#  Symplectic Camel theorems and $\mathbb{\mathcal{C}}^{0}$-rigidity of
coisotropic submanifolds

Emmanuel Opshtein

###### Abstract

This paper deals with the $\mathbb{\mathcal{C}}^{0}$-rigidity of the reduction
of coiostropic submanifolds under the action of symplectic homeomorphism. More
precisely, we exhibit several situations where a symplectic homeomorphism that
takes a coisotropic submanifold to a smooth submanifold (which are then known
to be coisotropic by a result of Humilière-Leclercq-Seyfaddini) abides to the
non-squeezing property in the reduction.

##  1 Introduction

This paper continues the study of the action of the symplectic homeomorphisms
on smooth submanifolds as initiated in [Ops09, HLS15, BO16]. Recall that a
symplectic homeomorphism between symplectic manifolds $M$ and $M^{\prime}$ is
a homeomorphism that can be approximated by symplectic diffeomorphisms between
$M$ and $M^{\prime}$. In [HLS15], it was observed that when the image of a
coisotropic submanifold by a symplectic homeomorphism is a smooth submanifold,
then this image is coisotropic and the symplectic homeomorphism intertwines
the charateristic foliations of the source and target submanifolds. As a
result, it acts on the reduction, which is locally a symplectic manifold, and
one may ask which symplectic properties of the reduction are preserved. In
other terms, the set of coisotropic submanifolds is
$\mathbb{\mathcal{C}}^{0}$-rigid, as well as the characteristic foliations,
and we ask which further symplectic invariants of coisotropic submanifolds are
$\mathbb{\mathcal{C}}^{0}$-rigid. As far as we know, three results are already
known, prior to this paper:

  * $\bullet$

When the reduction homeomorphism is smooth, it is symplectic [BO16].

  * $\bullet$

Let $\Sigma,\Sigma^{\prime}$ be hypersurfaces in symplectic manifolds, $h$ a
symplectic homeomorphism that takes $\Sigma$ to $\Sigma^{\prime}$ and
$\textnormal{Red}\,h$ its action at the level of the reduction. Then
$\textnormal{Red}\,h$ preserves the stable displacement energy of subsets of
$\textnormal{Red}\,\Sigma$ [BO16].

  * $\bullet$

Let $h$ be a symplectic homeomorphism of $\mathbb{T}^{2n}$ that preserves the
split coisotropic subtorus $\mathbb{T}^{2k+l}\times\\{0\\}$. Then
$\textnormal{Red}\,h:\mathbb{T}^{2k}\to\mathbb{T}^{2k}$ preserves the spectral
capacity of open subsets of $\mathbb{T}^{2k}$ [HLS16].

In these situations, the symplectic homeomorphism therefore also inherits
symplectic properties at the level of the reduction. In this paper, we study
the (non)-squeezing properties of the reduction of symplectic homeomorphisms
in this setting, namely:

######  Conjecture 1.

Let $h$ be a symplectic homeomorphism that takes some coisotropic submanifold
to a smooth, hence coisotropic, submanifold. Then the reduction of $h$ does
not squeeze balls into cylinders of smaller capacities.

Our first result generalizes the last result cited above.

######  Theorem 1.

Let $N$ be a closed manifold that admits a metric with non-negative scalar
curvature. Assume the existence of a symplectic homeomorphism defined in a
neighbourhood of $B^{2k}(a)\times N\subset\mathbb{C}^{k}\times T^{*}N$ such
that $h(B^{2k}(a)\times N)\subset Z^{2k}(A)\times N$. Then $A\geq a$.

In other terms, when the characteristic foliation consists of closed leaves
and is trivial, and provided the leaves satisfy some topological property made
explicit in remark 2.2, the reduction of a symplectic homeomorphism that
preserves this submanifold verifies the non-squeezing property. Theorem 1
relies on the classical non-squeezing theorem via a short and easy argument.

By contrast, our study of the general case is less conclusive. In a model
situation, we still get the following result:

######  Theorem 2.

Let $h$ be a symplectic homeomorphism defined in a neighbourhood of
$B^{2k}(1)\times[-1,1]^{n-k}\subset\mathbb{C}^{n}$, with values in
$\mathbb{C}^{n}$, that takes the coisotropic submanifold
$B^{2k}(1)\times[-1,1]^{n-k}$ to $\mathbb{C}^{k}\times\mathbb{R}^{n-k}$. Then
there exists $\delta(h)>0$ such that for all $a<\delta(h)$, if
$h\big{(}B^{2k}(a)\times[-1,1]^{n-k}\big{)}\subset
Z^{2k}(A)\times\mathbb{R}^{n-k}$, $a\leq A$.

The dependence of the constant $\delta(h)$ with $h$ prevents the previous
theorem to provide a satisfactory answer to conjecture 1. The best we would
get in the general framework of conjecture 1, within the techniques of the
present paper, would be that the balls of small diameters in the reduction of
the source are either not-squeezed or highly distorted. When the reduction has
dimension $2$ however, non-squeezing means area-preservation, and theorem 2 is
enough for a complete answer.

######  Theorem 3.

Conjecture 1 holds when the dimension of the reduction is $2$. In other terms,
if $C$ is a coisotropic submanifold of dimension $n+1$ in a symplectic
manifold $M^{2n}$, and if some symplectic homeomorphism takes $C$ to a smooth
submanifold $C^{\prime}$, then $C^{\prime}$ is coisotropic, $h$ conjugates the
characteristic foliations of $C,C^{\prime}$ and
$\textnormal{Red}\,h:\textnormal{Red}\,C\to\textnormal{Red}\,C^{\prime}$ is
area preserving.

Notice that for a coisotropic submanifold, having a reduction of dimension $2$
means being of minimal dimension above the lagrangian case. Thus, this
corollary is in some sense orthogonal to the previous local result, obtained
in [BO16], that concerned hypersurfaces, since these have maximal dimension
below the open case. Theorem 2 relies on a slightly stronger version of the
following coisotropic Camel theorem (see also [Bus19] for a very close
statement):

######  Theorem 4.

Let $\varphi:B^{2n}(a)\times\mathbb{R}^{l}\to\mathbb{R}^{2n}$ be a smooth map
such that:

  * i)

$\forall t\in\mathbb{R}^{l}$, $\varphi(\cdot,t)$ is a symplectic embedding,

  * ii)

$\varphi(\cdot,t)=\textnormal{Id}\,+\sum_{1}^{l}t_{i}\frac{\partial}{\partial
y_{i}}$ for $|t|\gg 1$,

  * iii)

$\textnormal{Im}\,\varphi\cap\\{y_{1}=\dots=y_{l}=0\\}\subset[-1,1]^{n-k}\times
Z^{2(n-l)}(A)$.

Then $A\geq a$.

For $l=1$, this is due to Eliashberg [Eli90], and the generalization above is
fairly straightforward. The strenghtened version necessary for our
application, theorem 3.1, requires more work, but is still not sufficient to
handle conjecture 1. In fact, this conjecture would follow if we could
localize the symplectic camel theorem in the following sense:

######  Conjecture 2.

Theorem 4 holds when iii) is replaced by:

  * iii)’

$\varphi$ is knotted with $\partial Z_{A,R}^{l}$, where

$Z_{A,R}^{l}:=\left\\{(x_{1},\dots,x_{n},y_{1},\dots,y_{n})\;\left|\begin{array}[]{l}y_{1}=\dots=y_{l}=0,\\\
\pi(x_{n}^{2}+y_{n}^{2})<A,\\\
\|(x_{1},\dots,x_{n-1},y_{l+1},\dots,y_{n-1})\|_{\infty}<R\end{array}\right.\right\\}.$

In other terms, $\varphi$ provides a symplectic isotopy of the ball
$B^{2n}(a)$ through the window $Z_{A,R}^{l}$, without any further assumption
that the isotopy avoids the coisotropic wall
$\mathbb{R}^{l}\times\mathbb{R}^{2(n-l)}$ away from this window.

#### Organization of the paper:

The short section 2 proves theorem 1 and is independent from the rest of the
paper. In section 3, we state and prove two versions of the coisotropic
versions of the Camel theorem that we have stated above. We provide an
extensive argument, although some good references are already available [MT93,
Nie14]. The reason is that these references either insist on the dimension
$4$, or on the pseudoconvexity of the Camel space, which are not relevant in
our context. Finally, we prove theorems 2 and 3 in section 4.

#### Notation:

Throughout this paper, we adopt the following notation:

  * $\bullet$

$B(a)$ denotes the ball of capacity $a$ in $\mathbb{C}^{n}$.

  * $\bullet$

Given two sets $A\subset B$, $\textnormal{Op}\,(A,B)$ stands for an arbitrary
but fixed neighbourhood of $A$ in $B$.

  * $\bullet$

A continuous map $\Phi:\mathbb{R}^{l}\times B^{2n}(a)\to\mathbb{C}^{n}$ is a
parametric symplectic embedding if $\Phi(t,\cdot):B^{2n}(a)\to\mathbb{C}^{n}$
is a symplectic embedding for all $t\in\mathbb{R}^{l}$. We then denote
$\Phi_{t}:=\Phi(t,\cdot)$

  * $\bullet$

We split $\mathbb{C}^{n}$ as
$\mathbb{C}^{n}=\mathbb{C}^{k+l}=\mathbb{C}\times\mathbb{C}^{k-1}\times\mathbb{R}^{l}\oplus
i\mathbb{R}^{l}$ and we use coordinates $(z_{1},z^{\prime},x,y)$ on these four
factors, respectively.

  * $\bullet$

We endow $\mathbb{C}^{k-1}\times\mathbb{C}^{l}$ with the $L_{\infty}$-norm :
$\|(z^{\prime},x,y)\|:=\max\\{\|z^{\prime}\|_{\infty},\|x\|_{\infty},\|y\|_{\infty}\\}$
(and
$\|z^{\prime}\|_{\infty}:=\max\\{\|\textnormal{Re}\,z^{\prime}\|_{\infty},\|\textnormal{Im}\,z^{\prime}\|_{\infty}\\}$).

  * $\bullet$

$Z_{A,R}^{k,l}:=D(A)\times\\{(z^{\prime},0,y)\,|\,\|(z^{\prime},0,y)\|<R\\}\subset\mathbb{C}^{k}\times
i\mathbb{R}^{l}$. When $k,l$ are clear from the context, we omit the
subscripts for easier readability.

  * $\bullet$

$\Gamma_{A,R}^{k,l}:=S^{1}(A)\times\\{(z^{\prime},0,y)\,|\,\|(z^{\prime},0,y)\|<R\\}\subset\mathbb{C}^{k}\times
i\mathbb{R}^{l}$ (this is the ”horizontal” part of $\partial Z_{A,R}^{k,l}$),

  * $\bullet$

$F_{A,R,M}^{k,l}:=D(A+\pi
M^{2})\times\\{(z^{\prime},x,y)\,|\,R-M<\|(z^{\prime},0,y)\|<R+M\text{ and
}\|x\|<M\\}\subset\mathbb{C}^{k}\times i\mathbb{R}^{l}$. This is an
$M$-neighbourhood of the ”vertical” part of $\partial Z_{A,R}^{k,l}$. As a
result, notice that any point in $F^{k,l}_{A,R,M}$ is at euclidean distance no
less than $M^{\prime}$ of $\partial F^{k,l}_{A,R,M+M^{\prime}}$.

The following figure illustrates our notation. It is rather accurate when
$k=1$ and $l=0$. When $k+l\geq 2$, the figure is hopefully usefull, but it is
only partially representative. The main difference is that $F_{A,R,M}$ becomes
connected.

$\Gamma_{A,R}$$\mathbb{C}^{k}\times\mathbb{R}^{l}$$B_{\infty}(R)$$F_{A,R,M}$$\mathbb{C}$$D(A)$$D(A+M)$

Figure 1: The sets $\Gamma_{A,R}$ and $F_{A,R,M}$ when $k=1$.

#### Aknowledgements:

Lev Buhovsky was initially engaged in this project and contributed to
uncovering a crucial flaws in the initial argument. This paper therefore owes
him a lot.

