# Index bounded relative symplectic cohomology

Yuhan Sun  Hill Center, Department of Mathematics, Rutgers University, 110
Frelinghuysen Rd, Piscataway NJ 08854<EMAIL_ADDRESS>
###### Abstract.

We study the relative symplectic cohomology with the help of an index bounded
contact form. For a Liouville domain with an index bounded boundary, we
construct a spectral sequence which starts from its classical symplectic
cohomology and converges to the relative symplectic cohomology of it inside a
Calabi-Yau manifold. In the appendix we compare the relative symplectic
cohomology of a Liouville domain inside its completion with its classical
symplectic cohomology. As an application, we obtain a version of the Viterbo
isomorphism.

###### Contents

  1. 1 Introduction
  2. 2 Backgrounds
    1. 2.1 Hamiltonian Floer theory on closed manifolds
    2. 2.2 Liouville domain and contact cylinder
    3. 2.3 Relative symplectic cohomology of a Liouville domain
    4. 2.4 Hamiltonian Floer theory on manifolds with convex boundary
    5. 2.5 Lower semi-continuous Hamiltonian functions
  3. 3 A filtration on the completed telescope
    1. 3.1 Auxiliary symplectic forms
    2. 3.2 A filtration which is not exhaustive
  4. 4 The induced spectral sequence
    1. 4.1 Ignoring upper orbits and convergence
    2. 4.2 The first page of the spectral sequence
    3. 4.3 Further remarks
  5. 5 Examples and extensions
    1. 5.1 Lagrangian spheres in Calabi-Yau manifolds
    2. 5.2 Perturbation of Morse-Bott orbits
    3. 5.3 Filled Liouville cobordisms
  6. A Local computations

##  1\. Introduction

Given a closed symplectic manifold $(M,\omega)$ and a compact subset $D$ of
$M$, the relative symplectic cohomology $SH_{M}(D)$ is a Floer-theoretic
invariant, which captures both dynamical and topological information of the
pair $(M,D)$. Its construction by Varolgunes [28] is based on the Hamiltonian
Floer theory of $M$ with more algebraic ingredients. Roughly speaking, one
considers a family of increasing Hamiltonian functions that goes to zero on
$D$ while going to positive infinity on $M-D$. Then $SH_{M}(D)$ is defined as
the homology of a completed telescope of Floer complexes, given by this family
of Hamiltonian functions.

The idea of using Hamiltonian functions “localized at a subset” may date back
to Cieliebak-Floer-Hofer [5] and Viterbo [30]. Recently there are several new
versions of Hamiltonian Floer theories related to this idea and aimed at
various local-to-global problems. Besides [28], let us mention an incomplete
list: Groman [14], McLean [20] and Venkatesh [29]. A priori, the definitions
in [14, 20, 28, 29] are different, depending on whether $M$ is open or closed,
taking completion with the action filtration or the Novikov filtration, and
the orders of taking different limits. It would be interesting to compare
these theories to attack particular problems. But in this article we mainly
focus on the version of $SH_{M}(D)$ by Varolgunes, and just touch one
comparison result about the action-complete symplectic cohomology by
Venkatesh.

Along its definition in [28], several good properties of this invariant
$SH_{M}(D)$ have been established, including the Hamiltonian isotopy
invariance, a Künneth formula, a displaceability criterion and the Mayer-
Vietoris property. These properties indicate that this invariant would play an
important role in symplectic topology and mirror symmetry. One motivation of
it comes from mirror symmetry suggested by Seidel [24] and the family Floer
program. On the other hand, some symplectic topological applications have
already appeared in [26, 10]. Also see its relation with quantum cohomology by
Borman-Sheridan-Varolgunes [4].

Here we study a computational side of this invariant, and provide some
applications to symplectic topology. The main goal is to construct a
filtration on the underlying complex of this cohomology and look at the
induced spectral sequence. Now we set up notations and state our results. Let
$(M,\omega)$ be a smooth symplectic manifold, closed or convex at infinity. We
say $M$ is symplectic Calabi-Yau if $c_{1}(TM)=0$. A convex domain
$(D,\theta)$ in $M$ is a compact codimension zero symplectic submanifold of
$M$, with a boundary $\partial D$ and a one-form $\theta$ locally defined near
$\partial D$, such that the restriction of $\theta$ to $\partial D$ is a
contact form and the local Liouville vector field points outward. A Liouville
domain $(D,\theta)$ in $M$ is a convex domain in $M$ such that the one-form
$\theta$ is defined on all of $D$ and the restriction of $\omega$ on $D$ is
$d\theta$. We will focus on a special family of convex domains, where the
restriction of $\theta$ to $\partial D$ are index bounded contact forms, see
Definition 2.2.

Consider the exact sequence

$\cdots\rightarrow H^{1}(D)\rightarrow
H^{2}(M,D)\xrightarrow{j}H^{2}(M)\xrightarrow{i}H^{2}(D)\rightarrow\cdots.$

Since $D$ is exact, we have that $i([\omega])=0$ hence $[\omega]$ admits at
least one lift $[\tilde{\omega}]\in H^{2}(M,D)$. This auxiliary form
$\tilde{\omega}$ will be used to characterize how far a Floer solution travels
outside $D$.

######  Theorem 1.1.

Let $(M,\omega)$ be a closed symplectic Calabi-Yau manifold and $D$ be a
Liouville domain in $M$ with an index bounded boundary. Suppose that
$[\omega]$ admits an integral lift $[\tilde{\omega}]\in
H^{2}(M,D;\mathbb{Z})$. Given any positive number $E$, there is a truncated
invariant $SH_{M}(D;\Lambda_{E})$ such that

  1. (1)

there is a spectral sequence that starts from the classical symplectic
cohomology $SH(D;\Lambda_{E})$ with coefficient $\Lambda_{E}$, and converges
to $SH_{M}(D;\Lambda_{E})$;

  2. (2)

for an increasing sequence $E_{1}<E_{2}<\cdots$ that goes to positive
infinity, the inverse limit of the truncated invariant recovers the relative
symplectic cohomology

$(\varprojlim_{E}SH^{k}_{M}(D;\Lambda_{E}))\otimes_{\Lambda_{0}}\Lambda\cong
SH^{k}_{M}(D)\otimes_{\Lambda_{0}}\Lambda;$

  3. (3)

if the class $[\tilde{\omega}]$ vanishes on $H_{2}(M,D)$, then the above
spectral sequence degenerates at the first page, which shows that
$SH_{M}(D;\Lambda_{E})\cong SH(D;\Lambda_{E})$.

Here $\Lambda_{E}=\Lambda_{0}/\Lambda_{\geq E}$ is the truncated Novikov ring,
see Section 2 for our conventions on Novikov rings.

######  Remark 1.2.

In the definition of an index bounded contact form, we assume that it is non-
degenerate. We will give a perturbative method in Section 5 that works for
Morse-Bott non-degenerate contact forms.

Initially, the ambient space $M$ is assumed to be closed in the definition. It
is not hard to extend the definition to the case where $M$ is open and convex
at infinity. We give such a definition in the Appendix A. The definition is
almost the same as the case when the ambient space is closed. However, in the
open case to define cofinal families of $S$-shape Hamiltonian functions we
need to prescribe their slopes at infinity. The main result in the appendix is
that for different choices of slopes at infinity, we actually get the same
invariant. And we also deduce a relation between the relative symplectic
cohomology $SH_{\hat{D}}(D)$ and the usual symplectic cohomology $SH(D)$,
which says that $SH_{\hat{D}}(D)$ is the homology of “completed chains”.

######  Proposition 1.3.

(See Proposition A.7 for related definitions and the proof.) Let $D$ be a
Liouville domain such that the Liouville form is a non-degenerate contact form
on $\partial D$. There is a graded complex of $\Lambda_{0}$-modules $(C,d)$
which admits a $\mathbb{C}$-reduction $(\bar{C},\bar{d})$ such that

  1. (1)

$SH(D;\mathbb{C})=H(\bar{C},\bar{d})$ where $SH(D;\mathbb{C})$ is the
classical symplectic cohomology with $\mathbb{C}$-coefficients, defined by
Viterbo [31];

  2. (2)

$SH_{\hat{D}}(D)=H(\varprojlim_{r\geq
0}C\otimes_{\Lambda_{0}}\Lambda_{0}/\Lambda_{\geq r},d)$.

Combined with Abbondandolo-Schwarz’s proof of the Viterbo isomorphism, we get
a Viterbo isomorphism relating the relative symplectic cohomology and the
homology of “completed chains” in the free loop space, see (A.2).

In the above proposition, we don’t need to assume the index bounded condition.
The infinite-long cylindrical coordinate of $\hat{D}$ is used to ignore upper
orbits of $S$-shape Hamiltonian functions. If we use the index bounded
assumption, we have the following corollary.

######  Corollary 1.4.

For an index bounded Liouville domain $D$ and its completion $\hat{D}$, we
have the following isomorphism

$SH_{\hat{D}}(D)\otimes_{\Lambda_{0}}\Lambda\cong
SH(D;\Lambda_{0})\otimes_{\Lambda_{0}}\Lambda$

where the right hand side term $SH(D;\Lambda_{0})$ is the classical symplectic
cohomology of $D$ with $\Lambda_{0}$-coefficients.

###### Proof.

The index bounded condition tells us in the chain level we have finitely many
generators. Hence in Proposition A.7 the complex $C$ is finite-dimensional in
each degree. So the completion procedure has no extra effect. See also
Proposition 4.12 for an alternative proof by using the spectral sequence. ∎

Sine there is an absolute action for Hamiltonian orbits on a Liouville domain,
we can define the classical symplectic cohomology $SH(D)$ with either
$\mathbb{C}$-coefficients or $\Lambda_{0}$-coefficients. We will use both
versions. However, $SH(D;\Lambda_{0})$ is not canonically invariant since its
torsion parts encodes the information of the defining Hamiltonian function. It
is interesting to extract quantitative invariants from the torsion parts and
relate it with some energy-capacity invariants or spectral invariants, but we
will not discuss this perspective here. So we will use
$SH(D;\Lambda_{0})\otimes_{\Lambda_{0}}\Lambda$ more often.

After killing the torsion part, the invariant
$SH_{\hat{D}}(D)\otimes_{\Lambda_{0}}\Lambda$ does not see the size of an
index bounded Liouville domain. Because we can interpolate between domains of
different sizes, as shown in [26]. On the other hand, the action-completed
symplectic cohomology by Venkatesh usually captures the size of a domain in
its completion when it is monotone. See the examples in [29] about
tautological line bundles over projective spaces. As a contrast, we show a
result about the independence of the radius coordinate when it is index
bounded.

######  Proposition 1.5.

(Proposition 4.13) For an index bounded convex domain $D$ and its completion
$\hat{D}$, we have the following isomorphism

$\widehat{SH}(D;\Lambda)\cong SH(D;\Lambda)$

where the left hand side term $\widehat{SH}(D;\Lambda)$ is the action-
completed symplectic cohomology of $D$ in $\hat{D}$ by Venkatesh [29].

In the above proposition, we only assume that $D$ is a convex domain with an
index bounded boundary, without using the exactness of the symplectic form.

Without the index bounded assumption, similar conclusions of above theorems
may not hold. We recall the following example computed in [28]. Let $D$ be a
disk in the 2-sphere $S$ with an area form. If the disk has area more than
half the area of $S$, then $SH_{S}(D)\otimes_{\Lambda_{0}}\Lambda\neq 0$,
while the usual symplectic cohomology of $D$ is zero. This indicates that
$SH_{M}(D)$ is not totally local nor totally global. And a spectral sequence
helps us to understand this local-to-global phenomenon.

Note that a Liouville domain is a special case of a filled Liouville
cobordism. The argument to prove Theorem 1.1 also works for the relative
symplectic cohomology of a filled Liouville cobordism, with index bounded
boundaries.

######  Theorem 1.6.

Let $M$ be a closed symplectic Calabi-Yau manifold and $W$ be a Liouville
cobordism in $M$ with a filling $D$, such that the boundaries of $W$ are index
bounded. Suppose that $[\omega]$ admits an integral lift $[\tilde{\omega}]\in
H^{2}(M,W;\mathbb{Z})$. Given any positive number $E$, there is a truncated
invariant $SH_{M}(W;\Lambda_{E})$ such that

  1. (1)

there is a spectral sequence that starts from the symplectic cohomology
$SH(W;\Lambda_{E})$ of a filled Liouville cobordism defined by Cieliebak-
Oancea [8], and converges to $SH_{M}(W;\Lambda_{E})$;

  2. (2)

for an increasing sequence $E_{1}<E_{2}<\cdots$ that goes to positive
infinity, the inverse limit of the truncated invariant recovers the relative
symplectic cohomology

$(\varprojlim_{E}SH^{k}_{M}(W;\Lambda_{E}))\otimes_{\Lambda_{0}}\Lambda\cong
SH^{k}_{M}(W)\otimes_{\Lambda_{0}}\Lambda;$

  3. (3)

if the class $[\tilde{\omega}]$ vanishes on $H_{2}(M,W)$, then the above
spectral sequence degenerates at the first page, which shows that
$SH_{M}(W;\Lambda_{E})\cong SH(W;\Lambda_{E})$.

Similarly, there is a relative Lagrangian Floer cohomology, which is defined
in [26] as an open-string analogue of the relative symplectic cohomology. The
above constructions of spectral sequences also work in some situations for the
relative Lagrangian Floer cohomology, under suitable index bounded conditions.

The proof of the above theorems draws much inspiration from the work [20] of
McLean and its usage of the index bounded condition. Roughly speaking, the
Hamiltonian functions we are using to compute $SH_{M}(D)$ are approximately
zero on $D$ and positive infinity outside $D$, as a direct limit. And the
periodic orbits of our interest lie around the boundary of $D$. The levels of
these orbits can go to either zero or positive infinity. We first define a
valuation on the free module generated by these orbits. However, due to the
completion procedure, there will be elements with a negative infinite
valuation which comes from limits of orbits going to infinite high level. Then
we use the index bounded condition to ignore these high limits of orbits.
Therefore the original underlying complex of this relative invariant is quasi-
isomorphic to a new complex without the high limits. And the same valuation
gives an exhaustive filtration, which will induce a convergent spectral
sequence.

An important class of examples fitting into the above theorems comes from
simply-connected Lagrangian submanifolds in Calabi-Yau manifolds. Let $L$ be a
simply-connected Lagrangian submanifold in a Calabi-Yau manifold $M$. Take $D$
as a Weinstein neighborhood of $L$, which is isomorphic to a disk bundle
$D_{r}T^{*}L$ of the cotangent bundle of $L$, with respect to some Riemannian
metric $g$ on $L$. Note that there is a correspondence between the geodesics
of $g$ and the Reeb orbits on the co-sphere bundle of $T^{*}L$. Hence the
index bounded condition for the contact form on the co-sphere bundle will be
satisfied if the metric $g$ satisfies some relations between the length of
closed geodesics and their Morse indices. For many simply-connected manifolds,
the existence of such a nice Riemannian metric is known. (In Section 5 we show
it is true when $g$ has a positive Ricci curvature.) Then we obtain a spectral
sequence starting from $SH(D)$ and converging to $SH_{M}(D)$. Note that
$SH_{M}(D)$ detects the displaceability of $D$ inside $M$, letting
$r\rightarrow 0$ we can detect the displaceability of $L$ itself inside $M$.
On the other hand, the usual invariant to detect the displaceability of $L$,
the self-Lagrangian Floer cohomology $HF(L)$ may not be defined due to
possible holomorphic disks on $L$ with Maslov index zero. We state one
following application.

######  Proposition 1.7.

Let $M$ be a symplectic Calabi-Yau manifold with dimension greater than four.
For a Lagrangian sphere $S$ in $M$ and a Weinstein neighborhood $U$ of $S$
induced by the round metric on $S$, we have that

  1. (1)

$U$ is not stably-displaceable in $M$;

  2. (2)

$M-U$ is not stably-displaceable in $M$.

###### Proof.

(See a more detailed proof in Subsection 5.1.) By using our spectral sequence
we can show that $SH_{M}(U)\otimes_{\Lambda_{0}}\Lambda$ is neither zero nor
the homology of $M$. Hence $(1)$ follows the stable displacement criterion and
$(2)$ follows the same criterion and the Mayer-Vietoris property of the
relative symplectic cohomology. ∎

######  Remark 1.8.

The item $(1)$ in above proposition is known by using Lagrangian Floer theory,
since $HF(S)$ is non-zero in this case. The item $(2)$ can be regarded as an
analogue of a theorem of Ishikawa [17] (also see [26]): if $U$ is a round ball
in a Calabi-Yau manifold $M$ then $M-U$ is not stably-displaceable.

The examples about a round ball and a neighborhood of a sphere tell that if
the indices of Reeb orbits on the boundary of a Liouville domain are positive
enough, then the complement of this Liouville domain in any Calabi-Yau
manifold has some symplectic rigidity. In particular the complement cannot be
a small neighborhood of a divisor. This contrasts to the monotone case. Then
the Mayer-Vietoris property of the relative symplectic cohomology is a
powerful tool to study this phenomenon of rigidity. See Subsection 5.1 for
some other examples about negatively-curved Lagrangian submanifolds and
Brieskorn manifolds.

The outline of this article is as follows. In Section 2 we review backgrounds
about Hamiltonian Floer theories. In Section 3 and 4 we construct the
filtration and show its properties to prove Theorem 1.1. At the end of Section
4, we give two speculations about how the neck-stretching technique and a
weaker index bounded condition may upgrade our work. In Section 5 we discuss
some extensions of the theorems as well as applications. In the appendix we
discuss the relative symplectic cohomology of a Liouville domain inside its
completion and prove a version of the Viterbo isomorphism.

######  Remark 1.9.

We focus on the case that $M$ is Calabi-Yau and $D$ is index bounded. More
local-to-global results of $SH_{M}(D)$ in other interesting cases can be found
in the work by Borman-Sheridan-Varolgunes [4] and by Groman [15].

We only use the additive structure of $SH_{M}(D)$ as a $\Lambda_{0}$-module.
There is also a multiplicative structure on it defined in [26]. The relation
between $SH_{M}(D)$ and the quantum cohomology of $M$, as rings, can be found
in [4].

### Acknowledgments

The author acknowledges Mark McLean for his generous guidance on this project.
The author also acknowledges Umut Varolgunes for helpful discussions.

