathrm{Spin}}\nolimits^{c}$-connections on
${\mathbb{S}}^{+}$, and the space of unitary connections on
$\det{\mathbb{S}}^{+}$. Given a 3-manifold $Y$ with spinor bundle
${\mathbb{S}}$, let $\mathop{\mathrm{Conn}}\nolimits({\mathbb{S}})$,
$\mathop{\mathrm{Conn}}\nolimits(\det{\mathbb{S}})$ be similarly defined. A
$\mathop{\mathrm{Spin}}\nolimits^{c}$ connection induces a unitary connection
on its associated determinant line bundle, and therefore we have a map from
$\mathop{\mathrm{Conn}}\nolimits({\mathbb{S}}^{+})$ to
$\mathop{\mathrm{Conn}}\nolimits(\det{\mathbb{S}}^{+})$ (or
$\mathop{\mathrm{Conn}}\nolimits({\mathbb{S}})$ to
$\mathop{\mathrm{Conn}}\nolimits(\det{\mathbb{S}})$ in the case of
3-manifolds). This map is an isomorphism when $H^{1}$ of the 3- or 4-manifold
in question has no 2-torsion; in particular,
$\mathop{\mathrm{Conn}}\nolimits({\mathbb{S}})\simeq\mathop{\mathrm{Conn}}\nolimits(\det{\mathbb{S}})$
in the case of 3-manifolds. Abusing notation, we often use the same notation
($A$ in the case of 4-manifolds and $B$ in the case of 3-manifolds) to denote
either a $\mathop{\mathrm{Spin}}\nolimits^{c}$ connection or its induced
connection on the associated determinant bundle. It should be clear from the
context which is meant. E.g. $A-A_{0}$ and $F_{A}$ are $i{\mathbb{R}}$-valued
differential forms; so the connection $A$ in these cases must refer to a
connection on $\det{\mathbb{S}}^{+}$. The $A$ appearing in covariant
derivatives $\nabla_{A}$ or Dirac operators
$\mbox{\mbox{$\partial\mkern-7.5mu\mbox{/}\mkern 2.0mu$}}_{A}^{+}$ on a
spinor-bundle necessarily refer to a
$\mathop{\mathrm{Spin}}\nolimits^{c}$-connection. In the same token,
$\mathop{\mathrm{Conn}}\nolimits(X)$ (resp.
$\mathop{\mathrm{Conn}}\nolimits(Y)$) is used to denote either
$\mathop{\mathrm{Conn}}\nolimits({\mathbb{S}}^{+})$ or
$\mathop{\mathrm{Conn}}\nolimits(\det{\mathbb{S}}^{+})$ (resp.
$\mathop{\mathrm{Conn}}\nolimits({\mathbb{S}})$ or
$\mathop{\mathrm{Conn}}\nolimits(\det{\mathbb{S}})$).

  20. 20.

We frequently use the shorthand $\partial_{s}$ for $\frac{d}{ds}$.

  21. 21.

“LHS” and “RHS” respectively stand for “left hand side” and “right hand side”.

  22. 22.

Given a function $f$, $(f)_{+}$ denotes the function $\max\,(f,0)$. Given a
2-form $\mu$ on a 4-manifold, $\mu^{+}$ denotes its self-dual part.

  23. 23.

$(A_{r},\Psi_{r})$ is used to denote a solution to the $r$-th version of the
Seiberg-Witten equations; i.e.
${\mathfrak{S}}_{\mu_{r},{\mathfrak{p}}}(A_{r},\Psi_{r})=0$. However, in the
context of a sequence $\\{(A_{n},\Psi_{n})\\}_{n\in{\mathbb{Z}}^{+}}$,
$(A_{n},\Psi_{n})$ is often used as a short hand for
$(A_{r_{n}},\Psi_{r_{n}})$, where $\\{r_{n}\\}_{n\in{\mathbb{Z}}^{+}}$ is a
sequence of positive numbers approaching $\infty$.

  24. 24.

The book [T] contains four previously published articles. When referring to
theorems or equations therein, we use roman numerical to label the specific
article we refer to. For example: Theorem I.1.2 stands for Theorem 1.2 of the
first article in [T].

This article frequently refers to formulas in various literature, which
unfortunately use different conventions. For the reader’s convenience, we
clarify some of their relations here. The Seiberg-Witten equations in this
article follow the convention of [KM]. In Taubes’s articles, $F_{A}/2$ above
is replaced by $F_{A}$. This results in a difference of factor 2 in many
expressions below from their analogs in Taubes’s articles. To sum up,

$\begin{split}\Psi&=\Psi_{KM}=\Psi_{PFH}/\sqrt{2}=\Psi_{har}/\sqrt{2};\\\
\frac{i\mu}{4}&=irw_{f}|_{PFH}=-2w|_{KM};\\\
\frac{i\mu^{+}}{4}&=\frac{ir\omega}{8}|_{har},\end{split}$

where the first expressions in all three lines are in the notation used in
this article, and the subscripts $KM$, $PFH$, $har$ refer respectively to
their counterparts in [KM], [LT], and [Ts].

The rest of this article is organized as follows: Sections 2 and 3
respectively contain prerequisites regarding $Gr$ and $SW$ sides of the story
in our context, in the sense of Taubes’s $SW=Gr$ in [T]. Section 3.6
enumerates some key steps of Taubes’s arguments in the $SW\Rightarrow Gr$ part
of [T], which basically all subsequent literature in this direction also go
through, including the present article. It also indicates where each step is
carried out in the remainder of the article. Section 4 is the crux of the
proof, and contains a preliminary version of the crucial “energy bound”. The
latter is iteratively improved in Sections 5 and 6 to serve our purposes.
Sections 5 and 6 describe how some of Taubes’s original argument should be
modified in our context. The final section, Section 7, brings everything
together to complete the proofs of the main theorems.

Lastly, a word of caution is in order: For the sake of brevity, we often quote
minute details from [Ts] and [T] directly without preparation. The reader is
thus strongly advised to keep copies of [Ts] and [T] at hand.

##### Acknowledgements.

The author’s debt to Taubes is abundantly evident. This work is supported in
part by Hong Kong Research Grant Counsel grants GRF-401913, 14316516,
143055419, and in earlier stages, by U.S. NSF grants.

##  2 Preliminaries: the $Gr$ side

Some basic notions and facts about the right hand side of our variant of
$SW\Rightarrow Gr$ are gathered in this section.

###  2.1 t-orbits and t-curves

Recall (weighted) subvarieties and related notions and conventions from
Section 1.4; in particular, items (11)-(14). The notion of t-orbits and
t-curves are special cases of weighted subvarieties.

Let $Y$ be a closed oriented riemannian 3-manifold and $\theta$ be a harmonic,
regular Morse-Novikov 1-form on $Y$.

######  Definition 2.1.

A t-orbit with respect to $(Y,\theta)$ is a (possibly empty) finite set
$\boldsymbol{\gamma}=\\{(\gamma_{a},m_{a})\\}_{a}$ where:

  * •

Each $\gamma_{a}\subset Y-\theta^{-1}(0)$ is a finite-length flow line of the
vector field dual to $\theta$, and $\gamma_{a}\neq\gamma_{b}$ when $a\neq b$.
$m_{a}\in{\mathbb{Z}}^{+}$.

  * •

$\partial\,(\bigcup_{a}\gamma_{a})=\theta^{-1}(0)$ as oriented manifolds,

$\gamma_{a}$ is called a constituent flow line of $\boldsymbol{\gamma}$, and
$m_{a}$ its weight.

A multi-orbit of $(Y,\theta)$ is a union $\cup_{a}\gamma_{a}\subset Y$, where
$\gamma_{a}$ are disjoint finite-length flow lines of the vector field dual to
$\theta$, and the index set $\\{a\\}$ has finite (possibly 0) elements.

Let $({\mathscr{X}},\omega)$ be a (possibly noncompact) symplectic 4-manifold,
and $J$ be a compatible almost complex structure. A subvariety
$C\subset{\mathscr{X}}$ is said to be a pseudo-holomorphic (or
$J$-holomorphic) subvariety if the representing map $u_{0}\colon\thinspace
C_{0}\to{\mathscr{X}}$ of $C$ satisfies then following additional conditions:
$C_{0}$ (possibly non-compact, non-connected or empty) is $C^{\infty}$ complex
curve, and $u_{0}\colon\thinspace C_{0}\to{\mathscr{X}}$ is a proper
$J$-holomorphic map. A weighted pseudo-holomorphic (or $J$-holomorphic)
subvariety in ${\mathscr{X}}$ is a weighted subvariety
$\\{(C_{a},m_{a})\\}_{a}$ such that $C_{a}$ are irreducible $J$-holomorphic
subvarieties.

Let $X$ be an oriented MCE, and $\nu$ be an admissible form on $X$. Let $J$ be
the almost complex structure on

$X_{0}:=X-\nu^{-1}(0)$

determined by $g$ and $\omega=2\nu^{+}$.

######  Definition 2.2.

A t-curve on $X_{\bullet}$ is a weighted $J$-holomorphic subvariety
$\mathbf{C}=[C,\tilde{C}]$ in $X_{0}\cap X_{\bullet}$ such that the
intersection number of $\mathbf{C}$ with each linking 2-sphere of
$\nu^{-1}(0)$ is one.

Note that when $(X,\nu)=({\mathbb{R}}\times Y,\pi_{2}^{*}(*_{3}\theta))$ is
cylindrical, a translation (by $\tau_{L}$) of a t-curve/pseudo-holomorphic
subvariety gives another t-curve/pseudo-holomorphic subvariety. Moreover,
every multi-orbit $\gamma$ on $(Y,\theta)$ corresponds to a translation-
invariant pseudo-holomorphic subvariety ${\mathbb{R}}\times\gamma$ in
$(X,\nu)$, and vice versa. Such a translation-invariant pseudo-holomorphic
subvariety is said to be constant. The notion of constant t-curves in
cylindrical admissible pairs is similarly defined.

###  2.2 Topology on the space of t-curves

Following Taubes [Td], we define

######  Definition 2.3.

Given two subsets $S,S^{\prime}$ in a riemannian manifold ${\mathscr{X}}$, the
geometric distance between $S,S^{\prime}$ in ${\mathscr{X}}$ is

$\mathop{\mathrm{dist}\,}\nolimits_{\mathscr{X}}(S,S^{\prime}):=\Big{(}\sup_{x\in
S}\mathop{\mathrm{dist}\,}\nolimits(x,S^{\prime})+\sup_{x\in
S^{\prime}}\mathop{\mathrm{dist}\,}\nolimits(x,S)\Big{)}.$

Let $({\mathscr{X}},\omega)$ be a (possibly noncompact) symplectic 4-manifold,
and $J$ be a compatible almost complex structure. Let $C_{n}$,
$n\in{\mathbb{Z}}^{\geq 0}$ be pseudo-holomorphic subvarieties in
${\mathscr{X}}$ with $\int_{C_{n}}\omega<\infty$. The sequence
$\\{C_{n}\\}_{n\in\Z^{+}}$ is said to converge geometrically to $C_{0}$ if the
following hold:

  * •

Regard each $C_{n}$ and its irreducible components as 2-dimensional
rectifiable currents. Then $\\{C_{n}\\}_{n\in\Z^{+}}$ converges weakly as
currents to a current of the form $\sum_{a}m_{a}C^{\prime}_{a}$, where
$m_{a}\in\Z^{+}$, and $C^{\prime}_{a}$ are irreducible components of $C_{0}$.

  * •

For any compact subset $K\subset{\mathscr{X}}$,
$\lim_{n\to\infty}\mathop{\mathrm{dist}\,}\nolimits_{K}(C_{n}|_{K},C_{0}|_{K})=0$.

There is a version of Gromov compactness result for pseudo-holomorphic
subvarieties in 4-dimensional symplectic manifolds that hold simultaneously
for all genera:

######  Proposition 2.4.

([Td] Propositions 3.8) Let $({\mathscr{X}},\omega),J$ be as in Definition
2.3. Suppose there is a countable exhaustion
$\\{{\mathscr{X}}_{n}\\}_{n\in{\mathbb{Z}}^{+}}$ of ${\mathscr{X}}$ by open
subsets with compact closure,
${\mathscr{X}}_{1}\subset\cdots{\mathscr{X}}_{n}\subset{\mathscr{X}}_{n+1}\subset\cdots\subset{\mathscr{X}}$,
and let $\\{(\omega_{n},J_{n})\\}_{n}$ be a corresponding sequence whose
typical element $(\omega_{n},J_{n})$ consists of a symplectic form
$\omega_{n}$ and a compatible almost complex structure $J_{n}$ on
${\mathscr{X}}_{n}$. Suppose that $\\{(\omega_{n},J_{n})\\}_{n}$ converges to
$(\omega,J)$ in the $C^{\infty}$ topology. Let $z_{E}>0$, and let
$\\{C_{n}\\}_{n\in\Z^{+}}$ be a sequence of $J_{n}$-holomorphic subvarieties
in $({\mathscr{X}}_{n},\omega_{n})$ such that $\int_{C_{n}}\omega<z_{E}$
$\forall n$. Then there is a $J$-holomorphic subvariety
$C\subset{\mathscr{X}}$ with $\int_{C}\omega<z_{E}$ and a subsequence of
$\\{C_{n}\\}_{n}$ which converges geometrically to $C$.