##  2 Proof of theorem 1

Let $g$ be a metric on $N$ with non-positive sectional curvature,
$\varepsilon>0$, and $h:\textnormal{Op}\,(B^{2k}(a)\times
N,\mathbb{C}^{k}\times T^{*}N)\to\mathbb{C}^{k}\times T^{*}N$ be a symplectic
homeomorphism such that $h(B^{2k}(a)\times N)\subset Z^{2k}(A)\times N$.
Consider an $\varepsilon$-approximation of $h$ in the
$\mathbb{\mathcal{C}}^{0}$-norm by a symplectic diffeomorphism $f$. Thus, $f$
is defined in a neighbourhood of $B^{2k}(a)\times N$ in $B^{2k}(a)\times
T^{*}N$ and provided this neighbourhood is small enough and the approximation
fine enough, we have

$f(B^{2k}(a)\times T_{\delta}^{*}N)\subset Z(A+\varepsilon)\times
T_{2\varepsilon}^{*}N,$

where $\delta,\varepsilon$ are small positive numbers. Since $g$ has non-
positive curvature, Cartan-Hadamard’s theorem guarantees that the universal
cover $\tilde{N}$ of $N$ is diffeomorphic to $\mathbb{R}^{l}$. The covering
$\pi:\tilde{N}\to N$ lifts to symplectic coverings $T^{*}\pi:T^{*}\tilde{N}\to
T^{*}N$ and $\Pi:=\textnormal{Id}\,\times T^{*}\pi:\mathbb{C}^{k}\times
T^{*}\tilde{N}\to\mathbb{C}^{k}\times T^{*}N$. The map $\pi$ obviously being a
Riemanian covering with respect to the metrics $(\tilde{g}:=\pi^{*}g,g)$ on
the pair $(\tilde{N},N)$, it is easy to see that $\Pi^{-1}(U\times
T_{\delta}^{*}N)=U\times T_{\delta}^{*}\tilde{N}$, where
$T^{*}_{\delta}\tilde{N}$ stands for the $\delta$-neighbourhood of the zero
section with respect to the lifted metric $\tilde{g}$. Thus, $f$ lifts through
$\Pi$ to a symplectic map

$\tilde{f}:B(a)\times T^{*}_{\delta}\tilde{N}\longrightarrow
Z(A+\varepsilon)\times T^{*}\tilde{N}.$

Since $h$ is a homeomorphism that sends $\\{0\\}\times N$ to $\\{*\\}\times
N$, the projection $\text{pr}\circ h_{|{\\{0\\}\times N}}:\\{0\\}\times N\to
N$ has degree $1$. This property also holds for $f$, and as a result,
$\tilde{f}$ is injective, hence an embedding

$\tilde{f}:B^{2k}(a)\times T^{*}_{\delta}\tilde{N}\longrightarrow
Z^{2k}(A+\varepsilon)\times T^{*}\tilde{N}\approx Z^{2k}(A+\varepsilon)\times
T^{*}\mathbb{R}^{l}=Z^{2n}(A+\varepsilon).$

######  Claim 2.1.

$T^{*}_{\delta}\widetilde{N}$ contains a symplectic ball of capacity $a$.

Let thus $\varphi:B^{2l}(a)\hookrightarrow T^{*}_{\delta}\mathbb{R}^{l}$ be a
symplectic embedding. The map $\tilde{f}\circ(\textnormal{Id}\,\times\varphi)$
therefore provides a symplectic embedding of $B^{2k}(a)\times B^{2l}(a)$ into
$Z(A+\varepsilon)$. By Gromov’s non-squeezing theorem, we get
$A+\varepsilon\geq a$ and, letting $\varepsilon$ to $0$, $A\geq a$. $\square$

Proof of claim 2.1: Let $(N,g)$ be a non-positively curved Riemannian manifold
and $(\widetilde{N},\tilde{g})$ its universal cover. Fix any point
$p\in\widetilde{N}$. By Cartan-Hadamard’s theorem, the map
$f:=\exp_{p}:T_{p}\widetilde{N}\to\widetilde{N}$ is a diffeomorphism.
Moreover, the negative curvature implies that the map

$D_{\lambda}(x):=f(\lambda f^{-1}(x))$

verifies $\|(D_{\lambda}^{\prime})^{-1}\|^{-1}_{\tilde{g}}\geq\sqrt{\lambda}$.
Indeed, given $u\in T_{x}\tilde{N}$ of norm $1$, $U(t):=t\mapsto
D_{t}^{\prime}(x)u$ is the Jacobi vector field along the geodesic issued from
$p$ and passing at $x$ at time $1$, that vanishes at $0$ and equals $u$ at
$1$. A classical computation shows that the function $h(t):=g(U(t),U(t))$ is
convex in non-positive curvature. Since it verifies $h(0)=0$ and $h(1)=1$,
$h(t)\geq t$ for $t\geq 1$, so $\|D_{\lambda}^{\prime}(x)u\|^{2}\geq\lambda$,
which is our assertion. Given now any small ball
$\mathcal{D}\subset\widetilde{N}$, $T^{*}\mathcal{D}$ contains a relatively
compact symplectic ball $\varphi:B(a)\hookrightarrow T^{*}\mathcal{D}$. The
image of $\varphi$ lies in $T^{*}_{\tilde{g},K}\widetilde{N}$ for some large
$K$. Putting $\lambda:=\frac{\delta}{K}$, the natural lift of $D_{\lambda}$ to
$T^{*}\widetilde{N}$ then takes $\varphi(B(a))$ into
$T^{*}_{\delta}\tilde{N}$. $\square$

######  Remark 2.2.

The proof above shows that theorem 1 holds whenever $N$ has the property that
any neighbourhood $T^{*}_{\delta,\tilde{g}}\tilde{N}$ of the universal cover
of $N$ equipped with some pull-back metric has infinite Gromov capacity. It is
however slightly unclear to us whether there is a more conventional
characterization of this natural symplectic property. The existence of a non-
positive curvature on $N$ is enough, as explained above.

##  3 Two Camel theorems

The aim of this section is to state and prove two Camel-like theorems, that we
use in our proof of theorem 2.

######  Theorem 3.1.

There exists a universal constant $C>0$ for which the following holds. Let
$\Phi:\mathbb{R}^{l}\times B^{2n}(a)\to\mathbb{C}^{n}=\mathbb{C}^{k+l}$ be a
parametric symplectic embedding which verifies:

  * (i)

$\Phi$ is standard at infinity: $\exists K>0$ such that for $|t|>K$,
$\Phi_{t}(z)=z+t\frac{\partial}{\partial t}$, where we see
$t\in\mathbb{R}^{l}\simeq\\{0\\}\times\mathbb{R}^{l}\subset\mathbb{C}^{k}\times\mathbb{C}^{l}$,

  * (ii)

$\Phi$ is knotted with $\partial Z_{A,R}^{k,l}$.

  * (iii)

$\textnormal{Im}\,\Phi$ does not intersect $F_{A,R,CA}^{k,l}$.

Then $A\geqslant a$.

######  Theorem 3.2.

There exists a universal constant $C>0$ for which the following holds. Let $L$
be a Lagrangian embedding of $(S^{1})^{k}\times\mathbb{R}^{l}$ into
$\mathbb{C}^{n}=\mathbb{C}^{k+l}$ which verifies:

  * (i)

$L$ is properly embedded, standard at infinity: there is a compact set
$K\subset\mathbb{C}^{n}$ such that

$L\cap{}^{c}K=\big{(}S^{1}(a)^{k}\times\mathbb{R}^{l}\big{)}\cap{}^{c}K=\\{\pi|z_{i}|^{2}=a,i=1\dots
k\\}\times\mathbb{R}^{l}\cap{}^{c}K\subset\mathbb{C}^{k}\times\mathbb{C}^{l}.\vspace*{-,2cm}$

  * (ii)

$L$ is knotted with $\partial Z_{A,R}^{k,l}$.

  * (iii)

$L$ does not meet $F_{A,R,CA}^{k,l}$.

Then $a\leqslant A$.

Obivously, the assumptions on theorem 3.1 are less restrictive than the usual
ones, so theorem 3.1 implies theorem 4. Theorem 3.2 is also a slightly
localized version of an existing Lagrangian camel theorem [Bus19]. As already
explained, getting a genuinely localized version of both results would consist
in removing assumption (iii). We prove both theorems below, because both seem
interesting in their own sake, and that none of them seems to follow from the
other, but their proofs are very similar. For our application to the
$\mathbb{\mathcal{C}}^{0}$-rigidity of the reduction of a symplectic
homeomorphism however, only theorem 3.1 is relevant, through the following
corollary:

######  Corollary 3.3.

If there is a compactly supported symplectic diffeomorphism $f$ of
$\mathbb{C}^{n}=\mathbb{C}^{k+l}$ such that

  * (i)

$f(B^{2k}(a)\times\mathbb{R}^{l})$ is knotted with $\partial Z_{A,R}^{k,l}$
for some $R$,

  * (ii)

$f(B^{2k}(a)\times\mathbb{R}^{l})\cap F_{A,R,CA}^{k,l}=\emptyset$,

then $A\geq a$.

Proof: Let $K,\varepsilon>0$ to be taken very large and very small,
respectively. Let $\rho:\mathbb{R}^{+}\to[\varepsilon,1]$ be a smooth function
with $\rho(s)\equiv\varepsilon$ for $s<K$ and $\rho(s)\equiv 1$ for $s>K+1$.
The map

$\begin{array}[]{rcccl}\varphi&:&B^{2k}(a)\times
B^{2l}(a)\times\mathbb{R}^{l}&\longrightarrow&\mathbb{C}^{k+l}\\\
&&(z_{1},z_{2},t)&\longmapsto&(z_{1},\frac{1}{\rho(\|t\|)}\textnormal{Re}\,z_{2}+t,\rho(\|t\|)\textnormal{Im}\,z_{2})\end{array}$

verifies:

  * (i)

$\varphi$ is smooth,

  * (ii)

for all $t\in\mathbb{R}^{l}$, $\varphi(\cdot,t)$ is a symplectic embedding of
$B^{2k}(a)\times B^{2l}(a)\supset B^{2n}(a)$,

  * (iii)

for $|t|>K+1$, $\varphi(z_{1},z_{2},t)=(z_{1},z_{2})+t$,

  * (iv)

for $|t|<K$, $\varphi(z_{1},z_{2},t)$ belongs to an
$\varepsilon$-neighbourhood of
$B^{2k}(a)\times\mathbb{R}^{l}\subset\mathbb{C}^{k}\times\mathbb{C}^{l}$.

Let now $f$ be the symplectic diffeomorphism considered by our statement. For
$\varepsilon\ll 1$ and $K\gg 1$, the restriction of the map
$\Phi:=f\circ\varphi$ to $B^{2n}(a)\times\mathbb{R}^{l}\subset B^{2k(a)}\times
B^{2l}(a)\times\mathbb{R}^{l}$ verifies all the assumptions of theorem 3.1, so
$A\geq a$. $\square$

Before proving theorems 3.1 and 3.2, let us unify their statements. First, we
fix $k,l,n$ so we drop all these indices from our notations. Notice also that
scaling the symplectic form allows to assume that $A=1$, which we do
henceforth, removing from our notations any reference to $A$ (hence
$Z_{R},\;F_{R,M},\Gamma_{R}$ stand for
$Z_{1,R}^{k,l},F_{1,R,M}^{k,l},\Gamma_{1,R}^{k,l}$). Let
$f:X\times\mathbb{R}^{l}\to\mathbb{C}^{n}=\mathbb{C}^{k}\times\mathbb{C}^{l}$
be a map that is either a Lagrangian embedding if $X=S^{1}(a)^{k}$, or a
parametric symplectic embedding of the ball if $X=B^{2n}(a)$. It is also
standard at infinity, knotted with $\partial Z_{R}$, and its image avoids
$F_{R,M}$. We need to find a constant $C$ such that $M\geq C$ implies the
inequality $a\leq 1$. Since $f$ is standard at infinity in both cases, its
image is contained in $\\{(z,x,y)\;|\;\|(z,y)\|<K\\}$ for some large $K$, that
we take larger than $M$. Moreover, still because $f$ is standard at infinity,
we can compactify the triple $(f,\textnormal{Dom}\,f,\mathbb{C}^{k}\times
i\mathbb{R}^{l}\oplus\mathbb{R}^{l})$ by compactifying the
$\mathbb{R}^{l}$-factor to a very large torus, which we choose in any case to
contain $\\{|x|\geq\max\\{1,M\\}\\}$. We get a new triple
$(\tilde{f},\textnormal{Dom}\,\tilde{f},\mathbb{C}^{k}\times
i\mathbb{R}^{l}\times\mathbb{T}^{l})$, where
$\textnormal{Dom}\,\tilde{f}=X\times\mathbb{T}^{l}$, $X=S^{1}(a)^{k}$ or
$X=B^{2n}(a)$, and $\tilde{f}$ is either a Lagrangian embedding or a
parametric symplectic embedding. Moreover, $\textnormal{Im}\,\tilde{f}$ is
still knotted with $\partial Z_{R}$, and avoids $F_{R,M}$, both sets being
unaffected by the compactification because $Z_{R}\subset\\{x=0\\}$ and $K>M$.
In order to keep light notation, we drop the tilde from $\tilde{f}$. We call
$Y:=\mathbb{C}^{k}\times i\mathbb{R}^{l}\times\mathbb{T}^{l}$ our ambient
space. It is equipped with the standard symplectic form $\omega$, and with a
standard complex structure $J_{\textnormal{st}}$. The almost complex
structures that we will consider in this analysis belong to the set of
$\omega$-compatible almost complex structure $\mathcal{J}(\omega)$ on $Y$ and
are of the following type. First, fix two disjoint neighbourhoods
$V_{\Lambda}\Subset\\{\|(z,y)\|<K\\}$ of $\Lambda:=\textnormal{Im}\,(f)$ and
$V_{\Gamma}$ of $\Gamma_{R}\cup F_{R,M}$, and consider some
$\omega$-compatible almost complex structure $J_{\Lambda}$ on $V_{\Lambda}$.
We define

$\mathcal{J}(R,M,J_{\Lambda}):=\\{J\in\mathcal{J}(\omega)\,|\,J_{|V_{\Lambda}}\equiv
J_{\Lambda}\text{ and }J_{|V_{\Gamma}\cup\\{\|(z,y)\|>K\\}}\equiv
J_{\textnormal{st}}\\}.$

The idea is the same as for the classical Camel theorem: we must fill
$\Gamma_{R}$ by holomorphic discs of area $1$. Since $\Gamma_{R}$ is not
totally real, the problem is not in a good setting. We therefore introduce for
$c\in\mathbb{R}^{k-1}$

$\Gamma_{R,c}:=\Gamma_{R}\cap\\{\textnormal{Re}\,z^{\prime}=c\\}=S^{1}\times\\{(z^{\prime},0,y)\;|\;\textnormal{Re}\,z^{\prime}=c,\;\|(\textnormal{Im}\,z^{\prime},y)\|<R\\}$

which provides a foliation of $\Gamma_{R}$ by Lagrangian leaves, which we will
each fill. The main difficulty here is that we do not assume that the map $f$
avoids the coisotropic wall $\mathbb{C}^{k}\times i\mathbb{R}^{l}$ away from
$Z(1)\times i\mathbb{R}^{l}$, so the classical filling technique must be
adapted. For the reader aware of the classical proof, it might be worth having
in mind that the main issue in this setting, compared to the more classical
one, is the compactness. Since $J=J_{\textnormal{st}}$ on $F_{R,M}$, the
circles $S^{1}\times\\{(z^{\prime},0,y)\\}$ are filled by the vertical discs
$\mathbb{D}\times\\{(z^{\prime},0,y)\\}$ for $\|(z^{\prime},0,y)\|\approx R$.
We need to guarantee that the boundaries of the non-vertical holomorphic discs
do not approach $\partial\Gamma_{R,c}$, which will be realized by taking $M$
large enough (see lemma 3.8).

