# Constructing symplectic forms on trisections

Peter Lambert-Cole

(Date: December 2022)

###### Abstract.

Previously work of the author with Meier and Starkston showed that every
closed symplectic manifold $(X,\omega)$ with a rational symplectic form admits
a trisection compatible with the symplectic topology. In this paper, we
describe the converse direction and give explicit criteria on a trisection of
a closed, smooth 4-manifold $X$ that allows one to construct a symplectic
structure on $X$. Combined, these give a new characterization of 4-manifolds
that admit symplectic structures. This construction motivates several problems
on taut foliations, the Thurston norm and contact geometry in 3-dimensions by
connecting them to questions about the existence, classification and
uniqueness of symplectic structures on 4-manifolds.

##  1\. Introduction

Let $X$ be a smooth, closed, oriented 4-manifold. It is of great interest in
low-dimensional topology to determine whether $X$ admits a symplectic
structure and if so, to classify all such symplectic structures. For example,
work of Donaldson and Gompf showed that $X$ admits a symplectic structure if
and only if it admits a compatible Lefschetz pencil. This essentially reduces
the problem to algebra and mapping class groups of surfaces.

Suppose that $\mathcal{T}$ is a trisection of $X$, which is a particular
decomposition into three 4-dimensional 1-handlebodies.

######  Definition 1.1.

Let $X$ be a closed, smooth,oriented 4-manifold. A trisection of $X$ is a
decomposition $X=Z_{1}\cup Z_{2}\cup Z_{3}$ such that

  1. (1)

each sector $Z_{\lambda}$ is a 4-dimensional 1-handlebody,

  2. (2)

each double intersection $H_{\lambda}=Z_{\lambda}\cap Z_{\lambda+1}$ is a
3-dimensional 1-handlebody of genus $g$, and

  3. (3)

the triple intersection $\Sigma=Z_{1}\cap Z_{2}\cap Z_{3}$ is a closed,
orientable surface of genus $g$.

The trisection $\mathcal{T}$ and the 4-manifold $X$ itself can be encoded (up
to diffeomorphism) by a trisection diagram
$(\Sigma;\mathbf{\alpha},\mathbf{\beta},\mathbf{\gamma})$, which consists of a
closed, oriented surface and three cut systems of curves. This data determines
how to build a trisected 4-manfiold from the ‘inside-out’. The cut systems
determine how to attach each double intersection $H_{\lambda}$ to the central
surface to produce the spine $H_{1}\cup H_{2}\cup H_{3}$. Capping off by
attaching the 4-dimensional sectors $Z_{1},Z_{2},Z_{3}$ is unique up to
diffeomorphism by the well-known result of Laudenbach and Poenaru [LP72].

The following is a natural question to ask.

######  Question 1.2.

Is there a criterion on a trisection that ensures $X$ admits a symplectic
structure?

However, answering this question has proven surprising difficult. The first
problem is to find the correct notion of compatibility between a symplectic
structure and a trisected 4-manifold. For example, the central surface is
nullhomologous, therefore it cannot be symplectic, and in general has negative
Euler characteristic, therefore it cannot be Lagrangian. The notions of
Weinstein trisections of symplectic 4-manifolds and Stein trisections of
complex manifolds were introduced in [LCM22]. Some recent progress has been
made in constructing Stein trisections [LCM22, LC21, Zup22]. Moreover, by
using Auroux’s work on representing symplectic 4-manifolds as branched
coverings of $\mathbb{CP}^{2}$ [Aur00], it was shown that every symplectic
4-manifold admits a Weinstein trisection [LCMS21]. This construction is
evidence that the latter structure is the conceptually correct one to look
for.

A second difficulty is that the compatibility between trisections and
symplectic topology does not appear to fit into existing conceptual
frameworks. For example, the boundary of a relative trisection inherits an
open book decomposition [GK16, Theorem 20]. This suggests that for a compact,
symplectic manifold with contact boundary, the open book induced by the
relative trisection should support the contact structure. However, consider
the complement of a complex curve $\mathcal{C}_{d}$ in $\mathbb{CP}^{2}$. The
contact structure on the boundary of
$\mathbb{CP}^{2}\setminus\nu(\mathcal{C}_{d})$ is Stein-fillable, hence tight.
Naively applying the method of Miller-Kim [KM20] to the curves in bridge
position given in [LCM22] can yield an induced open book that is not right-
veering. In particular, this open book supports a contact structure that is
not tight.

In this paper, we describe how to construct a symplectic structure from
geometric data on the spine of a trisected 4-manifold.

###  1.1. Representing cohomology via cocycles

The construction of the closed 2-form $\omega$ on the trisected 4-manifold in
Theorem 1.3 inherently determines the class $[\omega]\in H^{2}_{DR}(X)$.
Before stating the theorem, we review how to interpret cohomology classes on
trisected 4-manifolds.

Feller-Klug-Schirmer-Zemke [FKSZ18] and Florens-Moussard [FM22] gave
presentations of the singular homology groups of a 4-manifold in terms of a
trisection diagram. This was reinterpreted in [LC22] as the Cech cohomology
groups of certain presheaves with respect to an open cover determined by the
trisection.

In particular, let $\mathcal{H}^{i}_{DR}$ denote the presheaf on $X$ that
assigns to each open set $U$ the group $H^{i}_{DR}(U)$. By abuse of notation,
let $\mathcal{T}=\\{U_{1},U_{2},U_{3}\\}$ denote the open cover of $X$ by
three sets obtained by slightly thickening the three sectors of the
trisection. In particular, $U_{\lambda}$ is an open neighborhood of
$X_{\lambda}$; $U_{\lambda}\cap U_{\lambda+1}$ is an open neighborhood of the
handlebody $H_{\lambda}$, and $U_{1}\cap U_{2}\cap U_{3}$ is an open
neighborhood of the central surface $\Sigma$.

There is an isomorphism ([LC22])

$H^{2}_{DR}(X)\cong\check{H}^{1}(\mathcal{T},\mathcal{H}^{1}_{DR})$

In words, this states that a class $[\omega]\in H^{2}_{DR}(X)$ can be
represented by a 1-cocycle of closed 1-forms. Specifically, there is a triple
$(\beta_{1},\beta_{2},\beta_{3})$, where $\beta_{\lambda}$ is a closed 1-form
on $H_{\lambda}$, and

$\beta_{1}+\beta_{2}+\beta_{3}=0$

when restricted to the central surface $\Sigma$.

###  1.2. Main theorem

######  Theorem 1.3.

Let $X$ be a closed, oriented 4-manifold and let $\mathcal{T}$ be a trisection
of $X$ with central surface $\Sigma$. Given the following data

  1. (1)

a pair $(C,\beta)$ where $C=\\{\lambda^{2}g\\}$ is a conformal class of
Riemannian metrics on $\Sigma$ and $\beta$ is a $(\lambda^{2}g)$-harmonic
1-form with nondegenerate zeros,

  2. (2)

a smooth function $f:\Sigma\rightarrow\mathbb{R}$ such that, for each
$i=1,\dots,2g-2$, there is a connected open neighborhood $U_{i}$ of $b_{i}$
such that $f|_{U_{i}}\in\\{\pm 1\\}$ and $\sum_{i}f(b_{i})=0$,

  3. (3)

extensions
$\\{\widetilde{\beta}_{1},\widetilde{\beta}_{2},\widetilde{\beta}_{3}\\}$ of

$\displaystyle\beta_{1}$ $\displaystyle=\beta$ $\displaystyle\beta_{3}$
$\displaystyle=-\beta_{1}-\beta_{2}$ $\displaystyle\beta_{2}$
$\displaystyle=-(\ast_{\lambda^{2}g}\beta)$

across the handlebodies $H_{1},H_{2},H_{3}$, respectively, as closed and
nonvanishing 1-forms that have the form

$\widetilde{\beta}_{\lambda}=\beta_{\lambda}+d(sf)$

on a collar neighborhood of $\partial H_{\lambda}=\Sigma$,

there is a symplectic form on $\nu(H_{1}\cup H_{2}\cup H_{3})$ representing
the class

$(\beta_{1},\beta_{2},\beta_{3})\in\check{H}^{1}(\mathcal{T},\mathcal{H}^{1}_{DR})\cong
H^{2}_{DR}(X)\cong H^{2}_{DR}(\nu(H_{1}\cup H_{2}\cup H_{3}))$

In addition, there exists inward-pointing Liouville vector fields for $\omega$
along each of the three boundary components
$\widehat{Y}_{1},\widehat{Y}_{2},\widehat{Y}_{3}$.

If each of the induced contact structures
$(\widehat{Y}_{\lambda},\widehat{\xi}_{\lambda})$ are tight, then there exists
a symplectic form $\omega$ on $X$ representing the class

$(\beta_{1},\beta_{2},\beta_{3})\in\check{H}^{1}(\mathcal{T},\mathcal{H}^{1}_{DR})\cong
H^{2}_{DR}(X)$

that is well-defined up to diffeomorphsim.

Finally, an $n$-parameter family of data extends to an $n$-parameter family of
symplectic forms. Furthermore, if the class
$(\beta_{1},\beta_{2},\beta_{3})\in\check{H}^{1}(\mathcal{T},\mathcal{H}^{1}_{DR})\cong
H^{2}_{DR}(X)$ remains constant in the family, then the constructed symplectic
forms are isotopic.

###  1.3. Conformal structures

The inclusion of a conformal structure is something of a red herring. It is
included as a suggestive observation that may be insightful when parametrizing
these structures and investigating questions of existence and uniqueness.
However, it has no geometric consequences in terms of the symplectic form
constructed. We could equivalently describe the required data as either

  1. (1)

a pair $(j,\zeta)$ where $j$ is a complex structure on $\Sigma$ and $\zeta$ is
a holomorphic 1-form. In this case, we can recover $\beta$ as the real part of
$\zeta$.

  2. (2)

a singular flat metric on $\Sigma$ with $2g-2$ conical singularities, each of
total angle $4\pi$

The key information we require from this data is

  1. (1)

the classes $[\beta_{1}],[\beta_{2}],[\beta_{3}]\in H^{1}_{DR}(\Sigma)$,

  2. (2)

the kernel foliations $\mathcal{F}_{1},\mathcal{F}_{2},\mathcal{F}_{3}$ of
these three 1-forms, which are pairwise transverse,

  3. (3)

that the 1-forms $\beta_{1},\beta_{2},\beta_{3}$ are intrinsically harmonic,
meaning they are harmonic for some Riemannian metric $g$.

###  1.4. How can the construction fail?

Many smooth 4-manifolds are known to not admit symplectic structures. Where
might the obstructions lie when considered in light of Theorem 1.3?

  1. (1)

Some $\beta_{\lambda}$ does not extend as a closed, nonvanishing 1-form. This
may be for cohomological reasons, as $[\beta_{\lambda}]$ is not in the image
of the restriction map $H^{1}_{DR}(H_{\lambda})\rightarrow
H^{1}_{DR}(\Sigma)$. The 1-cocycle $(\beta_{1},\beta_{2},\beta_{3})$ encodes a
class $[\omega]$ in $H^{2}_{DR}(X)$ and $[\beta_{1}]\wedge[\beta_{2}]$ is
positive in $H^{2}_{DR}(\Sigma)$ if and only if $[\omega]\wedge[\omega]$ is
positive in $H^{4}_{DR}(X)$. In particular, if $b_{2}^{+}(X)=0$, then the
required cocycle cannot exists. Secondly, if $\beta_{\lambda}$ extends as a
closed 1-form, the form may be forced to vanish in some points.

  2. (2)

The induced contact structure $\widehat{\xi}_{\lambda}$ is not tight. This may
happen at the level of homotopy. The contact structure is a 2-plane field,
which are classified up to homotopy by their Euler class in $H^{2}(Y)$ and a
3-dimensional obstruction. In addition, the contact structure is homotopic to
the field of $J$-complex tangencies along $Y$, where $J$ is an $\omega$-tame
almost-complex structure. This extends to an almost-complex structure on the
4-dimensional sector if and only if $\xi$ has vanishing Euler class and is
homotopic to the unique tight contact structure. Consequently, if $X$ does not
admit an almost-complex structure, then at least one of the constructed
contact structures must be overtwisted. Secondly, in the unique correct
homotopy class of 2-plane fields, there exists one tight and one overtwisted
contact structure. Therefore, the construction may simply yield the
overtwisted one.

The 4-manifold $3\mathbb{CP}^{2}$ has $b_{2}^{+}>0$ and admits an almost-
complex structure, but is known not to admit a symplectic structure. Moreover,
it admits a genus 3 trisection. It would be interesting to understand how no
possible choice of data on this trisection can satisfy the conditions of
Theorem 1.3. Consequently, this motivates the task of classifying the grafted
contact structures that can be constructed on the genus 3 Heegaard splitting
of $S^{3}$.

###  1.5. Symplectic structures and trisection diagrams

A trisected 4-manifold can be encoded by a trisection diagram, which is a
tuple $(\Sigma;\mathbf{\alpha},\mathbf{\beta},\mathbf{\gamma})$ consisting of
the central surface $\Sigma$ and three cut systems of curves
$\mathbf{\alpha},\mathbf{\beta},\mathbf{\gamma}$ that determine how the
handlebodies $H_{1},H_{2},H_{3}$ are glued in.

The data $(C,\beta)$ in Theorem 1.3 determines a different picture on the
central surface, namely three transverse foliations
$\mathcal{F}_{1},\mathcal{F}_{2},\mathcal{F}_{3}$. Moreover, if we suppose
that $\beta_{1},\beta_{2},\beta_{3}$ are integral 1-forms, then they define
maps

$\beta_{1},\beta_{2},\beta_{3}:\Sigma\longrightarrow S^{1}$

with $2g-2$ nondegenerate critical points. The preimage of a regular value is
a multicurve and for a fixed $\beta_{\lambda}$, the union of these multicurves
determine a pants decomposition of $\Sigma$ – i.e. a maximal system of $3g-3$
curves on $\Sigma$.

