# Periodic leaf-wise intersection points from Lagrangians

Sara Venkatesh

###### Abstract

We investigate leaf-wise intersection points on hypersurfaces of contact type
in monotone symplectic manifolds. We show that monotone Floer-essential
Lagrangians detect periodic leaf-wise intersection points in hypersurfaces of
contact type whose Reeb flow is Zoll. Examples include the prequantization
bundles appearing in monotone toric negative line bundles. Generalizing, we
prove the existence of leaf-wise intersection points for certain annulus
subbundles in weak+-monotone negative line bundles, not necessarily toric. The
proofs combine reduced symplectic cohomology with the original methods
employed by Albers-Frauenfelder to prove global existence results of this
kind.

###### Contents

  1. 1 Introduction
    1. 1.1 Outline of paper
    2. 1.2 Acknowledgements
  2. 2 Floer theory and symplectic cohomology
    1. 2.1 Hamiltonians with truncated support
    2. 2.2 Leaf-wise intersections in Floer theory
  3. 3 Auxiliary Hamiltonians and a proof sketch
    1. 3.1 The Hamiltonians
      1. 3.1.1 $H_{n}$ and $K_{n}$
      2. 3.1.2 $\check{H}_{m,n}$ and $\check{K}_{m,n}$
    2. 3.2 Sketch of proof
  4. 4 Proof of Theorem 1
    1. 4.1 A chain of isomorphisms
      1. 4.1.1 The isomorphism (1)
      2. 4.1.2 The isomorphisms (2) and (5)
      3. 4.1.3 The isomorphisms (3) and (4)
    2. 4.2 Vanishing computations
      1. 4.2.1 Proof that $\widehat{SH_{*}}(K)=0$
    3. 4.3 Proof of Theorem 1
  5. 5 Other line bundles
  6. 6 Leaf-wise intersection points from Lagrangians
    1. 6.1 Lagrangian quantum cohomology
    2. 6.2 Proof of Theorem 8
  7. A Hamiltonians with smaller time-support

##  1 Introduction

The concept of a leaf-wise intersection point was first introduced by Moser in
his 1978 paper [10], where he studied certain types of fixed points that occur
when one perturbs a dynamical system. Imagine a particle moving along a
trajectory, as prescribed by some dynamical system, when suddenly a
perturbation disturbs the system: the particle is thrown off course. After
some finite time, the perturbation ceases and the dynamical system returns to
its original form. Is there any chance that, miraculously, the particle now
finds itself back on its original trajectory? In the rare case that this
phenomenon occurs, it is characterized by two points: the point at which the
particle first leaves the unperturbed trajectory, and the point at which the
particle returns to it. In other words, the two points are intersection points
between a trajectory of the original dynamical system and a trajectory of the
system described by the finite-time perturbation. In the case where two such
intersection points exist, we say that a leaf-wise intersection point exists.

In symplectic geometry, leaf-wise intersection points naturally occur between
symplectic flows. Moser himself worked in an exact symplectic framework,
studying hypersurfaces of contact type in Liouville domains equipped with a
dynamical system induced by choice of contact form. He showed that, for a
sufficiently small perturbation given by an exact symplectomorphism, such a
hypersurface always contains a leaf-wise intersection point.

Moser’s framework has been extended in many directions. In [3], Albers-
Frauenfelder studied leaf-wise intersection points of hypersurfaces $\Sigma$
of restricted contact type in Liouville domains. They showed that the number
of leaf-wise intersection points of a sufficiently-small Hamiltonian occurring
on $\Sigma$ is bounded below by the sum of the Betti numbers of $\Sigma$. More
generally, they showed that the existence of a leaf-wise intersection point of
a fixed Hamiltonian, regardless of size, is guaranteed if an algebraic
invariant called Rabinowitz Floer homology is non-vanishing on $\Sigma$.

This paper states the first results for leaf-wise intersection points on
hypersurfaces of contact type within non-exact symplectic manifolds. Following
Albers-Frauenfelder, we probe for leaf-wise intersection points using various
fixed-point Floer theories. The only examples of such manifolds for which
fixed-point Floer theory has been carefully examined are negative line bundles
[4][12]. The first calculations are due to Ritter, who calculated another
Floer invariant, called _symplectic cohomology_ , for these line bundles in
[12]. It is for these manifolds, through the lens of symplectic cohomology,
that we prove our first result.

A negative line bundle $(E,\Omega)$ with negativity constant $k$ over a
symplectic manifold $(M,\omega)$ is a complex line bundle

$\mathfrak{p}:E\rightarrow M$

whose Chern class satisfies

$c_{1}^{E}=-k[\omega]$

for some $k>0$. A choice of Hermitian metric picks out a family of contact
hypersurfaces $\\{\Sigma_{r}\\}_{r>0}$, where $\Sigma_{r}$ is the circle
subbundle appearing at radius $r$. Recall that a symplectic manifold
$(M,\omega)$ is monotone of monotonicity $c$ if

$c_{1}^{TM}=c[\omega]$

for some $c>0$.

###### Theorem 1

Let $E\rightarrow M$ be a negative line bundle with negativity constant $k$
over a monotone symplectic manifold with monotonicity constant $c>k$. If the
symplectic cohomology of $E$ is non-zero, then for any compactly-supported
Hamiltonian, the radius-$\frac{1}{\sqrt{k(c-k)\pi}}$ circle subbundle contains
a leaf-wise intersection point.

  * Remark 1)

Ritter showed in [12] that the symplectic cohomology of $E$ is non-zero
precisely when the action of $\mathfrak{p}^{*}c_{1}^{E}$ by quantum cup
product on the quantum cohomology $QH^{*}(E)$ is not nilpotent. In particular,
this action is non-nilpotent when the base $M$ is a toric symplectic manifold
[13].

###### Corollary 1

Let $E\rightarrow M$ be a negative line bundle with negativity constant $k$
over a monotone toric symplectic manifold with monotonicity constant $c>k$.
For any compactly-supported Hamiltonian, the
radius-$\frac{1}{\sqrt{k(c-k)\pi}}$ circle subbundle contains a leaf-wise
intersection point.

  * Example 1)

The line bundle $\mathcal{O}(-n)\rightarrow\mathbb{C}P^{m}$ satisfies the
conditions of Corollary 1 whenever $0<n\leq m$.

The proof is inspired by the methods of Albers-Frauenfelder; it uses the non-
vanishing of a Rabinowitz-Floer-esque invariant on
$\Sigma_{\frac{1}{\sqrt{k(c-k)\pi}}}$. This invariant vanishes on all other
circle subbundles, and we conjecture that Theorem 1 does not hold at any other
radius: for any other radius $r$ there exist Hamiltonians with no leaf-wise
intersection points on $\Sigma_{r}$.

The radius-$\frac{1}{\sqrt{k(c-k)\pi)}}$ circle bundle has appeared in other
contexts as a “special” circle bundle. In [14], Ritter-Smith showed that, if
the base $M$ is toric, the wrapped Fukaya category of $E$ is split-generated
by a single Lagrangian torus inside $\Sigma_{\frac{1}{\sqrt{k(c-k)\pi}}}$,
equipped with particular choices of local systems. The presence of this
Lagrangian implies that the Rabinowitz Floer invariant is non-zero [18],
which, as we will see in the proof of Theorem 1, in turn implies the existence
of a leaf-wise intersection point. Thus, Floer-essential Lagrangians detect
leaf-wise intersection points. A technical requirement for these arguments is
that the Reeb flow of a circle subbundle is periodic of constant period, that
is, that each circle subbundle is Zoll.

###### Theorem 2

Let $E$ be a monotone symplectic manifold that is either compact or convex at
infinity. Let $\Sigma\subset E$ be a compact hypersurface of contact type with
Zoll Reeb flow, containing a compact monotone Lagrangian $L$. If there exists
a local system $\gamma$ such that the Lagrangian Floer homology
$HF^{*}(L,\gamma)$ is non-vanishing, then for any compactly-supported
Hamiltonian, $\Sigma$ contains a leaf-wise intersection point.

  * Example 2)

A silly example of this phenomenon comes from the Clifford torus $L$ in
$\mathbb{C}P^{1}$. This is a monotone, Floer essential Lagrangian, and so it
cannot be displaced from itself by a Hamiltonian flow $\phi_{H}^{1}$
corresponding to a given Hamiltonian $H$. $L$ is also a contact hypersurface,
with periodic Reeb flow. The intersection points of $\phi_{H}^{1}(L)$ with $L$
are precisely the leaf-wise intersection points of $H$.

A corollary to the proof of Theorem 2 removes the Zoll condition, at the
expense of a precise statement.

###### Corollary 2

Let $\Sigma\subset E$ be a compact hypersurface of contact type in a monotone
symplectic manifold that is either compact or convex at infinity. If $\Sigma$
contains a compact monotone Lagrangian $L$ such that $HF^{*}(L,\gamma)\neq 0$
for some local system $\gamma$, then for any compactly-supported Hamiltonian,
there exists a hypersurface of contact type arbitrarily close to $\Sigma$ with
a leaf-wise intersection point.

We restate Theorem 2 and Corollary 2 together in Section 6 as Theorem 8.

The proofs of Theorems 1 and 2 require monotonicity of the underlying manifold
in different ways. In Theorem 2, monotonicity is required to define a closed-
open map from the Rabinowitz Floer invariant to the Lagrangian quantum
cohomology of $L$. The latter has only been rigorously defined for monotone
Lagrangians. Using the machinery of [8], it should be possible to amend
Theorem 2 to more general classes of symplectic manifolds and Floer-essential
Lagrangians. The proof of Theorem 1 uses monotonicity as a technical
assumption that allows us to precisely specify which hypersurface of contact-
type contains a leaf-wise intersection point (see Lemma 11).

The constant $\frac{1}{\sqrt{k(c-k)\pi}}$ has an interesting alternative
characterization. As mentioned in Remark Remark 1), the class
$\mathfrak{p}^{*}c_{1}^{E}$ acts on the quantum cohomology $QH^{*}(E)$ via
quantum cup product. The largest eigenvalue $\lambda$ of this action has a
valuation

$ev(\lambda)=\frac{1}{{(c-k)}}.$

Relaxing the monotonicity condition in Theorem 1 yields an imprecise result in
terms of this eigenvalue.

###### Theorem 3

Let $E$ be a weak+-monotone negative line bundle. Let $\lambda$ be a largest
non-zero eigenvalue of the action of $\mathfrak{p}^{*}c_{1}^{E}$ by quantum
cup product on $QH^{*}(E)$. For any compactly-supported Hamiltonian, there
exists a circle subbundle $\Sigma_{r}$ with
$r\in\left(\sqrt{\frac{ev(\lambda)}{k\pi}},\infty\right)$ such that
$\Sigma_{r}$ has a leaf-wise intersection point.

###  1.1 Outline of paper

In Section 2 we recall the definitions of Hamiltonian Floer theory and
symplectic cohomology and set notation. We discuss capturing leaf-wise
intersection points through Floer theory. In Section 3 we introduce auxiliary
Hamiltonians and sketch the proof of Theorem 1. In Section 4 we prove Theorem
1. In Section 5 we discuss a modified Floer theory and prove Theorem 3. In
Section 6 we introduce Lagrangian quantum cohomology and prove Theorem 2 and
Corollary 2, repackaged as Theorem 8.

###  1.2 Acknowledgements

This work is supported by the National Science Foundation under Award No.
1902679.

##  2 Floer theory and symplectic cohomology

Let $\Sigma$ be a contact manifold with a fixed contact form $\alpha$. The
symplectization of $\Sigma$ is the manifold $(0,\infty)\times\Sigma$, equipped
with the symplectic form $d(r\alpha)$, where $r$ is the coordinate on
$(0,\infty)$. Let $(E,\Omega)$ be a symplectic manifold, and assume that $E$
is either closed or there is a subset of the symplectization
$[C,\infty)\times\Sigma$ such that $E$ is symplectomorphic to
$[C,\infty)\times\Sigma$ outside of a compact set. We define three Floer
theories on $E$ over the Novikov field over $\mathbb{Z}/2\mathbb{Z}$, denoted
by $\Lambda$ and defined by

$\Lambda=\left\\{\sum_{i=0}^{\infty}c_{i}T^{\alpha_{i}}\hskip
5.69046pt\bigg{|}\hskip
5.69046ptc_{i}\in\mathbb{Z}/2\mathbb{Z},\mathbb{R}\ni\alpha_{i}\rightarrow\infty\right\\}.$

$T$ is a formal variable of degree equal to two times the minimum Chern number
of $E$ on spheres:

$|T|=2\cdot\min_{\begin{subarray}{c}\beta\in\pi_{2}(E)\\\
c_{1}^{TE}(\beta)\neq 0\end{subarray}}|c_{1}^{TE}(\beta)|.$

$\Lambda$ has a valuation $ev:\Lambda\rightarrow\mathbb{R}\cup{\infty}$, given
by

$ev\left(\sum_{i=0}^{\infty}c_{i}T^{\alpha_{i}}\right)=\min_{c_{i}\neq
0}\alpha_{i}.$

A Hamiltonian $H:E\times S^{1}\rightarrow\mathbb{R}$ has an associated
Hamiltonian vector field $X_{H}$, defined by

$dH(-)=\Omega(-,X_{H}).$

We denote the set of time-one orbits $x:S^{1}\rightarrow E$ of $X_{H}$ by
${\cal P}(H)$. If the time-one orbits of $X_{H}$ satisfy a non-degeneracy
condition, then an additional generic choice of $\Omega$-compatible almost-
complex structure $J\in End(TE)$ produces a graded vector space generated by
these orbits. The graded vector space is called the Floer complex and is
denoted by

$CF^{*}(H,J):=\bigoplus_{\begin{subarray}{c}x\in{\cal P}(H)\\\
\dot{x}=X_{H}(x)\end{subarray}}\Lambda\langle x\rangle.$

A choice of grading is determined by the Conley-Zehnder index $\mu_{CZ}$ on a
lift of ${\cal L}E$ (see [5] for a definition of the Conley-Zehnder index).
Choose a capping disk $\tilde{x}$ for every $X_{H}$-periodic orbit $x$, and
define $|x|=\mu_{CZ}(\tilde{x})$. Then, for example,

$|T^{\alpha}x|=\alpha\cdot|T|+\mu_{CZ}(\tilde{x})$

gives an $\mathbb{R}$-grading on $CF^{*}(H,J)$. Up to chain isomorphism, the
Floer complex is independent of choice of capping disk.

Let $x_{+},x_{-}$ be two periodic orbits of $X_{H}$. Fix a homotopy class
$\beta\in\pi_{2}(E)$. Let $\widehat{{\cal M}}_{\beta}(x_{-},x_{+})$ be the
moduli space of maps $u:\mathbb{R}\times S^{1}\rightarrow E$ satisfying
Floer’s equation

(1) $\frac{\partial u}{\partial s}+J\left(\frac{\partial u}{\partial
t}-X_{H}\right)=0,$

with asymptotes

$\lim_{s\rightarrow\pm\infty}u(s,t)=x_{\pm}(t),$

and such that the sphere formed by the connect sum
$\tilde{x}_{+}\\#u\\#(-\tilde{x}_{-})$ represents the homotopy class $\beta$:

$[\tilde{x}_{+}\\#u\\#(-\tilde{x}_{-})]=\beta\in\pi_{2}(E).$

$\mathbb{R}$ acts by translation on each $u$; let ${\cal
M}_{\beta}(x_{-},x_{+})$ be the quotient of $\widehat{{\cal
M}}_{\beta}(x_{-},x_{+})$ by this $\mathbb{R}$ action. ${\cal
M}_{\beta}(x_{+},x_{-})$ is a zero-dimensional manifold when

$|x_{-}|-|x_{+}|+2c_{1}^{TE}(\beta)=1.$

If Gromov compactness is satisfied, ${\cal M}_{\beta}(x_{+},x_{-})$ is
compact. This happens under our assumptions if, for instance, $E$ is monotone.
The Floer differential is defined as

$\partial^{fl}(x_{+})=\sum_{\begin{subarray}{c}x_{-}\in{\cal P}(H)\\\
\beta\in\pi_{2}(E)\\\
|x_{-}|-|x_{+}|+2c_{1}^{TE}(\beta)=1\end{subarray}}\sum_{u\in{\cal
M}_{\beta}(x_{+},x_{-})}T^{\beta}x_{-}$

and satisfies $(\partial^{fl})^{2}=0$. Thus, $CF^{*}(H,J)$ is a cochain
complex.