Although the proof of the Camel theorem is folklore, we could not find a
complete proof in the litterature, except in dimension $4$ [MT93]. We
therefore provide a complete proof, and not only of the compactness issue.

###  3.1 Preliminaries for the compactness

Here is the main statement we will need in order to address the compactness
issues.

######  Proposition 3.4.

There exists $M>0$ such that for any $R$, $c$,
$J_{\Lambda}\in\mathcal{J}(\omega)$ and $J\in\mathcal{J}(R,4M,J_{\Lambda})$,
all the $J$-holomorphic discs with boundary on $\Gamma_{R,c}$, of area $1$,
that intersect $F_{R,M}$ are vertical discs.

This statement relies on classical area estimation of analytic sets, provided
by the next lemmas.

######  Lemma 3.5.

There exists a relative holomorphic embedding
$\varphi:(B^{2n}(\varepsilon_{0}),\mathbb{R}^{n})\hookrightarrow(Y,\Gamma_{0})$,
with $\varphi(0)=(z_{1}=1,z^{\prime}=x=y=0)\in\Gamma_{0}$.

Proof: There exists a holomorphic embedding of
$(\mathbb{D},\mathbb{R})\to(\mathbb{C},S^{1})$ because $S^{1}$ is real-
analytic. Taking product with the identity provides the desired holomorphic
map. $\square$

In the next lemma, we say that an analytic subset has real boundary if every
local branch of $X$ near a point $x\in\partial X$ can be parametrized by a
holomorphic map defined on $\mathbb{D}\cap\\{\textnormal{Im}\,z\geq 0\\}$. For
us, the main example of an analytic subset with real boundary on $\Gamma_{c}$
is the image of a holomorphic disc
$u:(\mathbb{D},\partial\mathbb{D})\to(Y,\Gamma_{c})$.

######  Lemma 3.6.

Let $\omega$ be a symplectic form on $B^{2n}(1)\subset\mathbb{C}^{n}$,
compatible with $J_{\textnormal{st}}$. There exists a constant $\hbar$, that
depends only on $\omega$, such that:

  1. (i)

For any proper analytic subset $X$ of $B^{2n}(1)$ of complex dimension $1$,
with $0\in X$, $\mathcal{A}_{\omega}(X)\geq\hbar$.

  2. (ii)

For any proper analytic subset $X$ of $B^{2n}(1)$ of complex dimension $1$,
with real boundary on $\mathbb{R}^{n}$ and $0\in\partial X$,
$\mathcal{A}_{\omega}(X)\geq\frac{\hbar}{2}$.

Proof: Both assertions are well-known when $\omega=\omega_{\textnormal{st}}$
(this is the monotonicity lemma). Since $\omega_{\textnormal{st}}$ and
$\omega_{0}:=\omega(0)$ are $J_{\textnormal{st}}$-compatible, they have
symplectic basis of the form
$(e_{1},J_{\textnormal{st}}e_{1},\dots,e_{n},J_{\textnormal{st}}e_{n})$,
$(f_{1},J_{\textnormal{st}}f_{1},\dots,f_{n},J_{\textnormal{st}}f_{n})$. The
$J_{\textnormal{st}}$-linear map $A$ that takes $e_{i}$ to $f_{i}$ therefore
transports $\omega_{0}$ to $\omega_{\textnormal{st}}$. Then $A(X)$ is an
analytic subset passing through $0$, proper in $A(B(1))\supset B(r)$, for some
$r>0$ that depends only on $A$, hence on $\omega_{0}$.

Let us prove (i). The montonicity lemma guarantees that
$\mathcal{A}_{\omega_{\textnormal{st}}}(AX\cap
B(\varepsilon))\geq\pi\varepsilon^{2}$ for all $\varepsilon<r$. Since moreover
$A^{*}\omega_{0}=\omega_{\textnormal{st}}$, we have
$A^{*}\omega=\omega_{\textnormal{st}}+R$, where $R\in O(\varepsilon)$ on
$B(\varepsilon)$, so

$\mathcal{A}_{A^{*}\omega}(AX\cap B(\varepsilon))=\int_{AX\cap
B(\varepsilon)}A^{*}\omega=\int_{AX\cap
B(\varepsilon)}\omega_{\textnormal{st}}+R\geq\mathcal{A}_{\omega_{\textnormal{st}}}(AX\cap
B(\varepsilon))-C\varepsilon\mathcal{A}_{g_{\textnormal{st}}}(AX\cap
B(\varepsilon)),$

where $\mathcal{A}_{g_{\textnormal{st}}}$ stands for the euclidean area, and
$C$ depends only on $\omega$. Now since $AX$ is an analytic set,
$\mathcal{A}_{g_{\textnormal{st}}}(AX\cap
B_{\varepsilon})=\mathcal{A}_{\omega_{\textnormal{st}}}(AX\cap
B_{\varepsilon})$, so we get

$\mathcal{A}_{A^{*}\omega}(AX\cap
B(\varepsilon))\geq\mathcal{A}_{\omega_{\textnormal{st}}}\big{(}AX\cap
B(\varepsilon)\big{)}(1-C\varepsilon)\geq\pi\varepsilon^{2}(1-C\varepsilon).$

For some small enough $\varepsilon_{0}<r$, we therefore see that
$\mathcal{A}_{A^{*}\omega}(AX\cap
B(\varepsilon_{0}))\geq\frac{\pi}{2}\varepsilon_{0}^{2}$. Finally, since
$\varepsilon_{0}<r$, $B(r)\subset A(B(1))$ and $X$ is analytic,

$\mathcal{A}_{\omega}(X\cap B(1))\geq\mathcal{A}_{\omega}(X\cap
A^{-1}(B(\varepsilon_{0})))=\mathcal{A}_{A^{*}\omega}(AX\cap
B(\varepsilon_{0}))\geq\frac{\pi}{2}\varepsilon_{0}^{2}=\hbar.$

In order to prove (ii), we consider a proper analytic set with real boundary
on $\mathbb{R}^{n}$. Recall that this means that $(X,\partial
X)\subset(B^{2n}(1),\mathbb{R}^{n})$, and that every local branch $B$ of $X$
near a point $x\in\partial X$ can be parametrized by a holomorphic function
$f_{B}$ defined on a neighbourhood of $0$ in $\mathbb{H}$, and we assume here
that $f_{B}(\mathbb{R})\subset\mathbb{R}^{n}$. Then $X\cup\sigma(X)$ (where
$\sigma(z_{1},\dots,z_{n})=(\overline{z}_{1},\dots,\overline{z}_{n})$) is
analytic on $X$, on $\sigma(X)$, and around the points of $\partial
X=\partial\sigma(X)$ by the reflection principle, applied to each component of
$f_{B}$. Thus $X\cup\sigma(X)$ is a proper analytic set of $B^{2n}(1)$,
without boundary, which passes through $0$, so its area is at least $\hbar$ by
the first part of the argument. The first part of the argument also shows that

$\mathcal{A}_{\omega}(X)\geq(1-C\varepsilon_{0})\mathcal{A}_{\omega_{\textnormal{st}}}(X\cap
B(\varepsilon_{0}))=\frac{1}{2}(1-C\varepsilon_{0})\mathcal{A}_{\omega_{\textnormal{st}}}(X\cup\sigma(X)\cap
B(\varepsilon_{0}))\geq\frac{1}{2}\hbar.\hskip 28.45274pt\square$

######  Lemma 3.7.

A non-constant $J_{\textnormal{st}}$-holomorphic disc with boundary on
$\Gamma_{c}$ is a branched covering of a vertical disc, where the vertical
discs are understood to be the discs
$\mathbb{D}\times\\{(\textnormal{Re}\,z^{\prime}=c,\textnormal{Im}\,z^{\prime}=c_{1},y=c_{2})\\}$.

Proof: We can write $u=(u_{1},u^{\prime},u_{2})$, where
$u_{1}:(\mathbb{D},\partial\mathbb{D})\to(\mathbb{C},S^{1})$,
$u^{\prime}:(\mathbb{D},\partial\mathbb{D})\to(\mathbb{C}^{k-1},\\{\textnormal{Re}\,z^{\prime}=c\\})$,
$u_{2}:(\mathbb{D},\partial\mathbb{D})\to(i\mathbb{R}^{l}\times\mathbb{T}^{l},\\{x=0\\})$
are $J_{\textnormal{st}}$-holomorphic discs. Then $u^{\prime},u_{2}$ are
easily seen to be constant discs by Stokes formula for instance (the integrals
of $\lambda_{\textnormal{st}}$ over their boundaries vanish). And $u_{1}$
takes values in $\mathbb{D}$ by the maximum principle. Finally,
$u_{1}:\mathbb{D}\to\mathbb{D}$ is a proper map, so is a branched covering of
$\mathbb{D}$. $\square$

Proof of proposition 3.4: Let $M,R>0$, $J_{\Lambda}\in\mathcal{J}(\omega)$,
$J\in\mathcal{J}(R,4M,J_{\Lambda})$ and
$u:(\mathbb{D},\partial\mathbb{D})\to(Y,\Gamma_{R,c})$ be a $J$-holomorphic
disc of area $1$ that intersects $F_{R,M}$. Notice that if this disc lies in
$F_{R,4M}$, it is $J_{\textnormal{st}}$-holomorphic hence vertical by lemma
3.7. We therefore assume throughout this proof that $u(\mathbb{D})$ meets both
$F_{R,M}$ and $Y\backslash F_{R,4M}$. We are then in one of three possible
situations:

  * $\bullet$

Either $u(\partial\mathbb{D})\subset F_{R,3M}$. Then since
$\textnormal{Im}\,u$ meets $Y\backslash F_{R,4M}$, there is a euclidean ball
$B$ of radius ${\nicefrac{{M}}{{2}}}$ centered on $\textnormal{Im}\,u$ that
lies in $F_{R,4M}\backslash F_{R,3M}$. Then $u(\mathbb{D})\cap B$ is a proper
analytic subset of $B$ (because $J=J_{\textnormal{st}}$ on $F_{R,4M}$), so has
area at least ${\nicefrac{{\pi M^{2}}}{{4}}}$ by the monotonicity lemma. Since
$u$ has total area $1$, this situation does not occur if we choose $M\geq 2$.

  * $\bullet$

Or $u(\partial\mathbb{D})\cap F_{R,2M}=\emptyset$. Since $u$ visits $F_{R,M}$,
the same argument shows that $u$ has area at least ${\nicefrac{{\pi
M^{2}}}{{4}}}$, so does not occur again if $M\geq 2$.

  * $\bullet$

Or $u(\partial\mathbb{D})$ meets both $F_{R,2M}$ and $Y\backslash F_{R,3M}$.
Denote by $\delta_{0}$ the euclidean diameter of the
$J_{\textnormal{st}}$-holomorphic ball
$\varphi:(B(\varepsilon_{0}),\mathbb{R}^{n})\to(Y,\Gamma_{0})$ centered at
$p_{0}:=(z_{1}=1,x=y=z^{\prime}=0)$ provided by lemma 3.5. By assumption,
$u(\partial\mathbb{D})$ is connected and meets both $F_{R,2M}$ and
$Y\backslash F_{R,3M}$ so its intersection with $F_{R,3M}\backslash F_{R,2M}$
has diameter at least $M$. Thus, we can center at least
$k:=\lfloor{\nicefrac{{M}}{{2\delta_{0}}}}\rfloor$ disjoint euclidean balls of
radii $\delta_{0}$ on points $p_{j}\in u(\partial\mathbb{D})\cap
F_{R,3M}\backslash F_{R,2M}$ ($j\in[1,k]$), and these balls lie in $F_{R,4M}$
if $M\geq\delta_{0}$. Denote by $\tau_{j}$ the composition of a translation in
the $(z^{\prime},y)$-factor and rotation in the $z_{1}$-factor that brings
$p_{0}$ to $p_{j}$. Then, the $\tau_{j}\circ\varphi(B(\varepsilon_{0}))$
provide $k$ disjoint $J_{\textnormal{st}}$-balls centered on the $p_{j}$.
Moreover, since $\omega$ is invariant by the $\tau_{j}$, the pull-backs
$(\tau_{j}\circ\varphi)^{*}\omega$ are a fixed symplectic form on
$B^{2n}(\varepsilon_{0})$. As a result, the area of $u$ is at least the sum of
the areas of the intersections of $u(\mathbb{D})$ with these $k$ balls, each
of which being at least some constant $\hbar$ by lemma 3.6.(ii) (this constant
depends only on $\varphi$ by the discussion above). Thus the area of $u$ is at
least $k\hbar\geq\big{(}{\nicefrac{{M}}{{2\delta_{0}}}}-1\big{)}\hbar$. Taking
$M>2\delta_{0}\big{(}1+{\nicefrac{{1}}{{\hbar}}}\big{)}$, the area of $u$
exceeds $1$, so this situation does not happen neither.