The foliations $\mathcal{F}_{\lambda}$ are not transverse to the boundary
$\partial H_{\lambda}=\Sigma$, but they do satisfy a condition analogous to
being taut, namely the existence of a closed transversal intersecting every
leaf. The leaves of a taut foliation are known to minimize the Thurston norm
[Thu86]. If $\widetilde{\beta}_{\lambda}$ is integral, then compact leaves of
$\mathcal{F}_{\lambda}$ also minimize the generalized Thurston norm
(specifically, Scharlemann’s generalization [Sch89]).

This begs the question of whether the extensions
$\\{\widetilde{\beta}_{\lambda}\\}$ are necessary pieces of data or are
uniquely determined by data defined in terms of the central surface $\Sigma$.

######  Question 1.4.

Given the data of

  1. (1)

an intrinsically harmonic, integral 1-form $\beta$ on $\Sigma$ with
nondegenerate zero set $B$,

  2. (2)

a Lagrangian subspace $L\subset H^{1}(\Sigma,\mathbb{Z})$ containing
$[\beta]$, and

  3. (3)

a class $c_{1}\in H_{1}(\Sigma,B;\mathbb{Z})$

Is there (a unique) handlebody $H$ with $\partial H=\Sigma$ such that

  1. (1)

$L=\text{image}(H^{1}(H)\rightarrow H^{1}(\Sigma))$, and

  2. (2)

for each regular value $\theta$ for $\beta:\Sigma\rightarrow S^{1}$, the
Thurston norm of the unique element $[F]\in H_{2}(H,\beta^{-1}(\theta))$ is
given by the pairing $\langle c_{1},\beta^{-1}(\theta)\rangle$?

###  1.6. Grafted contact structures

In the course of the proof of Theorem 1.3, three contact structures are
constructed in Theorem 1.3. In order to cap off the 4-dimensional sectors with
strong symplectic fillings, it is necessary that these contact structures be
tight.

Intuitively, these contact structures are obtained from ‘taut’ foliations by a
so-called grafting process. This is a very natural operation that has
nonetheless been hardly studied. The inspiration arises from complex analysis
and geometry. Let $X$ be a complex surface. A subset $P\subset X$ is an
analytic polyhedron if there exists an open set $U$ containing the closure of
$P$ and holomorphic functions $f_{1},\dots,f_{n}$ on $U$ such that

$P=\\{z\in U:|f_{1}(z)|,\dots,|f_{n}(z)|\leq 1\\}$

For a generic choice of $\\{f_{1},\dots,f_{n}\\}$, the boundary of $P$ is a
smooth manifold with corners.

The top-dimensional faces of $\partial P$ are Levi-flat, which means they are
foliated by complex codimension 1 holomorphic submanifolds. The function

$g=\text{max}(|f_{1}|,\dots,|f_{n}|)$

is a continuous, plurisubharmonic function on $U$ and $P$ is the sublevel set
$g^{-1}([0,1])$ If $X$ is a Stein surface, then $g$ can be approximated by a
smooth, strictly plurisubharmonic function.$\widetilde{g}$. The sublevel set
$\widetilde{P}=\widetilde{g}^{-1}([0,1])$ is a compact Stein domain and the
field of complex tangencies along
$\partial\widetilde{P}=\widetilde{g}^{-1}(1)$ is a contact structure.

Given the importance of tightness in Theorem 1.3, it would be very interesting
to produce criteria that ensures the resulting grafted contact structure is
tight.

######  Problem 1.5.

Give topological criteria so that the contact structures obtained in Theorem
1.3 are tight.

Establishing overtwistedness criteria may prove easier, as there may be an
analogous criteria to non-right-veering monodromy for an open book
decomposition.

###  1.7. Classification of symplectic structures

There are several natural equivalence relations on symplectic structures on a
4-manifold $X$:

  1. (1)

the pair $\omega_{1},\omega_{2}$ are isotopic if they are connected by a path
of cohomologous symplectic forms,

  2. (2)

the pair $\omega_{1},\omega_{2}$ are homotopic if they are connected by a path
of symplectic forms,

  3. (3)

the pair $\omega_{1},\omega_{2}$ are diffeomorphic if there exists a
diffeomorphism $\phi:X\rightarrow X$ such that
$\phi^{*}(\omega_{2})=\omega_{1}$, and

  4. (4)

the pair $\omega_{1},\omega_{2}$ are deformation-equivalent if there exists a
diffeomorphism $\phi:X\rightarrow X$ such that $\omega_{1}$ and
$\phi^{*}(\omega_{2})$ are homotopic.

We refer the reader to the excellent survey of Salamon on the classification
and uniqueness of symplectic forms on 4-manifolds [Sal13], which has also been
incorporated into the most recent edition of [MS17].

######  Question 1.6.

Suppose that $[\omega_{1}]=[\omega_{2}]$. Are they diffeomorphic?

The construction here gives one potential approach to this question: The
symplectic structure obtained (a) encodes the cohomology class of $[\omega]$,
(b) is well-defined by the data up to diffeomorphism, and (c) extends to
1-parameter families of the data. Hence studying the moduli of data may be
profitable in classifying symplectic structures. In particular, consider the
space $\Omega\mathcal{M}_{g}$ of holomorphic 1-forms on Riemann surfaces of
genus $g$. A point $(C,\zeta)$ in $\Omega\mathcal{M}_{g}$ is one necessary
piece of data in Theorem 1.3. Isoperiodic deformations of $(C,\zeta)$, which
are deformations that preserve $[\zeta]\in H^{1}_{DR}(\Sigma,\mathbb{C})$,
then lead to a family of symplectic forms in the same DeRham cohomology class
in $H^{2}_{DR}(X)$. The connectivity and dynamics of leaves of the isoperiodic
foliation on $\Omega\mathcal{M}_{g}$ was determined by Calsamiglia, Deroin and
Francaviglia [CDF15].

Moreover, this construction means that different symplectic forms can be
directly compared.

######  Theorem 1.7 ([LCMS21]).

Let $\omega_{1},\omega_{2}$ be rational symplectic forms on $X$. Then there
exists a single smooth trisection $\mathcal{T}$ of $X$ such that
$\omega_{1},\omega_{2}$ can both be constructed via Theorem 1.3 on
$\mathcal{T}$.

In no way is this possible when considering symplectic structures via
Lefschetz pencils or branched coverings of $\mathbb{CP}^{2}$. While the
Lefschetz pencils constructed by Donaldson are well-defined for sufficiently
large $k$, there is no stabilization operation that connects any two Lefschetz
pencils on a given 4-manifold.

##  2\. Inspiration

We describe the inspiration for Theorem 1.3 by working out the geometry of
$(\mathbb{CP}^{2},\omega_{FS})$ with respect to the standard trisection and
then describing how the geometry pulls back to an arbitrary projective surface
via a holomorphic branched covering.

###  2.1. The Fubini-Study form on $\mathbb{CP}^{2}$

The construction for a general trisection is analogous to the construction of
the Fubini-Study form on $\mathbb{CP}^{2}$.

In particular, we have that

$\omega_{FS}=-\frac{i}{2}\partial\overline{\partial}\log(|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2})$

The Kaḧler potential $\phi=\log(|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2})$ is well-
defined on homogeneous coordinates, up to adding a constant, hence its
exterior derivatives are well-defined. In the chart $z_{\lambda-1}=1$, we can
equivalently describe the form by

$\omega_{FS}=dd^{\mathbb{C}}\log(1+|z_{\lambda}|^{2}+|z_{\lambda+1}|^{2})$

where $d^{\mathbb{C}}f:=df(J-)$ for a function $f$ and
$J:T\mathbb{CP}^{2}\rightarrow\mathbb{CP}^{2}$ the almost-complex structure on
the tangent bundle. In particular, the form

$\alpha_{\lambda}=d^{\mathbb{C}}\log(1+|z_{\lambda}|^{2}+|z_{\lambda+1}|^{2})$

is a primitive for $\omega_{FS}$ in this coordinate chart. In polar
coordinates $z_{j}=r_{j}e^{i\theta_{j}}$, we have that

$\displaystyle\alpha_{\lambda}$
$\displaystyle=\phi(r_{\lambda},r_{\lambda+1})\left(2r_{\lambda}dr_{\lambda}(J-)+2r_{\lambda+1}dr_{\lambda+1}(J-)\right)$
$\displaystyle=\phi(r_{\lambda},r_{\lambda+1})\left(2r_{\lambda}^{2}d\theta_{\lambda}+2r_{\lambda+1}^{2}d\theta_{\lambda+1}\right)$

where

$\phi(x,y)=\frac{1}{1+x^{2}+y^{2}}$

Now, consider the change of coordinates from $\\{z_{\lambda-1}=1\\}$ to
$\\{z_{\lambda+1}=1\\}$:

$\displaystyle s_{\lambda}$ $\displaystyle=\frac{r_{\lambda}}{r_{\lambda+1}}$
$\displaystyle\psi_{\lambda}$
$\displaystyle=\theta_{\lambda}-\theta_{\lambda+1}$ $\displaystyle
s_{\lambda-1}$ $\displaystyle=\frac{1}{r_{\lambda+1}}$
$\displaystyle\psi_{\lambda-1}$ $\displaystyle=-\theta_{\lambda+1}$

Then

$\displaystyle\alpha_{\lambda-1}$
$\displaystyle=\phi(s_{\lambda-1},s_{\lambda})(2s^{2}_{\lambda-1}d\psi_{\lambda-1}+2s^{2}_{\lambda}d\psi_{\lambda})$
$\displaystyle=r^{2}_{\lambda+1}\phi(r_{\lambda},r_{\lambda+1})\left(-2\frac{1}{r^{2}_{\lambda+1}}d\theta_{\lambda+1}+2\frac{r^{2}_{\lambda}}{r^{2}_{\lambda+1}}(d\theta_{\lambda}-d\theta_{\lambda+1})\right)$
$\displaystyle=\phi(r_{\lambda},r_{\lambda+1})\left(-2d\theta_{\lambda+1}+2r^{2}_{\lambda}(d\theta_{\lambda}-d\theta_{\lambda+1})\right)$

So that

$\displaystyle\alpha_{\lambda}-\alpha_{\lambda-1}$
$\displaystyle=\phi\left(2r^{2}_{\lambda+1}d\theta_{\lambda+1}+2d\theta_{\lambda+1}+2r^{2}_{\lambda}d\theta_{\lambda+1}\right)=2r^{2}_{\lambda}d\theta_{\lambda}$

Therefore, along $H_{\lambda}=Z_{\lambda}\cap
Z_{\lambda-1}=\\{r_{\lambda}=1\\}$, we obtain the closed 1-form
$\beta_{\lambda}=2d\theta_{\lambda}$. In summary, by tracing through the
$\check{\text{C}}$ech-DeRham isomorphism, we can replace a closed 2-form with
a 1-cocycle of closed 1-forms:

$\omega_{FS}\mapsto(2d\theta_{1},2d\theta_{2},2(-d\theta_{1}-d\theta_{2}))$

Now consider the restriction to the central torus. The pair
$\\{d\theta_{1},d\theta_{2}\\}$ define a framing of $T^{*}T^{2}$. Moreover,
they are Hodge dual for some flat metric $C$ on $T^{2}$ that makes this an
orthonomal framing. The three 1-forms induce three kernel foliations
$\mathcal{F}_{1},\mathcal{F}_{2},\mathcal{F}_{3}$ that are linear with respect
to the coordinates $\theta_{1},\theta_{2}$. The condition
$d\theta_{\lambda}\wedge d\theta_{\lambda+1}>0$ implies that these foliations
are pairwise transverse.

###  2.2. Branched covering

Now consider a projective surface $X$ with a holomorphic branched covering

$p:X\rightarrow\mathbb{CP}^{2}$

that is generic with respect to the standard trisection $\mathbb{CP}^{2}$.
Such a branched covering always exists by choosing an embedding
$X\hookrightarrow\mathbb{CP}^{N}$ and projecting onto a generic projective
plane. We can then pull back the standard trisection via $p$ to get a
decomposition

$X=\widetilde{Z}_{1}\cup\widetilde{Z}_{2}\cup\widetilde{Z}_{3}$

into three analytic polyhedra. In general, the smooth decomposition is not a
trisection, as the sectors may have 2-handles. However, it accurately
illustrates the geometric picture in the holomorphic setting.

Let $\mathcal{K}\subset X$ be the ramification locus of $\pi$. Assume that
$p(\mathcal{K})$ is immersed and self-transverse and in general position with
respect to the stratification induced by the trisection. In particular, it
intersects the central $T^{2}$ in $2b$ points and each
$H_{\lambda}=S^{1}\times\mathbb{D}^{2}$ along arcs. Positivity of intersection
implies that these arcs are transverse to the foliation by holomorphic disks.

Now, consider the sector $Z_{\lambda}$ in $\mathbb{CP}^{2}$. It lies in the
open chart $U_{\lambda}=\mathbb{CP}^{2}\setminus L_{\lambda}$, where
$L_{\lambda}=\\{z_{\lambda-1}=0\\}$ denotes the projective line at infinity
with respect to this chart. In addition, it is the analytic polyhedron defined
by the holomorphic coordinate functions $z_{\lambda},z_{\lambda+1}$ in the
chart $U_{\lambda}$.