Now let $X$ be a 4-dimensional MCE and $\nu$ be an admissible form on $X$.

######  Definition 2.5.

A sequence of t-curves $\\{\mathbf{C}_{n}\\}_{n\in{\mathbb{Z}}^{+}}$ in
$X^{{}^{\prime}a}$ is said to (locally) converge to a t-curve $\mathbf{C}_{0}$
in $X^{{}^{\prime}a}$ if the underlying subvarieties of $\mathbf{C}_{n}$
converge geometrically to the underlying subvariety of $\mathbf{C}_{0}$, and
the associated currents of $\mathbf{C}_{n}$ converge to the associated current
of $\mathbf{C}_{0}$.

The notion of convergence of t-orbits is similarly defined.

######  Definition 2.6.

Let $i\in{\mathfrak{Y}}_{m}$. A t-curve $\mathbf{C}$ on $X^{{}^{\prime}a}$ is
said to be asymptotic on the $Y_{i}$-end to some t-orbit
$\boldsymbol{\gamma}_{i}$ with respect to the pair $(Y_{i},\nu_{i})$ if:

  * (i)

Let $\gamma_{i}$ and $C$ be respectively the underlying subvarieties of
$\boldsymbol{\gamma}_{i}$ and $\mathbf{C}$. Given any sequence of positive
numbers $\\{L_{n}\\}_{n\in\Z^{+}}$ with $\lim_{n}L_{n}\to\infty$,
$\\{(\tau_{-L_{n}}C)|_{\hat{Y}_{i}}\\}_{n}$ geometrically converges to
$[0,\infty)\times\gamma_{i}$ in $\hat{Y}_{i}$. In fact, for all sufficiently
small $\varepsilon>0$, there exists an $R(\varepsilon)>0$, such that
$\mathop{\mathrm{dist}\,}\nolimits_{\hat{Y}_{i}}(\tau_{-L}C,[0,\infty)\times\gamma_{i})<\varepsilon$,
$\forall L>R(\varepsilon)$.

  * (ii)

The sequence of currents $\\{\tau_{-L_{n}}\tilde{C}\\}_{n}$ converges weakly
to $\pi_{2}^{*}\tilde{\gamma}_{i}$, where
$\pi_{2}\colon\thinspace\hat{Y}_{i}={\mathbb{R}}\times Y_{i}\to Y_{i}$ is the
projection.

The t-orbit $\boldsymbol{\gamma}_{i}$ above is said to be the limiting t-orbit
of $\mathbf{C}$ on the $Y_{i}$-end. In the case when $X={\mathbb{R}}\times Y$
is cylindrical, the limiting t-orbit on $(+\infty)$-end is called the
$+\infty$-limit of $\mathbf{C}$, and the limiting t-orbit on the
$(-\infty)$-end, as a t-orbit in $Y$, is called the $-\infty$-limit of
$\mathbf{C}$. If ${\bf C}$ has a limiting t-orbit on each of its Morse ends,
${\bf C}$ is said to be asymptotically constant.

Replace the t-curve ${\bf C}$ and the limiting t-orbits
$\boldsymbol{\gamma}_{i}$ above respectively by a pseudo-holomorphic
subvariety $P$, and $\textsl{p}_{i}$, where $\textsl{p}_{i}$ is a union of
flow lines of the vector field dual to $*\nu_{i}$. The notion of $P$ being
asymptotic to $\textsl{p}_{i}$ on the $Y_{i}$, is similarly defined.

###  2.3 $\mathop{\mathrm{Spin}}\nolimits^{c}$ structures

#### (a) $\mathop{\mathrm{Spin}}\nolimits^{c}$-structure of t-orbits

A t-orbit $\boldsymbol{\gamma}=\\{(\gamma_{i},m_{i})\\}_{i}$ is assigned a
$\mathop{\mathrm{Spin}}\nolimits^{c}$ structure as follows. Let $B_{\theta}$
be a small tubular neighborhood of $\theta^{-1}(0)\subset Y$. Let

$[\boldsymbol{\gamma}]:=[\tilde{\gamma}]=\sum_{i}m_{i}[\gamma_{i}]\in
H_{1}(Y,\theta^{-1}(0);\mathbb{Z})\simeq H_{1}(Y,B_{\theta};\mathbb{Z})$

and let $E$ be a complex line bundle on $Y-B_{\theta}$ with
$c_{1}(E)=p.d.[\boldsymbol{\gamma}]$. Let $K^{-1}$ denote the subbundle
$\mathop{\mathrm{Ker}\,}\nolimits\theta$ of $TY|_{Y-B_{\theta}}$. This is an
oriented plane bundle; choose a complex structure on $K^{-1}$ compatible with
the orientation. Then $\mathbb{S}=E\oplus E\otimes K^{-1}$ is trivial over
$\partial(Y-B_{\theta})$. Extend $\mathbb{S}$ trivially over $B_{\theta}$ to
get a complex rank 2 vector bundle on $Y$, also denoted by $\mathbb{S}$. Then
the $\mathop{\mathrm{Spin}}\nolimits^{c}$-structure of $\boldsymbol{\gamma}$,
denoted $\mathfrak{s}(\boldsymbol{\gamma})$, is that with $\mathbb{S}$ as its
spinor bundle, and with $E$, $E\otimes K^{-1}$ being eigen-bundles under the
Clifford action of $*\theta/|\theta|$ corresponding to eigenvalues $-i$ and
$i$ respectively.

Let $S_{Y,*\theta}\subset H_{1}(Y,\theta^{-1}(0);\mathbb{Z})$ denote the
subset that maps to the fundamental class of $\theta^{-1}(0)$ under the
connecting map $\delta\colon\thinspace H_{1}(Y,\theta^{-1}(0);\mathbb{Z})\to
H_{0}\,(\theta^{-1}(0);\mathbb{Z})$ in the relative exact sequence of the pair
$(Y,\theta^{-1}(0))$. Note that the relative homology class
$[\boldsymbol{\gamma}]\in H_{1}(Y,\theta^{-1}(0);\mathbb{Z})$ of any t-orbit
falls in $S_{Y,*\theta}$, and in fact the procedure described above defines an
isomorphism from $S_{Y,*\theta}$ to the space of
$\mathop{\mathrm{Spin}}\nolimits^{c}$ structures on $Y$,

$\varsigma_{(Y,*\theta)}\colon\thinspace
S_{Y,*\theta}\stackrel{{\scriptstyle\sim}}{{\to}}\mathop{\mathrm{Spin}}\nolimits^{c}(Y),$

as affines spaces over $H_{1}(Y;\mathbb{Z})\simeq H^{2}(Y;\mathbb{Z})$.

Note that the assumption that $\theta$ is regular (cf. Definition 1.4) implies
that for any fixed ${\mathfrak{s}}\in\mathop{\mathrm{Spin}}\nolimits(Y)$,
there are finitely many t-orbits (with respect to $(Y,\theta)$),
$\boldsymbol{\gamma}$, with
${\mathfrak{s}}(\boldsymbol{\gamma})=\boldsymbol{\gamma}$. Denote this set by
${\mathbb{P}}(Y,\theta;{\mathfrak{s}})$. ($Y$, $\theta$, or both are sometimes
omitted from the notation when they are clear from the context. For example,
${\mathbb{P}}(Y,\theta;{\mathfrak{s}})={\mathbb{P}}(Y,{\mathfrak{s}})={\mathbb{P}}({\mathfrak{s}})$.)

#### (b) $\mathop{\mathrm{Spin}}\nolimits^{c}$-structure of t-curves

Let $H_{*,closed}$ denote the Borel-Moore homology222also called the “closed
homology” or “homology with closed support”, cf. [BT] Remark 5.17;
https://en.m.wikipedia.org/wiki/Borel-Moore_homology#CITEREF1960 and
references therein. It is dual to the cohomology of compact support.. It is
useful to note that in our setting,

$\begin{split}&H_{*,closed}\,(X;{\mathbb{K}})\simeq
H_{*,closed}\,(X^{{}^{\prime}a},\partial X^{{}^{\prime}a};{\mathbb{K}})\simeq
H_{*}(X_{c},\partial X_{c};{\mathbb{K}});\\\
&H_{*,closed}\,\big{(}X^{{}^{\prime}a},\partial X^{{}^{\prime}a}\cup
Z^{{}^{\prime}a}_{0};{\mathbb{K}}\big{)}\simeq
H_{*,closed}\,\big{(}\overline{X^{{}^{\prime}a}},\partial\overline{X^{{}^{\prime}a}}\cup
Z_{0}^{{}^{\prime}a};{\mathbb{K}})\simeq
H_{*}(\overline{X^{{}^{\prime}a}_{\delta}},\partial\overline{X^{{}^{\prime}a}_{\delta}};{\mathbb{K}}\big{)},\end{split}$

where $X^{{}^{\prime}a}_{\delta}$, $Z^{{}^{\prime}a}_{\delta}$ are as defined
in Section 1.4 (9).

A t-curve $\mathbf{C}=\\{(C_{k},m_{k})\\}_{k}$ in $X^{{}^{\prime}a}$
determines a relative homology class through its associated current

$\begin{split}[\mathbf{C}]&=[\tilde{C}]\in
H_{2,closed}\,(X^{{}^{\prime}a},\partial X^{{}^{\prime}a}\cup
Z^{{}^{\prime}a}_{\delta};\mathbb{Z})\\\ &\quad\simeq
H_{2,closed}\,(X^{{}^{\prime}a},\partial X^{{}^{\prime}a}\cup
Z^{{}^{\prime}a}_{0};\mathbb{Z})\simeq
H_{2}(\overline{X^{{}^{\prime}a}_{\delta}},\partial\overline{X^{{}^{\prime}a}_{\delta}};\mathbb{Z}).\end{split}$

The last condition in Definition 2.2 implies that $[\mathbf{C}]$ maps to the
fundamental class $[Z_{0}^{{}^{\prime}a}]\in
H_{1,closed}\,(Z_{0}^{{}^{\prime}a},\partial
Z_{0}^{{}^{\prime}a};\mathbb{Z}^{{}^{\prime}a})$ under the connecting map
$\delta$ in the relative exact sequence

$\begin{split}\cdots 0\to&H_{2,closed}\,(X^{{}^{\prime}a},\partial
X^{{}^{\prime}a};\mathbb{Z})\stackrel{{\scriptstyle
p}}{{\to}}H_{2,closed}\,(X^{{}^{\prime}a},\partial X^{{}^{\prime}a}\cup
Z^{{}^{\prime}a}_{0};\mathbb{Z})\\\
&\quad\quad\stackrel{{\scriptstyle\delta}}{{\to}}H_{1,closed}\,\big{(}Z^{{}^{\prime}a}_{0},\partial
Z^{{}^{\prime}a}_{0};\mathbb{Z}^{{}^{\prime}a}\big{)}\cdots.\end{split}$

Let

$S_{X,\nu}:=\delta^{-1}[Z_{0}^{{}^{\prime}a}]\subset
H_{2,closed}\,(X^{{}^{\prime}a},\partial X^{{}^{\prime}a}\cup
Z^{{}^{\prime}a}_{0};\mathbb{Z}).$

In parallel to the case with t-orbits, there is an isomorphism

$\varsigma_{X,\nu}\colon\thinspace
S_{X,\nu}\stackrel{{\scriptstyle\sim}}{{\to}}\mathop{\mathrm{Spin}}\nolimits^{c}(X)\simeq\mathop{\mathrm{Spin}}\nolimits^{c}(X^{{}^{\prime}a})$

as affine spaces over the abelian group

$\mathop{\mathrm{Ker}\,}\nolimits\delta=\mathop{\mathrm{Im}\,}\nolimits
p\simeq H_{2,closed}\,(X^{{}^{\prime}a},\partial
X^{{}^{\prime}a};\mathbb{Z})\simeq H^{2}(X;\mathbb{Z}).$

(In this article, we do not distinguish between the set of
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structures and $X$ and the set of
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structures on $X^{{}^{\prime}a}$, as
they are canonically isomorphic.) The inverse of $\varsigma_{X,\nu}$ can be
described more concretely as follow: Given a
$\mathop{\mathrm{Spin}}\nolimits^{c}$ structure $\mathfrak{s}$ on $X$, take
${\mathbb{S}}_{X}=\mathbb{S}=\mathbb{S}^{+}\oplus\mathbb{S}^{-}$ to be the
associated spinor bundle. Over $X^{{}^{\prime}a}_{0}$, the Clifford action of
$\nu^{+}/|\nu^{+}|$ splits $\mathbb{S}^{+}|_{X_{0}}$ as a direct sum $E\oplus
E\otimes K^{-1}$ of eigenbundles, where $E$ is a complex line bundle
corresponds to eigenvalue $-i$. Then $\varsigma_{X,\nu}^{-1}(\mathfrak{s})\in
H_{2,closed}\,(X^{{}^{\prime}a},\partial X^{{}^{\prime}a}\cup
Z_{0};\mathbb{Z})$ is the Poincaré dual of $c_{1}(E)\in
H^{2}(X^{{}^{\prime}a}-\nu^{-1}(0);{\mathbb{Z}})$.

Notice that the definition of $\mathop{\mathrm{Spin}}\nolimits^{c}$ structures
of t-orbits and t-curves only depend on their associated currents. So are the
various notions of energy defined in the upcoming subsection. Also, when
$(X,\nu)$ is cylindrical, two t-curves related by translation have the same
$\mathop{\mathrm{Spin}}\nolimits^{c}$ structure.