We finally conclude that if
$M>\max\\{2,\delta_{0},2\delta_{0}\big{(}1+\frac{1}{\hbar}\big{)}\\}$, the
only discs of area $1$ with boundary on $\Gamma_{R,c}$ that intersect
$F_{R,M}$ are the vertical ones. $\square$

###  3.2 Filling by holomorphic discs

Recall our notations: $\mathcal{J}(\omega)$ is the set of $\omega$-compatible
almost complex structures on $Y=\mathbb{C}^{k}\times
i\mathbb{R}^{l}\times\mathbb{T}^{l}$,
$Z_{R}=\mathbb{D}\times\\{(z^{\prime},0,y),\;\|(z^{\prime},y\|\leq R)\\}$,
$\Gamma_{R}=S^{1}\times\\{(z^{\prime},0,y),\;\|(z^{\prime},y\|\leq R)\\}$ is
the ”horizontal” part of $\partial Z_{R}$, while
$F_{R,M}=\mathbb{D}\times\\{(z^{\prime},x,y),\;\|x\|\leq
M,\;R-M\leq\|(z^{\prime},y)\|\leq R+M\\}$ is the $M$-neighbourhood of the
vertical part of $\partial Z_{R}$. The constant $M$ is chosen large enough so
that the conclusion of proposition 3.4 holds, and $R$ is then chosen large
compared to $M$, so that $F_{R,M}$ does not cover $Z_{R}$. A map
$f:X\times\mathbb{T}^{l}\to Y$ is given, whose image is surrounded by a small
neighbourhood $V_{\Lambda}$. We consider $M$ for which proposition 3.4 holds,
and we assume that $V_{\Lambda}\subset Y\backslash(\Gamma_{R}\cup F_{R,4M})$.
We aim at filling $\Gamma_{R}$ with holomorphic discs for good complex
structures in $\mathcal{J}(\omega)$. In order to apply the Fredholm theory, we
rather fill each Lagrangian leaf $\Gamma_{R,c}$,
$c\in\mathcal{D}:={\overline{B}}^{k-1}(R)\subset\mathbb{R}^{k-1}$ in a
consistent way. To prove theorem 3.1, we will further need a parametric
version of this filling. Throughout this section, $T$ stands for a parameter
space, which is a smooth closed manifold,
$J_{\Lambda}:T\times\mathcal{D}\to\mathcal{J}(\omega)$ is any
$\mathbb{\mathcal{C}}^{l}$-smooth map, and
$J:T\times\mathcal{D}\to\mathcal{J}(\omega)$ is a smooth map that verifies
$J_{(c,t)|V_{\Lambda}}\equiv J_{\Lambda}$ and $J\equiv J_{\textnormal{st}}$ on
$F_{R,4M}$. Formally, $J$ is a $\mathbb{\mathcal{C}}^{\ell}$-section of the
bundle $\mathcal{J}(R,4M,J_{\Lambda})\to T\times\mathcal{D}$, whose fiber is

$\mathcal{J}(R,4M,J_{\Lambda}(t,c)):=\\{J\in\mathcal{J}(\omega),\;J_{|V_{\Gamma}\cup\\{\|(z,y)\|>K\\}}\equiv
J_{\textnormal{st}},\;J_{|V_{\Lambda}}\equiv J_{\Lambda}(t,c)\\},$

where $V_{\Gamma}$ is any fixed neighbourhood of $\Gamma\cup F_{R,4M}$. Also,
$\pi_{2}(Y,\Gamma_{c})$ is generated by the class of vertical discs
(parametrized injectively and holomorphically). The image of this class by the
Hurewitz morphism $\pi_{2}(Y,\Gamma_{c})\to H_{2}(Y,\Gamma_{c})$ is denoted by
$E_{c}$. The spaces of interest for us are

$\mathcal{M}(J):=\\{(t,c,u),\;(t,c)\in
T\times\mathcal{D},\,u:(\mathbb{D},\partial\mathbb{D})\to(Y,\Gamma_{R,c}),\,\overline{\partial}_{J_{(t,c)}}u=0,\,[u]=E_{c}\\}$

and what will turn out to be the interior of $\mathcal{M}(J)$, obtained by
replacing the closed ball $\mathcal{D}$ by the open ball
$\overset{\vspace*{-1pt}\circ}{\mathcal{D}}:=B^{k-1}(R)$ and
$\Gamma_{R,c}=S^{1}\times\\{(\textnormal{Im}\,z^{\prime},c,y),\;\|(\textnormal{Im}\,z^{\prime},y)\|\leq
R\\}$ by
$\overset{\vspace*{-1pt}\circ}{\Gamma}_{R,c}:=S^{1}\times\\{(\textnormal{Im}\,z^{\prime},c,y),\;\|(\textnormal{Im}\,z^{\prime},y)\|<R\\}$:

$\mathcal{M}^{\prime}(J):=\\{(t,c,u),\;(t,c)\in
T\times\overset{\vspace*{-1pt}\circ}{\mathcal{D}},\,u:(\mathbb{D},\partial\mathbb{D})\to(Y,\overset{\vspace*{-1pt}\circ}{\Gamma}_{R,c}),\,\overline{\partial}_{J_{(t,c)}}u=0,\,[u]=E_{c}\\}$

Denoting $\tau_{c}$ the translation of vector $c\in\mathbb{R}^{k-1}$, the
spaces $\mathcal{M}(J)$ and $\mathcal{M}^{\prime}(J)$ are in one-to-one
correspondence via the map $(t,c,u)\to(t,c,\tau_{-c}\circ u)$ with the
following spaces, more suited to our analysis:

$\begin{array}[]{l}\mathcal{M}_{0}(J):=\\{(t,c,u),\;(t,c)\in
T\times\mathcal{D},\,u:(\mathbb{D},\partial\mathbb{D})\to(Y,\Gamma_{R,0}),\,\overline{\partial}_{\tau_{c}^{*}J_{(t,c)}}u=0,\,[u]=E_{0}\\}.\\\
\mathcal{M}_{0}^{\prime}(J):=\\{(t,c,u),\;(t,c)\in
T\times\overset{\vspace*{-1pt}\circ}{\mathcal{D}},\,u:(\mathbb{D},\partial\mathbb{D})\to(Y,\overset{\vspace*{-1pt}\circ}{\Gamma}_{R,0}),\,\overline{\partial}_{\tau_{c}^{*}J_{(t,c)}}u=0,\,[u]=E_{0}\\}.\end{array}$

Notice that the almost complex structures $\tau_{c}^{*}J_{(t,c)}$ do not
belong anymore to $\mathcal{J}(R,4M,J_{\Lambda}(t,c))$ but to

$\mathcal{J}(R,4M,c,J_{\Lambda}):=\\{J\in\mathcal{J}(\omega)\;|\;J_{|V_{\Lambda}}\equiv\tau_{c}^{*}J_{\Lambda}(t,c)),J_{|\tau_{c}^{*}(V_{\Gamma}\cup\\{\|(y,z)\|>K\\})}\equiv
J_{\textnormal{st}}\\}.$

Also, $\mathcal{M}(J)$ is endowed with an action of
$G:={\textnormal{PSL}}_{2}(\mathbb{R})\simeq{\textnormal{Aut}}(\mathbb{D},j)$
by source reparametrization of $u$.

######  Lemma 3.8.

If $(t,c,u)\in\mathcal{M}_{0}(J)$ and
$\textnormal{Im}\,u\subset\tau_{c}^{-1}(F_{4R,M})$, $u$ is a vertical disc,
and $\tau_{c}^{*}J_{(t,c)}$ is Fredholm-regular with respect to $u$.

The Fredhlom theory and its usual notations are assumed in the following
proof. The reader can consult [MS12], or look at the proof of lemma 3.10 for
few details.

Proof: $\tau_{c}^{*}J_{(t,c)}=J_{\textnormal{st}}$ on
$\tau_{c}^{-1}(F_{4R,M})$, so the first part of the assertion directly follows
from lemma 3.7. Moreover $\tau_{c}^{*}J_{(t,c)}$ is linear (constant) on this
region, so the derivative

$D_{u}\xi:=D\overline{\partial}_{\tau_{c}^{*}J_{(t,c)}}(u)\xi=\overline{\partial}_{\tau_{c}^{*}J_{(t,c)}}\xi=\overline{\partial}\xi.$

Therefore, if $\xi$ belongs to $\ker D_{u}$, $\xi$ is easily seen to be a
constant in an $n-1$-real parameter space, once modded out the source
reparametrization. It follows that $\dim\ker
D\overline{\partial}_{\tau_{c}^{*}J_{(t,c)}}(u)=n-1$, while the index of
$D\overline{\partial}_{\tau_{c}^{*}J_{(t,c)}}(u)$ is $n-1$, so
$D\overline{\partial}_{\tau_{c}^{*}J_{(t,c)}}(u)$ is indeed surjective.
$\square$

######  Lemma 3.9.

If $(t,c,u)\in\mathcal{M}_{0}(J)$, then $u$ is somewhere injective, and almost
all points of $\textnormal{Im}\,u$ have exactly one preimage.

This point is pefectly clear because the class $E$ has least area in
$H_{2}(M,\Gamma_{c})$. Here is nonetheless a full proof.

Proof: We first recall that the critical set ${\cal C}(u)$ of a non-constant
$J$-holomorphic disc $u$ is a discrete subset of $\mathbb{D}$, hence
negligible (see [MS12, Lemma 2.4.1]). Let
$(t,c,\tilde{u})\in\mathcal{M}_{0}(J)$, so $u:=\tau_{c}\circ\tilde{u}$ is
$J_{(t,c)}$-holomorphic. By [Laz00], there exists a holomorphic disc
$v:(\mathbb{D},\partial\mathbb{D})\to(Y,\Gamma_{c})$ with
$\textnormal{Im}\,v\subset\textnormal{Im}\,u$, and $v^{-1}(v(z))=\\{z\\}$ for
almost all points $z\in\mathbb{D}$. Then,

$0<\mathcal{A}_{\omega}(v)\leq\mathcal{A}_{\omega}(u)=[\omega][E]=1.$

Since moreover $\text{Hur}_{2}(Y,\Gamma_{c})$ is generated by the class $[E]$,
it follows that $\mathcal{A}_{\omega}(v)=A=\mathcal{A}_{\omega}(u)$. Let
$D^{\prime}:=u^{-1}(v(\mathbb{D}))$. This is an open subset of $\mathbb{D}$
because each local branch of $u^{-1}\circ v$ is holomorphic (its derivatives
are $\mathbb{C}$-linear maps) and non-constant. We also claim that it has full
measure in $\mathbb{D}$. Indeed,

$1=\mathcal{A}_{\omega}(u)=\int_{D^{\prime}}u^{*}\omega+\int_{\mathbb{D}\backslash
D^{\prime}}u^{*}\omega\geqslant\mathcal{A}_{\omega}(v)+\int_{\mathbb{D}\backslash
D^{\prime}}u^{*}\omega\geq\mathcal{A}_{\omega}(v)=1.$

This chain of inequalities consists therefore of equalities only, so

$0=\int_{\mathbb{D}\backslash
D^{\prime}}u^{*}\omega=\int_{\mathbb{D}\backslash D^{\prime}\cup\,{\cal
C}(u)}u^{*}\omega.$

On $\mathbb{D}\backslash(D^{\prime}\cup{\cal C}(u))$, $u$ is a local
diffeomorphism and $u^{*}\omega>0$. Thus

$0={\cal L}\textnormal{eb}\big{(}(\mathbb{D}\backslash
D^{\prime})\backslash{\cal C}(u)\big{)}\geq{\cal
L}\textnormal{eb}(\mathbb{D}\backslash D^{\prime})-{\cal
L}\textnormal{eb}({\cal C}(u))={\cal L}\textnormal{eb}(\mathbb{D}\backslash
D^{\prime}),$

which shows that $D^{\prime}$ has full measure. Define now

$\textnormal{NI}:=\\{p\in v(\mathbb{D})\,|\,\\#v^{-1}(\\{p\\})\neq 1\\},\hskip
8.5359ptN_{v}:=v^{-1}(\overline{\textnormal{NI}}),\hskip
8.5359ptN_{u}:=u^{-1}(\overline{\textnormal{NI}}).$

It is clear that $\overline{\textnormal{NI}}\subset\textnormal{NI}\cup v({\cal
C}(v))=v(v^{-1}(\textnormal{NI})\cup{\cal C}(v))$. Since ${\cal C}(v)$ and
$v^{-1}(\textnormal{NI})$ are both negligible sets, and $v$ is smooth, it
follows that $\overline{\textnormal{NI}}$ has vanishing $2$-dimensional
Hausdorff measure. Then

$N_{u}=u^{-1}(\overline{\textnormal{NI}})=\left(u^{-1}(\overline{\textnormal{NI}})\cap{\cal
C}(u)\right)\cup\left(u^{-1}\big{(}\overline{\textnormal{NI}}\backslash
u({\cal C}(u))\big{)}\right).$