##  2\. Backgrounds

Now we review the construction of symplectic cohomology theories. First we
specify the ring and field that will be used. The Novikov ring $\Lambda_{0}$
and its field $\Lambda$ of fractions are defined by

$\Lambda_{0}=\\{\sum_{i=0}^{\infty}a_{i}T^{\lambda_{i}}\mid
a_{i}\in\mathbb{C},\lambda_{i}\in\mathbb{R}_{\geq
0},\lambda_{i}<\lambda_{i+1},\lim_{i\rightarrow\infty}\lambda_{i}=+\infty\\}$

and

$\Lambda=\\{\sum_{i=0}^{\infty}a_{i}T^{\lambda_{i}}\mid
a_{i}\in\mathbb{C},\lambda_{i}\in\mathbb{R},\lambda_{i}<\lambda_{i+1},\lim_{i\rightarrow\infty}\lambda_{i}=+\infty\\}$

where $T$ is a formal variable. The maximal ideal of $\Lambda_{0}$ is defined
by

$\Lambda_{+}=\\{\sum_{i=0}^{\infty}a_{i}T^{\lambda_{i}}\mid
a_{i}\in\mathbb{C},\lambda_{i}\in\mathbb{R}_{>0},\lambda_{i}<\lambda_{i+1},\lim_{i\rightarrow\infty}\lambda_{i}=+\infty\\}.$

There is a valuation $v:\Lambda\rightarrow\mathbb{R}\cup\\{+\infty\\}$ by
setting

$v(\sum_{i=0}^{\infty}a_{i}T^{\lambda_{i}}):=\min_{i}\\{\lambda_{i}\mid
a_{i}\neq 0\\}\quad\text{and}\quad v(0):=+\infty$

which make $\Lambda_{0}$ to be a complete valuation ring. And when we say the
completion of a $\Lambda_{0}$-module we mean the completion with respect to
this valuation. We write

$\Lambda_{\geq
r}:=v^{-1}([r,+\infty])\quad\text{and}\quad\Lambda_{>r}:=v^{-1}((r,+\infty]),\quad\forall
r\in(-\infty,+\infty),$

which are ideals of $\Lambda_{0}$. So $\Lambda_{0}$ is a short notation for
$\Lambda_{\geq 0}$ and $\Lambda_{+}$ for $\Lambda_{>0}$. Later when we fix an
energy bound $E>0$, we just write $\Lambda_{E}:=\Lambda_{0}/\Lambda_{\geq E}$.

###  2.1. Hamiltonian Floer theory on closed manifolds

Now we set up the background on Hamiltonian Floer theory. We work with
symplectic Calabi-Yau manifolds. Hence most foundational details can be found
in [16] and [25].

Let $(M,\omega)$ be a closed smooth symplectic manifold. A smooth function
$H:M\rightarrow\mathbb{R}$ determines a smooth vector field $X_{H}$ such that
$dH(\cdot)=\omega(X_{H},\cdot)$. We say $H$ is a Hamiltonian function and
$X_{H}$ is the associated Hamiltonian vector field. Let
$\mathcal{L}M:=C^{\infty}(S^{1},M)$ be the space of free loops in $M$, where
we always view $S^{1}=\mathbb{R}/\mathbb{Z}$ and write $t$ as the coordinate
of $S^{1}$. We call $t$ the time variable.

Moreover, we can consider a family of functions $H_{t}:S^{1}\times
M\rightarrow\mathbb{R}$ parameterized by $S^{1}$. Then we have time-dependent
Hamiltonian vector fields $X_{H_{t}}$. Integrating it we obtain a family of
Hamiltonian symplectic diffeomorphisms $\phi^{t}:M\rightarrow M$. A loop
$\gamma\in\mathcal{L}M$ is called a time-one Hamiltonian orbit if
$\gamma^{\prime}(t)=X_{H_{t}}$. We only consider the component
$\mathcal{L}_{0}M$ which contains contractible loops. So we write

$\mathcal{CP}_{H_{t}}=\\{\gamma:S^{1}\rightarrow
M\mid\gamma^{\prime}(t)=X_{H_{t}}\\}$

as the set of contractible one-periodic orbits of $H_{t}$. An orbit is non-
degenerate if the Poincaré return map

$d\phi^{1}:T_{\gamma(0)}M\rightarrow T_{\gamma(0)}M$

does not have eigenvalue one. And we say a Hamiltonian $H_{t}$ is non-
degenerate if all of its one-periodic orbits are non-degenerate.

Next we assume that $c_{1}(TM)=0$ and assign an index $CZ(\gamma)$ to each
orbit $\gamma$, the Conley-Zehnder index, see Lecture 2 in [25]. And we grade
our orbits by setting

(2.1) $\mu_{H_{t}}(\gamma):=n+CZ(\gamma)$

where $2n$ is the real dimension of $M$. (Our grading is different from that
in [25], since we use cohomology instead of homology.) When we say a
degree-$k$ or an index-$k$ orbit we mean an orbit $\gamma$ with
$\mu_{H_{t}}(\gamma)=k$. We remark that for a constant orbit of a Morse
function, its degree defined above equals its Morse index.

Therefore $\mathcal{CP}_{H_{t}}$ becomes a graded set

$\mathcal{CP}_{H_{t}}=\bigoplus_{k\in\mathbb{Z}}\mathcal{CP}_{H_{t}}^{k}$

where $\mathcal{CP}_{H_{t}}^{k}$ is the set of orbits with index $k$.

Then for a non-degenerate Hamiltonian $H_{t}$, we define

(2.2) $CF^{k}(H_{t};\Lambda_{0})=\\{\sum_{i=1}^{\infty}c_{i}\gamma_{i}\mid
c_{i}\in\Lambda_{0},\gamma_{i}\in\mathcal{CP}_{H_{t}}^{k}\\}$

which is the free $\Lambda_{0}$-module generated by index-$k$ orbits.
Similarly we can define

(2.3) $CF^{k}(H_{t};\Lambda)=\\{\sum_{i=1}^{\infty}c_{i}\gamma_{i}\mid
c_{i}\in\Lambda,\gamma_{i}\in\mathcal{CP}_{H_{t}}^{k}\\}$

which is the $\Lambda$-vector space generated by index-$k$ orbits. We don’t
grade the formal variable $T$.

By using a family of compatible almost complex structures $J_{t}$, we can
study the solutions of the Floer equation

(2.4) $\partial_{s}u+J_{t}(\partial_{t}u-X_{H_{t}})=0$

for $u:\mathbb{R}\times S^{1}\rightarrow\hat{M}$. Here $s$ is the
$\mathbb{R}$-coordinate and $t$ is the $S^{1}$-coordinate on the domain.
Consider the solution space

(2.5) $\displaystyle\mathcal{M}(\gamma_{-},\gamma_{+};A)=$
$\displaystyle\\{u:\mathbb{R}\times S^{1}\rightarrow
M\mid\partial_{s}u+J_{t}(\partial_{t}u-X_{H_{t}})=0,$ $\displaystyle
u(-\infty,t)=\gamma_{-},u(+\infty,t)=\gamma_{+},[u]=A\in\pi_{2}(M;\gamma_{-}\cup\gamma_{+})\\}/\sim.$

Here is an $\mathbb{R}$-action on this space by translating the $s$-coordinate
of a solution $u$, and $\sim$ is the quotient of this $\mathbb{R}$-action. The
$L^{2}$-energy of a solution $u$ is

$0\leq E(u)=\int\lvert\partial_{s}u\rvert^{2}=\int
u^{*}\omega+\int_{\gamma_{+}}H_{t}-\int_{\gamma_{-}}H_{t}.$

For generic pairs $(H_{t},J_{t})$, the solution space is an $l$-dimensional
manifold where

$\mu_{H_{t}}(\gamma_{+})-\mu_{H_{t}}(\gamma_{-})=l+1.$

By the Gromov-Floer compactness theorem, when $\gamma_{-},\gamma_{+},A$ are
fixed, the above solution spaces admit compactifications by adding broken
Floer trajectories and $J$-holomorphic sphere bubbles. The bubbles can be
ruled out by using the Calabi-Yau condition when the moduli space is zero or
one-dimensional, see [16]. Here are also coherent orientations on these moduli
spaces. In particular, when the moduli space is zero-dimensional, we can count
the signed number of elements as $n(\gamma_{-},\gamma_{+};A)$.

Then we define an operator

$d:CF^{k}(H_{t})\rightarrow CF^{k+1}(H_{t})$

with either $\Lambda_{0}$ or $\Lambda$ coefficients, by setting

(2.6)
$d(\gamma_{-}):=\sum_{\gamma_{+}}\sum_{[u]=A}n(\gamma_{-},\gamma_{+};A)\cdot\gamma_{+}\cdot
T^{\int u^{*}\omega+\int_{\gamma_{+}}H_{t}-\int_{\gamma_{-}}H_{t}}.$

The right-hand side is summed over all $\gamma_{+}$ with
$\mu_{H_{t}}(\gamma_{+})-\mu_{H_{t}}(\gamma_{-})=1$ and all classes
$A\in\pi_{2}(M;\gamma_{-}\cup\gamma_{-})$. It may not be a finite sum, but it
converges as an element in $CF^{k}(H_{t})$, by the Gromov compactness theorem.
Then we extend this operator $\Lambda_{0}$ or $\Lambda$-linearly to
$CF^{k}(H_{t})$.

By the analysis of codimension one boundaries of
$\mathcal{M}(\gamma_{-},\gamma_{+};A)$ with

$\mu_{H_{t}}(\gamma_{+})-\mu_{H_{t}}(\gamma_{-})=2,$

a big theorem in Hamiltonian Floer theory shows that $d^{2}=0$. Then we write
the resulting cohomology groups as $HF^{k}(H_{t},J_{t};\Lambda_{0})$ and
$HF^{k}(H_{t},J_{t};\Lambda)$.

Another theorem shows that $HF^{k}(H_{t},J_{t};\Lambda)$ is independent of the
choices of generic pairs $(H_{t},J_{t})$. Hence we can call it the Hamiltonian
Floer cohomology of $M$. This invariance result is proved by considering
continuation maps between different choices of $(H_{t},J_{t})$. We sketch it
here since we will use it later to define symplectic cohomology.

For simplicity we only vary $H_{t}$. The case for $J_{t}$ can be handled in
the same way. Let $H^{\alpha}_{t}$ and $H^{\beta}_{t}$ be two non-degenerate
Hamiltonians. Assume that both $(H^{\alpha}_{t},J_{t})$ and
$(H^{\beta}_{t},J_{t})$ are regular for a fixed $J_{t}$. Then we choose a
homotopy $H^{\alpha\beta}_{s,t}$ of Hamiltonians to connect $H^{\alpha}_{t}$
and $H^{\beta}_{t}$. That is,

(2.7) $\displaystyle H^{\alpha\beta}_{s,t}:\mathbb{R}\times S^{1}\times
M\rightarrow\mathbb{R};$ $\displaystyle
H^{\alpha\beta}_{s,t}=\begin{cases}H^{\alpha}_{t},\quad s\leq-1,\\\
H^{\beta}_{t},\quad s\geq 1.\end{cases}$

Then we consider the $s$-dependent Floer equation

(2.8) $\partial_{s}u+J_{t}(\partial_{t}u-X_{H^{\alpha\beta}_{s,t}})=0$

and the moduli space

$\displaystyle\mathcal{M}(\gamma^{\alpha}_{-},\gamma^{\beta}_{+};A)=$
$\displaystyle\\{u:\mathbb{R}\times S^{1}\rightarrow
M\mid\partial_{s}u+J_{t}(\partial_{t}u-X_{H^{\alpha\beta}_{s,t}})=0,$
$\displaystyle
u(-\infty,t)=\gamma^{\alpha}_{-},u(+\infty,t)=\gamma^{\beta}_{+},[u]=A\in\pi_{2}(M;\gamma^{\alpha}_{-}\cup\gamma^{\beta}_{+})\\}.$

Note that now the equation is $s$-dependent hence there is no
$\mathbb{R}$-action. For a generic path $H^{\alpha\beta}_{s,t}$, the above
moduli space is a manifold of dimension
$\mu_{H^{\beta}_{t}}(\gamma_{+})-\mu_{H^{\alpha}_{t}}(\gamma_{-})$. And it
admits a similar compactification by adding broken trajectories. When
$\mu_{H^{\beta}_{t}}(\gamma_{+})=\mu_{H^{\alpha}_{t}}(\gamma_{-})$, we define
an operator

(2.9) $f^{\alpha\beta}:CF^{k}(H^{\alpha}_{t})\rightarrow
CF^{k}(H^{\beta}_{t})$

by setting

$f^{\alpha\beta}(\gamma^{\alpha}_{-}):=\sum_{\gamma^{\beta}_{+}}\sum_{[u]=A}n^{\alpha\beta}(\gamma^{\alpha}_{-},\gamma^{\beta}_{+};A)\cdot\gamma^{\beta}_{+}\cdot
T^{\int u^{*}\omega+\int\partial_{s}(H^{\alpha\beta}_{s,t}(u(s,t)))},$

where $n^{\alpha\beta}(\gamma^{\alpha}_{-},\gamma^{\beta}_{+};A)$ is a signed
count of elements of the above moduli space. Note that the weight

(2.10) $\int
u^{*}\omega+\int\partial_{s}(H^{\alpha\beta}_{s,t}(u(s,t)))=\int\lvert\partial_{s}u\rvert^{2}+\int\partial_{s}(H^{\alpha\beta}_{s,t})$

is not necessarily non-negative, hence we need to use $\Lambda$-coefficients
now. We call this weight as the topological energy of a Floer solution. (It is
non-negative if the family of Hamiltonian functions satisfies that
$\int\partial_{s}(H^{\alpha\beta}_{s,t})\geq 0$.) Then one can show that
$f^{\alpha\beta}$ is a chain map and moreover $f^{\alpha\beta}\circ
f^{\beta\alpha}$ is chain homotopy equivalent to the identity map. This
completes the proof of the independence of Hamiltonians, see [25] for more
details.

We add one more lemma here which will be used frequently in later sections.

######  Lemma 2.1.

Let $M$ be a Calabi-Yau manifold and $H_{t}$ be a non-degenerate Hamiltonian
function on $M$. For a constant $\Delta>0$, suppose that we have a homotopy
$\\{H^{s}_{t}\\}_{s\in[0,1]}$ such that $H^{s}_{t}=H_{t}+s\Delta$. Then the
continuation map between $CF(H^{0}_{t})$ and $CF(H^{1}_{t})$ is a
multiplication by $T^{\Delta}$. In particular, when $\Delta=0$ this shows that
the continuation map of a constant homotopy is the identity map.

###### Proof.

Since our Hamiltonian functions are just a translation of a fixed one, the
Floer equation of the continuation map does not depend on $s$. Therefore any
solution of it still carries an $\mathbb{R}$-action. By using the Calabi-Yau
condition, we can pick a regular $J$ for $H_{t}$. For two different orbits
$\gamma$ and $\gamma^{\prime}$ with index $k$, any Floer solution of the
continuation map equation connecting $\gamma$ and $\gamma^{\prime}$ carries an
$\mathbb{R}$-action. So the corresponding moduli space is at least one-
dimensional, contradicting to that both $\gamma$ and $\gamma^{\prime}$ have
index $k$.

Then the continuation maps only exist when $\gamma=\gamma^{\prime}$ and it
will be an constant map. One can directly check that the constant map is
regular and has contribution one. Hence the continuation map between
$CF(H^{0}_{t})$ and $CF(H^{1}_{t})$ is an identity matrix, weighted by the
change of the Hamiltonian function which is $T^{\Delta}$. ∎

###  2.2. Liouville domain and contact cylinder

Let $(C,\alpha)$ be a contact manifold with a contact form $\alpha$. The Reeb
vector field of $\alpha$ is the unique vector field $R_{\alpha}$ on $C$ such
that

$d\alpha(R_{\alpha},\cdot)=0,\quad\alpha(R_{\alpha})=1.$

Then a Reeb orbit of length $\lambda>0$ is a map

$\gamma(t):\mathbb{R}/\lambda\mathbb{Z}\rightarrow
C,\quad\frac{d}{dt}\gamma(t)=R_{\alpha}.$

We write $\Gamma_{\alpha,\lambda}$ be the set of Reeb orbits of length
$\lambda$ and $\Gamma_{\alpha}:=\cup_{\lambda>0}\Gamma_{\alpha,\lambda}$. We
say a Reeb orbit is non-degenerate if the Poincaé return map of the Reeb flow
does not have eigenvalue one. And we say a contact form $\alpha$ is non-
degenerate if all of its orbits are non-degenerate.

For a convex domain $D$ in $M$, the symplectic form $\omega$ is exact near
$\partial D$. We write $\omega=d\theta$ and hence
$\alpha:=\theta\mid_{\partial D}$ is a contact form so that $\partial D$
becomes a contact hypersurface in $M$. For small $\epsilon>0$ there is an
embedding

$\bar{C}=[1-\epsilon,1+\epsilon]\times\partial D\rightarrow M$

such that the $\partial D$ is the image of $\\{1\\}\times\partial D$. We do
not distinguish $C$ from its image. Then $\bar{C}$ is called a contact
cylinder associated to the Liouville domain $D$ if
$\omega\mid_{\bar{C}}=d(r\alpha)$ where $r$ is the coordinate on
$[1-\epsilon,1+\epsilon]$. And we will write $D_{1+\epsilon}=D\cup\bar{C}$ as
a compact neighborhood of $D$. The one-form $\theta$ on $D$ smoothly extends
to a one-form $\theta$ on $D_{1+\epsilon}$ such that
$\omega\mid_{D_{1+\epsilon}}=d\theta$.

Similarly, a closed smooth hypersurface $C\subset M$ is called of contact type
if $C$ carries a contact form $\alpha$ such that $d\alpha\mid_{C}=\omega$.
Suppose that $C$ separates $M$ into two components. If one of the components
is a Liouville domain $D$, then $\bar{C}$ is also a contact cylinder
associated to $D$.

For a fixed contact cylinder associated to a Liouville domain $D$. We are
interested in the contractible Reeb orbits

$\gamma(t):\mathbb{R}/\lambda\mathbb{Z}\rightarrow\bar{C}\subset M$

since later we will see that those Reeb orbits correspond to Hamiltonian
orbits of a special class of Hamiltonian functions. For a contractible Reeb
orbit $\gamma$, we can also define its Conley-Zehnder index $CZ(\gamma)$ and
shift it to get a grading

$\mu(\gamma):=n+CZ(\gamma)$

Since $c_{1}(TM)=0$ it is a well-defined index, independent of the choice of a
capping disk. And we write $\Gamma_{\alpha}^{0}$ be the set of Reeb orbits
which are contractible in $M$.

######  Definition 2.2.

Let $\mathcal{N}\subset\mathbb{Z}$ be a subset of integers. The contact
cylinder $(\bar{C},\alpha)$ with a non-degenerate contact form $\alpha$ is
called $\mathcal{N}$-index bounded if for any finite subset
$I\subset\mathcal{N}$ there exists $\mu_{I}>0$ so that

$\\{length(\gamma)\mid\gamma\in\Gamma_{\alpha}^{0},\mu(\gamma)\in
I\\}\subset(0,\mu_{I}).$

When $\mathcal{N}=\mathbb{Z}$ we just say $(\bar{C},\alpha)$ is index bounded.