The preimage $\widetilde{U}_{\lambda}=p^{-1}(U_{\lambda})$ is the complement
of a hyperplane divisor in $X$, therefore it is an open Stein surface in $X$.
The preimage $\widetilde{Z}_{\lambda}=p^{-1}(Z_{\lambda})$ is an analytic
polyhedron

$\widetilde{Z}_{\lambda}=\\{w\in\widetilde{U}_{\lambda}:\text{max}\left(|z_{\lambda}\circ
p(w)|,|z_{\lambda+1}\circ p(w)|\right)\leq 1\\}$

in $\widetilde{U}_{\lambda}$, where $f_{1}=z_{\lambda}\circ p$ and
$f_{2}=z_{\lambda+1}\circ p$ are holomorphic functions on the Stein surface
$\widetilde{U}_{\lambda}$.

The boundary of $\widetilde{Z}_{\lambda}$ is piecewise-smooth, with a corner
at $p^{-1}(T^{2})$. Over the codimension-0 faces $\widetilde{H}_{\lambda}$ and
$\widetilde{H}_{\lambda+1}$, the boundary is Levi-flat and foliated by compact
holomorphic curves. In particular,

$\displaystyle\widetilde{H}_{\lambda}$
$\displaystyle=\bigcup_{\theta}\left\\{w\in\widetilde{U}_{\lambda}:z_{\lambda}\circ
p(w)=e^{2\pi i\theta}\right\\}\cap\widetilde{Z}_{\lambda}$
$\displaystyle\widetilde{H}_{\lambda+1}$
$\displaystyle=\bigcup_{\theta}\left\\{w\in\widetilde{U}_{\lambda}:z_{\lambda+1}\circ
p(w)=e^{2\pi i\theta}\right\\}\cap\widetilde{Z}_{\lambda}$

If the branch locus $p(\mathcal{K})$ intersects $H_{\lambda}$ transversely,
then these curves are nonsingular.

Next, the branch locus $p(\mathcal{K})$ intersects $T^{2}$ transversely. Since
$T^{2}$ is nullhomologous, the signed intersection of $p(\mathcal{K})$ with
$T^{2}$ vanishes and there exist $b$ positive intersections and $b$ negative
intersections for some $b$. If $p$ has degree $n$ as a finite holomorphic map,
then $p^{-1}(T^{2})$ is a simple $n$-fold branched covering of $T^{2}$ over
the $2b$ points. In general, it may be disconnected. In $\mathbb{CP}^{2}$, the
torus $T^{2}$ is Lagrangian with respect to the Fubini-Study Kähler form. The
pullback $p^{*}(\omega_{FS})$ also vanishes along $\Sigma=p^{-1}(T^{2})$,
although this form is not symplectic. Its square vanishes precisely along the
ramification locus $\mathcal{K}$. To modify it to a symplectic form, it is
necessary to add an exact 2-form $d\phi$ as in [Aur00, Proposition 10]. The
restriction of this 2-form to $\Sigma$ will have nonzero support in a
neighborhood of the ramification locus $B=\Sigma\cap\mathcal{K}$.
Consequently, this surface is no longer Lagrangian.

Finally, a flat metric $g$ on $T^{2}$ pulls back to a singular flat metric on
$\Sigma$. The closed 1-forms
$(d\theta_{2},-d\theta_{1},d\theta_{1}-d\theta_{2})$ pull back to harmonic
1-forms with respect to the conformal structure determined by the singular
flat metric. Moreover, the form $p^{*}(\beta_{\lambda})$ extends across
$H_{\lambda}$ as the 1-form defining the Levi-flat foliation by holomorphic
curves. The fact that

$p^{*}(\beta_{\lambda})\wedge
p^{*}(\beta_{\lambda+1})=p^{*}(\beta_{\lambda}\wedge\beta_{\lambda+1})\geq 0$

implies that these foliations meet transversely as well. Thinking
4-dimensionally, this transversality condition is equivalent to the statement
that $\Sigma$ is a convex corner of the analytic polyhedron
$\widetilde{Z}_{\lambda}$.

The singular points $\Sigma\cap\mathcal{K}$ are the preimages of the branch
points on $T^{2}$. We can describe this branching in local coordinates by
identifying $T^{2}$ with $i\mathbb{R}^{2}\subset\mathbb{C}^{2}$ and consider
the complex surface

$\\{z^{2}=w_{1}\mp iw_{2}\\}\subset\mathbb{C}^{3}$

for a positive (resp. negative) intersection point. The surface $\Sigma$ is
the intersection of $X$ with the subset
$\\{\text{Re}(w_{1})=\text{Re}(w_{2})=0\\}$. The tangent plane at
$0\subset\Sigma$ is an isolated complex point of $\Sigma$, meaning that
$T_{0}\Sigma$ is a complex line with respect to the almost-complex structure
on $X$ (and not simply a copy of $\mathbb{R}^{2}$). Complex lines come with an
orientation, therefore the singular point is oriented and this agrees with the
orientation of $p(\mathcal{K})\cap T^{2}$ downstairs. But since this
orientation is induced by the complex structure, it must extend in a
compatible way with the foliation on $\widetilde{H}_{\lambda}$ by holomorphic
curves.

Finally, the boundary of $\widetilde{Z}_{\lambda}$ can be smoothed to obtain a
Stein-fillable contact structure. The defining function
$\phi_{\lambda}=\text{max}(|f_{1}|,|f_{2}|)$ is a continuous, plurisubharmonic
function on $\widetilde{U}_{\lambda}$. It can be $C^{0}$-approximated by a
smooth, strictly plurisubharmonic function $\widetilde{\phi}_{\lambda}$. The
level set $\widetilde{Y}_{\lambda}=\widetilde{\phi}^{-1}_{\lambda}(1)$
naturally admits a Stein-fillable contact structure.

##  3\. Handlebody setup

In order to construct the required symplectic structure over each handlebody
$H_{\lambda}$, we need to construct an exact 2-form $d\mu_{\lambda}$ that
orients the leaves of the foliation
$\mathcal{F}_{\lambda}=\text{ker}(\widetilde{\beta}_{\lambda})$.

###  3.1. Morse form foliations

We restrict attention to foliations that are defined as the kernel of a
smooth, closed 1-form $\beta$. In particular, a closed 1-form is Morse if near
each singular point, there exists local coordinates and a function $f$ that is
quadratic in these coordinates such that $\beta=df$. Moreover, the closed
1-form $\beta$ is intrinsically harmonic if it is harmonic for some Riemannian
metric.

We can apply various results on the topology of Morse form foliations using
the following doubling argument. Recall that
$\widetilde{\beta}_{\lambda}=\beta_{\lambda}+d(sf)$ near the boundary. We can
take the double $\\#_{g}S^{1}\times S^{2}$ of $H_{\lambda}$ and extend
$\widetilde{\beta}_{\lambda}$ as a closed 1-form
$D(\widetilde{\beta}_{\lambda})$ on the total space by assuming

$D(\widetilde{\beta}_{\lambda})=\beta_{\lambda}+d(s^{2}f)$

on a tubular neighborhood of $\Sigma$. This 1-form has Morse singularities at
the singular points $B\subset\Sigma$ that are of index 1 or 2, according to
the sign of $f$. In addition, the induced foliation is topologically conjugate
to $\mathcal{F}_{\lambda}$.

There are significant restrictions on the topology of a foliation defined by a
closed, Morse 1-form. For example, the foliation has trivial holonomy. The
leaves of a closed Morse 1-form are either compact or noncompact. A finite
number of singular leaves intersect the vanishing locus of the 1-form, while
all remaining leaves are nonsingular. Given a closed Morse 1-form on a compact
$n$-manifold $M$, there is a decomposition $M=M_{c}\cup M_{\int}$ into two
compact $n$-manifolds such that the interior $M_{c}$ is the union of all
compact leaves, the interior of $M_{\infty}$ is the union of all noncompact
leaves, and $M_{c}\cap M_{\infty}=\partial M_{c}\cap\partial M_{\infty}$ is a
subvariety that is the union of some compactified components of noncompact,
singular leaves [FKL98, Proposition 1].

A closed transversal to a foliation is closed, smooth curve that is everywhere
positively-transverse to the foliation.

###  3.2. Tautness of the foliation

Recall that a foliation $\mathcal{F}$ on $M$ is taut if for every point $p\in
M$, there is a closed transversal to $\mathcal{F}$ through $p$. In this
subsection, we will prove that the foliation $\mathcal{F}_{\lambda}$ on
$H_{\lambda}$ satisfy the following stronger version of tautness.

######  Proposition 3.1.

For each point $p\in H_{\lambda}$, there exists

  1. (1)

an embedded, closed transversal $\gamma_{p}$ to $\mathcal{F}_{\lambda}$
through $p$,

  2. (2)

an embedded annulus $A_{p}\cong S^{1}\times[0,1]$ consisting of a family of
embedded, closed transversals to $\mathcal{F}_{\lambda}$ such that the
unoriented boundary is $\partial A_{p}=\gamma_{p}\cup\gamma$, where $\gamma$
is an embedded curve in $\partial H_{\lambda}=\Sigma$.

The proposition follows easily from the following two results.

######  Lemma 3.2.

Through every point $q\in\Sigma\setminus B$, there exists an embedded curve
$\gamma\subset\Sigma$ that is a closed transversal to the kernel foliation of
$\beta_{\lambda}$.

###### Proof.

By assumption, $\beta_{\lambda}$ is a harmonic 1-form on $\Sigma$ and
therefore Calabi’s condition ensures there is an immersed closed transversal
through each nonsingular point [Cal69]. By a perturbation, we can assume that
the curve is self-transverse and resolve the self-intersections preserving the
fact that is a (possibly disconnected) embedded closed transversal. If this
closed transversal is disconnected, choose the component through $p$. ∎

######  Proposition 3.3.

Every leaf of $\mathcal{F}_{\lambda}$ intersects the boundary $\partial
H_{\lambda}=\Sigma$.

This follows immediately by combining Lemmas 3.7 and 3.8. We defer these
proofs until after the following corollary and the proof of the main
proposition.

######  Corollary 3.4.

The doubled 1-form $D(\widetilde{\beta}_{\lambda})$ on $D(H_{\lambda})$ is
intrinsically harmonic.

###### Proof.

Since each leaf intersects $\Sigma$ and $\beta_{\lambda}$ is intrinsically
harmonic on $\Sigma$, then $D(\widetilde{\beta}_{\lambda})$ satisfies Calabi’s
condition. ∎

###### Proof of Proposition 3.1.

The boundary foliation admits an embedded closed transversal that intersects
every leaf (Lemma 3.2). Since each leaf of $\mathcal{F}_{\lambda}$ is
connected to the boundary (Proposition 3.3), we can find $\gamma_{p}$ by a
leaf-preserving isotopy. The trace of this isotopy is the required annulus
$A_{p}$. ∎

We now address the case of compact leaves.

######  Lemma 3.5.

The foliation $\mathcal{F}_{\lambda}$ on $H_{\lambda}$ has no $T^{2}$ leaves.

###### Proof.

Every torus in a 1-handlebody is compressible and moreover bounds a solid
torus $Q$. Then $[\widetilde{\beta}_{\lambda}]=[0]\in H^{1}_{DR}(Q,\partial
Q)$ and so there is a function $\phi:Q\rightarrow\mathbb{R}$ such that
$\phi(\partial Q)=0$ and $\widetilde{\beta}_{\lambda}=d\phi$ on $Q$. This
function attains a maximum at some point $x\in Q$ and therefore
$(\widetilde{\beta}_{\lambda})_{x}=d\phi_{x}=0$, which is a contradiction. ∎

Next, we can use the Euler class of the foliation $\mathcal{F}_{\lambda}$ to
show that every closed, compact leaf must be an (excluded) torus. Using the
doubling argument, we get isomorphisms

$\displaystyle H_{1}(H,B)$ $\displaystyle\cong H_{1}(Y,B)$ $\displaystyle
H_{2}(H,\Sigma\setminus B)$ $\displaystyle\cong H_{2}(Y\setminus B)$

We can therefore use the intersection pairing

$H_{1}(Y,B)\times H_{2}(Y\setminus B)\rightarrow H_{0}(Y\setminus B)\cong
H_{0}(Y)\cong\mathbb{Z}$

to define an intersection pairing

$\langle,\rangle:H_{1}(H_{\lambda},B)\times H_{2}(H_{\lambda},\Sigma\setminus
B)\rightarrow\mathbb{Z}$

Choose a vector field $v_{\lambda}$ directing the kernel of $\beta_{\lambda}$
on $\Sigma$ and extend it to a section of
$\text{ker}(\widetilde{\beta}_{\lambda})$ over $H_{\lambda}$. Generically,
this section will vanish along a smooth 1-complex representing a well-defined
class Euler class $e(\mathcal{F}_{\lambda})\in H_{1}(H_{\lambda},B)$ of the
foliation.

######  Lemma 3.6.

Let $L\in H_{\lambda}$ be a compact leaf $L$ of $\mathcal{F}_{\lambda}$. Then
the Euler characteristic of $L$ is given by

$\langle e(\mathcal{F}_{\lambda}),[L]\rangle$

###### Proof.

A section $s$ of $\text{ker}(\beta_{\lambda})$, restricted to $L$, is a vector
field on $L$ and the Euler characteristic of $L$, which is therefore given by
the signed count of the zeros of $s|_{L}$, is given by the intersection of
$e(\mathcal{F}_{\lambda})$ with $L$. ∎

######  Lemma 3.7.

Every compact leaf of the foliation $\mathcal{F}_{\lambda}$ intersects
$\Sigma$.

###### Proof.

If $L$ is a compact, closed, co-oriented hence oriented leaf, then by Lemma
3.6 its Euler characteristic vanishes since $[L]=0\in
H_{2}(H_{\lambda},\Sigma\setminus B)$. But this implies it is a torus, which
is excluded by Lemma 3.5. ∎

We now finish the proof of Proposition 3.3 by excluding noncomapct leaves.

######  Lemma 3.8.

Every noncompact leaf of the foliation $\mathcal{F}_{\lambda}$ intersects
$\Sigma$.