We now specialize to monotone negative line bundles. Let $(E,\Omega)$ be a
complex line bundle over a closed symplectic manifold $(B,\omega)$ with bundle
map denoted by

$\mathfrak{p}:E\longrightarrow B.$

The line bundle $E$ is negative if there exists a constant $k>0$ such that

$c_{1}^{E}=-k[\omega].$

Assume that $B$ is monotone, so that there exists a constant $c>0$ (the
monotonicity constant) such that

$c_{1}^{TB}=c[\omega].$

Oancea proved in [11] that there exists a Hermitian metric on $E$ with
curvature form $\alpha$ such that

$d\alpha=\mathfrak{p}^{*}\omega.$

Let $r$ be the radial coordinate on the complex fibers with respect to this
metric. $E$ has a symplectic form $\Omega$ that, on the complement of the zero
section, is written in coordinates as

$\Omega=d((1+k\pi r^{2})\alpha).$

$\Omega$ extends smoothly over the zero section as the area form in the fiber
direction and $\omega$ in the direction of the base. It is straight-forward to
check that this extension, also denoted by $\Omega$, is a symplectic form. If
$k<c$, $\Omega$ is monotone with monotonicity constant

$\kappa:=c-k.$

Fix a sequence $\\{0=R_{0}<R_{1}<R_{2}<\dots\\}$ converging to some fixed
$R\in\mathbb{R}_{>0}$. Let
$\\{h_{i}:[0,\infty)\rightarrow\mathbb{R}\\}_{i\in\mathbb{N}}$ be functions
defined inductively by convexly smoothing the piecewise linear function

$h_{i}(r)=\left\\{\begin{array}[]{cc}h_{i-1}(r)&r<R_{i}\\\
(i+\frac{1}{2})r+h_{i-1}(R_{i})-(i+\frac{1}{2})R_{i}&r\geq
R_{i}\end{array}\right..$

We define

$h_{0}(r)=\frac{1}{2}r.$

Assume without loss of generality that the sequence $\\{R_{i}\\}$ converges
quickly enough that the family $\\{h_{i}\big{|}_{[0,R]}\\}$ is universally
bounded. See Figure 1.

$\mathbb{R}$$r$$R_{1}$$R_{2}$$R_{3}$$h_{0}$$h_{1}$$h_{2}$$h_{3}$... Figure 1:
The family of functions $\\{h_{i}(r)\\}$

Let $G:M\rightarrow\mathbb{R}$ be a Morse function. On the complement of the
zero section, $E$ is symplectomorphic to $(1,\infty)\times\Sigma$, where
$\Sigma$ is the radius-one $S^{1}$-subbundle over $B$. Let $(r,x)$ be
coordinates on $(1,\infty)\times\Sigma$. For each $i\in\mathbb{N}$ define a
Hamiltonian $H_{i}:(1,\infty)\times\Sigma\rightarrow\mathbb{R}$ by

$H_{i}((r,x))=h_{i}(k\pi r^{2})+(1+k\pi r^{2})\mathfrak{p}^{*}G((r,x)).$

Extend $H_{i}$ to $E$ by setting

$H_{i}=G$

on the zero-section. For each $i\in\mathbb{Z}_{<0}$, similarly define
$H_{i}:E\rightarrow\mathbb{R}$ through

$H_{i}\big{|}_{(1,\infty)\times\Sigma}=-h_{-i-1}(k\pi r^{2})+(1+k\pi
r^{2})\mathfrak{p}^{*}G((r,x)).$

The periodic orbits of $H_{i}$ are not all non-degenerate; they are either

  1. 1.

constant, non-degenerate orbits on the zero section, or

  2. 2.

transversally non-degenerate $S^{1}$-orbits: orbits that come in $S^{1}$
families.

The Floer differential $\partial^{fl}$ is modified by Morse-Bott methods to
create a well-defined complex $CF^{*}(H_{i})$, as in the methods defined for
autonomous Hamiltonians in [6]. We refer there for details, or to [17] for the
study of autonomous Hamiltonians in the context of negative line bundles.

There are continuation maps $c_{i}:CF^{*}(H_{i})\rightarrow CF^{*}(H_{i+1})$
for each $i\in\mathbb{Z}$ producing a directed system

$\dots\xrightarrow{c_{i-2}}CF^{*}(H_{i-1})\xrightarrow{c_{i-1}}CF^{*}(H_{i})\xrightarrow{c_{i}}CF^{*}(H_{i+1})\xrightarrow{c_{i+1}}\dots.$

Continuation maps commute with the differential; thus, both the direct limit
and the inverse limit of the directed system are chain complexes. The
following useful Lemma is proved in [17].

###### Lemma 1

When $i\geq 0$, the continuation map $c_{i}$ can be chosen to be the canonical
inclusion, taking an $S^{1}$-orbit $x:S^{1}\rightarrow E$ to itself. When
$i<-1$, the continuation map $c_{i}$ can be chosen to be the canonical
projection.

Define the symplectic cohomology of the Hamiltonians $\\{H_{i}\\}$, denoted by
$SH^{*}(H)$, as the homology of the direct limit

$SC^{*}(H)=\lim_{0\leq i\rightarrow\infty}CF^{*}(H_{i}).$

The condition $0\leq i$ is superfluous, but it will allow for cleaner
formulations in proofs.

The Floer complexes $CF^{*}(H_{i})$ have a filtration by action. Recall that
the index of a periodic orbit $x$ of the Hamiltonian vector field $X_{H_{i}}$
is defined by fixing a capping disk $\tilde{x}$. With this choice made, the
action of $x$ is defined to be

${\cal A}_{H_{i}}(x)=-\int_{D}\tilde{x}^{*}\Omega+\int_{0}^{1}H_{i}(x(t))dt.$

The action of a sum $X=\sum\limits_{j=0}^{n}C_{j}x_{j}\in CF^{*}(H_{i})$ is
given by

(2) ${\cal A}_{H_{i}}(X)=\min_{C_{j}\neq 0}\left(ev(C_{j})+{\cal
A}(x_{j})\right).$

Action is increased by the Floer differential, and so

$CF^{*}_{(a,\infty)}(H_{i}):=\left\\{X\in CF^{*}(H_{i})\hskip
5.69046pt\bigg{|}\hskip 5.69046pt{\cal A}_{H_{i}}(X)>a\right\\}$

is a subcomplex of $CF^{*}(H_{i})$. Continuation maps may be chosen to
increase action; with these choices,

$SC_{(a,\infty)}^{*}(H):=\lim_{0\leq
i\rightarrow\infty}CF^{*}_{(a,\infty)}(H_{i})$

is a subcomplex of $SC^{*}(H)$.

Similarly, taking inverse limits, the complex

$SC^{(a,\infty)}_{*}(H):=\lim_{\begin{subarray}{c}\leftarrow\\\
i<0\end{subarray}}CF^{*}_{(a,\infty)}(H_{i})$

is a subcomplex of

$SC_{*}(H):=\lim_{\begin{subarray}{c}\leftarrow\\\
i<0\end{subarray}}CF^{*}(H_{i}).$

If $a>a^{\prime}$, there are inclusion maps

$SC^{*}_{(a,\infty)}(H)\hookrightarrow SC^{*}_{(a^{\prime},\infty)}(H)$

and

$SC^{(a,\infty)}_{*}(H)\hookrightarrow SC^{(a^{\prime},\infty)}_{*}(H).$

Thus, the action-filtered subcomplexes form directed systems. We take the
direct limit over all action windows $(a,\infty)$. As direct limits commute
with each other,

$\lim_{\begin{subarray}{c}\rightarrow\\\ a\end{subarray}}\lim_{0\leq
i\rightarrow\infty}CF^{*}_{(a,\infty)}(H_{i})=\lim_{0\leq
i\rightarrow\infty}\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}CF^{*}_{(a,\infty)}(H_{i})\simeq SC^{*}(H).$

However,

(3) $\widehat{SC_{*}}(H):=\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
i<0\end{subarray}}CF^{*}_{(a,\infty)}(H_{i})$

is not necessarily equal to $SC_{*}(H)$. Denote the homology of (3) by
$\widehat{SH^{*}}(H)$, and call it completed symplectic homology.

The following computation was performed by Ritter in [12].

######  Theorem 4 (Ritter [13])

The symplectic cohomology of a monotone negative line bundle over a toric base
is non-vanishing:

$SH^{*}(H)\neq 0.$

Ritter’s method of proof implies the dual theorem:

$SH_{*}(H)\neq 0.$

However, these non-vanishing theorems do not always hold for completed
symplectic homology.

###### Theorem 5

The completed symplectic homology of a monotone negative line bundle vanishes
for small enough radii:

$\widehat{SH_{*}}(H)=0$

whenever $R<\frac{1}{\sqrt{k\kappa\pi}}$.

  * Remark 2)

Theorem 5 is reminiscent of Theorem 1.2 in [4], which proved a non-vanishing
result for a related homology theory termed Rabinowitz Floer homology. Theorem
5 is also the natural extension of Theorem 1 in [19]. The proof employs, with
very little modification, the methods developed in [4].

###  2.1 Hamiltonians with truncated support

In preparation for setting up Floer theory for leaf-wise intersection points,
consider the following modification to the Floer chain complex. Let
$\rho:\mathbb{R}\rightarrow\mathbb{R}$ be a bump function with support in
$[0,1/2]$ that integrates to one:

$\int_{\mathbb{R}}\rho(t)dt=1.$

Define Hamiltonians

${\cal H}_{n}=\rho(t)H_{n}.$

The periodic orbits of ${\cal H}_{n}$ are precisely those of $H_{n}$. Standard
results in Floer theory, which we relegate to the Appendix, show that the
symplectic chain complex $SC^{*}({\cal H})$ is well-defined, and the non-
vanishing of $SH^{*}(H)$ in Theorem 4 implies the non-vanishing

(4) $SH^{*}({\cal H})\neq 0$

where $SH^{*}({\cal H})$ is the symplectic cohomology defined using the
Hamiltonians $\\{{\cal H}_{n}\\}$.

Analogously to Theorem 5,

###### Theorem 6

The completed symplectic homology of ${\cal H}$ vanishes whenever
$R<\frac{1}{\sqrt{k\kappa\pi}}:$

$\widehat{SH_{*}}({\cal H})=0.$

###  2.2 Leaf-wise intersections in Floer theory

Leaf-wise intersection points were first characterized through Floer theory by
Albers-Frauenfelder in [3]. Kang subsequently adjusted their framework to
study leaf-wise intersection points through symplectic cohomology [9]. In this
section, we recall their methods and prove some necessary, albeit technical,
Lemmas on perturbing away degenerate orbits.

Let $F:E\times\mathbb{R}\rightarrow\mathbb{R}$ be a compactly supported
Hamiltonian. Let $\Sigma_{R}=\\{R\\}\times\Sigma$ be the circle subbundle at
radius $R$. A leafwise intersection point of $F$ on $\Sigma_{R}$ is given by a
flow line of the Hamiltonian flow corresponding to $F$

$x:[0,1]\rightarrow E$

where $x(0)$ and $x(1)$ lie on the same Reeb trajectory in $\Sigma_{R}$. The
Reeb trajectories transcribe the fibers of the subbundle $\Sigma_{R}$; $x(1)$
is a leaf-wise intersection point if and only if $x(0)$ and $x(1)$ lie on the
same circle fiber. We are interested in leaf-wise intersection points on the
fixed contact hypersurface $\Sigma_{\frac{1}{\sqrt{k\kappa\pi}}}$.

Assume without loss of generality that the support of $F$ lies in
$E\times[1/2,1]$. As in Subsection 2.1, let
$\rho:\mathbb{R}\rightarrow\mathbb{R}$ be a bump function with support in
$[0,1/2]$ that integrates to one:

$\int_{\mathbb{R}}\rho(t)dt=1.$

Define Hamiltonians
$\left\\{H_{i}^{F}:E\times[0,1]\rightarrow\mathbb{R}\right\\}_{i\in\mathbb{Z}}$
by

$H_{i}^{F}=\rho(t)h_{i}(k\pi r^{2})+F.$

The dynamics of $H_{i}^{F}$ are governed by $\rho(t)h_{i}(k\pi r^{2})$ when
$t\in[0,1/2]$ and by $F$ when $t\in[1/2,1]$. Denoting by $R_{\alpha}$ the Reeb
vector field of $\alpha$, the Hamiltonian vector field of $H_{i}^{F}$ is

$\left(X_{H_{i}^{F}}\right)_{t}=\left\\{\begin{array}[]{cc}\rho(t)h^{\prime}_{i}(k\pi
r^{2})R_{\alpha}&t\in[0,1/2]\\\
\left(X_{F}\right)_{t}&t\in[1/2,1]\end{array}\right..$

The periodic orbits of $H_{i}^{F}$ that begin on a hypersurface $\Sigma_{R}$
correspond to leaf-wise intersection points on $\Sigma_{R}$ (see Figure 2).
Denoting the Reeb flow of $(\Sigma_{R},\alpha)$ by $\phi_{R}^{t}$, these
periodic orbits are in bijection precisely with the leaf-wise intersection
points $x:[0,1]\rightarrow E$ satisfying

$x(1)=\phi_{1}^{h^{\prime}_{i}(k\pi r)}(x(0)).$

Note that the periodic orbits of $H_{i}^{F}$ that begin on the zero section
are periodic orbits of $F$.

Figure 2: A leaf-wise intersection point exhibited as a periodic orbit $x(t)$
of $X_{H_{i}^{F}}$. The purple orbit $x(t)$ travels backwards along a Reeb
trajectory, pauses at $x(1/2)$, and then travels according to the flow
$\phi_{F}^{t}$ of the Hamiltonian vector field $(X_{F})_{t}$.

We want to show that $F$ has a leaf-wise intersection point on
$\Sigma_{\frac{1}{\sqrt{k\kappa\pi}}}$. Abusing terminology, call such points
the leaf-wise intersection points of $F$, dropping mention of the specific
fixed hypersurface. If the periodic orbits of $H_{i}^{F}$ beginning on
$\Sigma_{R}$ are degenerate for any $i$, it follows that these periodic orbits
exist, and we conclude that $F$ has a leaf-wise intersection point. So, for
the rest of the paper, we will assume that all such periodic orbits are non-
degenerate.

To set up Hamiltonian Floer theory for $H_{i}^{F}$, all periodic orbits of
$H_{i}^{F}$ must be non-degenerate, not just those beginning on $\Sigma_{R}$.
While this does not hold a priori for $F$, it holds for a suitable
perturbation of $F$. The existence of such a suitable perturbation follows
from a standard result in Floer theory.

###### Lemma 2

Fix a Hamiltonian $H$. Let $U\subset E\times\mathbb{R}$ be an open set. Let
${\cal H}(U)$ be the set of Hamiltonians with support in $U$. The subset of
Hamiltonians $K\in{\cal H}(U)$ for which every period orbit of $x(t)$ of $K+H$
is non-degenerate if $(x(t),t)$ passes through $U$ is comeager in ${\cal
H}(U)$.

###### Proof.

See the Appendix in [2]. ∎

###### Lemma 3

There exists a perturbation $\tilde{F}$ of $F$ such that

  1. 1.

the support of $\tilde{F}$ lies in $E\times[1/2,1]$,

  2. 2.

the leaf-wise intersection points are unchanged,

  3. 3.

the periodic orbits of $\tilde{F}+\rho(t)(n-\frac{1}{2})k\pi r^{2}$ are non-
degenerate for any $n\in\mathbb{Z}$, and

  4. 4.

the periodic orbits of $\tilde{F}$ contained in $\text{supp}(\tilde{F})$ are
non-degenerate.

###### Proof.

Let $\phi_{t}^{n}$ be the time-$t$ flow of the Hamiltonian vector field of
$F+\rho(t)(n-\frac{1}{2})k\pi r^{2}$, and let $B$ be the subset

$B=\left\\{(\phi_{t}^{n}(x),t)\in E\times[0,1]\hskip 5.69046pt\bigg{|}\hskip
5.69046ptx\in\Sigma_{\frac{1}{\sqrt{k\kappa\pi}}}\right\\}\cup\bigg{(}E\times[0,1/2]\bigg{)}.$

Note that $B$ does not depend on $n$.

Let $U_{B}$ be the open subset

$U_{B}=\big{(}E\times[0,1]\big{)}\setminus B.$

By Lemma 2, there exists a comeager set of perturbations ${\cal M}(n)$ of
$F+\rho(t)(n-\frac{1}{2})k\pi r^{2}$ such that every periodic orbit passing
through $U_{B}$ is non-degenerate. By assumption, each perturbation in ${\cal
M}(n)$ occurs for $t\in[1/2,1]$, the interval on which
$F=F+\rho(t)(n-\frac{1}{2})k\pi r^{2}$. We therefore view elements of ${\cal
M}(n)$ as direct perturbations of the Hamiltonian $F$. Define the intersection

${\cal M}(U_{B})=\bigcap_{n\in\mathbb{Z}}{\cal M}(n).$

As a countable intersection of comeager sets, ${\cal M}(U_{B})$ is also
comeager. Let ${\cal M}(F)$ be a comeager set of perturbations of $F$ with
non-degenerate periodic orbits. The periodic orbits of $F$ contained in
$B\bigcap\text{supp}(F)$ are leaf-wise intersection points, and so are non-
degenerate by assumption. There therefore exists a comeager subset of ${\cal
M}(F)$ comprised of perturbations of $F$ with support in $U_{B}$. It follows
that ${\cal M}(U_{B})\cap{\cal M}(F)$ is itself comeager. Choose a
perturbation

$\tilde{F}\in{\cal M}(U_{B})\cap{\cal M}(F).$

By construction, the support of $\tilde{F}$ is in $E\times[1/2,1]$.

