The inequality

$2{\mathrm{wind}}_{\pi}({\widetilde{u}})\leq{\mathrm{ind}}_{\delta}({\widetilde{u}})-2+2g+\\#\Gamma-\\#\Gamma^{\rm
odd}$

holds. In particular, if $g=0$, ${\mathrm{ind}}_{\delta}({\widetilde{u}})=2$
and $\\#\Gamma\leq\\#\Gamma^{\rm odd}+1$ then
${\mathrm{wind}}_{\pi}({\widetilde{u}})=0$ and $u$ is an immersion transverse
to the Reeb vector field. Moreover, if $g=0$,
${\mathrm{ind}}_{\delta}({\widetilde{u}})=2$ and $\\#\Gamma=\\#\Gamma^{\rm
odd}$ then

$\displaystyle\sum_{i=1}^{m_{+}}{\mathrm{wind}}_{\tau_{i}^{+}}(\mu_{i}^{+})=\sum_{i=1}^{m_{+}}{\mathrm{wind}}_{\tau_{i}^{+}}(\nu^{<-\delta}(\gamma^{+}_{i}))$
$\displaystyle\sum_{i=1}^{m_{-}}{\mathrm{wind}}_{\tau_{i}^{-}}(\mu_{i}^{-})=\sum_{i=1}^{m_{-}}{\mathrm{wind}}_{\tau_{i}^{-}}(\nu^{\geq\delta}(\gamma^{-}_{i}))$

###### Proof.

Note that the identities

$\displaystyle 2{\mathrm{wind}}_{\tau_{i}^{-}}(\mu^{-}_{i})\geq
2{\mathrm{wind}}_{\tau_{i}^{-}}(\nu^{\geq\delta}(\gamma_{i}^{-}))={\rm
CZ}^{\delta}_{\tau_{i}^{-}}(\gamma_{i}^{-})+p^{\delta}(\gamma_{i}^{-})$
$\displaystyle 2{\mathrm{wind}}_{\tau_{i}^{+}}(\mu^{+}_{i})\leq
2{\mathrm{wind}}_{\tau_{i}^{-}}(\nu^{<-\delta}(\gamma_{i}^{+}))={\rm
CZ}^{-\delta}_{\tau_{i}^{+}}(\gamma_{i}^{+})-p^{-\delta}(\gamma_{i}^{+})$

follow from the definitions of the quantities involved, and from the basic
properties of spectra of asymptotic operators described before. It follows
that

$\displaystyle\sum_{i=1}^{m_{+}}2{\mathrm{wind}}_{\tau_{i}^{+}}(\mu^{+}_{i})$
$\displaystyle\leq-\\#\Gamma^{\rm odd}_{+}+\sum_{i=1}^{m_{+}}{\rm
CZ}^{-\delta}_{\tau_{i}^{+}}(\gamma_{i}^{+})\,,$
$\displaystyle-\sum_{i=1}^{m_{-}}2{\mathrm{wind}}_{\tau_{i}^{-}}(\mu_{i}^{-})$
$\displaystyle\leq-\\#\Gamma^{\rm odd}_{-}-\sum_{i=1}^{m_{-}}{\rm
CZ}^{\delta}_{\tau_{i}^{-}}(\gamma_{i}^{-})\,.$

Using these identities we estimate:

(4.9) $\displaystyle
2{\mathrm{wind}}_{\pi}({\widetilde{u}})=2{\mathrm{wind}}_{\infty}({\widetilde{u}})-4+4g+2\\#\Gamma$
$\displaystyle=\left[\sum_{i=1}^{m_{+}}2{\mathrm{wind}}_{\tau_{i}^{+}}(\mu^{+}_{i})-\sum_{i=1}^{m_{-}}2{\mathrm{wind}}_{\tau_{i}^{-}}(\mu^{-}_{i})\right]-4-4g+2\\#\Gamma$
$\displaystyle\leq\left[\sum_{i=1}^{m_{+}}{\rm
CZ}^{-\delta}_{\tau_{i}^{+}}(\gamma_{i}^{+})-\sum_{i=1}^{m_{-}}{\rm
CZ}^{\delta}_{\tau_{i}^{-}}(\gamma_{i}^{-})\right]-\\#\Gamma^{\rm
odd}-4+4g+2\\#\Gamma$
$\displaystyle={\mathrm{ind}}_{\delta}({\widetilde{u}})+2-2g-\\#\Gamma-\\#\Gamma^{\rm
odd}-4+4g+2\\#\Gamma$
$\displaystyle={\mathrm{ind}}_{\delta}({\widetilde{u}})-2+2g+\\#\Gamma-\\#\Gamma^{\rm
odd}$

as desired. If conditions $g=0$, ${\mathrm{ind}}_{\delta}({\widetilde{u}})=2$
and $\\#\Gamma\leq\\#\Gamma^{\rm odd}+1$ hold then
${\mathrm{ind}}_{\delta}({\widetilde{u}})-2+2g+\\#\Gamma-\\#\Gamma^{\rm
odd}\leq 1$, hence ${\mathrm{wind}}_{\pi}({\widetilde{u}})=0$. The fact that
$u$ is an immersion transverse to the Reeb vector field follows from (4.4).
Finally, assume that $g=0$, ${\mathrm{ind}}_{\delta}({\widetilde{u}})=2$ and
$\\#\Gamma=\\#\Gamma^{\rm odd}$. The right-hand side in (4.9) vanishes and we
get

$\displaystyle\sum_{i=1}^{m_{+}}2{\mathrm{wind}}_{\tau_{i}^{+}}(\mu^{+}_{i})-\sum_{i=1}^{m_{-}}2{\mathrm{wind}}_{\tau_{i}^{-}}(\mu^{-}_{i})$
$\displaystyle\qquad=\sum_{i=1}^{m_{+}}2{\mathrm{wind}}_{\tau_{i}^{+}}(\nu^{<-\delta}(\gamma^{+}_{i}))-\sum_{i=1}^{m_{-}}2{\mathrm{wind}}_{\tau_{i}^{-}}(\nu^{\geq\delta}(\gamma^{-}_{i}))$

Dividing by two we arrive at

$\displaystyle\sum_{i=1}^{m_{+}}{\mathrm{wind}}_{\tau_{i}^{+}}(\mu^{+}_{i})-{\mathrm{wind}}_{\tau_{i}^{+}}(\nu^{<-\delta}(\gamma^{+}_{i}))$
$\displaystyle\qquad=\sum_{i=1}^{m_{-}}{\mathrm{wind}}_{\tau_{i}^{-}}(\mu^{-}_{i})-{\mathrm{wind}}_{\tau_{i}^{-}}(\nu^{\geq\delta}(\gamma^{-}_{i}))$

The left-hand side is nonpositive and the right-hand side is nonnegative.
Hence both sides are equal to zero. ∎

Next we discuss Fredholm theory. Let $g$ be the genus of $S$. Consider the
moduli space

(4.10)
$\mathcal{M}_{J,g}(\gamma_{1}^{+},\dots,\gamma_{m_{+}}^{+};\gamma_{1}^{-},\dots,\gamma_{m_{-}}^{-})$

consisting of equivalence classes of pairs $({\widetilde{u}},j)$, where $j$ is
a conformal structure on $S$, and
${\widetilde{u}}:(\dot{S},j)\to(\mathbb{R}\times Y,{\widetilde{J}})$ is
nonconstant pseudo-holomorphic map with finite Hofer energy, as in (4.2), and
such that:

  * •

For every $i\in\\{1,\dots,m_{+}\\}$, $z_{i}^{+}$ is a positive nondegenerate
puncture where ${\widetilde{u}}$ is asymptotic to $\gamma^{+}_{i}$.

  * •

For every $i\in\\{1,\dots,m_{-}\\}$, $z_{i}^{-}$ is a negative nondegenerate
puncture where ${\widetilde{u}}$ is asymptotic to $\gamma^{-}_{i}$.

Two such pairs $({\widetilde{u}}=(a,u),j)$ and
$({\widetilde{u}}^{\prime}=(a^{\prime},u^{\prime}),j^{\prime})$ are declared
equivalent if there exists a biholomorphism $\phi:(S,j)\to(S,j^{\prime})$ that
fixes each point of $\Gamma$, and a constant $c\in\mathbb{R}$ such that
$(c+a^{\prime}\circ\phi,u^{\prime}\circ\phi)=(a,u)$. Once $g$, $\delta$,
$\Gamma_{\pm}$ and $\gamma_{i}^{\pm}$ are all fixed, we write
$\mathcal{M}_{J}$ to denote the above moduli space, for simplicity. A curve
$C\in\mathcal{M}_{J}$ is said to be embedded, immersed, or somewhere injective
if it can be represented by a pair $({\widetilde{u}},j)$ where
${\widetilde{u}}$ is an embedding, an immersion, or a somewhere injective map,
respectively. The equivalence class of a pair $({\widetilde{u}},j)$ will be
denoted by $[{\widetilde{u}},j]$.

######  Remark 4.10.

If $g=0$ and $m_{+}+m_{-}\leq 3$ then we can disregard the conformal structure
$j$ since in this case, up to a diffeomorphism, there is only one conformal
structure. This becomes relevant in later sections where we will be dealing
with cylinders.

If $C=[{\widetilde{u}},j]\in\mathcal{M}_{J}$ is immersed then one can follow
[22] and build the functional analytic set-up for a Fredholm theory based on a
certain space of sections of the normal bundle $N_{\widetilde{u}}$ of
${\widetilde{u}}$. This Fredholm theory can then be used to model a
neighborhood of $C$ in $\mathcal{M}_{J}$. Let us describe the details we need.

Fix choices of positive/negative holomorphic polar coordinates $(s,t)$
centered at the positive/negative punctures. Up to homotopy, we can assume
that $N_{\widetilde{u}}$ is a ${\widetilde{J}}$-invariant bundle, and that

(4.11) $N_{\widetilde{u}}$ agrees with $u^{*}\xi$ near the punctures.

Let $\tau_{\xi}$ be a global symplectic trivialization of $u^{*}\xi$, and
$\tau_{N}$ be a global complex trivialization of $N_{\widetilde{u}}$. Up to
homotopy we can assume that at a given puncture $z_{*}$, of sign $\epsilon$,
the trivializations $\tau_{\xi}$ and $\tau_{N}$ converge as $\epsilon
s\to\infty$ to trivializations $\tau_{\xi}(\gamma^{\pm}_{i})$ and
$\tau_{N}(\gamma^{\pm}_{i})$ of $\xi_{\gamma^{\pm}_{i}}$, respectively. We
write

${\mathrm{wind}}(\tau_{\xi}(\gamma^{\pm}_{i}),\tau_{N}(\gamma^{\pm}_{i}))\in\mathbb{Z}$

to denote the winding number of a constant vector in
$\tau_{\xi}(\gamma^{\pm}_{i})$ represented in $\tau_{N}(\gamma^{\pm}_{i})$. To
compute this winding number the periodic orbit is traversed along the Reeb
flow. In other words, this is the winding number of
$\tau_{\xi}(\gamma^{\pm}_{i})$ relative to $\tau_{N}(\gamma^{\pm}_{i})$. The
next lemma is proved in [22].

######  Lemma 4.11.

Under the above assumptions, the identity

$\sum_{i=1}^{m_{+}}{\mathrm{wind}}(\tau_{\xi}(\gamma^{+}_{i}),\tau_{N}(\gamma^{+}_{i}))-\sum_{i=1}^{m_{-}}{\mathrm{wind}}(\tau_{\xi}(\gamma^{-}_{i}),\tau_{N}(\gamma^{-}_{i}))=-2+2g+\\#\Gamma$

holds.

###### Proof.

Near $z_{i}^{+}$ we consider positive holomorphic polar coordinates, and near
$z_{i}^{-}$ we consider negative holomorphic polar coordinates. As usual,
there is a sign $\epsilon=\pm 1$ associated to each puncture $z^{\pm}_{i}$.
All these sets of coordinates are denoted by $(s,t)$ with no fear of
ambiguity. For $r\gg 1$ large enough we denote by $K_{r}\subset\dot{S}$ the
complement of the union of the sets described by $\\{(s,t)\mid\epsilon s>r\\}$
near each puncture. Then $K_{r}$ is compact, and its boundary consists of
circles. The component $\partial_{i}^{\pm}K_{r}$ of $\partial K_{r}$ near
$z_{i}^{\pm}$ is oriented by the parametrization $t\mapsto(\epsilon r,t)$.
Hence, if $\partial K_{r}$ is oriented as the boundary of $K_{r}$ we have
$\partial
K_{r}=\sum_{i=1}^{m_{+}}\partial_{i}^{+}K_{r}-\sum_{i=1}^{m_{-}}\partial_{i}^{-}K_{r}$.

We see $\xi$ and $R$ as $\mathbb{R}$-invariant objects in $\mathbb{R}\times
Y$. By Definition 4.5, near each puncture the vector
$\partial_{s}{\widetilde{u}}$ becomes arbitrarily close to
$T\partial_{a}|_{\widetilde{u}}$, and the vector $\partial_{t}{\widetilde{u}}$
becomes arbitrarily close to $TR|_{\widetilde{u}}$, respectively, as $\epsilon
s\to+\infty$. Hence, if $r\gg 1$ is large enough, there is a vector subbundle
$E_{\widetilde{u}}\subset{\widetilde{u}}^{*}T(\mathbb{R}\times Y)$ that
complements $u^{*}\xi$, agrees with ${\rm
span}\\{{\partial_{a}}|_{\widetilde{u}},R|_{\widetilde{u}}\\}$ on $K_{r-1}$,
and agrees with $T{\widetilde{u}}$ on $\dot{S}\setminus K_{r}$. Moreover,
$E_{\widetilde{u}}$ has a global frame $\tau_{E}$ that agrees with
$\\{{\partial_{a}}|_{\widetilde{u}},R|_{\widetilde{u}}\\}$ on $K_{r-1}$, and
with $\\{\partial_{s}{\widetilde{u}},\partial_{t}{\widetilde{u}}\\}$ on
$\dot{S}\setminus K_{r}$. By (4.11), there is no loss of generality to assume
that $N_{\widetilde{u}}$ agrees with $u^{*}\xi$ on $\dot{S}\setminus K_{r}$.
Hence, we have splittings

${\widetilde{u}}^{*}T(\mathbb{R}\times Y)=E_{\widetilde{u}}\oplus
u^{*}\xi=T{\widetilde{u}}\oplus N_{\widetilde{u}}$

whose corresponding summands coincide near the punctures. Consider a global
trivialization $\tau_{T}$ of $T{\widetilde{u}}$. Note that

$\sum_{i=1}^{m_{+}}{\mathrm{wind}}_{\partial^{+}_{i}K_{r}}(\tau_{E},\tau_{T})-\sum_{i=1}^{m_{-}}{\mathrm{wind}}_{\partial^{-}_{i}K_{r}}(\tau_{E},\tau_{T})=2-2g-\\#\Gamma\,,$

where ${\mathrm{wind}}_{\partial^{\pm}_{i}K_{r}}$ denotes the winding number
along the curve $\partial^{\pm}_{i}K_{r}$ as oriented by $t$. This is true
because near each puncture the vector $\epsilon\partial_{s}{\widetilde{u}}$
points radially towards the puncture.

Recall the global trivializations $\tau_{\xi}$ and $\tau_{N}$ of $u^{*}\xi$
and of $N_{\widetilde{u}}$, respectively. Hence, we have two competing global
trivializations $\tau_{E}\oplus\tau_{\xi}$ and $\tau_{T}\oplus\tau_{N}$ of
${\widetilde{u}}^{*}T(\mathbb{R}\times Y)$. It follows that

$\displaystyle 0$
$\displaystyle=\left(\sum_{i=1}^{m_{+}}{\mathrm{wind}}_{\partial^{+}_{i}K_{r}}(\tau_{E},\tau_{T})+\sum_{i=1}^{m_{+}}{\mathrm{wind}}_{\partial^{+}_{i}K_{r}}(\tau_{\xi},\tau_{N})\right)$
$\displaystyle\qquad-\left(\sum_{i=1}^{m_{-}}{\mathrm{wind}}_{\partial^{-}_{i}K_{r}}(\tau_{E},\tau_{T})+\sum_{i=1}^{m_{-}}{\mathrm{wind}}_{\partial^{-}_{i}K_{r}}(\tau_{\xi},\tau_{N})\right)$
$\displaystyle=2-2g-\\#\Gamma+\left(\sum_{i=1}^{m_{+}}{\mathrm{wind}}_{\partial^{+}_{i}K_{r}}(\tau_{\xi},\tau_{N})-\sum_{i=1}^{m_{-}}{\mathrm{wind}}_{\partial^{-}_{i}K_{r}}(\tau_{\xi},\tau_{N})\right)\,.$

The desired conclusion follows from taking the limit as $r\to+\infty$. ∎

To construct a weighted Fredholm theory, Hofer, Wysocki and Zehnder consider
in [22] a certain Banach space of sections of $N_{\widetilde{u}}$. A section
in this space can be represented near a puncture of sign $\epsilon$, with
respect to holomorphic polar coordinates and trivialization $\tau_{N}$ as
above, as a $\mathbb{C}$-valued function $(s,t)\mapsto w(s,t)$ that decays,
together with its first derivatives, like $e^{-\epsilon\delta s}$ as $\epsilon
s\to+\infty$. A smooth Fredholm map is then defined on a neighborhood of zero
of this Banach space, in such a way that exponentiating sections in its zero
locus gives rise to images of curves in $\mathcal{M}_{J}$ near $C$ in a
suitable topology. The linearization at zero is called the linearized Cauchy-
Riemann operator at ${\widetilde{u}}$, and will be denoted by
$D_{\widetilde{u}}$. Its Fredholm index is equal to
${\mathrm{ind}}_{\delta}({\widetilde{u}})$. For precise definitions we refer
to [22, Section 6].