###### Proof.

The boundary of the closure of a noncompact leaf $L$ contains a compactified
component $L^{\prime}$ of a singular leaf (by [FKL98, Proposition 1]). The set
of accumulation points of $L$ in $L^{\prime}$ is open in $L^{\prime}$; in
particular, the closure of the set of accumulation points contains the
singular point. The singularities all lie in $\Sigma$, hence $L^{\prime}$
intersects $\Sigma$ and in particular it intersects $\Sigma$ along a
compactified component $\widetilde{L}^{\prime}$ of a singular, noncompact leaf
of the kernel foliation of $\beta_{\lambda}$ on $\Sigma$.

Choose some point $x$ in $\widetilde{L}^{\prime}$. Every neighborhood of $x$
contains $L$ and moreover, since $x\in\Sigma\setminus B$, the foliation is
locally a product with the foliation on $\Sigma$. In particular, $L$ must
intersect this neighborhood as a leaf intersecting the boundary. ∎

###  3.3. Thurston norm

Just as compact leaves of a taut foliation are known to minimize the Thurston
norm [Thu86], the nonsingular compact leaves of the foliation
$\mathcal{F}_{\lambda}$ minimize the (generalized) Thurston norm defined by
Scharlemann [Sch89].

To prove this, we first establish an embedding result that reduces the problem
to the original case.

######  Proposition 3.9.

Let $M$ be a compact 3-manifold with boundary and let $f:M\rightarrow S^{1}$
be a smooth map such that

  1. (1)

$f$ has no critical points in $M$, and

  2. (2)

$f|_{\partial M}$ has only nondegenerate critical points of index 1.

. Then there exists a fibered 3-manifold $\pi:Y\rightarrow S^{1}$ and an
embedding $i:M\hookrightarrow Y$ such that $\pi\circ i=f$.

###### Proof.

Without loss of generality, assume that $0$ is a regular value of
$f|_{\partial M}$. Let $F_{0}=f^{-1}(0)$ be the compact, regular level set. As
$\theta$ varies in $[0,2\pi]$, the topology of the regular level set
$f^{-1}(\theta)$ changes at the critical points of $f|_{\partial M}$ by either
adding or removing a 1-handle. Fix a Morse-Smale metric $g$ on $\Sigma$. If
$p\in\text{Crit}(f|_{\partial M})$ corresponds to adding a 1-handle to the
regular level set, then we can use the unstable manifold of $p$ to isotope the
attaching $S^{0}$ for this down to $\partial F_{0}$. On the other hand, if $p$
is a critical point corresponding to removing a 1-handle, then we can use the
stable manifold of $p$ to isotope the attaching $S^{0}$ for the cancelling
1-handle up to $F_{2\pi}$.

We therefore have two surfaces

$\widehat{F}_{0}=F_{0}\cup\\{\text{1-handles}\\}\qquad\widehat{F}_{2\pi}=F_{2\pi}\cup\\{\text{1-handles}\\}$

Let $M^{0}$ be the compact 3-manifold obtained by cutting $M$ along
$f^{-1}(0)$. Then $M^{0}$ embeds into $\widehat{F}_{0}\times[0,2\pi]$ in a
fiber-preserving manner. Similarly, it also embeds into
$\widehat{F}_{2\pi}\times[0,2\pi]$.

Now choose a closed surface $F$ and embeddings
$\widehat{F}_{0},\widehat{F}_{2\pi}\hookrightarrow F$. To obtain $Y$, we
choose any homeomorphism $\phi:F\rightarrow F$ such that
$\phi(F_{2\pi})=F_{0}$ and take the mapping torus

$Y=F\times[0,2\pi]/(x,2\pi)\sim(\phi(x),0)$

The embedding $M_{0}\subset\widehat{F}_{0}\times[0,2\pi]\subset
F\times[0,2\pi]$ then descends to an embedding of $M$ into $Y$. ∎

######  Theorem 3.10.

Suppose that $\widetilde{\beta}_{\lambda}$ is integral. Then each nonsingular
leaf $L$ minimizes the Thurston norm for the class $[L]\in
H_{2}(H_{\lambda},\partial L)$.

###### Proof.

Since $\widetilde{\beta}_{\lambda}$ is integral, there exists a map
$f:H_{\lambda}\rightarrow S^{1}$ such that
$f^{*}(d\theta)=\widetilde{\beta}_{\lambda}$. By Proposition 3.9, we obtain an
embedding $H_{\lambda}\hookrightarrow Y$ into a fibered 3-manifold that
extends $f$.

Let $L=f^{-1}(\theta_{0})$ be a nonsingular, compact leaf. The $L$ embeds in
the fiber $\pi^{-1}(\theta_{0})$. Let $F_{\theta_{0}}$ denote the complement
of $L$ in the fiber. Let $L^{\prime}$ be any embedded surface representing
$[L]\in H_{2}(H_{\lambda},\partial L)$. Then $L^{\prime}\cup F_{\theta_{0}}$
is a closed surface representing the class of the fiber in $H_{2}(Y)$. Since
$\pi^{-1}(\theta_{0})$ minimizes the Thurston norm in $Y$, we consequently
have the inequality

$-\chi(L\cup F_{\theta_{0}})\leq-\chi(L^{\prime}\cup F_{\theta_{0}})$

which implies that $-\chi(L)\leq-\chi(L^{\prime})$. ∎

###  3.4. Extending $B$ to a 1-complex $\mathcal{S}$

The construction of $\omega$ in the next subsection requires a small
perturbation near the vanishing locus $B\subset\Sigma$. To extend this
perturbation across the spine, it is necessary to extend $B$ to a 1-complex
$\mathcal{S}$ that is transverse to the foliations
$\\{\mathcal{F}_{\lambda}\\}$. We can accomplish this via the following lemma.

######  Lemma 3.11.

For each $\lambda$, there exists a collection of disjoint, embedded paths
$\gamma_{\lambda,1},\dots,\gamma_{\lambda,g-1}$ in $H_{\lambda}$ with boundary
on $B$ that are transverse to the foliation $\mathcal{F}_{\lambda}$.

For brevity, let $\gamma_{\lambda}$ denote the union $\cup\gamma_{\lambda,i}$
of the $g-1$ arcs constructed in $H_{\lambda}$. Then the 1-complex is the
union

$\mathcal{S}:=\bigcup_{\lambda=1,2,3}\gamma_{\lambda}$

###### Proof of Lemma 3.11.

Choose a pairing on the points of $B$, where each pair has one positive and
one negative point (with respect to the orientation on $B$ induced by the
function $f$). In particular, we can index the positive points as
$p_{1},\dots,p_{g-1}$ and the negative points as $q_{1},\dots,q_{g-1}$.

By assumption, the 1-form $\beta_{\lambda}$ is harmonic on $\Sigma$. Calabi
has shown that, consequently, for any two points $p_{i},q_{i}\in\Sigma$, there
is an embedded path transverse to $\text{ker}(\beta_{\lambda})$ in $\Sigma$
from $q_{i}$ to $p_{i}$ [Cal69]. Let
$\gamma_{\lambda,1},\dots,\gamma_{\lambda,g-1}$ be a collection of such
transverse paths. These paths can be pushed into $H_{\lambda}$ as a collection
of embedded paths transverse to $\mathcal{F}_{\lambda}$. ∎

By the Morse lemma, we can choose local coordinates $(x,y)$ near each $b\in B$
such that

$\beta=xdx-ydy\qquad\ast\beta=xdy+ydx$

This further implies that

$\beta\wedge\ast\beta=(x^{2}+y^{2})dx\wedge dy$

We set $R=x^{2}+y^{2}$ to be the square of the radius. Then near $B$ we have

$\beta_{1}\wedge\beta_{2}=\beta_{2}\wedge\beta_{3}=\beta_{3}\wedge\beta_{1}=Rdx\wedge
dy$

We now extend these local coordinates to coordinates on each cylinder
$\nu(\gamma_{\lambda,i})$. The foliation $\mathcal{F}_{\lambda}$ induces a
foliation by disks on each $\nu(\gamma_{\lambda,i})$ and therefore determines
a product splitting $\nu(\gamma_{\lambda,i})=\gamma_{\lambda,i}\times D^{2}$.
We can view $(x,y)$ as local coordinates on the leaves of
$\mathcal{F}_{\lambda}$ that intersect $\gamma_{\lambda,i}$. By construction,
each arc $\gamma_{\lambda,i}$ of $\mathcal{S}$ connects two points where $f$
takes opposite signs. At the endpoints of $\gamma_{\lambda,i}$, the surface
$\Sigma$ inherits two orientations: one as the boundary of $H_{\lambda}$ and
the other as the boundary of $\nu(\gamma_{\lambda,i})$. We can assume these
agree at the endpoint where $f=1$ and disagree at the endpoint where $f=-1$.
In particular, at the negative endpoint of $\gamma_{\lambda,i}$, the 2-form
$dx\wedge dy$ induces the negative orientation on $\Sigma$.

In the discussion that follows, we will let $R=x^{2}+y^{2}$ in this local
coordinate system and let $D_{R}$ denote the disk of radius $\sqrt{R}$. We
also set constants

$0<R_{0}<R_{1}<R_{2}$

where $\sqrt{R_{2}}$ is less than the radius of definition of the coordinates
$(x,y)$ on $\nu(\mathcal{S})$. We will let $\nu_{R}(B)$ and
$\nu_{R}(\mathcal{S})$ denote the tubular neighborhoods of radius $\sqrt{R}$
around $B$ in $\Sigma$ and $\mathcal{S}$ in $H_{\lambda}$, respectively.

###  3.5. Handlebody primitive

The goal of this section is to construct an exact 2-form on $H_{\lambda}$ that
orients the leaves of $\mathcal{F}_{\lambda}$. Since $\mathcal{F}_{\lambda}$
is much like a taut foliation, we adapt a standard argument (eg [Cal07,
Section 4.4]) for constructing 2-forms that orient taut foliations.
Throughout, we will consider a collar neighborhood $\Sigma\times[0,8\epsilon]$
of $\partial H_{\lambda}$, where $\epsilon$ is an undefined, small constant to
be chosen later and $s$ is a normal coordinate to $\Sigma=\partial
H_{\lambda}$.

######  Lemma 3.12.

There exists a collection of paths $\\{\gamma_{q_{1}},\dots,\gamma_{q_{k}}\\}$
in $\Sigma\setminus B$ and closed 1-forms
$PD(\gamma_{q_{1}}),\dots,PD(\gamma_{q_{k}})$ such that

  1. (1)

$\beta_{\lambda}\wedge PD(\gamma_{q_{i}})\geq 0$ for all $i=1,\dots,k$,

  2. (2)

the union $\bigcup\text{Supp}(\beta_{\lambda}\wedge PD(\gamma_{q_{i}}))$ of
the supports covers the complement of $\nu_{R_{0}}(B)$.

###### Proof.

Let $q$ be a point in the complement of a $\sqrt{R}_{0}$-neighborhood of $B$.
Since $\beta_{\lambda}$ is harmonic, we can find a closed transversal through
$q$. A tubular neighborhood $\nu(\gamma_{q})$ splits as a product
$\gamma_{q}\times[-\delta,\delta]$. Let $s$ be a normal coordinate and
$\phi(s)$ a bump function supported on $[-\delta,\delta]$. Define
$PD(\gamma_{q})=\phi(s)ds$ in these local coordinates. Since $\gamma_{q}$ is a
closed transversal, we have that $\beta_{\lambda}\wedge PD(\gamma_{q})\geq 0$.

The set $\\{\text{Supp}(\beta_{\lambda}\wedge PD(\gamma_{q}))\\}$ as $q$
ranges through the points in $\Sigma\smallsetminus\nu_{\sqrt{R}_{0}}(B)$ is an
open cover of a compact set, hence it has a finite subcover that is indexed by
a finite set of points $q_{1},\dots,q_{k}$. ∎

Choose a cutoff function satisfying

$\psi_{q}=\begin{cases}1&\text{if $s\geq 5\epsilon$}\\\ 0&\text{if $s\leq
2\epsilon$}\end{cases}$

such that $\psi^{\prime}_{q}>0$ for $s\in[3\epsilon,5\epsilon]$. Define the
1-form

$\mu_{q}=\psi_{q}\sum_{i=1}^{k}PD(\gamma_{q_{i}})$

where $PD(\gamma_{q_{i}})$ also denotes (by abuse of notation) the pullback of
$PD(\gamma_{q_{i}})$ via the projection map
$\pi:\Sigma\times[0,8\epsilon]\rightarrow\Sigma$.

######  Lemma 3.13.

For $\epsilon>0$ sufficiently small, we have that
$\widetilde{\beta}_{\lambda}\wedge d\mu_{q}\geq 0$ and is strictly positive
when $s\in[3\epsilon,5\epsilon]$ and $R\geq R_{0}$.

###### Proof.

We have that

$\displaystyle\widetilde{\beta}_{\lambda}\wedge d\mu_{q}$
$\displaystyle=\sum_{i=1}^{k}(\beta_{\lambda}+fds+sdf)\wedge\psi^{\prime}_{q}ds\wedge
PD(\gamma_{q_{i}})$
$\displaystyle=-\psi^{\prime}_{q}\sum\left(ds\wedge\beta_{\lambda}\wedge
PD(\gamma_{q_{i}})+sds\wedge df\wedge PD(\gamma_{q_{i}})\right)$

The derivative $-\psi^{\prime}_{q}$ is positive and the first term in the sum
is nonnegative everywhere and strictly positive on the support of
$PD(\gamma_{q_{i}})$. Moreover, it dominates the second term in the sum for
$|s|\leq 5\epsilon$ sufficiently small. ∎

We can choose an annulus $A_{q_{i}}$ with boundary
$\gamma_{q_{i}}\times\\{4\epsilon\\}\cup\gamma^{\prime}_{q_{i}}\times\\{0\\}$
such that

  1. (1)

$\gamma^{\prime}_{q_{i}}$ is transversely isotopic to $\gamma_{q_{i}}$ in
$\Sigma\setminus B$ and disjoint from an $\sqrt{R}_{2}$-neighborhood of $B$,

  2. (2)

the intersection $A_{q_{i}}\cap\Sigma\times[0,8\epsilon]$ is the union of
$\gamma_{q_{i}}\times[4\epsilon,8\epsilon]$ and
$\gamma^{\prime}_{q_{i}}\times[0,8\epsilon]$.