If the contact cylinder $(\bar{C},\alpha)$ is just Morse-Bott non-degenerate
and satisfies the above condition, we say it is Morse-Bott index bounded.

We remark that the index bounded property is a dynamical property for the
contact form $\alpha$ of $C$. Since in the contact cylinder $\bar{C}$ the
contact form is just rescaled by the coordinate $r\in[1-\epsilon,1+\epsilon]$,
we can still talk about the index bound for all Reeb orbits in $\bar{C}$.
However, this property might not be preserved under small perturbations other
than the scaling.

###  2.3. Relative symplectic cohomology of a Liouville domain

The relative symplectic cohomology is the homology of a suitable limit of
complexes derived from Hamiltonian Floer theory of $M$. The whole construction
[28] of relative symplectic cohomology defines a module $SH_{M}(D)$ over the
universal Novikov ring $\Lambda_{0}$, for any compact subset $D$ of a closed
symplectic manifold $M$. Now we briefly review its definition when $D$ is a
convex domain in a symplectic Calabi-Yau manifold $M$.

First we specify the Hamiltonian functions we will use to define $SH_{M}(D)$.
The following set

$C^{\infty}_{S^{1}\times D\subset S^{1}\times M}:=\\{H_{t}\in
C^{\infty}(S^{1}\times M,\mathbb{R)}\mid H_{t}(x)<0,\forall(t,x)\in
S^{1}\times D\\}$

is a directed set, with the relation $H_{t}\geq H_{t}^{\prime}$ if
$H_{t}(x)\geq H_{t}^{\prime}(x)$ for all $x\in S^{1}\times M$.

######  Lemma 2.3.

(Lemma 3.3.1, [28]) Let $H_{1,t}\leq H_{2,t}\leq\cdots$ be elements of
$C^{\infty}_{S^{1}\times D}$. They form a cofinal family if and only if
$H_{i,t}(x)\rightarrow 0$ for $(t,x)\in S^{1}\times D$, and
$H_{i,t}(x)\rightarrow+\infty$ for $(t,x)\in S^{1}\times(M-D)$, as
$i\rightarrow+\infty$.

Next we call the following data an acceleration data for $D$:

  1. (1)

$H_{1,t}\leq H_{2,t}\leq\cdots$ a cofinal family in $C^{\infty}_{D\times
S^{1}\subset M\times S^{1}}$;

  2. (2)

Monotone one-cube of Hamiltonians $\\{H_{s,t}\\}_{s\in[i,i+1]}$ for all $i$,
which means that $H_{s,t}(x)\geq H_{s^{\prime},t}(x)$ if $s\geq s^{\prime}$
and $H_{s,t}=H_{i,t}$ if $s=i$.

From an acceleration data, we obtain a one-ray of chain complexes over
$\Lambda_{0}$:

$\mathcal{CF}^{k}=CF^{k}(H_{1,t})\rightarrow CF^{k}(H_{2,t})\rightarrow\cdots$

connected by continuation maps. Here each $CF^{k}(H_{i,t})$ is the degree-$k$
Floer complex of the Hamiltonian $H_{i,t}$. Since $H_{i,t}\leq H_{i+1,t}$ are
connected by a monotone family of Hamiltonians, the weight (2.10) in the
continuation map is non-negative.

Then the relative symplectic cohomology module $SH^{k}_{M}(D;\Lambda_{0})$ is
defined as the cohomology

(2.11) $H(\widehat{tel}(\mathcal{CF}^{k});\Lambda_{0})$

of the completion $\widehat{tel}(\mathcal{CF}^{k})$ of the telescope

$tel(\mathcal{CF}^{k})=\bigoplus_{n\in\mathbb{Z}_{+}}(C_{n}[1]\oplus C_{n}).$

Here we write $C_{n}=CF^{k}(H_{n,t})$. Another algebraic way to define it is
that

(2.12) $\displaystyle SC^{k}_{M}(D;\Lambda_{0}):=\varprojlim_{r\geq
0}\varinjlim_{n}C_{n}\otimes_{\Lambda_{0}}\Lambda_{0}/\Lambda_{\geq r}$
$\displaystyle SH^{k}_{M}(D;\Lambda_{0}):=H(SC^{k}_{M}(D);\Lambda_{0}).$

For the equivalence of these two definitions, see Section 2 in [28]. And it is
also shown that the definition of $SH^{k}_{M}(D;\Lambda_{0})$ is independent
of various choices.

######  Proposition 2.4.

(Proposition 3.3.4, [28])

  1. (1)

Let $H_{s}$ and $H_{s}^{\prime}$ be two different acceleration data, then
$H(SC_{M}(D,H_{s}))\cong H(SC_{M}(D,H_{s}^{\prime}))$ canonically. Therefore
we simply denote this invariant by $SH_{M}(D)$.

  2. (2)

Let $\phi:M\rightarrow M$ be a symplectomorphism. There exists a canonical
isomorphism $SH_{M}(D)=SH_{M}(\phi(D))$ by relabeling an acceleration data by
the map $\phi$.

  3. (3)

For $D\subset D^{\prime}$, there exists canonical restriction maps
$SH_{M}(D^{\prime})\rightarrow SH_{M}(D)$. This satisfies the presheaf
property.

Hence we can just write this invariant as $SH_{M}(D):=SH_{M}(D;\Lambda_{0})$
and its torsion-free part $SH_{M}(D;\Lambda_{0})\otimes_{\Lambda_{0}}\Lambda$.
This invariant has many good properties. Notably it satisfies the Mayer-
Vietoris exact sequence in some settings. Another property we will keep using
here is the stably displaceability condition.

######  Theorem 2.5.

(Theorem 4.0.1 [28]) If the compact subset $D\subset M$ is stably Hamiltonian
displaceable then $SH_{M}(D;\Lambda_{0})\otimes_{\Lambda_{0}}\Lambda=0$.

In practice when $D$ is a convex domain, with a non-degenerate contact form on
its boundary, we will use a particular class of acceleration data to compute
the relative symplectic cohomology. For small $\epsilon>0$ we fix a contact
cylinder $\bar{C}$ associated to $D$ and write $D_{1+\epsilon}=D\cup\bar{C}$
as a compact neighborhood of $D$. We introduce the notion of $S$-shape
Hamiltonian functions, see Figure 1.

$r$$1-\frac{\epsilon}{4}$$1$$1+\frac{\epsilon}{4}$$1+\frac{\epsilon}{2}$$1+\frac{3\epsilon}{4}$$1+\epsilon$
Figure 1. Hamiltonian functions in the cylindrical coordinate.

######  Definition 2.6.

A time-independent Hamiltonian function $H:M\rightarrow\mathbb{R}$ is called
an $S$-shape Hamiltonian if

  1. (1)

$H$ is cylindrical on the region
$[1-\frac{\epsilon}{4},1+\frac{3\epsilon}{4}]\times\partial D\subset\bar{C}$.
That is, $H(x)=H(x^{\prime})$ if $r(x)=r(x^{\prime})$ where $r$ is the
cylindrical coordinate. So we can write $H(x)=h(r(x))$ for some
$h(r):[1-\frac{\epsilon}{4},1+\frac{3\epsilon}{4}]\rightarrow\mathbb{R}$ on
the cylinder region;

  2. (2)

$h^{\prime}(r)$ is concave and $h(r)=\lambda r+m$ on
$[1+\frac{\epsilon}{4},1+\frac{\epsilon}{2}]\times\partial D\subset\bar{C}$
for some constants $\lambda>0$ and $m$;

  3. (3)

the linear slope $\lambda$ is not in the action spectrum of the contact form;

  4. (4)

$H$ is a $C^{2}$-small Morse function on $D_{1-\frac{\epsilon}{4}}$, and it is
a Morse function on $M-D_{1+\frac{3\epsilon}{4}}$ with small derivatives, such
that it only has constant orbits outside
$[1-\frac{\epsilon}{4},1+\frac{3\epsilon}{4}]\times\partial D$.

By this special shape of our Hamiltonian functions, we have further
description of their one-periodic orbits. First in the region where the
Hamiltonians has small derivatives, there are only constant orbits. Hence all
non-constant orbits lie in the contact cylinder $\bar{C}$. By a direct
computation we have that

$X_{H_{n,t}}(r_{0},c_{0})=-\partial_{r}H_{n,t}(r_{0})\cdot
R_{\alpha}(c_{0}),\quad\forall(r_{0},c_{0})\in\bar{C}=[1-\epsilon,1+\epsilon]\times\partial
D.$

That is, the Hamiltonian vector field is proportional to the Reeb vector field
with the (negative) slope as the ratio. Hence any $\tau$-periodic Reeb orbit
gives rise to a one-periodic Hamiltonian orbit in $r_{0}\times\partial D$ if
and only if

$\tau=\lvert\int_{0}^{1}\partial_{r}H_{n,t}(r_{0})dt\rvert.$

And since the Hamiltonian is linear in the middle of the cylindrical region,
with a slope which is not in the action spectrum of the contact form, there
are no orbits in this region. Hence all the non-constant Hamiltonian orbits
can be separated into two groups. One group is located in the region
$[1-\frac{\epsilon}{4},1+\frac{\epsilon}{4}]\times\partial D$ and we call them
lower orbits. The other group is located in the region
$[1+\frac{\epsilon}{2},1+\frac{3\epsilon}{4}]\times\partial D$ and we call
them upper orbits.

Previously, an $S$-shape Hamiltonian function is time-independent. So the non-
constant orbits appear in $S^{1}$-families. Now we use small time-dependent
perturbations to make them nondegenerate, by using the technique in [6].

######  Proposition 2.7.

(Lemma 2.1 and Proposition 2.2 in [6]) Let $H$ be a time-independent $S$-shape
Hamiltonian function and let $\gamma$ be a non-constant one-periodic orbit of
$H$ such that $\gamma$ is transversally non-degenerate. Pick $U$ to be a
neighborhood of $\gamma$ which does not contain other one-periodic orbits,
then there exists a time-dependent function $H_{t}$ such that

  1. (1)

the support of $H-H_{t}$ is in $U$;

  2. (2)

there are exactly two one-periodic orbits $\gamma_{\pm}$ of $H_{t}$ in $U$;

  3. (3)

$\int\gamma^{*}\tilde{\theta}=\int\gamma_{+}^{*}\tilde{\theta}=\int\gamma_{-}^{*}\tilde{\theta}$,
where $\tilde{\theta}$ is the Liouville one-form in the cylindrical region;

  4. (4)

the difference between the Conley-Zehnder indices of $\gamma$ and
$\gamma_{\pm}$ is bounded by one.

Note that the orbit $\gamma$ is both a Hamiltonian orbit of $H$ and a Reeb
orbit of the contact form. One can compute its index in two ways. The
following lemma relates these two indices.

######  Lemma 2.8.

(Lemma 5.25, [20]) Let $\bar{C}=[1-\epsilon,1+\epsilon]\times C$ be an index
bounded contact cylinder with cylindrical coordinate $r$ and associated
contact form $\alpha$ and let $\pi:\bar{C}\rightarrow C$ be the natural
projection map. Let $f:[1-\epsilon,1+\epsilon]\rightarrow\mathbb{R}$ be a
smooth function, $R_{\lambda,[-m,m]}$ be the set of Reeb orbits of length
$\lambda$ and index in $[-m,m]$ and $O_{\lambda,[-m,m]}$ be the set of one-
periodic orbits of $f(r)$ contained in $\\{r=(f^{\prime})^{-1}(\lambda)\\}$ of
index in $[-m,m]$ which are null homologous in M. Then the map

$O_{\lambda,[-m,m]}\rightarrow R_{\lambda,[-m-\frac{1}{2},m+\frac{1}{2}]}$

sending $\gamma:\mathbb{R}/\mathbb{Z}\rightarrow\bar{C}$ to
$\pi\circ\gamma\circ b_{\lambda}$ is well-defined where

$b_{\lambda}:[0,\lambda]\rightarrow[0,1],\quad
b_{\lambda}(t):=t/\lambda,\quad\forall t\in[0,\lambda].$

Hence we can translate the index bounded condition, which previously works for
Reeb orbits, to the setting of Hamiltonian orbits. Let $D$ be a convex domain
with an index bounded boundary in a Calabi-Yau manifold. We start with a time-
independent $S$-shape Hamiltonian function and perturb it. Before
perturbation, a non-constant orbit $\gamma$ satisfies the index-length
relation in Definition 2.2 since it comes from a Reeb orbit. After
perturbation, by the above proposition and lemma, the index-length relation
still holds for new orbits $\gamma_{\pm}$.

Now we say a time-dependent function $H_{t}$ is a time-dependent $S$-shape
Hamiltonian function if it is a perturbation of a time-independent $S$-shape
Hamiltonian function. Note that we only perturb the region where non-constant
orbits lie, so we can still talk about upper and lower orbits after the
perturbation.

######  Definition 2.9.

Let $D$ be a convex domain in a Calabi-Yau manifold $M$. A time-dependent
$S$-shape Hamiltonian function $H_{t}$ is called index bounded if for any
integer $k$, there exists a constant $\mu_{k}>0$ such that
$\int\gamma^{*}\tilde{\theta}<\mu_{k}$ for all degree-$k$ one-periodic orbits
of $H_{t}$.

And the above discussion says that if $D$ be a convex domain with an index
bounded boundary, then we can always find time-dependent non-degenerate
$S$-shape Hamiltonian functions which are index bounded. In practice we will
use families of time-dependent non-degenerate $S$-shape Hamiltonian functions
to compute $SH_{M}(D)$.

###  2.4. Hamiltonian Floer theory on manifolds with convex boundary

Now we review the construction of Hamiltonian Floer theory on convex
manifolds, and fix our notation along the way. Symplectic cohomology was first
introduced by Cieliebak-Floer-Hofer [5] and Viterbo [31] in the exact setting,
and by Ritter [22] in the non-exact setting.

Let $(M,\omega=d\theta)$ be a Liouville domain, and let
$\alpha:=\theta\mid_{\partial M}$ be the contact form. Then we can attach a
cylindrical end to $M$ to get an open symplectic manifold

$\hat{M}:=M\cup(\partial M\times[1,+\infty)).$

Let $r$ be the coordinate on $[1,+\infty)$. We equip the manifold $\hat{M}$
with a smooth symplectic form $\hat{\omega}$, where $\hat{\omega}=\omega$ on
$M$ and $\hat{\omega}=d(r\alpha)$ on $\partial M\times[1,+\infty)$. And
$(\hat{M},\hat{\omega})$ will be called as the completion of $M$. In the
following we assume that $\alpha$ on $\partial M$ is non-degenerate.

Now we define admissible Hamiltonian functions we will use. A Hamiltonian
function $H_{t}:S^{1}\times\hat{M}\rightarrow\mathbb{R}$ is called admissible
if

  1. (1)

it is a negative time-independent Morse function on $M$.

  2. (2)

all its one-periodic orbits in $\hat{M}$ are non-degenerate.

  3. (3)

it is a linear function just depending on $r$ with a positive slope on
$\partial M\times[R_{0},+\infty)$ for some $R_{0}>1$.

  4. (4)

the slope of the linear part is not an element in $Spec(\alpha)$.

For an admissible Hamiltonian function, there are only finitely many one-
periodic orbits. Next with an admissible Hamiltonian $H_{t}$, we consider the
degree-$k$ Floer complex $CF^{k}(H_{t})$ as in (2.2) and (2.3). Then for
suitably chosen almost complex structures, we can use moduli spaces of Floer
solutions to define differentials and continuation maps as in the closed case.

For a monotone family $H_{i,t}$ such that $H_{i,t}\leq H_{i,t}$ and the linear
slope of $H_{i,t}$ goes to positive infinity, we have a one-ray of complexes

$\mathcal{CF}^{k}=CF^{k}(H_{1,t})\rightarrow CF^{k}(H_{2,t})\rightarrow\cdots$

connected by continuation maps. Here all Floer differentials and continuation
maps are weighted by the topological energy, see (2.10). Note that our
Liouville domain has an absolute action for Hamiltonian orbits, the images of
an orbit under Floer differentials and continuation maps are finite sums of
other orbits. Hence we can both define the classical symplectic cohomology
over $\mathbb{C}$ or over $\Lambda_{0}$. The former theory can be defined by
the later one by setting $T=1$.

The classical symplectic cohomology of $M$ over $\Lambda_{0}$ is defined as

$SH^{k}(M;\Lambda_{0}):=H(tel(\mathcal{CF}^{k})).$

An essential difference between this definition and the relative symplectic
cohomology is that the classical one does not complete $tel(\mathcal{CF}^{k})$
before taking homology. Note that here in the definition we do not require
that $H_{i,t}$ converge to zero in $M$. Hence the torsion part of
$SH^{k}(M;\Lambda_{0})$ encodes some information of the defining functions
$H_{i,t}$. It would be interesting to construct spectral invariants by
requiring that $H_{i,t}$ converge to a particular function in $M$. But in this
article we will focus on the torsion-free part of $SH^{k}(M;\Lambda_{0})$.

And the classical symplectic cohomology of $M$ over $\mathbb{C}$ is defined as

$SH^{k}(M;\mathbb{C}):=H(tel(\mathcal{CF}^{k})\mid_{T=1}).$

In other words, all differentials and continuation maps are defined without
weights.

###  2.5. Lower semi-continuous Hamiltonian functions

It is known that defining $SH^{k}(M;\mathbb{C})$ via taking a direct limit is
equivalent to the definition via a single Hamiltonian function that are
quadratic at infinity. Similarly, the definition of the relative symplectic
cohomology $SH_{M}(D)$ can be viewed as a Hamiltonian Floer theory of a lower
semi-continuous Hamiltonian function $F^{D}_{0}$, where $F^{D}_{0}\mid_{D}=0$
and $F^{D}_{0}\mid_{M-D}=+\infty$. A general study of Hamiltonian Floer theory
of lower semi-continuous functions can be found in [14] and [20]. Now we will
discuss a special case of it for our purpose.

Let $M$ be a closed symplectic manifold and let $F:M\rightarrow\mathbb{R}$ be
a lower semi-continuous function. Pick a monotone one-ray $\\{H_{n,t}\\}$ of
non-degenerate Hamiltonian functions such that

$H_{1,t}\leq H_{2,t}\leq\cdots\leq H_{n,t}\leq\cdots\rightarrow F.$

The Hamiltonian Floer cohomology $HF(F)$ is defined as

(2.13) $H(\widehat{tel}(CF(H_{1,t})\rightarrow
CF(H_{2,t})\rightarrow\cdots)).$

Also we have an equivalent definition in terms of (2.12).