######  Lemma 4.12 (Automatic transversality).

Assume that $g=0$, ${\mathrm{ind}}_{\delta}({\widetilde{u}})=2$ and
$p^{\mp\delta}(\gamma^{\pm}_{i})=1$ for all $\gamma_{i}^{\pm}$. Then $C$ is
immersed and $D_{\widetilde{u}}$ is surjective.

###### Proof.

We know from Lemma 4.9 that $C$ is immersed. Hence, we can consider the
functional analytic set-up from [22] as outlined above. Let $\zeta\in\ker
D_{\widetilde{u}}$ be a section that does not vanish identically. In [22] it
is shown that $D_{\widetilde{u}}$ is a Cauchy-Riemann type operator. Moreover,
it can be represented near a puncture $z^{\pm}_{i}$ using holomorphic polar
coordinates $(s,t)$ as above, and the trivialization $\tau_{N}$, by
$\partial_{s}+J(s,t)\partial_{t}+A(s,t)$ where $-J(s,t)\partial_{t}-A(s,t)$
converges as $\epsilon s\to+\infty$ to a representation of the asymptotic
operator $A_{\gamma^{\pm}_{i}}$ in the frame $\tau_{N}(\gamma^{\pm}_{i})$.
Hence, if we represent $\zeta$ as a $\mathbb{C}$-valued function $w(s,t)$ near
$z^{\pm}_{i}$ using $(s,t)$ and $\tau_{N}$, then we can apply [45, Theorem
A.1] or [25, Theorem 6.1] to obtain an asymptotic formula

$w(s,t)=e^{\kappa s}(v(t)+R(s,t))\qquad(\epsilon s\sim+\infty)\,.$

Here $\kappa\in\sigma(A_{\gamma^{\pm}_{i}})$ satisfies $\epsilon\kappa<0$, and
$v(t)$ is the representation in $\tau_{N}(\gamma^{\pm}_{i})$ of a
corresponding eigensection. A crucial, but simple, remark at this point is
that the inequalities below follow from the definition of $\delta$

$\displaystyle\epsilon=+1\quad\Rightarrow\quad{\mathrm{wind}}_{\tau_{N}(\gamma^{+}_{i})}(\kappa)\leq{\mathrm{wind}}_{\tau_{N}(\gamma^{+}_{i})}(\nu^{<-\delta}(\gamma_{i}^{+}))$
$\displaystyle\epsilon=-1\quad\Rightarrow\quad{\mathrm{wind}}_{\tau_{N}(\gamma^{-}_{i})}(\kappa)\geq{\mathrm{wind}}_{\tau_{N}(\gamma^{-}_{i})}(\nu^{\geq\delta}(\gamma_{i}^{-}))\,.$

Standard degree theory and Lemma 4.11 together imply that the total algebraic
count $\\#\zeta^{-1}(0)$ of zeros of $\zeta$ satisfies

$\displaystyle 2\\#\zeta^{-1}(0)$
$\displaystyle\leq\sum_{i=1}^{m_{+}}2{\mathrm{wind}}_{\tau_{N}(\gamma^{+}_{i})}(\nu^{<-\delta}(\gamma_{i}^{+}))-\sum_{i=1}^{m_{+}}2{\mathrm{wind}}_{\tau_{N}(\gamma^{-}_{i})}(\nu^{\geq\delta}(\gamma_{i}^{-}))$
$\displaystyle=-4+2\\#\Gamma+\sum_{i=1}^{m_{+}}2{\mathrm{wind}}_{\tau_{\xi}(\gamma^{+}_{i})}(\nu^{<-\delta}(\gamma_{i}^{+}))-\sum_{i=1}^{m_{+}}2{\mathrm{wind}}_{\tau_{\xi}(\gamma^{-}_{i})}(\nu^{\geq\delta}(\gamma_{i}^{-}))$
$\displaystyle=-4+2\\#\Gamma-\\#\Gamma^{\rm odd}+\sum_{i=1}^{m_{+}}{\rm
CZ}_{\tau_{\xi}(\gamma^{+}_{i})}^{-\delta}(\gamma_{i}^{+})-\sum_{i=1}^{m_{+}}{\rm
CZ}_{\tau_{\xi}(\gamma^{-}_{i})}^{\delta}(\gamma_{i}^{-})$
$\displaystyle=-2-2+\\#\Gamma+\sum_{i=1}^{m_{+}}{\rm
CZ}_{\tau_{\xi}(\gamma^{+}_{i})}^{-\delta}(\gamma_{i}^{+})-\sum_{i=1}^{m_{+}}{\rm
CZ}_{\tau_{\xi}(\gamma^{-}_{i})}^{\delta}(\gamma_{i}^{-})$
$\displaystyle=-2+{\mathrm{ind}}_{\delta}({\widetilde{u}})=0\,.$

Hence, sections in $\ker D_{\widetilde{u}}$ never vanish, or vanish
identically. Assume, by contradiction, that the cokernel of
$D_{\widetilde{u}}$ is not zero. Then there are at least three linearly
independent sections in $\ker D_{\widetilde{u}}$ because the index of
$D_{\widetilde{u}}$ is two. Since $N_{\widetilde{u}}$ has rank two, this
implies that a linear combination of them would vanish at some point. But
above we proved that the only way in which a section can vanish at some point
is by vanishing identically. This contradiction shows that $D_{\widetilde{u}}$
has a trivial cokernel or, in other words, that $D_{\widetilde{u}}$ is
surjective. ∎

##  5\. Asymptotic analysis and compactness

In this section, we collect the information we glean from the particular kind
of exponential convergence of each $C_{n}$. More precisely, we extract a
limiting cylinder with nondegenerate punctures, and apply some basic
asymptotic analysis to establish some very useful facts about the image of its
projection in the three-manifold; see Proposition 5.1. We also prove a
corresponding compactness statement. Our arguments here are valid for any
sequence of curves satisfying certain axioms, see (*) and (A)-(G) below, and
we structure our proofs in this level of generality.

We draw freely from the notation and definitions established in Section 2.
Consider a sequence $g_{n}\in C^{\infty}(Y)$ such that

$\text{$g_{n}\to 1$ in $C^{\infty}$}.$

The Reeb vector fields of $\lambda_{n}=g_{n}\lambda$ and $\lambda$ are denoted
by $R_{n}$ and $R$, respectively. It should be stressed that we make no
genericity assumptions on $\lambda_{n}$ or on $\lambda$. Consider
$d\lambda_{n}$-compatible, equivalently $d\lambda$-compatible, complex
structures $J_{n},J$ on $\xi$ satisfying

$J_{n}\to J\ \text{in}\ C^{\infty}\,.$

The induced $\mathbb{R}$-invariant almost complex structures on
$\mathbb{R}\times Y$ induced by $\lambda_{n},J_{n}$ and $\lambda,J$ are
denoted by ${\widetilde{J}}_{n}$ and ${\widetilde{J}}$, respectively.

The cylinder $\mathbb{R}\times\mathbb{R}/\mathbb{Z}$ is equipped with
coordinates $s\in\mathbb{R}$, $t\in\mathbb{R}/\mathbb{Z}$, and with its
standard conformal structure obtained from the identification
$\mathbb{R}\times\mathbb{R}/\mathbb{Z}\simeq\mathbb{C}\setminus\\{0\\}$,
$(s,t)\simeq e^{2\pi(s+it)}$. Assume that we are given nonconstant finite-
energy pseudo-holomorphic maps

(5.1)
${\widetilde{u}}_{n}:\mathbb{R}\times\mathbb{R}/\mathbb{Z}\to(\mathbb{R}\times
Y,{\widetilde{J}}_{n})$

and

(5.2)
${\widetilde{u}}:\mathbb{R}\times\mathbb{R}/\mathbb{Z}\to(\mathbb{R}\times
Y,{\widetilde{J}})$

with a positive puncture at $s=+\infty$ and a negative puncture at
$s=-\infty$. Assume that

  * ($*$)

${\widetilde{u}}_{n}$ has nondegenerate punctures.

Let ${\widetilde{u}}_{n}$ be asymptotic to
$\gamma^{+}_{n}=(x^{+}_{n},T^{+}_{n})$ at $s=+\infty$, and to
$\gamma^{-}_{n}=(x^{-}_{n},T^{-}_{n})$ at $s=-\infty$, where
$\gamma^{\pm}_{n}$ are Reeb orbits of $\lambda_{n}$. Assume further that that

  * (A)

${\widetilde{u}}_{n}\to{\widetilde{u}}$ in $C^{\infty}_{\rm
loc}(\mathbb{R}\times\mathbb{R}/\mathbb{Z})$.

  * (B)

There are Reeb orbits $\gamma^{\pm}=(x^{\pm},T^{\pm})$ of $\lambda$ such that
$\gamma^{\pm}_{n}\to\gamma^{\pm}$ as $n\to\infty$. More precisely,
$x^{\pm}_{n}(T^{\pm}_{n}(\cdot+t^{\pm}_{n}))\to x^{\pm}(T^{\pm}\cdot)$ in
$C^{\infty}(\mathbb{R}/\mathbb{Z},Y)$ as $n\to\infty$, for suitable choices of
$t^{\pm}_{n}\in\mathbb{R}/\mathbb{Z}$.

  * (C)

If $c:\mathbb{R}\to Y$ is a periodic trajectory of $R$ with primitive period
$\leq T^{+}$ then $g_{n}$ satisfies $g_{n}|_{c(\mathbb{R})}\equiv 1$ and
$dg_{n}|_{c(\mathbb{R})}\equiv 0$. In particular, $R_{n}$ coincides with $R$
on $c(\mathbb{R})$, and $c$ is also a periodic Reeb trajectory of
$\lambda_{n}$, for all $n$.

  * (D)

The following hold for every $n$:

    * –

$u_{n}(\mathbb{R}\times\mathbb{R}/\mathbb{Z})\subset
Y\setminus(x^{+}_{n}(\mathbb{R})\cup x^{-}_{n}(\mathbb{R}))$.

    * –

If $\gamma=(x,T)$ is a periodic $\lambda_{n}$-Reeb orbit in
$Y\setminus(x^{+}_{n}(\mathbb{R})\cup x^{-}_{n}(\mathbb{R}))$ with $T\leq
T_{+}$ then the intersection number of the loop $x(T\cdot)$ with the map
$u_{n}$ is strictly positive.

    * –

If $\beta$ is a loop in $\mathbb{R}\times\mathbb{R}/\mathbb{Z}$ then
$u_{n}\circ\beta$ has intersection number zero with $u_{n}$.

  * (E)

Let $T_{\min}(\lambda)>0$ be the minimal period of a $\lambda$-Reeb orbit.
There exists $r>0$ such that

$r\leq\int_{\mathbb{R}\times\mathbb{R}/\mathbb{Z}}u_{n}^{*}d\lambda_{n}\leq\frac{1}{2}T_{\min}(\lambda)\qquad\forall
n\,.$

  * (F)

The point $u(0,0)$ does not belong to a $\lambda$-Reeb orbit.

  * (G)

Denote asymptotic operators by $A^{\pm}_{n}=A_{\gamma^{\pm}_{n}}$,
$A^{\pm}=A_{\gamma^{\pm}}$, and asymptotic eigenvalues of
${\widetilde{u}}_{n}$ at $s=\pm\infty$ by
$\mu^{\pm}_{n}\in\sigma(A^{\pm}_{n})$. We assume that there exists $\delta>0$
such that: $[-\delta,0)\cap\sigma(A^{+})=\emptyset$,
$(0,\delta]\cap\sigma(A^{-})=\emptyset$,
${\mathrm{ind}}_{\delta}({\widetilde{u}}_{n})=2$, $\mu^{+}_{n}<-\delta$,
$\mu^{-}_{n}>\delta$,
$p^{-\delta}(\gamma_{n}^{+})=p^{\delta}(\gamma_{n}^{-})=1$ $\forall n$.

We now state the asymptotic analysis result that is a main goal of this
section.

######  Proposition 5.1.

Let $g_{n},\lambda_{n},\lambda,J_{n},J,{\widetilde{u}}_{n},{\widetilde{u}}$ be
as above. Assume that ($*$) and (A)-(G) hold. Then:

  * (a)

${\widetilde{u}}$ has a nontrivial asymptotic formula at both punctures,
${\widetilde{u}}$ is asymptotic to $\gamma^{+}$ at $s=+\infty$, and
${\widetilde{u}}$ is asymptotic to $\gamma^{-}$ at $s=-\infty$.

  * (b)

${\mathrm{ind}}_{\delta}({\widetilde{u}})=2$ and
$p^{-\delta}(\gamma^{+})=p^{\delta}(\gamma^{-})=1$.

  * (c)

The asymptotic eigenvalues $\mu^{+}$ and $\mu^{-}$ of ${\widetilde{u}}$ at
$s=+\infty$ and at $s=-\infty$, respectively, satisfy $\mu^{+}<-\delta$,
$\mu^{-}>\delta$ and

$\displaystyle{\mathrm{wind}}(\mu^{+})={\mathrm{wind}}(\nu^{<-\delta}(\gamma^{+}))\qquad{\mathrm{wind}}(\mu^{-})={\mathrm{wind}}(\nu^{\geq\delta}(\gamma^{-}))$

in any trivialization of $\gamma^{+}$ and $\gamma^{-}$.

  * (d)

The following holds:

    * –

$u(\mathbb{R}\times\mathbb{R}/\mathbb{Z})\subset
Y\setminus(x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R}))$.

    * –

If $\gamma=(x,T)$ is a $\lambda$-Reeb orbit in
$Y\setminus(x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R}))$ such that $T\leq T^{+}$,
then the intersection number of the loop $x(T\cdot)$ with the map $u$ is
strictly positive.

    * –

If $\beta$ is a loop in $\mathbb{R}\times\mathbb{R}/\mathbb{Z}$ then
$u\circ\beta$ has intersection number zero with $u$.

Proving the above proposition will take the bulk of this section, and we split
the proof into several steps. We start with a simple lemma.

######  Lemma 5.2.

$\pi_{\lambda}\circ du$ does not vanish.

###### Proof.

From (E) we know that $\pi_{\lambda}\circ du$ does not vanish identically.
Hence zeros of $\pi_{\lambda}\circ du$ are isolated. If there is a zero then
standard degree theory allows one to find zeros of $\pi_{\lambda_{n}}\circ
du_{n}$ for $n$ large enough. But (G) and Lemma 4.9 together imply that
$\pi_{\lambda_{n}}\circ du_{n}$ does not vanish. This contradiction shows that
$\pi_{\lambda}\circ du$ does not vanish. ∎

###  5.1. Localizing the ends

Next, we prove the following lemma, which allows us to localize the argument,
and we collect some consequences; this is the topic of this subsection.

######  Lemma 5.3.

For every $(\mathbb{R}/\mathbb{Z})$-invariant open $C^{\infty}$-neighborhood
$\mathcal{U}^{+}$ of $x^{+}(T^{+}\cdot)$ there exist $n_{+},s_{+}$ such that
for every $n\geq n_{+}$ and $s\geq s_{+}$ we have
$u_{n}(s,\cdot)\in\mathcal{U}^{+}$. Similarly, for every
$(\mathbb{R}/\mathbb{Z})$-invariant open $C^{\infty}$-neighborhood
$\mathcal{U}^{-}$ of $x^{-}(T^{-}\cdot)$ there exist $n_{-},s_{-}$ such that
for every $n\geq n_{-}$ and $s\leq s_{-}$ we have
$u_{n}(s,\cdot)\in\mathcal{U}^{-}$.