Since ${\cal C}(u)$ is discrete, the measure of the first set in the right
hand side vanishes. On the other hand, $u^{-1}$ is smooth on
$\overline{\textnormal{NI}}\backslash u({\cal C}(u))$, so it preserves the
vanishing of the $2$-dimensional Hausdorff measure, so the latter is also
Lebesgue-negligible. We therefore conclude that ${\cal
L}\textnormal{eb}(N_{u})=0$. Now $\varphi:=v^{-1}\circ u:D^{\prime}\backslash
N_{u}\longrightarrow\mathbb{D}\backslash N_{v}$ is a holomorphic map,
surjective on $\mathbb{D}\backslash N_{v}$ because
$\textnormal{Im}\,v\subset\textnormal{Im}\,u$, so the map
$\deg\varphi:\mathbb{D}\backslash N_{v}\to\mathbb{Z}$ that associates to each
point of $\mathbb{D}\backslash N_{v}$ the algebraic count of its preimages
takes values in $\mathbb{N}^{*}$. Thus,

$1=\int_{\mathbb{D}}u^{*}\omega=\int_{D^{\prime}\backslash
N_{u}}u^{*}\omega=\int_{D^{\prime}\backslash
N_{u}}(v\circ\varphi)^{*}\omega=\int_{\mathbb{D}\backslash
N_{v}}\deg\varphi\;v^{*}\omega\geq\int_{\mathbb{D}}v^{*}\omega=1$

(the last inequality holds because $v^{*}\omega>0$). Thus $\deg\varphi=1$
almost everywhere in $\mathbb{D}\backslash N_{v}$, therefore constantly equals
$1$ because $\varphi$ is holomorphic. Thus $\varphi=v^{-1}\circ u$ is
injective on $D^{\prime}\backslash N_{u}$, so also is
$u_{|D^{\prime}\backslash N_{u}}$. This proves the lemma because
$D^{\prime}\backslash N_{u}$ has full measure and $u$ is smooth. $\square$

######  Lemma 3.10.

For every smooth section $J\in\Gamma^{\ell}(\mathcal{J}(R,M,J_{\Lambda}))$,
and for every $\varepsilon>0$, there exists
$J^{\prime}\in\Gamma^{\ell}(\mathcal{J}(R,M,J_{\Lambda}))$, with
$d_{\mathbb{\mathcal{C}}^{\ell}}(J,J^{\prime})<\varepsilon$, such that
$\mathcal{M}_{0}^{\prime}(J^{\prime})$ is a smooth submanifold.

Proof : This is standard and lengthy, so we only sketch the proof briefly,
referring to [MS12, §3] for details. One defines the universal moduli space

$\mathcal{M}:=\\{(u,J,t,c),\;(t,c)\in
T\times\overset{\vspace*{-1pt}\circ}{\mathcal{D}},\;J\in\mathcal{J}(R,M,J_{\Lambda}),\,u:(\mathbb{D},\partial\mathbb{D})\to(Y,\overset{\vspace*{-1pt}\circ}{\Gamma}_{R,0}),\,\overline{\partial}_{\tau_{c}^{*}J_{(t,c)}}u=0,\,[u]=E_{0}\\}.$

The first point is to see that $\mathcal{M}$ is a Banach manifold. Let
$\mathbb{\mathcal{B}}^{k,p}:=W^{k,p}((\mathbb{D},\partial\mathbb{D})\to(Y,\overset{\vspace*{-1pt}\circ}{\Gamma}_{R,0}))$,
$\mathcal{E}^{k-1,p}$ the Banach vector bundle over
$\mathbb{\mathcal{B}}^{k,p}\times\mathcal{J}(R,M,J_{\Lambda})\times
T\times\overset{\vspace*{-1pt}\circ}{\mathcal{D}}$ whose fiber is

$\mathcal{E}^{k-1,p}(u,J,t,c):=W^{k-1,p}\big{(}(\mathbb{D},\partial\mathbb{D})\to(\Lambda^{(0,1)}_{\tau_{c}^{*}J}(u^{*}TY),\Lambda^{(0,1)}_{\tau_{c}^{*}J}(u^{*}T\Gamma_{R,0}))\big{)}.$

Then $\mathcal{F}(u,J,t,c):=\overline{\partial}_{\tau_{c}^{*}J}u$ defines a
differentiable section of this Banach bundle, whose zero-section is precisely
$\mathcal{M}$. The derivative of $\mathcal{F}$ at the zero-section can be
computed explicitely (using any connection on $\mathcal{J}(R,M,J_{\Lambda})$,
we identify $T_{(t,c,J)}\mathcal{J}(M,J_{\Lambda})$ with a subset of
$T_{t}T\times T_{c}\mathbb{R}^{k-1}\times T_{J}\mathcal{J}(\omega)$):

$d\mathcal{F}(\xi,Y,\delta t,\delta
c)=D_{u}\xi+\frac{1}{2}\tau_{c}^{*}\left(Y(u)+\frac{\partial J}{\partial
c}(u)\delta c+\frac{\partial J}{\partial t}(u)\delta t\right)\circ du\circ j,$

where $D_{u}(\xi):=D\overline{\partial}_{\tau_{c}^{*}J}(u)\xi$. We claim that
this differential is surjective at all $(u,t,c,J)\in\mathcal{M}$. Since
$D_{u}$ is onto for the elements $u\in\mathcal{M}$ which are contained in
$\tau_{c}^{-1}(F_{R,4M})$ by lemma 3.8, $d\mathcal{F}(u,J,t,c)$ is onto for
all these elements. Let $(u,J,t,c)$ be another element of $\mathcal{M}$. To
show that $d\mathcal{F}(u,J,t,c)$ is surjective, consider an element $\eta$ of
the dual that anihiliates the image of $d\mathcal{F}(u,J,t,c)$. Then in
particular, for all elements $(\xi,Y)\in
T_{(u,J)}\mathbb{\mathcal{B}}^{k,p}\times\mathcal{J}(M,J_{\Lambda}(t,c))$,

$\langle D_{u}\xi,\eta\rangle_{L^{2}}=0\text{ and }\langle Y(u)\circ du\circ
j,\eta\rangle_{L^{2}}=0.$

The first equation guarantees that if $\eta$ vanishes on some region, it
vanishes identically. Now $\textnormal{Im}\,u$ intersects the region
$Y\backslash(F_{R,4M}\cup V_{\Lambda})$, where, by lemma 3.9 there exists a
somewhere injective point of $u$, and where $J$ is unconstrained by definition
of $\mathcal{J}(M,J_{\Lambda}(t,c))$. Using the freedom on $Y$ at such a
point, the classical argument shows that $\eta\equiv 0$. It follows that
$d\mathcal{F}(u,J,t,c)$ is surjective also at this point
$(u,J,t,c)\in\mathcal{M}$. Thus, $0$ is a regular value of $\mathcal{F}$ and
$\mathcal{M}=\mathcal{F}^{-1}(0)$ is a Banach manifold.

Now the map $\pi:\mathcal{M}\to\mathcal{J}(R,M,J_{\Lambda})$ is a Fredholm
map, so for any smooth section
$J:T\times\overset{\vspace*{-1pt}\circ}{\mathcal{D}}\to\mathcal{J}(R,M,J_{\Lambda})$,
there is a $\mathbb{\mathcal{C}}^{\ell}$ $\varepsilon$-small deformation
$P^{\prime}$ of the submanifold $P:=\\{(J(t,c),t,c)\\}$, such that $\pi$ is
transverse to $P^{\prime}$ [Sma65]. Such a perturbation is obviously the graph
over $T\times\mathcal{D}$ of a section $J^{\prime}$, which verifies
$d_{\mathbb{\mathcal{C}}^{\ell}}(J,J^{\prime})<\varepsilon$. The
transversality ensures that
$\pi^{-1}(P^{\prime})=\mathcal{M}_{0}^{\prime}(J^{\prime})$ is a smooth
manifold. $\square$
We will call henceforth generic the sections
$J:T\times\mathcal{D}\to\mathcal{J}(R,M,J_{\Lambda})$ such that
$\mathcal{M}_{0}^{\prime}(J)$, hence $\mathcal{M}^{\prime}(J)$ is a smooth
submanifold. We now address the compactness.

######  Lemma 3.11.

Let $J\in\mathcal{J}(R,4M,J_{\Lambda})$ and
$(u_{n},t_{n},c_{n})\in\mathcal{M}(J)$. After extracting a subsequence,

  * $\bullet$

either there exists a sequence $g\in G={\textnormal{PSL}}_{2}(\mathbb{R})$
such that $(u_{n}\circ g_{n},t_{n},c_{n})$ converges in
$\mathcal{M}^{\prime}(J)$,

  * $\bullet$

or the $u_{n}$ are vertical discs for $n\gg 1$,
$\textnormal{Im}\,u_{n}=\mathbb{D}\times\\{(z^{\prime}_{n},y_{n})\\}$, with
$(z^{\prime}_{n},y_{n})\to\partial B_{\infty}(R)$.

Proof: We argue throughout this proof modulo extraction of subsequence. Since
$T$ is assumed to be compact, we can assume that $(t_{n},c_{n})$ converges to
$(t,c)\in T\times\mathcal{D}$ so $J_{(t_{n},c_{n})}$ converges to $J_{(t,c)}$.
Then $(u_{n})$ is a sequence of $J_{(t_{n},c_{n})}$-holomorphic discs with
boundaries in $\Gamma_{R,c_{n}}$, representing the class $E_{c_{n}}$. By
proposition 3.4, either the $u_{n}$ meet $F_{R,M}$ and then are vertical discs
so the conclusion of the lemma is achieved, or their images remain in
$Y\backslash F_{R,M}$ (and obviously in a compact set of $Y$ by maximum
principle since $J=J_{\textnormal{st}}$ at infinity). In the latter case,
$(u_{n})$ is a sequence of $J_{(t_{n},c_{n})}$-holomorphic discs with
boundaries in a subset of $\Gamma_{R,c_{n}}$ contained in a neighbourhood of a
compact subset of $\Gamma_{R,c}$. Their images are moreover contained in a
compact subset of $Y$, with bounded symplectic area. Then Gromov’s compactness
theorem guarantees the existence of an accumulation point of $(u_{n})$ (in the
sense of Gromov), which is a non-constant $J_{(t,c)}$-bubble tree. Since
however $\pi_{2}(Y)=0$ and $[u_{n}]=E_{c}$ has least area among the non-
constant symplectic discs with boundary on $\Gamma_{c}$, this means that there
is $g_{n}\in G$ such that $u_{n}\circ g_{n}$ converges to an element of
$\mathcal{M}(J)$. $\square$

We fix a generic section
$J:T\times\mathcal{D}\to\mathcal{J}(R,4M,J_{\Lambda})$ until the end of this
section, and we define

$\begin{array}[]{l}W:=\mathcal{M}(J)\times_{G}\overline{\mathbb{D}}:=(\mathcal{M}(J)\times\overline{\mathbb{D}})_{/G},\\\
W^{\prime}:=\mathcal{M}^{\prime}(J)\times_{G}\mathbb{D},\end{array}$

where $g\in G$ acts on $(u,t,c,w)$ by $g\cdot(u,t,c,w):=(u\circ
g,t,c,g^{-1}(w))$. We also put

$W^{\partial}:=W\backslash
W^{\prime}=\big{(}\mathcal{M}(J)\backslash\mathcal{M}^{\prime}(J)\big{)}\times_{G}\overline{\mathbb{D}}\,\cup\,\mathcal{M}^{\prime}(J)\times_{G}\partial\mathbb{D}=W^{\partial}_{\text{V}}\cup
W^{\partial}_{\text{H}}.$

We now explain that $W$ has the structure of a smooth compact manifold with
boundary $W^{\partial}$ (and corners). Notice first that lemma 3.7 obviously
shows the following:

######  Lemma 3.12.

The map

$\begin{array}[]{rcccl}i&:&T\times\\{\|(z^{\prime},y)\|=R\\}\times\overline{\mathbb{D}}&\longrightarrow&W^{\partial}_{\text{V}}\\\
&&(t,z^{\prime},y,w)&\longmapsto&[t,\textnormal{Re}\,z^{\prime},w\mapsto(w,z^{\prime},y,0),w]\end{array}$

is a one-to one correspondence.

Also, lemma 3.10 provides $\mathcal{M}^{\prime}(J)$ with a structure of smooth
manifold, for which the action of $G$ on
$\mathcal{M}^{\prime}(J)\times\overline{\mathbb{D}}$ is smooth, proper and
free. As a result, $W^{\prime}$ is a smooth manifold,
$W^{\partial}_{\text{H}}$ lies in the closure of $W^{\prime}$, as a boundary
of $W$.

######  Lemma 3.13.

$W$ is a compact manifold with $W^{\partial}$ as boundary and corners along
$W^{\partial}_{\text{H}}\cap W^{\partial}_{\text{V}}$.