We can then use the following lemma to extend $\mu_{q_{i}}$ to a 1-form
supported in a neighborhood of $A_{q_{i}}$.

######  Lemma 3.14.

Let $p\in H_{\lambda}$ be a point and let $\gamma_{p},A_{p}$ be a closed
transversal and annulus satisfying the conclusions of Proposition 3.1. There
exist tubular neighborhoods $U_{p}\subset U^{\prime}_{p}$ of $\gamma_{p}$ and
$V_{p}$ of $A_{p}$ and an exact 2-form $d\mu_{p}$ such that

  1. (1)

the support of $\mu_{p}$ is contained in $\nu(A_{p})$,

  2. (2)

the form $\widetilde{\beta}_{\lambda}\wedge d\mu_{p}$ is positive on $U_{p}$
and its positive support is contained in $U^{\prime}_{p}$,

  3. (3)

the negative support of $\widetilde{\beta}_{\lambda}\wedge d\mu_{p}$ is
contained in $\nu(A_{p})\cap\Sigma\times[\epsilon,2\epsilon]$

###### Proof.

We can choose coordinates $(s,t,\theta)$ on
$\nu(A_{p})\cong[0,1+2\delta]\times[-\delta,\delta]\times S^{1}$ so that

  1. (1)

$s$ agrees with the collar coordinate on $\partial H_{\lambda}$ for $0\leq
s\leq 6\epsilon$,

  2. (2)

$\widetilde{\beta}_{\lambda}=d\theta$

Let $\phi_{1}(t)$ be a bump function on $[-\delta,\delta]$ and choose a second
bump function satisfying

$\phi_{2}(s)=\begin{cases}0&\text{for $s\geq 1+\delta$}\\\ 1&\text{for
$1-\delta\geq s\geq 2\epsilon$}\\\ 0&\text{for $s\leq\epsilon$}\end{cases}$

and $\phi^{\prime}_{2}>0$ on the interval $1-\delta/2\leq s\leq 1+\delta/2$.
Define

$\mu_{p}=\phi_{2}(s)\phi_{1}(t)dt$

Then

$d\mu_{p}=\phi_{1}\left(\phi^{\prime}_{2}ds\wedge dt\right)$

and

$\widetilde{\beta}_{\lambda}\wedge
d\mu_{p}=\phi_{1}\phi^{\prime}_{2}\widetilde{\beta}_{\lambda}\wedge ds\wedge
dt$

This is nonzero precisely on the support of $\phi_{1}\phi^{\prime}_{2}$. The
function $\phi_{1}$ is nonnegative everywhere, while $\phi^{\prime}_{2}$ is
positive on the interval $1-\delta/2\leq s\leq 1+\delta/2$, the positive
support of $\phi^{\prime}_{2}$ is contained in the interval $1-\delta\leq
s\leq 1+\delta$ and the negative support is contained in the interval
$\epsilon\leq s\leq 2\epsilon$. We can therefore choose
$U_{p}=S^{1}\times[1-\delta/2,1+\delta/2]\times[-\delta/2,-\delta/2]$ and
$U^{\prime}_{p}=S^{1}\times[1-\delta,1+\delta]\times[-\delta,\delta]$. ∎

Define

$W_{\epsilon}=H_{\lambda}\setminus(\Sigma\times[0,5\epsilon])$

######  Proposition 3.15.

There exists a collection of points $\\{p_{1},\dots,p_{k}\\}$ in
$W_{\epsilon}$ such that

  1. (1)

$\widetilde{\beta}_{\lambda}\wedge d\mu_{p_{i}}\geq 0$ for all $i=1,\dots,k$,

  2. (2)

the union $\bigcup_{i=1}^{k}\text{Supp}(\widetilde{\beta}_{\lambda}\wedge
d\mu_{p_{i}})$ of the supports covers $W_{\epsilon}$.

  3. (3)

the 3-form

$\widetilde{\beta}_{\lambda}\wedge\sum_{i=1}^{k}d\mu_{p_{i}}$

is a volume form on $W_{\epsilon}$

###### Proof.

For each $p\in W_{\epsilon}$, we can find some closed transversal $\gamma_{p}$
and annulus $A_{p}$ and use Lemma 3.14 to define $\mu_{p}$. As in the proof of
Lemma 3.12, the union of the supports of $d\mu_{p}$ as $p$ ranges over the set
$W_{\epsilon}$ is an open cover, which by compactness has a finite subcover
indexed by a finite collection of points $p_{1},\dots,p_{n}$. The required
properties follow immediately by the construction of Lemma 3.14. ∎

We now define the 1-form

(1) $\mu_{H_{,}\lambda}=\mu_{q}+\sum_{i=1}^{k}\mu_{p_{i}}$

######  Lemma 3.16.

For $C>0$ sufficently large,

  1. (1)

$\alpha_{+}=C\widetilde{\beta}_{\lambda}+\mu_{H,\lambda}$ is a positive
contact form on $W_{\epsilon}$,

  2. (2)

$\alpha_{-}=-C\widetilde{\beta}_{\lambda}+\mu_{H,\lambda}$ is a negative
contact form on $W_{\epsilon}$.

###### Proof.

By construction,

$\widetilde{\beta}_{\lambda}\wedge
d\mu_{H,\lambda}=\widetilde{\beta}_{\lambda}\wedge
d\mu_{q}+\widetilde{\beta}_{\lambda}\wedge\sum d\mu_{p_{i}}>0$

Furthermore, we have that

$\alpha_{+}\wedge d\alpha_{+}=C\widetilde{\beta}_{\lambda}\wedge
d\mu_{H,\lambda}+\mu_{H,\lambda}\wedge d\mu_{H,\lambda}$

and

$\alpha_{-}\wedge d\alpha_{-}=C(-\beta_{\lambda}\wedge
d\mu_{H,\lambda})+\mu_{H,\lambda}\wedge d\mu_{H,\lambda}$

In both cases, the first term dominates for $C>0$ sufficiently large. ∎

##  4\. Constructing the form

In this section, we describe the symplectic form $\omega$ in a neighborhood of
each stratum of the trisection decomposition and how to patch these forms
together to get a global symplectic form.

###  4.1. Coordinates on the spine

Let $(p_{1},p_{2})$ denote coordinates on $D^{2}$ and let $(x,y)$ denote local
coordinates on $\Sigma$. We assume that locally $dx\wedge dy$ orients the
surface $\Sigma$ and the orientation on $\nu(\Sigma)=D^{2}\times\Sigma$ is
given by the 4-form

$dp_{1}\wedge dx\wedge dp_{2}\wedge dy=dp_{2}\wedge dp_{1}\wedge dx\wedge dy$

In a neighborhood of the central surface, we can assume that locally in
$\Sigma\times D^{2}$ the pieces of the trisection are given by:

$\displaystyle H_{1}$ $\displaystyle=\\{p_{1}\geq 0,p_{2}=0\\}$ $\displaystyle
Z_{1}$ $\displaystyle=\\{p_{1}\geq 0,p_{2}\leq 0\\}$ $\displaystyle H_{2}$
$\displaystyle=\\{p_{1}+p_{2}=0,p_{2}\geq 0\\}$ $\displaystyle Z_{2}$
$\displaystyle=\\{p_{2}\geq 0,p_{1}+p_{2}\geq 0\\}$ $\displaystyle H_{3}$
$\displaystyle=\\{p_{1}=0,p_{2}\leq 0\\}$ $\displaystyle Z_{3}$
$\displaystyle=\\{p_{1}\leq 0,p_{1}+p_{1}\leq p_{1}\\}$

In particular, if we set

$\phi_{1}=p_{2}\qquad\phi_{2}=-p_{1}-p_{2}\qquad\phi_{3}=p_{1}$

then

$H_{\lambda}=\\{\phi_{\lambda-1}\geq 0,\phi_{\lambda}=0\\}\qquad
Z_{\lambda}=\\{\text{max}(-\phi_{\lambda-1},\phi_{\lambda})\leq 0\\}$

and in particular, $Z_{\lambda}$ is locally the sublevel set of
$g_{\lambda}=\text{max}(\phi_{\lambda},-\phi_{\lambda-1})$.

We attach $H_{\lambda}\times I$ to $\Sigma\times D^{2}$ as follows. Fix some
$\epsilon>0$. Let $s\in[0,6\epsilon]$ be a coordinate on a collar neighborhood
of $\partial H_{\lambda}$ and $t$ a coordinate on the $I$-factor. Glue
$H_{\lambda}\times I$ to $\Sigma\times D^{2}$ using the following
identifications for $\lambda=1,2,3$:

$s=\phi_{\lambda-1}\qquad t=\phi_{\lambda}$

Finally, we obtain a smooth manifold $\nu(H_{1}\cup H_{2}\cup H_{3})$ with
three smooth boundary components as follows. Recall that the sector
$Z_{\lambda}$ is locally defined near $\Sigma$ by $g_{\lambda}=0$. We can
extend $g_{\lambda}$ across $H_{\lambda}\times I\cup H_{\lambda-1}\times I$.
Now let $\widehat{g}_{\lambda}$ denote some $C^{0}$-close smoothing of
$g_{\lambda}$ (as a function of $p_{1},p_{2}$), that agrees with $g_{\lambda}$
outside of a tubular neighborhood of $\Sigma$. Let $\widehat{Y}_{\lambda}$ be
the level set of $\widehat{g}^{-1}_{\lambda}(-\epsilon)$ for some small
$\epsilon>0$. We can assume that the smoothing is chosen so that the form

$\zeta_{\lambda}=d\phi_{\lambda}-d\phi_{\lambda-1}=dt-ds$

coorients $\widehat{Y}_{\lambda}$ in $\Sigma\times D^{2}$.

###  4.2. Constructing $\omega$ in a neighborhood of $\Sigma$

The first step is to construct a 2-form in a neighborhood of the central
surface.

Let $(\beta_{1},\beta_{2},\beta_{3})$ be the 1-cocycle of 1-forms. By
assumption, they have extensions
$\widetilde{\beta}_{1},\widetilde{\beta}_{2},\widetilde{\beta}_{3}$ across the
three handlebodies that take the form

$\displaystyle\widetilde{\beta}_{\lambda}$
$\displaystyle=\beta_{\lambda}+d(sf)$

near $\Sigma$ for a common function $f$. We can extend these to 1-forms on the
4-dimensional $\Sigma\times D^{2}$ as

$\widetilde{\beta}_{\lambda}=\beta_{\lambda}+d(\phi_{\lambda-1}f)$

Define the starting form

$\omega_{0}=dp_{2}\wedge\widetilde{\beta}_{1}-dp_{1}\wedge\widetilde{\beta}_{2}$

and the 1-forms

$\mu_{\Sigma,\lambda}=(t+2)\widetilde{\beta}_{\lambda}-(s-1)\widetilde{\beta}_{\lambda+1}$

for $\lambda=1,2,3$.

######  Lemma 4.1.

For each $\lambda=1,2,3$, we have that

$\displaystyle\omega_{0}$
$\displaystyle=dt\wedge\widetilde{\beta}_{\lambda}-ds\wedge\widetilde{\beta}_{\lambda+1}$
$\displaystyle\mu_{\lambda+1}$
$\displaystyle=\mu_{\lambda}-3\widetilde{\beta}_{\lambda}$

Consequently, each 1-form $\mu_{\Sigma,\lambda}$ is a primitive for
$\omega_{0}$.

###### Proof.

This follows by a straightforward computation using the cocycle relations

$\displaystyle\phi_{1}+\phi_{2}+\phi_{3}$ $\displaystyle=0$
$\displaystyle\beta_{1}+\beta_{2}+\beta_{3}$ $\displaystyle=0$

∎

######  Proposition 4.2.

In a sufficiently small neighborhood of $\Sigma$ we have

  1. (1)

$\omega_{0}\wedge\omega_{0}\geq 0$, and

  2. (2)

for each $\lambda=1,2,3$, we have that

$\zeta_{\lambda}\wedge\mu_{\Sigma,\lambda}\wedge\omega_{0}\geq 0$

  3. (3)

for each $\lambda=1,2,3$, we have that

$\displaystyle dt\wedge\mu_{\Sigma,\lambda}\wedge\omega_{0}$
$\displaystyle\geq 0$ $\displaystyle
dt\wedge\mu_{\Sigma,\lambda+1}\wedge\omega_{0}$ $\displaystyle\leq 0$

In addition, all 4-forms vanish exactly along $B\times D^{2}$, where
$B=\beta^{-1}(0)=\beta_{\lambda}^{-1}(0)$, and

###### Proof.

First, by multiplying out, we have that

$\displaystyle\omega_{0}\wedge\omega_{0}$
$\displaystyle=-2dp_{2}\wedge\widetilde{\beta}_{1}\wedge
dp_{1}\wedge\widetilde{\beta}_{2}=2dp_{2}\wedge
dp_{1}\wedge\widetilde{\beta}_{1}\wedge\widetilde{\beta}_{2}$
$\displaystyle=2dp_{2}\wedge
dp_{1}\wedge(\beta_{1}\wedge\beta_{2}+d(f(\phi_{1}+\phi_{2}))\wedge(\beta_{2}-\beta_{1}))$
$\displaystyle=2dp_{2}\wedge
dp_{1}\wedge\beta_{1}\wedge\beta_{2}+2(\phi_{1}+\phi_{2})dp_{2}\wedge
dp_{1}\wedge df\wedge(\beta_{2}-\beta_{1})$

We need to check positivity in two cases: inside and outside a neighborhood of
radius $R_{0}$ of $B\subset\Sigma$. Outside the neighborhood, then
$\beta_{1}\wedge\beta_{2}$ is strictly positive and bounded below, therefore
the 4-form is positive provided that $|\phi_{1}+\phi_{2}|$ is sufficiently
small. Inside this neighborhood, then $df=0$ and the form is nonnegative and
equal to 0 if and only if $\beta_{1}=\beta_{2}=0$, which occurs precisely
along $B\times D^{2}$.