###### Proof.

We will prove the statement for the positive end, the argument for the
negative end is analogous and will be left to the reader. By Stokes theorem,
(B) and (E)

(5.3)
$T^{+}-T^{-}=\lim_{n\to\infty}T^{+}_{n}-T^{-}_{n}=\lim_{n\to\infty}\int_{\mathbb{R}\times\mathbb{R}/\mathbb{Z}}u_{n}^{*}d\lambda_{n}\leq\frac{1}{2}T_{\min}(\lambda)$

and by Stokes theorem together with $(u_{n}^{*}d\lambda_{n})/(ds\wedge dt)\geq
0$

(5.4)
$T^{-}_{n}\leq\int_{\mathbb{R}/\mathbb{Z}}u_{n}(s,\cdot)^{*}\lambda_{n}\leq
T^{+}_{n}\qquad\forall s\in\mathbb{R}\,.$

We now claim that for every pair of sequences $n_{j},s_{j}\to\infty$ the
sequence
$\widetilde{v}_{j}:\mathbb{R}\times\mathbb{R}/\mathbb{Z}\to\mathbb{R}\times Y$
defined by

$\widetilde{v}_{j}(s,t)=(a_{n_{j}}(s+s_{j},t)-a_{n_{j}}(s_{j},0),u_{n_{j}}(s+s_{j},t))$

has a subsequence that $C^{\infty}_{\rm loc}$-converges to a trivial cylinder
over $\gamma^{+}$, i.e. to a map of the form
$(s,t)\mapsto(T^{+}s+a_{0},x^{+}(T^{+}(t+t_{0}))$. With this claim at hand one
can finish the proof of the lemma as follows. Let $\mathcal{U}^{+}$ be an open
$(\mathbb{R}/\mathbb{Z})$-invariant $C^{\infty}$-neighborhood of
$x^{+}(T^{+}\cdot)$. Assume, by contradiction, that there are sequences
$n_{j},s_{j}\to\infty$ such that
$u_{n_{j}}(s_{j},\cdot)\not\in\mathcal{U}^{+}$ for all $j$. Then
$\widetilde{v}_{j}(s,t)=(a_{n_{j}}(s+s_{j},t)-a_{n_{j}}(s_{j},0),u_{n_{j}}(s+s_{j},t))$
has a $C^{\infty}_{\rm loc}$-convergent subsequence to a trivial cylinder
$\widetilde{v}=(b,v)$ over $\gamma^{+}$. Hence,
$\lim_{j}u_{n_{j}}(s_{j},\cdot)=v(0,\cdot)\in\mathcal{U}^{+}$, and $j\gg
1\Rightarrow u_{n_{j}}(s_{j},\cdot)\in\mathcal{U}^{+}$ since $\mathcal{U}^{+}$
is open. This contradiction concludes the argument.

We now argue to establish the above claim. Note that $d\widetilde{v}_{j}$ is
$C^{0}_{\rm loc}$-bounded since, otherwise, we would find a bubbling-off point
and, after suitably rescaling and taking a limit, one would get a nonconstant
${\widetilde{J}}$-holomorphic plane with finite Hofer energy and total
$d\lambda$-area at most $(1/2)T_{\min}(\lambda)$, but this is impossible.
Since $\widetilde{v}_{j}(0,0)$ belongs to $\\{0\\}\times Y$, we get
$C^{1}_{\rm loc}$-bounds for $\widetilde{v}_{j}$. Elliptic bootstrapping
implies $C^{\infty}_{\rm loc}$-bounds. Hence, we obtain the desired
subsequence convergent to some ${\widetilde{J}}$-holomorphic map
$\widetilde{v}$ satisfying
$E(\widetilde{v})\leq\sup_{n}E({\widetilde{u}}_{n})<\infty$. We still denote
the convergent subsequence by $\widetilde{v}_{j}$. The map $\widetilde{v}$ is
not constant since by (5.4)

(5.5)
$\int_{\mathbb{R}/\mathbb{Z}}v(s,\cdot)^{*}\lambda=\lim_{j\to\infty}\int_{\mathbb{R}/\mathbb{Z}}u_{n_{j}}(s_{j}+s,\cdot)^{*}\lambda_{n_{j}}\in[T^{-},T^{+}]\qquad\forall
s\in\mathbb{R}\,.$

This also implies that $s=+\infty$ is a positive puncture and $s=-\infty$ is a
negative puncture of $\widetilde{v}$; otherwise
$\int_{\mathbb{R}/\mathbb{Z}}v(s_{*},\cdot)^{*}\lambda<T^{-}$ would hold for
some $s_{*}$.

Consider $\gamma=(x,T)$ an asymptotic limit of $\widetilde{v}$ at the negative
puncture. By the above estimate we know that $T\in[T^{-},T^{+}]$. If $T=T^{-}$
then for every $\alpha>0$ we can find some $\underline{s}\in\mathbb{R}$ such
that
$\int_{\mathbb{R}/\mathbb{Z}}v(\underline{s},\cdot)^{*}\lambda<T^{-}+\alpha$.
With $s$ fixed arbitrarily we could take $j\gg 1$ such that
$\underline{s}+s_{j}>s$ and estimate

$T^{-}_{n_{j}}\leq\int_{\mathbb{R}/\mathbb{Z}}u_{n_{j}}(s,\cdot)^{*}\lambda_{n_{j}}\leq\int_{\mathbb{R}/\mathbb{Z}}u_{n_{j}}(\underline{s}+s_{j},\cdot)^{*}\lambda_{n_{j}}\to\int_{\mathbb{R}/\mathbb{Z}}v(\underline{s},\cdot)^{*}\lambda<T^{-}+\alpha\,.$

Passing to the limit as $j\to\infty$

$T^{-}\leq\int_{\mathbb{R}/\mathbb{Z}}u(s,\cdot)^{*}\lambda\leq
T^{-}+\alpha\quad\forall s$

where (B) was used. Using Stokes, and passing to the limit as $\alpha\to 0$ we
get

$\int_{\mathbb{R}\times\mathbb{R}/\mathbb{Z}}u^{*}d\lambda=0$

which means that ${\widetilde{u}}$ is a trivial cylinder over a periodic
orbit, in contradiction to (F). We proved that $T>T^{-}$. It follows from
(5.3) and (5.5) that $\gamma$ is geometrically distinct from $\gamma^{-}$.

We now argue by contradiction to show that $x(\mathbb{R})\subset
x^{+}(\mathbb{R})$. If we have $x(\mathbb{R})\subset Y\setminus
x^{+}(\mathbb{R})$ then $x(\mathbb{R})\subset Y\setminus(x^{+}(\mathbb{R})\cup
x^{-}(\mathbb{R}))$. If $n$ is large enough then $S=\\{x(T\cdot+t_{0})\mid
t_{0}\in\mathbb{R}/\mathbb{Z}\\}$ has an open neighborhood $\mathcal{W}\subset
C^{\infty}(\mathbb{R}/\mathbb{Z},Y)$ such that every loop in $\mathcal{W}$
does not intersect the link $x^{+}_{n}(\mathbb{R})\cup x^{-}_{n}(\mathbb{R})$.
Otherwise, by the Arzelà-Ascoli theorem, we would find a sequence of loops
$c_{n_{j}}$ converging to a loop in $S$, such that each $c_{n_{j}}$ intersects
$x^{+}_{n_{j}}(\mathbb{R})\cup x^{-}_{n_{j}}(\mathbb{R})$. Passing to the
limit as $j\to\infty$, and using (B), we would find a loop in $S$ that
intersects $x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R})$, but we work under the
assumption that this is not the case. After shrinking $\mathcal{W}$, we can
assume in addition that $\mathcal{W}$ is path-connected. By assumptions (C)
and (D) every loop in $S$ has positive intersection number with $u_{n}$. Since
loops in $\mathcal{W}$ do not intersect the asymptotic limits of
${\widetilde{u}}_{n}$, we can use the path-connectedness of $\mathcal{W}$, and
the homotopy invariance of intersection numbers, to conclude that every loop
in $\mathcal{W}$ has positive intersection number with $u_{n}$. We can find
$\underline{s}\sim-\infty$ such that $v(\underline{s},\cdot)$ is close to $S$,
hence belongs to $\mathcal{W}$. Hence, for $j$ large enough
$u_{n_{j}}(s_{j}+\underline{s},\cdot)$ also belongs to $\mathcal{W}$ and,
consequently, has positive intersection number with $u_{n_{j}}$. But, by (D),
this intersection number is zero. This contradiction shows that
$x(\mathbb{R})\subset x^{+}(\mathbb{R})$. It follows from this, (5.3) and
(5.5) that $\gamma=\gamma^{+}$. By (5.5) we get
$\int_{\mathbb{R}/\mathbb{Z}}v(s,\cdot)^{*}\lambda=T^{+}$ for every $s$, hence
$v^{*}d\lambda$ vanishes identically, and $\widetilde{v}$ is the desired
trivial cylinder over $\gamma^{+}$. ∎

The above proof made only use of assumptions (A)-(F).

######  Corollary 5.4.

If $U_{+},U_{-}\subset Y$ are open neighborhoods of
$x^{+}(\mathbb{R}),x^{-}(\mathbb{R})$, respectively, then there exist
$N\in\mathbb{N}$ and $\underline{s}>0$ such that if $n\geq N$ then

$\displaystyle u([\underline{s},+\infty)\times\mathbb{R}/\mathbb{Z})$
$\displaystyle\cup\bigcup_{n}u_{n}([\underline{s},+\infty)\times\mathbb{R}/\mathbb{Z})\subset
U_{+}\,,$ $\displaystyle
u((-\infty,-\underline{s}]\times\mathbb{R}/\mathbb{Z})$
$\displaystyle\cup\bigcup_{n}u_{n}((-\infty,-\underline{s}]\times\mathbb{R}/\mathbb{Z})\subset
U_{-}\,.$

######  Corollary 5.5.

If $\gamma^{+}$ is nondegenerate then ${\widetilde{u}}$ is asymptotic to
$\gamma^{+}$ at $s=+\infty$, and has an asymptotic formula at $s=+\infty$.
Similarly, if $\gamma^{-}$ is nondegenerate then ${\widetilde{u}}$ is
asymptotic to $\gamma^{-}$ at $s=-\infty$, and has an asymptotic formula at
$s=-\infty$.

###### Proof.

Assume that $\gamma^{+}$ is nondegenerate. By Corollary 5.4 the only
asymptotic limit of ${\widetilde{u}}$ at $s=+\infty$ is $\gamma^{+}$. The
conclusion at $s=+\infty$ follows from a direct application of Theorem 4.6.
The same argument applies to $s=-\infty$ if $\gamma^{-}$ is nondegenerate. ∎

###  5.2. Special coordinates

To proceed, it will be helpful to construct special coordinates to analyze the
ends of the cylinders ${\widetilde{u}}_{n}$; this is the goal of this
subsection. We only analyze positive punctures, the construction for the
negative punctures is analogous. Let
$\underline{\gamma}^{+}=(x^{+},\underline{T}^{+})$ and
$\underline{\gamma}^{+}_{n}=(x^{+}_{n},\underline{T}^{+}_{n})$ be the
primitive Reeb orbits underlying $\gamma^{+}$ and $\gamma^{+}_{n}$,
respectively. Let $k_{+}\in\mathbb{N}$ denote the covering multiplicity of
$\gamma^{+}$. Assume that ${\gamma}^{+}$ is degenerate. A direct application
of [2, Proposition 1] implies that for $n$ large enough there is a divisor
$j_{+}$ of $k_{+}=l_{+}j_{+}$ such that

(5.6) $\gamma^{+}_{n}=(\underline{\gamma}^{+}_{n})^{l_{+}}\ \ \forall
n\,,\qquad\underline{\gamma}^{+}_{n}\to(\underline{\gamma}^{+})^{j_{+}}\,.$

Moreover, by [2, Proposition 1], $j_{+}$ is equal to one or to the
multiplicity of the first degenerate iterate of $\underline{\gamma}^{+}$. In
particular, we have $\underline{T}^{+}_{n}\to j_{+}\underline{T}^{+}$.

######  Lemma 5.6.

There is an open tubular neighborhood $N_{+}$ of $x^{+}(\mathbb{R})$ in $Y$, a
smooth covering map $P_{+}:\mathbb{R}/j_{+}\mathbb{Z}\times B\to N_{+}$ where
$B\subset\mathbb{C}$ is an open ball centered at the origin, and local
diffeomorphisms $P_{n}:\mathbb{R}/j_{+}\mathbb{Z}\times B\to N_{+}$ with the
following properties.

  * (a)

The action of the group of deck transformations of $P_{+}$ is the action by
$\mathbb{Z}/j_{+}\mathbb{Z}$ on the first component, and $P_{n}\to P_{+}$ in
$C^{\infty}_{\rm loc}$ as $n\to+\infty$.

  * (b)

There are lifts
$\tilde{x}^{+}_{n}:\mathbb{R}\to\mathbb{R}/j_{+}\mathbb{Z}\times B$ of
$x^{+}_{n}:\mathbb{R}\to Y$ by $P_{n}$, and
$\tilde{x}^{+}:\mathbb{R}\to\mathbb{R}/j_{+}\mathbb{Z}\times B$ of
$x^{+}:\mathbb{R}\to Y$ by $P_{+}$ such that

$\tilde{x}^{+}_{n}(t)=(j_{+}t/\underline{T}^{+}_{n},0)\qquad\text{and}\qquad\tilde{x}^{+}(t)=(t/\underline{T}^{+},0)\,.$

  * (c)

There are smooth functions
$\tilde{f}_{n},\tilde{f}:\mathbb{R}/j_{+}\mathbb{Z}\times B\to\mathbb{R}$ such
that $\tilde{\lambda}_{n}=P_{n}^{*}\lambda_{n}$ and
$\tilde{\lambda}=P_{+}^{*}\lambda$ can be written as

$\tilde{\lambda}_{n}=\tilde{f}_{n}(\theta,z)(d\theta+x_{1}dx_{2})\,,\qquad\tilde{\lambda}=\tilde{f}(\theta,z)(d\theta+x_{1}dx_{2})\,.$

Moreover, $\tilde{f}_{n},\tilde{f}$ satisfy

$\displaystyle\tilde{f}_{n}(\theta,0)=\underline{T}^{+}_{n}/j_{+},\
\tilde{f}(\theta,0)=\underline{T}^{+}\ \forall\theta$ $\displaystyle
d\tilde{f}_{n}(\theta,0)=d\tilde{f}(\theta,0)=0\ \forall\theta$
$\tilde{f}_{n}\to\tilde{f}$ in $C^{\infty}_{\rm
loc}(\mathbb{R}/j_{+}\mathbb{Z}\times B)$.

###### Proof.

Consider a Martinet tube $\Psi:V\to\mathbb{R}/\mathbb{Z}\times B$ for
$\lambda$ and $\gamma^{+}$. This means that $\Psi$ is a diffeomorphism defined
on an open neighborhood $V$ of $x^{+}(\mathbb{R})$ such that

$\displaystyle\Psi(x^{+}(\underline{T}^{+}t))=(t,0)$
$\displaystyle\Psi_{*}\lambda=h(\theta,z)(d\theta+x_{1}dx_{2})$ $\displaystyle
h(\theta,0)=\underline{T}^{+},\ dh(\theta,0)=0\ \forall\theta\,.$

Here $B\subset\mathbb{C}$ is an open ball centered at the origin. The contact
structure $\xi=\ker\lambda$ gets represented as
$\xi_{0}=\Psi_{*}\xi=\ker(d\theta+x_{1}dx_{2})$. It follows from (C) that

$\Psi_{*}\lambda_{n}=h_{n}(\theta,z)(d\theta+x_{1}dx_{2})$

where $h_{n}(\theta,0)=\underline{T}^{+}$, $dh_{n}(\theta,0)=0$
$\forall\theta$, and $h_{n}\to h$ in $C^{\infty}_{\rm loc}$. Consider the
$(\mathbb{Z}/j_{+}\mathbb{Z})$-invariant covering map

$P:\mathbb{R}/j_{+}\mathbb{Z}\times B\to\mathbb{R}/\mathbb{Z}\times
B\,,\quad(\theta+j_{+}\mathbb{Z},z)\mapsto(\theta+\mathbb{Z},z)\,.$

Use this map to lift $\Psi_{*}\lambda,\Psi_{*}\lambda_{n}$ to contact forms
$\tilde{\lambda}=P^{*}(\Psi_{*}\lambda)$,
$\tilde{\lambda}_{n}=P^{*}(\Psi_{*}\lambda_{n})$ on
$\mathbb{R}/j_{+}\mathbb{Z}\times B$, respectively, written as

$\tilde{\lambda}_{n}=\tilde{h}_{n}(\theta,z)(d\theta+x_{1}dx_{2})\qquad\tilde{\lambda}=\tilde{f}(\theta,z)(d\theta+x_{1}dx_{2})$

where $d\tilde{f},d\tilde{h}_{n}$ vanish on
$\mathbb{R}/j_{+}\mathbb{Z}\times\\{0\\}$, $\tilde{f},\tilde{h}_{n}$ are equal
to $\underline{T}^{+}$ on $\mathbb{R}/j_{+}\mathbb{Z}\times\\{0\\}$, and
$\tilde{h}_{n}\to\tilde{f}$ in $C^{\infty}_{\rm loc}$. The loops

$c_{n}:t\in\mathbb{R}/j_{+}\mathbb{Z}\mapsto\Psi\circ
x^{+}_{n}(\underline{T}^{+}_{n}t/j_{+})$

are constant-speed reparametrizations of simply covered Reeb orbits of
$\Psi_{*}\lambda_{n}$. Perhaps replacing $t$ by $t+t_{n}$ for suitable
$t_{n}$, there is no loss of generality to assume that $c_{n}$ converges to
$t\in\mathbb{R}/j_{+}\mathbb{Z}\mapsto(t,0)\in\mathbb{R}/\mathbb{Z}\times B$
in $C^{\infty}$. Here we used assumption (B). If we denote the Reeb vector
field of $\Psi_{*}\lambda_{n}$ by $R_{\Psi_{*}\lambda_{n}}$ then