From its definition, we can see that the lower semi-continuous Hamiltonian
Floer cohomology $HF(F)$ is a generalization of the relative symplectic
cohomology. For a domain $D$ in $M$, we have that

$SH_{M}(D)\cong HF(F^{D}_{0}).$

This indicator-type function $F^{D}_{0}$ is a very degenerate one, since it is
identically zero on $D$. In practice, it is often more handy to replace its
part on $D$ by a fixed non-degenerate Hamiltonian function $f$. And our main
result of this subsection is the following isomorphism, which shows that the
choice $f$ does not matter if we work over the Novikov field.

######  Proposition 2.10.

Let $D$ be a domain in $M$. Let $F:M\times S^{1}\rightarrow\mathbb{R}$ be a
lower semi-continuous function such that $F$ is a negative non-degenerate
Hamiltonian function $f$ on $D\times S^{1}$ and $F=+\infty$ otherwise. Then we
have that

$SH_{M}(D)\otimes_{\Lambda_{0}}\Lambda\cong
HF(F)\otimes_{\Lambda_{0}}\Lambda.$

###### Proof.

Pick a cofinal family $\\{H_{n,t}\\}$ of non-degenerate Hamiltonian functions
such that

$H_{1,t}\leq H_{2,t}\leq\cdots\leq H_{n,t}\leq\cdots\rightarrow F^{D}_{0}$

and a cofinal family $\\{K_{n,t}\\}$ of non-degenerate Hamiltonian functions
such that

$K_{1,t}\leq K_{2,t}\leq\cdots\leq K_{n,t}\leq\cdots\rightarrow F.$

By the non-degeneracy of $f$ in $D$, we can choose $K_{n,t}$ such that
$K_{n,t}=f$ for all $n\in\mathbb{N},(x,t)\in D\times S^{1}$. Moreover, since
$F$ is negative on $D$ we can choose above families such that $H_{n,t}\geq
K_{n,t}$ for all $n$. And we will need another condition

$\max_{(x,t)\in M\times S^{1}}(H_{n,t}-K_{n,t})=\max_{(x,t)\in D\times
S^{1}}(H_{n,t}-K_{n,t}),\quad\forall n\in\mathbb{N}.$

Hence there is a continuation map

$CF(K_{i,t})\rightarrow CF(H_{i,t})$

over $\Lambda_{0}$ for each $i$. These continuation maps induce a
$\Lambda_{0}$-module map

$HF(F)\rightarrow HF(F^{D}_{0})\cong SH_{M}(D).$

Next we set $\Delta:=-\min_{(x,t)\in D\times S^{1}}K_{1,t}$, and set
$F+\Delta$ to be the lower semi-continuous function such that it is $f+\Delta$
on $D\times S^{1}$ and positive infinity otherwise. Note that the family
$\\{K_{n,t}+\Delta\\}$ is a cofinal family for the function $F+\Delta$, so on
$D\times S^{1}$ we have that $K_{n,t}+\Delta\geq K_{1,t}+\Delta\geq 0\geq
H_{n,t}$ for all $n$. On the other hand, since $H_{n,t}-K_{n,t}$ takes maximum
on $D\times S^{1}$, we have $K_{n,t}+\Delta\geq H_{n,t}$ globally on $M\times
S^{1}$, see Figure 2. This gives the second $\Lambda_{0}$-module map in the
following

$HF(F)\rightarrow HF(F^{D}_{0})\rightarrow HF(F+\Delta)\rightarrow
HF(F^{D}_{0}+\Delta^{\prime}).$

Similarly we can find some other constant $\Delta^{\prime}$ to define the
third map. Since the data to define $HF(F+\Delta)$ is just a translation of
the data to define $HF(F)$, the composition of the first two maps is the
multiplication by $T^{\Delta}$, by Lemma 2.1. By the same reason, the
composition of the last two maps is the multiplication by
$T^{\Delta^{\prime}}$. After tensoring with the Novikov field, these
compositions become isomorphisms, which show that the three
$\Lambda_{0}$-modules $HF(F),HF(F^{D}_{0})$ and $HF(F+\Delta)$ are all
isomorphic over $\Lambda$. ∎

$K_{n,t}$$K_{n,t}+\Delta$$H_{n,t}$ Figure 2. A sandwich of Hamiltonian
functions

Therefore if we only care about the relative symplectic cohomology over the
Novikov field, we can relax the condition of the acceleration data of
Hamiltonian functions we used for computations. That is, the Hamiltonian
functions converge to a fixed negative Morse function on $D$, instead of
converging to zero.

##  3\. A filtration on the completed telescope

Let $(M,\omega)$ be a closed symplectic manifold and $D\subset M$ be a
Liouville domain. Recall that the relative symplectic cohomology
$SH_{M}(D;\Lambda_{0})$ is the cohomology of the completion
$\widehat{tel}(\mathcal{CF})$ of the telescope

$tel(\mathcal{CF})=\bigoplus_{i\in\mathbb{N}}(C_{i}[1]\oplus C_{i}).$

Now we will define a filtration on $\widehat{tel}(\mathcal{CF})$ which is
compatible with the differentials.

###  3.1. Auxiliary symplectic forms

First we define some auxiliary symplectic forms with respect to our Liouville
domain $D$. The boundary $\partial D$ is a contact hypersurface in $M$ and for
small $\epsilon>0$ we write $D_{1+\epsilon}=D\cup(\partial
D\times[1,1+\epsilon])$ as a compact neighborhood of $D$. That is, we fix a
choice of a contact cylinder associated to $D$. On $D_{1+\epsilon}$ there is a
1-form $\theta$ such that $d\theta=\omega$. Then we can interpolate $\omega$
from $D_{1+\frac{3}{4}\epsilon}$ to $M-D_{1+\epsilon}$.

######  Lemma 3.1.

There exists a global 2-form $\tilde{\omega}$ and a 1-form $\tilde{\theta}$ on
$M$ such that

  1. (1)

$\omega=\tilde{\omega}+d\tilde{\theta}$ on $M$;

  2. (2)

Support of $\tilde{\theta}$ is in $D_{1+\epsilon}$ and $\tilde{\theta}=\theta$
in $D_{1+\frac{3}{4}\epsilon}$;

  3. (3)

Support of $\tilde{\omega}$ is in $M-D_{1+\frac{3}{4}\epsilon}$ and
$\tilde{\omega}=\omega$ in $M-D_{1+\epsilon}$.

###### Proof.

Let $\rho(r):[1,2]\rightarrow\mathbb{R}$ be a smooth increasing function such
that

$\rho(r)=\left\\{\begin{aligned} &0,\quad 1\leq r\leq
1+\frac{3}{4}\epsilon;\\\ &1,\quad 1+\epsilon\leq r.\end{aligned}\right.$

Next we define that

$\tilde{\omega}\mid_{x}=\left\\{\begin{aligned} &0,\quad x\in
D_{1+\frac{3}{4}\epsilon};\\\ &d(\rho(r)\theta),\quad x\in
D_{1+\epsilon}-D_{1+\frac{3}{4}\epsilon};\\\ &\omega,\quad x\in
M-D_{1+\epsilon}\end{aligned}\right.$

and

$\tilde{\theta}\mid_{x}=\left\\{\begin{aligned} &\theta,\quad x\in
D_{1+\frac{3}{4}\epsilon};\\\ &(1-\rho(r))\theta,\quad x\in
D_{1+\epsilon}-D_{1+\frac{3}{4}\epsilon};\\\ &0,\quad x\in
M-D_{1+\epsilon}\end{aligned}\right.$

Then we can check that $\tilde{\omega}$ and $\tilde{\theta}$ satisfy the
conditions we need, see Figure 3. ∎

$\tilde{\omega}$$1$$1+\frac{3}{4}\epsilon$$1+\epsilon$$\tilde{\theta}$$1$$1+\frac{3}{4}\epsilon$$1+\epsilon$
Figure 3. Cut-off functions for $\tilde{\omega}$ and $\tilde{\theta}$.

From the following exact sequence

$\cdots\rightarrow H^{1}(D)\rightarrow
H^{2}(M,D)\xrightarrow{j}H^{2}(M)\rightarrow H^{2}(D)\rightarrow\cdots$

we have that $j([\tilde{\omega}])=[\omega]$. For later purposes we assume that
$[\tilde{\omega}]$ admits an integral lift in $H^{2}(M,D;\mathbb{Z})$.

For a Hamiltonian vector field which is small on
$M-D_{1+\frac{3\epsilon}{4}}$, the Floer equation becomes very close to the
genuine Cauchy-Riemann equation on that region. This implies the positivity of
the $\omega$-energy of solutions on that region. To prove this first we recall
the following computation for a solution to the Floer equation. Let $H$ be a
Hamiltonian function and $J$ be a compatible almost complex structure. For a
solution $u$ with finite energy we have that

$\omega([u])=\int
u^{*}\omega=\int\omega(\partial_{s}u,\partial_{t}u)=\int\lvert\partial_{s}u\rvert_{g}^{2}+\int\omega(\partial_{s}u,X_{H}).$

For the second term we have that

$\displaystyle\int\omega(\partial_{s}u,X_{H})=-\int dH(\partial_{s}u)$
$\displaystyle=-\int_{-\infty}^{+\infty}\frac{\partial}{\partial
s}\int_{0}^{1}H(u)dtds=-(\int_{0}^{1}H(u(+\infty,t))dt-\int_{0}^{1}H(u(+\infty,t))dt)$
$\displaystyle\geq-\lvert\lvert H\rvert\rvert$

where $\lvert\lvert H\rvert\rvert$ is the Hofer norm of $H$.

######  Lemma 3.2.

Let $(M,\omega)$ be a closed symplectic manifold and $D$ be a Liouville domain
in $M$. We assume that $[\tilde{\omega}]$ admits an integral lift. Then for
any non-degenerate Hamiltonian function $H$ with $\lvert\lvert
H\rvert\rvert_{M-D_{1+\frac{3\epsilon}{4}}}<1$, for any finite-energy solution
$u:S^{1}\times\mathbb{R}\rightarrow M$ of the Floer equation

$\partial_{s}u+J(\partial_{t}u-X_{H})=0,$

the integral $\int u^{*}\tilde{\omega}\geq 0$. Here $J$ is a cylindrical
almost complex structure compatible with $D$.

###### Proof.

We prove this lemma by computation. By the definitions of $\tilde{\omega}$ and
a cylindrical almost complex structure $J$, we know that
$\tilde{g}(\cdot,\cdot):=\tilde{\omega}(\cdot,J\cdot)$ defines a Riemannian
metric on the region where $\tilde{\omega}\neq 0$. Then by the above
computation we have that

$\int
u^{*}\tilde{\omega}=\int\tilde{\omega}(\partial_{s}u,\partial_{t}u)\geq\int\lvert\partial_{s}u\rvert_{\tilde{g}}^{2}-\lvert\lvert
H\rvert\rvert_{M-D_{1+\frac{3\epsilon}{4}}}\geq-\lvert\lvert
H\rvert\rvert_{M-D_{1+\frac{3\epsilon}{4}}}.$

Since $\tilde{\omega}$ is supported in $M-D_{1+\frac{3\epsilon}{4}}$, we only
need to consider the relative Hofer norm $\lvert\lvert
H\rvert\rvert_{M-D_{1+\frac{3\epsilon}{4}}}$. In particular, if the image of
$u$ does not intersect $M-D_{1+\frac{3\epsilon}{4}}$, then $\int
u^{*}\tilde{\omega}=0$.

By our assumption that $[\tilde{\omega}]$ admits an integral lift, the number
$\int u^{*}\tilde{\omega}$ is an integer. Hence when $\lvert\lvert
H\rvert\rvert_{M-D_{1+\frac{3\epsilon}{4}}}<1$, the number $\int
u^{*}\tilde{\omega}$ is non-negative. ∎

Note that Lemma 3.2 also works for a family of Hamiltonian $\\{H_{t,s}\\}$
under small perturbations. That is, if $\lvert\lvert
H_{t,s}-H_{t,s^{\prime}}\rvert\rvert_{M-D_{1+\frac{3\epsilon}{4}}}<1$ for any
$s,s^{\prime}$ then we still have positivity of solutions of the parameterized
Floer equation.

######  Remark 3.3.

The positivity of the $\tilde{\omega}$-energy of Floer trajectories traveling
outside may be also proved by using Abouzaid-Seidel’s integrated maximal
principle [2]. See a similar situation in Proposition 1.12 [4]. It would be
interesting to compare the $Q$-filtration therein and $\tilde{\omega}$-energy
here.

###  3.2. A filtration which is not exhaustive

By the computation in the last subsection, we know that when the Hamiltonian
functions have small relative Hofer norms outside the Liouville domain, the
outside part of the corresponding Floer solutions carry non-negative energy.
Now we use this fact to define a filtration on the telescope, which is the
underlying complex of the relative symplectic cohomology.

First we consider the case of a single $S$-shape Hamiltonian $H$ such that
$\lvert\lvert H\rvert\rvert<\frac{1}{2}$ on $M-D_{1+\frac{3}{4}\epsilon}$. We
define a valuation of a single element

$a\cdot\gamma\in
CF^{k}(H)=\bigoplus_{\gamma\in\mathcal{CP}^{k}(H)}\Lambda_{0}\cdot\gamma$

by setting

(3.1) $\sigma(a\cdot\gamma)=v(a)-\int_{\gamma}H-\int_{\gamma}\tilde{\theta}$

where $v(a)$ is the valuation on $\Lambda_{0}$. If $\gamma$ is a constant
orbit, the integral $\int_{\gamma}\tilde{\theta}$ is zero. For a general sum
$x=\sum_{i}a_{i}\gamma_{i}$ we define the valuation as

(3.2)
$\sigma(x):=\inf_{i}\\{\sigma(a_{i}\gamma_{i})\\}=\inf_{i}\\{v(a_{i})-\int_{\gamma_{i}}H_{i}-\int_{\gamma_{i}}\tilde{\theta}\\}.$

######  Lemma 3.4.

For the valuation $\sigma$ we have that

$\sigma(d(a\cdot\gamma))\geq\sigma(a\cdot\gamma).$

Hence it induces a filtration

(3.3) $\mathcal{F}^{\lambda}CF^{k}(H)=\\{x\in
CF^{k}(H)\mid\sigma(x)\geq\lambda\\}$

on the complex $CF^{k}(H)$.

###### Proof.

We prove this lemma by a direct computation. First from our definition

$d(a\cdot\gamma)=a\cdot\sum_{\gamma^{\prime},A}n(\gamma,\gamma^{\prime};A)\cdot\gamma^{\prime}\cdot
T^{\omega(A)-\int_{\gamma}H+\int_{\gamma^{\prime}}H}$

where $A$ is the homotopy class represented by a solution $u$ connecting
$\gamma$ and $\gamma^{\prime}$. Hence we have that

(3.4) $\displaystyle\sigma(d(a\cdot\gamma))=$
$\displaystyle\sigma(a\cdot\sum_{\gamma^{\prime}}n(\gamma,\gamma^{\prime};A)\cdot\gamma^{\prime}\cdot
T^{\omega(A)-\int_{\gamma}H+\int_{\gamma^{\prime}}H})$ $\displaystyle=$
$\displaystyle\inf_{\gamma^{\prime}}\\{v(a)+\omega(A)-\int_{\gamma}H+\int_{\gamma^{\prime}}H-\int_{\gamma^{\prime}}H-\int_{\gamma^{\prime}}\tilde{\theta}\\}$
$\displaystyle=$
$\displaystyle\inf_{\gamma^{\prime}}\\{v(a)+\omega(A)-\int_{\gamma}H-\int_{\gamma^{\prime}}\tilde{\theta}\\}$
$\displaystyle=$
$\displaystyle\inf_{\gamma^{\prime}}\\{v(a)+\tilde{\omega}(A)+d\tilde{\theta}(A)-\int_{\gamma}H+\int_{\gamma^{\prime}}\\}$
$\displaystyle=$
$\displaystyle\inf_{\gamma^{\prime}}\\{v(a)+\tilde{\omega}(A)+\int_{\gamma^{\prime}}\tilde{\theta}-\int_{\gamma}\tilde{\theta}-\int_{\gamma}H-\int_{\gamma^{\prime}}\tilde{\theta}\\}$
$\displaystyle=$
$\displaystyle\inf_{\gamma^{\prime}}\\{v(a)+\tilde{\omega}(A)-\int_{\gamma}\tilde{\theta}-\int_{\gamma}H\\}$
$\displaystyle=$
$\displaystyle\inf_{\gamma^{\prime}}\\{\sigma(a\cdot\gamma)+\tilde{\omega}(A)\\}$
$\displaystyle\geq$ $\displaystyle\sigma(a\cdot\gamma).$

Here we use that $\omega=\tilde{\omega}+d\tilde{\theta}$ and the Stokes
formula to compute $d\tilde{\theta}(A)$. The last inequality uses Lemma 3.2.

Similarly we can check that $\sigma(dx)\geq\sigma(x)$ for a sum
$x=\sum_{i}a_{i}\gamma_{i}$. Hence this valuation $\sigma$ induces a
decreasing filtration on $CF(H)$. ∎

Note that the above computation also works for a one-parameter family of
Hamiltonian functions, if the variation of these functions is sufficiently
small outside the Liouville domain. However, to construct the whole telescope
to compute the relative symplectic cohomology we need to let the Hamiltonian
functions go to positive infinity. Hence we can not directly use the above
lemma to give a filtration on the telescope. Next we will discuss how to cut
the whole telescope into pieces such that we can construct a filtration on
each piece, then we put all the pieces together to get a filtration on the
whole telescope.

More precisely, for two $S$-shape Hamiltonian functions $H_{0,t}\leq H_{1,t}$,
we can construct a monotone family $\\{H_{s,t}\\}_{s\in[0,1]}$ to connect
them. Pick a division $0=s_{0}<s_{1}<\cdots<s_{l}=1$ such that all
$H_{s_{i},t}$ are non-degenerate $S$-shape Hamiltonian functions, and the
Hofer distance between $H_{s_{i},t}$ and $H_{s_{i+1},t}$ is less than one.
Then we have a family of continuation maps

$CF(H_{0,t})=CF(H_{s_{0},t})\rightarrow
CF(H_{s_{1},t})\rightarrow\cdots\rightarrow CF(H_{s_{l},t})=CF(H_{1,t}).$

Each of these continuation maps satisfies the above lemma. Next we define the
continuation map $f_{01}:CF(H_{0,t})\rightarrow CF(H_{1,t})$ as the
compositions of the continuation maps between the division. Hence the map
$f_{01}$ satisfies that
$\sigma(f_{01}(a\cdot\gamma))\geq\sigma(a\cdot\gamma)$. Similarly, for a
monotone one-parameter family of smooth Hamiltonian functions $H_{n,t}$, we
define the continuation map $f_{n(n+1)}:CF(H_{n,t})\rightarrow CF(H_{n+1,t})$
by using a sufficiently fine division.