We then compute that

$\displaystyle dt\wedge\mu_{\Sigma,\lambda}\wedge\omega_{0}$
$\displaystyle=(t+2)dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}$
$\displaystyle=(t+2)\omega_{0}\wedge\omega_{0}$ $\displaystyle
ds\wedge\mu_{\Sigma,\lambda}\wedge\omega_{0}$ $\displaystyle=(s-1)dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}$
$\displaystyle=(s-1)\omega_{0}\wedge\omega_{0}$
$\displaystyle\zeta_{\lambda}\wedge\mu_{\Sigma,\lambda}\wedge\omega_{0}$
$\displaystyle=(t-s+3)\omega_{0}\wedge\omega_{0}$ $\displaystyle
dt\wedge(\mu_{\Sigma,\lambda}-3\widetilde{\beta}_{\lambda})\wedge\omega_{0}$
$\displaystyle=(t-1)dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}$
$\displaystyle=(t-1)\omega_{0}\wedge\omega_{0}$

∎

Recall that a (positive) confoliation on a 3-manifold is a 2-plane field
defined as the kernel of a 1-form $\alpha$ satisfying $\alpha\wedge
d\alpha\geq 0$. A negative confoliation is a 2-plane field satisfying the
converse inequality everywhere.

######  Corollary 4.3.

For $\lambda=1,2,3$,

  1. (1)

The form $\mu_{\Sigma,\lambda}$ defines a (positive) confoliation on
$\widehat{Y}_{\lambda}$,

  2. (2)

the form $\mu_{\Sigma,\lambda}$ defines a positive confoliation on
$H_{\lambda}$ (with its coorientation given by $dt$),

  3. (3)

The form $\mu_{\Sigma,\lambda+1}$ defines a negative confoliation on
$H_{\lambda}$ (with its coorientation given by $dt$)

###### Proof.

The form $\zeta_{\lambda}$ coorients $\widehat{Y}_{\lambda}$, so the from
$\mu_{\lambda,\Sigma}$ defines a confoliation if and only if

$\zeta_{\lambda}\wedge\mu_{\Sigma,\lambda}\wedge d\mu_{\Sigma,\lambda}\geq 0$

which holds by the previous lemma since $\mu_{\Sigma,\lambda}$ is a primitive
for $\omega_{0}$. ∎

Recall that we have chosen Morse coordinates $(x,y)$ on a neighborhood $U$ of
$B=\beta^{-1}(0)$. We can assume that $f|_{U}$ is constant. Let
$R=x^{2}+y^{2}$ and choose some fixed values $0<R_{0}<R_{1}<R_{2}$. Choose a
cutoff function $\phi_{B}$ satisfying

$\phi_{B}=\begin{cases}1&\text{for }R\leq R_{1}\\\ 0&\text{for }R\geq
R_{2}\end{cases}$

and define the 1-form

$\mu_{B}=\phi_{B}f(xdy-ydx)$

######  Proposition 4.4.

For $\delta>0$ sufficiently small, the form

$\omega_{\Sigma}=\omega_{0}+\delta d\mu_{B}$

is symplectic on $\Sigma\times D^{2}$. Furthermore, for each $\lambda=1,2,3$,
we have that

$\displaystyle\zeta_{\lambda}\wedge(\mu_{\Sigma,\lambda}+\delta\mu_{B})\wedge\omega_{\Sigma}$
$\displaystyle>0$ $\displaystyle
dt\wedge(\mu_{\Sigma,\lambda}+\delta\mu_{B})\wedge\omega_{\Sigma}$
$\displaystyle>0$ $\displaystyle
dt\wedge(\mu_{\Sigma,\lambda+1}+\delta\mu_{B})\wedge\omega_{\Sigma}$
$\displaystyle<0$

###### Proof.

By Proposition 4.2, we know that the forms are strictly positive away from
$B\times D^{2}$. Let $dvol_{\Sigma}$ be an area form on $\Sigma$. Then we have
that

$\omega_{0}\wedge\omega_{0}=hdp_{2}\wedge dp_{1}\wedge dvol_{\Sigma}$

where $h$ is a function on $\Sigma$ that is nonnegative and vanishes precisely
at $B$. On the complement of the subset $\\{R\leq R_{0}\\}$, the function $h$
is strictly positive and bounded below. Therefore, for $\delta>0$ sufficiently
small, the form $\omega_{\Sigma}$ remains symplectic on the complement of
$\\{R\leq R_{0}\\}$. Furthemore, we have that

$\displaystyle\omega_{0}$
$\displaystyle=dp_{2}\wedge(\beta_{1}+fdp_{1})-dp_{1}\wedge(\beta_{2}+fdp_{2})$
$\displaystyle=dp_{2}\wedge\beta_{1}-dp_{1}\wedge\beta_{2}+2fdp_{2}\wedge
dp_{1}$
$\displaystyle=dt\wedge\beta_{\lambda}-ds\wedge\beta_{\lambda+1}+2fdt\wedge
ds$ $\displaystyle\mu_{\Sigma,\lambda}$
$\displaystyle=(t+2)(\beta_{\lambda}+fds)-(s-1)(\beta_{2}+f(dt-ds))$
$\displaystyle\mu_{\Sigma,\lambda+1}$
$\displaystyle=(t-1)(\beta_{\lambda}+fds)-(s-1)(\beta_{2}+f(dt-ds))$
$\displaystyle d\mu_{B}$ $\displaystyle=fdx\wedge dy$

In addition, since $\mu_{B}$ is a 1-form on a surface, we have that

$\mu_{B}\wedge d\mu_{B}=0\qquad d\mu_{B}\wedge d\mu_{B}=0$

Therefore, to establish the proposition, we need to check the following four
inequalities:

$\displaystyle\omega_{0}\wedge d\mu_{B}$ $\displaystyle>0$
$\displaystyle\zeta_{\lambda}\wedge(\mu_{B}\wedge\omega_{0}+\mu_{\Sigma,\lambda}\wedge
d\mu_{B})$ $\displaystyle>0$ $\displaystyle
dt\wedge(\mu_{B}\wedge\omega_{0}+\mu_{\Sigma,\lambda}\wedge d\mu_{B})$
$\displaystyle>0$ $\displaystyle
dt\wedge(\mu_{B}\wedge\omega_{0}+\mu_{\Sigma,\lambda+1}\wedge d\mu_{B})$
$\displaystyle<0$

After expanding, we have the following individual terms:

$\displaystyle\omega_{0}\wedge d\mu_{B}$ $\displaystyle=2f^{2}dp_{2}\wedge
dp_{1}\wedge dx\wedge dy$ $\displaystyle dt\wedge\mu_{B}\wedge\omega_{0}$
$\displaystyle=dt\wedge ds\wedge\mu_{B}\wedge\beta_{\lambda+1}$
$\displaystyle-ds\wedge\mu_{B}\wedge\omega_{0}$ $\displaystyle=dt\wedge
ds\wedge\beta_{\lambda}\wedge\mu_{B}$ $\displaystyle
dt\wedge\mu_{\Sigma,\lambda}\wedge d\mu_{B}$
$\displaystyle=f^{2}(t+s+1)dt\wedge ds\wedge dx\wedge dy$ $\displaystyle-
ds\wedge\mu_{\Sigma,\lambda}\wedge d\mu_{B}$ $\displaystyle=f^{2}(1-s)dt\wedge
ds\wedge dx\wedge dy$ $\displaystyle dt\wedge\mu_{\Sigma,\lambda+1}\wedge
d\mu_{B}$ $\displaystyle=f^{2}(t+s-2)dt\wedge ds\wedge dx\wedge dy$

All except for the second and third terms are strictly positive for $|s|,|t|$
small. For these terms, note that we can find some quadratic forms
$Q_{1},Q_{2}$ such that

$\displaystyle\beta_{\lambda}\wedge\mu_{b}$ $\displaystyle=Q_{1}dx\wedge dy$
$\displaystyle\mu_{B}\wedge\beta_{\lambda+1}$ $\displaystyle=Q_{2}dx\wedge dy$

Therefore, for any nonzero constant $A$, we have that

$\displaystyle
A\omega_{0}\wedge\omega_{0}+\delta_{1}dt\wedge\mu_{B}\wedge\omega_{0}-\delta_{2}ds\wedge\mu_{B}\wedge\omega_{0}$
$\displaystyle=(A(x^{2}+y^{2})+\delta_{1}Q_{1}+\delta_{2}Q_{2})dt\wedge
ds\wedge dx\wedge dy$

In particular, for $\delta_{1},\delta_{2}$ sufficiently small we have that
$A(x^{2}+y^{2})+\delta_{1}Q_{1}+\delta_{2}Q_{2}$ remains a definite quadratic
form. Consequently, the second and third terms do not affect any of the
positivity calculations. ∎

######  Corollary 4.5.

For $\lambda=1,2,3$ and $\delta>0$ sufficiently small,

  1. (1)

the form $\mu_{\Sigma,\lambda}$ defines a positive contact structure on
$\widehat{Y}_{\lambda}$,

  2. (2)

the form $\mu_{\Sigma,\lambda}$ defines a positive contact structure on
$H_{\lambda}$ (with its coorientation given by $dt$),

  3. (3)

the form $\mu_{\Sigma,\lambda+1}$ defines a negative contact structure on
$H_{\lambda}$ (with its coorientation given by $dt$),

###  4.3. Extending over the handlebodies

In the previous subsection, we constructed an exact symplectic form
$\omega=d\mu_{\Sigma,\lambda}+\delta d\mu_{B}$ in a neighborhood of the
central surface. We now describe how to extend it as a symplectic form across
the handlebody $H_{\lambda}$. Furthermore we will show that the form

$\alpha_{\lambda}=\mu_{\Sigma,\lambda}+\delta\mu_{B}$

extends as a positive contact form and

$\alpha_{\lambda+1}=\mu_{\Sigma,\lambda+1}+\delta\mu_{B}=\mu_{\Sigma,\lambda}-3\widetilde{\beta}_{\lambda}+\delta\mu_{B}$

extends as a negative contact form across $H_{\lambda}$.

We extend $\mu_{\Sigma,\lambda}+\delta\mu_{B}$ as follows:

  1. (1)

Recall that

$\mu_{\Sigma,\lambda}=(t+2)\widetilde{\beta}_{\lambda}+(-s+1)\widetilde{\beta}_{\lambda+1}$

To extend $\omega_{0}=d\mu_{\Sigma,\lambda}$ across $H_{\lambda}$, choose a
smooth function $\Phi(s)$ satisfying

$\Phi(s)=\begin{cases}1-s&\text{if }s\leq 2\epsilon\\\ 0&\text{if }s\geq
4\epsilon\end{cases}$

that furthermore satisfies $\Phi^{\prime}<0$ for $s\in[2\epsilon,3\epsilon]$
and $\Phi^{\prime}\leq\Phi\leq 0$ for $s\in[3\epsilon,4\epsilon]$. We can
extend the primitive as

$\mu_{\Sigma,\lambda}=(t+2)\widetilde{\beta}_{\lambda}+\Phi(s)\widetilde{\beta}_{\lambda+1}$

  2. (2)

Recall that $B$ can be extended to a 1-complex $\mathcal{S}$ and the Morse
coordinates on a neighborhood of $B$ can be extended across a tubular
neighborhood of the 1-complex $\mathcal{S}$. In these coordinates, the form
$\mu_{B}=f\phi_{B}(xdy-ydx)$ extends across $\nu(\mathcal{S})$.

Finally, in the previous section, we constructed an exact 2-form
$d\mu_{H,\lambda}$ that orients the leaves of $\mathcal{F}_{\lambda}$ for
$s\geq 5\epsilon$.

The main technical calculation of this section is the following result.

######  Proposition 4.6.

Consider the form

$\alpha_{\lambda}=\mu_{\Sigma,\lambda}+\frac{1}{C}\mu_{H,\lambda}+\delta\mu_{B}$

For $C>0$ sufficiently large and $\delta>0$ sufficiently small, we have that

  1. (1)

$d\alpha_{\lambda}$ is a symplectic form on a neighborhood of $H_{\lambda}$,

  2. (2)

$\alpha_{\lambda}$ is a positive contact form on $H_{\lambda}$, and

  3. (3)

$\alpha_{\lambda}-3\widetilde{\beta}_{\lambda}$ is a negative contact form on
$H_{\lambda}$

###### Proof.

To prove part (1), we need to check that

$d\alpha_{\lambda}\wedge d\alpha_{\lambda}>0$

and to prove part (2), we need to check that

$dt\wedge\alpha_{\lambda}\wedge d\alpha_{\lambda}>0$

and to prove part (3), we need to check that

$dt\wedge(\alpha_{\lambda}-3\beta_{\lambda})\wedge d\alpha_{\lambda}<0$

We will break up the calculation into cases according to intervals in the
$s$-coordinate, which are determined by where the various cutoff functions are
defined. Furthermore, in each case, we will first assume that $\delta=0$ and
prove that each term is nonnegative everywhere and strictly positive for
$R\geq R_{0}$. Then we will show that by choosing $\delta>0$ sufficiently
small, we can achieve strict positivity for $R\leq R_{0}$ without compromising
it for $R\geq R_{0}$.