Let

$B^{\prime}=\left\\{(\phi_{t}^{n}(x),t)\in E\times[0,1]\hskip
5.69046pt\big{|}\hskip
5.69046ptx\in\Sigma_{\frac{1}{\sqrt{k\kappa\pi}}}\right\\}.$

Define ${\cal M}(U_{B^{\prime}})$ analogously to ${\cal M}(U_{B})$. Note that
a periodic orbit $x(t)$ of any Hamiltonian of the form
$F+\rho(t)(n-\frac{1}{2})k\pi r^{2}$ that passes through $U_{B^{\prime}}$ also
passes through $U_{B}$. Therefore, $\tilde{F}$ is an element of the larger set
${\cal M}(U_{B^{\prime}})$.

By construction the perturbation $\tilde{F}$ satisfies the conditions of the
Lemma.

∎

###### Lemma 4

Define Hamiltonians $H_{n}^{\tilde{F}}=\tilde{F}+\rho(t)h_{n}(k\pi r^{2})$.
Recall the data of radii $\\{R_{1},R_{2},\dots\\}$ in the definition of the
functions $\\{h_{n}\\}$. The periodic orbits of $H_{n}^{\tilde{F}}$ with
starting point within

$E\setminus\bigg{(}[R_{1},R_{n}]\times\Sigma_{\frac{1}{\sqrt{k\kappa\pi}}}\bigg{)}.$

are non-degenerate.

###### Proof.

By construction, the function $h_{n}(k\pi r^{2})$ is linear in $k\pi r^{2}$ on
the cylindrical end modeled as $[R_{n},\infty)\times\Sigma$ and on the
radius-$R_{1}$ disk bundle. The dynamics of $\tilde{F}+\rho(t)h_{n}(k\pi
r^{2})$ coincide with the dynamics of $\tilde{F}+\rho(t)(n-\frac{1}{2})k\pi
r^{2}$ on any trajectory beginning in $[R_{n},\infty)\times\Sigma$, and they
coincide with the dynamics of $\tilde{F}+\rho(t)\frac{1}{2}k\pi r^{2}$ within
the disk bundle of radius $R_{1}$. By Lemma 3, the periodic orbits in these
sets are non-degenerate.

∎

##  3 Auxiliary Hamiltonians and a proof sketch

Recall Theorem 1.

###### Theorem 1

Let $E\rightarrow M$ be a negative line bundle with negativity constant $k$
over a monotone symplectic manifold with monotonicity constant $c>k$. If the
symplectic cohomology of $E$ is non-zero, then for any compactly-supported
Hamiltonian, the radius-$\frac{1}{\sqrt{k\kappa\pi}}$ circle subbundle
contains a leaf-wise intersection point.

The proof of Theorem 1 proceeds in two steps:

  1. 1.

show that, for any annulus subbundle $A\subset E$ containing
$\Sigma_{\frac{1}{\sqrt{k\kappa\pi}}}$, there exists some circle subbundle
$\Sigma\subset A$ containing a leaf-wise intersection point; and

  2. 2.

show that there exists a sequence of leaf-wise intersection points whose limit
exists and is a leaf-wise intersection point on
$\Sigma_{\frac{1}{\sqrt{k\kappa\pi}}}$.

The second step is straight-forward, and is given by Proposition 7. The first
step proceeds by a contradiction, arising through the subtle maneuverings of
different families of Hamiltonians. In this section, we collect the families
of Hamiltonians, and we sketch the proof of Step 1.

###  3.1 The Hamiltonians

####  3.1.1 $H_{n}$ and $K_{n}$

Fix a radius $P$ and a radius $R$, with $0<P<\frac{1}{\sqrt{k\kappa\pi}}<R$.
Fix monotone-increasing sequences of real numbers $P_{1},P_{2},\dots$ and
$R_{1},R_{2},\dots$ with

$0<P_{1}<P_{2}<\dots\rightarrow P<R_{1}<R_{2}<\dots\rightarrow R$

and converging to $P$, respectively $R$.

For $n\geq 0$, define $H_{n}$ and $H_{n}^{F}$ using the functions described in
Section 2. We set

$H_{n}=\rho(t)\left(h_{n}(k\pi r^{2})+\mathfrak{p}^{*}G\right)$

and

$H_{n}^{F}=\rho(t)h_{n}(k\pi r^{2})+F.$

Thus, $H_{n}^{F}$ is precisely as defined in Section 2, and, in the notation
of Subsection 2.1, $H_{n}$ is now the Hamiltonian ${\cal H}_{n}$. See Figure 1
to recall the definition of $h_{n}$.

For $n\geq 0$, define $K_{n}$ and $K_{n}^{F}$ analogously to the Hamiltonians
$H_{n}$ and $H_{n}^{F}$, except

  * •

the radial functions $k_{n}$ are negated and

  * •

are defined through the radii sequence $P_{1},P_{2},\dots$ in lieu of
$R_{1},R_{2},\dots$.

Precisely, there are functions $k_{n}:\mathbb{R}\rightarrow\mathbb{R}$, as
depicted in Figure 3, and a Morse function $G$ on the base of the bundle $B$
such that

$K_{n}=\rho(t)\left(k_{n}(k\pi r^{2})+(1+k\pi r^{2})\mathfrak{p}^{*}G\right)$

and

$K_{n}^{F}=\rho(t)k_{n}(k\pi r^{2})+F.$

$\mathbb{R}$$r$$k_{1}$$k_{2}$$k_{3}$$k_{4}$...$P_{1}$$P_{2}$$P_{3}$ Figure 3:
The family of functions $\\{k_{n}(r)\\}$

Fix continuation maps

$\displaystyle CF^{*}(H_{n})\rightarrow CF^{*}(H_{n+1}),$ $\displaystyle\hskip
28.45274ptCF^{*}(H_{n}^{F})\rightarrow CF^{*}(H_{n+1}^{F}),$ $\displaystyle
CF^{*}(K_{m})\rightarrow CF^{*}(K_{m-1}),$ $\displaystyle\hskip
28.45274ptCF^{*}(K_{m}^{F})\rightarrow CF^{*}(K_{m-1}^{F}).$

####  3.1.2 $\check{H}_{m,n}$ and $\check{K}_{m,n}$

Fix one more radius $Q$, with $P<Q<R$. Let $\check{k}_{m,n}$ be a function
that is

  1. 1.

equal to $k_{m}$ on $[0,P_{m}]$,

  2. 2.

convex on $[P,\infty)$,

  3. 3.

and a translation of $h_{n}$ on $[Q,\infty)$.

See Figure 4(a). Define functions $\check{K}_{m,n}$ and $\check{K}_{m,n}^{F}$
by

$\check{K}_{m,n}=\rho(t)\left(\check{k}_{m,n}(k\pi r^{2})+(1+k\pi
r^{2})\mathfrak{p}^{*}G\right)$

and

$\check{K}_{m,n}^{F}=\rho(t)\check{k}_{m,n}(k\pi r^{2})+F.$

Let

$\check{h}_{m,n}=\check{k}_{m,n}+\min_{r\geq 0}\check{k}_{m,n}(r),$

as in Figure 4(b). Define functions $\check{H}_{m,n}$ and
$\check{H}_{m,n}^{F}$ by

$\check{H}_{m,n}=\rho(t)\left(\check{h}_{m,n}(k\pi r^{2})+(1+k\pi
r^{2})\mathfrak{p}^{*}G\right)$

and

$\check{H}_{m,n}^{F}=\rho(t)\check{h}_{m,n}(k\pi r^{2})+F.$

These are variations on the “v-shaped” Hamiltonians appearing in [7].

$\mathbb{R}$$r$$P_{1}$$P_{2}$$P_{3}$$R_{1}$$Q$$P$ (a) $\check{k}_{1,4}$

$\mathbb{R}$$r$$P_{1}$$P_{2}$$P_{3}$$R_{1}$$Q$$P$ (b) $\check{h}_{1,4}$

Figure 4: “v-shaped” Hamiltonians

For each $m\in\mathbb{Z}_{<0}$ and $n\in\mathbb{N}$, define a generic homotopy
$\check{K}^{s}_{m,n}$ between $\check{K}_{m,n}$ and $\check{K}_{m-1,n}$ that
is

  1. 1.

monotone increasing everywhere and

  2. 2.

is equal to $\check{K}_{m,n}+k(s)$ for an $s$-dependent function $k$ on
$E\setminus\left([P_{m-1},\infty)\times\Sigma\right)$ and on
$[Q,\infty)\times\Sigma$.

Let

$c_{m}:CF^{*}(\check{K}_{m,n})\rightarrow CF^{*}(\check{K}_{m-1,n})$

be the continuation map induced by $\check{K}^{s}_{m,n}$. We suppress the
$n$-index from the notation.

Similarly define continuation maps

$c_{n}:CF^{*}(\check{K}_{m,n})\rightarrow CF^{*}(\check{K}_{m,n+1})$

The continuation maps $c_{n}$ and $c_{m}$ induce a barrage of continuation
maps

$\displaystyle CF^{*}(\check{H}_{m,n})\rightarrow
CF^{*}(\check{K}_{m-1,n}^{F}),$ $\displaystyle\hskip
28.45274ptCF^{*}(\check{K}_{m,n}^{F})\rightarrow
CF^{*}(\check{K}_{m,n+1}^{F})$ $\displaystyle
CF^{*}(\check{H}_{m,n})\rightarrow CF^{*}(\check{H}_{m-1,n}),$
$\displaystyle\hskip 28.45274ptCF^{*}(\check{H}_{m,n})\rightarrow
CF^{*}(\check{H}_{m,n+1})$ $\displaystyle
CF^{*}(\check{H}_{m,n}^{F})\rightarrow CF^{*}(\check{H}_{m-1,n}^{F}),$
$\displaystyle\hskip 28.45274ptCF^{*}(\check{H}_{m,n}^{F})\rightarrow
CF^{*}(\check{H}_{m,n+1}^{F}).$

###  3.2 Sketch of proof

Assume for contradiction that there does not exist a circle subbundle
contained inside the annulus subbundle $[P_{1},R]\times\Sigma$ with a leaf-
wise intersection point of $F$. By Lemma 4, we may assume that the periodic
orbits of $H_{n}^{F},K_{m}^{F},\check{H}_{m,n}^{F},$ and $\check{K}_{m,n}^{F}$
are all non-degenerate for every $n$ and $m$.

Denote the disk bundle of radius $R$ by $D_{R}$. Let

$C=\sup_{x\in D_{R}}|F(x)|+\sup_{x\in D_{R}}(1+k\pi
R^{2})|\mathfrak{p}^{*}G(x)|.$

We show in Subection 4.1 the following chain of quasi-isomorphisms and quasi-
inclusions

$\displaystyle\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(K_{m})$
$\displaystyle\overset{(1)}{\simeq}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(\check{K}_{m,n})\overset{(2)}{\hookrightarrow}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a+C,\infty)}(\check{K}_{m,n}^{F})$
$\displaystyle\overset{(3)}{\simeq}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a+C,\infty)}(\check{H}_{m,n}^{F})\overset{(4)}{\hookrightarrow}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a+2C,\infty)}(\check{H}_{m,n})\overset{(5)}{\simeq}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}CF^{*}_{(a+2C,\infty)}(H_{n}).$

Taking the direct limit over action window, the quasi-inclusions become quasi-
isomorphisms, so that there is a quasi-isomorphism

$\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(K_{m})\simeq\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}CF^{*}_{(a+2C,\infty)}(H_{n}).$

Taking homology yields an isomorphism

$\widehat{SH_{*}}(K):=H\left(\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(K_{m})\right)\simeq
H\left(\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}CF^{*}_{(a+2C,\infty)}(H_{n})\right)=:SH^{*}(H).$

We show in Proposition 6 that $\widehat{SH_{*}}(K)=0$. Thus, if the symplectic
cohomology $SH^{*}(H)$ is non-zero, we reach a contradiction. We conclude that
there exists a leaf-wise intersection point somewhere within the annulus
$[P_{1},R]\times\Sigma$.

Choose a sequence of nested annuli

$[P_{1},R]\times\Sigma\supset[P_{1}^{1},R^{1}]\times\Sigma\supset[P_{1}^{2},R^{2}]\times\Sigma\supset\dots$

whose intersection is

$\Sigma_{\frac{1}{\sqrt{k\kappa\pi}}}=\bigcap\limits_{i=1}^{\infty}[P_{1}^{i},R^{i}]\times\Sigma.$

Let $\\{x_{i}(t)\\}$ be a sequence of leaf-wise intersection points, such that
$x_{i}(0)\in[P_{1}^{i},R^{i}]\times\Sigma$. We show in Proposition 7 that the
sequence $\\{x_{i}\\}$ has a subsequence that converges to a leaf-wise
intersection point on $\Sigma_{\frac{1}{\sqrt{k\kappa\pi}}}$.

##  4 Proof of Theorem 1

###  4.1 A chain of isomorphisms

In this subsection we fix an action window $(a,\infty)$ and prove the
following chain of isomorphisms and inclusions.

$\displaystyle\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(K_{m})$
$\displaystyle\overset{(1)}{\simeq}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n})\overset{(2)}{\hookrightarrow}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a+C,\infty)}(\check{K}_{m,n}^{F})$
$\displaystyle\overset{(3)}{\simeq}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a+C,\infty)}(\check{H}_{m,n}^{F})\overset{(4)}{\simeq}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}HF^{*}_{(a+C,\infty)}(H_{n}^{F})\overset{(5)}{\hookrightarrow}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}HF^{*}_{(a+2C,\infty)}(H_{n}).$

####  4.1.1 The isomorphism (1)

We prove the following Proposition, illustrated in Figure 5.

###### Proposition 1

There is a quasi-isomorphism

$\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(K_{m})\cong\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(\check{K}_{m,n})$

for any $a\in\mathbb{R}$.

The proof of Proposition 1 proceeds through the study of an intermediary
subcomplex, $CF^{*}_{P}(\check{K}_{m,n})$. Define
$CF^{*}_{P}(\check{K}_{m,n})$ as the vector space generated by the subset of
orbits of $CF^{*}_{P}(\check{K}_{m,n}$ contained in
$E\setminus[P,\infty)\times\Sigma$. The Floer differential and continuation
map on $CF^{*}_{P}(\check{K}_{m,n})$ are characterized by Lemma 5 and the
immediate Corollary 3.

Figure 5: The subcomplex $CF^{*}_{P}(\check{K}_{m,n})$, and the quasi-
isomorphism of Proposition 1

  * Remark 3)

Lemma 5, Lemma 6, and Lemma 7, as well as Corollary 3 and Corollary 4 are
completely analogous to Lemmas 1, 2, and 3 appearing in [18]. The Hamiltonians
$\check{K}_{m,n}$ are of a slightly different flavor, but the only thing to
check is the integrated maximum principle, which we reprove in the Appendix.
With the reproven result, these Lemmas follow immediately from the results in
[18], and we therefore omit the proofs. See also Lemma 3 in [19] for a maximum
principle that is very closely related to the result we need.

###### Lemma 5

Any Floer solution of $\check{K}_{m,n}$ or $\check{K}_{m,n}^{s}$ with negative
asymptote contained in $E\setminus[P,\infty)\times\Sigma$ remains entirely in
$E\setminus[P,\infty)\times\Sigma$.

###### Corollary 3

The following inclusions are inclusions of well-defined subcomplexes:

$CF^{*}_{P}(\check{K}_{m,n})\subset CF^{*}(\check{K}_{m,n}),$
$\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{P}(\check{K}_{m,n})\subset\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}(\check{K}_{m,n}),$

and

$\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{P}(\check{K}_{m,n})\subset\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}(\check{K}_{m,n}),$

Relate $\check{K}_{m,n}$ to $K_{m}$ by defining a homotopy $K_{m,0}^{s}$
between the Hamiltonian $K_{m}$ and the Hamiltonian $\check{K}_{m,0}$ that is
monotone increasing everywhere in $s$. Let

$\iota_{m}:CF^{*}(K_{m})\rightarrow CF^{*}(\check{K}_{m,0})$

be the continuation map induced by $K_{m,0}^{s}$.