Then we have a one-ray

$\mathcal{CF}:=CF(H_{1,t})\rightarrow CF(H_{2,t})\rightarrow\cdots\rightarrow
CF(H_{n,t})\rightarrow\cdots$

such that its differentials and continuation maps are compatible with
$\sigma$. We write $\widehat{tel}(\mathcal{CF})$ as the completion of the
telescope of this one-ray. For a general element
$x=\sum_{i}a_{i}\gamma_{i}\in\widehat{tel}(\mathcal{CF})$ we define
$\sigma:\widehat{tel}(\mathcal{CF})\rightarrow\mathbb{R}\cup\\{-\infty\\}$ by

(3.5)
$\sigma(x):=\inf_{i}\\{\sigma(a_{i}\gamma_{i})\\}=\inf_{i}\\{v(a_{i})-\int_{\gamma_{i}}H_{i}-\int_{\gamma_{i}}\tilde{\theta}\\}$

This valuation gives a filtration on $\widehat{tel}(\mathcal{CF})$. The
differentials of the completed telescope are sums of the differentials and
continuation maps in the Floer complex. Hence the differential of
$\widehat{tel}(\mathcal{CF})$ is compatible with this filtration, which makes
$\widehat{tel}(\mathcal{CF})$ a filtered differential graded module. The
homology of $\widehat{tel}(\mathcal{CF})$ is the relative symplectic
cohomology, which is independent of the choices of divisions between each pair
of adjacent Hamiltonian functions by a homotopy argument.

Note that for a fixed Hamiltonian $H$ this valuation takes value strictly in
$\mathbb{R}$ since $H$ is bounded and there are finitely many periodic orbits.
But for a general element in the completion of the telescope the valuation can
be negative infinity. Therefore the induced filtration is not exhaustive. That
is,

$\bigcup_{\lambda}\mathcal{F}^{\lambda}\widehat{tel}(\mathcal{CF})\neq\widehat{tel}(\mathcal{CF})$

which is one main reason that the induced spectral sequence sometimes does not
converge. We also remark that this filtration is weakly convergent since

$\bigcap_{\lambda}\mathcal{F}^{\lambda}\widehat{tel}(\mathcal{CF})=\\{0\\}.$

Now we recall two foundational theorems on spectral sequences from [19], one
on the existence and the other on the convergence.

######  Definition 3.5.

Let $R$ be a commutative ring with unit. An $R$-module $A$ is called a
filtered differential graded module if

  1. (1)

$A$ is a direct sum of submodules, $A=\oplus_{n=0}^{\infty}A^{n}$.

  2. (2)

There is an $R$-linear map $d:A\rightarrow A$, satisfying $d\circ d=0$.

  3. (3)

$A$ has a filtration $F$ and the differential $d$ respects the filtration,
that is, $d:F^{p}A\rightarrow F^{p}A$.

######  Theorem 3.6.

(Theorem 2.6, [19]) Each filtered differential graded module $(A,d,F)$
determines a spectral sequence, $\\{E_{r}^{*,*},d_{r}\\},r=1,2,\cdots$ with
$d_{r}$ of bidegree $(r,1-r)$ and

$E_{1}^{p,q}\cong H^{p+q}(F^{p}A/F^{p+1}A).$

######  Theorem 3.7.

(Theorem 3.2, [19]) Let $(A,d,F)$ be a filtered differential graded module
such that the filtration is exhaustive and weakly convergent. Then the
associated spectral sequence with $E_{1}^{p,q}\cong H^{p+q}(F^{p}A/F^{p+1}A)$
converges to $H(A,d)$, that is,

$E_{\infty}^{p,q}\cong F^{p}H^{p+q}(A,d)/F^{p+1}H^{p+q}(A,d).$

##  4\. The induced spectral sequence

Previously we constructed a filtration on $\widehat{tel}(\mathcal{CF})$.
However the induced spectral sequence does not always converge since the
filtration is not exhaustive. Now we study in what geometric conditions this
induced spectral sequence converges. In this section we assume that $D$ is a
Liouville domain with an index bounded boundary in a symplectic Calabi-Yau
manifold $M$. How to generalize this condition to other settings will be
discussed in the last section.

###  4.1. Ignoring upper orbits and convergence

For practical reasons, we will consider a truncated relative symplectic
cohomology. There are two kinds of truncations: one for degree and the other
for energy. Note that the relative symplectic cohomology $SH^{k}_{M}(D)$ is a
$\mathbb{Z}$-graded module defined over $\Lambda_{0}$. These two truncations
both happen on the chain level. The degree-truncation means that we only focus
on the chain complex with a finite degree range, and the energy-truncation
means that we only consider the chain complex over $\Lambda_{E}$ for some
positive number $E$.

Now for a monotone family of $S$-shape Hamiltonian functions, the orbits form
two groups: upper orbits and lower orbits. And the Floer
differentials/continuation maps have four types: upper-to-upper, upper-to-
lower, lower-to-upper and lower-to-lower. The next lemma says that, under the
index bounded condition, any lower-to-upper Floer trajectory has a big
topological energy.

######  Lemma 4.1.

Given any energy bound $E>0$ and an integer $k$, there exists an $S$-shape
Hamiltonian function such that any lower-to-upper Floer differential with
degree-$k$ input has topological energy greater than $E$.

###### Proof.

For a Floer differential with degree-$k$ input $\gamma$ and degree-$(k+1)$
output $\gamma^{\prime}$, the energy weight is

$\displaystyle\int u^{*}\omega-\int_{\gamma}H_{t}+\int_{\gamma^{\prime}}H_{t}$
$\displaystyle=$
$\displaystyle-\int_{\gamma}\tilde{\theta}+\int_{\gamma^{\prime}}\tilde{\theta}+\int
u^{*}\tilde{\omega}-\int_{\gamma}H_{t}+\int_{\gamma^{\prime}}H_{t}.$

By the index bounded condition, the difference
$-\int_{\gamma}\tilde{\theta}+\int_{\gamma^{\prime}}\tilde{\theta}$ is a
bounded number which only depends on $k$. And by the computation in previous
subsections, the term $\int u^{*}\tilde{\omega}$ is non-negative. Therefore if
we choose an $S$-shape Hamiltonian function such that its upper level is high
enough, then the above energy weight is larger than $E$. ∎

Note that this estimate is uniform for all degree $k$ orbits, hence we can
make a subcomplex which only contains lower orbits. Let $H$ be a Hamiltonian
function which satisfies the above lemma for all three integers $k-1,k,k+1$
and an energy bound $E$. Then we have that

######  Lemma 4.2.

Let $CF^{k,L}(H)$ be the subspace of $CF^{k}(H)$ which only contains lower
orbits. Then

$0\rightarrow
CF^{k-1,L}(H)\xrightarrow{d}CF^{k,L}(H)\xrightarrow{d}CF^{k+1,L}(H)\rightarrow
0$

satisfies that $d\circ d=0$ over $\Lambda_{E}$.

###### Proof.

Take $\gamma\in CF^{k-1,L}(H)$, the usual argument to show $d\circ
d(\gamma)=0$ is to look at broken Floer trajectories. In our case, if it
breaks along an upper orbit, then the energy weight is greater than $E$ by the
above lemma, which is automatically zero over $\Lambda_{E}$. Hence we can
ignore the upper orbits contributions. ∎

The above two lemmas tell us for any fixed energy bound $E$ and degree-$k$, we
can use only lower orbits to form a homology theory. Next we construct a
family of Hamiltonian functions, such that the upper-to-upper continuation
maps also have big topological energy.

######  Lemma 4.3.

Given any energy bound $E>0$ and an integer $k$, there exists a family of non-
degenerate $S$-shape Hamiltonian functions $\\{H_{n}\\}_{n\in\mathbb{N}}$ such
that

  1. (1)

$H_{1}\leq H_{2}\leq\cdots H_{n}\leq H_{n+1}\leq\cdots$;

  2. (2)

$H_{1}=H_{2}=\cdots H_{n}=\cdots$ on $S^{1}\times D$;

  3. (3)

$H_{1}$ satisfies the above two lemmas;

  4. (4)

any lower degree-$k$ orbits are inside $D$;

  5. (5)

any continuation map between upper degree-$k$ orbits has topological energy
greater than $E$.

###### Proof.

Take $H_{1}$ as a function which satisfies the above two lemmas. Then we fix
the lower part of $H_{1}$ and raise the upper part of it high enough, to get
$H_{2}$. If we have a continuation map from upper degree-$k$ orbits of $H_{1}$
to upper degree-$k$ orbits of $H_{2}$, then it is weighted by an energy

$\int
u^{*}\omega-\int_{\gamma}H_{1,t}+\int_{\gamma^{\prime}}H_{2,t}+\int(\partial_{s}H_{s,t}).$

By the monotonicity of the family $\\{H_{n}\\}_{n\in\mathbb{N}}$ and the
computation in the above lemma, this energy is larger than $E$ if the
difference between the upper levels of $H_{1}$ and $H_{2}$ are big enough.
Next we can construct $\\{H_{n}\\}_{n\in\mathbb{N}}$ inductively. ∎

Therefore by using the above family of Hamiltonian functions, we are computing
the Hamiltonian Floer cohomology of a semi lower-continuous function, which is
a non-degenerate Hamiltonian function on $D$ and is positive infinity outside
$D$. By Proposition 2.10, the resulting invariant is isomorphic to the
relative symplectic cohomology over the Novikov field.

Now we consider two direct systems

$\mathcal{CF}^{k}=CF^{k}(H_{1,t})\rightarrow CF^{k}(H_{2,t})\rightarrow\cdots$

and

$\mathcal{CF}^{k,L}=CF^{k,L}(H_{1,t})\rightarrow
CF^{k,L}(H_{2,t})\rightarrow\cdots$

over $\Lambda_{E}$, induced by Hamiltonian functions defined in the above
lemma. By the special shapes of the Hamiltonian functions, any high-to-high
continuation maps are zero over $\Lambda_{E}$. Hence all high orbits are
equivalent to some lower orbits or zero, in the direct limit. In conclusion,
we have the following chain level isomorphism

$\varinjlim_{n}\mathcal{CF}^{k}\cong\varinjlim_{n}\mathcal{CF}^{k,L}$

over $\Lambda_{E}$. This induces a quasi-isomorphism

$H(\varinjlim_{n}\mathcal{CF}^{k};\Lambda_{E})\cong
H(\varinjlim_{n}\mathcal{CF}^{k,L};\Lambda_{E})$

We call the process to get this quasi-isomorphism ignoring upper orbits. And
we define the truncated symplectic cohomology as

$SH^{k}_{M}(D;\Lambda_{E}):=H(\varinjlim_{n}\mathcal{CF}^{k,L};\Lambda_{E}).$

The truncated symplectic cohomology has the advantage that all generators of
the underlying complex are local and the differentials are global. This is
more suitable to realize the local-to-global philosophy that the global
picture is a deformation of the local picture. And the “deformation” will be
characterized by our spectral sequence. In other words, the
$\tilde{\omega}$-energy captures how “far” a Floer solution goes out $D$.
Moreover, under the index bounded condition, the underlying complex of
$SH^{k}_{M}(D;\Lambda_{E})$ is finite-dimensional.

######  Lemma 4.4.

For a fixed degree $k$, the direct limit $\varinjlim_{n}\mathcal{CF}^{k,L}$
which contains lower orbits is a finite-dimensional free $\Lambda_{E}$-module.

###### Proof.

By the index bounded condition and non-degeneracy of the contact form, for
each fixed Hamiltonian $CF^{k,L}(H_{n,t})$ is a finite-dimensional free
$\Lambda_{E}$-module, generated by degree-$k$ orbits. On the other hand, the
lower parts of the Hamiltonian functions $H_{n,t}$ are fixed. Hence we have a
canonical identification between $CF^{k,L}(H_{n,t})$ for different $n$. Next
we study the continuation maps between $CF^{k,L}(H_{n,t})$. After identifying
the generators of $CF^{k,L}(H_{n,t})$ for different $n$, the continuation maps
can be written as $l\times l$ matrices $\\{a^{n}_{ij}\\}$ with entries
$a^{n}_{ij}\in\Lambda_{E}$, where $l$ is the dimension of $CF^{k,L}(H_{n,t})$.

Let $u$ be a continuation maps which is contained in the region where the
Hamiltonian function is fixed. Then by regularity it can only be the identity
map, see Lemma 2.1. There may be other continuation maps that travels outside
$D$, which have non-trivial topological energy. Hence the matrices of
continuation maps have the following properties:

  1. (1)

the entries on the diagonal are $a^{n}_{jj}=1+b^{n}_{jj}$ with
$v(b^{n}_{jj})>0$;

  2. (2)

the off-diagonal entries have strictly positive valuations.

Note that the determinant of the matrix $\\{a^{n}_{ij}\\}$ has a constant term
one, which says that the matrix is invertible. Hence each continuation map is
an isomorphism of $\Lambda_{0}$-modules. So the direct limit
$\varinjlim_{n}\mathcal{CF}^{k,L}$ is isomorphic to $CF^{k,L}(H_{n,t})$, which
is a finite-dimensional free $\Lambda_{E}$-module. ∎

To effectively use the truncated symplectic cohomology, there are two options:
to show it is an invariant with good properties, or to relate it with the
original relative symplectic cohomology $SH^{k}_{M}(D)$.

For the first option, we expect the following is true

######  Proposition 4.5.

The truncated symplectic cohomology $SH^{k}_{M}(D;\Lambda_{E})$ is a finite-
dimensional $\Lambda_{E}$-module. Let $\lambda>0$ be the smallest number such
that

$T^{\lambda}\cdot SH^{k}_{M}(D;\Lambda_{E})=0.$

Then the displacement energy of $D$ is not less than $\lambda$.

###### Proof.

The finite-dimensionality follows from the above lemma. We expect the proof of
the energy relation is similar to Remark 4.2.8 in [28] in the original
relative symplectic cohomology setting. The full proof will be pursued in the
future. ∎

We call the number $\lambda$ the torsion threshold of
$SH^{k}_{M}(D;\Lambda_{E})$. So this proposition is an analogue of the
relation between the torsion threshold of the Lagrangian Floer cohomology and
the displacement energy of a Lagrangian submanifold. Since the energy bound
$E$ can be as large as one needs, this proposition is useful for most
displacement problems.

For the second option, we have the following

######  Proposition 4.6.

The inverse limit of the truncated symplectic cohomology recovers the original
relative symplectic cohomology. That is,

$(\varprojlim_{E}SH^{k}_{M}(D;\Lambda_{E}))\otimes_{\Lambda_{0}}\Lambda\cong
SH^{k}_{M}(D)\otimes_{\Lambda_{0}}\Lambda.$

Here the inverse limit is taken as $E$ goes to infinity.

Note that for a general $\Lambda_{0}$-module $K$, we do not always have the
relation

$K\cong\varprojlim_{E}(K\otimes_{\Lambda_{0}}\Lambda_{E}).$

Hence the above proposition is not a purely algebraic result. The index
bounded condition is essentially used to obtain some finiteness in the chain
level. We will compute the $\varprojlim^{1}$ functor in both chain level and
homology level to show the homology commute with the inverse limit. First we
recall some results in homological algebra.

######  Definition 4.7.

An inverse system

$\mathcal{C}=C_{1}\leftarrow C_{2}\leftarrow\cdots$

is said to satisfy the Mittag-Leffler condition if for each $n\in\mathbb{N}$,
there exists $i\geq n$ such that for all $j\geq i$, we have that

$\text{Im}(C_{j}\rightarrow C_{n})\cong\text{Im}(C_{i}\rightarrow C_{n}).$

The Mittag-Leffler condition shows the vanishing of the $\varprojlim^{1}$ of
an inverse system.

######  Proposition 4.8.

(Proposition 3.5.7 [27]) If an inverse system $\mathcal{C}$ satisfies the
Mittag-Leffler condition, then $\varprojlim^{1}(\mathcal{C})=0$.

######  Proposition 4.9.

(Proposition 3.5.8 [27]) For an inverse system

$\mathcal{C}=C_{1}\leftarrow C_{2}\leftarrow\cdots$

of complexes, which satisfies the degree-wise Mittag-Leffler condition, we
have an short exact sequence

$0\rightarrow\varprojlim\nolimits^{1}H^{*}(C_{n})\rightarrow H^{*}(\varprojlim
C_{n})\rightarrow\varprojlim H^{*}(C_{n})\rightarrow 0.$

Now we begin the proof of Proposition 4.6.

###### Proof.

For an energy bound $E>0$, we already showed that

$\varinjlim_{n}\mathcal{CF}^{k}=\varinjlim_{n}CF^{k}(H_{n,t};\Lambda_{E})\cong\varinjlim_{n}CF^{k,L}(H_{n,t};\Lambda_{E})=\varinjlim_{n}\mathcal{CF}^{k,L}$

over $\Lambda_{E}$. The first and third equalities are definitions. The second
isomorphism is given by the process of ignoring upper orbits. So
$\varinjlim_{n}\mathcal{CF}^{k}$ is a finite-dimensional $\Lambda_{E}$-module,
freely generated by lower orbits. Now let $E_{1}<E_{2}<\cdots$ be an
increasing sequence which goes to infinity. Abusing the notation, we just
write

$\mathcal{CF}(\Lambda_{E_{l}}):=\varinjlim_{n}CF^{k}(H_{n,t};\Lambda_{E_{l}}).$

Consider an inverse system

$\mathcal{CF}(\Lambda_{E_{1}})\leftarrow\mathcal{CF}(\Lambda_{E_{2}})\leftarrow\cdots$

Here $\mathcal{CF}(\Lambda_{E_{l}})$ is the one-ray of Floer complexes with
coefficients in $\Lambda_{E_{l}}$. And the maps in the inverse system are
projections. Hence in the chain level all maps are surjective which satisfies
the Mittag-Leffler condition. Then we have the exact sequence

$0\rightarrow\varprojlim\nolimits^{1}H^{*}(\mathcal{CF}(\Lambda_{E_{l}}))\rightarrow
H^{*}(\varprojlim\mathcal{CF}(\Lambda_{E_{l}}))\rightarrow\varprojlim
H^{*}(\mathcal{CF}(\Lambda_{E_{l}}))\rightarrow 0.$

Next we will show that

$H^{*}(\varprojlim\mathcal{CF}(\Lambda_{E_{l}}))\cong\varprojlim
H^{*}(\mathcal{CF}(\Lambda_{E_{l}})).$

It suffices to show $\varprojlim^{1}H^{*}(\mathcal{CF}(\Lambda_{E_{l}}))=0$,
which will be implied by that the homology
$H^{*}(\mathcal{CF}(\Lambda_{E_{l}}))$ satisfies the degree-wise Mittag-
Leffler condition. This is where the finite dimensionality of
$\mathcal{CF}(\Lambda_{E_{l}})$ plays the key role.