$\frac{d}{dt}\
c_{n}(t)=(\underline{T}^{+}_{n}/j_{+})R_{\Psi_{*}\lambda_{n}}(c_{n}(t))$

A sequence of lifts of $c_{n}$ by $P$

$\tilde{c}_{n}:\mathbb{R}/j_{+}\mathbb{Z}\to\mathbb{R}/j_{+}\mathbb{Z}\times
B$

can be chosen in such a way that it $C^{\infty}$-converges to the map
$t\mapsto(t,0)$. If we denote the Reeb vector field of
$\tilde{\lambda}_{n}=P^{*}(\Psi_{*}\lambda_{n})$ by $R_{\tilde{\lambda}_{n}}$
then

$\frac{d}{dt}\
\tilde{c}_{n}(t)=(\underline{T}^{+}_{n}/j_{+})R_{\tilde{\lambda}_{n}}(\tilde{c}_{n}(t))$

One can now find co-orientation preserving contactomorphisms

$\phi_{n}:(\mathbb{R}/j_{+}\mathbb{Z}\times
B,\xi_{0})\to(\mathbb{R}/j_{+}\mathbb{Z}\times B,\xi_{0})$

such that

$\phi_{n}\to id\ \ \text{in $C^{\infty}$,}\quad{\rm
supp}(\phi_{n})\to\mathbb{R}/j_{+}\mathbb{Z}\times\\{0\\},\quad\phi_{n}\circ\tilde{c}_{n}(t)=(t,0)\
\forall n,t\,.$

The contact forms

(5.7)
$(\phi_{n})_{*}\tilde{\lambda}_{n}=(\phi_{n})_{*}(\tilde{h}_{n}(\theta,z)(d\theta+x_{1}dx_{2}))=\tilde{f}_{n}(\theta,z)(d\theta+x_{1}dx_{2})$

have
$t\in\mathbb{R}/j_{+}\mathbb{Z}\mapsto\phi_{n}\circ\tilde{c}_{n}(t)=(t,0)$ as
a Reeb orbit of primitive period $\underline{T}^{+}_{n}$. Moreover, if we
denote the Reeb vector field of $(\phi_{n})_{*}\tilde{\lambda}_{n}$ by
$R_{(\phi_{n})_{*}\tilde{\lambda}_{n}}$ then

$\displaystyle(1,0)$ $\displaystyle=\frac{d}{dt}\
\phi_{n}\circ\tilde{c}_{n}(t)=(\underline{T}^{+}_{n}/j_{+})R_{(\phi_{n})_{*}\tilde{\lambda}_{n}}(\phi_{n}\circ\tilde{c}_{n}(t))$
$\displaystyle=(\underline{T}^{+}_{n}/j_{+})R_{(\phi_{n})_{*}\tilde{\lambda}_{n}}(t,0)$

Thus $R_{(\phi_{n})_{*}\tilde{\lambda}_{n}}(t,0)$ is, for each $n$, a constant
multiple of $(1,0)$ and

(5.8) $d\tilde{f}_{n}(\theta,0)=0\,,\
\tilde{f}_{n}(\theta,0)=\underline{T}^{+}_{n}/j_{+}\
\forall\theta\,,\qquad\tilde{f}_{n}\to\tilde{f}\ \text{in}\ C^{\infty}_{\rm
loc}$

Set $P_{+}=\Psi^{-1}\circ P$, $P_{n}=\Psi^{-1}\circ P\circ(\phi_{n})^{-1}$
defined on $\mathbb{R}/j_{+}\mathbb{Z}\times B$, and
$N_{+}=\Psi^{-1}(\mathbb{R}/\mathbb{Z}\times B)$. ∎

###  5.3. Uniform exponential decay

We now begin the process of putting all of this together to prove Proposition
5.1; the goal of this section is to prove item (a).

Consider the neighborhood $N_{+}$ of $x^{+}(\mathbb{R})$ and the smooth maps
$P_{+},P_{n}$ given by Lemma 5.6. The lift of the contact structure by $P_{n}$
or by $P_{+}$ is equal to $\ker(d\theta+x_{1}dx_{2})$. The lifts of $J_{n}$
and of $J$ by $P_{n}$ and by $P_{+}$, respectively, can be represented as
smooth functions

$\hat{J}_{n}:\mathbb{R}/j_{+}\mathbb{Z}\times B\to\mathbb{R}^{2\times
2}\qquad\hat{J}:\mathbb{R}/j_{+}\mathbb{Z}\times B\to\mathbb{R}^{2\times 2}$

with respect to the frame
$\\{\partial_{x_{1}},-x_{1}\partial_{\theta}+\partial_{x_{2}}\\}$. Note that
$\hat{J}$ is invariant by the action of $\mathbb{Z}/j_{+}\mathbb{Z}$ on the
first coordinate. We get $\hat{J}_{n}\to\hat{J}$ in $C^{\infty}_{\rm loc}$
from $J_{n}\to J$ in $C^{\infty}_{\rm loc}$. Similarly, write

$\tilde{R}_{n}:\mathbb{R}/j_{+}\mathbb{Z}\times
B\to\mathbb{R}^{3}\qquad\tilde{R}:\mathbb{R}/j_{+}\mathbb{Z}\times
B\to\mathbb{R}^{3}$

for the lifts of the Reeb vector fields of $\lambda_{n}$ and $\lambda$ by
$P_{n}$ and by $P_{+}$, respectively. The function $\tilde{R}$ is
$(\mathbb{Z}/j_{+}\mathbb{Z})$-invariant and $\tilde{R}_{n}\to\tilde{R}$ in
$C^{\infty}_{\rm loc}$. We write $D_{2}\tilde{R}_{n},D_{2}\tilde{R}$ for the
partial derivative in the second component.

Let $\underline{s}>0$ be given by an application of Corollary 5.4 to the
neighborhood $N_{+}$ of $x^{+}(\mathbb{R})$. It follows from Lemma 5.6 that a
loop in $N_{+}$ can be lifted by $P_{n}$ or by $P$ to
$\mathbb{R}/j_{+}\mathbb{Z}\times B$ as a loop if, and only if, it is
homotopic to $t\in\mathbb{R}/\mathbb{Z}\mapsto
x^{+}(mj_{+}\underline{T}^{+}t)$, for some $m\in\mathbb{Z}$. For every
$s\geq\underline{s}$ and $n$ the loop $t\mapsto\mathbb{R}/\mathbb{Z}\mapsto
u_{n}(s,t)$ is freely homotopic to $t\in\mathbb{R}/\mathbb{Z}\mapsto
x^{+}(k_{+}\underline{T}^{+}t)$. Hence, since $k_{+}=j_{+}l_{+}$,
$(s,t)\in[\underline{s},+\infty)\times\mathbb{R}/\mathbb{Z}\mapsto
u_{n}(s,t)\in N_{+}$ can be lifted by $P_{n}$ as smooth maps

$(s,t)\in[\underline{s},+\infty)\times\mathbb{R}/\mathbb{Z}\mapsto(\theta_{n}(s,t),z_{n}(s,t))\in\mathbb{R}/j_{+}\mathbb{Z}\times
B.$

Similarly, $(s,t)\in[\underline{s},+\infty)\times\mathbb{R}/\mathbb{Z}\mapsto
u(s,t)\in N_{+}$ can be lifted by $P_{+}$ as a smooth map

$(s,t)\in[\underline{s},+\infty)\times\mathbb{R}/\mathbb{Z}\mapsto(\theta(s,t),z(s,t))\in\mathbb{R}/j_{+}\mathbb{Z}\times
B.$

Moreover, since ${\widetilde{u}}_{n}\to{\widetilde{u}}$ in $C^{\infty}_{\rm
loc}$, these lifts can be arranged to satisfy

$(\theta_{n}(s,t),z_{n}(s,t))\to(\theta(s,t),z(s,t))\qquad C^{\infty}_{\rm
loc}([\underline{s},+\infty)\times\mathbb{R}/\mathbb{Z})\,.$

Write in components

$z_{n}(s,t)=x_{1,n}(s,t)+ix_{2,n}(s,t)\qquad z(s,t)=x_{1}(s,t)+ix_{2}(s,t)\,.$

Abbreviate

$j_{n}(s,t)=\hat{J}_{n}(\theta_{n}(s,t),z_{n}(s,t))\quad
j(s,t)=\hat{J}(\theta(s,t),z(s,t))$

for simplicity. Denote

$\displaystyle D_{n}(s,t)$
$\displaystyle=\int_{0}^{1}D_{2}\tilde{R}_{n}(\theta_{n}(s,t),vz_{n}(s,t))dv$
$\displaystyle D(s,t)$
$\displaystyle=\int_{0}^{1}D_{2}\tilde{R}(\theta(s,t),vz(s,t))dv$

and

(5.9) $\displaystyle S(s,t)$ $\displaystyle=[\partial_{t}a\ I-\partial_{s}a\
j]D(s,t)$ $\displaystyle S_{n}(s,t)$ $\displaystyle=[\partial_{t}a_{n}\
I-\partial_{s}a_{n}\ j_{n}]D_{n}(s,t)$

where $a_{n}(s,t),a(s,t)$ are the $\mathbb{R}$-components of
${\widetilde{u}}_{n}(s,t),{\widetilde{u}}(s,t)$. The Cauchy-Riemann equations
for the lifts of ${\widetilde{u}}_{n},{\widetilde{u}}$ by $P_{n},P_{+}$ get
written as

(5.10)
$\displaystyle\partial_{s}a_{n}-\tilde{f}_{n}(\theta_{n},z_{n})(\partial_{t}\theta_{n}+x_{1,n}\partial_{t}x_{2,n})=0\,,$
$\displaystyle\partial_{t}a_{n}+\tilde{f}_{n}(\theta_{n},z_{n})(\partial_{s}\theta_{n}+x_{1,n}\partial_{s}x_{2,n})=0\,,$
$\displaystyle\partial_{s}z_{n}+j_{n}\partial_{t}z_{n}+S_{n}z_{n}=0\,,$

and, similarly,

(5.11)
$\displaystyle\partial_{s}a-\tilde{f}(\theta,z)(\partial_{t}\theta+x_{1}\partial_{t}x_{2})=0\,,$
$\displaystyle\partial_{t}a+\tilde{f}(\theta,z)(\partial_{s}\theta+x_{1}\partial_{s}x_{2})=0\,,$
$\displaystyle\partial_{s}z+j\partial_{t}z+Sz=0\,.$

Above, and also in what follows, we see derivatives of $\theta_{n},\theta$ as
$\mathbb{R}$-valued functions.

######  Lemma 5.7.

We have

(5.12) $\displaystyle\lim_{\tau\to+\infty}\sup_{n}\
\|D^{\beta}z_{n}(\tau,\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}=0\quad\forall\beta$
(5.13) $\displaystyle\lim_{\tau\to+\infty}\sup_{n}\ \left(\begin{aligned}
&\|D^{\beta}[a_{n}-T^{+}_{n}s](\tau,\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}\\\
&\quad+\|D^{\beta}[\theta_{n}-k_{+}t](\tau,\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}\end{aligned}\right)=0\quad\text{$\forall\beta\neq(0,0)$}$

where $\beta=(\beta_{1},\beta_{2})\in\mathbb{N}_{0}\times\mathbb{N}_{0}$ and
$D^{\beta}=\partial^{\beta_{1}}_{s}\partial^{\beta_{2}}_{t}$.

###### Proof.

By Lemma 5.3, (5.12) holds for all $\beta=(0,\beta_{2})$. We claim that

(5.14)
$\sup_{s\in[\underline{s},+\infty),\,t\in\mathbb{R}/\mathbb{Z},\,n}|\nabla{\widetilde{u}}_{n}(s,t)|<\infty\,.$

If not then, up to a subsequence, we find bubbling-off points around which we
could rescale and find a nonconstant holomorphic plane with finite Hofer
energy. Such a plane would need to have $d\lambda$-area at least equal to
$T_{\min}(\lambda)$, but (E) forces the $d\lambda$-area of this plane to be at
most $\frac{1}{2}T_{\min}(\lambda)$. This contradiction proves (5.14).
Standard boot-strapping arguments show that (5.14) implies uniform bounds

(5.15)
$\sup_{s\in[\underline{s},+\infty),\,t\in\mathbb{R}/\mathbb{Z},\,n}|D^{\alpha}\nabla{\widetilde{u}}_{n}(s,t)|<\infty\quad\forall\alpha$

for the higher-order derivatives, see [23, Lemma 7.1]. It follows from (5.15),
(5.9), and (5.10), that (5.12) holds when $|\beta|\leq 1$.

Choose lifts $\tilde{\theta}_{n}:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ of
$\theta_{n}$. If (5.13) does not hold then one finds $\beta\neq(0,0)$,
$\epsilon>0$ and sequences $s_{m}\in\mathbb{R}$,
$t_{m}\in\mathbb{R}/\mathbb{Z}$, $n_{m}\in\mathbb{N}$ satisfying
$s_{m}\to+\infty$, $n_{m}\to+\infty$ and

(5.16)
$|D^{\beta}[a_{n_{m}}-T^{+}_{n_{m}}s](s_{m},t_{m})|+|D^{\beta}[\tilde{\theta}_{n_{m}}-k_{+}t](s_{m},t_{m})|\geq\epsilon\,.$

Define $h_{m}:\mathbb{C}\to\mathbb{C}$ by

$\displaystyle h_{m}(s+it)$
$\displaystyle=\frac{1}{T_{n_{m}}^{+}}(a_{n_{m}}(s+s_{m},t+t_{m})-a_{n_{m}}(s_{m},t_{m}))$
$\displaystyle\quad+\frac{i}{k_{+}}(\tilde{\theta}_{n_{m}}(s+s_{m},t+t_{m})-\tilde{\theta}_{n_{m}}(s_{m},t_{m}))\,.$

By (5.15) $\nabla h_{m}$ is $C^{\infty}_{\rm loc}$-bounded. Together with
$h_{m}(0)=0$ we get $C^{\infty}_{\rm loc}$-bounds for $h_{m}$. Hence, up to a
taking a subsequence, we can assume without loss of generality that $h_{m}$ is
$C^{\infty}_{\rm loc}$ convergent to some $h:\mathbb{C}\to\mathbb{C}$.
Equations (5.10), together with (5.12) for $|\beta|\leq 1$, imply that $h$ is
holomorphic. Moreover, (5.15) implies that $h$ has bounded derivatives. By
Lemma 5.3, $\partial_{t}h\equiv 1$ and $h$ is not constant. By Liouville’s
theorem, $h(s+it)$ must be a degree-one complex polynomial in $s+it$
satisfying $h(0)=0$ and $\partial_{t}h\equiv 1$. Hence, $h(s+it)=s+it$. By
(5.16) we have $|D^{\beta}[h_{m}-s-it](0)|$ bounded away from zero, but we
have just proved that $h_{m}-s-it$ $C^{\infty}_{\rm loc}$-converges to $0$.
This contradiction concludes the proof that (5.13) holds.

The proof that (5.12) holds for all $\beta$ can now be completed by
differentiating the equation for $z_{n}$ in (5.10) and plugging the bounds for
$S_{n}$ that follow from (5.13). ∎

Let

$\hat{M}_{n}:\mathbb{R}/j_{+}\mathbb{Z}\times B\to
Sp(2)\qquad\hat{M}:\mathbb{R}/j_{+}\mathbb{Z}\times B\to Sp(2)$

be smooth functions satisfying $\hat{M}_{n}\hat{J}_{n}=i\hat{M}_{n}$,
$\hat{M}\hat{J}=i\hat{M}$ pointwise. Since $\hat{J}_{n}\to\hat{J}$, we can
arrange these functions to satisfy

$\hat{M}_{n}\to\hat{M}\quad\text{in}\quad C^{\infty}_{\rm loc}\,.$

Consider the smooth functions $\mathbb{R}/\mathbb{Z}\to\mathbb{R}^{2\times 2}$

$\displaystyle j^{\infty}(t)$ $\displaystyle=\hat{J}(k_{+}t,0)$ $\displaystyle
D^{\infty}(t)$ $\displaystyle=D_{2}\tilde{R}(k_{+}t,0)$ $\displaystyle
S^{\infty}(t)$ $\displaystyle=-T_{+}j^{\infty}(t)D^{\infty}(t)$ $\displaystyle
M^{\infty}(t)$ $\displaystyle=M(k_{+}t,0)\,.$

As before, we abbreviate

$\displaystyle M_{n}(s,t)=\hat{M}_{n}(\theta_{n}(s,t),z_{n}(s,t))\,,\qquad
M(s,t)=\hat{M}(\theta(s,t),z(s,t))\,.$

The next lemma is [27, Lemma 4.12] or [28, Lemma 3.33].