###### Lemma 6

The map $\iota_{m}$ induces an isomorphism of complexes

$CF^{*}(K_{m})\cong CF^{*}_{P}(K_{m,0}).$

More generally, define

$\phi_{m,n}=c_{n}\circ...\circ c_{0}\circ\iota_{m}:CF^{*}(K_{m})\rightarrow
CF^{*}(\check{K}_{m,n}).$

Because each continuation map $c_{n}$ acts as the canonical inclusion of a
subcomplex, Lemma 6 directly extends to show that $\phi_{n}$ induces an
isomorphism of complexes

$CF^{*}(K_{m})\cong CF^{*}_{P}(\check{K}_{m,n}).$

Let

$\Phi_{n}=\lim_{\begin{subarray}{c}\leftarrow\\\ m\end{subarray}}\phi_{m,n}$

and

$\Phi=\lim_{\begin{subarray}{c}\rightarrow\\\ n\end{subarray}}\Phi_{n}.$

###### Lemma 7

$\Phi_{n}$ induces an isomorphism of complexes

$\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}(K_{m})\cong\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{P}(\check{K}_{m,n})$

and $\Phi$ induces an isomorphism of complexes

$\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}(K_{m})\cong\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{P}(\check{K}_{m,n}).$

The maps $\Phi_{n}$ and $\Phi$ are built out of the continuation maps
$\\{c_{n}\\}$ and the inclusions $\iota_{m,n}$. By construction, both $c_{n}$
and $\iota_{m,n}$ are action-preserving. As a consequence, Lemma 7 has an
action-truncated version. Let

$\Phi_{n,a}:\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(K_{m})\rightarrow\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{P,(a,\infty)}(\check{K}_{m,n})$

be the restriction of $\Phi_{n}$ to the action-truncated pieces. Similarly
define $\Phi_{a}$.

###### Corollary 4

$\Phi_{n,a}$ induces an isomorphism of complexes

$\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(K_{m})\cong\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{P,(a,\infty)}(\check{K}_{m,n})$

and $\Phi_{a}$ induces an isomorphism of complexes

$\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(K_{m})\cong\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{P,(a,\infty)}(\check{K}_{m,n}).$

We are now ready to prove Proposition 1.

Proof of Proposition 1: From Corollary 4 it suffices to show that the
inclusion

$\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{P,(a,\infty)}(\check{K}_{m,n})\hookrightarrow\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(\check{K}_{m,n})$

is a quasi-isomorphism. This is equivalent to showing that the quotient
complex

(5) ${\raisebox{1.99997pt}{$\lim\limits_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(\check{K}_{m,n})$}\left/\raisebox{-1.99997pt}{$\lim\limits_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{P,(a,\infty)}(\check{K}_{m,n})$}\right.}$

is acyclic. We will show that, in fact, (5) is identically zero. As direct
limits commute with quotients, it suffices to show that

(6) ${\raisebox{1.99997pt}{$\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(\check{K}_{m,n})$}\left/\raisebox{-1.99997pt}{$\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{P,(a,\infty)}(\check{K}_{m,n})$}\right.}$

is identically 0 for any $n\in\mathbb{N}$.

All connecting maps are surjections, and so the Mittag-Leffler condition is
satisfied. This implies that (6) is isomorphic to the inverse limit

(7) $\lim_{\begin{subarray}{c}\leftarrow\\\ m\end{subarray}}\hskip
5.69046pt{\raisebox{1.99997pt}{$CF^{*}_{(a,\infty)}(\check{K}_{m,n})$}\left/\raisebox{-1.99997pt}{$CF^{*}_{P,(a,\infty)}(\check{K}_{m,n})$}\right.}.$

Thus, it suffices to show that

$0\hskip 5.69046pt=\hskip
5.69046pt{\raisebox{1.99997pt}{$CF^{\ell}_{(a,\infty)}(\check{K}_{m,n})$}\left/\raisebox{-1.99997pt}{$CF^{\ell}_{P,(a,\infty)}(\check{K}_{m,n})$}\right.}$

for fixed $\ell\in\mathbb{R}$ and for sufficiently large $m$. Note that we may
indeed consider each graded component individually, as the inverse system
respects the grading. Define a splitting

$CF^{\ell}_{(a,\infty)}(\check{K}_{m,n})=CF^{\ell}_{P,(a,\infty)}(\check{K}_{m,n})\oplus{\cal
C}^{\ell}_{(a,\infty),m,n},$

where ${\cal C}^{\ell}_{(a,\infty),m,n}$ is generated by Novikov-weighted
periodic orbits in $[P,R]\times\Sigma$ of index ${\ell}$ and action at least
$a$.

Let $Cx$ be a Novikov-weighted periodic orbit in ${[P,\infty]}\times\Sigma$ of
$\check{K}_{m,n}$ and of index ${\ell}$. We want to show that ${\cal A}(Cx)<a$
for $m>>0$.

Let $x$ have winding number $\mathfrak{w}$. As shown in, for example, [17],
the grading of $Cx$ has the formula

${\ell}=-2\mathfrak{w}+{2}{\kappa}ev(C)+C^{\prime},$

where $C^{\prime}$ is a universally-bounded constant. Rearranging,

(8) $ev(C)=\frac{1}{2\kappa}\left({\ell}+2\mathfrak{w}-C^{\prime}\right).$

There exists a constant $B$ independent of $m,n$ such that, if $x$ is an orbit
of $\check{K}_{m,n}$ that starts in $[P,R]\times\Sigma$,

$\check{K}_{m,n}(x(t))\leq-(Q-P)m+B.$

The action of $Cx$ is therefore bounded above by

(9) ${\cal A}(Cx)<-\mathfrak{w}P_{\mathfrak{w}}+ev(C)-(Q-P)m+B.$

Substituting (8) into (9) yields

${\cal
A}(Cx)<\mathfrak{w}\left(-P_{\mathfrak{w}}+\frac{1}{\kappa}\right)-(Q-P)m+B+\frac{1}{2\kappa}\left({\ell}-C^{\prime}\right).$

By assumption either $P_{\mathfrak{w}}<\frac{1}{\kappa}$ and $\mathfrak{w}\leq
0$ or $P_{\mathfrak{w}}>\frac{1}{\kappa}$ and $\mathfrak{w}\geq 0$. Either
way,

$\mathfrak{w}\left(-P_{\mathfrak{w}}+\frac{1}{\kappa}\right)<0.$

As $m\rightarrow\infty$, $-(Q-P)m+B\rightarrow-\infty$. Thus, there exists
some value $M$ such that, for all $m>M$,

${\cal A}(Cx)<a.$

It follows that for $m>>0$ there is a $\Lambda_{0}$-module isomorphism

${\raisebox{1.99997pt}{$CF^{\ell}_{(a,\infty)}(\check{K}_{m,n})$}\left/\raisebox{-1.99997pt}{$CF^{\ell}_{P,(a,\infty)}(\check{K}_{m,n})$}\right.}\hskip
5.69046pt\cong\hskip 5.69046pt{\cal C}^{\ell}_{(a,\infty),m,n}=0.$

∎

###### Corollary 5

There is an isomorphism

$\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(K_{m})\cong\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n})$

###### Proof.

Recalling that direct limits preserve exact sequences, the quasi-isomorphism
of Proposition 1 induces a map of Milnor exact sequences

${0}$${\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}\text{\textsuperscript{$1$}
}HF^{*}_{(a,\infty)}(K_{m})}$${H\left(\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(K_{m})\right)}$${\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(K_{m})}$${0}$${0}$${\lim\limits_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}\text{\textsuperscript{$1$}
}HF^{*}_{(a,\infty)}(\check{K}_{m,n})}$${H\left(\lim\limits_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(\check{K}_{m,n})\right)}$${\lim\limits_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n})}$${0}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

The right-hand vertical arrow is an isomorphism because the Mittag-Leffler
condition is satisfied: the continuation maps, by construction, act as
canonical surjections. Taking the direct limit $a\rightarrow-\infty$ over the
right-most isomorphism yields the Corollary.

∎

####  4.1.2 The isomorphisms (2) and (5)

We begin with a general result. Let $F_{1}:E\times S^{1}\rightarrow\mathbb{R}$
and $F_{2}:E\times S^{1}\rightarrow\mathbb{R}$ be two Hamiltonians with
compact support in $E$ and support in $[0,1/2]\subset S^{1}$. Assume that
$F_{1}\geq F_{2}$ everywhere. Let

$\check{K}^{F_{1},G}_{m,n}=\rho(t)(\check{k}_{m}(k\pi
r^{2})+\mathfrak{p}^{*}G)+F_{1}\hskip 28.45274pt\text{and}\hskip
28.45274ptK^{F_{2},G}_{m,n}=\rho(t)(\check{k}_{m}(k\pi
r^{2})+\mathfrak{p}^{*}G)+F_{2}.$

Denote by $\mathcal{S}$ the supremum of $F_{1}-F_{2}$:

$\mathcal{S}=\sup_{x\in E\times S^{1}}F_{1}(x)-F_{2}(x).$

The following Lemma appears in a slightly different guise in [9].

###### Lemma 8

There are chain maps

$CF^{*}_{(a,\infty)}(\check{K}^{F_{2},G}_{m,n})\rightarrow
CF^{*}_{(a,\infty)}(\check{K}^{F_{1},G}_{m,n})\rightarrow
CF^{*}_{(a,\infty)}(\check{K}^{F_{2}+\mathcal{S},G}_{m,n})\cong
CF^{*}_{(a-\mathcal{S},\infty)}(\check{K}^{F_{2},G}_{m,n})$

whose composition is chain-homotopic to the inclusion of action-filtered
pieces

$CF^{*}_{(a,\infty)}(\check{K}^{F_{2},G}_{m,n})\hookrightarrow
CF^{*}_{(a-\mathcal{S},\infty)}(\check{K}^{F_{2},G}_{m,n}).$

Kang considers Hamiltonians that are truly linear at infinity: Hamiltonians of
the form $h_{m}(k\pi r^{2})$, rather than $K_{m}=k_{m}(k\pi r^{2})+(1+k\pi
r^{2})\mathfrak{p}^{*}G$. However, as with all other similar results in this
paper, the proof of Lemma 8 is identical, given that the integrated maximum
principle holds. We therefore refer to [9] for the proof of Lemma 8.

Specializing to our set-up,

###### Lemma 9

There is an isomorphism

$\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{0,G})\rightarrow\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F,G}).$

###### Proof.

Assume without loss of generality that $F\geq 0$ everywhere. Taking $F_{2}=0$,
$F_{1}=F$, and

$\mathcal{S}=\sup_{x\in E\times S^{1}}F(x),$

Lemma 8 produces maps

$\mu_{m,n}:CF^{*}_{(a,\infty)}(\check{K}_{m,n}^{0,G})\rightarrow
CF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F,G}).$

and

$\nu_{m,n}:CF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F,G})\rightarrow
CF^{*}_{(a-\mathcal{S},\infty)}(\check{K}_{m,n}^{0,G}).$

such that $\nu_{m,n}\circ\mu_{m,n}$ is chain-homotopic to the inclusion

$\iota_{a,a-M}:CF^{*}_{(a,\infty)}(\check{K}_{m,n}^{0,G})\hookrightarrow
CF^{*}_{(a-\mathcal{S},\infty)}(\check{K}_{m,n}^{0,G}).$

Let $\mu_{m,n}^{*}$ and $\nu_{m,n}^{*}$ be the induced maps on homology. As
continuation maps commute up to homology, there are induced maps on limits

$M^{*}_{a}=\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}\mu_{m,n}^{*}:\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{0,G})\rightarrow\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F,G})$

and

$N^{*}_{a}=\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}\nu_{m,n}^{*}:\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F,G})\rightarrow\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a-\mathcal{S},\infty)}(\check{K}_{m,n}^{0,G})$

such that

$N^{*}_{a}\circ M^{*}_{a}=\iota^{*}_{a,a-\mathcal{S}}$

is the inclusion of action-filtered subcomplexes. Taking the limit over action
window $(a,\infty)$ produces maps $M^{*}$ and $N^{*}$ with

(10) $N^{*}\circ M^{*}=\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\iota^{*}_{a,a-\mathcal{S}}=id.$

Conversely, there are maps

${\cal M}^{*}_{a}:\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{\mathcal{S},G})\rightarrow\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a-\mathcal{S},\infty)}(\check{K}_{m,n}^{F,G})$

and

${\cal N}^{*}_{a}:\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F,G})\rightarrow\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{\mathcal{S},G})$

satisfying

(11) ${\cal M}^{*}_{a}\circ{\cal N}^{*}_{a}=\iota_{a,a-\mathcal{S}}.$

The identification
$CF^{*}_{(a,\infty)}(\check{K}_{m,n}^{M,G})=CF^{*}_{(a-\mathcal{S},\infty)}(\check{K}_{m,n}^{0,G})$
identifies the maps ${\cal M}_{a}$ and $M_{a-\mathcal{S}}$ and the maps
$N_{a}^{*}$ and ${\cal N}_{a-\mathcal{S}}^{*}$. Under this identification,
Equation (11) becomes

$M_{a-M}^{*}\circ N_{a}^{*}=\iota_{a,a-\mathcal{S}}.$

Taking the limit over action window $(a,\infty)$ yields the identity

(12) $M^{*}\circ N^{*}=id.$

Equations (10) and (12) shows that $M^{*}$ and $N^{*}$ are inverses.

∎

Note that the Hamiltonian $\check{K}_{m,n}$ is precisely the Hamiltonian
$\check{K}_{m,n}^{0,G}$. Thus, the isomorphism (2) is shown by the following
Proposition.

###### Proposition 2

There is an isomorphism

$\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{0,G})\rightarrow\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F}).$

###### Proof.

By Lemma 9 it suffices to show that there is an isomorphism

$\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F,G})\rightarrow\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F}).$

The proof of this isomorphism is very similar to the proof of Lemma 9. We
therefore outline the strategy and refer to the above proof for details.

Assume without loss of generality that $G\geq 0$. Let

$\mathcal{S}=\max_{x\in M}G(x)$

and

$\mathcal{R}=\min_{\begin{subarray}{c}\text{supp}(F)\subset
D_{r}\end{subarray}}r$

Let $G_{s}^{1}$ be a monotone homotopy such that $G_{s}^{1}=0$ when $s>>0$ and
$G_{s}^{1}=G$ when $s<<0$. Let $G_{s}^{2}$ be a monotone homotopy such that
$G_{s}^{2}=G$ when $s>>0$ and $G_{s}^{2}=\mathcal{S}$ when $s<<0$. As in the
proof of Lemma 9, the Hamiltonians

$\rho(t)(\check{k}_{m,n}(k\pi r^{2})+(1+k\pi
r^{2})\mathfrak{p}^{*}G_{s}^{1})+F\hskip 28.45274pt\text{and}\hskip
28.45274pt\rho(t)(\check{k}_{m,n}(k\pi r^{2})+(1+k\pi
r^{2})\mathfrak{p}^{*}G_{s}^{2})+F$

induce maps

$CF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F,0})\rightarrow
CF^{*}_{(a,\infty})(\check{K}_{m,n}^{F,G})\rightarrow
CF^{*}_{(a-(1+k\pi\mathcal{R}^{2})\mathcal{S},\infty})(\check{K}_{m,n}^{F,0}).$

whose composition is homotopic to the inclusion map

$CF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F,0})\hookrightarrow
CF^{*}_{(a-(1+k\pi\mathcal{R}^{2})\mathcal{S},\infty})(\check{K}_{m,n}^{F,0}).$

Note that compactness of Floer trajectories is achieved by the integrated
maximum principle. Similarly, all periodic orbits remain inside the disk
bundle of radius $k\pi\mathcal{R}^{2}$; thus, the maximum bound on a change in
action-window is given by $(1+k\pi\mathcal{R}^{2})\mathcal{S}$.

Similarly, there are maps

$CF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F,G})\rightarrow
CF^{*}_{(a,\infty})(\check{K}_{m,n}^{F,\mathcal{S}})\rightarrow
CF^{*}_{(a-(1+k\pi\mathcal{R}^{2})\mathcal{S},\infty})(\check{K}_{m,n}^{F,G}).$

whose composition is homotopic to the inclusion map

$CF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F,G})\hookrightarrow
CF^{*}_{(a-(1+k\pi\mathcal{R}^{2})\mathcal{S},\infty})(\check{K}_{m,n}^{F,G}).$

As in the proof of Lemma 9, taking homology and limits over $m,n$ and $a$
yields inverse maps

$\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F,G})\rightleftharpoons\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F}).$

The Proposition follows.

∎

The isomorphism (5) proceeds in the same manner.

###### Proposition 3

There is an isomorphism

$\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(H_{n}^{F})\cong\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(H_{n}).$

###### Proof.

The proof is completely analogous to the proof of Proposition 2, and so we
omit it.

∎

####  4.1.3 The isomorphisms (3) and (4)

We prove the following Propositions.

###### Proposition 4

There is an isomorphism

$\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F})\simeq\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{H}_{m,n}^{F})$

###### Proposition 5

There is an isomorphism

$\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{H}_{m,n}^{F})\simeq\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}HF^{*}_{(a,\infty)}(H_{n}^{F}).$

These propositions require a careful analysis of the action of orbits on the
conical end of $E$, which we package into Lemma 10 and Lemma 11.

###### Lemma 10

Suppose $m>0$. Away from the collar neighborhood $[P_{n},P_{n+1}]\times\Sigma$
there is a set bijection between periodic orbits of $K_{m}^{F}$ and periodic
orbits of $K_{m+1}^{F}$.

Suppose $n\geq 0$. Away from the collar neighborhood
$[R_{n},R_{n+1}]\times\Sigma$ there is a set bijection between periodic orbits
of $H_{n}^{F}$ and periodic orbits of $H_{n+1}^{F}$.