Without losing generality, we assume that $\mathcal{CF}(\Lambda_{E_{l}})$ is
one-dimensional for some degree $k$. That is, we only have one generator
$\alpha$ of $\mathcal{CF}(\Lambda_{0})$. And $\mathcal{CF}(\Lambda_{E_{l}})$
is just the quotient of $\mathcal{CF}(\Lambda_{0})$ by reducing the
coefficient. By Gromov compactness, the Floer differential acting on $\alpha$
has the following decomposition

$d\alpha=d_{0}\alpha+d_{1}\alpha\cdot T^{\lambda_{1}}+d_{2}\alpha\cdot
T^{\lambda_{2}}+\cdots$

with $\lambda_{n}$ an increasing sequence of real numbers going to infinity.
Now for a fixed $l$, we need show that $H^{k}(\mathcal{CF}(\Lambda_{E_{l}}))$
satisfies the Mittag-Leffler condition. That is, find $i\geq l$ such that

$\text{Im}(H^{k}(\mathcal{CF}(\Lambda_{E_{j}}))\rightarrow
H^{k}(\mathcal{CF}(\Lambda_{E_{l}})))\cong\text{Im}(H^{k}(\mathcal{CF}(\Lambda_{E_{i}}))\rightarrow
H^{k}(\mathcal{CF}(\Lambda_{E_{l}})))$

for all $j\geq i$.

Let $z$ be the least integer such that $d_{z}\alpha\neq 0$. If $z=+\infty$,
which means that $d\alpha=0$, then

$H^{k}(\mathcal{CF}(\Lambda_{E_{i}}))\rightarrow
H^{k}(\mathcal{CF}(\Lambda_{E_{l}}))$

is a projection for all $i\geq l$. Hence all above maps are surjective, which
satisfy the Mittag-Leffler condition. If $z\neq+\infty$, then there exists $i$
such that $\lambda_{z}<E_{i}$. Therefore for any $j\geq i$, the element
$\alpha$ is not closed in $\Lambda_{E_{j}}$-coefficient. So all
$H^{k}(\mathcal{CF}(\Lambda_{E_{i}}))\cong 0$, which satisfy the Mittag-
Leffler condition.

This completes the proof when $\mathcal{CF}(\Lambda_{E_{l}})$ is one-
dimensional. When it is finite-dimensional, a similar proof by taking the
maximum of finitely many possibilities of $\lambda_{z}$’s works.

Hence we have showed that

$H^{*}(\varprojlim\mathcal{CF}(\Lambda_{E_{l}}))\cong\varprojlim
H^{*}(\mathcal{CF}(\Lambda_{E_{l}})).$

The left-hand side term is the non-truncated Hamiltonian Floer cohomology of a
lower semi-continuous function, which is a fixed Morse function on $D$ and is
positive infinity on $M-D$, see definition (2.12). By Proposition 2.10, it is
isomorphic to the non-truncated relative symplectic cohomology over the
Novikov field. In conclusion, we have that

$\displaystyle SH_{M}(D)\otimes_{\Lambda_{0}}\Lambda\cong
H^{*}(\varprojlim\mathcal{CF}(\Lambda_{E_{l}}))\otimes_{\Lambda_{0}}\Lambda$
$\displaystyle\cong$ $\displaystyle(\varprojlim
H^{*}(\mathcal{CF}(\Lambda_{E_{l}})))\otimes_{\Lambda_{0}}\Lambda=(\varprojlim_{E}SH_{M}(D;\Lambda_{E}))\otimes_{\Lambda_{0}}\Lambda.$

This completes the proof of Proposition 4.6, which is $(2)$ in Theorem 1.1. ∎

###  4.2. The first page of the spectral sequence

Now we will study a particular filtration on
$\varinjlim_{n}\mathcal{CF}^{k,L}$, which will give us a standard first page
in the spectral sequence. First we observe that by using the special family of
$S$-shape Hamiltonians, all Floer trajectories which are not contained in the
Liouville domain pair with $\tilde{\omega}$ positively.

######  Proposition 4.10.

For an $S$-shape Hamiltonian function $H$, let $\gamma_{-},\gamma_{+}$ be two
one-periodic orbits of $H$ which are lower orbits. Let $u$ be a Floer solution
connecting $\gamma_{-},\gamma_{+}$. If $im(u)\cap
M-D_{1+\frac{3\epsilon}{4}}\neq\emptyset$, then $\tilde{\omega}(u)>\hbar$
where $\hbar$ is a positive number independent of $H$.

###### Proof.

Note that if $\textit{Im}(u)\cap M-D_{1+\frac{3\epsilon}{4}}\neq\emptyset$
then $\textit{Im}(u)\cap M-D_{1+\epsilon}\neq\emptyset$ by the maximal
principle. We pick a number $\epsilon^{\prime}$ which is slightly greater than
$\epsilon$ and $\partial D_{1+\epsilon^{\prime}}$ intersects $Im(u)$
transversally. The intersection is a disjoint union of circles in
$S^{1}\times\mathbb{R}$, which is non-empty when $\epsilon^{\prime}$ is close
to $\epsilon$. And we write $\textit{Im}(u_{out}):=\textit{Im}(u)\cap
M-D_{1+\epsilon^{\prime}}$. Then we compute
$\tilde{\omega}(u_{out})=\omega(u_{out})$ since $\tilde{\omega}=\omega$ on
$M-D_{1+\epsilon^{\prime}}$.

$\omega(u_{out})=\int
u_{out}^{*}\omega=\int_{out}\omega(\partial_{s}u,\partial_{t}u)=\int_{out}\lvert\partial_{s}u\rvert_{g}^{2}+\int_{out}\omega(\partial_{s}u,X_{H}).$

For the second term we have that

$\int_{out}\omega(\partial_{s}u,X_{H})=-\int_{out}dH(\partial_{s}u)=-\int_{\partial
out}H(u(s,t))dt.$

Note that $H(u(s,t))$ is a constant function on $\partial u_{out}$ since on
the cylindrical region our Hamiltonian only depends on the radial coordinate.
Hence we have $\int_{\partial out}H(u(s,t))dt=0$ by a topological fact that

$[\partial u_{out}]=0\in H_{1}(S^{1}\times\mathbb{R};\mathbb{Z}).$

Therefore $\omega(u_{out})=\int_{out}\lvert\partial_{s}u\rvert_{g}^{2}>0$. If
we assume that $\tilde{\omega}$ admits an integral lift in
$H^{2}(M,D;\mathbb{Z})$ then $\omega(u_{out})\geq 1$.

Next we write $u_{middle}$ as the part of $u$ in the cylindrical region
between $1+\frac{3\epsilon}{4}$ and $1+\epsilon^{\prime}$. In this region the
form $\tilde{\omega}$ is a product of $\omega$ and a (non-negative) cut-off
function. Then a similar computation shows that
$\tilde{\omega}(u_{middle})\geq 0$. In conclusion we have that
$\tilde{\omega}(u)=\tilde{\omega}(u_{middle})+\tilde{\omega}(u_{out})\geq 1$.
∎

We include a lemma of the topological fact used above.

######  Lemma 4.11.

Let $u$ be a Floer solution connecting two Hamiltonian orbits that are
contained in $D_{1+\frac{3\epsilon}{4}}$. Then we have that

$[\partial u_{out}]=0\in H_{1}(S^{1}\times\mathbb{R};\mathbb{Z}).$

###### Proof.

The asymptotic boundaries of $u$ are inside $D_{1+\frac{3\epsilon}{4}}$. Hence
$\partial u_{out}$ bounds $u_{out}$ in $S^{1}\times\mathbb{R}$. Then we
consider the exact sequence for relative homology groups

$\cdots H_{2}(S^{1}\times\mathbb{R},\partial
u_{out};\mathbb{Z})\xrightarrow{\partial}H_{1}(\partial
u_{out};\mathbb{Z})\rightarrow
H_{1}(S^{1}\times\mathbb{R};\mathbb{Z})\rightarrow\cdots$

The class $[\partial u_{out}]$ is in the image of the map $\partial$ hence it
maps to zero in $H_{1}(S^{1}\times\mathbb{R};\mathbb{Z})$. ∎

To apply the above positivity estimate to the direct limit, we also need to
consider continuation maps, which involves family of Hamiltonian functions
$H_{s}$. We will use the argument in the previous subsection which cuts a one-
ray of Hamiltonian functions into small intervals. Let $H_{0},H_{1}$ be two
Hamiltonian functions with fixed lower parts and $\\{H_{s}\\}_{s\in[0,1]}$
connecting them. Take $s,s^{\prime}\in[0,1]$ such that the Hofer norm
$\lvert\lvert
H_{s}-H_{s^{\prime}}\rvert\rvert_{[1+\frac{3\epsilon}{4},1+\epsilon]\times\partial
D}<\frac{1}{2}$, then we have also a positive energy $0<\hbar<\frac{1}{2}$ for
continuation maps connecting orbits of $H_{s}$ and $H_{s^{\prime}}$. Now we
choose a fine division $0=s_{0}<s_{1}<\cdots<s_{n}=1$ such that for each pair
$s_{l},s_{l+1}$, the continuation maps connecting orbits of $H_{s_{l}}$ and
$H_{s_{l+1}}$ has a positive $\tilde{\omega}$-energy, if it travels outside.
Then the continuation maps connecting orbits of $H_{0}$ and $H_{1}$ will be
defined as the composition of the continuation maps between the division,
hence they also have such a property.

Finally, the filtration $\mathcal{F}$ defined in Section 3 descends to an
exhaustive filtration on $\varinjlim_{n}\mathcal{CF}^{k,L}$. Next we make this
$\mathbb{R}$-valued filtration into a $\mathbb{Z}$-valued filtration:

(4.1)
$\mathcal{F}_{\hbar}^{l}(\varinjlim_{n}\mathcal{CF}^{k,L}):=\\{x\in\varinjlim_{n}\mathcal{CF}^{k,L}\mid\sigma(x)\geq
l\hbar\\}$

for any integer $l$. Then the filtration $\mathcal{F}_{\hbar}$ is a
$\mathbb{Z}$-valued filtration which is exhaustive and weakly convergent.
Hence by Theorem 3.6 and Theorem 3.7 we have a convergence spectral sequence.

In the construction of this spectral sequence, its first page is calculated by
using all differentials of which the change of the $\tilde{\omega}$-energy are
less than $\hbar$. On the other hand, Floer solutions that are not contained
in $D$ are weighted by a positive $\tilde{\omega}$-energy greater than
$\hbar$. Hence the first page of this spectral sequence is calculated by all
differentials and continuation maps that are in $D$. Since in $D$ our
Hamiltonian function is a fixed function, the continuation maps are identity
maps and the differentials give the classical symplectic cohomology
$SH^{k}(D;\Lambda_{E})$. In particular, when $[\tilde{\omega}]=0$, there will
not be outside contribution hence the spectral sequence degenerates at its
first page. Now we have completed the proof of $(1)$ and $(3)$ in Theorem 1.1.

One direct application is that $SH^{k}_{\hat{D}}(D;\Lambda_{E})\cong
SH^{k}(D;\Lambda_{E})$ for any $E>0$. By taking the inverse limit over $E$, we
have the following.

######  Proposition 4.12.

For an index bounded Liouville domain $D$ and its completion $\hat{D}$, we
have the following isomorphism

$SH_{\hat{D}}(D)\otimes_{\Lambda_{0}}\Lambda\cong SH(D;\Lambda)$

where the right hand side term $SH(D;\Lambda)$ is the classical symplectic
cohomology of $D$ with $\Lambda$ coefficient, defined by Viterbo [31].

The key idea of the construction of the spectral sequence is that, under the
index bounded condition, a direct limit of Floer complexes can be replaced by
the Floer complex of a single Hamiltonian function. This reduces a possibly
infinite-dimensional complex to a finite-dimensional complex. Hence the
completion procedure does not matter. Now we use this idea to present another
application, which relates the action-completed symplectic cohomology and the
classical symplectic cohomology.

######  Proposition 4.13.

For an index bounded convex domain $D$ and its completion $\hat{D}$, we have
the following isomorphism

$\widehat{SH}(D)\otimes_{\Lambda_{0}}\Lambda\cong SH(D;\Lambda)$

where the left hand side term $\widehat{SH}(D)$ is the action-completed
symplectic cohomology by Venkatesh [29].

First we briefly recall the definition of the action-completed symplectic
cohomology. We identify $\hat{D}$ with $D\cup([1,+\infty)\times\partial D)$,
and write $r$ as the coordinate on $[1,+\infty)$.

######  Definition 4.14.

For a fixed small positive number $\epsilon$, a Hamiltonian function
$H:\hat{D}\rightarrow\mathbb{R}$ is called admissible if

  1. (1)

it is negative on the region where $r<1$;

  2. (2)

it is a $C^{2}$-small negative Morse function on the region where
$r<1-\epsilon$;

  3. (3)

it only depends on $r$ and it is a convex function on the region where
$r>1-\epsilon$;

  4. (4)

all the non-constant one-periodic orbits of $H$ are in the region where
$1-\epsilon<r<1$;

  5. (5)

it is linear with a positive slope on the region where $1+\epsilon<r$;

Then for an admissible function $H$, we can take the time-dependent
perturbation to get a non-degenerate Hamiltonian function $H_{t}$. Let
$\gamma$ be a one-periodic orbit of $H_{t}$, a capping of $\gamma$ is a map
$w$ from a disk to $\hat{D}$ such that its restriction on the boundary is
$\gamma$.

The Floer complex $CF^{k}(H_{t})$ is the free $\Lambda$-module generated by
contractible one-periodic orbits with index $k$. For each generator $\gamma$
we choose a capping $w$. And the differential is defined as

$d(\gamma):=\sum_{\gamma^{\prime}}\sum_{[u]\in\pi_{2}(\gamma,\gamma^{\prime})}n(\gamma,\gamma^{\prime};[u])\gamma^{\prime}\cdot
T^{\omega([u\\#w\\#(-w^{\prime})])}$

where $n(\gamma,\gamma^{\prime};[u])$ is the count of Floer solutions
connecting $\gamma$ and $\gamma^{\prime}$ representing the class $[u]$. We
extend $d$ to $CF^{k}(H_{t})$ linearly over $\Lambda$.

For an element $x=\lambda\cdot\gamma$ where $\lambda\in\Lambda$, we define an
action

$\mathcal{A}_{H_{t}}(x)=v(\lambda)+\int_{\gamma}H_{t}+\int w^{*}\omega.$

And for $x=\sum_{i=1}^{n}\lambda_{i}\gamma_{i}$, we define
$\mathcal{A}_{H_{t}}(x)=\min_{i}\\{\mathcal{A}_{H_{t}}(\lambda_{i}\gamma_{i})\\}$.
Then one can directly check that

$\mathcal{A}_{H_{t}}(dx)\geq\mathcal{A}_{H_{t}}(x).$

Hence

$CF^{k}_{a}(H_{t}):=\\{x\in CF^{k}(H_{t})\mid\mathcal{A}_{H_{t}}(x)>a\\}$

form a subcomplex. And we define
$CF^{k}_{(a,b)}(H_{t})=CF^{k}_{a}(H_{t})/CF^{k}_{b}(H_{t})$ for any numbers
$a<b$. We remark that the definitions of the differential and the action both
depend on the choices of cappings, however the filtered complex
$CF^{k}_{(a,b)}(H_{t})$ does not depend on the choice of cappings.

Next suppose that we have a sequence $\\{H_{n,t}\\}_{n=1}^{+\infty}$ of
admissible Hamiltonian functions, such that $H_{n,t}\leq H_{n+1,t}$ and the
linear slope of $H_{n,t}$ goes to infinity. For a fixed degree $k$, the
underlying complexes to compute $SH(\hat{D})$ and $\widehat{SH}(D)$ are

$SC(\hat{D})=\varinjlim_{n}\varprojlim_{a}\varinjlim_{b}CF^{k}_{(a,b)}(H_{n,t})$

and

$\widehat{SC}(D)=\varprojlim_{a}\varinjlim_{b}\varinjlim_{n}CF^{k}_{(a,b)}(H_{n,t})$

where $a\rightarrow-\infty$ and $b\rightarrow+\infty$. In the following we
will show that $SC(\hat{D})\cong\widehat{SC}(\hat{D})$ in the chain level,
which is actually stronger than $SH(\hat{D})\cong\widehat{SH}(D)$.

When the boundary of $D$ is index bounded, we can use a special family of
admissible Hamiltonian $\\{H_{n}\\}_{n=1}^{+\infty}$ in the above definitions,
for a fixed $k$. Let $G$ be an admissible Hamiltonian function of which the
slope is large when $r>1$. Hence all degree-$k$ orbits are in the region where
$r<1$. Let $\\{H_{n}\\}_{n=1}^{+\infty}$ be a sequence of admissible
Hamiltonian functions, then we have the following.

######  Lemma 4.15.

For any real numbers $a,b$ we have that

$\varinjlim_{n}CF^{k}_{(a,b)}(H_{n})\cong CF^{k}_{(a,b)}(G).$

###### Proof.

From $G$ we can construct a new cofinal family of admissible Hamiltonian
functions $\\{G_{n}\\}_{n=1}^{+\infty}$ such that

  1. (1)

$G_{1}=G$.

  2. (2)

$G_{n}=G$ if $r<1$.

  3. (3)

the slopes of $G_{n}$ goes to infinity as $n$ goes to infinity.

Since $\\{G_{n}\\}_{n=1}^{+\infty}$ is cofinal, we have that

$\varinjlim_{n}CF^{k}_{(a,b)}(H_{n})\cong\varinjlim_{n}CF^{k}_{(a,b)}(G_{n}).$

Next we show that $\varinjlim_{n}CF^{k}_{(a,b)}(G_{n})\cong
CF^{k}_{(a,b)}(G)$.

We focus on the degree-$k$ orbits of $G_{n}$. By the index bounded condition
and the special shape of $G_{n}$, there is a canonical basis of
$CF^{k}_{(a,b)}(G_{n})$. Geometrically this basis corresponds to the
degree-$k$ orbits of $G$, with action in $(a,b)$. Then we study the
continuation maps between $CF^{k}_{(a,b)}(G_{n})$ and
$CF^{k}_{(a,b)}(G_{n+1})$. Since the choice of $G_{n}$ is that they all equal
to $G$ on this region, the continuation maps are identity maps, by Lemma 2.1.
So we have that

$\varinjlim_{n}CF^{k}_{(a,b)}(G_{n})\cong
CF^{k}_{(a,b)}(G_{1})=CF^{k}_{(a,b)}(G).$

∎

By using the above Lemma, we have that

$\displaystyle SC^{k}(\hat{D})$
$\displaystyle=\varinjlim_{n}\varprojlim_{a}\varinjlim_{b}CF^{k}_{(a,b)}(H_{n})\cong\varinjlim_{n}\varprojlim_{a}\varinjlim_{b}CF^{k}_{(a,b)}(G_{n})$
$\displaystyle\cong\varinjlim_{n}CF^{k}_{(-\infty,+\infty)}(G_{n})\cong
CF^{k}_{(-\infty,+\infty)}(G)$

and

$\widehat{SC^{k}}D)=\varprojlim_{a}\varinjlim_{b}\varinjlim_{n}CF^{k}_{(a,b)}(H_{n})\cong\varprojlim_{a}\varinjlim_{b}CF^{k}_{(a,b)}(G)=CF^{k}_{(-\infty,+\infty)}(G).$

So we get an isomorphism on the chain level, which induces an isomorphism
between $SH(\hat{D})$ and $\widehat{SH}(D)$.