######  Lemma 5.8.

For every pair of sequences $s_{m}\to\infty$, $n_{m}\to\infty$ there exist
$c\in[0,1)$ and a sequence of indices $m_{l}\to\infty$ such that

$\displaystyle\lim_{l\to+\infty}\|D^{\beta}[j_{n_{m_{l}}}(s,t)-j^{\infty}(t+c)](s_{m_{l}},\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}=0$
$\displaystyle\lim_{l\to+\infty}\|D^{\beta}[D_{n_{m_{l}}}(s,t)-D^{\infty}(t+c)](s_{m_{l}},\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}=0$
$\displaystyle\lim_{l\to+\infty}\|D^{\beta}[S_{n_{m_{l}}}(s,t)-S^{\infty}(t+c)](s_{m_{l}},\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}=0$
$\displaystyle\lim_{l\to+\infty}\|D^{\beta}[M_{n_{m_{l}}}(s,t)-M^{\infty}(t+c)](s_{m_{l}},\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}=0$

hold for every partial derivative
$D^{\beta}=\partial^{\beta_{1}}_{s}\partial^{\beta_{2}}_{t}$.

Define functions

$\displaystyle\zeta_{n}(s,t)$ $\displaystyle=M_{n}z_{n}\,,$
$\displaystyle\Lambda_{n}(s,t)$
$\displaystyle=(M_{n}S_{n}-\partial_{s}M_{n}-i\partial_{t}M_{n})M_{n}^{-1}\,,$
$\displaystyle\Lambda^{\infty}(t)$
$\displaystyle=(M^{\infty}S^{\infty}-i\partial_{t}M^{\infty})(M^{\infty})^{-1}\,.$

######  Lemma 5.9.

The following hold:

  * (a)

$\partial_{s}\zeta_{n}+i\partial_{t}\zeta_{n}+\Lambda_{n}\zeta_{n}=0$.

  * (b)

For every pair of sequences $s_{m}\to\infty$, $n_{m}\to\infty$ there exist
$c\in[0,1)$ and a sequence of indices $m_{l}\to\infty$ such that

$\lim_{l\to+\infty}\|D^{\beta}[\Lambda_{n_{m_{l}}}(s,t)-\Lambda^{\infty}(t+c)](s_{m_{l}},\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}=0\,.$

###### Proof.

A direct computation shows (a), and (b) is a direct consequence of Lemma 5.8.
∎

######  Remark 5.10.

Consider $r\in\mathbb{N}_{0}$, $\alpha\in(0,1)$, $d<0$ and a finite-
dimensional vector space $V$. Consider the space
$C^{r,\alpha,d}_{0}([\underline{s},+\infty)\times\mathbb{R}/\mathbb{Z},V)$ of
functions $h:[\underline{s},+\infty)\times\mathbb{R}/\mathbb{Z}\to V$ of class
$C^{r,\alpha}_{\rm loc}$ such that for every $\beta$ satisfying $|\beta|\leq
r$ the $C^{0,\alpha}$-norm of $e^{-ds}D^{\beta}h$ on
$[\rho,+\infty)\times\mathbb{R}/\mathbb{Z}$ is finite and decays to zero as
$\rho\to+\infty$. The norm
$\|h\|_{r,\alpha,d}=\|e^{-ds}h\|_{C^{r,\alpha}([\underline{s},+\infty)\times\mathbb{R}/\mathbb{Z})}$
turns this space into a Banach space.

######  Proposition 5.11 (Proposition 4.15 in [27]).

Suppose that

$K_{n}:[0,+\infty)\times\mathbb{R}/\mathbb{Z}\to\mathbb{R}^{2k\times 2k}\qquad
K^{\infty}:\mathbb{R}/\mathbb{Z}\to\mathbb{R}^{2k\times 2k}$

are smooth maps satisfying

  * •

$K^{\infty}(t)$ is symmetric, for all $t$.

  * •

For every pair of sequences $s_{m}\to\infty$, $n_{m}\to\infty$ there exist
$c\in[0,1)$ and a sequence of indices $m_{l}\to\infty$ such that

$\lim_{l\to+\infty}\|D^{\beta}[K_{n_{m_{l}}}(s,t)-K^{\infty}(t+c)](s_{m_{l}},\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}=0\,.$

Consider the unbounded self-adjoint operator $L=-J_{0}\partial_{t}-K^{\infty}$
on $L^{2}(\mathbb{R}/\mathbb{Z},\mathbb{R}^{2k})$. Consider $r\in\mathbb{N}$,
$\alpha\in(0,1)$. Let $\delta>0$ be such that $-\delta$ does not belong to the
spectrum of $L$. Consider a $C^{\infty}_{\rm loc}$-bounded sequence of maps

$X_{n}\in
C^{r,\alpha,-\delta}_{0}([0,+\infty)\times\mathbb{R}/\mathbb{Z},\mathbb{R}^{2k})\cap
C^{\infty}([0,+\infty)\times\mathbb{R}/\mathbb{Z},\mathbb{R}^{2k})$

satisfying

$\partial_{s}X_{n}+J_{0}\partial_{t}X_{n}+K_{n}X_{n}=0\,.$

Then $X_{n}$ has a convergent subsequence in
$C^{r,\alpha,-\delta}_{0}([0,+\infty)\times\mathbb{R}/\mathbb{Z},\mathbb{R}^{2k})$.

Fix $\alpha\in(0,1)$. Let $r\in\mathbb{N}$ be arbitrary, and recall the number
$\delta>0$ from hypothesis (G). We write $C^{r,\alpha,-\delta}_{0}$ instead of
$C^{r,\alpha,-\delta}_{0}([\underline{s},+\infty)\times\mathbb{R}/\mathbb{Z})$
for simplicity. By assumption, ${\widetilde{u}}_{n}$ has nondegenerate
punctures in the sense of Definition 4.7. By Theorem 4.8,
${\widetilde{u}}_{n}$ has an asymptotic formula at both punctures. This
implies that $z_{n}\in C^{r,\alpha,-\delta}_{0}$ since, by (G), the
corresponding asymptotic eigenvalue $\mu_{n}^{+}$ of ${\widetilde{u}}_{n}$ at
the positive puncture satisfies $\mu^{+}_{n}<-\delta$. Moreover, from Lemma
5.7 it follows that

(5.17)
$\limsup_{\tau\to+\infty}\left(\sup_{n}\|D^{\beta}M_{n}(\tau,\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}\right)<\infty\qquad\forall\beta\,.$

This and the fact that $z_{n}$ belongs to $C^{r,\alpha,-\delta}_{0}$ together
imply that $\zeta_{n}\in C^{r,\alpha,-\delta}_{0}$. Note that for each $c$ the
operator $-i\partial_{t}-M^{\infty}(t+c)$ is representation of $A^{+}$ in a
suitable frame and, consequently, does not have $-\delta$ in the spectrum.
Finally, since $z_{n}\to z$ in $C^{\infty}_{\rm loc}$, we get

$\zeta_{n}\to\zeta:=M(s,t)z(s,t)\quad\text{in}\quad C^{\infty}_{\rm loc}\,.$

Consequently, $\zeta_{n}$ is $C^{\infty}_{\rm loc}$-bounded. We have checked
all assumptions of Proposition 5.11, which implies $\zeta\in
C^{r,\alpha,-\delta}_{0}$ and $\zeta_{n}\to\zeta$ in
$C^{r,\alpha,-\delta}_{0}$. With (5.17) this gives that $z(s,t)\in
C^{r,\alpha,-\delta}_{0}$ and that $z_{n}\to z\text{ in
}C^{r,\alpha,-\delta}_{0}$. Since $r\in\mathbb{N}$ is arbitrary, we conclude
the proof of the following statement.

######  Lemma 5.12.

We have:

$\displaystyle\lim_{\tau\to+\infty}e^{\delta\tau}\left(\|D^{\beta}z(\tau,\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}+\sup_{n}\|D^{\beta}z_{n}(\tau,\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}\right)=0\qquad\forall\beta\,.$

Since each ${\widetilde{u}}_{n}$ has an asymptotic formula, we can find
constants $c_{n},d_{n}\in\mathbb{R}$ such that the function
$V_{n}:[\underline{s},+\infty)\times\mathbb{R}/\mathbb{Z}\to\mathbb{R}^{2}$
defined by

$V_{n}(s,t)=\begin{pmatrix}a_{n}(s,t)/T^{+}_{n}-s-c_{n}\\\
\tilde{\theta}_{n}(s,t)/k_{+}-t-d_{n}\end{pmatrix}$

satisfies $\|V_{n}(s,\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}\to 0$ as
$s\to+\infty$, for each $n$. Equations (5.10) can be written as

(5.18)
$\partial_{s}V_{n}(s,t)+J_{0}\partial_{t}V_{n}(s,t)+B_{n}(s,t)z_{n}(s,t)=0$

where $B_{n}(s,t)$ is a $2\times 2$ matrix-valued function whose entries
consist of smooth functions defined on $\mathbb{R}/j_{+}\mathbb{Z}\times B$
composed with the map $(s,t)\mapsto(\theta_{n}(s,t),z_{n}(s,t))$. From the
bounds given by Lemma 5.7 we get

$\limsup_{\tau\to+\infty}\
\sup_{n}\|D^{\beta}B_{n}(\tau,\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}<\infty\qquad\forall\beta\,.$

Together with Lemma 5.12 this yields

(5.19) $\limsup_{\tau\to+\infty}\
\sup_{n}e^{\delta\tau}\|D^{\beta}[B_{n}z_{n}](\tau,\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}=0\qquad\forall\beta\,.$

It follows as in [28, Lemma 3.36], which is a version of [25, Lemma 6.3] for
sequences, that one can use (5.18) and (5.19) to find $b>0$ such that

(5.20) $\lim_{\tau\to+\infty}\
\sup_{n}e^{b\tau}\|D^{\beta}V_{n}(\tau,\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}=0\quad\text{if}\quad|\beta|\geq
1\,.$

With $\beta=(1,0)$ we get

$\lim_{\tau\to+\infty}\
\sup_{n}e^{b\tau}\left(\|\partial_{s}a_{n}(\tau,\cdot)-T_{n}^{+}\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}+\|\partial_{s}\tilde{\theta}_{n}(\tau,\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}\right)=0$

Taking the limit as $n\to\infty$

$\lim_{\tau\to+\infty}e^{b\tau}\left(\|\partial_{s}a(\tau,\cdot)-T^{+}\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}+\|\partial_{s}\tilde{\theta}(\tau,\cdot)\|_{L^{\infty}(\mathbb{R}/\mathbb{Z})}\right)=0\,.$

It follows from this exponential decay that the integrals

$\displaystyle
c=a(\underline{s},0)-T^{+}\underline{s}+\int_{\underline{s}}^{+\infty}\partial_{s}a(s,0)-T^{+}\
ds$ $\displaystyle
d=\tilde{\theta}(\underline{s},0)+\int_{\underline{s}}^{+\infty}\partial_{s}\tilde{\theta}(s,0)\
ds$

are convergent, and

$a(s,0)-T^{+}s-c\to 0\qquad\tilde{\theta}(s,0)-d\to 0\,.$

Together with Lemma 5.2 and Lemma 5.12, this concludes the proof that
$s=+\infty$ is a nondegenerate puncture of ${\widetilde{u}}$ in the sense of
Definition 4.7, in case $\gamma^{+}$ is a degenerate periodic orbit. The case
where $\gamma^{+}$ is nondegenerate was taken care by Corollary 5.5. The
analysis at the puncture $s=-\infty$ is identical. An application of Theorem
4.8 concludes the proof of (a) in Proposition 5.1.

###  5.4. Asymptotic data of the cylinder ${\widetilde{u}}$

Having proved (a), we now prove (b) and (c) in Proposition 5.1.

######  Lemma 5.13.

Choose symplectic trivializations of $\xi$ along $\gamma^{+},\gamma^{-}$ that
are homotopic to symplectic trivializations of $\xi$ along
$\gamma^{+}_{n},\gamma^{-}_{n}$, respectively. In these trivializations, if
$n$ is large enough then

$\displaystyle{\mathrm{wind}}(\nu^{<-\delta}(\gamma^{+}))$
$\displaystyle={\mathrm{wind}}(\nu^{<-\delta}(\gamma^{+}_{n}))$
$\displaystyle{\mathrm{wind}}(\nu^{\geq\delta}(\gamma^{-}))$
$\displaystyle={\mathrm{wind}}(\nu^{\geq\delta}(\gamma^{-}_{n}))\,.$

Moreover, if $n$ is large enough then

$p^{-\delta}(\gamma^{+})=p^{-\delta}(\gamma^{+}_{n})=1\,,\qquad
p^{\delta}(\gamma^{-})=p^{\delta}(\gamma^{-}_{n})=1\,.$

###### Proof.

We only prove the statements for $\gamma^{+},\gamma_{n}^{+}$, the
corresponding statements for $\gamma^{-},\gamma_{n}^{-}$ can be proved
analogously. Represent asymptotic operators $A^{+}_{n},A^{+}$ in the
coordinates given by Lemma 5.6, with respect to the frame
$\\{\partial_{x_{1}},\partial_{x_{2}}\\}$ of $\xi$ along
$\mathbb{R}/j_{+}\mathbb{Z}\times\\{0\\}$. By (b) and (c) of Lemma 5.6,
$A^{+}_{n}$ gets represented as $-j_{n}(t)\partial_{t}-B_{n}(t)$, $A^{+}$ gets
represented as $-j(t)\partial_{t}-B(t)$, where $j_{n}\to j$ and $B_{n}\to B$
in $C^{\infty}$. Hence, the spectrum of $A^{+}_{n}$ is a small perturbation of
the spectrum of $A^{+}$, and winding numbers of nearby eigenvalues in any
given frame coincide. In particular, $\sigma(A^{+}_{n})$ is bounded away from
$-\delta$, $\nu^{<-\delta}(A^{+}_{n})\to\alpha\in\sigma(A^{+})$,
$\nu^{\geq-\delta}(A^{+}_{n})\to\beta\in\sigma(A^{+})$ with $\alpha<-\delta$
and $\beta>-\delta$. We used that, by (G), $-\delta\not\in\sigma(A^{+})$. If
$\alpha<\nu^{<-\delta}(A^{+})$ then there is an eigenvalue of $A^{+}$ in
$(\alpha,-\delta)$, and for $n$ large enough there is an eigenvalue of
$A^{+}_{n}$ in $(\nu^{<-\delta}(A^{+}_{n}),-\delta)$ which is impossible. This
shows that $\alpha=\nu^{<-\delta}(A^{+})$. Similarly, one proves that
$\beta=\nu^{\geq-\delta}(A^{+})$. Hence

$n\gg 1\qquad\Rightarrow\qquad\begin{aligned}
&{\mathrm{wind}}(\nu^{<-\delta}(A^{+}))={\mathrm{wind}}(\nu^{<-\delta}(A^{+}_{n}))\\\
&{\mathrm{wind}}(\nu^{\geq-\delta}(A^{+}))={\mathrm{wind}}(\nu^{\geq-\delta}(A^{+}_{n}))\end{aligned}$

from where it follows that $n\gg 1\Rightarrow
p^{-\delta}(\gamma^{+})=p^{-\delta}(\gamma^{+}_{n})$. ∎

######  Corollary 5.14.

${\mathrm{ind}}_{\delta}({\widetilde{u}})={\mathrm{ind}}_{\delta}({\widetilde{u}}_{n})=2$.

The above lemma and corollary give (b) in Proposition 5.1. Let $\mu^{+}$ be
the asymptotic eigenvalue of ${\widetilde{u}}$ at $s=+\infty$, and let
$\mu^{-}$ be the asymptotic eigenvalue of ${\widetilde{u}}$ at $s=-\infty$.
These exist since we prove in (a) that ${\widetilde{u}}$ has a nontrivial
asymptotic formula at both punctures. Apply Lemma 4.9 to the curve
${\widetilde{u}}$ to get
${\mathrm{wind}}(\mu^{+})={\mathrm{wind}}(\nu^{<-\delta}(\gamma^{+}))$ in some
(hence any) trivialization along $\gamma^{+}$, and
${\mathrm{wind}}(\mu^{-})={\mathrm{wind}}(\nu^{\geq\delta}(\gamma^{-}))$ in
some (hence any) trivialization along $\gamma^{-}$. This proves (c).

To conclude, we need to prove (d), and we now do this. Suppose, by
contradiction, that $\exists(s_{0},t_{0})$ such that $u(s_{0},t_{0})\in
x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R})$. For simplicity assume
$u(s_{0},t_{0})\in x^{+}(\mathbb{R})$, the case $u(s_{0},t_{0})\in
x^{-}(\mathbb{R})$ is handled analogously. By (a), (b) and (c) we can apply
Lemma 4.9 to conclude that $u$ is an immersion transverse to $R$. We find
$0<\epsilon<1/2$ and a compact neighborhood $K$ of $x^{+}(\mathbb{R})$ such
that, if we denote by $\Delta_{\epsilon}$ the closed $\epsilon$-disk centered
at $(s_{0},t_{0})$, then the following holds:

  * (i)

$u(\partial\Delta_{\epsilon})\cap K=\emptyset$.

  * (ii)

$\exists n_{0}\in\mathbb{N}$ such that $n\geq n_{0}\Rightarrow
x_{n}(\mathbb{R})\subset K$.

  * (iii)

$\exists n_{1},N\in\mathbb{N}$ such that $n\geq n_{1}\Rightarrow 0<{\rm
int}(x_{n}^{+}(T^{+}_{n}\cdot),u|_{\Delta_{\epsilon}})\leq N$.