For $R\leq R_{0}$, we have that $\phi_{B}=1$ and $f=\pm 1$. Thereforem

$\mu_{B}=f(xdy-ydx)\qquad d\mu_{B}=2fdx\wedge dy$

Moreover, since $\mu_{B}$ is a 1-form on a surface, we have

$\mu_{B}\wedge\mu_{B}=0\qquad\mu_{B}\wedge d\mu_{B}=0$

Therefore,

$\displaystyle(d\alpha_{\lambda}+\delta
d\mu_{B})\wedge(d\alpha_{\lambda}+\delta d\mu_{b})$
$\displaystyle=d\alpha_{\lambda}\wedge
d\alpha_{\lambda}+2\delta(d\alpha_{\lambda}\wedge d\mu_{B})$ $\displaystyle
dt\wedge(\alpha_{\lambda}+\delta\mu_{B})\wedge(d\alpha_{\lambda}+\delta
d\mu_{B})$ $\displaystyle=dt\wedge\alpha_{\lambda}\wedge d\alpha_{\lambda}+$
$\displaystyle+\delta(dt\wedge\mu_{B}\wedge
d\alpha_{\lambda}+dt\wedge\alpha_{\lambda}\wedge d\mu_{B})$ $\displaystyle
dt\wedge(\alpha_{\lambda}-3\widetilde{\beta}_{\lambda}+\delta\mu_{B})\wedge(d\alpha_{\lambda}+\delta
d\mu_{B})$
$\displaystyle=dt\wedge(\alpha_{\lambda}-3\widetilde{\beta}_{\lambda})\wedge
d\alpha_{\lambda}$ $\displaystyle+\delta(dt\wedge\mu_{B}\wedge
d\alpha_{\lambda}+dt\wedge\alpha_{\lambda}\wedge d\mu_{B})$
$\displaystyle-3\delta dt\wedge\widetilde{\beta}_{\lambda}\wedge d\mu_{B}$

Consequently, it suffices to check that for $R\leq R_{0}$:

$\displaystyle d\alpha_{\lambda}\wedge d\mu_{B}$ $\displaystyle>0$
$\displaystyle dt\wedge(\alpha_{\lambda}+\delta\mu_{B})\wedge
d\alpha_{\lambda}$ $\displaystyle>0$ $\displaystyle
dt\wedge(\alpha_{\lambda}-3\beta_{\lambda}+\delta\mu_{B})\wedge
d\alpha_{\lambda}$ $\displaystyle<0$ $\displaystyle
dt\wedge\alpha_{\lambda}\wedge d\mu_{B}$ $\displaystyle>0$ $\displaystyle
dt\wedge(\alpha_{\lambda}-3\beta_{\lambda})\wedge d\mu_{B}$ $\displaystyle<0$

Case 0: $s\in[0,\epsilon]$. This is a tubular neighborhood of $\Sigma$ and the
computations from the previous subsection apply.

Case 1: $s\in[\epsilon,2\epsilon]$. This is where the functions
$\\{\phi^{\prime}_{p}\\}$ are supported. Here, the primitive is

$\alpha_{\lambda}=(t+2)\widetilde{\beta}_{\lambda}+(1-s)\widetilde{\beta}_{\lambda+1}+\frac{1}{C}\sum_{p}\phi_{p}(s)PD(\gamma_{p})$

and so

$d\alpha_{\lambda}=dt\wedge\widetilde{\beta}_{\lambda}-ds\wedge\widetilde{\beta}_{\lambda+1}+\frac{1}{C}\sum_{p}\phi^{\prime}_{p}ds\wedge
PD(\gamma_{p})$

Therefore

$\displaystyle d\alpha_{\lambda}\wedge d\alpha_{\lambda}$
$\displaystyle=dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\left(\widetilde{\beta}_{\lambda+1}-\frac{1}{C}\sum_{p}\phi^{\prime}_{p}PD(\gamma_{p})\right)$
$\displaystyle=dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}$
$\displaystyle-\frac{1}{C}\sum_{p}\phi^{\prime}_{p}dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge PD(\gamma_{p})$ $\displaystyle
dt\wedge\alpha_{\lambda}\wedge d\alpha_{\lambda}$ $\displaystyle=(t+2)dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}$
$\displaystyle-\frac{t+2}{C}\sum_{p}\phi^{\prime}_{p}dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge PD(\gamma_{p})$
$\displaystyle+\frac{1}{C}\sum_{p}\left(-\phi_{p}-\phi^{\prime}_{p}(1-s)\right)dt\wedge
ds\wedge\widetilde{\beta}_{\lambda+1}\wedge PD(\gamma_{p})$
$\displaystyle-\frac{1}{C^{2}}\sum_{p,p^{\prime}}\left(\phi_{p}\phi^{\prime}_{p^{\prime}}\right)dt\wedge
ds\wedge PD(\gamma_{p})\wedge PD(\gamma_{p^{\prime}})$ $\displaystyle
dt\wedge(\alpha_{\lambda}-3\widetilde{\beta}_{\lambda})\wedge
d\alpha_{\lambda}$ $\displaystyle=(t-1)dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}$
$\displaystyle-\frac{t-1}{C}\sum_{p}\phi^{\prime}_{p}dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge PD(\gamma_{p})$
$\displaystyle+\frac{1}{C}\sum_{p}\left(-\phi_{p}-\phi^{\prime}_{p}(1-s)\right)dt\wedge
ds\wedge\widetilde{\beta}_{\lambda+1}\wedge PD(\gamma_{p})$
$\displaystyle-\frac{1}{C^{2}}\sum_{p,p^{\prime}}\left(\phi_{p}\phi^{\prime}_{p^{\prime}}\right)dt\wedge
ds\wedge PD(\gamma_{p})\wedge PD(\gamma_{p^{\prime}})$

Fix some volume form $dvol_{\Sigma}$. Then

$dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}=h_{\lambda}dt\wedge
ds\wedge dvol_{\Sigma}$

for some nonnegative function $h_{\lambda}$ that vanishes precisely at $B$.
Note that $PD(\gamma_{p})$ is supported outside of the cylinder $R\leq R_{0}$.
On the complement of this cylinder, there is a lower bound on $h_{\lambda}$
independent of $C$. Therefore, the positive 4-form $dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}$
dominates the remaining terms for $C$ sufficiently small on the entire
complement of $R\leq R_{0}$. Moreover, it is the only nonzero term for $R\leq
R_{0}$, where it is nonnegative for all $R$ and strictly positive for $R>0$.

Finally, since $\psi^{\prime}_{p}$ is not supported in $R\leq R_{0}$, we can
leave this computation to Case 2 below.

Case 2: $s\in[2\epsilon,3\epsilon]$. Here, the primitive is

$\alpha_{\lambda}=(t+2)\widetilde{\beta}_{\lambda}+\Psi(s)\widetilde{\beta}_{\lambda+1}$

with exterior derivative

$d\alpha_{\lambda}=dt\wedge\widetilde{\beta}_{\lambda}+\Phi^{\prime}ds\wedge\widetilde{\beta}_{\lambda+1}$

Therefore we have

$\displaystyle d\alpha_{\lambda}\wedge d\alpha_{\lambda}$
$\displaystyle=-\Phi^{\prime}dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}$
$\displaystyle dt\wedge\alpha_{\lambda}\wedge d\alpha_{\lambda}$
$\displaystyle=-\Phi^{\prime}(t+2)dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}$
$\displaystyle dt\wedge(\alpha_{\lambda}-3\widetilde{\beta}_{\lambda})\wedge
d\alpha_{\lambda}$ $\displaystyle=-\Phi^{\prime}(t-1)dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}$

which satisfy the required inequalities since $\Phi^{\prime}<0$.

For $R\leq R_{0}$, since $df=0$ for $R\leq R_{0}$, we have that

$\displaystyle\widetilde{\beta}_{\lambda}$ $\displaystyle=\beta_{\lambda}+fds$
$\displaystyle\widetilde{\beta}_{\lambda+1}$
$\displaystyle=\beta_{\lambda+1}+f(dt-ds)$

so therefore

$\displaystyle\alpha_{\lambda}$
$\displaystyle=(t+2)(\beta_{\lambda}+fds)+\Phi(\beta_{\lambda+1}+f(dt-ds))$
$\displaystyle d\alpha_{\lambda}$
$\displaystyle=dt\wedge\beta_{\lambda}-\Phi^{\prime}ds\wedge\beta_{\lambda+1}+2fdt\wedge
ds$

As a result

$d\alpha_{\lambda}\wedge d\mu_{B}=2f^{2}dt\wedge ds\wedge dx\wedge dy>0$

and

$\displaystyle dt\wedge(\alpha_{\lambda}+\delta\mu_{B})\wedge
d\alpha_{\lambda}$ $\displaystyle=-\Phi^{\prime}dt\wedge
ds\wedge((t+2)\beta_{\lambda}+\delta\mu_{B})\wedge\beta_{\lambda+1}$
$\displaystyle=-\Phi^{\prime}((t+2)R+\delta Q)dt\wedge ds\wedge dx\wedge
dy\geq 0$ $\displaystyle
dt\wedge(\alpha_{\lambda}-3\beta_{\lambda}+\delta\mu_{B})\wedge
d\alpha_{\lambda}$ $\displaystyle=-\Phi^{\prime}dt\wedge
ds\wedge((t-1)\beta_{\lambda}+\delta\mu_{B})\wedge\beta_{\lambda+1}$
$\displaystyle=-\Phi^{\prime}((t-1)R+\delta Q)dt\wedge ds\wedge dx\wedge
dy\leq 0$

where $\mu_{B}\wedge\beta_{\lambda+1}=Qdx\wedge dy$.

Finally

$\displaystyle dt\wedge\alpha_{\lambda}\wedge d\mu_{B}$
$\displaystyle=\left((t+2-\Phi)f^{2}\right)dt\wedge ds\wedge dx\wedge dy>0$
$\displaystyle dt\wedge(\alpha_{\lambda}-3\beta_{\lambda})\wedge d\mu_{B}$
$\displaystyle=\left((t-1-\Phi)f^{2}\right)dt\wedge ds\wedge dx\wedge dy<0$

This suffices to ensure the required inequalities for $R\leq R_{0}$ in this
case.

Case 3: $s\in[3\epsilon,4\epsilon]$. This is where $\Phi,\Phi^{\prime}$ go to
0 and contains the support of $\psi^{\prime}_{q}$ (but not
$\phi^{\prime}_{p}$). Here the primitive is

$\alpha_{\lambda}=(t+2)\widetilde{\beta}_{\lambda}+\Phi(s)\widetilde{\beta}_{\lambda+1}+\frac{1}{C}\sum_{p}\phi_{p}PD(\gamma_{p})-\frac{1}{C}\sum_{q}\psi_{q}PD(\gamma_{q})$

and

$d\alpha_{\lambda}=dt\wedge\widetilde{\beta}_{\lambda}+\Phi^{\prime}ds\wedge\widetilde{\beta}_{\lambda+1}-\frac{1}{C}\sum_{q}\psi^{\prime}_{q}ds\wedge
PD(\gamma_{q})$

since each $\phi_{p}$ is constant in this region.

Therefore

$\displaystyle d\alpha_{\lambda}\wedge d\alpha_{\lambda}$
$\displaystyle=(-\Phi^{\prime})dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}+\frac{1}{C}\sum_{q}\psi^{\prime}_{q}dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge PD(\gamma_{q})$

The first term is positive on the intersection of the supports of
$\Phi^{\prime}$ and
$\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}$. Similarly
the second term is positive on the intersection of the supports of
$\psi^{\prime}_{q}$ and $PD(\gamma_{q})$, since $\phi^{\prime}_{q}\geq 0$ and
$\gamma_{q}$ is positively transverse to the foliation
$\mathcal{F}_{\lambda}$. In particular, this is strictly positive when
$s\in[3\epsilon,4\epsilon]$ and $R\geq R_{0}$.