###### Proof.

We prove the Lemma for $n\geq 0$. The case $m>0$ is completely analogous.

The Hamiltonians vector fields of $H_{n}^{F}$ and $H_{n+1}^{F}$ coincide on
$\left(E\setminus[R_{n},\infty)\times\Sigma\right)\times[0,\frac{1}{2}]$ and
on $E\times[\frac{1}{2},1]$. Any periodic orbit that starts in
$E\setminus[R_{n},\infty)\times\Sigma$ remains in
$E\setminus[R_{n},\infty)\times\Sigma$ through the time interval
$[0,\frac{1}{2}]$. Thus, there is a set bijection between the subset of orbits
starting in $E\setminus[R_{n},\infty)\times\Sigma$.

The integral curves corresponding to ${H_{n}^{F}}$, respectively
${H_{n+1}^{F}}$, on $[R_{n+1},\infty)\times\Sigma\times[0,\frac{1}{2}]$ is
fiberwise $re^{2\pi(i+1/2)nt}$, respectively $re^{2\pi i(n+3/2)t}$. If
$x_{n}(t)$ is an integral curve of ${H_{n}^{F}}$ and $x_{n+1}(t)$ is an
integral curve of ${H_{n+1}^{F}}$ with
$x_{n}(0)=x_{n+1}(0)\in[R_{n+1},\infty)\times\Sigma$ then it follows that

$x_{n}(1/2)=x_{n+1}(1/2).$

As the Hamiltonian vector fields of ${H_{n}^{F}}$ and ${H_{n+1}^{F}}$ coincide
on $E\times[\frac{1}{2},1]$, it follows that $x_{n}(t)=x_{n+1}(t)$ for all
$t\in[\frac{1}{2},1]$. In particular, $x_{n}(1)=x_{n+1}(1)$. Thus, $x_{n}$ is
a periodic orbit if and only if $x_{n+1}$ is a periodic orbit.

∎

###### Lemma 11

Fix an action window $(a,\infty)$ and a degree $q$. For $n>>0$ large enough,
all periodic orbits appearing in $CF^{q}_{(a,\infty)}({H_{n}^{F}})$ begin in
$E\setminus[R_{n},\infty)\times\Sigma$. Similarly, for $m<<0$ negative enough,
all periodic orbits appearing in $CF^{q}_{(a,\infty)}(K_{m}^{F})$ begin in
$E\setminus[P_{m},\infty)\times\Sigma$.

###### Proof.

Let $x(t)$ be a periodic orbit of $F+\rho(t)\frac{1}{2}k\pi r^{2}$. With the
orientation induced from the complex structure, let $z$ be the arc segment
starting at $x(1/2)$ and continuing to $x(1)$. Fix a capping disk
$\tilde{v}_{x}$ of the loop $\ell=(-z)\\#x\big{|}_{[1/2,1]}$.

Figure 6: The capping disk in the fiber

Let $\tilde{u}$ be the fiber disk of the loop $\gamma=x\big{|}_{[0,1/2]}\\#z$.
See Figure 6. Then $x\simeq\gamma\\#\ell$ has a capping disk
$\tilde{x}\simeq\tilde{u}\\#\tilde{v}$ such that

$\int_{D}\tilde{x}^{*}\Omega=\int_{D}(\tilde{u}\\#\tilde{v})^{*}\Omega=-2k\pi
r(x)^{2}+\int_{D}\tilde{v}^{*}\Omega,$

where $r(x)$ is the radius at which $x\big{|}_{[0,1/2]}$ lives. The Conley-
Zehnder index $\mu_{CZ}$ of $\tilde{x}$ is, by the Additivity Axiom,

$\mu_{CZ}(\tilde{x})=\mu_{CZ}(\tilde{u})+\mu_{CZ}(\tilde{v})=-2+\mu_{CZ}(\tilde{v})$

(see, for example, [15]).

Now let $n\geq 0$, and let $x_{n}(t)$ be a periodic orbit of $H_{n}^{F}$
living in $[R_{n},\infty)\times\Sigma$. By definition of ${H_{n}^{F}}$,
$x_{n}(t)$ is a periodic orbit of $F+\rho(t)(nk\pi r^{2}+\frac{1}{2}k\pi
r^{2})$. Thus, by Lemma 10, $x_{n}(t)$ corresponds to some periodic orbit
$x(t)$ of $F+\rho(t)\frac{1}{2}k\pi r^{2}$. Note that $x$ and $x_{n}$ begin at
the same point, and therefore begin at the same radius $r(x)$; similarly,
$x\big{|}_{[1/2,1]}=x_{n}\big{|}_{[1/2,1]}$. As above, let $z$ be the arc
segment starting at $x(1/2)$ and continuing widdershins to $x(1)$ and let
$\tilde{v}$ be the fixed capping disk of
$(-z)\\#x\big{|}_{[1/2,1]}=(-z)\\#x_{n}\big{|}_{[1/2,1]}$. Let $\tilde{u}_{n}$
be the fiber disk of the loop $x_{n}\big{|}_{[0,1/2]}\\#z$, so that $x_{n}$
has a capping disk $\tilde{x}_{n}\simeq\tilde{u}_{n}\\#\tilde{v}$, satisfying

$\int_{D}\tilde{x}_{n}^{*}\Omega=\int_{D}(\tilde{u}_{n}\\#\tilde{v})^{*}\Omega=-2k\pi
nr(x)^{2}+\int_{D}\tilde{v}^{*}\Omega$

Note that

$\mu_{CZ}(\tilde{x}_{n})=\mu_{CZ}(\tilde{u}_{n})+\mu_{CZ}(\tilde{v})=-2n+\mu_{CZ}(\tilde{v}).$

Away from the support of $F$, the function $F+\rho(t)\frac{1}{2}\kappa\pi
r^{2}$ has flow $re^{(1/2)\pi it}$, which has no time-one orbits. All periodic
orbits are therefore contained in a compact set. By the non-degeneracy
assumption, there are finitely many periodic orbits. The symplectic area of
each chosen cap $\tilde{v}$ is bounded by some constant $C_{\Omega}$:

$\bigg{|}\int_{D}\tilde{v}^{*}\Omega\bigg{|}\leq C_{\Omega}.$

Similarly, the Conley-Zehnder index of each chosen cap $\tilde{v}$ is bounded
by some constant $C_{\mu}$:

$|\mu_{CZ}(\tilde{v})|\leq C_{\mu}.$

Now let $X=T^{\alpha}x_{n}(t)$ be the orbit $x_{n}$ with a Novikov weight, and
suppose $X$ lies in degree $q$. As shown in, for example, [17], $X$ has
grading

(13) $q=\mu(X)=2\kappa\alpha-2n+\mu_{CZ}(\tilde{v}).$

The bounds on $\mu_{CZ}(\tilde{v})$ imply that

$q-C_{\mu}\leq 2\kappa\alpha-2n\leq q+C_{\mu},$

so that

(14)
$\frac{1}{2\kappa}\left(q-C_{\mu}+2n\right)\leq\alpha\leq\frac{1}{2\kappa}\left(q+C_{\mu}+2n\right)$

The action of $X$ is, by definition,

${\cal
A}(X)=-k\pi(n+\frac{1}{2})r(x)^{2}+\alpha+\int_{D}\tilde{v}^{*}\Omega+\int_{0}^{1}H_{n}^{F}(x_{n})dt$

Let $(a,\infty)$ be a fixed action window. $X$ is in
$CF^{\ell}_{(a,\infty)}({H_{n}^{F}})$ if and only if

$a<\alpha-k\pi(n+\frac{1}{2})r(x)^{2}+\int_{D}\tilde{v}^{*}\Omega+\int_{0}^{1}H_{n}^{F}(x_{n})dt,$

for which

$a-C_{\Omega}<\alpha-k\pi(n+\frac{1}{2})r(x)^{2}+\int_{0}^{1}H_{n}^{F}(x_{n})dt$

must be true. Let

$C_{F}=\max_{p\in E}|{F}(p)|.$

There exists a bound $B$, independent of $n$, such that

$\int_{0}^{1/2}{H_{n}^{F}}(k\pi r(x(t))^{2})dt<(n+\frac{1}{2})\left(k\pi
r(x)^{2}-R_{n}\right)+B.$

Altogether,

$\int_{0}^{1}H_{n}^{F}(x_{n})dt\leq C_{F}+(n+\frac{1}{2})k\pi
r(x)^{2}-(n+\frac{1}{2})R_{n}+B.$

Thus,

(15) $a-C_{\Omega}-C_{F}-B+(n+\frac{1}{2})R_{n}<\alpha$

Suppose $n>>0$. More precisely, suppose that $n$ is large enough that

$\left(R_{n}-\frac{1}{\kappa}\right)n>-a+C_{\Omega}+C_{F}+B+\frac{q}{2\kappa}-\frac{C_{\mu}}{2\kappa}-\frac{1}{2}R_{n}.$

Note that $R_{n}-\frac{1}{\kappa}$ is bounded below by some positive number by
construction, so this is always possible. Combining equations (14) and (15)
yields a contradiction:

$\alpha\leq\frac{1}{2\kappa}\left(q-C_{\mu}+2n\right)<a-C_{\Omega}-C_{F}-B+(n+\frac{1}{2})R_{n}<\alpha.$

Now let $m<0$, and suppose that $x_{m}(t)$ is a periodic orbit of $K_{m}^{F}$
living in $[P_{n},\infty)\times\Sigma$. Running the same program, suppose that
$m$ is negative enough that

$\left(P_{m}-\frac{1}{\kappa}\right)m>-a+C_{\Omega}+C_{F}+B+\frac{q}{2\kappa}-\frac{C_{\mu}}{2\kappa}-\frac{1}{2}P_{m}.$

Note that $P_{m}-\frac{1}{\kappa}$ is bounded above by some negative number by
construction, so this is always possible. Combining equations (14) and (15)
again yields a contradiction:

$\alpha\leq\frac{1}{2\kappa}\left(q-C_{\mu}+2m\right)<a-C_{\Omega}-C_{F}-B+(m+\frac{1}{2})P_{m})<\alpha.$

The Lemma follows.

∎

The proofs of Proposition 4 and Proposition 5 now follow as corollaries of two
Lemmas. To state the first Lemma, let $C_{m}=\check{H}_{m,n}-\check{K}_{m,n}$.

###### Lemma 12

There is a quasi-isomorphism

$\lim_{\begin{subarray}{c}\leftarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F})\cong\lim_{\begin{subarray}{c}\leftarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a+C_{m},\infty)}(\check{H}_{m,n}^{F}).$

###### Proof.

By construction,

$\displaystyle\check{H}_{m,n}^{F}$ $\displaystyle=\rho(t)\check{H}_{m,n}+F(t)$
$\displaystyle=\rho(t)\left(\check{K}_{m,n}+C_{m}\right)+F(t).$

The periodic orbits of $\check{H}_{m,n}^{F}$ and $\check{K}_{m,n}^{F}$
coincide. Let $x(t)$ be one such periodic orbit. The action of $x(t)$ as a
periodic orbit of $\check{H}_{m,n}^{F}$ is

$\displaystyle{\cal A}_{\check{H}_{m,n}^{F}}(x)$
$\displaystyle=-\int_{D}\tilde{x}^{*}\Omega+\int_{0}^{1}\check{H}_{m,n}^{F}(x(t))dt$
$\displaystyle=-\int_{D}\tilde{x}^{*}\Omega+\int_{0}^{1}\rho(t)\check{H}_{m,n}(x(t))+F(x(t))dt$
$\displaystyle=-\int_{D}\tilde{x}^{*}\Omega+\int_{0}^{1}\rho(t)\left(\check{K}_{m,n}+C_{m}\right)(x(t))+F(x(t))dt$
$\displaystyle=-\int_{D}\tilde{x}^{*}\Omega+\int_{0}^{1}\rho(t)\check{K}_{m,n}(x(t))+F(x(t))dt+C_{m}$
$\displaystyle={\cal A}_{\check{K}_{m,n}^{F}}(x)+C_{m}.$

The Lemma follows.

∎

###### Lemma 13

Suppose that $F$ has no leafwise intersection point in $[P,R]\times\Sigma$.
Then there is a quasi-isomorphism

$\lim_{\begin{subarray}{c}\leftarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a+C_{m},\infty)}(\check{H}_{m,n}^{F})\cong\lim_{\begin{subarray}{c}\leftarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(\check{H}_{m,n}^{F})$

###### Proof.

Fix $\ell\in\mathbb{R}$. It suffices to show that there is an isomorphism

(16) $CF^{\ell}_{(a+C_{m},\infty)}(\check{H}_{m,n}^{F})\cong
CF^{\ell}_{(a,\infty)}(\check{H}_{m,n}^{F})$

for large enough values of $n$ and negative enough values of $a$. Equation
(16) will hold on the level of cochain complexes if it holds on the level of
sets. Thus, it suffices to show that if $X\in CF^{\ell}(\check{H}^{F}_{m,n})$
has action ${\cal A}(X)>a$, the action of $X$ also satisfies ${\cal
A}(X)>a+C_{m}$.

As $\check{H}_{m,n}^{F}=\check{H}_{m^{\prime},n^{\prime}}^{F}$ on
$E\setminus[P_{1},\infty)\times\Sigma$, there is a fixed finite set

$\left\\{C_{1}x_{1},\dots,C_{N}x_{N}\right\\}$

to which the generators of $CF^{\ell}_{(a+C_{m},\infty)}(\check{H}_{m,m})$
belong, for any $m,n$. Let

$C=\min_{1\leq i\leq N}{\cal A}_{\check{H}^{F}_{-1,0}}(C_{i}x_{i})$

and choose $a<C$.

By Lemma 11, if $n$ is large enough, all periodic orbits of
$\check{H}_{m,n}^{F}$ with action greater than $a$ are contained inside
$E\setminus[R,\infty)\times\Sigma$. By assumption, this means that all
periodic orbits are contained inside $E\setminus[P_{1},\infty)\times\Sigma$.
We may therefore view $X$ as an element of $CF^{\ell}(\check{H}_{-1,n}^{F})$
with action

${\cal A}_{\check{H}_{-1,n}^{F}}(X)\geq C>a.$

Since $\check{H}_{m,n}^{F}=\check{H}_{-1,n}^{F}+C_{m}$,

$\displaystyle{\cal A}_{\check{H}^{F}_{m,n}}(X)={\cal
A}_{\check{H}^{F}_{-1,n}}(X)+C_{m}>a+C_{m},$

and the claim follows.

∎

Proof of Proposition 4: As in the proof of Proposition 1, Lemma 12 implies
that there is an isomorphism

$\lim_{\begin{subarray}{c}\leftarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{K}_{m,n}^{F})\cong\lim_{\begin{subarray}{c}\leftarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a+C_{m},\infty)}(\check{H}_{m,n}^{F}).$

Lemma 13 provides an isomorphism

$\lim_{\begin{subarray}{c}\leftarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a+C_{m},\infty)}(\check{H}_{m,n}^{F})\cong\lim_{\begin{subarray}{c}\leftarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(\check{H}_{m,n}^{F}).$

The Mittag-Leffler solution is satisfied because the dimension of
$CF^{\ell}_{(a,\infty)}(\check{H}_{m,n}^{F})$ is degree-wise universally
bounded. Thus, there is an isomorphism

$\lim_{\begin{subarray}{c}\leftarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a+C_{m},\infty)}(\check{H}_{m,n}^{F})\cong\lim_{\begin{subarray}{c}\leftarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(\check{H}_{m,n}^{F}).$

Stringing these isomorphisms together yields the Proposition.

∎

Proof of Proposition 5: By the assumption that there are no leaf-wise
intersection points on a circle subbundle within $[P_{1},R]\times\Sigma$, the
periodic orbits of any $\check{H}_{m,n}^{F}$ are naturally divided into:

  1. 1.

a finite set $\\{x_{1},\dots,x_{N}\\}$ of orbits that begin inside
$E\setminus[P_{1},\infty)\times\Sigma$ that is the same for every choice of
$m,n$, and

  2. 2.

orbits that begin in $[R,\infty)\times\Sigma$.