  * (iv)

$\exists n_{2}\in\mathbb{N}$ such that if $n\geq n_{2}$ then
$u_{n}|_{\partial\Delta_{\epsilon}}$ is homotopic to
$u|_{\partial\Delta_{\epsilon}}$ in $Y\setminus K$.

These properties follows from the transversality between $u$ and $R$, and from
assumptions (A) and (B). From homotopy invariance of intersection numbers we
get

$n\geq\max\\{n_{0},n_{1},n_{2}\\}\Rightarrow{\rm
int}(u_{n}|_{\Delta_{\epsilon}},x_{n}^{+}(T^{+}_{n}\cdot))={\rm
int}(u|_{\Delta_{\epsilon}},x_{n}^{+}(T^{+}_{n}\cdot))>0\,.$

It follows that $u_{n}(\mathbb{R}\times\mathbb{R}/\mathbb{Z})$ intersects
$x^{+}_{n}(\mathbb{R})$, in contradiction to (D). We proved that

$u(\mathbb{R}\times\mathbb{R}/\mathbb{Z})\cap(x^{+}(\mathbb{R})\cup
x^{-}(\mathbb{R}))=\emptyset\,.$

Now let $\gamma=(x,T)$ be a Reeb orbit of $\lambda$ in
$Y\setminus(x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R}))$ that satisfies $T\leq
T_{+}$. Choose a compact neighborhood $K$ of $x^{+}(\mathbb{R})\cup
x^{-}(\mathbb{R})$ such that $x(\mathbb{R})\cap K=\emptyset$. By (B) we find
$n_{0}\in\mathbb{N}$ such that

$n\geq n_{0}\Rightarrow x^{+}_{n}(\mathbb{R})\cup x^{-}_{n}(\mathbb{R})\subset
K\,.$

By Corollary 5.4 we find $h>0$ such that

$|s|\geq h\Rightarrow u(\\{s\\}\times\mathbb{R}/\mathbb{Z})\subset K\
\text{and}\ u_{n}(\\{s\\}\times\mathbb{R}/\mathbb{Z})\subset K\ \forall n\,.$

By (C), $\gamma=(x,T)$ is a Reeb orbit of $\lambda_{n}$. By (D), ${\rm
int}(x(T\cdot),u_{n})>0$ for all $n$. We find $(s_{n},t_{n})$ such that
$u_{n}(s_{n},t_{n})\in x(\mathbb{R})$. We get $|s_{n}|<h\ \forall n$ and, up
to a subsequence, we may assume $(s_{n},t_{n})\to(s_{*},t_{*})$. From (A) we
get $u(s_{*},t_{*})\in x(\mathbb{R})$. From the transversality between $u$ and
$R$ provided by Lemma 4.9 we get ${\rm int}(x(T\cdot),u)>0$, as desired.

Consider a loop $\beta\subset\mathbb{R}\times\mathbb{R}/\mathbb{Z}$. Since
$u(\mathbb{R}\times\mathbb{R}/\mathbb{Z})\cap(x^{+}(\mathbb{R})\cap
x^{-}(\mathbb{R}))=\emptyset$ we find a compact neighborhood $F$ of
$u(\beta)$, a compact neighborhood $K$ of of $x^{+}(\mathbb{R})\cap
x^{-}(\mathbb{R})$ and $n_{0}\in\mathbb{N}$ such that $F\cap K=\emptyset$ and

$n\geq n_{0}\Rightarrow\text{$u_{n}(\beta)\subset F$ and $u_{n}\circ\beta$ is
homotopic to $u\circ\beta$ in $F$.}$

Here we used (A). By Corollary 5.4 we find $h>0$ such that

$|s|\geq h\Rightarrow
u(\\{s\\}\times\mathbb{R}/\mathbb{Z})\subset\mathring{K}\ \text{and}\
u_{n}(\\{s\\}\times\mathbb{R}/\mathbb{Z})\subset\mathring{K}\ \forall n\,.$

Hence, there exists $n_{1}\in\mathbb{N}$ such that

$n\geq n_{1}\Rightarrow\text{$u_{n}(\pm h,\cdot)$ is homotopic to $u(\pm
h,\cdot)$ in $K$.}$

Using the homotopy invariance of intersection numbers

$\displaystyle n\geq\max\\{n_{0},n_{1}\\}\quad\Rightarrow\quad{\rm
int}(u\circ\beta,u)$ $\displaystyle={\rm
int}(u\circ\beta,u|_{[-h,h]\times\mathbb{R}/\mathbb{Z}})$ $\displaystyle={\rm
int}(u_{n}\circ\beta,u_{n}|_{[-h,h]\times\mathbb{R}/\mathbb{Z}})$
$\displaystyle=0$

as desired. The proof of Proposition 5.1 is completed.

###  5.5. Compactness

Before moving on, we also prove the compactness statement we will need.
Continue with the notation $g_{n},\lambda_{n},\lambda,J_{n},J$ as above, and
consider nonconstant finite-energy ${\widetilde{J}}_{n}$-holomorphic maps
${\widetilde{u}}_{n}=(a_{n},u_{n})$ as in (5.1).

######  Lemma 5.15.

Assume that hypotheses ($*$), (B) and (E) hold. There exist sequences
$n_{j}\in\mathbb{N}$, $c_{j}\in\mathbb{R}$,
$(s_{j},t_{j})\in\mathbb{R}\times\mathbb{R}/\mathbb{Z}$ such that
$n_{j}\to\infty$ and the sequence

$(s,t)\mapsto(a_{n_{j}}(s+s_{j},t+t_{j})+c_{j},u_{j}(s+s_{j},t+t_{j}))$

is $C^{\infty}_{\rm loc}$-convergent to a nonconstant finite-energy
${\widetilde{J}}$-holomorphic map

(5.21)
${\widetilde{u}}=(a,u):\mathbb{R}\times\mathbb{R}/\mathbb{Z}\to\mathbb{R}\times
Y$

with a positive puncture at $s=+\infty$ and a negative puncture at
$s=-\infty$. Moreover, if $\lambda$ has finitely many simple Reeb orbits then
$u(0,0)$ does not belong to a Reeb orbit of $\lambda$.

###### Proof.

From (B) we know that $\\{{\widetilde{u}}_{n}\\}$ has uniformly bounded Hofer
energy. From (E) it follows that $d{\widetilde{u}}_{n}$ is $C^{0}_{\rm
loc}$-bounded: otherwise we would find a nonconstant finite-energy
${\widetilde{J}}$-holomorphic plane with total $d\lambda$-area strictly less
than $T_{\min}(\lambda)$, which is impossible. Define
$\widetilde{v}_{n}=(a_{n}-a_{n}(0,0),u_{n})$. Then $\widetilde{v}_{n}$ is
$C^{1}_{\rm loc}$-bounded. Elliptic boot-strapping implies $C^{\infty}_{\rm
loc}$-bounds. Hence there is a $C^{\infty}_{\rm loc}$-convergent subsequence
$\widetilde{v}_{n_{j}}\to{\widetilde{u}}=(a,u):\mathbb{R}\times\mathbb{R}/\mathbb{Z}\to\mathbb{R}\times
Y$ to a ${\widetilde{J}}$-holomorphic map with finite Hofer energy. Since

$T^{-}_{n_{j}}\leq\int_{\mathbb{R}/\mathbb{Z}}u_{n_{j}}(s,\cdot)^{*}\lambda_{n_{j}}\leq
T^{+}_{n_{j}}\quad\Rightarrow\quad
T^{-}\leq\int_{\mathbb{R}/\mathbb{Z}}u(s,\cdot)^{*}\lambda\leq
T^{+}\quad\forall s$

one gets that ${\widetilde{u}}$ is nonconstant, has a positive puncture at
$s=+\infty$ and a negative puncture at $s=-\infty$.

Assume that $\lambda$ has finitely many simple Reeb orbits. From this, (B) and
(E) we get $0<T^{+}-T^{-}<T_{\min}(\lambda)$. Hence $\gamma^{+},\gamma^{-}$
are geometrically distinct. By (B), the distance between $\gamma^{+}_{n}$ and
$\gamma^{-}_{n}$ is bounded away from zero, uniformly in $n\gg 1$. Hence, one
finds $(s_{n},t_{n})$ such that $u_{n}(s_{n},t_{n})$ converges to a point in
the complement of the set of Reeb orbits of $\lambda$. One can now run the
previous argument with ${\widetilde{u}}_{n}(s+s_{n},t+t_{n})$ in the place of
${\widetilde{u}}_{n}$ to get the desired conclusion. ∎

##  6\. Two or infinitely many simple Reeb orbits

We now put everything together to prove our main theorem. To prove our
theorem, we can assume that there are finitely many simple Reeb orbits, and
this will be our standing assumption throughout this section.

Recall the contact structure $\xi=\ker\lambda$. The following lemma collects
what we need to know from Proposition 3.1.

######  Lemma 6.1.

There exists a $d\lambda$-compatible complex structure $J:\xi\to\xi$ and
sequences $g_{n},J_{n},{\widetilde{u}}_{n}$ with the following properties.

  1. (a)

$g_{n}\in C^{\infty}(Y)$, $g_{n}\to 1$ in $C^{\infty}$,
$\lambda_{n}=g_{n}\lambda$ is nondegenerate $\forall n$.

  2. (b)

$J_{n}:\xi\to\xi$ are $d\lambda_{n}$-compatible (equivalently
$d\lambda$-compatible) and $J_{n}\to J$ in $C^{\infty}$.

  3. (c)

${\widetilde{u}}_{n}=(a_{n},u_{n}):\mathbb{R}\times\mathbb{R}/\mathbb{Z}\to\mathbb{R}\times
Y$ are non-constant finite-energy ${\widetilde{J}}_{n}$-holomorphic maps, with
a positive puncture at $s=+\infty$ and a negative puncture at $s=-\infty$.

  4. (d)

The asymptotic limits $\gamma^{\pm}_{n}=(x^{\pm}_{n},T^{\pm}_{n})$ of
${\widetilde{u}}_{n}$ at $s=\pm\infty$ satisfy $\sup_{n}T^{+}_{n}<+\infty$,
and $r\leq T^{+}_{n}-T^{-}_{n}\leq\frac{1}{2}T_{\min}(\lambda)$ for some
$r>0$. Moreover, there are Reeb orbits $\gamma^{\pm}=(x^{\pm},T^{\pm})$ of
$\lambda$ such that $\gamma^{\pm}_{n}\to\gamma^{\pm}$, in the sense that
$x^{\pm}_{n}(T^{\pm}_{n}(\cdot+t^{\pm}_{n}))\to x^{\pm}(T^{\pm}\cdot)$ in
$C^{\infty}(\mathbb{R}/\mathbb{Z},Y)$ for suitable
$t^{\pm}_{n}\in\mathbb{R}/\mathbb{Z}$.

  5. (e)

If $x:\mathbb{R}\to Y$ is a periodic $\lambda$-Reeb trajectory with period at
most $T^{+}$ then $g_{n}|_{x(\mathbb{R})}\equiv 1$,
$dg_{n}|_{x(\mathbb{R})}\equiv 0$ $\forall n$.

  6. (f)

For every $n$, ${\widetilde{u}}_{n}(\mathbb{R}\times\mathbb{R}/\mathbb{Z})$
defines a Birkhoff section for the Reeb flow of $\lambda_{n}$. In particular,
$u_{n}$ is a proper embedding into $Y\setminus(x^{+}_{n}(\mathbb{R})\cup
x^{-}_{n}(\mathbb{R}))$ transverse to the Reeb vector field of $\lambda_{n}$.

  7. (g)

Denote asymptotic operators by $A^{\pm}_{n}=A_{\gamma^{\pm}_{n}}$,
$A^{\pm}=A_{\gamma^{\pm}}$, and asymptotic eigenvalues of
${\widetilde{u}}_{n}$ at $s=\pm\infty$ by
$\mu^{\pm}_{n}\in\sigma(A^{\pm}_{n})$. For some $\delta>0$ we have
$[-\delta,0)\cap\sigma(A^{+})=\emptyset$,
$(0,\delta]\cap\sigma(A^{-})=\emptyset$,
${\mathrm{ind}}_{\delta}({\widetilde{u}}_{n})=2$, $\mu^{+}_{n}<-\delta$,
$\mu^{-}_{n}>\delta$,
$p^{-\delta}(\gamma_{n}^{+})=p^{\delta}(\gamma_{n}^{-})=1$ $\forall n$.

Before giving the proof, a couple of remarks are in order. The almost complex
structures ${\widetilde{J}}_{n}$ are constructed from $J_{n},\lambda_{n}$ as
in Section 4. By (a), the cylinders ${\widetilde{u}}_{n}$ have an asymptotic
formula at both punctures. By (d), $\gamma^{+}_{n}$ is geometrically distinct
from $\gamma^{-}_{n}$, and $\gamma^{+}$ is geometrically distinct from
$\gamma^{-}$ for all $n$ large enough. Hence, each ${\widetilde{u}}_{n}$ is
not a trivial cylinder and has a non-trivial asymptotic formula at both
punctures when $n$ is large enough. This allows one to consider asymptotic
eigenvalues in (g).

###### Proof of Lemma 6.1.

This follows from Proposition 3.1; all but $(g)$ are essentially immediate so
we only explain (g). The reasoning to obtain (g) from Proposition 3.1 is as
follows. We analyze only the positive puncture, the analysis of the negative
puncture is left to the reader. Consider a trivialization of $\xi$ along the
simple Reeb orbit underlying $\gamma^{+}$. This trivialization can be
iterated, thus inducing a trivialization of $\xi$ along $\gamma^{+}$, and also
homotopy classes of trivializations of $\xi$ along $\gamma^{+}_{n}$ for all
$n$ large enough. In the argument that follows, winding numbers and rotation
numbers are computed with respect to these trivializations. By the asymptotic
formula for ${\widetilde{u}}_{n}$ at both punctures, the winding number
${\mathrm{wind}}(\mu^{+}_{n})$ of $\mu^{+}_{n}$ is smaller than the rotation
number $\theta_{n}$ of $\gamma^{+}_{n}$. Note that $\theta_{n}\to\theta$ where
$\theta$ is the rotation number of $\gamma^{+}$. Moreover, up to a
subsequence, there exists an eigenvalue of $A^{+}$ such that
$\mu^{+}_{n}\to\mu$. From $\mu^{+}_{n}<0$ we get $\mu\leq 0$. If $\gamma^{+}$
is nondegenerate then $\theta\not\in\mathbb{Z}$, $\mu<0$, and existence of the
gap $[-\delta,0)$ in the spectrum of $A^{+}$ for some $0<\delta<-\mu$ follows.
If $\gamma^{+}$ is degenerate then $\theta\in\mathbb{Z}$ is the winding number
${\mathrm{wind}}(0)$ of the eigenvalue $0$ of $A^{+}$. In this case
Proposition 3.1 tells us that $\theta_{n}-\theta<0$, and from
${\mathrm{wind}}(\mu^{+}_{n})<\theta_{n}$ we get
${\mathrm{wind}}(\mu^{+}_{n})\leq\theta-1$. From this we can conclude that
$\mu<0$, since otherwise ${\mathrm{wind}}(\mu^{+}_{n})$ would converge to
${\mathrm{wind}}(0)=\theta$ and ${\mathrm{wind}}(\mu^{+}_{n})$ would be equal
to the integer $\theta$ for $n$ large enough. Again, existence of the gap
$[-\delta,0)$ in the spectrum of $A^{+}$ for some $0<\delta<-\mu$ follows. The
only statement left to be proved for (g) is
$p^{-\delta}(\gamma_{n}^{+})=p^{\delta}(\gamma_{n}^{-})=1$. The cylinders
given by Proposition 3.1 have high multiplicity ends; this follows from Lemma
3.11 as explained in Part 2 of the proof of Proposition 3.1. Since these
cylinders are part of a curve counted by the U-map in ECH, and such a curve
connects orbit sets that are ECH generators and can only have hyperbolic
orbits with multiplicity one, we get that $\gamma^{+}_{n}$ is elliptic. ∎

Recall the moduli space

$\mathcal{M}=\mathcal{M}_{J,0}(\gamma^{+},\gamma^{-})$

defined in Section 4.2. It is crucial to keep in mind that, by definition,
finite-energy maps representing curves in $\mathcal{M}$ have nondegenerate
punctures. Hence, by Theorem 4.8, these maps have an asymptotic formula at
both punctures. Since $\gamma^{+}$ is geometrically distinct from
$\gamma^{-}$, curves in $\mathcal{M}$ are not trivial cylinders, and we
further get a non-trivial asymptotic formula at both punctures. In particular,
at the positive puncture of any curve in $\mathcal{M}$ there is an asymptotic
eigenvalue in $(-\infty,-\delta)$, and at the negative puncture there is an
asymptotic eigenvalue in $(\delta,+\infty)$. In what follows $\mathcal{M}$ is
equipped with the quotient topology induced by the $C^{\infty}_{\rm
loc}$-topology on the space of smooth maps
$\mathbb{R}\times\mathbb{R}/\mathbb{Z}\to\mathbb{R}\times Y$.