###  2.4 Energy of t-curves and chains of t-curves

Let $X$ be an MCE and $\nu$ an admissible form on $X$.

######  Definition 2.7.

Given an oriented 2-dimensional subvariety $C$ in $X^{{}^{\prime}a}$ and
$X_{\bullet}\subset X^{{}^{\prime}a}$, the $\nu$-energy of $C|_{X_{\bullet}}$
is:

${\mathscr{F}}_{\nu}(C|_{X_{\bullet}}):=\int_{C|_{X_{\bullet}}}\nu.$

When $X_{\bullet}$ is compact, the following is called the $*\nu$-energy of
$C|_{X_{\bullet}}$:

${\mathscr{F}}^{*}_{\nu}(C|_{X_{\bullet}}):=\int_{C|_{X_{\bullet}}}*\nu.$

Finally, setting $\omega=2\nu^{+}$, the $\omega$-energy of $C|_{X_{\bullet}}$
is:

${\mathscr{F}}_{\omega}(C|_{X_{\bullet}}):={\mathscr{F}}_{\nu}(C|_{X_{\bullet}})+\sup_{X_{*}\in{\mathfrak{X}}_{1}}{\mathscr{F}}^{*}_{\nu}(C|_{X_{*}}),$

where ${\mathfrak{X}}_{1}=\\{X_{*}\,|\,X_{*}\subset X_{\bullet},|X_{*}|\leq
1\\}$. (Cf. §1.4 for the definition of $|X_{\bullet}|$).

When $\omega^{-1}(0)=\nu^{-1}(0)=\emptyset$, the finite-energy condition above
agrees with that in [BEHWZ]. It also straightforwardly from the definitions
that in the cylindrical case, two t-curves that are related by translation
have the same $\nu$-energy, $*\nu$-energy, and $\omega$-energy.

######  Remark 2.8.

Just as in [BEHWZ], a bound on ${\mathscr{F}}_{\omega}$ is the prerequisite
for local compactness results, while a bound on ${\mathscr{F}}_{\nu}$ is used
to establish global compactness (weak convergence to broken-trajectories or
“buildings”). In the context of [Ts] where $X$ is closed,
${\mathscr{F}}_{\omega}$ is simply the LHS of its Equation (7.10), which also
give a bound on this in terms of the $\mathop{\mathrm{Spin}}\nolimits^{c}$
structure of $C$, the cohomology class $[\omega]$, and the metric on $X$.

The following useful lemma is an immediate consequence of the compatibility
condition of $J$.

######  Lemma 2.9.

Under the assumptions of the preceding definition, if $C$ is a pseudo-
holomorphic subvariety in $X_{\bullet}$, then
${\mathscr{F}}_{\omega}(C),{\mathscr{F}}_{\nu}(C),{\mathscr{F}}_{\nu}^{*}(C)\geq
0$. Moreover, when $(X_{\bullet}=I\times Y,\nu=\pi_{2}^{*}\underline{\nu})$ is
cylindrical, ${\mathscr{F}}_{\nu}(C)=0$ iff $C$ is constant. Namely,
$C=I\times\gamma$, where $\gamma$ is a union of flow lines of the vector dual
to $*\underline{\nu}$ in $Y$.

Given a t-curve $\mathbf{C}=\\{(C_{k},m_{k})\\}_{k}$ in $X_{\bullet}$, its
$\nu$-energy, $*\nu$-energy, $\omega$-energy are respectively defined to be

$\begin{split}{\mathscr{F}}_{\nu}(\mathbf{C}):=\sum_{k}m_{k}{\mathscr{F}}_{\nu}(C_{k}|_{X_{\bullet}}),\\\
{\mathscr{F}}^{*}_{\nu}(\mathbf{C}):=\sum_{k}m_{k}{\mathscr{F}}^{*}_{\nu}(C_{k}|_{X_{\bullet}}),\\\
{\mathscr{F}}_{\omega}(\mathbf{C}):=\sum_{k}m_{k}{\mathscr{F}}_{\omega}(C_{k}|_{X_{\bullet}}).\end{split}$

We will see in Proposition 2.16 below that by the preceding lemma, any t-curve
$\mathbf{C}$ with finite $\omega$-energy in $X^{{}^{\prime}a}$ is asymptotic
to a t-orbit $\boldsymbol{\gamma}_{i}$ on each of its Morse end $\hat{Y}_{i}$.
It also follows from the preceding lemma and Proposition 2.4 that

######  Proposition 2.10.

Let $X$ be an MCE and $\nu$ be an admissible form on $X$. Let
$\\{\mathbf{C}_{n}\\}_{n\in{\mathbb{Z}}^{+}}$ be a sequence of t-curves on
$X^{{}^{\prime}a}$, with a uniform upper bound on their $\omega$-energy. Then
there is a t-curve $\mathbf{C}_{0}$ and a subsequence of
$\\{\mathbf{C}_{n}\\}_{n\in{\mathbb{Z}}^{+}}$ that locally converges to
$\mathbf{C}_{0}$.

######  Definition 2.11.

Let $(X,\nu)$ be an admissible pair.

  * (i)

Suppose the admissible pair $(X={\mathbb{R}}\times Y,\nu)$ is cylindrical.
(And thus $X$ has no vanishing ends and
$X^{{}^{\prime}a}=X^{\prime\prime}=X$.) Let ${\mathfrak{s}}_{Y}$ denote the
restriction of the $\mathop{\mathrm{Spin}}\nolimits^{c}$ structure
${\mathfrak{s}}$ to $\\{s\\}\times Y$, for any $s\in{\mathbb{R}}$. For
${\mathfrak{K}}\in{\mathbb{Z}}^{\geq 0}$, a (${\mathfrak{K}}$-component) chain
of t-curves ${\mathfrak{C}}$ on $X^{{}^{\prime}a}$ consists of:

    * –

a sequence of ${\mathfrak{K}}+1$ t-orbits with
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure ${\mathfrak{s}}_{Y}$,
$\\{\boldsymbol{\gamma}_{0},\boldsymbol{\gamma}_{1},\ldots,\boldsymbol{\gamma}_{{\mathfrak{K}}}\\}$,
which we call the rest orbits of ${\mathfrak{C}}$; and

    * –

a sequence (possibly empty) of non-constant, asymptotically constant t-curves
$\mathbf{C}_{k}$ on $X={\mathbb{R}}\times Y$ with
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure ${\mathfrak{s}}$, with
$-\infty$\- and $+\infty$-limits respectively $\boldsymbol{\gamma}_{k-1}$ and
$\boldsymbol{\gamma}_{k}$.

We say that $\mathfrak{C}$ has $\mathop{\mathrm{Spin}}\nolimits^{c}$-structure
${\mathfrak{s}}$. The ($-\infty$)-limit of $\mathfrak{C}$ is
$\boldsymbol{\gamma}_{0}$; the $+\infty$-limit of $\mathfrak{C}$ is
$\boldsymbol{\gamma}_{{\mathfrak{K}}}$. The $k$-th component of
${\mathfrak{C}}$ is ${\bf C}_{k}$. We typically write ${\mathfrak{C}}$ as an
ordered set, ${\mathfrak{C}}=\\{{\bf C}_{k}\\}_{k=1}^{\mathfrak{K}}$, when
${\mathfrak{K}}>0$. Two chains of t-curves, ${\mathfrak{C}}$,
${\mathfrak{C}}^{\prime}$, are said to be equivalent and denoted
${\mathfrak{C}}\sim_{\tau}{\mathfrak{C}}^{\prime}$, if they have the same
number of components and the same rest orbits, and when ${\mathfrak{K}}>0$,
each pair of corresponding components ${\bf C}_{k}$, ${\bf C}^{\prime}_{k}$
are related by translation. I.e. ${\bf C}^{\prime}_{k}=\tau_{L}{\bf C}_{k}$
for a certain $L\in{\mathbb{R}}$.

  * (ii)

Suppose the admissible pair $(X,\nu)$ is not cylindrical. Denote the
$\mathop{\mathrm{Spin}}\nolimits^{c}$ structure on $X$ by ${\mathfrak{s}}$,
and the restriction ${\mathfrak{s}}|_{\text{{\small
s}}_{i}^{-1}(0)}=:{\mathfrak{s}}_{i}$ for each $i\in\mathfrak{Y}_{X}$. A chain
of t-curves on $X^{{}^{\prime}a}$ is a pair
$\mathfrak{C}=\\{\mathbf{C}_{0},\\{\mathfrak{C}_{i}\\}_{i\in\mathfrak{Y}_{m}}\\}$
such that:

    * (1)

$\mathbf{C}_{0}$ is an asymptotically constant t-curve with
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure ${\mathfrak{s}}$ on
$X^{{}^{\prime}a}$. For each $i\in{\mathfrak{Y}}_{m}$, let
$\boldsymbol{\gamma}_{0,i}$ denote the limiting t-orbit of $\mathbf{C}_{0}$ on
the $Y_{i}$-end.

    * (2)

Let $(Y_{i},\nu_{i})$ denote the ending pair of the end $\hat{Y}_{i}\subset
X$. For each $i\in{\mathfrak{Y}}_{m}$, $\mathfrak{C}_{i}$ is a chain of
t-curves of $\mathop{\mathrm{Spin}}\nolimits^{c}$-structure
$\hat{{\mathfrak{s}}}_{i}$ on the cylindrical admissible pair
$(\mathbb{R}\times Y_{i},\hat{\nu}_{i})$, where $\hat{{\mathfrak{s}}}_{i}$,
$\hat{\nu}_{i}$ are respectively the pull-back of ${\mathfrak{s}}_{i}$ and
$\nu_{i}$ under the projection ${\mathbb{R}}\times Y_{i}\to Y_{i}$.

    * (3)

For each $i\in{\mathfrak{Y}}_{m}$, the $-\infty$-limit of $\mathfrak{C}_{i}$
is $\boldsymbol{\gamma}_{0,i}$.

We say that the chain $\mathfrak{C}$ has
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure ${\mathfrak{s}}$. The rest
orbits of ${\mathfrak{C}}_{i}$ are said to be the rest orbits of
${\mathfrak{C}}$ (in the $\hat{Y}_{i}$-end). Two chains of t-curves,
${\mathfrak{C}}=\\{{\bf
C}_{0},\\{{\mathfrak{C}}_{i}\\}_{i\in{\mathfrak{Y}}_{m}}\\}$,
${\mathfrak{C}}^{\prime}=\\{{\bf
C}^{\prime}_{0},\\{{\mathfrak{C}}^{\prime}_{i}\\}_{i\in{\mathfrak{Y}}_{m}}\\}$,
are said to be equivalent and denoted
${\mathfrak{C}}\sim_{\tau}{\mathfrak{C}}^{\prime}$, if ${\bf C}_{0}={\bf
C}^{\prime}_{0}$ and
${\mathfrak{C}}_{i}\sim_{\tau}{\mathfrak{C}}^{\prime}_{i}$ $\forall
i\in{\mathfrak{Y}}_{m}$.

An equivalence class of chains of t-curves is an analog of what is called an
“(unparametrized) broken trajectory” or “broken $X$-trajectory” in [KM]’s
Definitions 16.1.2 and 24.6.1.

The $\nu$-energy and $\omega$-energy of chains of t-curves on
$(X^{{}^{\prime}a},\nu)$ are defined as follows: When $(X,\nu)$ is cylindrical
and ${\mathfrak{C}}=\\{\mathbf{C}_{k}\\}_{k}$ has at least one component,

${\mathscr{F}}_{\nu}({\mathfrak{C}}):=\sum_{k}{\mathscr{F}}_{\nu}(\mathbf{C}_{k});\quad{\mathscr{F}}_{\omega}({\mathfrak{C}}):=\sum_{k}{\mathscr{F}}_{\omega}(\mathbf{C}_{k}).$

If ${\mathfrak{C}}$ has 0 components and a single rest orbit
$\boldsymbol{\gamma}$, set
${\mathscr{F}}_{\nu}({\mathfrak{C}}):={\mathscr{F}}_{\nu}({\mathbb{R}}\times\boldsymbol{\gamma})=0$
and
${\mathscr{F}}_{\omega}({\mathfrak{C}}):={\mathscr{F}}_{\omega}({\mathbb{R}}\times\boldsymbol{\gamma})$.

When $(X,\nu)$ is non-cylindrical and
$\mathfrak{C}=(\mathbf{C}_{0},\\{\mathfrak{C}_{i}\\}_{i\in\mathfrak{Y}_{m}})$,

${\mathscr{F}}_{\nu}({\mathfrak{C}}):=\sum_{k}{\mathscr{F}}_{\nu}(\mathbf{C}_{0})+\sum_{i}{\mathscr{F}}_{\nu}({\mathfrak{C}}_{i});\quad{\mathscr{F}}_{\omega}({\mathfrak{C}}):=\sum_{k}{\mathscr{F}}_{\omega}(\mathbf{C}_{0})+\sum_{i}{\mathscr{F}}_{\omega}({\mathfrak{C}}_{i}).$

Note that two equivalent chains of t-curves have the same $\nu$-energy and
$\omega$-energy.