Fix $\ell\in\mathbb{R}$. By Lemma 11, for sufficiently large $n$ and any $m$,
all summands of an element in $CF^{\ell}_{(a,\infty)}(\check{H}_{m,n}^{F})$
are of type (1). Let $C_{1},\dots C_{N}$ be Novikov coefficients such that
$|C_{i}x_{I}|=\ell$. Since

$\check{H}_{m,n}^{F}\big{|}_{E\setminus[P_{1},\infty)\times\Sigma}=\check{H}_{0,n}^{F}\big{|}_{E\setminus[P_{1},\infty)\times\Sigma}$

for all $n,n^{\prime}$, $C_{i}x_{i}$ can be viewed as an element of
$CF^{\ell}(\check{H}_{0,n}^{F})$ (of smaller action). Fix any

(17) $a<\min_{1\leq i\leq N}{\cal
A}_{\check{H}_{0,n}^{F}}(C_{i}x_{i})=\min_{\begin{subarray}{c}1\leq i\leq N\\\
m\in\mathbb{Z}_{<0}\end{subarray}}{\cal A}_{\check{H}_{m,n}^{F}}(C_{i}x_{i})$

Fix $m\in\mathbb{Z}_{<0}$. Since all elements have action greater than $a$,
the quasi-inverse maps

$CF^{\ell}(\check{H}_{m,n}^{F})\xleftrightharpoons{\rule{14.22636pt}{0.0pt}}CF^{\ell}(\check{H}_{0,n}^{F})$

restrict to quasi-isomorphisms

$CF^{\ell}_{(a,\infty)}(\check{H}_{m,n}^{F})\xleftrightharpoons{\rule{14.22636pt}{0.0pt}}CF^{\ell}_{(a,\infty)}(\check{H}_{0,n}^{F})$

We conclude that there are isomorphisms

$HF^{\ell}_{(a,\infty)}(\check{H}_{m,n}^{F})\simeq
HF^{\ell}_{(a,\infty)}(\check{H}_{0,n}^{F})$

compatible with continuation.

∎

###  4.2 Vanishing computations

####  4.2.1 Proof that $\widehat{SH_{*}}(K)=0$

###### Proposition 6

The completed symplectic homology of the family $\\{K_{m}\\}$ vanishes:

$\widehat{SH_{*}}(K)=0.$

The proof of Proposition 6 closely follows the work of Albers-Kang [4]. The
tangent bundle of $E$ has a pointwise splitting $T_{x}E=T_{x}\mathbb{C}\oplus
T_{x}B$. Let ${\cal J}$ be the set of $S^{1}$-families of almost-complex
structures that are $\Omega$-tame and upper-triangular with respect to this
splitting. Thus, a $J\in{\cal J}$ looks like

$J=\left[\begin{array}[]{cc}i_{t}&B\\\ 0&j\end{array}\right].$

We state without proof a few Lemmas, referring to [4] and [17] instead.

###### Lemma 14

The set of $J\in{\cal J}$ that are regular, and whose components $i_{t}$ and
$j$ are regular as well, is comeager and non-empty. Gromov compactness holds.

###### Lemma 15

The complex $\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(K_{m})$ has a filtration given by
$\mu_{CZ}(\mathfrak{p}_{*}(x))$. In other words, the projection
$\mathfrak{p}_{*}(x)$ corresponds to a periodic orbit of $G$, the projection
$\mathfrak{p}_{*}(\tilde{x})$ corresponds to a capping of
$\mathfrak{p}_{*}(x)$, and the Conley-Zehnder index of
$\mathfrak{p}_{*}(\tilde{x})$ lifts to a filtration on
$\lim\limits_{\begin{subarray}{c}\leftarrow\\\ m\end{subarray}}CF^{*}(K_{m})$.

###### Lemma 16

A Floer trajectory $u$ of $(K_{m},J)$ descends to a Floer trajectory
$\mathfrak{p}_{*}u$ of $(G,j)$. The Floer differential decomposes as

$\partial^{fl}=\partial_{0}+\partial_{1}+\partial_{2}+\dots$

where $\partial_{\ell}$ counts Floer trajectories $u$ such that
$\mathfrak{p}_{*}u$ sits in a $k$-dimensional strata of the moduli space of
Floer trajectories of $(G,j)$. The decomposition of $\partial^{fl}$ satisfies

$(\partial_{0}+\partial_{1}+\dots+\partial_{k})^{2}=0$

for all $k\geq 0$.

###### Lemma 17

If $\partial_{0}(X)=0$, then there exists
$Y\in\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}(K_{m})$ such that $\partial_{0}(Y)=X$. The action of
$Y$ is

${\cal A}(Y)\geq{\cal A}(X)-A.$

###### Lemma 18

A sequence $\\{X_{1},X_{2},X_{3},...\\}$ of elements in
$\lim\limits_{\begin{subarray}{c}\leftarrow\\\ m\end{subarray}}CF^{*}(K_{m})$
satisfies

$\mu_{CZ}(\mathfrak{p}_{*}(X_{i}))\rightarrow\pm\infty$

if and only if

${\cal A}(X_{i})\rightarrow\pm\infty.$

We mimic the proof of Theorem 1 in [4]

Proof of Proposition 6: Let
$X\in\lim\limits_{\begin{subarray}{c}\begin{subarray}{c}\rightarrow\\\
a\end{subarray}\end{subarray}}\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(K_{m})$ be a cocycle. By design the
continuation maps $c:CF^{*}(K_{m})\rightarrow CF^{*}(K_{m-1})$ are canonical
surjections. Because of this, $X$ can be written as a sum of weighted periodic
orbits

$X=\sum_{i=0}^{\infty}C_{i}x_{i},$

where $C_{i}\in\Lambda$, $x_{i}\in{\cal P}(K)$, and
$\lim\limits_{i\rightarrow\infty}{\cal A}(C_{i}x_{i})\neq-\infty$. Lemma 18
states that there exists a lower bound $P$ such that

$\mu_{CZ}(C_{i}\mathfrak{p}_{*}(x_{i}))\geq P$

for all $i$. Let

$X_{p}=\sum_{\mu_{CZ}(C_{i}\mathfrak{p}_{*}(x_{i}))=p}C_{i}x_{i}.$

Suppose there exists
$Y\in\lim\limits_{\begin{subarray}{c}\begin{subarray}{c}\rightarrow\\\
a\end{subarray}\end{subarray}}\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(K_{m})$, decomposed into filtered pieces
as

$Y=\sum_{p=P}^{\infty}Y_{p},$

and satisfying

(18)
$\partial_{0}(Y_{p})=-\partial_{1}(Y_{p-1})-\partial_{2}(Y_{p-2})-\dots-\partial_{p-P}(Y_{P})+X_{p}.$

Then

$X_{p}=\partial_{0}(Y_{p})+\partial_{1}(Y_{p-1})+\partial_{2}(Y_{p-2})+\dots+\partial_{p-P}(Y_{P}),$
$X=\sum_{p=P}^{\infty}X_{p}=\sum_{p=P}^{\infty}\partial_{0}(Y_{p})+\partial_{1}(Y_{p-1})+\partial_{2}(Y_{p-2})+\dots+\partial_{p-P}(Y_{P}),$

and so, rearranging,

$X=\sum_{p=P}^{\infty}(\partial_{0}+\partial_{1}+\partial_{2}\dots)(Y_{p})=\sum_{p=P}^{\infty}\partial(Y_{p})=\partial(Y).$

It remains to construct such a $Y$. We do so inductively.

As $X$ is a cocycle, the filtration on $\partial$ yields the equation

(19) $\partial_{k}(X_{P})+\partial_{1}(X_{P+1})\dots+\partial_{0}(X_{P+k})=0$

for all $k$. In particular, $\partial_{0}(X_{P})=0$. By Lemma 17, there is a
unique $Y_{P}$ satisfying $\partial_{0}(Y_{P})=X_{P}$. So suppose that there
exists $Y_{p}$ such that equation (18) holds. Lemma 16 states that

(20) $(\partial_{0}+\partial_{1}+...+\partial_{k})^{2}=0$

for all $k$.

Consider the equation

(21)
$\partial_{1}(Y_{p})+\partial_{2}(Y_{p})+\dots+\partial_{p+1-P}(Y_{P})+X_{p+1}.$

Applying $\partial_{0}$ to (21), then manipulating via (20) and (19),

(22)
$\displaystyle-\partial_{0}\partial_{1}(Y_{p})-\partial_{0}\partial_{2}(Y_{p-1})$
$\displaystyle-\dots-\partial_{0}\partial_{p+1+P}(Y_{P})+\partial_{0}X_{p+1}$
(23)
$\displaystyle=-\partial_{0}\partial_{0}(Y_{p})-\partial_{1}(\partial_{0}+\partial_{1})(Y_{p})$
(24) $\displaystyle\hskip
11.38092pt-\partial_{0}(\partial_{0}+\partial_{1})(Y_{p-1})-\partial_{1}(\partial_{0}+\partial_{1}+\partial_{2})(Y_{p-1})-\partial_{2}(\partial_{0}+\partial_{1}+\partial_{2})(Y_{p-1})$
(25) $\displaystyle\hskip 11.38092pt-\dots$ (26) $\displaystyle\hskip
11.38092pt-\partial_{1}(X_{p})-\dots-\partial_{p+1}(X_{P})$ (27)
$\displaystyle=\partial_{0}(\partial_{0}(Y_{p})+\partial_{1}(Y_{p-1})+\partial_{2}(Y_{p-2})+\dots+\partial_{p-P}(Y_{P}))$
(28) $\displaystyle\hskip
11.38092pt+(\partial_{0}+\partial_{1})(\partial_{0}(Y_{p-1})+\partial_{1}(Y_{p-2})+\dots+\partial_{p-1-P}(Y_{P}))$
(29) $\displaystyle\hskip 11.38092pt+\dots$ (30) $\displaystyle\hskip
11.38092pt+(\partial_{0}+\dots+\partial_{p+1-P})\partial_{0}(Y_{P})$

Use the induction hypothesis (18) to replace each statement about $Y$ in (30)
with one about $X$:

$\displaystyle=\partial_{0}(X_{p})+(\partial_{0}+\partial_{1})(X_{p-1})+\dots+(\partial_{0}+\dots+\partial_{p+1-P})(X_{P})$
$\displaystyle=\sum_{k=P}^{p}\partial_{0}(X_{k})+\partial_{1}(X_{k-1})+\dots+\partial_{k}(X_{P})$
$\displaystyle=0,$

where the final equality is equation (19).

By Lemma 17 there exists $Y_{p+1}$ satisfying

$\partial_{0}(Y_{p+1})=\partial_{1}(Y_{p})+\partial_{2}(Y_{p})+\dots+\partial_{p+1-P}(Y_{P})+X_{p+1}.$

By construction, $\mu_{CZ}(\mathfrak{p}_{*}(Y_{p}))=p$. By Lemma 18, the
action of $Y_{p}$ grows like $p$. In particular,

$\lim_{p\rightarrow\infty}{\cal A}(Y_{p})\neq-\infty.$

Thus, $Y$ is indeed an element of
$\lim\limits_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(K_{m})$. This proves the induction
hypothesis, and so proves the proposition.

∎

###  4.3 Proof of Theorem 1

Recall that the Reeb flow of a contact one-form is Zoll if it is periodic, and
if all simple Reeb orbits have the same period. We prove the following
proposition.

###### Proposition 7

Let $\Sigma$ be a hypersurface of contact type with Zoll Reeb flow in a
monotone symplectic manifold $E$. Choose a sequence of positive numbers
$\epsilon_{1},\epsilon_{2},\dots\rightarrow 0$ such that

$U_{\epsilon_{i}}\simeq(-\epsilon_{i},\epsilon_{i})\times\Sigma\subset E$

is a tubular neighborhood. Suppose that for every $i$ there exists
$\delta_{i}\in(-\epsilon_{i},\epsilon_{i})$ such that
$\\{\delta_{i}\\}\times\Sigma$ contains a leaf-wise intersection point of some
Hamiltonian $F$. Then $\Sigma$ contains a leaf-wise intersection point of $F$
as well.

###### Proof.

The proof is standard; we sketch the argument and refer to [5] for details.
Let $\\{z_{i}\\}$ be the set of leaf-wise intersection points in the statement
of the proposition. Without loss of generality, assume that the time support
of $z_{i}$ occurs in the interval $[1/2,1]$. Fix embedded paths
$\\{y_{i}:[0,1/2]\rightarrow\\{\delta_{i}\\}\times\Sigma\\}$ that are flow
lines of the vector field $\rho(t)\sigma_{i}R_{\alpha}$ for some
$\sigma_{i}\in[0,1)$ and satisfy $y_{i}(0)=z_{i}(1)$ and
$y_{i}(1/2)=z_{i}(1/2)$. Let $x_{i}=y_{i}\\#z_{i}$ be the composition. Thus,
there is a Hamiltonian vector field $X_{i}$ defined by

$X_{i}=\left\\{\begin{array}[]{cc}\rho(t)\sigma_{i}R_{\alpha}&t\in[0,1/2]\\\
X_{F}&t\in[1/2,1]\end{array}\right.$

such that

$\dot{x}_{i}=X_{i}(x_{i}).$

As $supp(F)$ is compact, $y_{i}$ is embedded, and the Reeb flow is Zoll, the
$L^{2}$ norm of $x_{i}$ is uniformly bounded and the family $\\{x_{i}\\}$ is
equicontinuous. There therefore exists a subsequence of $x_{i}$ with a ${\cal
C}^{0}$ limit $x$ of $x_{i}$. Abusing notation, we replace the sequence
$x_{i}$ with the convergent subsequence.

Necessarily, there is a constant $\sigma\in[0,1]$ such that

$\lim_{i\rightarrow\infty}\sigma_{i}=\sigma.$

Let

$X=\left\\{\begin{array}[]{cc}\rho(t)\sigma R_{\alpha}&t\in[0,1/2]\\\
X_{F}&t\in[1/2,1]\end{array}\right..$

Then

$\displaystyle x(t)-x(0)-\int_{0}^{t}X(x(\tau))d\tau$
$\displaystyle=\lim_{i\rightarrow\infty}\left(x_{i}(t)-x_{i}(0)-\int_{0}^{t}X(x(\tau))d\tau\right)$
$\displaystyle=\lim_{i\rightarrow\infty}\left(\int_{0}^{t}\dot{x}_{i}(\tau)d\tau-\int_{0}^{t}X(x(\tau))d\tau\right)$
$\displaystyle=\lim_{i\rightarrow\infty}\left(\int_{0}^{t}\dot{x}_{i}(\tau)-X(x_{i}(\tau))d\tau\right)+\lim_{\rightarrow\infty}\left(\int_{0}^{t}X(x_{i}(\tau))-X(x(\tau))d\tau\right)$
$\displaystyle=\lim_{i\rightarrow\infty}\left(\int_{0}^{t}\dot{x}_{i}(\tau)-X_{i}(x_{i}(\tau))+(\sigma_{i}-\sigma)\rho(t)R_{\alpha}(x_{i})d\tau\right)+\lim_{i\rightarrow\infty}\left(\int_{0}^{t}X(x_{i}(\tau))-X(x(\tau))d\tau\right)$
$\displaystyle=0.$

Boot-strapping, $x$ is ${\cal C}^{\infty}$ satisfying

$\dot{x}=X(x(t)),$

with $x(0)=x(1)\in\Sigma$.

The periodic orbits of $X$ correspond to leaf-wise intersection points of $F$;
since $x(0)\in\Sigma$, $x$ corresponds to a leaf-wise intersection point of
$F$ on $\Sigma$.

∎

We are finally ready to prove the Theorem.

Proof of Theorem 1: Let $U_{\epsilon}=(R-\epsilon,R+\epsilon)\times\Sigma$.
Suppose that there does not exist a circle subbundle inside $U_{\epsilon}$
with a leaf-wise intersection point of $F$. Set $P_{1}=R-\epsilon$. Stringing
together Propositions 1, 2, 4, 5, and 3, there is an isomorphism

$\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(K_{m})\simeq\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}HF^{*}_{(a,\infty)}(H_{n})$

There is a surjective map

$H\left(\lim\limits_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}CF^{*}_{(a,\infty)}(K_{m})\right)\rightarrow\lim\limits_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(K_{m}).$

The left-hand side is isomorphic to $\widehat{SH_{*}}(K)$, which, by
Proposition 6 is zero:

$\widehat{SH_{*}}(K)=0.$

Thus,

$\lim_{\begin{subarray}{c}\rightarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}HF^{*}_{(a,\infty)}(K_{m})=0.$

The right-hand side is $SH^{*}(H)$, which, by Lemma 22, is non-vanishing under
the assumption that symplectic cohomology is non-vanishing.

We reach a contradiction and conclude that there exists a circle subbundle in
$U_{\epsilon}$ with a leaf-wise intersection point of $F$. Let
$\epsilon_{1},\epsilon_{2},\dots\rightarrow 0$ be a sequence defining a nested
sequence of neighborhoods $U_{\epsilon_{i}}$. Each $U_{\epsilon_{i}}$ contains
a circle subbundle with a leaf-wise intersection point. By Proposition 7,
$\Sigma=\\{R\\}\times\Sigma$ contains a leaf-wise intersection point.

∎

Examples of negative monotone line bundles with non-vanishing symplectic
cohomology were found by Ritter [12][13]. He showed that the symplectic
cohomology is non-vanishing whenever the base $M$ is toric.

###### Corollary 1

Let $E\rightarrow M$ be a monotone negative line bundle with negativity
constant $k$ and monotonicity constant $\kappa$ over a toric symplectic
manifold. For any compactly-supported Hamiltonian, the
radius-$\frac{1}{\sqrt{k\kappa\pi}}$ circle subbundle contains a leaf-wise
intersection point.

##  5 Other line bundles

Monotone line bundles have exactly one “special” circle subbundle; the
hypersurface for which Theorem 1 produces a leaf-wise intersection point. This
is the circle subbundle on which Floer-essential Lagrangian tori appear (in
toric line bundles) and it is the only circle subbundle on which a Rabinowitz-
Floer-type invariant $RFH^{*}(\Sigma)$ is non-zero [14][19].