Proof: Let $[t_{n},c_{n},u_{n},w_{n}]$ be a sequence in $W^{\prime}$. We can
extract a subsequence for which $(t_{n},c_{n})\to(t,c)\in T\times\mathcal{D}$.
Lemma 3.11 leaves us with an alternative, up to extracting a subsequence.
Either $\exists g_{n}\in G$ such that $u_{n}\circ g_{n}$ converges to
$u\in\mathcal{M}^{\prime}(J)$. We can further extract so that
$g_{n}^{-1}(w_{n})$ converges to $w\in\overline{\mathbb{D}}$, and
$[t_{n},c_{n},u_{n},w_{n}]\to[t,c,u,w]\in W^{\prime}\cup
W^{\partial}_{\text{H}}$. Or the $u_{n}$ are vertical discs of the form
$u_{n}(z)=(g_{n}(z),z^{\prime}_{n},y_{n}^{\prime})$, where
$(z^{\prime}_{n},y^{\prime}_{n})\to\partial B(R)$ and
$g_{n}:\mathbb{D}\to\mathbb{D}$ are ramified coverings. Since the $u_{n}$ have
area $1$ by assumption, these coverings are in fact automorphisms of the
discs. Extracting again from $g_{n}^{-1}(w_{n})$ a converging subsequence, we
see that $[t_{n},c_{n},u_{n},w_{n}]$ converge to an element of
$W^{\partial}_{\text{V}}$. This discussion also provide a parametrization of a
neighbourhood of $W^{\partial}_{\text{V}}$ by
$\mathbb{D}\times\\{R-\varepsilon<\|(z^{\prime}_{n},y_{n}^{\prime})\|\leq
R\\}$, hence the fact that the points of $W^{\partial}_{\text{V}}\backslash
W^{\partial}_{\text{H}}$ are boundary points, and the points of
$W^{\partial}_{\text{V}}\cap W^{\partial}_{\text{H}}$ are corners. $\square$

The space $\mathcal{M}(J)\times_{G}\overline{\mathbb{D}}$ comes with a natural
evaluation

$\begin{array}[]{rcccl}\sigma&:&W&\longrightarrow&T\times Y\\\
&&[u,t,c,w]&\longmapsto&(t,u(w)).\end{array}$

As a quotient of a $G$-invariant smooth map on
$\mathcal{M}(J)\times\overline{\mathbb{D}}$, $\sigma$ is smooth. It also sends
$W^{\partial}_{\text{H}}=\mathcal{M}(J)\times_{G}\partial\mathbb{D}$ to
$T\times\Gamma_{R}$ by construction. Notice that lemma 3.11 implies that
$W^{\partial}_{\text{H}}$ is a smooth manifold with boundary
$W^{\partial}_{\text{H}}\cap W^{\partial}_{\text{V}}$. By lemma 3.12,
$\sigma_{|\partial W^{\partial}_{\text{H}}}:\partial
W^{\partial}_{\text{H}}\to T\times\partial\Gamma_{R}$ is a degree $1$ map,
hence so is $\sigma_{|W^{\partial}_{\text{H}}}:W^{\partial}_{\text{H}}\to
T\times\Gamma_{R}$. Lemma 3.12 also states that
$\sigma_{|W^{\partial}_{\text{V}}}:W^{\partial}_{\text{V}}\to
T\times\mathbb{D}\times\\{\|(z^{\prime},y)\|=R,\;x=0\\}$ has degree $1$. A
consequence of these two points is that $\textnormal{Im}\,\sigma$ is
homologous to $T\times Z_{R}$, relative to $T\times\partial Z_{R}$. Indeed,
the concatenation $(T\times Z_{R})\star\textnormal{Im}\,\sigma$, provides an
element of $H_{\dim Z_{R}+\dim T}(T\times Y)$, well-defined because
$\partial\sigma$ has degree $1$, and because $T\times Z_{R}$ and
$\textnormal{Im}\,\sigma$ coincide over $T\times\partial Z_{R}$. But this
homology group is

$H_{2k+l+\dim T}(T\times Y)=H_{2k+l+\dim T}(T\times\mathbb{C}^{k}\times
i\mathbb{R}^{l}\times\mathbb{T}^{l})\simeq H_{2k+l+\dim
T}(T\times\mathbb{T}^{l})=0\text{ (because $k>0$).}$

The image of $\sigma$ is moreover covered by $J_{(t,c)}$-holomorphic discs,
for $(t,c)\in T\times\mathbb{R}^{k}$. Summarizing the discussion of this
paragraph, we have obtained:

######  Theorem 3.14.

Let $M>0$ be such that the conclusion of proposition 3.4 holds, and

  * $\bullet$

$Y:=(\mathbb{C}^{k}\times
i\mathbb{R}^{l}\times\mathbb{T}^{l},\omega=\pi_{*}\omega_{\textnormal{st}})$,
$\mathcal{D}_{R}:=\\{\|(z^{\prime},y)\|_{\infty}\leq
R\\}\subset\mathbb{C}^{k-1}\times\mathbb{R}^{l}$,

  * $\bullet$

$\Gamma_{R}:=S^{1}\times\mathcal{D}_{R}\times\\{x=0\\}\subset Y$,
$Z_{R}=\mathbb{D}\times\mathcal{D}_{R}\times\\{x=0\\}\subset Y$,

  * $\bullet$

$F_{R,M}$ an $M$-neighbourhood of
$\mathbb{D}\times\partial\mathcal{D}_{R}\times\\{x=0\\}$ in the
$\|(z_{1},z^{\prime},x,y)\|_{\infty}$-norm,

  * $\bullet$

$\Lambda\Subset Y\backslash(\Gamma_{R}\cup F_{R,4M})$,
$V_{\Lambda}\in\textnormal{Op}\,(\Lambda,Y\backslash(\Gamma_{R}\cup
F_{R,4M}))$, $J_{\Lambda}:T\times Y\to\mathcal{J}(\omega)$,

  * $\bullet$

$\mathcal{J}(R,4M,J_{\Lambda}):=\\{(t,c,J)\in
T\times\mathcal{D}\times\mathcal{J}(\omega)\,|\,J_{|V_{\Lambda}}\equiv
J_{\Lambda}(t,c),\,J\equiv J_{\textnormal{st}}\text{ on }\\{|(z,y)|>K\\}\cup
F_{R,4M}\\}$ (for some $K\gg 1$).

For a generic section
$J:T\times\mathcal{D}_{R}\to\mathcal{J}(R,4M,J_{\Lambda})$ \- which exists in
any $\mathbb{\mathcal{C}}^{\ell}$-neighbourhood of a given smooth section -,
there exists a manifold with boundary and corners $W$ and a smooth map
$\sigma:(W,\partial W)\to(T\times Y,T\times\partial Z_{R})$ such that:

  * (i)

$[\textnormal{Im}\,\sigma]=[T\times Z(R)]\in H(T\times Y,T\times\partial
Z_{R},\mathbb{Z})$,

  * (ii)

$\forall p\in\textnormal{Im}\,\sigma$, there exist $(t,c)\in
T\times\mathcal{D}_{R}$,
$u:(\mathbb{D},\partial\mathbb{D})\to(Y,\Gamma_{R,c})$ such that
$p\in\\{t\\}\times\textnormal{Im}\,u$, $\mathcal{A}_{\omega}(u)=1$,
$\overline{\partial}_{J(c,t)}u=0$.

There is nothing to prove here, since all proofs have been done thoughout this
section. A remark is however in order. In order to get the point (i), one
needs to know the orientability of the moduli spaces, which have not been
addressed here. It is a well-known issue in our specific situation, but if one
wishes to forget it so as to consider this paper mostly self-contained, the
point (i) has to be replaced by

$[\textnormal{Im}\,\sigma]=[T\times\mathbb{Z}_{R}]\in H(T\times
Y,T\times\partial Z_{R},\mathbb{Z}_{2}).$

This is a weaker statement, which is enough for our applications to the
$\mathbb{\mathcal{C}}^{0}$-rigidity of the reduction (see below), and to
theorems 3.2 and 3.1, provided the knotting holds in the
$\mathbb{Z}_{2}$-coefficient setting. For theorem 3.1 for instance, it amounts
to assuming that $\Phi(\mathbb{R}^{l}\times\\{0\\})$ intersects $Z_{R}$
transversally, an odd number of times.

###  3.3 Proof of theorem 3.1

Let $\Phi:\mathbb{R}^{l}\times B^{2n}(a)\to\mathbb{C}^{n}$ be a parametric
symplectic embedding, which satisfies the assumption of theorem 3.1 with
$C:=4M$, $M$ being the constant obtained in the previous paragraph, in
particular theorem 3.14. Thus, (i) for $|t|\geq K$,
$\Phi(t,\cdot)=\textnormal{Id}\,+t$, (ii) $\Phi$ is knotted with $\partial
Z_{R}=S^{1}\times\mathcal{D}_{R}\times\\{x=0\\}$ (recall that we rescaled the
space so that $A=1$), and (iii) $\textnormal{Im}\,\Phi$ does not intersect
$F_{R,4M}$. Since $B^{2n}(a)$ is contractible, (ii) amounts to saying that
$\textnormal{Im}\,\Phi\cap\partial Z_{R}=\emptyset$ and
$\Phi(\mathbb{R}^{l}\times\\{0\\})$ has a non-vanishing homological
intersection with $Z_{R}$, relative to $\partial Z_{R}$ (say with coefficients
in $\mathbb{Z}$, or in $\mathbb{Z}_{2}$ if one decides to forget about the
orientability of the moduli spaces). Restricting $\Phi$ to
$\mathbb{R}^{l}\times B^{2n}(a^{\prime})$, $a^{\prime}<a$, we can also assume
that $\overline{\textnormal{Im}\,\Phi}\cap\partial Z_{R}=\emptyset$.

By (i), we can compactify the domain of $\Phi$ to
$\mathbb{T}^{l}\times\overline{B^{2n}(a^{\prime})}$, getting a parametric
symplectic embedding
$\Phi:\mathbb{T}^{l}\times\overline{B^{2n}(a^{\prime})}\to\mathbb{C}^{k}\times
i\mathbb{R}^{l}\times\mathbb{T}^{l}=Y$ which still satisfies (ii), meaning
that the homological intersection of $[\Phi(\mathbb{T}^{l}\times\\{0\\})]$
with $[Z_{R}]$ relative to $\partial Z_{R}$ does not vanish. Since
$\textnormal{Dom}\,\Phi$ is now compact, there exists $K>0$ such that
$\textnormal{Im}\,\Phi\Subset\\{|(z,y)|<K\\}$. Taking $T:=\mathbb{T}^{l}$, we
define $\Lambda(t):=\Phi(\\{t\\}\times\overline{B^{2n}(a^{\prime})})$ and
$J_{\Lambda}(t):=\Phi(t,\cdot)_{*}J_{\textnormal{st}}$. Since
$\mathcal{J}(\omega)$ is a fiber bundle over $Y$ with contractible fiber,
there exists a smooth path $J(t)\in\mathcal{J}(R,4M,\Lambda(t))$. Applying
theorem 3.14, we get a filling $\sigma:(W,\partial W)\to(T\times
Y,T\times\partial Z_{R})$ with holomorphic discs. Since
$[\Phi(T\times\\{0\\})]\cdot[Z_{R}]\neq 0$, the graph
Graph$(\Phi):=\\{(t,\Phi(t,0)),t\in T\\}$ has non-vanishing homological
intersection with $T\times Z_{R}$. Moreover, since by theorem 3.14 (i), our
filling $\textnormal{Im}\,\sigma$ is homologous to $T\times Z_{R}$ relative to
$T\times\partial Z_{R}$,
$[\text{Graph}(\Phi)]\cdot[\textnormal{Im}\,\sigma]\neq 0$ and there exists
$t\in\mathbb{T}^{l}$ such that $(t,\Phi(t,0))\in\textnormal{Im}\,\sigma$. By
theorem 3.14 (ii), there exists $t^{\prime}\in\mathbb{T}^{l}$ and a disc
$u:(\mathbb{D},\partial\mathbb{D})\to(Y,\Gamma_{R})$ such that
$(t,\Phi(t,0))\in\\{t^{\prime}\\}\times\textnormal{Im}\,(u)$ (hence
$t=t^{\prime}$ and $\Phi(t,0)\in\textnormal{Im}\,(u)$),
$\mathcal{A}_{\omega}(u)=1$, and $\overline{\partial}_{J}u=0$ for some
$J\in\Gamma^{\ell}(\mathcal{J}(M,J_{\Lambda}))$. This disc is therefore
symplectic, has total area $1$, and its trace on the symplectic ball
$\Phi(\\{t\\}\times B^{2n}(a^{\prime}))$ is
$\Phi(t,\cdot)_{*}J_{\textnormal{st}}$-holomorphic. The classical monotonicity
argument then shows that $1\geq a^{\prime}$. Since this holds for all
$a^{\prime}<a$, we get $a\leq 1$. $\square$

###  3.4 Proof of theorem 3.2

Let $\Phi:\mathbb{T}^{k}\times\mathbb{R}^{l}\hookrightarrow\mathbb{C}^{k+l}$
be a Lagrangian embedding that satisfies the assumptions of theorem 3.2: (i)
$\Lambda:=\textnormal{Im}\,\Phi$ coincides with
$S^{1}(a)^{k}\times\mathbb{R}^{l}$ outside a compact set, and (ii) $L$ is
knotted with $\partial Z_{R}$, meaning that
$[\Phi(\\{*\\}\times\mathbb{R}^{l})]\cdot[Z_{R}]\neq 0$, where coefficients
belong to $\mathbb{Z}$ (or $\mathbb{Z}_{2}$) and homology is understood as
always in these paragraphs relative to the boundary. Finally, $\Lambda$ does
not meet $F_{R,4M}$.