######  Definition 2.12.

Let $\Gamma=\\{r_{n}\\}_{n\in{\mathbb{Z}}^{+}}\subset[1,\infty)$ be a sequence
of strictly increasing numbers such that $r_{n}\to\infty$ as $n\to\infty$, and
$\\{(A_{r},\Psi_{r})\\}_{r}\in\Gamma$ be a corresponding sequence of
admissible solutions to the Seiberg-Witten equation
${\mathfrak{S}}_{\mu_{r},\hat{{\mathfrak{p}}}_{r}}(A_{r},\Psi_{r})=0$, where
$\mu_{r}$, $\hat{{\mathfrak{p}}}_{r}$ are as in the statement of Theorem 1.7.
We say that the sequence $\\{(A_{r},\Psi_{r})\\}_{r\in\Gamma}$ weakly
t-converges to a chain of t-curves ${\mathfrak{C}}$ if the following holds:

  * (i)

When $(X,\mu_{r},\hat{{\mathfrak{p}}}_{r})$ is cylindrical: If
${\mathfrak{C}}$ has 0 components, namely, it has a single rest orbit
$\boldsymbol{\gamma}$, then $\\{(A_{r},\Psi_{r})\\}_{r\in\Gamma}$ t-converges
to the constant t-curve ${\mathbb{R}}\times\boldsymbol{\gamma}$. If
${\mathfrak{C}}$ has at least one component, write
$\mathfrak{C}=\\{\mathbf{C}_{k}\\}_{k=1}^{\mathfrak{K}}$. There is a sequence
$\\{{\bf L}_{r}=(L_{1,r},\cdots,L_{{\mathfrak{K}},r})\\}_{r\in\Gamma}$ in
$\R^{\mathfrak{K}}$, with $\lim_{r\to\infty}(L_{k_{1},r}-L_{k_{2},r})=\infty$
for any fixed pair $k_{1},k_{2}$ with $k_{1}>k_{2}$, such that
$\\{\tau_{-L_{k,r}}(A_{r},\Psi_{r})\\}_{r\in\Gamma}$ t-converges to
$\mathbf{C}_{k}$ for each $k$. (Note that the admissibility assumption on
$\nu$ implies that in this case $X$ has no vanishing ends, and
$X^{{}^{\prime}a}=X$.)

  * (ii)

When $(X,\mu_{r},\hat{{\mathfrak{p}}}_{r})$ is not cylindrical, write
$\mathfrak{C}=(\mathbf{C}_{0},\\{\mathfrak{C}_{i}\\}_{i\in\mathfrak{Y}_{m}}),\mathfrak{C}_{i}=\\{\mathbf{C}_{i,k}\\}_{k=1}^{{\mathfrak{K}}_{i}}$
in the notation of Definition 2.11 (ii). There exists a sequence $\\{{\bf
L}_{r}^{i}=(L^{i}_{1,r},\cdots,L^{i}_{{\mathfrak{K}}_{i},r})\\}_{r\in\Gamma}$
in $(\R^{+})^{{\mathfrak{K}}_{i}}$ for each $i\in\mathfrak{Y}_{m}$, with
$\lim_{r\to\infty}(L^{i}_{k_{1},r}-L^{i}_{k_{2},r})=\infty$ for any fixed pair
$k_{1},k_{2}$ with $k_{1}>k_{2}$, such that
$\\{\tau_{-L^{i}_{k,r}}(A_{r},\Psi_{r})\Big{|}_{\hat{Y}_{i}}\\}_{r\in\Gamma}$
t-converges to $\mathbf{C}_{i,k}$, and $\\{(A_{r},\Psi_{r})\\}_{r\in\Gamma}$
t-converges to $\mathbf{C}_{0}$.

Note that if a sequence $\\{(A_{r},\Psi_{r})\\}_{r}$ weakly t-converges to a
chain of t-curves ${\mathfrak{C}}$, and
${\mathfrak{C}}\sim_{\tau}{\mathfrak{C}}^{\prime}$, then
$\\{(A_{r},\Psi_{r})\\}_{r}$ also weakly t-converges to
${\mathfrak{C}}^{\prime}$. It therefore makes sense to talk about weakly
t-convergence to equivalence classes of chains of t-curves.

###  2.5 Relative homology classes of (chains of) t-curves

Let $(X,\nu)$ be an admissible pair as before.

Given two asymptotically constant t-curves $\mathbf{C},\mathbf{C}^{\prime}$ in
$X^{{}^{\prime}a}$ with the same
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure and limiting t-orbits on all
its Morse ends,
$[\mathbf{C^{\prime}}-\mathbf{C}]:=[\tilde{C}^{\prime}-\tilde{C}]$ defines a
class in $H_{2}(X^{{}^{\prime}a},\partial X^{{}^{\prime}a};{\mathbb{Z}})$. The
two t-curves $\mathbf{C}^{\prime}$, $\mathbf{C}$ are said to be of the same
relative homology class if $[\mathbf{C^{\prime}}-\mathbf{C}]=0$. The precise
value of $a$ does not matter, as all $H_{2}(X^{{}^{\prime}a},\partial
X^{{}^{\prime}a};{\mathbb{Z}})$ are mutually canonically isomorphic, and the
relative homology classes of two ${\bf C},{\bf C}^{\prime}$ in
$X^{{}^{\prime}a}$ and in $X^{{}^{\prime}b}$, $b\leq a$ are identified by such
a canonical isomorphism. We now describe the set of relative homology classes
for (chains of) t-curves in more detail. Let $\\{(Y_{i},\nu_{i})\\}_{i}$ be
the set of ending pairs of $(X,\nu)$, and let ${\mathfrak{s}}$ denote the
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure on $X$. Let
${\mathfrak{s}}_{i}:={\mathfrak{s}}|_{\text{{\small s}}_{i}^{-1}(1)}$.
Consider the following commutative diagram of relative exact sequences, and
recall from Section 2.3 (a) and (b) the definitions of the maps and spaces
$\varsigma_{X,\nu}\colon\thinspace
S_{X,\nu}\to\mathop{\mathrm{Spin}}\nolimits^{c}(X)$,
$\varsigma_{Y_{i},\nu_{i}}\colon\thinspace
S_{Y_{i},\nu_{i}}\to\mathop{\mathrm{Spin}}\nolimits^{c}(Y_{i})$ respectively
via the second and the third columns of this diagram:

(2.1) $\begin{CD}@ VVV@ VVV@ VVV\\\ \cdots H_{2}(X^{{}^{\prime}a},\partial
X^{{}^{\prime}a};{\mathbb{K}})@>{j_{\mathbb{K}}}>{}>H_{2,closed}\,(X^{{}^{\prime}a},\partial
X^{{}^{\prime}a};{\mathbb{K}})@>{\partial}>{}>\bigoplus_{i\in{\mathfrak{Y}}_{m}}H_{1}(Y_{i};{\mathbb{K}})\cdots\\\
@ VVV@ VpVV@ VVV\\\ \cdots H_{2}\,(X^{{}^{\prime}a},\partial
X^{{}^{\prime}a}\cup
Z^{{}^{\prime}a}_{0};{\mathbb{K}})@>{}>{}>H_{2,closed}\,(X^{{}^{\prime}a},\partial
X^{{}^{\prime}a}\cup
Z^{{}^{\prime}a}_{0};{\mathbb{K}})@>{\partial}>{}>\bigoplus_{i\in{\mathfrak{Y}}_{m}}H_{1}(Y_{i},\nu_{i}^{-1}(0);{\mathbb{K}})\cdots\\\
@ VVV@ V\delta VV@ VVV\\\ \cdots H_{1}(Z^{{}^{\prime}a}_{0},\partial
Z^{{}^{\prime}a}_{0};{\mathbb{K}})@>{}>{}>H_{1,closed}\,(Z^{{}^{\prime}a}_{0},\partial
Z^{{}^{\prime}a}_{0};{\mathbb{K}})@>{}>{}>\bigoplus_{i\in{\mathfrak{Y}}_{m}}H_{0}\,(\nu_{i}^{-1}(0);{\mathbb{K}})\cdots\\\
@ VVV@ VVV@ VVV\\\ \end{CD}$

Let $\boldsymbol{\gamma}_{i}$ be respectively t-orbits of $(Y_{i},\nu_{i})$
with $\mathop{\mathrm{Spin}}\nolimits^{c}$-structure ${\mathfrak{s}}_{i}$. The
relative homology class of a t-curve $\mathbf{C}$ with
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure ${\mathfrak{s}}$ and with
limiting t-orbits $\boldsymbol{\gamma}_{i}$ takes value in the space
${\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$
defined below.

For each Morse ending pair $(Y_{i},\nu_{i})$ $i\in{\mathfrak{Y}}_{m}$, define
the space ${\mathscr{Z}}_{(Y_{i},\nu_{i},{\mathfrak{s}}_{i})}$ to be the space
of all 1-dimensional integral currents $\tilde{\gamma}_{i}$ on $Y_{i}$ that
represents the class $\varsigma_{(Y_{i},\nu_{i})}^{-1}({\mathfrak{s}}_{i})\in
S_{Y_{i},\nu_{i}}\subset H_{1}(Y_{i},\nu_{i}^{-1}(0);{\mathbb{Z}})$.

Given ${\mathfrak{s}}\in\mathop{\mathrm{Spin}}\nolimits^{c}(X)$ and
$\tilde{\gamma}_{i}\in{\mathscr{Z}}_{(Y_{i},\nu_{i},{\mathfrak{s}}_{i})}$ for
each $i\in{\mathfrak{Y}}_{m}$, define

${\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}_{i}\\}_{i\in{\mathfrak{Y}}_{m}}\big{)}=\\{\tilde{C}\\}/\sim,$

where $\tilde{C}$ is a 2-dimensional integral current on $X^{{}^{\prime}a}$
satisfying the following conditions.

  * •

For each $i\in{\mathfrak{Y}}_{m}$, $\tilde{C}$ is asymptotic to
$\tilde{\gamma}_{i}$ on the $\hat{Y}_{i}$-end in the sense of item (ii) in
Definition 2.6, and the limit
$\lim_{l\to\infty}\int_{\tilde{C}|_{\hat{Y}_{i,l}}}\pi_{2}^{*}h<\infty$ exists
for any $h\in\Omega^{2}(Y_{i})$.

  * •

$\tilde{C}$ represents the element in
$\varsigma_{X,\nu}^{-1}({\mathfrak{s}})\in S_{X,\nu}\subset
H_{2,closed}\,(X^{{}^{\prime}a},\partial X^{{}^{\prime}a}\cup
Z_{0}^{{}^{\prime}a};{\mathbb{Z}})$.

Meanwhile, we write $\tilde{C}\sim\tilde{C}^{\prime}$ for two such currents
$\tilde{C}$, $\tilde{C}^{\prime}$ if $\tilde{C}-\tilde{C}^{\prime}$ is the
boundary of an integral current that is asymptotic to $0$ on all Morse ends of
$X$.

It follows from the preceding definition and (2.1) that the set
${\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}_{i}\\}_{i\in{\mathfrak{Y}}_{m}}\big{)}$
is an affine space over the abelian group

${\mathscr{H}}_{X}:=\mathop{\mathrm{Ker}\,}\nolimits j_{\mathbb{Z}}\subset
H_{2}(X^{{}^{\prime}a},\partial X^{{}^{\prime}a};{\mathbb{Z}}).$

An element in
${\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$
is said to be a (${\mathbb{Z}}$-) relative homology class in
$(X^{{}^{\prime}a},\nu,{\mathfrak{s}})$ (rel $\\{\tilde{\gamma}_{i}\\}_{i}$).
Later on, we sometimes omit ${\mathfrak{s}}$ from the notation when the
$\mathop{\mathrm{Spin}}\nolimits^{c}$ structure is fixed.

######  Remark 2.13.

Assertions (c) and (d) of Theorem 1.9 postulates that $b^{1}(Y_{i})=0$ for all
vanishing ends $i\in{\mathfrak{Y}}_{v}$. (In particular, this holds if
$(X,\nu)$ has no vanishing ends; especially, when $(X,\nu)$ is cylindrical.)
This constraint implies that $H_{2}(\partial
X^{{}^{\prime}a};{\mathbb{Z}})=\bigoplus_{i\in{\mathfrak{Y}}_{v}}H_{2}(Y_{i};{\mathbb{Z}})=0$,
and hence $H_{2}(X^{{}^{\prime}a},\partial
X^{{}^{\prime}a};{\mathbb{Z}})\simeq
H_{2}(X^{{}^{\prime}a};{\mathbb{Z}})\simeq H_{2}(X;{\mathbb{Z}})$. It also
implies that the $\nu$-energy of a chain ${\mathfrak{C}}$ depends only on its
relative homology class and its limiting t-orbits.