The following is a standard fact we will need:

######  Lemma 6.2.

Let $C_{n}\in\mathcal{M}$ be a convergent sequence. Let
$\widetilde{v}_{n}=(b_{n},v_{n})$ be representatives of $C_{n}$ that
$C^{\infty}_{\rm loc}$-converge to a representative of the limiting curve.
Then, for every neighborhood $N$ of $x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R})$
in $Y$ we find $s_{*}>0$ and $n_{*}$ such that if $|s|\geq s_{*}$ and $n\geq
n_{*}$ then $v_{n}(\\{s\\}\times\mathbb{R}/\mathbb{Z})\subset N$.

###### Proof.

The proof follows a standard recipe: if not then we would find
$n_{j}\to\infty$, $s_{j}$ with $|s_{j}|\to+\infty$, and
$t_{j}\in\mathbb{R}/\mathbb{Z}$ such that $v_{n_{j}}(s_{j},t_{j})$ converges
to a point in the complement of the union of the finitely many simple Reeb
orbits of $\lambda$. Reparametrizing to center the maps at $(s_{j},t_{j})$,
and using the fact no bubbling-off points can arise due to low
$d\lambda$-area, we would obtain a cylinder with zero $d\lambda$-area through
a point not in any periodic orbit, which is impossible. ∎

With the above preliminaries behind us, we can now prove the crucial fact that
the moduli space $\mathcal{M}$ has a compact component; we prove this, along
with a few other useful facts, in the following lemma.

######  Lemma 6.3.

The space $\mathcal{M}$ has a compact connected nonempty component
$\mathcal{Y}$ which is a smooth $1$-manifold whose elements are represented by
maps
${\widetilde{u}}=(a,u):\mathbb{R}\times\mathbb{R}/\mathbb{Z}\to\mathbb{R}\times
Y$ that are $\tilde{J}$-holomorphic and have the following properties:

  * (i)

$u$ is an immersion transverse to the Reeb vector field of $\lambda$.

  * (ii)

The map ${\widetilde{u}}$ has a non-trivial asymptotic formula at both
punctures, is asymptotic to $\gamma^{+}$ at $s=+\infty$, and to $\gamma^{-}$
at $s=-\infty$. The asymptotic eigenvalue $\mu^{+}$ at $+\infty$ satisfies

$\mu^{+}<-\delta\qquad{\mathrm{wind}}(\mu^{+})={\mathrm{wind}}(\nu^{<-\delta}(\gamma^{+}))$

in some (hence any) trivialization along $\gamma^{+}$. The asymptotic
eigenvalue $\mu^{-}$ at $-\infty$ satisfies

$\mu^{-}>\delta\qquad{\mathrm{wind}}(\mu^{-})={\mathrm{wind}}(\nu^{\geq\delta}(\gamma^{-}))$

in some (hence any) trivialization along $\gamma^{-}$.

  * (iii)

${\rm ind}_{\delta}({\widetilde{u}})=2$,
$p^{-\delta}(\gamma^{+})=p^{\delta}(\gamma^{-})=1$.

###### Proof.

In the course of the proof, it will be helpful to establish the following two
additional properties:

  * (iv)

$u(\mathbb{R}\times\mathbb{R}/\mathbb{Z})\subset
Y\setminus(x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R}))$ and every Reeb orbit of
$\lambda$ of period $\leq T^{+}$ has strictly positive intersection number
with $u$.

  * (v)

If $\beta$ is a loop in $\mathbb{R}\times\mathbb{R}/\mathbb{Z}$ then
$u\circ\beta$ has intersection number zero with $u$.

We now explain why all of (i) - (v) hold.

The curves ${\widetilde{u}}_{n}$ satisfy assumptions ($*$), (B), (C), (D), (E)
and (G) from Section 5. Lemma 5.15 implies that $\\{{\widetilde{u}}_{n}\\}$ is
$C^{\infty}_{\rm loc}$-convergent to a ${\widetilde{J}}$-holomorphic finite-
energy map ${\widetilde{u}}=(a,u)$ in (5.21) with a positive puncture at
$s=+\infty$ and a negative puncture at $s=-\infty$, and that $u(0,0)$ does not
belong to a Reeb orbit of $\lambda$. This shows that also hypotheses (A) and
(F) from Section 5 are satisfied. Proposition 5.1 implies that all properties
(i)-(v) as in the statement hold for the map ${\widetilde{u}}$. In particular,
${\widetilde{u}}$ represents a curve in $\mathcal{M}$.

We denote by $\mathcal{Y}$ the connected component of $\mathcal{M}$ containing
the curve represented by the map ${\widetilde{u}}$ obtained above. Properties
(iv)-(v) are shared by all curves in $\mathcal{Y}$; this follows from the
definition of $\mathcal{M}$ and from Lemma 4.9. Lemma 4.12 gives $\mathcal{Y}$
the structure of a smooth $1$-dimensional manifold when equipped with a
topology induced from the functional analytic set-up used in [22] to define
the Fredholm theory discussed in Section 4.2. It follows essentially from
Proposition 5.11 that the topology on $\mathcal{M}$ agrees with the quotient
topology induced from the $C^{\infty}_{{\rm loc}}$-topology. Thus
$\mathcal{Y}$ is a smooth $1$-manifold with the latter topology.

We will show that $\mathcal{Y}$ is compact. To this end we will first show
that every curve in $\mathcal{Y}$ is represented by a map
$\widetilde{v}=(b,v)$ satisfying properties (i)-(v). We already know this to
be the case for properties (iv)-(v). Lemma 4.9 gives (i) for all curves in
$\mathcal{Y}$.

We claim that every curve in $\mathcal{Y}$ projects to
$Y\setminus(x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R}))$. To this end, consider
the set $\mathcal{Z}\subset\mathcal{Y}$ of curves that project to
$Y\setminus(x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R}))$. We will show that
$\mathcal{Z}$ is open and closed.

Let us first show that $\mathcal{Z}$ is open. Let $C\in\mathcal{Z}$ be
represented by a map $\widetilde{v}=(b,v)$ such that
$v(\mathbb{R}\times\mathbb{R}/\mathbb{Z})$ does not intersect
$x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R})$. Suppose, by contradiction, that
there exists $C_{n}\in\mathcal{Y}$, $C_{n}\to C$ represented by maps
$\widetilde{v}_{n}=(b_{n},v_{n})$, and $(s_{n},t_{n})$ for which we have that
$v_{n}(s_{n},t_{n})\in x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R})$. Up to
translating in the $\mathbb{R}$-direction and reparametrizing, we can assume
$\widetilde{v}_{n}\to\widetilde{v}$ in $C^{\infty}_{\rm loc}$. Up to a
subsequence, we can further assume that one of the following holds:
$v_{n}(s_{n},t_{n})\in x^{+}(\mathbb{R})\ \forall n$ or $v_{n}(s_{n},t_{n})\in
x^{-}(\mathbb{R})\ \forall n$. Assume $v_{n}(s_{n},t_{n})\in
x^{+}(\mathbb{R})\ \forall n$, the other case is handled analogously. We claim
that $\inf_{n}s_{n}>-\infty$. If not, since $\gamma^{+}$ and $\gamma^{-}$ are
geometrically distinct, we find $s^{\prime}_{n}\to-\infty$, $t^{\prime}_{n}$
such that $v_{n}(s^{\prime}_{n},t^{\prime}_{n})\to p$ for some $p\in
Y\setminus(x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R}))$. This contradicts Lemma
6.2. Now we claim that $\sup_{n}s_{n}<+\infty$. Consider a tubular
neighborhood $N^{+}$ of $x^{+}(\mathbb{R})$ equipped with coordinates
$N^{+}\simeq\mathbb{R}/\mathbb{Z}\times\mathbb{C}$ such that
$x^{+}(T^{+}t)\simeq(t,0)$. We can define the linking number, relative to the
coordinates on $N^{+}$, of a loop $(c_{1}(t),c_{2}(t))$ in
$\mathbb{R}/\mathbb{Z}\times(\mathbb{C}\setminus\\{0\\})\simeq N^{+}\setminus
x^{+}(\mathbb{R})$ with $x^{+}(T^{+}\cdot)$ as the winding number of
$c_{2}(t)$. Since $\widetilde{v}$ has a non-trivial asymptotic formula at
$s=+\infty$, we find $h>0$ such that if $s\geq h$ then the loop $v(s,\cdot)$
lies in $N^{+}\setminus x^{+}(\mathbb{R})$ and its linking number is
${\mathrm{wind}}(\nu^{\leq-\delta}(\gamma^{+}))$, where the latter winding is
computed in a trivialization along $\gamma^{+}$ induced from the coordinates
on $N^{+}$. Since $\widetilde{v}_{n}\to\widetilde{v}$, we can find $n_{0}$
such that if $n\geq n_{0}$ then $v_{n}(h,\cdot)$ lies in $N^{+}\setminus
x^{+}(\mathbb{R})$ and has linking number
${\mathrm{wind}}(\nu^{\leq-\delta}(\gamma^{+}))$. However, if
$\sup_{n}s_{n}=+\infty$ then we can assume, up to selection of a subsequence,
that $s_{n}\to+\infty$, in particular there is $n_{1}$ such that $n\geq n_{1}$
implies that $s_{n}>h$. By the non-trivial asymptotic formula for
$\widetilde{v}_{n}$ and properties (iv)-(v) for $\widetilde{v}_{n}$, we find
for each $n\geq n_{1}$ some $\tau_{n}>s_{n}$ such that $v_{n}(\tau_{n},\cdot)$
lies in $N^{+}\setminus x^{+}(\mathbb{R})$ and has linking number
${\mathrm{wind}}(\nu^{\leq-\delta}(\gamma^{+}))$. Summarizing, if
$n\geq\max\\{n_{0},n_{1}\\}$ then both loops $v_{n}(h,\cdot)$,
$v_{n}(\tau_{n},\cdot)$ lie in $N^{+}\setminus x^{+}(\mathbb{R})$ and have the
same linking number with $x^{+}(T^{+}\cdot)$. By positivity of intersections,
the intersection number of
$\widetilde{v}_{n}|_{[h,\tau_{n}]\times\mathbb{R}/\mathbb{Z}}$ with the
trivial cylinder over $x^{+}(\mathbb{R})$ is strictly positive, when
$n\geq\max\\{n_{0},n_{1}\\}$. It follows that $v_{n}(h,\cdot)$ and
$v_{n}(\tau_{n},\cdot)$ have different linking numbers with
$x^{+}(T^{+}\cdot)$. This contradiction concludes the proof that
$\sup_{n}s_{n}<+\infty$. We have now proved that $s_{n}$ is bounded. Up to a
subsequence, $(s_{n},t_{n})\to(s_{*},t_{*})$ and $v(s_{*},t_{*})\in
x^{+}(\mathbb{R})$, which is a contradiction the assumption on
$\widetilde{v}$. Hence, $\mathcal{Z}$ is open.

The complement of $\mathcal{Z}$ in $\mathcal{Y}$ is obviously open, by the
transversality between projections to $Y$ of curves in $\mathcal{Y}$ and the
Reeb vector field (Lemma 4.9). By connectedness, $\mathcal{Z}=\mathcal{Y}$. We
have proved that all curves in $\mathcal{Y}$ project to
$Y\setminus(x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R}))$, and satisfy properties
(i), (iv) and (v). Properties (iv) and (v) for representatives of all curves
in $\mathcal{Y}$ now follow from the fact that there is one curve in
$\mathcal{Y}$ represented by a map satisfying (i)-(v).

Finally, compactness of $\mathcal{Y}$ is a direct application of Lemma 5.15
and Proposition 5.1 to the constant sequences $g_{n}=1$, $J_{n}=J$, in view of
the properties obtained above for the maps representing curves in
$\mathcal{Y}$. ∎

By the uniformization theorem, $\mathcal{M}$ can be described as the set of
equivalence classes of finite-energy ${\widetilde{J}}$-holomorphic maps
${\widetilde{u}}:\mathbb{R}\times\mathbb{R}/\mathbb{Z}\to\mathbb{R}\times Y$
asymptotic to $\gamma^{+}$ at the positive puncture $s=+\infty$ and to
$\gamma^{-}$ at the negative puncture $s=-\infty$, with an asymptotic formula
at both punctures. Here $\mathbb{R}\times\mathbb{R}/\mathbb{Z}$ is equipped
with the conformal structure obtained by pulling back multiplication by $i$ in
$\mathbb{C}\setminus\\{0\\}$ via the map $(s,t)\mapsto e^{2\pi(s+it)}$. Two
such maps ${\widetilde{u}}_{0}=(a_{0},u_{0})$,
${\widetilde{u}}_{1}=(a_{1},u_{1})$ are declared equivalent if there exist
$(\Delta s,\Delta t)\in\mathbb{R}\times\mathbb{R}/\mathbb{Z}$,
$c\in\mathbb{R}$ such that

(6.1) $(a_{1}(s,t)+c,u_{1}(s,t))=(a_{0}(s+\Delta s,t+\Delta t),u_{0}(s+\Delta
s,t+\Delta t))\quad\forall(s,t)\,.$

The moduli space of cylinders with one marked point $\mathcal{M}^{(1)}$ is
defined as a set of equivalence classes of pairs $({\widetilde{u}},z)$ where
${\widetilde{u}}$ is a finite-energy cylinder that represents a curve in
$\mathcal{M}$ and $z\in\mathbb{R}\times\mathbb{R}/\mathbb{Z}$. Two pairs
$({\widetilde{u}}_{0},z_{0}=(s_{0},t_{0}))$,
$({\widetilde{u}}_{1},z_{1}=(s_{1},t_{1}))$ are declared equivalent if there
exist $(\Delta s,\Delta t)\in\mathbb{R}\times\mathbb{R}/\mathbb{Z}$,
$c\in\mathbb{R}$ such that (6.1) holds and $(s_{0},t_{0})=(s_{1}+\Delta
s,t_{1}+\Delta t)$. The equivalence class of $({\widetilde{u}},z)$ will be
denoted by $[{\widetilde{u}},z]$. There is a forgetful map

$F:\mathcal{M}^{(1)}\to\mathcal{M}\,,\qquad
F([{\widetilde{u}},z])=[{\widetilde{u}}]$

and an evaluation map

${\rm ev}:\mathcal{M}^{(1)}\to Y\,,\qquad{\rm
ev}([{\widetilde{u}}=(a,u),z])=u(z)\,.$

In the following we consider the connected component
$\mathcal{Y}\subset\mathcal{M}$ given by Lemma 6.3. It follows from this
proposition that

${\rm ev}(F^{-1}(\mathcal{Y}))\subset Y\setminus(x^{+}(\mathbb{R})\cup
x^{-}(\mathbb{R}))\,.$

Moreover, the automatic transversality from Lemma 4.12 and Lemma 6.3 together
guarantee that $F^{-1}(\mathcal{Y})$ has the structure of a smooth
$3$-manifold such that $F$ is a trivial fibration over $\mathcal{Y}$ with
fiber $\mathbb{R}\times\mathbb{R}/\mathbb{Z}$. Note also that $\mathcal{Y}$ is
diffeomorphic to the circle, since it is a compact connected smooth
$1$-manifold without boundary.

######  Lemma 6.4.

The map ${\rm ev}|_{F^{-1}(\mathcal{Y})}:F^{-1}(\mathcal{Y})\to
Y\setminus(x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R}))$ is a local diffeomorphism
and a proper map.