Note that in Proposition 4.13, the domain $D$ is only convex, not necessarily
exact. (When $D$ is exact, the same conclusion is true, shown in [29].) Now we
sketch a way to study the relative symplectic cohomology when $D$ is not
exact. In this case, the auxiliary one form $\tilde{\theta}$ only lives in the
cylindrical region and cannot be extended to all of $D$. So there is no
absolute action functional for orbits and Floer solutions in $D$. But we can
still break the symplectic form $\omega$ into pieces by using cut-off
functions in the cylindrical region. Suppose that $D$ only has one boundary
component, what we will get is

$\omega=\tilde{\omega}_{-}+d\tilde{\theta}+\tilde{\omega}_{+}$

where the support of $\tilde{\omega}_{-}$ is in $D$, the support of
$\tilde{\theta}$ is in the cylindrical region and the support of
$\tilde{\omega}_{+}$ is outside of $D$. Ideally the
$\tilde{\omega}_{-}$-energy characterize how far a Floer solution goes inside
$D$ and the $\tilde{\omega}_{+}$-energy characterize how far a Floer solution
goes outside $D$. Then the topological energy of a Floer solution breaks into

$\int u^{*}\omega+\int_{\gamma_{+}}H-\int_{\gamma_{-}}H=\int
u^{*}\tilde{\omega}_{-}+\int u^{*}\tilde{\omega}_{+}+\int
u^{*}d\tilde{\theta}+\int_{\gamma_{+}}H-\int_{\gamma_{-}}H.$

Previously we use the positivity of $\int u^{*}\tilde{\omega}_{+}$ and the
bounds of $\int u^{*}d\tilde{\theta}$ to estimate the topological energy. Now
we have one more term $\int u^{*}\tilde{\omega}_{-}$. A priori it is not
necessarily positive. This difference between $\int u^{*}\tilde{\omega}_{-}$
and $\int u^{*}\tilde{\omega}_{+}$ comes from that the cut-off functions have
different derivatives for $\tilde{\omega}_{\pm}$ respectively, see an analogue
in Figure 5. Therefore if we have a priori lower bound for $\int
u^{*}\tilde{\omega}_{-}$ then there should be a similar spectral sequence when
$D$ is just a convex index bounded domain.

###  4.3. Further remarks

Now we give two speculations about possible generalizations of the spectral
sequence: higher energy differentials and the homologically index bounded
case. We first remark that the technique of neck-stretching should be very
helpful to analyze the high energy differentials in the spectral sequence. The
following is based on a compactness theorem of stretching Floer trajectories,
not holomorphic curves, which we don’t know how to prove at present. A
possible way is to mimic what was done in [3].

$r$$s$$\gamma$$\gamma^{\prime}$ Figure 4. A possible degeneration of a Floer
trajectory after neck-stretching.

Let $D$ be a Liouville domain in $M$, and $\partial
D\times[1-\epsilon,1+\epsilon]$ be a cylindrical neighborhood of $\partial D$.
By our choice of $S$-shape Hamiltonian functions, all non-constant orbits are
in the region where $r<1+\frac{3\epsilon}{4}-\delta\epsilon$. Now we choose a
contact hypersurface $\partial D\times\\{r_{0}\\}$ to stretch, where
$1+\frac{3\epsilon}{4}-\delta\epsilon<r_{0}<1+\frac{3\epsilon}{4}$. Note that
the $S$-shape Hamiltonian functions have small derivatives when
$r>1+\frac{3\epsilon}{4}$. We expect that there is a compactness theorem for
Floer trajectories that intersect $\partial D\times\\{r_{0}\\}$, which ideally
tells us that when the stretching parameter goes to infinity, a sequence of
Floer trajectories with uniformly bounded energy will converge to a “broken
Floer building”. We say the top-level of this building is its part in the
region where $r>r_{0}$.

Since our auxiliary form $\tilde{\omega}$ is supported in the region where
$r>1+\frac{3\epsilon}{4}>r_{0}$, only the top-level of the broken Floer
building carries a non-trivial $\tilde{\omega}$-energy. It is interesting to
extract useful information from this top-level building to describe the
differentials in further pages of the spectral sequence.

A simple case is that there is no top-level building, which says that the
spectral sequence degenerates at the first page. For example, the outside
$M-D$ is symplectic aspherical. Or we know some bounds on the indices of Reeb
orbits in $\partial D$. Assuming transversality, the dimension of the moduli
space of the top-level curve is

$(n-3)(2-p)-\sum_{j=1}^{p}CZ(\gamma_{j})$

where $\gamma_{1},\cdots,\gamma_{p}$ are Reeb orbits at the negative
punctures, see Section 2.8 [7]. In many concrete examples this dimension is
negative, particularly when $n=3$.

Another interesting case is that $D$ is the complement of a divisor in $M$.
Then we may expect that the higher energy differentials in the spectral
sequence are related to some kinds of Gromov-Witten invariants, see [23, 11,
13].

Next we give our second speculation to describe a possible way to relax the
index bounded condition. Note that we use the auxiliary form $\tilde{\omega}$
to separate inside and outside contributions to the homology. More
algebraically, one can introduce an extra formal variable $q$ to record the
$\tilde{\omega}$-energy. Consider the Novikov ring with a polynomial ring as
the ground ring

$\Lambda_{0}(\mathbb{C}[q])=\\{\sum_{i=0}^{\infty}a_{i}T^{\lambda_{i}}\mid
a_{i}\in\mathbb{C}[q],\lambda_{i}\in\mathbb{R}_{\geq
0},\lambda_{i}<\lambda_{i+1},\lim_{i\rightarrow\infty}\lambda_{i}=+\infty\\}.$

Then in the definition of Floer differentials and continuation maps, see
(2.6), we also weight them by $q^{\tilde{\omega}(A)}$. By the assumption of
integrality of $\tilde{\omega}$ and choice of $S$-shape Hamiltonian functions,
the number $\tilde{\omega}(A)$ is a non-negative integer. Hence we get a
homology theory over $\Lambda_{0}(\mathbb{C}[q])$. However in the general case
there are Floer solutions that travels outside $D$ but not with a positive
$q$-weight. Those are coming from Morse flow lines involving upper constant
orbits. Suppose that under some condition we can ignore upper constant orbits.
Then we may form a complex with all generators in $D$. The differentials
without $q$-weight are totally in $D$, and those with $q$-weight intersect
$M-D$. Note that the $q$ variable also induces a filtration which is
compatible with differentials. Hence we can study the spectral sequence given
by the $q$ variable.

Our expectation is that in this case the first page (which has no
$q$-information) is $SH_{\hat{D}}(D)$. And the spectral sequence converges to
$SH_{M}(D;\Lambda_{0}(\mathbb{C}[q]))$. In order to recover some information
about the original relative symplectic cohomology $SH_{M}(D)$, we need to set
$q=1$. A finite-dimensional condition, which we call homologically index
bounded, on $SH_{\hat{D}}(D)$ may help us to do this specialization. For
example, when $D$ is a Weinstein neighborhood of a Lagrangian submanifold, by
using the Viterbo isomorphism (A.2) one may get a spectral sequence that
starts from the homology of “completed chains” in the free loop space and
converges to $SH_{M}(D;\Lambda_{0}(\mathbb{C}[q]))$.

##  5\. Examples and extensions

We now discuss some applications of our spectral sequence. First we state some
general (abstract) corollaries, then we move to concrete examples.

######  Definition 5.1.

Let $C=(\bigoplus_{k\in\mathbb{Z}}C^{k},d)$ be a $\mathbb{Z}$-graded complex
of free $\Lambda_{0}$-modules, and
$\tilde{C}=\bigoplus_{k\in\mathbb{Z}}\tilde{C}^{k}$ be a $\mathbb{Z}$-graded
$\Lambda_{0}$-module. We say $C$ supports $\tilde{C}$ if the homology of
$(C,d)$ is the same as $\tilde{C}$ after tensoring $\Lambda$. That is,

$H^{k}(C,d)\otimes_{\Lambda_{0}}\Lambda\cong\tilde{C}^{k}\otimes_{\Lambda_{0}}\Lambda.$

Now let $D$ be an index bounded Liouville domain in a symplectic Calabi-Yau
$M$. Then we can use the complex $\mathcal{CF}^{L}_{M}(D)$ to compute the
relative symplectic cohomology of $D$. Hence we have following two
corollaries.

######  Corollary 5.2.

By ignoring the upper orbits and using the Mayer-Vietoris property, we have
that

  1. (1)

if $\mathcal{CF}^{L}_{M}(D)$ doesn’t support the empty set, then $D$ is not
stably-displaceable;

  2. (2)

if $\mathcal{CF}^{L}_{M}(D)$ doesn’t support $H(M;\Lambda_{0})$, then the
complement $M-D$ is not stably-displaceable.

######  Corollary 5.3.

By using the first page of the spectral sequence and using the Mayer-Vietoris
property, we have that

  1. (1)

if $SH(D)$ doesn’t support the empty set, then $D$ is not stably-displaceable;

  2. (2)

if $SH(D)$ doesn’t support $H(M;\Lambda_{0})$, then $M-D$ is not stably-
displaceable.

Note that the above conclusions are also true for $\mathcal{N}$-index bounded
Liouville domains, since we can only focus on several fixed degrees.

######  Remark 5.4.

In Section 4 we established the convergence of the spectral sequence for the
truncated invariant. However, since the truncation bound $E$ can be
arbitrarily large, we can show the stable-displacement energy is arbitrarily
large under the assumption of above corollaries. Then the stably non-
displacement follows.

A first example is a symplectic embedding of a ball $B$ into a Calabi-Yau
manifold $M$. This recovers a result by Ishikawa [17], also see [26].

######  Example 5.5.

Since the standard contact structure on an odd-dimensional sphere is (Morse-
Bott) index bounded, there is a spectral sequence with vanishing first page
$SH(B;\Lambda)$ converging to $SH_{M}(B)\otimes_{\Lambda_{0}}\Lambda$. Hence a
ball inside a Calabi-Yau manifold always has vanishing relative symplectic
cohomology. However, it may have large volume such that it is not
displaceable.

This contrasts the case that a large ball in $\mathbb{C}P^{n}$ does have a
non-vanishing relative invariant. Moreover, the fact that a symplectic ball is
the complement of a divisor in $\mathbb{C}P^{n}$, or in many other Fano
manifolds, can not be true in the Calabi-Yau setting. Suppose that $M=B\cup K$
where $K$ is a divisor or more generally, a partly stratified symplectic
subset which is known to be stably-displaceable. Then both $B$ and $K$ has
vanishing relative invariant, which contradicts the Mayer-Vietoris exact
sequence and $SH_{M}(M)=QH(M)$.

###  5.1. Lagrangian spheres in Calabi-Yau manifolds

Let $(L,g)$ be a Riemannian manifold and $T^{*}L$ be its cotangent bundle with
the standard symplectic form. The unit disk bundle $D_{1}T^{*}L$, with respect
to the metric $g$, is a Liouville domain with the unit sphere bundle $ST^{*}L$
being its contact boundary. A geodesic $q(t)$ in $L$ lifts to a Reeb orbit
$\gamma(t)=(q(t),q^{\prime}(t))$ in $ST^{*}L$. Pick a trivialization $\Phi$ of
the contact distribution along $\gamma$, there is a Conley-Zehnder index
$CZ_{\Phi}(\gamma)$. On the other hand, the trivialization $\Phi$ also gives a
trivialization of the symplectic vector bundle $TT^{*}L$ along $q$. Hence
there is a Maslov index $\mu_{\Phi}(q)$ of $q$, viewed as a loop in $L$. The
relation between these two indices is the following lemma in [30].

######  Lemma 5.6.

In the above notations, we have that

$CZ_{\Phi}(\gamma)+\mu_{\Phi}(q)=ind(q)$

where $ind(q)$ is the Morse index of $q$ as a geodesic.

Now we consider a spin Lagrangian submanifold $L$ in a Calabi-Yau manifold
$M$. Let $D$ be a Weinstein neighborhood of $L$, which is isomorphic to
$D_{1}T^{*}L$ for some metric $g$ on $L$. Then $\partial D$ is a contact
hypersurface in $M$. For a loop $\gamma$ in $D$ that is contractible in $M$,
both the Conley-Zehnder index and the Maslov index are well-defined and
independent of the choice of a trivialization. In particular, when $L$ is
simply-connected, the Maslov index is always zero and we have that
$CZ_{\Phi}(\gamma)=ind(q)$.

Therefore if $L$ admits an index bounded Riemannian metric $g$ then any
Lagrangian embedding of $L$ into a Calabi-Yau manifold admits an index bounded
neighborhood. Here the index bounded condition for a metric $g$ is the same
for a contact form.

######  Definition 5.7.

A Riemannian metric $g$ is called index bounded if for every $m>0$, there
exists $\mu_{m}>0$ so that

$\\{length(q)\mid-m<ind(q)<m\\}\subset(0,\mu_{m})$

for all closed geodesics. And the metric is called (Morse-Bott) non-degenerate
if the length functional is (Morse-Bott) non-degenerate.

Now let $S^{n}$ be a sphere with dimension $n\geq 3$, and let $g_{R}$ be the
round metric on $S^{n}$. It is known that $g_{R}$ is a Morse-Bott non-
degenerate index bounded metric. Next let $g_{\epsilon}$ be a
$C^{\infty}$-small generic perturbation of $g_{R}$. By the Bonnet-Myers
theorem, there exists a constant $C>0$ such that any closed geodesic with
length $\lambda$ has index greater than $C\lambda$. This will be explained
later in a more general case. Hence $g_{\epsilon}$ is a non-degenerate index
bounded metric.

The above discussion tells us that for any Lagrangian sphere $S=S^{n}$ in a
Calabi-Yau manifold $M$, a Weinstein neighborhood $U$ of $S$ induce by the
metric $g_{\epsilon}$ has a non-degenerate index bounded contact boundary.
Hence our spectral sequence works for computing $SH_{M}(U)$.

######  Lemma 5.8.

For $(M,S,U)$ as above, we have that
$SH_{M}(U)\otimes_{\Lambda_{0}}\Lambda\neq 0$.

###### Proof.

For a give energy bound $E>0$, there is a convergent spectral sequence which
starts from the symplectic cohomology $SH(T^{*}S^{n};\Lambda_{E})$ and
converges to $SH_{M}(U;\Lambda_{E})$. In our degree notation (2.1), the usual
symplectic cohomology $SH(T^{*}S^{n};\Lambda_{E})$ is non-zero and one-
dimensional in degrees

$n,1,0,\cdots,i(1-n)+n,i(1-n)+n-1,\cdots,\quad i=1,2,\cdots.$

(Note that our Hamiltonian flow is the reverse of the Reeb flow in the
cylindrical region.) The non-zero element in degree-$n$ cannot be killed in
the spectral sequence when $n\geq 3$, since the differential only changes the
degree by one.

Hence for any energy bound $E>0$, the truncated invariant satisfies that

$SH_{M}^{n}(U;\Lambda_{E})\cong\Lambda_{E}.$

Then by taking the inverse limit over $E$, we have that
$SH_{M}(U)\otimes_{\Lambda_{0}}\Lambda\neq 0$. ∎

So any Lagrangian sphere with dimension $n\geq 3$ in a Calabi-Yau manifold is
stably non-displaceable. This fact is known by using the Lagrangian Floer
cohomology of $S$, see Theorem L [12]. But by using the Mayer-Vietoris
property, we can get more from the above lemma. Note that $SH_{M}(U)$ is zero
in degree two hence cannot be isomorphic to the quantum cohomology of $M$.
Therefore, the Mayer-Vietoris property shows that the complement of $U$ in $M$
is stably non-displaceable. This result is new and it contrasts to the case
where the ambient space is not Calabi-Yau: a Weinstein neighborhood $U$ of a
Lagrangian sphere can be compactified to be a quadric hypersurface
$Q^{n}\subset\mathbb{C}P^{n+1}$. Note that $Q^{n}$ is a monotone symplectic
manifold, and the complement of $U$ in $Q^{n}$ is a small neighborhood of a
divisor $Q^{n-1}$, which is stably displaceable.

The case of a sphere can be generalized to Riemannian manifolds with a
positive Ricci curvature.

######  Lemma 5.9.

(see Theorem 19.4 [21]) Let $(L,g)$ be a closed Riemannian manifold of
dimension $n$ whose Ricci curvature satisfies $Ric_{g}\geq(n-1)C$ for some
positive real number $C$. Then any closed geodesic on $L$ with length
$\lambda$ has Morse index greater than $\lambda\sqrt{C}/\pi-1$. In particular,
$g$ is index bounded.

###### Proof.

Let $\gamma$ be a closed geodesic on $L$ with length $\lambda$, then there
exists an integer $l$ such that

$l\pi/\sqrt{C}<\lambda\leq(l+1)\pi/\sqrt{C}.$

We cut $\gamma$ into $l+1$ segments such that each of the first $l$ segments
has length slightly greater than $\pi/\sqrt{C}$. Then by the Bonnet-Myers
theorem, any geodesic segment with length greater than $\pi/\sqrt{C}$ is
unstable. Hence each of these $l$ segments has index at least one, the index
of $\gamma$ is at least $l$.

Now let $\gamma$ be a closed geodesic with index $k$ and length $\lambda$, we
know that $k>\lambda\sqrt{C}/\pi-1$ which shows that
$\lambda<(k+1)\pi/\sqrt{C}$. Hence $g$ is index bounded. ∎

######  Remark 5.10.

The positivity of the Ricci curvature of a metric $g$ is preserved under
$C^{\infty}$-small perturbations. Hence we obtain a non-degenerate index
bounded metric $g_{\epsilon}$ after perturbation.

Let $(L,g_{\epsilon})$ be a closed Riemannian manifold with a positive Ricci
curvature. Then $g_{\epsilon}$ is a non-degenerate index bounded metric.
Suppose that there is a Lagrangian embedding $L\rightarrow M$ into a Calabi-
Yau manifold $M$. Another consequence of the Bonnet-Myers theorem is that $L$
has a finite fundamental group. Hence $L$ is a Lagrangian submanifold of $M$
with a vanishing Maslov class. Now let $U$ be a Weinstein neighborhood of $L$
induced by the metric $g_{\epsilon}$, the above lemma tells us that $U$ is a
Liouville domain with a non-degenerate index bounded boundary in $M$. In
conclusion, our spectral sequence works for computing $SH_{M}(U)$.