By (i), we can compactify the domain of $\Phi$ to
$\mathbb{T}^{k}\times\mathbb{T}^{l}$, getting a Lagrangian embedding
$\Phi:\mathbb{T}^{n}\hookrightarrow\mathbb{C}^{k}\times
i\mathbb{R}^{l}\times\mathbb{T}^{l}=Y$, which still verifies (ii), and is
compactly contained into $\\{\|(y,z)\|<K\\}$ for some $K\in\mathbb{R}$. We
consider $T=\\{*\\}$, $\Lambda=\textnormal{Im}\,\Phi$, and an almost-complex
structure for the moment arbitrary in $\textnormal{Op}\,(\Lambda)$. Applying
theorem 3.14, we get for each $J_{\Lambda}$ a complex structure
$J\in\mathcal{J}(R,4M,J_{\Lambda})$ and a $J$-holomorphic disc
$u:(\mathbb{D},\partial\mathbb{D})\to(Y,\Gamma_{c})$ with $[u]=E_{c}$ and
$\textnormal{Im}\,u\cap\Lambda\neq\emptyset$. Moreover, since $J$ is standard
at infinity, this disc lies in the compact domain $\\{|(y,z)|\leq K\\}$, where
$K$ does not depend on $J_{\Lambda}$.

We now proceed to a neck-stretching argument. Recall that $\Lambda$ is the
image by an exact lagrangian embedding $\Phi$ of
$L:=S^{1}(a)^{k}\times\mathbb{T}^{l}$. Consider on $L$ the Euclidean metric
$g:=g_{\textnormal{st}}$. As usual in SFT, we consider a neighbourhood
$\mathcal{U}$ of $\Lambda$ symplectomorphic to a neighbourhood of the zero
section in $T^{*}L$, endow it with the metric $g$ induced canonically from $g$
on $L$, and consider $\Sigma:=\\{g=\varepsilon_{0}\\}\subset\mathcal{U}$. For
small enough $\varepsilon_{0}$, this is a contact type hypersurface which
splits $Y$ into two pieces $Y^{-}\cup Y^{+}$, where $Y^{-}$ has concave
boundary at $X$ (and contains $\partial Z_{R}$), and $Y^{+}$ has convex
boundary (and contains $\Lambda$). Now we consider a neck-stretching along
$\Sigma$. Notice that since $\Lambda$ is disjoint from $\Gamma_{R}\cup
F_{R,4M}$ and contained into $\\{|(y,z)|<K\\}$, the same holds for $\Sigma$
provided $\varepsilon_{0}$ is small enough. We can therefore consider a
degeneration $J_{\varepsilon}$ of the complex structure with the following
properties.

  * $\bullet$

$J_{\Lambda}(\varepsilon)$ is defined in $\textnormal{Op}\,(\Lambda)$ and
stretches the neck of $\Sigma$ when the parameter $\varepsilon$ goes to
$+\infty$.

  * $\bullet$

$J_{\varepsilon}$ is generic in $\mathcal{J}(R,4M,J_{\Lambda}(\varepsilon))$,

  * $\bullet$

$J_{\varepsilon}$ is uniformly bounded in the
$\mathbb{\mathcal{C}}^{\ell}$-topology outside the neck-stretching zone.

The previous discussion guarantees the existence of a disc
$u_{\varepsilon}:(\mathbb{D},\partial\mathbb{D})\to(Y,\Gamma_{c_{\varepsilon}})$
for some $c_{\varepsilon}\leq R$, with
$\overline{\partial}_{J_{\varepsilon}}u_{\varepsilon}=0$,
$[u_{\varepsilon}]=E_{c_{\varepsilon}}$, $u_{\varepsilon}$ meets $\Lambda$,
and $\textnormal{Im}\,u_{\varepsilon}\subset\\{|(y,z)|<K\\}$. The compactness
theorem in SFT thus implies that there is some $\varepsilon_{n}$ such that
$u_{n}:=u_{\varepsilon_{n}}$ converge to a holomorphic building $B$ [BEH+03],
whose main features are summarized below:

  * $\bullet$

It has a main component, in $Y^{-}$, which is a $J_{\infty}$-holomorphic map
$u_{\infty}^{\text{main}}:S\to Y^{-}$, where $S$ is a punctured disc (with $p$
punctures, $p\geqslant 1$ because $u_{n}$ intersects $\Lambda$, hence
$Y^{+}$), and whose boundary is sent to $\Gamma_{R}$ and has action $1$. The
action is computed with respect to the standard Liouville form $\lambda$ on
$Y\simeq\mathbb{C}^{k}\times T^{*}\mathbb{T}^{l}$.

  * $\bullet$

All other components on $Y^{-},Y^{+}$ are $J_{\infty}$-holomorphic maps whose
domains are punctured spheres (because the $u_{n}$ are discs).

  * $\bullet$

The components of the building in $Y^{-}$ are asymptotic at each puncture to a
negative Reeb orbit of $\partial Y^{-}=\Sigma$ (the boundary of the image is
oriented by the opposite of the Reeb flow). Similarly, the components in
$Y^{+}$ are asymptotic to positive Reeb orbits of $\Sigma$.

  * $\bullet$

There might be intermediate layers: components of $B$ in
$\Sigma\times\mathbb{R}$ (the symplectization of $\Sigma$). These components
are $J$-holomorphic for some cylindrical almost complex structure, again
punctured spheres, asymptotic to positive Reeb orbits at
$\Sigma\times\\{+\infty\\}$ and negative Reeb orbits at
$\Sigma\times\\{-\infty\\}$.

  * $\bullet$

The total symplectic area of these components is $1$, and they glue together
to form a topological disc (see figure 2). We will refer to this last property
by saying that the building $B$ forms a disc.

$Y^{-}$$\Sigma\times\mathbb{R}$$Y^{+}$$\Lambda$$\Sigma$$\partial
u\subset\Gamma_{R}$

Figure 2: A holomorphic building.

Denote by $-\gamma_{1},\dots,-\gamma_{p}$ the (negative) Reeb orbits to which
$u_{\infty}^{\text{main}}$ is asymptotic. Since $B$ forms a disc, its other
components glue together to form $p$ discs asymptotic to
$\gamma_{1},\dots,\gamma_{p}$. Since the $\gamma_{i}$ are Reeb orbits, they
project to closed geodesics of $L$ under the natural projection
$\pi:Y^{+}\simeq T^{*}_{\varepsilon_{0}}L\to L$ (recall that
$Y^{+}\simeq\\{g<\varepsilon_{0}\\}$). We write $[\pi\gamma_{i}]=\sum
k_{j}^{(i)}e_{j}\in H_{1}(L)$ for the homology classes of these geodesics,
where $e_{j}$ is the class of the $j$-th $S^{1}$-factor in
$L\simeq(S^{1})^{n}$, and $k_{j}^{(i)}\in\mathbb{Z}$. Observe at this point
that since $\gamma_{i}$ bounds a topological disc, $k_{j}^{(i)}=0$ for $j>k$
(because $Y\simeq\mathbb{C}^{k}\times T^{*}\mathbb{T}^{l}$). We will use the
following obvious fact:

######  Fact 3.15.

If $\gamma$ is a positive Reeb orbit of $\Sigma$, the cylinder

$\begin{array}[]{rcccl}\rho_{\gamma}&:&[0,1]\times
S^{1}&\longrightarrow&Y^{+}\simeq T^{*}_{\varepsilon_{0}}L\\\
&&(s,t)&\longmapsto&s\gamma(t)\end{array}$

is symplectic, with (oriented) boundaries $\gamma$ and $-\pi\gamma$.

######  Lemma 3.16.

For each $i$, $\sum k_{j}^{(i)}>0$.

Proof: Denote by $S_{i}$ the connected component of $B$ in $Y^{+}$ which is
glued to $u_{\infty}^{\text{main}}$ along $\gamma_{i}$. This component cannot
be a disc because $\pi\gamma_{i}$ is non-contractible in $Y^{+}\subset
T^{*}\mathbb{T}^{n}$. Therefore, $S_{i}$ has boundary components asymptotic to
positive Reeb orbits $\gamma_{i},\gamma_{i}^{1},\dots,\gamma_{i}^{k_{i}}$,
with $k_{i}\geqslant 1$. Observe that $\pi S_{i}$ provides a chain whose
boundary is $[\pi\gamma_{i}]+\sum[\pi\gamma_{i}^{k}]$, so
$[\pi\gamma_{i}]=-\sum[\pi\gamma_{i}^{k}]$. But since $B$ forms a disc, there
is a disc with positive area - composed of a gluing of different components of
$B$ \- which is asymptotic to each $-\gamma_{i}^{k}$. Gluing to these discs
the cylinders $\rho_{\gamma_{i}^{k}}$, we get discs with positive area, whose
boundaries lie in $L$ and represent the class $-[\pi\gamma_{i}^{k}]$ (see
figure 3). Therefore,

$\hskip
113.81102pt0<\sum\lambda[-\pi\gamma_{i}^{k}]=\lambda[\pi\gamma_{i}]=\sum_{i=1}^{k}k_{j}^{(i)}\,a.\hskip
128.0374pt\square$

$L$$S_{i}$$\gamma_{i}^{1}$$\Sigma$$\gamma_{i}$$\gamma_{i}^{2}$$\pi\gamma_{i}^{1}$$\pi\gamma_{i}^{2}$

Figure 3: Capping the class $[\pi\gamma_{i}]$.

The map obtained by gluing to $u_{\infty}^{\text{main}}$ the symplectic
cylinders $\rho_{\gamma_{i}}$ now gives a symplectic surface of area

$0<1-\sum\lambda[\pi\gamma_{i}]=1-\sum_{i=1}^{p}\left(\sum_{j}k_{j}^{(i)}\right)a\leqslant
1-a,$

where the last inequality holds because $p\geq 1$ and the previous lemma. We
get $1\geq a$. $\square$

##  4 Coisotropic $\mathbb{\mathcal{C}}^{0}$-rigidity

###  4.1 The reduction of a coisotropic submanifold

We now define precisely what we mean by reduction of a coisotropic
submanifold. This is just a generalization of the discussion for
hypersurfaces, which was held in [BO16].

######  Definition 4.1.

Let $\Sigma^{m}$ be a smooth manifold endowed with a foliation $\mathcal{F}$,
where the dimension of each leaf of the foliation equals $r$.

  1. 1.

We say that an open topological submanifold 111Recall that a topological
submanifold of a topological manifold $X$ is a subset $Y\subset X$, such that
there exists a topological manifold $Z$ and a map $i:Z\rightarrow X$ which is
a homeomorphism onto the image $i(Z)=Y$. $\;U^{m-r}\subset\Sigma$ is
(topologically) transverse to the foliation $\mathcal{F}$ on $\Sigma$ if $U$
has a neighbourhood $V\subset\Sigma$ such that $U$ intersects exactly once
each leaf of $V$ (where by a leaf of $V$ we mean a connected component of the
intersection of $V$ with a leaf of $\mathcal{F}$).

  2. 2.

Let $U,U^{\prime}\subset\Sigma$ be $(m-r)$-dimensional topological
submanifolds, that are transverse to the foliation $\mathcal{F}$. We say that
$U$ and $U^{\prime}$ are equivalent (denoting $U\sim U^{\prime}$) if there
exists a (continuous) homotopy $G:W\times[0,1]\rightarrow\Sigma$, $t\in[0,1]$,
of a topological manifold $W^{m-r}$, such that $G_{|W\times\\{0\\}}$ is a
homeomorphism onto $U$, $G_{|W\times\\{1\\}}$ is a homeomorphism onto
$U^{\prime}$, and such that for each $x\in W$, the trajectory $t\mapsto
G(x,t)$ goes along a leaf of $\mathcal{F}$.

  3. 3.

The reduction of a smooth manifold $\Sigma$ endowed with a foliation
$\mathcal{F}$, denoted by $\textnormal{Red}\,(\Sigma,\mathcal{F})$, is defined
as the set of open topological submanifolds $U^{m-r}\subset\Sigma$ which are
transverse to the characteristic foliation of $\Sigma$, considered modulo the
above equivalence relation. If $\Sigma$ is a coisotropic (or, more generally,
a pre-symplectic) submanifold of a symplectic manifold $M$, and $\mathcal{F}$
is the characteristic foliation on $\Sigma$, then we simplify the notation for
the reduction to $\textnormal{Red}\,(\Sigma)$.

Now let us address several points:

#### Smooth and symplectic structures

On a topological submanifold $U\subset\Sigma$ which is transverse to the
foliation $\mathcal{F}$, we have a natural structure of a smooth manifold.
Moreover, if $\Sigma$ is a coisotropic (or, more generally, a pre-symplectic)
submanifold of a symplectic manifold, and $\mathcal{F}$ is the characteristic
foliation on $\Sigma$ (we will call such situation a “symplectic setting”),
then $U$ moreover inherits a natural symplectic structure. Indeed, let $V$ be
a neighbourhood of $U$ in $\Sigma$ such that $U$ intersects exactly once each
characteristics of $V$, as in definition 4.1. Then any point $z\in U$ has a
neighbourhood $U_{1}\subset U$ such that $U_{1}$ lies inside a (smooth) flow-
box $\Phi:W_{1}\times(0,1)^{r}\rightarrow\Sigma$, where
$\textnormal{Im}\,\Phi\subset V$. Then the map
$\varphi:=\pi\circ\Phi^{-1}:U_{1}\rightarrow W_{1}$ is injective and hence, by
the Invariance of Domain theorem, is a homeomorphism onto the open image
$\varphi(U_{1})\subset W_{1}$ (here $\pi:W_{1}\times(0,1)^{r}\rightarrow
W_{1}$ is the natural projection). The map $\varphi$ induces a natural smooth
structure on $U_{1}$, in case of a general foliation, and moreover induces
natural symplectic structure in $U_{1}$ if we are in a symplectic setting.