Non-monotone line bundles can have multiple special subbundles. Although very
little is understood about their Floer-essential Lagrangian tori, we showed in
[19] that $RFH^{*}(\Sigma)$ is non-vanishing at a finite number of circle
subbundles in weak+ monotone line bundles. Precisely, the cohomological class
$\mathfrak{p}^{*}c_{1}^{E}$ acts on the quantum cohomology of $E$,
$QH^{*}(E)$, by quantum cup product. This action has a finite set of
eigenvalues $\\{\lambda_{1},\dots,\lambda_{n}\\}$.

######  Theorem 7 (Venkatesh [19])

The invariant $RFH^{*}(\Sigma_{R})$ of a circle subbundle of radius $R$ in a
weak+-monotone toric negative line bundle is non-vanishing precisely when

$R=\sqrt{\frac{ev(\lambda_{i})}{k\pi}}$

for some $\lambda_{i}\in\\{\lambda_{1},\dots,\lambda_{n}\\}$.

We conjecture that each of the hypersurfaces indicated in the Theorem contain
a leaf-wise intersection point for any compactly-supported Hamiltonian
function.

###### Conjecture 1

Let $E$ be a negative line bundle, and let
$F:E\times\mathbb{R}\rightarrow\mathbb{R}$ be a compactly-supported
Hamiltonian function. A circle subbundle $\Sigma_{R}$ of radius $R$ contains a
leaf-wise intersection point of $F$ if

$R=\sqrt{\frac{ev(\lambda_{i})}{k\pi}}.$

The invariant in Theorem 7 builds upon a completion of the symplectic cochain
complex $SC^{*}(H)$. Recall from equation (2) that $SC^{*}(H)$ has a valuation
by action, denoted by ${\cal A}:SC^{*}(H)\rightarrow\mathbb{R}$. This
valuation defines a non-Archimedean norm, via

$||X||=e^{-{\cal A}(X)}.$

Denote by $\partial$ the differential on $SC^{*}(H)$. Let

$\widehat{\ker(\partial)}$

be the completion of $\ker(\partial)$ with respect to the norm $||\cdot||$,
and let

$\overline{\textnormal{im}(\partial)}$

be the closure of $\textnormal{im}(\partial)\subset\widehat{\ker(\partial)}$.
Define completed symplectic cohomology by

$\widehat{SH^{*}}(H):={\raisebox{1.99997pt}{$\widehat{\ker(\partial)}$}\left/\raisebox{-1.99997pt}{$\overline{\textnormal{im}(\partial)}$}\right.}.$

A preliminary result from [19] shows that

(31)
$\widehat{SH^{*}}(H)\left\\{\begin{array}[]{cc}=0&R<\min\limits_{\lambda\in\\{\lambda_{1},\dots,\lambda_{n}\\}}\sqrt{\frac{ev(\lambda)}{k\pi}}\\\
\neq 0&\text{else}\end{array}\right..$

The proof of Theorem 1 relies on the monotonicity of the line bundle to prove
technical results on action bounds. Replacing monotonicity with weak+
monotonicity, our techniques applied to the result (31) yield a less precise
result.

###### Theorem 3

Let $E$ be a weak+-monotone negative line bundle. Let $\lambda$ be a largest
non-zero eigenvalue of the action of $\mathfrak{p}^{*}c_{1}^{E}$ by quantum
cup product on $QH^{*}(E)$. For any compactly-supported Hamiltonian, there
exists a circle subbundle $\Sigma_{r}$ with
$r\in\left(\sqrt{\frac{ev(\lambda)}{k\pi}},\infty\right)$ such that
$\Sigma_{r}$ has a leaf-wise intersection point.

###### Proof.

As in the proof of Proposition 2, there is a constant $C$, independent of $a$,
that produces chain maps

$SC^{*}_{(a,\infty)}(H)\rightarrow SC^{*}_{(a-C,\infty)}(H^{F})\rightarrow
SC^{*}_{(a-2C,\infty)}(H)$

such that the induced map on homology

$SH^{*}_{(a,\infty)}(H)\xrightarrow{\simeq}SH^{*}_{(a-2C,\infty)}(H)$

is the canonical inclusion. It follows that there are maps on the completions

$\widehat{SH^{*}}(H)\rightarrow\widehat{SH^{*}}(H^{F})\rightarrow\widehat{SH^{*}}(H)$

such that the composition is an isomorphism.

Similarly, there are maps

$\widehat{SH^{*}}(H^{F})\rightarrow\widehat{SH^{*}}(H)\rightarrow\widehat{SH^{*}}(H^{F})$

whose composition is an isomorphism. We conclude that there is an isomorphism

(32) $\widehat{SH^{*}}(H^{F})\xrightarrow{\simeq}\widehat{SH^{*}}(H).$

Choose
$R<\min\limits_{\lambda\in\\{\lambda_{1},.\dots,\lambda_{n}\\}}\sqrt{\frac{ev(\lambda)}{k\pi}}$.
By (31),

(33) $\widehat{SH^{*}}(H)=0.$

On the other hand, Ritter showed in [12] that if $\lambda\neq 0$,

(34) $SH^{*}(H)\neq 0.$

Standard results in Floer theory yield an isomorphism

(35) $SH^{*}(H^{F})\simeq SH^{*}(H).$

If there does not exist a circle subbundle $\Sigma_{r}$,
$r\in\left(\sqrt{\frac{ev(\lambda)}{k\pi}},\infty\right)$, with a leaf-wise
intersection point, then ${\cal P}(H^{F})$ is a finite set, and
$SC^{*}(H^{F})$ is finitely generated as a vector space. In particular,
$\ker(\partial)$ is finitely generated, and is thus already complete.
Likewise, $\textnormal{im}(\partial)$ is finitely generated and is thus
already closed. There is therefore an isomorphism

(36) $\widehat{SH^{*}}(H^{F})\xrightarrow{\simeq}SH^{*}(H^{F}).$

Stringing together the equations (32) – (36),

$0\neq SH^{*}(H)\simeq
SH^{*}(H^{F})\simeq\widehat{SH^{*}}(H^{F})\simeq\widehat{SH^{*}}(H)=0,$

and we reach a contradiction. We conclude that there exists a circle subbundle
$\Sigma_{r}$ with a leaf-wise intersection point.

∎

##  6 Leaf-wise intersection points from Lagrangians

The circle subbundles studied in this paper are conjecturally distinguished by
Floer-essential Lagrangian submanifolds living inside them. Such submanifolds
point to a variety of fixed-point results, including results about leaf-wise
intersection points. In this section, we prove a theorem about leaf-wise
intersection points on more general monotone symplectic manifolds.

###### Theorem 8

Let $(E,\Omega)$ be a monotone symplectic manifold that is either compact or
convex at infinity. Let $\Sigma\subset E$ be a hypersurface of contact type
containing a compact Lagrangian $L$, and let
$(-\epsilon,\epsilon)\times\Sigma$ be a Weinstein neighborhood of $\Sigma$. If
there exists a local system $\gamma$ such that the Lagrangian Floer homology
$HF^{*}(L,\gamma)$ is non-vanishing, then for any Hamiltonian perturbation,
there exists $r\in(-\epsilon,\epsilon)$ such that $\\{r\\}\times\Sigma$
contains a leaf-wise intersection point. If the Reeb flow of $\Sigma$ is Zoll,
$\Sigma$ itself contains a leaf-wise intersection point.

The proof of Theorem 8 follows a well-documented algorithm. Similarly to the
recipes of Sections 4 and 5 there exists a Floer invariant, denoted by
$\widehat{SH^{*}}(\Sigma,E)$, that encodes leaf-wise intersection points close
to $\Sigma$. Precisely, if $\widehat{SH^{*}}(\Sigma,E)\neq 0$ then there
exists a contact hypersurface arbitrarily close to $\Sigma$ with a leaf-wise
intersection point. If the Reeb flow of $\Sigma$ is Zoll, then this encoding,
combined with Proposition 7, shows that $\Sigma$ itself has a leaf-wise
intersection point. To prove the Theorem, it thus suffices to show that
$\widehat{SH^{*}}(\Sigma,E)\neq 0$.

Our method for proving that $\widehat{SH^{*}}(\Sigma,E)\neq 0$ originated in
[16] and was adapted to the current framework in our paper [18]. There is a
map

$\widehat{SH^{*}}(\Sigma,E)\rightarrow HF^{*}(L,\gamma)$

that fits into a commutative diagram with the quantum cohomology of $E$,
denoted by $QH^{*}(E)$.

(37) ${QH^{*}(E)}$${\widehat{SH^{*}}(\Sigma,E)}$${HF^{*}(L,\gamma)}$

$HF^{*}(L,\gamma)$ is a ring with unit, and the image of the map

$QH^{*}(E)\rightarrow HF^{*}(L,\gamma)$

always contains the unit. If $HF^{*}(L,\gamma)\neq 0$, then, in particular,
the image of this map is non-zero. It follows that

$\widehat{SH^{*}}(\Sigma,E)\neq 0$

as well.

The cohomology theory $\widehat{SH^{*}}(\Sigma,E)$ relies on a new family of
Hamiltonians, defined as follows. A contact hypersurface $(\Sigma,\alpha)$ has
a family of nested Weinstein neighborhood

$U_{1}\supset U_{2}\supset U_{3}\supset\dots\supset\Sigma$

whose intersection is $\Sigma$

$\bigcap_{i}U_{i}=\Sigma.$

We assume without loss of generality that there are positive real numbers
$\epsilon_{1}>\epsilon_{2}>\epsilon_{3}\dots>0$ with symplectomorphisms

$U_{i}\simeq(-\epsilon_{i},\epsilon_{i})_{r}\times\Sigma,$

when the latter is equipped with the symplectic form $d(e^{r}\alpha)$. Fix
$i\in\mathbb{N}$. Let $h_{n}^{i}:\mathbb{R}_{x}\rightarrow\mathbb{R}$ be a
smoothing of the function that is

  1. 1.

equal to $-nx$ on $(-\epsilon_{i},0]$,

  2. 2.

equal to $nx$ on $[0,\epsilon_{i})$,

  3. 3.

and constant everywhere else.

See Figure 7.

$\mathbb{R}$$r$$h_{n}^{i}$$-\epsilon_{i}$$\epsilon_{i}$ Figure 7: The
hamiltonians $h_{n}^{i}$

As in Subsection 2.1, let $\rho(t)$ be a bump function with support on
$[0,1/2]$ that integrates to 1, and let $F$ be a compactly-supported
Hamiltonian with time-support in $[1/2,1]$. Define

$H_{n}^{i}=\rho(t)h_{n}^{i}(e^{r})+F.$

Define quotient complexes

$CF^{*}_{a}(H_{n}^{i}):={\raisebox{1.99997pt}{$CF^{*}(H_{n}^{i})$}\left/\raisebox{-1.99997pt}{$CF^{*}_{(a,\infty)}(H_{n}^{i})$}\right.}.$

Choose monotone-increasing homotopies $CF^{*}_{a}(H_{n}^{i})\rightarrow
CF^{*}_{a}(H_{n+1}^{i})$. With these choices, the family
$\\{CF^{*}_{a}(H_{n}^{i})\\}$ is bidirected; once by $n\in\mathbb{N}_{>0}$ and
once by $a\in\mathbb{R}$.

Define the reduced symplectic cohomology of $U_{i}$, written
$\widehat{SH^{*}}(U_{i})$, to be the cohomology of the cochain complex

$\widehat{SC^{*}}(U_{i})=\lim_{\begin{subarray}{c}\leftarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}CF^{*}_{a}(H_{n}^{i}).$

As with Lemma 3 we may assume, without changing the leaf-wise intersection
points of $F$, that the periodic orbits of each $H_{n}^{i}$, and of $F$
itself, are non-degenerate.

###  6.1 Lagrangian quantum cohomology

In this subsection we recall a model of $HF^{*}(L,\gamma)$ called Lagrangian
quantum cohomology and we define and study the maps in Figure 37. This picture
was constructed in an analogous setup in [18]. Rather than repeat each
argument, we recall the results detailed there and argue that we can transfer
each argument to this setting. We refer to [18] for a detailed exposition.

Choose a Morse function $f:L\rightarrow\mathbb{R}$ and a metric $g$ on $L$.
Denote by $\Phi_{t}$ the gradient flow of $-\nabla_{g}(f)$. Choose critical
points $x$ and $y$ of $f$ and a relative homology class $\beta\in H^{2}(E,L)$,
and denote by ${\cal M}_{\ell}(f,g,x,y,\beta)$ the moduli space of tuples
$(u_{1},\dots,u_{\ell})$, where

  1. 1.

$u_{i}:(D^{2},\partial D^{2})\rightarrow(E,L)$ is a non-constant
$J$-holomorphic disk,

  2. 2.

there exists $t\in(-\infty,0)$ such that $u_{i}(-1)=\Phi_{t}(u_{i+1}(1))$
whenever $i\leq 1<\ell$,

  3. 3.

$u_{1}(1)$ is in the unstable manifold of $x$ and $u_{\ell}(-1)$ is in the
stable manifold of $y$,

  4. 4.

and $[u_{1}\\#u_{2}\\#\dots\\#u_{\ell}]\in\beta$.

Let $Aut(D^{2},\pm 1)$ be the automorphisms of the disk fixing $-1$ and $1$,
so that $Aut(D^{2},\pm 1)$ acts on ${\cal M}_{\ell}(f,g,x,y,\beta)$. Denote by
${\cal M}_{0}(f,g,x,y)$ the space of Morse trajectories from $x$ to $y$. Note
that $\mathbb{R}$ acts canonically on ${\cal M}_{0}(f,g,x,y)$. Define ${\cal
M}^{0}(f,g,x,y,\beta)$ to be the rigid elements of

${\raisebox{1.99997pt}{${\cal
M}_{\ell}(f,g,x,y,\beta)$}\left/\raisebox{-1.99997pt}{$Aut(D^{2},\pm
1)$}\right.}\cup{\raisebox{1.99997pt}{${\cal
M}_{0}(f,g,x,y)$}\left/\raisebox{-1.99997pt}{$\mathbb{R}$}\right.}.$

We call the elements of ${\cal M}^{0}(f,g,x,y)$ pearly trajectories.

Choose a local system $\gamma$ on $L$, and note that there is a pairing
$\langle\gamma,\sigma\rangle$ for each $\sigma\in H_{1}(L)$. Let
$CF^{*}(L,\gamma)$ be the cochain complex that is generated by critical points
of $f$

$CF^{*}(L,\gamma)=\bigoplus_{x\in Crit(f)}\Lambda\langle x\rangle$

and whose differential is given on critical points of $f$ by

$\partial(x)=\sum_{\begin{subarray}{c}y\in Crit(f)\\\ \ell\in\mathbb{N}\\\
\beta\in H^{2}(M,L)\end{subarray}}\sum_{{\bf u}\in{\cal
M}^{0}(f,g,x,y)}T^{-\Omega(\beta)}\langle\gamma,[\partial u_{1}\\#\partial
u_{2}\\#\dots\\#\partial u_{\ell}]\rangle y.$

Lagrangian Floer cohomology, denoted by $HF^{*}(L,\gamma)$, is the cohomology
of $CF^{*}(L,\gamma)$.

Analogously to symplectic cohomology, Lagrangian Floer cohomology has action
truncated versions $HF^{*}_{(a,\infty)}(L;\gamma)$ and $HF^{*}_{a}(L;\gamma)$.

######  Lemma 19 (Lemma 6 in [18])

There is a map

$\widehat{SH^{*}}(U_{i})\rightarrow HF^{*}(L,\gamma).$

###### Proof.

Lemma 6 in [18] and the discussion following the proof of Lemma 6 consider
autonomous Hamiltonians, which we denote here by $\\{H_{n}^{\prime}\\}$, on a
domain with convex boundary and produce a map

$\lim_{\begin{subarray}{c}\leftarrow\\\
a\end{subarray}}\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}HF^{*}_{a}(H_{n}^{\prime})\rightarrow HF^{*}_{a}(L;\gamma)$

for any $n$ and $a$. This map counts “half-cylinders”, amalgamated with pearly
trajectories.

The only facts used about the family $\\{H_{n}^{\prime}\\}$ to construct this
map are that

  1. 1.

$H_{n}\leq 0$ on a small neighborhood of $L$,

  2. 2.

and the continuation maps $H_{n}\rightarrow H_{n+1}$ are monotone-increasing.

These can be easily arranged for our family $\\{H_{n}\\}$ without changing the
Floer complexes by shifting each function $h_{n}$ and the Hamiltonian $F$ down
by a constant. After these adjustments, the proof translates verbatim.

∎

######  Lemma 20 (Lemma 7 in [18])

The map

$\Phi:\widehat{SH^{*}}(U_{i})\rightarrow HF^{*}(L,\gamma).$

is non-vanishing.