Next,

$\displaystyle dt\wedge\alpha_{\lambda}\wedge d\alpha_{\lambda}$
$\displaystyle=-(t+2)\Phi^{\prime}dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}$
$\displaystyle-\frac{t+2}{C}\sum_{q}(-\psi^{\prime}_{q})dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge PD(\gamma_{q})$
$\displaystyle-\frac{1}{C}\sum_{q}(\Phi^{\prime}\psi_{q}-\Phi\psi^{\prime}_{q})dt\wedge
ds\wedge\widetilde{\beta}_{\lambda+1}\wedge PD(\gamma_{q})$
$\displaystyle+\frac{1}{C}\sum_{p}(-\Phi^{\prime}\psi_{p})dt\wedge
ds\wedge\widetilde{\beta}_{\lambda+1}\wedge PD(\gamma_{p})$
$\displaystyle+\frac{1}{C^{2}}\sum_{p,q}\phi_{p}\psi^{\prime}_{q}dt\wedge
ds\wedge PD(\gamma_{p})\wedge PD(\gamma_{q})$
$\displaystyle+\frac{1}{C^{2}}\sum_{q,q^{\prime}}\psi_{q^{\prime}}\psi^{\prime}_{q}dt\wedge
ds\wedge PD(\gamma_{q^{\prime}})\wedge PD(\gamma_{q})$

By assumption, the supports of $PD(\gamma_{p}),PD(\gamma_{q})$ are contained
in the support of $\beta_{\lambda}\wedge\beta_{\lambda+1}$. Therefore, we can
write

$\displaystyle dt\wedge ds\wedge\widetilde{\beta}_{\lambda+1}\wedge
PD(\gamma_{p})$ $\displaystyle=g_{p}dt\wedge
ds\wedge\beta_{\lambda}\wedge\beta_{\lambda+1}$ $\displaystyle dt\wedge
ds\wedge\widetilde{\beta}_{\lambda+1}\wedge PD(\gamma_{q})$
$\displaystyle=g_{q}dt\wedge ds\wedge\beta_{\lambda}\wedge\beta_{\lambda+1}$

for some functions $g_{p},g_{q}$ on $\Sigma\times[-5\epsilon,0]$ and

$\displaystyle dt\wedge ds\wedge PD(\gamma_{p})\wedge PD(\gamma_{q})$
$\displaystyle=h_{p,q}dt\wedge ds\wedge\beta_{\lambda}\wedge PD(\gamma_{q})$
$\displaystyle dt\wedge ds\wedge PD(\gamma_{q})\wedge PD(\gamma_{q^{\prime}})$
$\displaystyle=h_{q,q^{\prime}}dt\wedge ds\wedge\beta_{\lambda}\wedge
PD(\gamma_{q^{\prime}})$

for some functions $h_{p,q}$ and $h_{q,q^{\prime}}$ on
$\Sigma\times[0,6\epsilon]$. Consequently, we can group the first, second and
fourth terms as

$\left(-\Phi^{\prime}(t+2)+\frac{1}{C}\sum_{q}g_{q}(\Phi^{\prime}\psi_{q}-\Phi\psi^{\prime}_{q})+\frac{1}{C}\sum_{p}g_{p}(-\Phi^{\prime}\psi_{p})\right)dt\wedge
ds\wedge\beta_{\lambda}\wedge\beta_{\lambda+1}$

Using the relation $\Phi^{\prime}\leq\Phi\leq 0$ and $\psi^{\prime}_{q}\leq
0$, this is bounded below by

$-\Phi^{\prime}\left((t+2)-\frac{1}{C}\sum_{q}|g_{q}|(|\psi_{q}|+|\psi^{\prime}_{q}|)+\frac{1}{C}\sum_{p}g_{p}\psi_{p}\right)dt\wedge
ds\wedge\beta_{\lambda}\wedge\beta_{\lambda+1}$

which is positive for $C$ sufficiently large and $s\leq 4\epsilon$.
Furthermore, we can group the third, fifth and sixth terms as

$-\frac{1}{C}\sum_{q}\psi^{\prime}_{q}\left(t+2-\frac{1}{C}\sum_{p}\phi_{p}h_{p,q}-\frac{1}{C}\sum_{q^{\prime}}\psi_{q^{\prime}}h_{q,q^{\prime}}\right)dt\wedge
ds\wedge\beta_{\lambda}\wedge PD(\gamma_{q})$

which is positive for $C$ sufficiently large.

Next, since

$\displaystyle dt\wedge\widetilde{\beta}_{\lambda}\wedge d\alpha_{\lambda}$
$\displaystyle=-\Phi^{\prime}dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge\widetilde{\beta}_{\lambda+1}-\frac{1}{C}\sum_{q}\psi^{\prime}_{q}dt\wedge
ds\wedge\widetilde{\beta}_{\lambda}\wedge PD(\gamma_{q})$

we have that

$\displaystyle dt\wedge(\alpha_{\lambda}-3\widetilde{\beta}_{\lambda})\wedge
d\alpha_{\lambda}$

is bounded above by

$\displaystyle-\Phi^{\prime}\left((t-1)+\frac{1}{C}\sum_{q}|g_{q}|(|\psi_{q}|+|\psi^{\prime}_{q}|)+\frac{1}{C}\sum_{p}g_{p}\psi_{p}\right)dt\wedge
ds\wedge\beta_{\lambda}\wedge\beta_{\lambda+1}$
$\displaystyle-\frac{1}{C}\sum_{q}\psi^{\prime}_{q}\left(t-1-\frac{1}{C}\sum_{p}\phi_{p}h_{p,q}-\frac{1}{C}\sum_{q^{\prime}}\psi_{q^{\prime}}h_{q,q^{\prime}}\right)dt\wedge
ds\wedge\beta_{\lambda}\wedge PD(\gamma_{q})$

which for $C$ sufficiently large is nonnegative everywhere and strictly
positive when $s\in[3\epsilon,4\epsilon]$ and $R\geq R_{0}$.

As in Case 1 above, since $\psi_{q}$ is not supported in $R\leq R_{0}$ and
$\psi_{p}$ is not supported when $R\leq R_{0}$ and
$s\in[3\epsilon,4\epsilon]$, this computation reduces to Case 2 above.

Case 4: $s\geq 4\epsilon$. This is where $\Phi$ vanishes. Here, the primitive
is

$\alpha_{\lambda}=(t+2)\widetilde{\beta}_{\lambda}+\frac{1}{C}\mu_{H}$

and

$d\alpha_{\lambda}=dt\wedge\widetilde{\beta}_{\lambda}+\frac{1}{C}d\mu_{H}$

So

$d\alpha_{\lambda}\wedge
d\alpha_{\lambda}=\frac{1}{C}dt\wedge\widetilde{\beta}_{\lambda}\wedge
d\mu_{H}$

which by construction is positive. In addition, we have that

$\displaystyle dt\wedge\alpha_{\lambda}\wedge d\alpha_{\lambda}$
$\displaystyle=\frac{1}{C}\left(dt\wedge\left((t+2)\widetilde{\beta}_{\lambda}+\frac{1}{C}\mu_{H}\right)\wedge
d\mu_{H}\right)$

which for $C$ sufficiently large is positive multiple of a perturbation of

$(t+2)dt\wedge\widetilde{\beta}_{\lambda}\wedge d\mu_{H}>0$

Similarly, the form

$dt\wedge(\alpha_{\lambda}-3\widetilde{\beta}_{\lambda})\wedge\alpha_{\lambda}=\frac{1}{C}\left(dt\wedge\left((t-1)\widetilde{\beta}_{\lambda}+\frac{1}{C}\mu_{H}\right)\wedge
d\mu_{H}\right)$

is a positive multiple of a perturbation of

$(t-1)dt\wedge\widetilde{\beta}_{\lambda}\wedge d\mu_{H}<0$

for $C$ sufficiently large.

For $R\leq R_{0}$, we can check that

$\displaystyle d\alpha_{\lambda}\wedge d\mu_{B}$
$\displaystyle=dt\wedge\widetilde{\beta}_{\lambda}\wedge dx\wedge dy$
$\displaystyle dt\wedge\alpha_{\lambda}\wedge d\mu_{B}$
$\displaystyle=(t+2)dt\wedge\widetilde{\beta}_{\lambda}\wedge dx\wedge
dy+\frac{1}{C}dt\wedge\mu_{H}\wedge dx\wedge dy$ $\displaystyle
dt\wedge(\alpha_{\lambda}+\mu_{B})\wedge d\alpha_{\lambda}$
$\displaystyle=\frac{1}{C}\left((t+2)dt\wedge\widetilde{\beta}_{\lambda}\wedge
d\mu_{H}+dt\wedge\mu_{B}\wedge d\mu_{H}+\frac{1}{C}dt\wedge\mu_{H}\wedge
d\mu_{H}\right)$ $\displaystyle
dt\wedge(\alpha_{\lambda}-3\widetilde{\beta}_{\lambda})\wedge d\mu_{B}$
$\displaystyle=(t-1)dt\wedge\widetilde{\beta}_{\lambda}\wedge dx\wedge
dy+\frac{1}{C}dt\wedge\mu_{H}\wedge dx\wedge dy$ $\displaystyle
dt\wedge(\alpha_{\lambda}-3\beta_{\lambda}+\mu_{B})\wedge d\alpha_{\lambda}$
$\displaystyle=\frac{1}{C}\left((t-1)dt\wedge\widetilde{\beta}_{\lambda}\wedge
d\mu_{H}+dt\wedge\mu_{B}\wedge d\mu_{H}+\frac{1}{C}dt\wedge\mu_{H}\wedge
d\mu_{H}\right)$

The first, second and fourth terms are positive for $C$ sufficiently large.
Moreover, the second dominates the third and the fourth dominates the fifth
for $C$ sufficiently large. ∎

##  5\. 4-dimensional sectors

###  5.1. Contact structures on the 3-manifold pieces

At this point, we have constructed a symplectic form on $\nu(H_{1}\cup
H_{2}\cup H_{3})$. We now want to construct an inward-pointing Liouville
vector field $\rho_{\lambda}$ along each boundary component
$\widehat{Y}_{\lambda}$. It is equivalent to constructing a primitive
$\alpha_{\lambda}$ for $\omega$ in a neighborhood of $\widehat{Y}_{\lambda}$
that restricts to a contact form. Then this 1-form $\alpha_{\lambda}$ is
$\omega$-dual to the required Liouville vector field.

We now define the following 1-form over $\widehat{Y}_{\lambda}$:

$\displaystyle\widehat{\alpha}_{\lambda}$
$\displaystyle:=\begin{cases}\alpha_{\lambda}&\text{over
$\nu(H_{\lambda})$}\\\ \mu_{\Sigma,\lambda}+\delta\mu_{B}&\text{over
$\nu(\Sigma)$}\\\ \alpha_{\lambda-1}-3\widetilde{\beta}_{\lambda-1}&\text{over
$\nu(H_{\lambda-1})$}\end{cases}$

and 2-form

$\omega:=\begin{cases}d\alpha_{\lambda}&\text{over $\nu(H_{\lambda})$}\\\
\omega_{0}+d\mu_{B}&\text{over $\nu(\Sigma)$}\end{cases}$

Combining Corollary 4.5 and Proposition 4.6 yields

######  Proposition 5.1.

The 2-form $\omega$ is well-defined and symplectic on $\nu(H_{1}\cup H_{2}\cup
H_{3})$. Moreover, the form $\widehat{\alpha}_{\lambda}$ is well-defined, is a
primitive for $\omega$, and is a contact form along $\widehat{Y}_{\lambda}$.

###  5.2. Capping off with symplectic fillings

The last remaining piece is to fill in the three concave ends of
$(\nu(H_{1}\cup H_{2}\cup H_{3}),\omega)$ with Weinstein 1-handlebodies to
obtain a closed symplectic 4-manifold.

We fix a standard model $(W_{k},\omega_{k},\rho_{k},\phi_{k})$ of a
4-dimensional Weinstein 1-handlebody, where

  1. (1)

$W_{k}$ has a handle decomposition consisting of one 0-handle and $k$
1-handles for $k\geq 0$,

  2. (2)

$\omega_{k}$ is an exact symplectic form,

  3. (3)

$\phi_{k}$ is a proper, exhausting Morse function with exactly one index-0
critical point and exactly $k$ index-1 critical points, and

  4. (4)

$\rho_{k}$ is a complete Liouville vector field that is gradient-like for the
function $\phi_{k}$.

By translation, we can assume that all critical values of $\phi_{k}$ are
negative, so that the level set $Y_{k}=\phi^{-1}_{k}(0)$ is homeomorphic to
either $S^{3}$ (if $k=0$) or $\\#_{k}S^{1}\times S^{2}$ (if $k\geq 1$). Let
$\alpha_{k}=\omega_{k}(\rho_{k},-)$ be the induced contact form and $\xi_{k}$
the corresponding contact structure. The symplectization $(\mathbb{R}\times
Y,d(e^{t}\alpha_{k}))$ embeds symplectically into $(W_{k},\omega_{k})$ by
flowing along the Liouville vector field $\rho_{k}$.

######  Proposition 5.2.

If the contact structure $(\widehat{Y}_{\lambda},\widehat{\xi}_{\lambda})$
induced by the contact form of Proposition 5.1 is tight, then it can be filled
in by a symplectic 1-handlebody.

###### Proof.

There is a unique tight contact structure on $S^{3}$ and on
$\\#_{k}S^{1}\times S^{2}$ for each $k\geq 1$, which is contactomorphic to the
standard model $(Y_{k},\xi_{k})$. Choose an identification
$(\widehat{Y}_{\lambda},\widehat{\xi}_{\lambda})$ with $(Y_{k},\xi_{k})$ for
the appropriate $k\geq 0$. There exists some positive function
$f:\widehat{Y}_{\lambda}\rightarrow\mathbb{R}$ such that

$\widehat{\alpha}_{\lambda}=f\alpha_{k}$

Consequently, we can symplectically identify a neighborhood of
$\widehat{Y}_{\lambda}\subset(X,\omega)$ with a neighborhood of the graph of
$\log f$ in the symplectization $(\mathbb{R}\times Y_{k},d(e^{t}\alpha_{k}))$.
We can then smoothly identify the 4-dimensional sector $\widehat{Z}_{\lambda}$
of the trisection with the sublevel set of the graph of $\log f$ to obtain a
symplectic structure extending the one we constructed on the spine. ∎

######  Proposition 5.3.

An $n$-parameter family of contact forms $\alpha_{\lambda}$ determines a
$n$-parameter family of symplectic fillings.

###### Proof.

An $n$-parameter family of contact forms $\widehat{\alpha}_{\lambda,t}$
corresponds to a family of positive functions of the form

$\widehat{\alpha}_{\lambda,t}=f_{t}\alpha_{k}$

This determines a family of graphs in $W_{k}$ and therefore a family of
diffeomorphisms of $\widehat{Z}_{\lambda}$ into $W_{k}$. Pulling back, we
obtain a family of symplectic forms on $\widehat{Z}_{\lambda}$. ∎

######  Remark 5.4.

The theorem of Laudenbach and Poenaru implies that each 4-dimensional sector
$Z_{\lambda}$ can be glued in uniquely up to diffeomorphism. Similarly, the
symplectic structure we construct on $Z_{\lambda}$ is essentially unique as
there is a unique tight contact structure on $Y_{k}$, which has a unique
Weinstein filling up to deformation [CE12, Theorem 16.9(c)].

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