When $(X^{{}^{\prime}a},\nu)=(X,\nu)=({\mathbb{R}}\times
Y_{i},\pi_{2}^{*}\nu_{i})$ is cylindrical,
${\mathscr{H}}_{X}=H_{2}(X;{\mathbb{Z}})=H_{2}(Y_{i};{\mathbb{Z}})$. We use
${\mathscr{H}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime})$
to denote ${\mathscr{H}}(({\mathbb{R}}\times
Y_{i},\pi_{2}^{*}\nu_{i},\hat{{\mathfrak{s}}}_{i}),\\{\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime}\\})$,
where
$\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime}\in{\mathscr{Z}}_{Y_{i},\nu_{i},{\mathfrak{s}}_{i}}$
are respectively the limiting currents at the $-\infty$\- and the
$+\infty$-ends of $X$. By concatenation, there is a composition map that sends
every pair
$(a,b)\in{\mathscr{H}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i}^{-},\tilde{\gamma}_{i})\times{\mathscr{H}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{+})$
to an element
$b*a\in{\mathscr{H}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i}^{-},\tilde{\gamma}_{i}^{+})$.
Also, every element $\\{h_{i}\\}_{i}$ in
$\prod_{i\in{\mathfrak{Y}}_{m}}{\mathscr{H}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime})$
defines an isomorphism

$\operatorname{c}_{\\{h_{i}\\}_{i}}\colon\thinspace{\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}_{i}\\}_{i\in{\mathfrak{Y}}_{m}})\to{\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}^{\prime}_{i}\\}_{i\in{\mathfrak{Y}}_{m}}).$

Let $\mathfrak{C}=\\{\mathbf{C}_{1},\ldots,\mathbf{C}_{{\mathfrak{K}}}\\}$ be
a chain of t-curves on $({\mathbb{R}}\times Y_{i},\pi_{2}^{*}\nu_{i})$ with
${\mathfrak{K}}>0$. The relative homology class of ${\mathfrak{C}}$ is the
element

$[{\mathfrak{C}}]:=[\tilde{C}_{\mathfrak{K}}]*\cdots*[\tilde{C}_{2}]*[\tilde{C}_{1}].$

When ${\mathfrak{C}}$ is a 0-component chain of t-curves with a single rest
orbit $\boldsymbol{\gamma}$, we set the relative homology class of
${\mathfrak{C}}$ to be the element
$[{\mathbb{R}}\times\boldsymbol{\gamma}]\in{\mathscr{H}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma},\tilde{\gamma})$,
which maps to the element $0\in H_{2}(Y_{i};{\mathbb{Z}})$ under the canonical
isomorphism
${\mathscr{H}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma},\tilde{\gamma})\simeq
H_{2}(Y_{i};{\mathbb{Z}})$.

Suppose the admissible pair $(X,\nu)$ is not cylindrical. Given a chain of
t-curves
$\mathfrak{C}=(\mathbf{C}_{0},\\{\mathfrak{C}_{i}\\}_{i\in{\mathfrak{Y}}_{m}})$
on $X^{{}^{\prime}a}$, the relative homology class of ${\mathfrak{C}}$ is
defined to be

$[{\mathfrak{C}}]=\operatorname{c}_{\\{[{\mathfrak{C}}_{i}]\\}_{i}}([{\bf
C}_{0}]).$

It follows directly from the definition that equivalent chains of t-curves
have the same relative homology class.

What follows are some observations for future reference. When the currents
$\tilde{\gamma}_{i}$, $\tilde{C}$ above are not required to be integral, the
same strategy can be employed to define the real-variants of the spaces
${\mathscr{Z}}_{(Y_{i},\nu_{i},{\mathfrak{s}}_{i})}$,
${\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$
and other related notions. These are respectively denoted by
${\mathscr{Z}}^{\mathbb{R}}_{(Y_{i},\nu_{i},{\mathfrak{s}}_{i})}$,
${\mathscr{H}}^{\mathbb{R}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$,
etc. The set of real relative homology classes,
${\mathscr{H}}^{\mathbb{R}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$,
is now an affine space under
${\mathscr{H}}^{\mathbb{R}}_{X}:=\mathop{\mathrm{Ker}\,}\nolimits
j_{\mathbb{R}}\subset H_{2}(X^{{}^{\prime}a},\partial
X^{{}^{\prime}a};{\mathbb{R}})$. Note that $\mathop{\mathrm{Ker}\,}\nolimits
j_{\mathbb{Z}}={\mathscr{H}}_{X}$ is torsion-free; so we may and will often
identify it with the integral lattice in $\mathop{\mathrm{Ker}\,}\nolimits
j_{\mathbb{R}}={\mathscr{H}}_{X}^{\mathbb{R}}$. Similarly, When
$\\{\tilde{\gamma}_{i}\\}_{i}$ consists of integral currents,
${\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$
embeds in
${\mathscr{H}}^{\mathbb{R}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$
as an orbit under the
${\mathscr{H}}_{X}\subset{\mathscr{H}}_{X}^{\mathbb{R}}$-action.
${\mathbb{R}}$-coefficient versions of composition maps $*$ and
$\operatorname{c}_{\\{h_{i}\\}_{i}}$ are similarly defined and have similar
properties.

######  Lemma 2.14.

Fix $i\in{\mathfrak{Y}}_{m}$. For any
$\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime}\in{\mathscr{Z}}^{\mathbb{R}}_{(Y_{i},\nu_{i},{\mathfrak{s}}_{i})}$,
the metric on $Y_{i}$ determines an isomorphism $I_{\mathscr{H}}$ from
${\mathscr{H}}^{\mathbb{R}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime})$
to $H_{2}(Y_{i};{\mathbb{R}})=H_{2}({\mathbb{R}}\times
Y_{i};{\mathbb{R}})={\mathscr{H}}_{{\mathbb{R}}\times Y}^{\mathbb{R}}$ as
affine spaces under $H_{2}(Y_{i};{\mathbb{R}})$. In particular, all the affine
spaces
${\mathscr{H}}^{\mathbb{R}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime})$
are identified via these isomorphisms. By concatenation (the real versions of
$\operatorname{c}_{\\{h_{i}\\}_{i}}$ above, which we shall denote by the same
notation), for any pair
$\\{\tilde{\gamma}_{i}\\}_{i},,\\{\tilde{\gamma}^{\prime}_{i}\\}_{i}$, the
metric on $X$ determines an isomorphism between the corresponding affine
spaces
${\mathscr{H}}^{\mathbb{R}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$,
${\mathscr{H}}^{\mathbb{R}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}^{\prime}_{i}\\}_{i}\big{)}$
as well. Moreover, given
$\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime},\tilde{\gamma}_{i}^{\prime\prime}\in{\mathscr{Z}}^{\mathbb{R}}_{(Y_{i},\nu_{i},{\mathfrak{s}}_{i})}$
and
$[\tilde{C}]\in{\mathscr{H}}^{\mathbb{R}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime})$,
$[\tilde{C}^{\prime}]\in{\mathscr{H}}^{\mathbb{R}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}^{\prime}_{i},\tilde{\gamma}_{i}^{\prime\prime})$,

(2.2)
$I_{\mathscr{H}}([\tilde{C}]*[\tilde{C^{\prime}}])=I_{\mathscr{H}}([\tilde{C}])+I_{\mathscr{H}}([\tilde{C^{\prime}}]).$

Proof. Let $\tilde{C}$ be a 2-current on ${\mathbb{R}}\times Y_{i}$
representing and element
$[\tilde{C}]\in{\mathscr{H}}^{\mathbb{R}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime})$,
and let $h$ be a harmonic 2-form on $Y_{i}$ representing an element in $[h]\in
H^{2}(Y_{i};{\mathbb{R}})$. The integral of $\pi^{*}_{2}h$ over $\tilde{C}$ is
a real number depending only on the class $[\tilde{C}]$; so in this way we
have a map

$I_{\mathscr{H}}\colon\thinspace{\mathscr{H}}^{\mathbb{R}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime})\to\mathop{\mathrm{Hom}\,}\nolimits(H^{2}(Y_{i};{\mathbb{R}});{\mathbb{R}})\simeq
H_{2}(Y_{i};{\mathbb{R}}).$

It is straightforward to verify that the $H_{2}(Y_{i};{\mathbb{R}})$-action on
${\mathscr{H}}^{\mathbb{R}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime})$
intertwines with that on $H_{2}(Y_{i};{\mathbb{R}})$, as well as the
composition rule (2.2). $\Box$

The ${\mathbb{Z}}$-coefficients version of statement in the preceding lemma
does not hold. However, since when
$\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime}\in{\mathscr{Z}}_{(Y_{i},\nu_{i},{\mathfrak{s}}_{i})}\subset{\mathscr{Z}}^{\mathbb{R}}_{(Y_{i},\nu_{i},{\mathfrak{s}}_{i})}$,
the space
${\mathscr{H}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime})$
embeds in
${\mathscr{H}}^{\mathbb{R}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime})$
as an orbit under the $H_{2}(Y_{i};{\mathbb{Z}})\subset
H_{2}(Y_{i};{\mathbb{R}})$ action, the isomorphism $I_{\mathscr{H}}$ above
also identifies it as an $H_{2}(Y_{i};{\mathbb{Z}})$-orbit in
$H_{2}(Y_{i};{\mathbb{R}})$; that is, an element
$t_{\mathscr{H}}(\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime})$ in the orbit
space
${\mathbb{T}}_{Y_{i}}:=H_{2}(Y_{i};{\mathbb{R}})/H_{2}(Y_{i};{\mathbb{Z}})$.
Equip the torus ${\mathbb{T}}_{Y_{i}}$ with the standard Euclidean metric.
This map,

$t_{\mathscr{H}}\colon\thinspace{\mathscr{Z}}_{(Y_{i},\nu_{i},{\mathfrak{s}}_{i})}\times{\mathscr{Z}}_{(Y_{i},\nu_{i},{\mathfrak{s}}_{i})}\to{\mathbb{T}}_{Y_{i}},$

is continuous with respect to the current topology on
${\mathscr{Z}}_{(Y_{i},\nu_{i},{\mathfrak{s}}_{i})}$, and is linear with
respect to
$\tilde{\gamma}_{i}^{\prime}-\tilde{\gamma}_{i}\in{\mathscr{Z}}(Y_{i})$, where
${\mathscr{Z}}(Y_{i})$ is the space of exact integral 1-currents on $Y_{i}$.
Thus, one has:

######  Lemma 2.15.

Suppose
$\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime}\in{\mathscr{Z}}_{(Y_{i},\nu_{i},{\mathfrak{s}}_{i})}$
are sufficiently close in the sense that
$t_{\mathscr{H}}(\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime})$ falls in the
ball $B_{0}(1/2)\subset{\mathbb{T}}_{Y_{i}}$. Then there is a distinguished
element in
${\mathscr{H}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime})$.
The notion of “sufficiently close” depends on the metric on $Y_{i}$, but the
distinguished element is independent of the metric on $Y_{i}$.

Suppose a pair
$\\{\tilde{\gamma}_{i}\\}_{i},,\\{\tilde{\gamma}^{\prime}_{i}\\}_{i}\in\prod_{i\in{\mathfrak{Y}}_{m}}{\mathscr{Z}}_{(Y_{i},\nu_{i},{\mathfrak{s}}_{i})}$
are sufficiently close in the sense described above, and let
$h_{i}\in{\mathscr{H}}(Y_{i},\nu_{i},{\mathfrak{s}}_{i};\tilde{\gamma}_{i},\tilde{\gamma}_{i}^{\prime})$
denote the aforementioned distinguished element. Then the concatenation map
$\operatorname{c}_{\\{h_{i}\\}_{i}}$ defines a canonical isomorphism from
${\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}_{i}\\}_{i}\big{)}$
to
${\mathscr{H}}\big{(}(X^{{}^{\prime}a},\nu,{\mathfrak{s}}),\\{\tilde{\gamma}^{\prime}_{i}\\}_{i}\big{)}$
as affine spaces under ${\mathscr{H}}_{X}$.

###  2.6 Some basic facts about t-curves

Analogs of the results below are well-known in the context of symplectic field
theory.

######  Proposition 2.16.

Let $(X,\nu)$ be an admissible pair, and let
$\\{(Y_{i},\nu_{i})\\}_{i\in{\mathfrak{Y}}}$ be the ending pairs. A t-curve
$\mathbf{C}=[C,\tilde{C}]$ with finite $\omega$-energy in $X^{{}^{\prime}a}$
is asymptotic to some t-orbit
$\boldsymbol{\gamma}_{i}=[\gamma_{i},\tilde{\gamma}_{i}]$ in the pair
$(Y_{i},\nu_{i})$, $\forall i\in{\mathfrak{Y}}_{m}$, in the following sense:

  * (i)

Given any sequence of positive numbers $\\{L_{n}\\}_{n\in\Z^{+}}$ with
$\lim_{n}L_{n}\to\infty$, $\\{(\tau_{-L_{n}}C)|_{\hat{Y}_{i}}\\}_{n}$
geometrically converges to $[0,\infty)\times\gamma_{i}$ in
$\hat{Y}_{i}\simeq[0,\infty)\times Y_{i}$. In fact, for all sufficiently small
$\varepsilon>0$, there exists an $R(\varepsilon)>0$, such that
$\mathop{\mathrm{dist}\,}\nolimits_{\hat{Y}_{i}}(\tau_{-L}C,[0,\infty)\times\gamma_{i})<\varepsilon$,
$\forall L>R(\varepsilon)$.

  * (ii)

The sequence of currents $\\{\tau_{-L_{n}}\tilde{C}\\}_{n}$ converges weakly
to $\pi_{2}^{*}\tilde{\gamma}_{i}$, where
$\pi_{2}\colon\thinspace\hat{Y}_{i}\simeq[0,\infty)\times Y_{i}\to Y_{i}$ is
the projection.

Proof. We first show that the statements (i) and (ii) hold for a subsequence
of $\\{L_{n}\\}_{n}$.