###### Proof.

First we prove that ${\rm ev}$ is a local diffeomorphism. Consider an
arbitrary element of $F^{-1}(\mathcal{Y})$, and represent it by a pair
$({\widetilde{u}}=(a,u),z_{0})$. The tangent space at ${\widetilde{u}}$ of the
corresponding space of holomorphic maps modulo reparametrization can be
identified with the $2$-dimensional vector space $\ker D_{\widetilde{u}}$,
where $D_{\widetilde{u}}$ is the linearized Cauchy-Riemann operator at
${\widetilde{u}}$ considered in Section 4.2. Differentiating the
$(\mathbb{R},+)$ action at ${\widetilde{u}}$ we get $\zeta_{0}\in\ker
D_{\widetilde{u}}$, $\zeta_{0}\neq 0$. The tangent space
$T_{[{\widetilde{u}}]}\mathcal{Y}$ of $\mathcal{Y}$ at
$[{\widetilde{u}}]=F([{\widetilde{u}},z_{0}])$ can be identified with $(\ker
D_{\widetilde{u}})/\mathbb{R}\zeta_{0}$. The proof of Lemma 4.12 shows that
sections in $\ker D_{{\widetilde{u}}}$ either vanish identically or are
nowhere vanishing. Any section $\zeta\in\ker D_{\widetilde{u}}$ has components
$\zeta=(\zeta^{\mathbb{R}},\zeta^{Y})$ according to the splitting
$T(\mathbb{R}\times Y)=T\mathbb{R}\times TY$. We claim that if
$z\in\mathbb{R}\times\mathbb{R}/\mathbb{Z}$, $\zeta\in\ker D_{\widetilde{u}}$,
$\zeta\neq 0$ and $\zeta^{Y}(z)\in{\rm im}\ du(z)$ then
$\zeta(z)=(\alpha,0)+d{\widetilde{u}}(z)v$ for some
$\alpha\in\mathbb{R}\setminus\\{0\\}$, $v\in
T_{z}(\mathbb{R}\times\mathbb{R}/\mathbb{Z})$. In fact, if
$\zeta^{Y}(z)=du(z)v$ for some $v$ then
$\zeta(z)=(\alpha,0)+d{\widetilde{u}}(z)v$ where
$\alpha=\zeta^{\mathbb{R}}(z)-da(z)v$. We must have $\alpha\neq 0$ since
$\zeta(z)\neq 0$ is in the normal bundle. The claim is proved. Let
$\zeta_{1}\in\ker D_{\widetilde{u}}$ be such that $\\{\zeta_{0},\zeta_{1}\\}$
is a basis of $\ker D_{\widetilde{u}}$. We claim that
$\zeta_{1}^{Y}(z)\not\in{\rm im}\ du(z)$ for every $z$. We argue indirectly
for this. Suppose we find $z_{*}$ such that $\zeta_{1}^{Y}(z_{*})\in{\rm im}\
du(z_{*})$. From the claim above $\zeta_{1}(z_{*})=(\alpha_{1},0)+V_{1}$ for
some $V_{1}\in{\rm im}\ d{\widetilde{u}}(z_{*})$ and some
$\alpha_{1}\in\mathbb{R}\setminus\\{0\\}$. Similarly, since $\zeta_{0}^{Y}(z)$
belongs to ${\rm im}\ du(z)$ for every $z$, we can write
$\zeta_{0}(z_{*})=(\alpha_{0},0)+V_{0}$ with $V_{0}\in{\rm im}\
d{\widetilde{u}}(z_{*})$ and $\alpha_{0}\in\mathbb{R}\setminus\\{0\\}$. It
follows that $\alpha_{0}\zeta_{1}(z_{*})-\alpha_{1}\zeta_{0}(z_{*})\in{\rm
im}\ d{\widetilde{u}}(z_{*})$, from where it follows that
$\alpha_{0}\zeta_{1}(z_{*})-\alpha_{1}\zeta_{0}(z_{*})=0$. Hence,
$\alpha_{0}\zeta_{1}-\alpha_{1}\zeta_{0}\equiv 0$ since it vanishes at one
point, and this is a contradiction. Surjectivity of the differential of ${\rm
ev}$ at $[{\widetilde{u}},z]$ follows since $T_{[{\widetilde{u}}]}\mathcal{Y}$
is spanned by $\zeta_{1}+\mathbb{R}\zeta_{0}$, and $u$ is an immersion. The
proof that ${\rm ev}$ is a local diffeomorphism is complete.

To prove that ${\rm ev}|_{F^{-1}(\mathcal{Y})}$ is a proper map into
$Y\setminus(x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R}))$, we need to consider a
sequence $[{\widetilde{u}}_{n}=(a_{n},u_{n}),z_{n}]\in\mathcal{Y}$ satisfying
$u_{n}(z_{n})\to p$, $p\not\in x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R})$, and
prove that some subsequence of $[{\widetilde{u}}_{n},z_{n}]$ converges in
$\mathcal{Y}$. From the compactness of $\mathcal{Y}$ given by Lemma 6.3 we can
assume, up to selection of a subsequence, that
$[{\widetilde{u}}_{n}]\to[{\widetilde{u}}]\in\mathcal{Y}$. Up to
reparametrizing and translating in the $\mathbb{R}$-direction, we can further
assume ${\widetilde{u}}_{n}\to{\widetilde{u}}$ in $C^{\infty}_{\rm loc}$. Let
$V$ be a closed neighborhood of $x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R})$ such
that $p\not\in V$. From Lemma 6.2 we find $\underline{s}>0$ such that
$|s|>\underline{s}\Rightarrow u_{n}(s,t)\in V\ \forall(n,t)$. Hence
$z_{n}=(s_{n},t_{n})$ satisfies $|s_{n}|\leq\underline{s}$ when $n\gg 1$ and,
up to a subsequence, we may further assume that $z_{n}\to z_{*}$ for some
$z_{*}\in\mathbb{R}\times\mathbb{R}/\mathbb{Z}$. Hence
$[{\widetilde{u}}_{n},z_{n}]\to[{\widetilde{u}},z_{*}]$. ∎

We can now put all of this together to prove the main theorem of our paper.

###### Proof of Theorem 1.1.

Recall that we are assuming that the Reeb flow of $\lambda$ has finitely many
simple Reeb orbits. We will now conclude that there are exactly two simple
Reeb orbits.

Recall that to each Reeb orbit $\gamma=(x,T)$ of $\lambda$, and each (homotopy
class of) symplectic trivialization $\tau$ of $\xi_{\gamma}$, there is an
associated rotation number ${\rm rot}_{\tau}(\gamma)\in\mathbb{R}$ uniquely
characterized by

${\rm rot}_{\tau}(\gamma)=\lim_{k\to\infty}\frac{{\rm
CZ}_{\tau^{k}}(\gamma^{k})}{2k}$

where $\tau^{k}$ denotes the trivialization of $\xi_{\gamma^{k}}$ induced from
$\tau$. It is not assumed here that $\tau$ comes from a trivialization over
the simple Reeb orbit underlying $\gamma$.

Denote $L=x^{+}(\mathbb{R})\cup x^{-}(\mathbb{R})$. Denote the primitive
period of $x^{+}$ by $\underline{T}^{+}$, and the multiplicity of $\gamma^{+}$
by $k_{+}$. The simple Reeb orbit underlying $\gamma^{+}$ is
$\underline{\gamma}^{+}=(x^{+},\underline{T}^{+})$, and we have
$\gamma^{+}=(\underline{\gamma}^{+})^{k_{+}}=(x^{+},T^{+}=k_{+}\underline{T}^{+})$.

Consider a Martinet tube $\Psi:U^{+}\to\mathbb{R}/\mathbb{Z}\times B$ around
$\gamma^{+}$, with coordinates $(\theta,x_{1}+ix_{2})$. If the argument of
$x_{1}+ix_{2}$ is denoted by $\varphi\in\mathbb{R}/2\pi\mathbb{Z}$ then, by
(iv) in Lemma 6.3, the slope of ${\widetilde{u}}$ at the positive puncture
with respect to coordinates
$(\theta,\frac{\varphi}{2\pi})\in\mathbb{R}/\mathbb{Z}\times\mathbb{R}/\mathbb{Z}$
is $(k_{+},{\mathrm{wind}}_{\tau^{k_{+}}}(\nu^{<-\delta}(\gamma^{+})))$, where
$\tau$ is the trivialization of $\xi_{\underline{\gamma}^{+}}$ induced by the
frame $\\{\partial_{x_{1}},\partial_{x_{2}}\\}$, and $\tau^{k_{+}}$ is the
trivialization of $\xi_{\gamma^{+}}$ induced by $\tau$.

The Reeb vector field $R$ of $\lambda$ is complete on $Y\setminus L$. For each
${\widetilde{u}}=(a,u)$ that represents a curve in $\mathcal{Y}$, the
$Y$-component $u$ is transverse to $R$. Hence, a trajectory of $R$ has non-
negative intersection number with $u$, and any intersection with
$u(\mathbb{R}\times\mathbb{R}/\mathbb{Z})$ counts positively to the
intersection number. We split the remaining arguments into a few claims.

We want to lift the Reeb flow via the evaluation map, and find a global cross-
section for the lifted flow. Our first claim collects what we need to know
away from the boundary:

  * (C1)

For each open neighborhood $W$ of $L$ there exists $\ell_{W}>0$ such that if
$b-a>\ell_{W}$ then for every $[{\widetilde{u}}=(a,u)]\in\mathcal{Y}$ and
every piece of trajectory $c:[a,b]\to Y\setminus W$ we have $c([a,b])\cap
u(\mathbb{R}\times\mathbb{R}/\mathbb{Z})\neq\emptyset$.

To prove (C1), consider the map ${\rm ev}:F^{-1}(\mathcal{Y})\to Y\setminus
L$. This map is a proper local diffeomorphism, by Lemma 6.4. Denote by
$\tilde{R}$ the lift of $R$ by ${\rm ev}$. Hence $\tilde{R}$ is a complete
vector field. Moreover, $F:F^{-1}(\mathcal{Y})\to\mathcal{Y}$ is a fibration
over $\mathcal{Y}$ and $\tilde{R}$ is transverse to the fibers. Therefore we
can find a global parameter $s\in\mathbb{R}/\mathbb{Z}$ on $\mathcal{Y}$
satisfying $\iota_{\tilde{R}}d(s\circ F)>0$. Since $Y\setminus W$ is compact
in $Y\setminus L$, ${\rm ev}^{-1}(Y\setminus W)$ is a compact subset of
$F^{-1}(\mathcal{Y})$. Let $\Delta$ be the infimum of
$\iota_{\tilde{R}}d(s\circ F)$ on ${\rm ev}^{-1}(Y\setminus W)$. By
compactness, this infimum is a minimum. Hence, $\Delta>0$. Let $c:[a,b]\to
Y\setminus W$ be a piece of a trajectory of $R$. Consider a lift
$\tilde{c}:[a,b]\to{\rm ev}^{-1}(Y\setminus W)$ of $c$ by ${\rm ev}$. Then
$\tilde{c}$ is a piece of trajectory $\tilde{R}$. Moreover, a lift
$[a,b]\to\mathbb{R}$ of $s\circ F\circ\tilde{c}$ oscillates more than
$\Delta(b-a)$. This oscillation is strictly larger than $1$ if
$b-a>\Delta^{-1}$. Hence, $\tilde{c}$ intersects every fiber of $F$ when
$b-a>\Delta^{-1}$. Claim (C1) is proved if we set $\ell_{W}=\Delta^{-1}$.

Our next claim collects what we need to know near the boundary.

  * (C2)

There is a neighborhood $W_{*}$ of $L$ and $\ell_{*}>0$ such that if
$b-a>\ell_{*}$ then for every $[{\widetilde{u}}=(a,u)]\in\mathcal{Y}$ and
every piece of Reeb trajectory $c:[a,b]\to W_{*}$ we have $c([a,b])\cap
u(\mathbb{R}\times\mathbb{R}/\mathbb{Z})\neq\emptyset$.

We prove (C2). For every $T>0$ there is an open neighborhood
$W_{*}^{+}=W_{*}^{+}(T)$ of $x^{+}(\mathbb{R})$ with coordinates
$(\theta,z=x_{1}+ix_{2}=|z|e^{i\varphi})\in\mathbb{R}/\mathbb{Z}\times\mathbb{C}$
such that:

  * •

$x^{+}(t)=(t/\underline{T}^{+},0)$.

  * •

$d\theta\wedge dx_{1}\wedge dx_{2}$ is positive relatively to $\lambda\wedge
d\lambda$.

  * •

For every piece Reeb trajectory of time-length $T$ contained in $W_{*}^{+}$, a
lift to $\mathbb{R}$ of $\varphi/2\pi$ along the trajectory oscillates not
less than ${\rm rot}(\underline{\gamma}^{+})\lfloor
T/\underline{T}^{+}\rfloor-1$.

Here ${\rm rot}(\underline{\gamma}^{+})$ is computed with respect to the frame
$\\{\partial_{x_{1}},\partial_{x_{2}}\\}$. As remarked above, for an arbitrary
element of $\mathcal{Y}$ represented by a pseudo-holomorphic map
${\widetilde{u}}=(a,u)$, if $\rho\gg 1$ the loop
$t\in\mathbb{R}/\mathbb{Z}\mapsto u(\rho,t)\in W_{*}^{+}\setminus
x^{+}(\mathbb{R})$ has slope
$(k_{+},{\mathrm{wind}}(\nu^{<-\delta}(\gamma_{+}))$ in the basis
$\\{d\theta,\frac{d\varphi}{2\pi}\\}$. Here the winding is computed in a
trivialization aligned with the frame
$\\{\partial_{x_{1}},\partial_{x_{2}}\\}$. Hence, the argument of a Reeb
trajectory of time-length $T$ contained in $W_{*}^{+}$ relative to
$u([\rho,+\infty)\times\mathbb{R}/\mathbb{Z})$ has a lift to $\mathbb{R}$ that
oscillates

$\displaystyle D$ $\displaystyle=2\pi\left({\rm
rot}(\underline{\gamma}^{+})-\frac{{\mathrm{wind}}(\nu^{<-\delta}(\gamma_{+}))}{k_{+}}\right)\lfloor
T/\underline{T}^{+}\rfloor-O(1)$ $\displaystyle=2\pi\left(\frac{{\rm
rot}(\gamma^{+})-{\mathrm{wind}}(\nu^{<-\delta}(\gamma_{+}))}{k_{+}}\right)\lfloor
T/\underline{T}^{+}\rfloor-O(1)\ \,.$

Here $O(1)$ denotes a positive constant independent of $T$ and of
${\widetilde{u}}$. We know that

${\rm rot}(\gamma^{+})>{\mathrm{wind}}(\nu^{<-\delta}(\gamma_{+}))$

since $\nu^{<-\delta}(\gamma_{+})$ is a negative eigenvalue of the asymptotic
operator at $\gamma_{+}$. Hence, we can fix $T$ such that $D>3\pi$, and
conclude that any piece of Reeb trajectory contained in
$W_{*}^{+}=W_{*}^{+}(T)$ of time-length $T$ intersects the image of $u$. An
analogous argument at the negative puncture will produce a neighborhood
$W_{*}^{-}$ of $x^{-}(\mathbb{R})$ with similar properties. Take
$W_{*}=W_{*}^{+}\cup W_{*}^{-}$ and (C2) is proved.

We now put (C1) and (C2) together to prove the following:

  * (C3)

There exists $\ell>0$ such that if $b-a>\ell$ then for every piece of Reeb
trajectory $c:[a,b]\to Y\setminus L$ and every
$[{\widetilde{u}}=(a,u)]\in\mathcal{Y}$ we have $c([a,b])\cap
u(\mathbb{R}\times\mathbb{R}/\mathbb{Z})\neq\emptyset$.

Let $\phi^{t}$ denote the Reeb flow of $\lambda$. To prove (C3), consider a
neighborhood $W_{*}$ of $L$ and $\ell_{*}>0$ given by (C2). Let $W\subset
W_{*}$ be a neighborhood of $L$ such that if $p\in W$ then
$\phi^{[0,\ell_{*}+1]}(p)\subset W_{*}$. Let $\ell_{W}>0$ be given by (C1).
Consider $p\in Y\setminus L$ arbitrary. If $\phi^{[0,\ell_{W}+1]}(p)\cap
W=\emptyset$ then, by (C1), $\phi^{[0,\ell_{W}+1]}(p)$ intersects the
projection to $Y$ of every curve in $\mathcal{Y}$. If
$\phi^{[0,\ell_{W}+1]}(p)\cap W\neq\emptyset$ then we have a piece of
trajectory inside $\phi^{[0,\ell_{W}+\ell_{*}+2]}(p)$ of time-length
$\ell_{*}+1$ that is contained in $W_{*}$. By (C2) it must intersect the
projection to $Y$ of every curve in $\mathcal{Y}$. We proved that one can take
$\ell=\ell_{W}+\ell_{*}+2$ for (C3).

Let us now complete the proof. Lift the Reeb flow $\phi^{t}$ of $\lambda$ from
$Y\setminus L$ to a flow $\psi^{t}$ on $F^{-1}(\mathcal{Y})$ via the proper
local diffeomorphism ${\rm ev}|_{F^{-1}(\mathcal{Y})}$. By Lemma 6.3, the flow
$\psi^{t}$ is transverse to the fibers of the fibration
$F:F^{-1}(\mathcal{Y})\to\mathcal{Y}$. By (C3) each fiber is a global cross-
section for $\psi^{t}$. Lemma 6.3 (i) also tells us that the
$\phi^{t}$-invariant $2$-form $d\lambda$ lifts to a $\psi^{t}$-invariant
$2$-form on $F^{-1}(\mathcal{Y})$ that defines an area-form of finite total
area on each fiber of the map $F$. Franks’ results from [13] guarantee that
$\psi^{t}$ has either infinitely many simple periodic orbits, or no simple
periodic orbits at all.

Periodic orbits of $\psi^{t}$ are mapped onto periodic orbit of $\phi^{t}$ by
$F$. Conversely, properness of the local diffeomorphism ${\rm
ev}|_{F^{-1}(\mathcal{Y})}$ implies that a periodic orbit of $\phi^{t}$ in
$Y\setminus L$ lifts to finitely many periodic orbits of $\psi^{t}$. It
follows that $\psi^{t}$ has finitely many simple periodic orbits, hence no
periodic orbits at all. This implies that $\phi^{t}$ has no periodic orbits on
$Y\setminus L$. ∎

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Dan Cristofaro-Gardiner

University of Maryland, College Park

<EMAIL_ADDRESS>
Umberto Hryniewicz

RWTH Aachen, Jakobstrasse 2, Aachen 52064, Germany

<EMAIL_ADDRESS>
Michael Hutchings

University of California, Berkeley

<EMAIL_ADDRESS>
Hui Liu

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei,
P.R. China

<EMAIL_ADDRESS>