###### Proof.

The proof of Lemma 7 in [18] translates verbatim. We briefly sketch it here.
By construction, the diagram

(38)
${HF^{*}(H_{0})}$${\widehat{SH^{*}}(\Sigma,E)}$${HF^{*}(L;\gamma)}$$\scriptstyle{\iota}$$\scriptstyle{\phi}$$\scriptstyle{\Phi}$

commutes. It thus suffices to show that the map

$\phi:HF^{*}(H_{0})\rightarrow HF^{*}(L;\gamma)$

is non-zero.

Assume without loss of generality that $f$ has a unique minimum
$\mathfrak{m}$. As $HF^{*}(L,\gamma)$ is a unital ring, $\mathfrak{m}$
represents the unit, and as $HF^{*}(L;\gamma)$ is non-zero by assumption,
$[\mathfrak{m}]\neq 0$ as well. Denote by ${\cal M}_{0}(\mathfrak{m})$ the
moduli space of all pearly trajectories $(u_{1},\dots,u_{\ell})$ with
$u_{\ell}$ now possibly constant, $u_{\ell}(-1)$ in the stable manifold of
$\mathfrak{m}$, and no restrictions on $u_{1}(1)$. Because $\mathfrak{m}$ is a
minimum, there is a single rigid element in ${\cal M}_{0}(\mathfrak{m})$: the
constant map $(u_{\ell}:D^{2}\rightarrow\mathfrak{m})$.

Let $H_{s}$ be a family of Hamiltonians that is identically zero when $s<<0$
and is equal to $H_{0}$ when $s>>0$. Choose a generic family $J_{s}$ of
$\Omega$-compatible almost-complex structures. Let ${\cal M}^{0}(x)$ be the
moduli space of rigid trajectories $w:\mathbb{R}\times S^{1}\rightarrow E$
that satisfy the Floer equation for $(H_{s},J_{s})$ and with
$\lim\limits_{s\rightarrow\infty}w(s,t)=x(t)$. Define

$Z=\sum_{\begin{subarray}{c}x\in{\cal P}(H_{0})\\\ w\in{\cal
M}^{0}(x)\end{subarray}}T^{\Omega([\tilde{x}\\#w])}x.$

Concatenating Floer cylinders, $\phi$ takes $[Z]$ to a cohomological-level sum
of the elements in ${\cal M}_{0}(\mathfrak{m})$. Thus,

$\phi([Z])=[\mathfrak{m}]\neq 0.$

∎

###### Corollary 6

The cohomology theory $\widehat{SH^{*}}(U_{i})$ is non-zero.

###  6.2 Proof of Theorem 8

The proof of Theorem 8 follows almost immediately from the previous section.

Proof of Theorem 8: Choose a neighborhood
$U_{i}\simeq(-\epsilon_{i},\epsilon_{i})\times\Sigma$ of $\Sigma$ and assume
for contradiction that there does not exist a hypersurface
$\\{r\\}\times\Sigma\subset U_{i}$ with a leaf-wise intersection point of $F$.
Then all elements of

${\cal P}({\cal H}^{i}):=\bigcup_{n}{\cal P}(H_{n}^{i})$

correspond to periodic orbits of $F$ with starting point in $E\setminus
U_{i}$. By assumption, the periodic orbits of $F$ are non-degenerate, and
therefore finite. Let $\\{\tilde{x}_{1},\dots,\tilde{x}_{\ell}\\}$ be the
elements of ${\cal P}({\cal H}^{i})$ equipped with fixed cappings. Thus, there
exists a constant $C_{\mu}>0$ with

$|\mu_{CZ}(\tilde{x}_{j})|<C_{\mu}$

for all $1\leq j\leq\ell$.

Similarly, there exists a constant $C_{F}$ with

$|{\cal A}_{F}(\tilde{x}_{j})|<C_{F}.$

By construction, $H_{n}^{i}(x(0))\simeq n\cdot\epsilon_{i}$. Therefore,
adjusting $C_{F}$ slightly if necessary,

${\cal A}_{H_{n}^{i}}(\tilde{x}_{j})>n\cdot\epsilon_{i}-C_{F}$

for all $n,j$.

Consider a graded component $\widehat{SH^{\ell}}(U_{i})$. By the monotonicity
of $E$, there exists a constant $\kappa>0$ with

$c_{1}^{TE}=\kappa[\Omega]$

on elements of the image of $\pi_{2}(E)\hookrightarrow H^{2}(E)$. Let
$\lambda\in\Lambda$. Then

$|\lambda x_{j}|={\ell}$

implies that

$\kappa ev(\lambda)>{\ell}-C_{\mu},$

or

$ev(\lambda)>\frac{{\ell}-C_{\mu}}{\kappa}.$

It follows that

${\cal
A}_{H_{n}^{i}}(\lambda\tilde{x}_{j})>n\cdot\epsilon_{i}-C_{F}+\frac{{\ell}-C_{\mu}}{\kappa}.$

Fix $a>>0$. For large enough $n$,

$n\cdot\epsilon_{i}-C_{F}+\frac{{\ell}-C_{\mu}}{\kappa}>a$

for all $j$, and so

$\lim_{\begin{subarray}{c}\rightarrow\\\
n\end{subarray}}HF^{\ell}_{a}(H_{n}^{i})=0.$

Taking the inverse limit over $a\rightarrow\infty$, we conclude that

$\widehat{SH^{*}}(U_{i})=0,$

contradicting Corollary 6. We conclude that there exists a contact
hypersurface $\\{r\\}\times\Sigma\subset U_{i}$ with a leaf-wise intersection
point of $F$. Taking the limit as $i\rightarrow\infty$, we conclude the first
statement of the theorem.

The second statement now follows immediately from Proposition 7.

∎

##  Appendix A Hamiltonians with smaller time-support

Let ${\cal H}_{n}$ be a Hamiltonian as in Subsection 2.1. Thus, there is a
Hamiltonian $H_{n}$ of the form

$H_{n}=h_{n}(\kappa\pi r^{2})+(1+\pi r^{2})\mathfrak{p}^{*}G,$

where $G:B\rightarrow\mathbb{R}$ is a Morse function and $h_{n}(\kappa\pi
r^{2})=n\kappa\pi r^{2}$ outside of a compact set $D$; there is a choice of
bump function $\rho:\mathbb{R}\rightarrow\mathbb{R}$ integrating to one, with

$supp(\rho)\subset[0,1/2];$

and

${\cal H}_{n}=\rho(t)\cdot H_{n}.$

The proof of the following Lemma also proves the integrated maximum principle
in our setting.

###### Lemma 21

For generic $\Omega$-tame $J$, conical outside of $D$, the symplectic cochain
complex $SC^{*}({\cal H}_{n})$ is well-defined.

###### Proof.

It suffices to show that the Floer solutions contributing to both the Floer
differential and the continuation maps remain in a compact set. We use the
integrated maximum principal of [1]. We prove that Floer solutions
contributing to the Floer differential remain in a compact set; the case of
continuation maps proceeds analogously. Let $u:\mathbb{R}\times
S^{1}\rightarrow E$ be a Floer solution for ${\cal H}_{n}$. Choose a generic
circle subbundle $\mathfrak{S}$ of radius $\sigma$ outside of $D$:

$D\cap\mathfrak{S}=\mathop{\varnothing},$

where genericity means that $\Sigma$ intersects $\mathfrak{S}$ transversally.
Let ${\cal D}$ be the region bounded by $\mathfrak{S}$, and denote
$\Sigma=u^{-1}(E\setminus{\cal D})$. Assume for contradiction that $\Sigma$ is
non-empty. Let $v\colon\Sigma\longrightarrow E\setminus{\cal D}$ be the
restriction of $u$. We will equate $\Sigma$ with its image under the inclusion
into $\mathbb{R}\times S^{1}$ and use the coordinates $(s,t)$ induced on
$\textnormal{Int}(\Sigma)$.

Let $c_{x}=h_{n}(k\pi x^{2})-h^{\prime}_{n}(k\pi x^{2})k\pi x^{2}$ be the
$y$-intercept of the tangent line to $h_{n}(k\pi r^{2})$ at $k\pi x^{2}$.
Denote by $X_{G}^{h}$ the unique lift to $\mathfrak{S}$ of the vector field
$X_{G}$ on $B$ that is induced by the connection form $\alpha$. Then on
$\mathfrak{S}$,

${\cal H}_{n}=\rho(t)\cdot
h_{n}^{\prime}(k\pi\sigma^{2})k\pi\sigma^{2}+\rho(t)\cdot(1+k\pi\sigma^{2})\mathfrak{p}^{*}G+\rho(t)\cdot
c_{\sigma}$

(and $(X_{{\cal
H}_{n}})_{t}=\rho(t)\cdot(h_{n}^{\prime}(k\pi\sigma^{2})+\mathfrak{p}^{*}G)R_{\alpha}+\rho(t)\cdot
X_{G}^{h}$). In particular,

${\cal H}_{n}\big{|}_{\mathfrak{S}}=(1+k\pi\sigma^{2})\alpha(X_{{\cal
H}_{n}})-\rho(t)\big{(}h_{n}^{\prime}(k\pi\sigma^{2})+c_{\sigma}\big{)}.$

Let $\partial\Sigma$ be the union of the boundary components of $\Sigma$
mapping into $\mathfrak{S}$. Note that we have chosen $J$ to be $\Omega$-tame,
so that

$E_{J}(v):=\frac{1}{2}\int_{\Sigma}\left(\Omega(\partial_{s}v,J\partial_{s}v\right)+\Omega\left(\partial_{t}v-X_{{\cal
H}_{n}}(v),J(\partial_{t}v-X_{{\cal H}_{n}}(v))\right)ds\wedge dt\geq 0$

Shuffling terms, we have

$\begin{split}E_{J}(v)&=\int_{\Sigma}v^{*}\Omega-v^{*}d{\cal H}_{n}\otimes
dt\\\ &=\int_{\partial\Sigma_{+}}(1+k\pi r^{2})v^{*}\alpha-{\cal
H}_{n}(v(s,t))dt\\\
&=\int_{\partial\Sigma_{+}}(1+k\pi\sigma^{2})v^{*}\alpha-(1+k\pi\sigma^{2})\alpha(X_{{\cal
H}_{n}})\otimes
dt+\rho(t)\cdot\left(h^{\prime}_{n}(k\pi\sigma^{2})-c_{\sigma}\right)dt.\end{split}$

A solution $v(s,t)$ of Floer’s equation satisfies

$(dv-X_{{\cal H}_{n}}\otimes dt)^{(0,1)}=0.$

The conical condition on $J$ says that, on $\mathfrak{S}$, $JR_{\alpha}$ is
proportional to $\partial_{r}$ and $JX_{F}^{h}$ lives in the horizontal
distribution. So altogether, $\alpha(JX_{{\cal H}_{n}})=0$. Thus,

$\displaystyle\int_{\partial\Sigma_{+}}(1+k\pi\sigma^{2})\alpha(dv-X_{{\cal
H}_{n}}\otimes dt)+$
$\displaystyle\rho(t)\left(h^{\prime}_{n}(k\pi\sigma^{2})-c_{\sigma}\right)dt$
$\displaystyle=\int_{\partial\Sigma_{+}}-(1+k\pi\sigma^{2})\alpha\circ J(dv-
X_{{\cal H}_{n}}\otimes dt)\circ
j+\rho(t)\left(h^{\prime}_{n}(k\pi\sigma^{2})-c_{\sigma}\right)dt$
$\displaystyle=\int_{\partial\Sigma_{+}}-\frac{1+k\pi\sigma^{2}}{2k\pi\sigma}dr\circ
dv\circ j+\rho(t)\left(h^{\prime}_{n}(k\pi\sigma^{2})-c_{\sigma}\right)dt$
$\displaystyle\leq\int_{\partial\Sigma_{+}}\rho(t)\left(h^{\prime}_{n}(k\pi\sigma^{2})-c_{\sigma}\right)dt,$

where the last inequality follows because a vector $\zeta$ that is positively-
oriented with respect to the boundary orientation satisfies $dr\circ
j(\zeta)\geq 0$.

$\Sigma$ is a collection of bounded regions in $\mathbb{C}^{*}$ on which
$\rho(t)\left(h^{\prime}_{n}(k\pi\sigma^{2})-c_{\sigma}\right)dt$ is exact:

$\int_{\partial\Sigma_{+}}\rho(t)\left(h^{\prime}_{n}(k\pi\sigma^{2})-c_{\sigma}\right)dt=0.$

Thus,

$E_{J}(v)\leq 0,$

implying that $v$ is constant. But $\Sigma$ is non-empty and intersects
$\mathfrak{S}$ transversally; and we reach a contradiction.

∎

###### Lemma 22

If symplectic cohomology is non-vanishing, the cohomology theory $SH^{*}({\cal
H})$ is also non-vanishing:

$SH^{*}({\cal H})\neq 0.$

###### Proof.

Choose any function $\rho:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$
satisfying

$\rho(s,t)=\rho(t)\hskip 28.45274pts>>0$

and

$\rho(s,t)=\max_{t}\rho(t)\hskip 28.45274pts<<0\text{ and }t\in[0,1].$

For each $n\in\mathbb{N}$, define a family of Hamiltonians $H_{n}^{s}$ by

$H_{n}^{s}=\rho(s,t)H_{n}.$

Then $\\{H_{n}^{s}\\}_{n\in\mathbb{N}}$ induces a continuation map

$SC^{*}({\cal H})\rightarrow SC^{*}(\max_{t}\rho(t)\cdot H),$

which is well-defined by the methods described in the proof of Lemma 21.
Standard methods imply that this induces a map of unital rings on the level of
homology

$SH^{*}(\mathcal{H})\rightarrow SH^{*}(\max_{t}\rho(t)\cdot H).$

As $SH^{*}(\max_{t}\rho(t)\cdot H)$ is non-zero, the unit in
$SH^{*}(\max_{t}\rho(t)\cdot H)$ is non-zero as well. It follows that
$SH^{*}(\mathcal{H})$ has a non-zero unit, and so must be non-zero.

∎

## References

  * [1] Mohammed Abouzaid.  Symplectic cohomology and Viterbo’s theorem.  In Free loop spaces in geometry and topology, page 271–485. Eur. Math. Soc., 2015.
  * [2] Peter Albers, Barney Braham, and Chris Wendl.  On non-separating contact hypersurfaces in symplectic 4-manifolds.  Algebraic & Geometric Topology, pages 697–737, 2010.
  * [3] Peter Albers and Urs Frauenfelder.  Leaf-wise intersections and rabinowitz floer homology.  J. Topol. Anal. 2, (1):77–98, 2010.
  * [4] Peter Albers and Jungsoo Kang.  Vanishing of Rabinowitz Floer homology on negative line bundles.  Mathematische Zeitschrift, pages 493–517, 2017.
  * [5] Michèle Audin and Mihai Damian.  Morse theory and Floer homology.  Springer, London and EDP Sciences, Les Ulis, 2014.
  * [6] Frèdèric Bourgeois and Alexandru Oancea.  Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces.  Duke Mathematics, (1):71–174, 2009.
  * [7] Kai Cieliebak, Urs Frauenfelder, and Alexandru Oancea.  Rabinowitz Floer homology and symplectic homology.  Ann. Sci. Éc. Norm. Supér (4), (43(6)):957–1015, 2010.
  * [8] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono.  Lagrangian Floer theory on compact toric manifolds I.  Duke Math Journal 151, (1):23–174, 2010.
  * [9] Jungsoo Kang.  Symplectic homology of displaceable Liouville domains and leafwise intersection points.  Geometriae Dedicata, 170.
  * [10] J. Moser.  A fixed point theorem in symplectic geometry.  Acta Math., (1-2):17–34, 1978.
  * [11] Alexandru Oancea.  Fibered symplectic cohomology and Leray-Serre spectral sequence.  Journal of symplectic geometry, (3):267–351, 2008.
  * [12] Alexander Ritter.  Floer theory for negative line bundles via Gromov-Witten invariants.  Adv. Math. 262 (2014), page 1035–1106, 2014.
  * [13] Alexander Ritter.  Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties.  Geom. Topol. 20, (4):1941–2052, 2016.
  * [14] Alexander Ritter and Ivan Smith.  The monotone wrapped Fukaya category and the open-closed string map.  Selecta Math. (N.S.) 23, (1):533–642, 2017.
  * [15] Dietmar Salamon.  Lectures on Floer homology. symplectic geometry and topology (Park City, UT, 1997).  Number 7. Amer. Math. Soc., Providence, RI, 1999.
  * [16] Paul Seidel.  A biased view of symplectic cohomology.  Current developments in mathematics Volume 2006 (2008), page 211–253, 2006.
  * [17] Sara Venkatesh.  Completed symplectic cohomology and Liouville cobordisms.  PhD thesis, Columbia University, 2018.
  * [18] Sara Venkatesh.  Rabinowitz Floer homology and mirror symmetry.  Journal of Topology, (No. 1):144–179, 2018.
  * [19] Sara Venkatesh.  The quantitative nature of reduced symplectic cohomology.  Advances in Mathematics, 2021.