The finite $\omega$-energy assumption in the definition of t-curves implies
that $\forall i\in{\mathfrak{Y}}_{m}$,

(2.3) $\sup_{L>0}\int_{C|_{\hat{Y}_{i,[L,L+1]}}}\omega<z_{i}\quad\text{for
some $z_{i}>0$.}$

Thus, by the admissibility condition on $\nu$ we may apply Proposition 2.4 to
find a subsequence $\\{R_{n}\\}_{n\in\Z^{+}}$ of $\\{L_{n}\\}_{n}$ with
$\lim_{n\to\infty}R_{n}=\infty$, such that $\\{\tau_{-R_{n}}C\\}_{n}$
converges geometrically over $[0,1]\times(Y_{i}\backslash\nu_{i}^{-1}(0))$ to
a $J_{i}$-holomorphic subvariety $C_{i}$. Here, $J_{i}$ is the almost complex
structure determined by the metric and the symplectic form
$\nu_{i}+ds\wedge*_{3}\nu_{i}$ on
$[0,1]\times(Y_{i}\backslash\nu_{i}^{-1}(0))$. Because of (2.3) and the fact
that ${\mathscr{F}}_{\nu}(C|_{\hat{Y}_{i}})<\infty$, we know that
$\int_{C_{i}}\omega<z_{i}$, and $\int_{C_{i}}\nu=0$. By Lemma 2.9 and the
conditions that define a t-curve, we know that $C_{i}=[0,1]\times\gamma_{i}$,
where $\gamma_{i}$ is the underlying subvariety of a certain t-orbit. (The
weights on closed orbits will be determined later).

Observe that statement (a) below implies statement (b), which in turn implies
statement (c).

  * (a)

the geometric convergence of $\\{\tau_{-R_{n}}C\\}_{n}$ over
$[0,1]\times(Y_{i}\backslash\nu_{i}^{-1}(0))$ to $[0,1]\times\gamma_{i}$.

  * (b)

the geometric convergence of $\\{\tau_{-R_{n}}C\\}_{n}$ over $[0,1]\times
Y_{i}$ to $[0,1]\times\gamma_{i}$.

  * (c)

the geometric convergence of (a subsequence of) $\\{\tau_{-R_{n}}C\\}_{n}$
over ${\mathbb{R}}^{+}\times Y_{i}$ to ${\mathbb{R}}\times\gamma_{i}$. We use
the same notation $\\{\tau_{-R_{n}}C\\}_{n}$ to denote the subsequence.

The statement (c) follows from statement (b) by a diagonalization argument. To
see why (a) implies (b), let $U_{\epsilon}\subset Y_{i}$ denote the set of
points with distance less or equal $\epsilon$ from $\nu_{i}^{-1}(0)$. By (a),
for any small $\varepsilon>0$, $\exists\,N(\varepsilon)$ such that

$\mathop{\mathrm{dist}\,}\nolimits_{[0,1]\times(Y_{i}\backslash
U_{\varepsilon/2})}(\tau_{-R_{n}}C,[0,1]\times\gamma_{i})<\varepsilon/2\quad\forall
n>N(\varepsilon).$

On the other hand, $\mathop{\mathrm{dist}\,}\nolimits_{[0,1]\times
U_{\varepsilon/2}}(\tau_{-R_{n}}C,[0,1]\times\gamma_{i})<\varepsilon/2$.
Together, this means $\mathop{\mathrm{dist}\,}\nolimits_{[0,1]\times
Y_{i}}(\tau_{-R_{n}}C,[0,1]\times\gamma_{i})<\varepsilon$ $\forall
n>N(\varepsilon)$.

We have previously established statement (a) for $\\{\tau_{-R_{n}}C\\}_{n}$.
Thus, the above observation establishes assertion (i) in the statement of the
Proposition for $\\{\tau_{-R_{n}}C\\}_{n}$. Next we establish assertion (ii)
for $\\{\tau_{-R_{n}}\mathbf{C}\\}_{n}$. Define the current
$\tilde{\gamma}_{i}$ such that $\pi_{2}^{*}\tilde{\gamma}_{i}$ is the (weak)
limit of the currents $\tau_{-R_{n}}\tilde{C}$ over
$[0,1]\times(Y_{i}\backslash\nu_{i}^{-1}(0))$. The currents
$\tau_{-R_{n}}\tilde{C}$, $\pi_{2}^{*}\tilde{\gamma}_{i}$ over
$[0,1]\times(Y_{i}\backslash\nu_{i}^{-1}(0))$ extend over $[0,1]\times Y_{i}$,
since $[0,1]\times\nu_{i}^{-1}(0)$ is a co-dimension 3 submanifold in
$[0,1]\times Y_{i}$. Set $\boldsymbol{\gamma}_{i}$ to be the t-orbit with
underlying subvariety $\gamma_{i}$, and with associated current
$\tilde{\gamma}_{i}$.

To complete the proof of this proposition, it now suffices to show that
$[0,1]\times\gamma_{i}$ is the geometric limit of
$\\{\tau_{-L_{n}}C|_{\hat{Y}_{i,[0,1]}}\\}_{n}$ for any sequence
$\\{L_{n}\\}_{n}$ with $\lim_{n\to\infty}L_{n}=\infty$. Suppose the contrary.
Then there is a sequence $\\{L_{n}\\}_{n}$, $\lim_{n\to\infty}L_{n}=\infty$
such that $\forall n$, $\tau_{-L_{n}}C|_{\hat{Y}_{i,[0,1]}}$ has distance
larger than $\varepsilon$ from $[0,1]\times\gamma_{i}$ for some small
$\varepsilon>0$. The same argument via Theorem 2.4 as above shows that there
is a subsequence $\\{L^{\prime}_{n}\\}_{n}$ of $\\{L_{n}\\}_{n}$, such that
$\lim_{n\to\infty}L_{n}^{\prime}\to\infty$ and
$\\{\tau_{-L^{\prime}_{n}}C|_{\hat{Y}_{i,[0,1]}}\\}_{n}$ converging
geometrically to $[0,1]\times\gamma_{i}^{\prime}$. Here, $\gamma_{i}^{\prime}$
is the underlying subvariety of a different t-orbit, and
$\mathop{\mathrm{dist}\,}\nolimits_{Y_{i}}(\gamma_{i},\gamma_{i}^{\prime})>\varepsilon$.
Without loss of generality, we may choose $\\{L_{n}^{\prime}\\}_{n}$ such that
$L^{\prime}_{n}>3R_{n}$. By Proposition 2.4 again,
$\\{\tau_{-(2R_{n})}C\\}_{n}$ converges geometrically to a pseudo-holomorphic
variety $C^{\prime}\subset{\mathbb{R}}\times Y_{i}$ asymptotic to
${\mathbb{R}}\times\gamma_{i}$ and ${\mathbb{R}}\times\gamma_{i}^{\prime}$
respectively on the $-\infty$\- and $+\infty$-ends. However,
$\int_{C^{\prime}}\nu=0$, and thus by Lemma 2.9,
$C^{\prime}={\mathbb{R}}\times\gamma_{i}$ and
$\gamma_{i}=\gamma^{\prime}_{i}$, contradicting our assumption on
$\gamma_{i}^{\prime}$. $\Box$

We say that a t-curve in a cylindrical admissible pair $({\mathbb{R}}\times
Y_{i},\pi_{2}^{*}\nu_{i})$ is constant if it is of the form
$\R\times\boldsymbol{\gamma}$ for some t-orbit $\boldsymbol{\gamma}$ of
$(Y_{i},\nu_{i})$.

######  Lemma 2.17.

Let $(Y,{\mathfrak{s}}_{Y})$ be a closed $\mathop{\mathrm{Spin}}\nolimits^{c}$
3-manifold. Let $g$ denote a riemannian metric on $Y$, and let $(X,\nu)$
denote a cylindrical admissible pair with $X={\mathbb{R}}\times Y$. Let
${\mathfrak{s}}$ be the $\mathop{\mathrm{Spin}}\nolimits^{c}$ structure on
$X={\mathbb{R}}\times Y$ induced by ${\mathfrak{s}}_{Y}$. There exists a
constant $\hbar>0$ such that any nonconstant t-curve $\mathbf{C}$ of
$\mathop{\mathrm{Spin}}\nolimits^{c}$-structure ${\mathfrak{s}}$ in $(X,\nu)$
has ${\mathscr{F}}_{\nu}(\mathbf{C})>\hbar$.

Proof. By assumption, $\nu=\pi_{2}^{*}\nu_{Y}$, where $\nu_{Y}$ is Hodge dual
to a regular Morse-Novikov 1-form. There are finitely many t-orbits with a
fixed $\mathop{\mathrm{Spin}}\nolimits^{c}$ structure in $Y$ (cf. e.g. [HL]).
Then without loss of generality, we may restrict to t-curves asymptotic to
fixed t-orbits $\boldsymbol{\gamma}_{\pm}$ respectively in the
$\pm\infty$-ends.

Suppose the lemma is false and there is a sequence of nontrivial t-curves
$\\{\mathbf{C}_{n}\\}_{n}$ asymptotic to t-orbits $\boldsymbol{\gamma}_{\pm}$
on the $\pm\infty$-ends, such that ${\mathscr{F}}_{\nu}(\mathbf{C}_{n})\to 0$.
We claim that $\forall\varepsilon>0$, $\exists N$ such that $\forall n>N$,
$C_{n}$ is of distance less than $\varepsilon$ from ${\mathbb{R}}\times\gamma$
for some t-orbit $\boldsymbol{\gamma}$. ($C_{n}$ and $\gamma$ respectively
denote the underlying subvariety of $\mathbf{C}_{n}$ and
$\boldsymbol{\gamma}$). It would then follow that $C_{n}$ has $\gamma$ as both
the $+\infty$\- and $-\infty$ ends, and $\pi_{2}(C_{n})$ represents a trivial
element in $H_{2}(Y;\Z)$, since $H_{2}(N_{\varepsilon};{\mathbb{Z}})$ is
trivial. Here, $N_{\varepsilon}$ denotes the tubular neighborhood of
${\mathbb{R}}\times\gamma$ consisting of points of distance less than
$\varepsilon$ from ${\mathbb{R}}\times\gamma$. Thus,
${\mathscr{F}}_{\nu}(C_{n})=0$ and we saw from Lemma 2.9 that $C_{n}$ must be
constant, contradicting our assumption.

To prove the claim, again suppose the contrary, that there exists of sequence
of t-curves $\\{\mathbf{C}_{n}\\}_{n}$, $C_{n}\not\subset N_{\varepsilon}$
$\forall n$ for any t-orbit $\boldsymbol{\gamma}$ with the prescribed
$\mathop{\mathrm{Spin}}\nolimits^{c}$ structure. Let $\Gamma\subset Y$ denote
the union of all underlying subvarieties of such t-orbits and let
$N^{\Gamma}_{\varepsilon}\subset X$ be the set consisting of points with
distance less than $\varepsilon$ from ${\mathbb{R}}\times\Gamma$. Any pseudo-
holomorphic variety $C_{n}$ from the above sequence is not included in
$N_{\varepsilon}^{\Gamma}$. We may then choose $L_{n}\in\R$ such that
$(\tau_{L_{n}}C_{n})\cap(\\{0\\}\times Y)$ contains a point with distance
larger than $\varepsilon$ from $\Gamma$. By Theorem 2.4,
$\\{\tau_{L_{n}}C_{n}\\}_{n}$ geometrically converges to a nonconstant t-curve
$\mathbf{C}$ with ${\mathscr{F}}_{\nu}(\mathbf{C})=0$, again contradicting
Lemma 2.9. $\Box$

##  3 Preliminaries: the $SW$ side

This section consists of a minimal review of Seiberg-Witten theory in the
context of MCE’s as well as some setups and definitions.

Recall the definitions of $\mathop{\mathrm{Conn}}\nolimits(M)$ and related
notions in Section 1.4 (19), where $M$ is a
$\mathop{\mathrm{Spin}}\nolimits^{c}$ 3- or 4-manifold, with spinor bundle
${\mathbb{S}}$ in the 3-dimensional case or
${\mathbb{S}}^{+}\oplus{\mathbb{S}}^{-}$ in the 4-dimensional case. Let
${\mathscr{C}}(M)$ denote
$\mathop{\mathrm{Conn}}\nolimits(M)\times\Gamma({\mathbb{S}})$ in the
3-dimensional case, and
$\mathop{\mathrm{Conn}}\nolimits(M)\times\Gamma({\mathbb{S}}^{+})$ in the
4-dimensional case. This is said to be the (Seiberg-Witten) configuration
space, and an element of ${\mathscr{C}}(M)$ is said to be a (Seiberg-Witten)
configuration. The gauge group over $M$ is
${\mathscr{G}}={\mathscr{G}}(M)=C^{\infty}(M;S^{1})$. It acts on the space
${\mathscr{C}}(M)$ by gauge transformations: Recall in particular that an
element $u\in C^{\infty}(M;S^{1})$ defines an automorphism on
$\mathop{\mathrm{Conn}}\nolimits(\det{\mathbb{S}}^{+})\times\Gamma({\mathbb{S}}^{+})$
(or
$\mathop{\mathrm{Conn}}\nolimits(\det{\mathbb{S}})\times\Gamma({\mathbb{S}})$)
by $u\cdot(A,\Psi)=(A-2u^{-1}du,u\cdot\Psi).$ We use ${\mathscr{B}}(M)$ to
denote the quotient space ${\mathscr{B}}(M):={\mathscr{C}}(M)/{\mathscr{G}}$.
This is called the quotient configuration space, and its elements quotient
configurations (or gauge equivalence classes of configurations). The gauge
equivalence of a configuration $(A,\Psi)$ is denoted by $[(A,\Psi)]$.