If we have two Riemannian manifolds both with positive Ricci curvature, then
the product manifold also admits a metric with positive Ricci curvature. Hence
the above argument can be applied to study Lagrangian submanifolds of product
type.

######  Example 5.11.

Let $L=S^{2}\times S^{3}$ be a Lagrangian submanifold in a Calabi-Yau manifold
$M$. Then there exists a Weinstein neighborhood of $L$ which is a Liouville
domain with a non-degenerate index bounded boundary. So we can use our
spectral sequence to compute $SH_{M}(L)$, which possibly determines the
displaceability of $L$. Note on the other hand, the Lagrangian Floer
cohomology of $L$ may have obstructions to be defined. The obstruction lies in
$H^{2}(L;\mathbb{Q})$, see Theorem L [12].

By using the geodesic-Reeb orbit correspondence, we can also study Lagrangian
submanifolds that carry a metric with negative sectional curvature. Note that
closed geodesics for such a metric are always non-degenerate, see Lemma 2.2
[7].

######  Example 5.12.

Let $L$ be a Maslov zero Lagrangian submanifold in a Calabi-Yau manifold $M$,
such that $L$ carries a metric with negative sectional curvature. Then this
metric induces a Weinstein neighborhood $U$ of $L$ such that all Reeb orbits
on $\partial U$ has Conley-Zehnder index zero. Therefore the non-constant
orbits of an $S$-shape Hamiltonian function have degree $n$. After the time-
dependent perturbation, the degree-changes are at most one. In particular when
$n\geq 3$, we focus on degrees in $\mathcal{N}=\\{2n-1,2n,2n+1\\}$ and apply
our spectral sequence to compute $SH^{2n}_{M}(U)$. The first page of the
spectral sequence is isomorphic to the usual symplectic cohomology of
$T^{*}L$, which is zero at degree $2n$. Hence $SH^{2n}_{M}(U)=0$ and it is not
isomorphic to the quantum cohomology of $M$, which shows that $M-U$ is not
stably-displaceable. In particular, the disk cotangent bundle of $L$ cannot be
compactified to be a Calabi-Yau manifold, by adding divisors at infinity while
keeping $L$ with a zero Maslov class. A similar conclusion without the Calabi-
Yau condition is proved by Côté, see Theorem 1.10 and Lemma 7.1 in [9].

The disk cotangent bundle can be regarded as a Liouville domain with a smooth
Lagrangian skeleton. On the other hand, Brieskorn manifolds are families of
Liouville domains of which the Lagrangian skeletons are chains of spheres
modeled by trees. In [18], the Reeb orbits of many Brieskorn manifolds have
been studied explicitly. Hence one can use the calculation of indices therein
to get more applications. Like the rigidity of symplectic embeddings of
Brieskorn manifolds into Calabi-Yau manifolds.

###  5.2. Perturbation of Morse-Bott orbits

Let $D$ be a Liouville domain with a contact boundary $(C,\alpha)$ in $M$ such
that $\alpha$ is Morse-Bott index bounded. Then a time-independent $S$-shape
Hamiltonian function $H$ has degenerate one-periodic orbits, given by the Reeb
orbits of $\alpha$. Now we will perturb $H$ to get a non-degenerate $S$-shape
Hamiltonian function $H_{t}$, such that it is index bounded. We remark that we
are perturbing the Hamiltonian function instead of perturbing the contact
form, since the index bounded condition may be destroyed by the later
perturbation.

Let $Y$ be the set of $l$-periodic Reeb orbits of $\alpha$. By the Morse-Bott
condition $Y$ is a closed smooth submanifold of $C$. It may have several
connected components, and we will construct our perturbation component-wisely.
For simplicity, we assume that $l=1$. The general case is similar. Our
perturbation is a modification of the case of a time-independent Hamiltonian
function with transversally non-degenerate orbits, where $Y=S^{1}$.

There is an $S^{1}$-action on $Y$ induced by the Reeb flow $\phi^{t}$. For a
Morse function $g:Y\rightarrow\mathbb{R}$, we twist it by the $S^{1}$-action
to get a time-dependent function on $Y$:

$g_{t}(y):=g(\phi^{1-t}(y)),\quad t\in[0,1],\quad y\in Y.$

Next let $N$ be the normal bundle of $Y$ in $M$. We extend $g_{t}$ to be a
function $\tilde{g}_{t}$ on $N$ which is supported near the zero section. We
also require that $\tilde{g}_{t}$ does not depend on the fiber direction in a
small neighborhood of the zero section.

Now for a time-independent $S$-shape Hamiltonian function
$H:M\rightarrow\mathbb{R}$, it has degenerate one-periodic orbits in the
cylindrical region, which forms the submanifold $Y$. Define
$G^{\epsilon}_{t}:S^{1}\times M\rightarrow\mathbb{R}$ as

$G^{\epsilon}_{t}(m):H(m)+\epsilon\tilde{g}_{t}(m),\quad m\in M.$

Our main result of this subsection is

######  Proposition 5.13.

For small $\epsilon>0$, the one-periodic orbits of $G^{\epsilon}_{t}$ in a
small neighborhood of $Y$ are in one-to-one correspondence with critical
points of $g$. Let $\gamma$ be a one-periodic orbit of $H$ on $Y$ and
$\gamma^{\epsilon}$ be a one-periodic orbits of $G^{\epsilon}_{t}$ near $Y$.
We have that

  1. (1)

$\int_{\gamma}\theta=\int_{\gamma^{\epsilon}}\theta$ where $\theta$ is the
Liouville one-form on $D$;

  2. (2)

$\lvert CZ(\gamma)-CZ(\gamma^{\epsilon})\rvert\leq\dim_{\mathbb{R}}Y$.

###### Proof.

This proposition, which is known to experts, is a modification of Lemma 2.1
and Proposition 2.2 in [6].

Let $J$ be a compatible almost complex structure on $M$ which is cylindrical
near $C$. By the construction, the Hamiltonian vector field of
$G^{\epsilon}_{t}$ is

$X_{G^{\epsilon}_{t}}(m)=X_{H}(m)+\epsilon J\nabla\tilde{g}_{t}(m).$

Here the gradient is computed with respect to the metric
$\omega(\cdot,J\cdot)$. Let $p\in Y$ be a critical point of $g$. Then
$\phi^{t}(p)$ is a one-periodic orbit of $H$ on $Y$. We can also check that it
is also a one-periodic orbit of $G^{\epsilon}_{t}$. Hence each critical point
of $g$ gives a one-periodic orbit of $G^{\epsilon}_{t}$. Next we will show
there is no other one-periodic orbit.

Let $U$ be a neighborhood of $Y$ in $M$, which does not contain other one-
periodic orbits of $H$ disjoint from $Y$. Then for any open set $V\subset U$
with $Y\subset V$, there exists $\epsilon_{0}>0$ such that for any
$0<\epsilon<\epsilon_{0}$ the one-periodic orbits of $G^{\epsilon}_{t}$ in $U$
are also in $V$. This is due to the compactness result in Lemma 2.2 [6]. Hence
when $\epsilon$ is small, any one-periodic orbit of $G^{\epsilon}_{t}$ is
close to an one-periodic orbit of $H$ on $Y$, particularly in the
$W^{1,2}$-topology.

Next consider a nonlinear operator

$A:W^{1,2}(S^{1},N)\rightarrow L^{2}(S^{1},TN)$

given by

$A(x(t)):=-J(x^{\prime}(t)-X_{H}(x(t))).$

By the Morse-Bott non-degeneracy, the linearization of $A$ is non-degenerate
in the normal direction of $Y$. More precisely, there exists a constant $c>0$
such that for any one-periodic orbit $x_{0}(t)$ of $H$ on $Y$ and a vector
field $y(t)$ along $x_{0}(t)$ with $y(t)\notin TY$ for some $t$, we have that

$\lvert\lvert DA(x_{0})\cdot y(t)\rvert\rvert\geq c\lvert\lvert
y(t)\rvert\rvert.$

Now define another operator

$f:W^{1,2}(S^{1},N)\rightarrow L^{2}(S^{1},TN)$

given by

$f(x(t)):=\tilde{g}_{t}^{\prime}(x(t)).$

Note that the kernel of the operator $A+\epsilon f$ is the set of all one-
periodic orbits of $G^{\epsilon}_{t}$ in $N$. Since any one-periodic orbit of
$G^{\epsilon}_{t}$ is close to an one-periodic orbit $x_{0}(t)$ of $H$ on $Y$,
we can write it as $x_{0}+y(t)$ with a vector field $y(t)$ along $x_{0}(t)$.

Then we use the Taylor expansion to calculate that

$(A+\epsilon f)(x_{0}+y)=A(x_{0})+DA(x_{0})\cdot y+\epsilon f(x_{0})+\epsilon
Df(x_{0})\cdot y+O(\lvert\lvert y\rvert\rvert^{2}).$

Note that $A(x_{0})=0$ and $\lvert\lvert DA(x_{0})\cdot y\rvert\rvert\geq
c\lvert\lvert y\rvert\rvert$. So we have that

$\displaystyle\lvert\lvert(A+\epsilon f)(x_{0}+y)\rvert\rvert$
$\displaystyle\geq c\lvert\lvert y\rvert\rvert+\epsilon\lvert\lvert
f(x_{0})\rvert\rvert-\epsilon c^{\prime}\lvert\lvert
y\rvert\rvert+O(\lvert\lvert y\rvert\rvert^{2})$ $\displaystyle\geq
c^{\prime\prime}\lvert\lvert y\rvert\rvert+\epsilon\lvert\lvert
f(x_{0})\rvert\rvert$

when $\epsilon$ and $\lvert\lvert y\rvert\rvert$ are sufficiently small. Hence
$(A+\epsilon f)(x_{0}+y)=0$ if and only if $y=0$ and $f(x_{0})=0$, which are
the orbits given by critical points of $g$. Geometrically these perturbed
orbits are the same orbits which start at critical points of $g$. So the
integrations of the Liouville one-form do not change.

The proof of $(2)$ will be a direct computation to relate the Conley-Zehnder
index with the Morse index, similar to that in [6]. ∎

Therefore, given a Liouville domain $D$ with a contact boundary $(C,\alpha)$
in $M$ such that $\alpha$ is Morse-Bott index bounded, we can create non-
degenerate index-bounded $S$-shape Hamiltonian functions associated with $D$.
Then we can use them to construct the spectral sequence as we did in the Morse
index bounded case.

###  5.3. Filled Liouville cobordisms

To compute the relative symplectic cohomology of a Liouville domain, we use
$S$-shape Hamiltonian functions. Note that in general, the relative symplectic
cohomology can be defined for any compact subset. Now we discuss the case
where the compact subset is a filled Liouville cobordism.

The symplectic (co)homology for filled Liouville cobordisms was studied in
[8]. We recall some definitions.

######  Definition 5.14.

A Liouville cobordism $(W^{2n},\omega=d\theta)$ is a compact exact symplectic
manifold with boundary, such that $\theta$ restricts to contact forms
$\alpha:=\theta\mid_{\partial W}$ on different components of $\partial W$.

We write $\partial_{+}W$ as the components of $\partial W$ where the
orientation induced by $\alpha\wedge(d\alpha)^{n-1}$ coincides with the
orientation of $\partial W$ as a boundary of $W$. We call $\partial_{+}W$ as
the convex boundary of $W$. Other components $\partial_{-}W:=\partial
W-\partial_{+}W$ will be called the concave boundary of $W$.

A Liouville cobordism $W$ is called index bounded if all its boundaries are
index bounded.

Note that a Liouville domain is a Liouville cobordism without concave
boundary.

######  Definition 5.15.

A filled Liouville cobordism is a Liouville cobordism $W$ together with a
Liouville domain $D$, such that

$(\partial_{-}W,\theta_{W})=(\partial D,\theta_{D}).$

Therefore we can glue them along the common boundary to get a new Liouville
domain $D\\#W$.

Given a symplectic manifold $(M,\omega)$, by a (filled) Liouville cobordism
$W$ in $M$ we mean a codimension zero symplectic embedding

$(W,\omega_{W}=d\theta)\rightarrow(M,\omega).$

We usually do not distinguish $W$ and its image under the embedding.

######  Example 5.16.

A cylindrical neighborhood of a contact hypersurface in $M$ is a Liouville
cobordism in $M$ with one convex boundary and a concave boundary, which are
diffeomorphic to each other.

Let $D$ be a Liouville domain in $M$, then a cylindrical neighborhood of
$\partial D$ is a filled Liouville cobordism in $M$ with a filling from $D$.

Now let $W$ be a Liouville cobordism in $M$. To compute the relative
symplectic cohomology $SH_{M}(W)$ one can use $U$-shape Hamiltonian functions.
That is, a Hamiltonian function which is a negative $C^{2}$-small Morse
function in $W$ and cylindrical near each components of $\partial W$, see
Figure 6.

$W$$\partial_{-}W$$\partial_{+}W$ Figure 5. $U$-shape Hamiltonian functions.

Then the non-constant orbits of a $U$-shape Hamiltonian function are near the
boundary of $W$. They also form two groups; upper orbits and lower orbits. Now
assume that $M$ is Calabi-Yau and $W$ is an index bounded filled Liouville
cobordism with a filling $D$. Then we have a spectral sequence which starts
from the symplectic cohomology for a filled Liouville cobordism, constructed
in [8], to the relative symplectic cohomology $SH_{M}(W)$.

The construction of this spectral sequence is the same to the case of an index
bounded Liouville domain. By using the index bounded condition we ignore upper
orbits and show the convergence of the spectral sequence. All Floer
trajectories that are not contain in $D\\#W$ change the
$\tilde{\omega}$-energy positively. Hence to compute the first page of the
spectral sequence, we only need contributions of Floer trajectories inside
$D\\#W$, which give the isomorphism to the local invariant constructed in [8].

With the argument in previous subsection, same conclusion holds when $W$ only
has a symplectic aspherical filling and $\partial_{-}W$ is simply-connected.

######  Remark 5.17.

Although the above spectral sequence describes some local-to-global
phenomenon, its first page still contains information which are not in $W$:
Floer solutions that intersect $D$. It would be interesting to know whether
there is a filtration such that we can view $SH_{M}(W)$ as a two-side
deformation: one from $D$ the other from $M-(D\\#W)$.

##  Appendix A Local computations

In this appendix we define the relative symplectic cohomology of a Liouville
domain inside its completion, namely $SH_{\hat{D}}(D)$. The definition is
almost the same as the case when the ambient space is a closed symplectic
manifold. However, to define cofinal families of $S$-shape Hamiltonian
functions we need to prescribe their slopes at infinity since Lemma 2.3 is not
true in the open case. The main result here is that for different choices of
slopes at infinity, we actually get the same invariant. And the key point in
the proof is to use the infinite volume of the cylindrical region to ignore
upper orbits. This method works for any Liouville domain with a non-degenerate
contact form on its boundary, without using the index bounded condition.

A direct corollary of the process of ignoring upper orbits is a relation
between $SH_{\hat{D}}(D)$ and $SH(D)$. Moreover, combined with Abbondandolo
and Schwarz’s proof [1] of the Viterbo isomorphism, we get a relation between
$SH_{T^{*}L}(L)$ and a completed version of the homology of loop space of a
smooth manifold $L$.

Now let $(D,\theta)$ be a Liouville domain. Then $\alpha:=\theta\mid_{\partial
D}$ is a contact form. Let $Spec(\alpha)$ be the set of periods of Reeb orbits
of $\alpha$. If $\alpha$ is non-degenerate then $Spec(\alpha)$, the so-called
action spectrum of $\alpha$ is a discrete subset of $\mathbb{R}_{+}$.

Recall that we can complete $(D,\theta)$ to be an open symplectic manifold
with infinite volume. That is,

$(\hat{D},\hat{\omega}):=(D\cup([1,+\infty)\times\partial D),\hat{\omega})$

where $\hat{\omega}=d\theta$ on $D$ and $\hat{\omega}=d(r\alpha)$ on the
cylindrical region. Here $r$ is the coordinate on $[1,+\infty)$. Note that
$\theta$ also extends to a smooth one-form $\hat{\theta}$ on $\hat{D}$ such
that $d\hat{\theta}=\hat{\omega}$.

######  Definition A.1.

For a positive number $\epsilon$ that is not in the period spectrum of
$\alpha$, a smooth function $H:\hat{D}\rightarrow\mathbb{R}$ is called
$\epsilon$-admissible if

  1. (1)

$H<0$ on $D$;

  2. (2)

there exists a positive number $R$ such that when $r>R$, $H$ is a linear
function in $r$ with slope $\epsilon$.

The set of $\epsilon$-admissible functions is denoted as
$\mathcal{H}(\epsilon)$.

For two $\epsilon$-admissible functions $H,H^{\prime}$, define $H>H^{\prime}$
if $H(x)>H^{\prime}(x)$ for any $x\in\hat{D}$. Then $\mathcal{H}(\epsilon)$
becomes a directed set.

######  Lemma A.2.

A sequence of $\epsilon$-admissible functions
$\\{H_{n}\\}_{n\in\mathbb{Z}_{+}}$ is a cofinal family in
$\mathcal{H}(\epsilon)$ if the following holds.

  1. (1)

$H_{n}(x)\leq H_{n+1}(x)$ for any $x\in\hat{D}$ and $n\in\mathbb{Z}_{+}$;

  2. (2)

$H_{n}(x)\rightarrow 0$ for $x\in D$ and $H_{n}(x)\rightarrow+\infty$ for
$x\in\hat{D}-D$;

  3. (3)

$H_{n}(x)=h_{n}(r)$ only depends on the cylindrical coordinate $r$ when
$x\in\hat{D}-D$ and $h^{\prime\prime}(r)\leq 0$ for all $n,r$.

###### Proof.

Suppose that $\\{H_{n}\\}_{n\in\mathbb{Z}_{+}}$ satisfies the above three
conditions. Given $G\in\mathcal{H}(\epsilon)$, we will find some
$n\in\mathbb{Z}_{+}$ such that $G<H_{n}$.

Since $D$ is compact, we have that $G<c<0$ on $D$ for some number $c$.
Moreover, $H_{n}$ converges to zero uniformly on $D$. Hence there is some
$i\in\mathbb{Z}_{+}$ such that $G<c\leq H_{i}$ on $D$. By the smoothness of
$G$ and $H_{i}$, there is also an open neighborhood $U$ of $D$ such that
$G<H_{i}$ on $U$.

Note that $G\in\mathcal{H}(\epsilon)$ there is $R>0$ such that $G$ is linear
with slope $\epsilon$ when $r>R$. On the other hand,
$H_{n}(R)\rightarrow+\infty$. So there is $j\in\mathbb{Z}_{+}$ such that