###  3.1 Setup and Assumptions

Let $Y$ be a closed connected $\mathop{\mathrm{Spin}}\nolimits^{c}$
3-manifold. We first consider the cylindrical case $X={\mathbb{R}}\times Y$,
or more generally, consider $Z=I\times Y\subset X$ for an interval
$I\subset{\mathbb{R}}$. Let
${\mathbb{S}}_{X}={\mathbb{S}}^{+}\oplus{\mathbb{S}}^{-}$ denote spinor bundle
over $X$, and $s\in{\mathbb{R}}$ be an affine coordinate for the first factor
of $X={\mathbb{R}}\times Y$. Let $\mu_{Y}$ be a closed 2-form on $Y$, and set
$\mu=\pi^{*}_{2}\mu_{Y}$, $\pi_{2}\colon\thinspace{\mathbb{R}}\times Y\to Y$
being the projection to the second factor. Let
${\mathfrak{q}}\in{\mathscr{P}}(Y)$, where ${\mathscr{P}}(Y)$ denote $Y$’s
version of the Banach space of large tame perturbatons ${\mathscr{P}}$
introduced in [KM]’s Theorem 11.6.1 and Definition 11.6.3. As explained in
II.4.3 of [KM], the Clifford action $\rho(\partial_{s})$ defines an
isomorphism ${\mathbb{S}}^{+}\simeq{\mathbb{S}}^{-}$, which in turn is
identified with the pull back of the spinor bundle
${\mathbb{S}}={\mathbb{S}}_{Y}$ on $Y$. Meanwhile, a
$\mathop{\mathrm{Spin}}\nolimits^{c}$-connection $A$ on ${\mathbb{S}}^{+}$ may
be written (via a gauge tranformation) in temporal gauge as a path of
$\mathop{\mathrm{Spin}}\nolimits^{c}$ connections $B(s)$ on ${\mathbb{S}}$ by
way of Equation (4.8) of [KM]. In this manner, an
$(A,\Psi)\in{\mathscr{C}}(X)$ in temporal gauge on the cylinder
${\mathbb{R}}\times Y$ can alternatively be expressed as a path
$(B(s),\Phi(s))$ in ${\mathscr{C}}(Y)$. By way of the latter interpretation of
$(A,\Psi)$, each ${\mathfrak{q}}\in{\mathscr{P}}(Y)$ determines a
4-dimensional perturbation denoted by $\hat{{\mathfrak{q}}}$. (Cf. Equation
(10.2) of [KM] and the subsequent discussion.) Also in terms of the
aforementioned temporal gauge expression, the Seiberg-Witten equation
${\mathfrak{S}}_{\mu,\hat{{\mathfrak{q}}}}(A,\Psi)=0$ can be written as a
gradient flow equation:

$\big{(}\frac{1}{2}\frac{dB}{ds},\frac{d\Phi}{ds}\big{)}=-{\mathfrak{F}}_{\mu_{Y}}(B,\Phi)-{\mathfrak{q}}(B,\Phi).$

In the above, ${\mathfrak{F}}_{\mu_{Y}}$ is as defined in (1.3), regarded as a
section of the tangent bundle over ${\mathscr{C}}(Y)$. The term
${\mathfrak{q}}$ appearing in the preceding equation is understood as another
section of this bundle: As in [KM], the same notation is used to denote both
an element ${\mathfrak{q}}\in{\mathscr{P}}(Y)$ and its image under the map
${\mathfrak{Q}}$ in [KM]’s Theorem 11.6.1. Write

${\mathfrak{F}}_{\mu_{Y},{\mathfrak{q}}}={\mathfrak{F}}_{\mu_{Y}}+{\mathfrak{q}}.$

This is the formal gradient of the (perturbed) Chern-Simons-Dirac functional
on ${\mathscr{C}}(Y)$:

(3.1)
$\operatorname{CSD}_{\mu_{Y},{\mathfrak{q}}}(B,\Phi):=\operatorname{CSD}_{\mu_{Y}}(B,\Phi)+f_{\mathfrak{q}}(B,\Phi),$

where

(3.2)
$\begin{split}\operatorname{CSD}_{\mu_{Y}}(B,\Phi)&:=-\frac{1}{8}\int_{Y}(B-B_{0})\wedge(F_{B}+F_{B_{0}})+\frac{1}{2}\int_{Y}\langle\mbox{\mbox{$\partial\mkern-7.5mu\mbox{/}\mkern
2.0mu$}}_{B}\Phi,\Phi\rangle\,d\operatorname{vol}\\\
&\qquad-\frac{i}{8}\int_{Y}\mu_{Y}\wedge(B-B_{0}).\\\ \end{split}$

In the above, $B_{0}\in{\mathscr{C}}(Y)$ is a reference connection, and
$f_{\mathfrak{q}}$ is the $\R$-valued, gauge invariant function on the space
${\mathscr{C}}(Y)$ with formal gradient ${\mathfrak{q}}$, and with
$f_{\mathfrak{q}}(B_{0},0)=0$. Given $(B,\Phi)\in{\mathscr{C}}(Y)$, we use
$(\hat{B},\hat{\Phi})$ to denote the element in
${\mathscr{C}}({\mathbb{R}}\times Y)$ corresponding to the constant path at
$(B,\Phi)$. In particular, a solution $(B,\Phi)$ to the 3-dimensional Seiberg-
Witten equation ${\mathfrak{F}}_{\mu_{Y},{\mathfrak{q}}}(B,\Phi)=0$ gives rise
to an ${\mathbb{R}}$-invariant Seiberg-Witten solution:
${\mathfrak{S}}_{\mu,\hat{{\mathfrak{q}}}}(\hat{B},\hat{\Phi})=0$ on
${\mathbb{R}}\times Y$.

Note that $\operatorname{CSD}_{\mu_{Y}}$ is in general not gauge invariant,
but is invariant under the identity component of the gauge group, denoted
${\mathscr{G}}_{0}\subset C^{\infty}(Y,S^{1})$. It also depends implicitly on
the choice of the reference connection $B_{0}$. It is written as
$\operatorname{CSD}^{B_{0}}_{\mu_{Y}}$ when we wish to emphasize this
dependence. Similarly for $\operatorname{CSD}_{\mu_{Y},{\mathfrak{q}}}$. A
further superscript $Y$ is added, e.g.
$\operatorname{CSD}_{\mu_{Y}}^{Y,B_{0}}=\operatorname{CSD}_{\mu_{Y}}$ when
needs for specifying the 3-manifold $Y$ arise.

Now let $X$ be a general $\mathop{\mathrm{Spin}}\nolimits^{c}$ MCE with spinor
bundle ${\mathbb{S}}_{X}={\mathbb{S}}^{+}\oplus{\mathbb{S}}^{-}$, and let
${\mathbb{S}}_{i}$ denote the spinor bundle on the ending
$\mathop{\mathrm{Spin}}\nolimits^{c}$ 3-manifold $Y_{i}$. Fix a reference
connection $A_{0}\in\mathop{\mathrm{Conn}}\nolimits(X)$ such that

(3.3) $A_{0}=\hat{B}_{0,i}$

for a reference connection $B_{0,i}\in\mathop{\mathrm{Conn}}\nolimits(Y_{i})$
on each end $\hat{Y}_{i}\subset X$.

######  Definition 3.1.

An admissible element $(A,\Psi)$ of ${\mathscr{C}}(X)$ is one with the
following property: there exists $(B_{i},\Phi_{i})\in{\mathscr{C}}(Y_{i})$
such that $(A,\Psi)-(\hat{B_{i}},\hat{\Phi}_{i})$ is in
$L^{2}_{1}(T^{*}\hat{Y}_{i})\times L^{2}_{1,A_{0}}({\mathbb{S}}^{+})$ over
each end $\hat{Y}_{i}$. In this case, $(B_{i},\Phi_{i})$ is said to be the
$Y_{i}$-end limit of $(A,\Psi)$.

All elements $(A,\Psi)$ of ${\mathscr{C}}(X)$ are assumed to be admissible in
this article.

######  Remarks 3.2.

(a) Note that the preceding definition of admissible $(A,\Psi)$ does not
depend on the choice of the reference connection $A_{0}$.

(b) When $(A,\Psi)$ is a Seiberg-Witten solution, the definition of
“$Y_{i}$-end limits of $(A,\Psi)$” above is consistent with the notion of
$Y_{i}$-end limits of general sections over $X$ given in Section 1.4, by way
of a well-known elliptic bootstrapping argument; see e.g. [KM].

This article concerns admissible solutions $(A,\Psi)=(A_{r},\Psi_{r})$ to the
Seiberg-Witten equation
${\mathfrak{S}}_{\mu_{r},\hat{{\mathfrak{p}}}_{r}}(A_{r},\Psi_{r})=0$, where
$\mu_{r}=r\nu+w_{r}$ and $\hat{{\mathfrak{p}}}_{r}$ are as described in
Theorem 1.7. The terms $w_{r}$ and $\hat{{\mathfrak{p}}}_{r}$ are regarded as
“additional perturbations” to the dominating perturbation $r\nu$ in the
Seiberg-Witten equation; they are ultimately inconsequential to the
$SW\Rightarrow Gr$ story told in this article. They are chosen to satisfy the
conditions spelled out in Assumption 3.4 below. These conditions are meant to
both simplify the arguments yet remain sufficiently general for the purposes
of Floer theory (e.g. transversality and compactness). The subscripts $r$ are
sometimes dropped from $w_{r}$ and $\hat{{\mathfrak{p}}}_{r}$, as their
dependence on $r$ is immaterial.

In order to describe the assumptions as well as for future reference, we list
some basic facts on [KM]’s nonlocal perturbations:

######  Facts 3.3.

1) Given $Y$, there exists a constant $m$ such that

(3.4) $\|{\mathfrak{q}}(B,\Phi)\|_{L^{2}}\leq
m\|{\mathfrak{q}}\|_{\mathscr{P}}\,(\|\Phi\|_{L^{2}}+1).$

for every ${\mathfrak{q}}\in{\mathscr{P}}(Y)$ and
$(B,\Phi)\in{\mathscr{C}}(Y)$. Here, $\|{\mathfrak{q}}\|_{\mathscr{P}}$ denote
the norm of ${\mathfrak{q}}$ as an element in the Banach space
${\mathscr{P}}(Y)$. (Cf. Equation (11.18) of [KM].) Given an compact interval
$I\subset{\mathbb{R}}$, let $Z$ denote the corresponding
$\mathop{\mathrm{Spin}}\nolimits^{c}$ cylindrical manifold $I\times Y$. Let
${\mathbb{S}}^{+}$ denote the spinor bundle over $Z$ and fix a reference
connection $A_{0}\in\mathop{\mathrm{Conn}}\nolimits({\mathbb{S}}^{+})$. Then
there exists a continuous function over ${\mathbb{R}}$, denoted $\hat{m}$,
such that

(3.5)
$\|\hat{{\mathfrak{q}}}(A,\Psi)\|_{L^{2}_{1,A}}\leq\|{\mathfrak{q}}\|_{\mathscr{P}}\,\hat{m}\,(\|(A-A_{0},\Psi)\|_{L^{2}_{1,A_{0}}}).$

(See Definition 10.5.1 and Theorem 11.6.1 of [KM]. Note that there is a
misprint in Theorem 11.6.1 (iv). By a Sobolev inequality, (3.5) is equivalent
to
$\|\hat{{\mathfrak{q}}}(A,\Psi)\|_{L^{2}_{1,A}}\leq\|{\mathfrak{q}}\|_{\mathscr{P}}\,\hat{{\rm
m}}\,(\|(A-A_{0},\Psi)\|_{L^{2}_{1,A}})$ for another continuous function
$\hat{{\rm m}}$.)

2) Let $Z$, ${\mathfrak{q}}$ be as before, and let the Banach manifolds
${\mathscr{C}}_{k}(Z)$ and the Banach bundles
${\mathscr{V}}_{k}(Z)\to{\mathscr{C}}_{k}(Z)$ be as in [KM]’s Definition
10.1.1. (The subscript $k$ signifies $L^{2}_{k}$-completion.) For $k\geq 1$,
the map ${\mathfrak{Q}}$ from ${\mathscr{P}}(Y)\times{\mathscr{C}}_{k}(Z)$ to
${\mathscr{V}}_{k}(Z)$ given by
$({\mathfrak{q}},c)\mapsto\hat{{\mathfrak{q}}}(c)$ is a continuous map of
Banach manifolds which vanishes over
$\\{0\\}\times{\mathscr{C}}_{k}(Z)\subset{\mathscr{P}}(Y)\times{\mathscr{C}}_{k}(Z)$.
For $k\geq 2$, it is a smooth map.

3) Let $X$ be a $\mathop{\mathrm{Spin}}\nolimits^{c}$ MCE with ending
3-manifolds $Y_{i}$. A nonlocal perturbation $\hat{{\mathfrak{p}}}$ on $X$, as
defined in Equation (24.2) of [KM], is of the following form: For every
$i\in{\mathfrak{Y}}$, there exists a compact interval
${\mathbb{I}}_{i}=[{\mathfrak{l}}_{i},{\mathfrak{l}}_{i}^{\prime}]\subset{\mathbb{R}}$
and ${\mathfrak{q}}_{i},{\mathfrak{p}}^{\prime}_{i}\in{\mathscr{P}}(Y_{i})$
such that $\hat{{\mathfrak{p}}}$ vanishes over
$X_{\boldsymbol{{\mathfrak{l}}}}$; it restricts to $\hat{{\mathfrak{q}}}_{i}$
over $\hat{Y}_{i,{\mathfrak{l}}_{i}^{\prime}}$, and over
$\hat{Y}_{i,{\mathbb{I}}_{i}}$ it takes the form

(3.6) $\hat{{\mathfrak{p}}}=\chi_{i}(\text{{\small
s}}_{i})\hat{{\mathfrak{q}}}_{i}+\lambda_{i}(\text{{\small
s}}_{i})\hat{{\mathfrak{p}}}^{\prime}_{i},$

where $\chi_{i}$ is a smooth cutoff function supported on
$({\mathfrak{l}}_{i},\infty)$ and equals $1$ as $\text{{\small
s}}_{i}>{\mathfrak{l}}^{\prime}_{i}$; $\lambda_{i}$ is a smooth bump function
supported on ${\mathbb{I}}_{i}$ and equals 1 on a strictly smaller interval
${\mathbb{I}}^{\prime}_{i}\subset{\mathbb{I}}_{i}$. As usual,
$\boldsymbol{{\mathfrak{l}}}$ above denotes a function from ${\mathfrak{Y}}$
to ${\mathbb{R}}^{\geq 0}$ given by $i\mapsto{\mathfrak{l}}_{i}$.

######  Assumption 3.4.

Let $X$ be a $\mathop{\mathrm{Spin}}\nolimits^{c}$ MCE with ending 3-manifolds
$Y_{i}$, $i\in{\mathfrak{Y}}$. Assume that $w_{r}$, $\hat{{\mathfrak{p}}}_{r}$
satisfy the following conditions:

  * (1)

$w_{r}$ is a smooth closed 2-form depending smoothly on $r$, and
$\|w_{r}\|_{C^{k+3}}<\varsigma_{w}$ for some $k\geq 3$ and an $r$-independent
positive constant $\varsigma_{w}$.

  * (2)

$w_{r}$ are in the cohomology class $4\pi c_{1}({\mathfrak{s}}_{X})$ for all
$r$;

  * (3)

Over each end $\hat{Y}_{i}\subset X$, $w_{r}=\pi_{2}^{*}w_{i,r}$ for a closed
2-forms $w_{i,r}$ on $Y_{i}$ . If $\hat{Y}_{i}$ is a vanishing end, the closed
form $w_{i,r}$ is independent of of $r$ and is written as $w_{i}$.

  * (4)

$\hat{{\mathfrak{p}}}_{r}=\hat{{\mathfrak{p}}}$ is of the type given in
Equation (24.2) of [KM]. Adopt the notations from Item 3) of Facts 3.3, with
the ingredients for constructing $\hat{{\mathfrak{p}}}$ (i.e.
${\mathfrak{q}}_{i}$, ${\mathfrak{p}}^{\prime}_{i}$, ${\mathfrak{l}}_{i}$,
${\mathfrak{l}}_{i}^{\prime}$, $\chi_{i}$, $\lambda_{i}$) understood to
(implicitly) depend on $r$. (A subscript $r$ is added, e.g.
${\mathfrak{q}}_{i}={\mathfrak{q}}_{i,r}$, when the $r$-dependence is
emphasized.) We require:

    * a)

${\mathfrak{q}}_{i}=0={\mathfrak{p}}^{\prime}_{i}$ when $\hat{Y}_{i}$ is a
Morse end. In this case we formally set
${\mathfrak{l}}_{i}={\mathfrak{l}}_{i}^{\prime}=\infty$ and
${\mathbb{I}}_{i}=\emptyset$.

    * b)

The triples ${\mathbb{I}}_{i}$, $\chi_{i}$, $\lambda_{i}$ corresponding to
different $i\in{\mathfrak{Y}}_{v}$ are related by translations. Namely, they
take the form of

${\mathbb{I}}_{i}=\tau_{{\mathfrak{l}}_{i}}{\mathbb{I}}_{0};\quad\chi_{i}=\tau_{{\mathfrak{l}}_{i}}^{*}\chi_{0};\quad\lambda_{i}=\tau_{{\mathfrak{l}}_{i}}^{*}\lambda_{0},$

where ${\mathbb{I}}_{0}:=[0,\zeta_{0}]$, $\chi_{0},\lambda_{0}$ are all
independent of both $r$ and $i$.

    * c)

${\mathfrak{l}}_{i}>\textup{l}_{i}^{+}$ $\forall i\in{\mathfrak{Y}}_{v}$ and
$r$, where $\textup{l}_{i}^{+}\in{\mathbb{R}}^{+}$ is as in Definition 1.5. In
particular, $\hat{{\mathfrak{p}}}$ vanishes over $X^{\prime\prime}$.

    * d)

Abusing notation, let
$\|\hat{{\mathfrak{p}}}\|_{{\mathscr{P}}}:=\sum_{i\in{\mathfrak{Y}}_{v}}\big{(}\|{\mathfrak{p}}^{\prime}_{i}\|_{{\mathscr{P}}}+\|{\mathfrak{q}}_{i}\|_{{\mathscr{P}}}\big{)}$.
There is an $r$-independent constant $\textsl{z}_{\mathfrak{p}}<1/8$ such that

$\|\hat{{\mathfrak{p}}}\|_{{\mathscr{P}}}=\sum_{i\in{\mathfrak{Y}}_{v}}\big{(}\|{\mathfrak{p}}^{\prime}_{i}\|_{{\mathscr{P}}}+\|{\mathfrak{q}}_{i}\|_{{\mathscr{P}}}\big{)}<\textsl{z}_{\mathfrak{p}}\quad\forall
r.$

  * (5)

Let $k$ be as in Item (1) above, and let
${\mathbb{I}}_{\epsilon}:=(-\epsilon,\zeta_{0}+\epsilon)\supset{\mathbb{I}}_{0}$
be a slightly larger open interval containing ${\mathbb{I}}_{0}$. Denote
$\upsilon_{i}(r):=\|\mu_{r}\|_{C^{k+3}(\hat{Y}_{i,{\mathfrak{l}}_{i}})}$. Fix
an $r$-independent constant $\zeta_{\mathfrak{p}}<\zeta_{w}/8$. With the
constants and functions $\zeta_{0}$, $\lambda_{0}$, $\chi_{0}$,
${\mathfrak{l}}_{i}$ from Assumption 3.4 4b)-4c) fixed, $\hat{{\mathfrak{p}}}$
is determined by ${\mathfrak{q}}_{i}$ and ${\mathfrak{p}}^{\prime}_{i}$. We
imposes a stronger, $r$-dependent constraint (cf. the upcoming remark) on
their sizes (controlled by $\|\hat{{\mathfrak{p}}}_{r}\|_{\mathscr{P}}$) than
that from Item 4d) above, such that the following holds for every
$i\in{\mathfrak{Y}}_{v}$ and $r\geq 1$: Fix $i\in{\mathfrak{Y}}_{v}$ and drop
the subscript $i$ for the rest of this item. Write $Z:={\mathbb{I}}_{0}\times
Y$; $Z_{\epsilon}:={\mathbb{I}}_{\epsilon}\times Y$, and let
$\hat{{\mathfrak{x}}}_{r}:=\chi_{0}\hat{{\mathfrak{q}}}_{r}+\lambda_{0}\hat{{\mathfrak{p}}}^{\prime}_{r}=\tau_{{\mathfrak{l}}_{r}}^{*}\hat{{\mathfrak{p}}}_{r}$.
Then give $r\geq 1$:

(3.7)
$\|\hat{{\mathfrak{x}}}_{r}(A,\Psi)\|_{C^{k}_{A}(Z)}<\zeta_{\mathfrak{p}}$

for any $(A,\Psi)\in{\mathscr{C}}(Z_{\epsilon})$ that satisfies:

    * (i)

$\|F_{A}\|^{2}_{L^{2}(Z_{\epsilon})}+\|\nabla_{A}\Psi\|^{2}_{L^{2}(Z_{\epsilon})}+\|\Psi\|^{4}_{L^{4}(Z_{\epsilon})}\leq
2^{8}\upsilon(r)^{2}$;

    * (ii)

Let ${\mathfrak{W}}$ denote the space of $C^{k+3}$ closed 2-forms on
$Z_{\epsilon}$ cohomologous to $\pi_{2}^{*}w$. Then ${\mathfrak{S}}_{{\rm
w},\hat{{\mathfrak{x}}}_{r}}(A,\Psi)=0$ for a ${\rm w}\in{\mathfrak{W}}$ with
$\|{\rm w}\|_{C^{k+3}(Z_{\epsilon})}\leq\upsilon(r)$.

Likewise,

(3.8)
$\|\hat{{\mathfrak{q}}}_{r}(A,\Psi)\|_{C^{k}_{A}(Z)}<\zeta_{\mathfrak{p}}$

for any $(A,\Psi)\in{\mathscr{C}}(Z_{\epsilon})$ that satisfies both item (i)
above and

    * (ii’)

${\mathfrak{S}}_{{\rm w},\hat{{\mathfrak{q}}}_{r}}(A,\Psi)=0$ for a ${\rm
w}\in{\mathfrak{W}}$ with $\|{\rm
w}\|_{C^{k+3}(Z_{\epsilon})}\leq\upsilon(r)$.

######  Remarks 3.5.

(a) Item (5) in the preceding assumption is invoked only in Section 5 to make
the contribution from $\hat{{\mathfrak{p}}}$ ignorable while performing
pointwise estimates. It may likely be removed or weakened with extra work,
appealing to the weak maximum principle in lieu of the strong-maximum-
principle arguments in Section 5. The stronger upper bound on
$\|\hat{{\mathfrak{p}}_{r}}\|_{{\mathscr{P}}}$ mentioned in Item (5) may also
be weakened by imposing stronger ($r$-dependent) lower bounds on
${\mathfrak{l}}_{i}$ than those required by Item 4c).

(b) Here is a more detailed description of the stronger bound on
$\hat{{\mathfrak{p}}}_{r}$ that is required by Item (5) to guarantee the
conditions (3.8) and (3.7). Fix $r\geq 1$. Given
${\mathfrak{q}}\in{\mathscr{P}}(Y)$, let ${\mathscr{M}}({\rm
w};{\mathfrak{q}})$ denote the space of gauge equivalences of $(A,\Psi)$
satisfying Condition (i) of Item (5) and the Seiberg-Witten equation
${\mathfrak{S}}_{{\rm w},\hat{{\mathfrak{q}}}}(A,\Psi)=0$. Given
${\mathfrak{q}},{\mathfrak{p}}^{\prime}\in{\mathscr{P}}(Y)$, let
${\mathscr{M}}({\rm w};{\mathfrak{q}},{\mathfrak{p}}^{\prime})$ denote the
space of gauge equivalences of $(A,\Psi)$ satisfying Condition (i) of Item (5)
and the Seiberg-Witten equation ${\mathfrak{S}}_{{\rm
w},\hat{{\mathfrak{x}}}}(A,\Psi)=0$, where
$\hat{{\mathfrak{x}}}:=\chi_{0}\hat{{\mathfrak{q}}}+\lambda_{0}\hat{{\mathfrak{p}}}^{\prime}$.
Then both the space $\tilde{{\mathscr{M}}}_{r}:=\bigcup_{{\rm w},\|{\rm
w}\|_{C^{k+3}}\leq\upsilon(r)}\bigcup_{{\mathfrak{q}},\|{\mathfrak{q}}\|_{\mathscr{P}}\leq\textsl{z}_{\mathfrak{p}}}{\mathscr{M}}({\rm
w};{\mathfrak{q}})$ and the space
$\tilde{{\mathscr{M}}}^{\prime}_{r}:=\bigcup_{{\rm w},\|{\rm
w}\|_{C^{k+3}}\leq\upsilon(r)}\bigcup_{{\mathfrak{q}},\|{\mathfrak{q}}\|_{\mathscr{P}}\leq\textsl{z}_{\mathfrak{p}}}\bigcup_{{\mathfrak{p}}^{\prime},\|{\mathfrak{p}}^{\prime}\|_{\mathscr{P}}\leq\textsl{z}_{\mathfrak{p}}}{\mathscr{M}}({\rm
w};{\mathfrak{q}},{\mathfrak{p}}^{\prime})$ are represented by compact
subspaces in ${\mathscr{C}}_{k+3}(Z)$. (To see this, repeat the elliptic
bootstrapping arguments in [KM]’s proofs of Theorems 10.7.1 and 5.2.1, with
the condition (i) of Item (5) playing the role of [KM]’s (10.15), and noting
the following: First, the $C^{k+3}$-bound on ${\rm w}$ in conditions (ii) and
(ii’) implies that such ${\rm w}$ lies in a compact subspace in the space of
$L^{2}_{j}$ closed 2-forms in cohomology class $\pi^{*}_{2}w$, $\forall\,0\leq