#### Naturality of the structures

If two topological submanifolds $U,U^{\prime}\subset\Sigma$ are equivalent
($U\sim U^{\prime}$), then they are diffeomorphic (and moreover
symplectomorphic if we are in a symplectic setting) via a homotopy
$G:W\times[0,1]\rightarrow\Sigma$, as in definition 4.1. Let us describe
explicitly the diffeomorphism (resp. symplectomorphism) between $U$ and
$U^{\prime}$. By continuity of $G$ and since $U$ is topologically transverse
to leaves of the foliation $\mathcal{F}$, for any point $z\in W$ and any
$t\in[0,1]$ there exists a neighbourhood $W_{1}\Subset W$ of $z$ such that the
closure of the image $G(W_{1}\times\\{t\\})$ lies inside a (smooth) flow-box
$\Phi:W_{2}\times(0,1)^{r}\rightarrow\Sigma$, and moreover the map
$\varphi_{t}:=\pi\circ\Phi^{-1}\circ G:W_{1}\times\\{t\\}\rightarrow W_{2}$ is
injective (here $\pi:W_{2}\times(0,1)^{r}\rightarrow W_{2}$ is the natural
projection, as before). Then, by the Invariance of Domain theorem,
$\varphi_{t}$ is a homeomorphism onto the open image
$W_{3}:=\varphi_{t}(W_{1})\subset W_{2}$. This induces a smooth (resp.
symplectic) structure on $W_{1}\times\\{t\\}$. Moreover, since $W_{1}\Subset
W$ and $G(W_{1}\times\\{t\\})\Subset\Phi(W_{2}\times(0,1)^{r})$, it follows
that we also have
$G(W_{1}\times\\{t^{\prime}\\})\Subset\Phi(W_{2}\times(0,1)^{r})$ as well and
moreover $\varphi_{t^{\prime}}(z,t^{\prime})=\varphi_{t}(z,t)$ for every $z\in
W_{1}$, whenever $t^{\prime}\in[0,1]$ is sufficiently close to $t$ (here
$\varphi_{t^{\prime}}=\pi\circ\Phi^{-1}\circ
G:W_{1}\times\\{t^{\prime}\\}\rightarrow W_{2}$). Hence the induced smooth
(resp. symplectic) structures on $W_{1}\times\\{t\\}$ and on
$W_{1}\times\\{t^{\prime}\\}$ coincide, when $t^{\prime}$ is sufficiently
close to $t$.

#### The induced map

Let $h:\Sigma\rightarrow\Sigma^{\prime}$ be a homeomorphism between smooth
manifolds $\Sigma$ and $\Sigma^{\prime}$ which are endowed with foliations
$\mathcal{F}$ and $\mathcal{F}^{\prime}$, such that $h$ maps $\mathcal{F}$ to
$\mathcal{F}^{\prime}$. Then $h$ defines a natural map
$\hat{h}:\textnormal{Red}\,(\Sigma,\mathcal{F})\rightarrow\textnormal{Red}\,(\Sigma^{\prime},\mathcal{F}^{\prime})$
by $\hat{h}([U]):=[h(U)]\subset\Sigma^{\prime}$. Clearly, the definition does
not depend on the representative $U$ of $[U]$. Recall that by a theorem of
Humilière, Leclercq and Seyfaddini [HLS15], if $h:M\rightarrow M^{\prime}$ is
a symplectic homeomorphism and if $h$ maps a smooth coisotropic submanifold
$\Sigma$ onto a smooth submanifold $\Sigma^{\prime}$, then $\Sigma^{\prime}$
is coisotropic, and the restriction
$h_{|\Sigma}:\Sigma\rightarrow\Sigma^{\prime}$ preserves the characteristic
foliation, and hence we get a natural induced map
$\hat{h}:\textnormal{Red}\,(\Sigma)\rightarrow\textnormal{Red}\,(\Sigma^{\prime})$.

###  4.2 Proof of theorem 2

Let us adopt the following notation for the sake of fluidity. Given two
submanifolds $A,B$ with boundaries, we say that $A$ is knotted with $\partial
B$, relative to $\partial A$, if any homotopy of $A$ relative to $\partial A$
that avoids $\partial B$ intersects $B$. Notice that here, the filling $B$ of
$\partial B$ is fixed, and not made explicit in the sentence. In the following
still, we will be interested in a knotting with $\partial Z(A,R)$, whose
filling is implicitely meant by $Z(A,R)$.

Let $h$ be a symplectic homeomorphism defined in a neighbourhood $U$ of
$B^{2k}(1)\times(-1,1)^{l}$ in $\mathbb{C}^{k+l}=\mathbb{C}^{n}$, with values
in $\mathbb{C}^{n}$, such that
$h\big{(}B^{2k}(1)\times(-1,1)^{l}\big{)}\subset\mathbb{C}^{k}\times\mathbb{R}^{l}$.
We also assume for later convenience that $U$ is contractible. We need to
prove that for $\delta$ small enough, for any $a<\delta$, if
$h(B^{2k}(a)\times(-1,1)^{l})\subset Z(A)\times\mathbb{R}^{l}$, then $A\geq
a$.

Since $h$ is a symplectic homeomorphism, we know by [HLS15] that it takes the
characteristic leaves $\\{*\\}\times(-1,1)^{l}$ into
$\\{*\\}\times\mathbb{R}^{l}$. In other terms, there exists a continuous
function $\hat{h}:B^{2k}(1)\to\mathbb{C}^{k}$ and an open subset
$\Omega(z)\subset\mathbb{R}^{l}$ for each point $z\in B^{2k}(1)$ such that

$h(\\{z\\}\times(-1,1)^{l})=\\{\hat{h}(z)\\}\times\Omega(z).$

Since $Z(A)$ is invariant by translation along the $z^{\prime}$-axis, we can
assume that $\hat{h}(0)=(z_{1},0)$, which we do henceforth (recall that we
split $\mathbb{C}^{k}=\mathbb{C}\times\mathbb{C}^{k-1}$, with
$(z_{1},z^{\prime})$ the split coordinates). We divide this proof in four
steps:

Step 1: Adjusting $\hat{h}$ and $\Omega$. As already observed in section 4.1,
the map $\hat{h}$ is a local homeomorphism. Thus, its restriction to
$B^{2k}(R)$ for $R$ small enough is injective, so after maybe restricting $h$
to $B^{2k}(R)\times(-1,1)^{l}$, we can assume that $\hat{h}$ is injective,
which we do henceforth. Moreover, since $h$ is a homeomorphism, the map
$z\mapsto\Omega(z)$ is continuous, so by a further restriction, we can also
ensure that

$\Omega(z)\supset\frac{1}{2}\Omega(0)\hskip 28.45274pt\forall z\in B^{2k}(R).$

Finally, since $\Omega(0)$ is homeomorphic to $(-1,1)^{l}$, there exists a
compactly supported diffeomorphism $\varphi:\mathbb{R}^{l}\to\mathbb{R}^{l}$
such that $\varphi(\Omega(0))$ is arbitrarily close to $(-4,4)^{l}$. Letting
$\Phi:T^{*}\mathbb{R}^{l}\to T^{*}\mathbb{R}^{l}$ the natural lift of
$\varphi$ and considering $(\textnormal{Id}\,\otimes\Phi)\circ h$ instead of
$h$, we can therefore also assume that $\Omega(0)$ is close to $(-4,4)^{l}$,
and that $\Omega(z)\supset(-2,2)^{l}$ for all $z$. Summarizing this
discussion, we see that we can assume without loss of generality that

  * $\bullet$

$h$ is a symplectic homeomorphism defined in a neighbourhood of
$B^{2k}(1)\times(-1,1)^{l}$ with values in $\mathbb{C}^{n}$, that takes
$B^{2k}(1)\times(-1,1)^{l}$ to a subset

$\\{(z,t)\in W\times\mathbb{R}^{l},\;t\in\Omega(z)\\},$

  * $\bullet$

$\hat{h}:B^{2k}(1)\to\mathbb{C}^{k}$ is injective,

  * $\bullet$

$\Omega(z)\supset(-2,2)^{l}$ for all $z$.

Obviously, $W$ is open, and if we fix any compact exhaustion $W_{n}$ of $W$,
there exists therefore $\delta_{n}>0$ such that

$\textnormal{Im}\,h\supset
Q_{n}:=W_{n}\times(-1,1)^{l}\times(-\delta_{n},\delta_{n})^{l}.$

Moreover, since $h$ is a homeomorphism, if $\delta_{n}$ is chosen small
enough,

$\textnormal{Im}\,h\cap
Q_{n}\cap\mathbb{C}^{k}\times\mathbb{R}^{l}=h(B^{2k}(1)\times(-1,1)^{l}).$

In other terms, $h$ intersects the coisotropic subset
$\mathbb{C}^{k}\times\mathbb{R}^{l}$ in $Q_{n}$ exactly along
$B^{2k}(1)\times(-1,1)^{l}$.

Step 2: Knotting $\hat{h}$. Let $n_{0}$ be such that $\hat{h}(0)\notin
W_{n_{0}}$, and put $R_{0}:=\frac{1}{2}d(\hat{h}(0),\partial W_{n_{0}})$
(measured in the infinity norm). For $\delta_{1}(h)\ll 1$,
$d\big{(}\hat{h}(\partial B(\delta_{1}(h))),\hat{h}(0)\big{)}<R_{0}$. Since
$\hat{h}(0)$ has the form $(z_{1},0)$, if $a\leq\delta_{1}(h)$ and
$\hat{h}(B(a))\subset Z(A)$, $\hat{h}(B(a))\subset D(A)\times
B_{\infty}(R_{0})$. Moreover, the first step above guarantees that

$h(\\{0\\}\times(-1,1)^{l})\cap D(A)\times B_{\infty}(R_{0})\times
B_{\infty}(\delta_{n})\cap Q_{n_{0}}=\\{\hat{h}(0)\\}\times(-1,1)^{l}$

Thus, provided that $R<R_{0}$ and $Z(A,R)\subset Q_{n_{0}}$,
$h(B(a)\times(-1,1)^{l})$ is knotted with $\partial Z(A,R)$ in the sense given
at the beginning of this section. This condition can be written:

$R<R_{0},\hskip 8.5359ptA\leq R_{0},\hskip 8.5359ptR<\delta_{n_{0}}.$ (4.2.1 )

Notice also that provided $A$ is chosen small enough compared with $R$,
$h\big{(}B(a)\times(-1,1)^{l}\big{)}$ is even knotted with $\partial
Z(A,R)\cup F(A,R,CA)$, where the filling of this last set is still by
$Z(A,R)$. For this to hold, it is necessary and sufficient that
$F(A,R,CA)\cap\mathbb{C}^{k}\times\mathbb{R}^{l}=\emptyset$, which holds as
soon as

$R>CA,$ (4.2.2 )

and $F(A,R,CA)\subset Q_{n_{0}}$, which needs in top of (4.2.1)

$CA<\delta_{n_{0}}.$ (4.2.3 )

To conclude this step, notice that the inequalities (4.2.1)-(4.2.3) can be
achieved as soon as

$A<\delta_{2}(h):=\min(1,\frac{1}{C})\min(R_{0},\delta_{n_{0}})$

because then $R$ can be chosen between $CA$ and $\min(R_{0},\delta_{n_{0}})$
and the inequalities are then all satisfied.

Step 3: from $h$ to a symplectomorphism. Let now assume that
$A<\delta(h):=\min(\delta_{1}(h),\delta_{2}(h))$. Since $h$ is a symplectic
homeomorphism, it can be approached in the uniform norm by symplectic
diffeomorphisms $f_{n}:U\to\mathbb{C}^{n}$. Using the contractibility
assumption on $U$, it is easy to see that $f_{n}$ can be extended to a
compactly supported Hamiltonian diffeomorphism of $\mathbb{C}^{n}$. Moreover,
since $f_{n}$ is close to $h$ in $U$, the following two points are satisfied
for large enough $n$:

  * $\bullet$

$f_{n}\big{(}B(a)\times(-1,1)^{l}\big{)}$ is knotted with $Z(A,R)\cup
F(A,R,CA)$ in $Q_{n_{0}}$,

  * $\bullet$

$f_{n}(U)\supset Q_{n_{0}}$, so
$f_{n}\big{(}B(a)\times\mathbb{R}^{l}\backslash(-1,1)^{l}\big{)}\cap
Q_{n_{0}}=\emptyset$.

As a result, and since $Z(A,R)\cup F(A,R,CA)\subset Q_{n_{0}}$,
$f_{n}\big{(}B(a)\times\mathbb{R}^{l}\big{)}$ is knotted with $Z(A,R)\cup
F(A,R,CA)$ so corollary 3.3 guarantees that $A\geq a$.

Step 4: from $A$ to $a$. We have therefore got our non-squeezing inequality
when $A<\delta(h)$. It only remains to get it for $a<\delta(h)$ in order to
establish our statement. But this is obvious. Indeed,

if $a<\delta(h)$, either $A\geq\delta(h)$, and then $A>a$, or $A<\delta(h)$
and then the previous analysis implies that $a\geq A$. $\square$

###  4.3 Proof of theorem 3

Theorem 3 can be seen as a corollary of theorem 2. Indeed, what we have to
prove is that if $\Sigma,\Sigma^{\prime}$ are two $n+1$-dimensional
coisotropic submanifolds of $2n$-dimensional symplectic manifolds, if there
exists a symplectic homeomorphism
$h:(\textnormal{Op}\,(\Sigma),\Sigma)\longrightarrow(\textnormal{Op}\,(\Sigma^{\prime}),\Sigma^{\prime})$,
the reduction $\hat{h}$ of $h$ is area preserving. Since area preservation is
a local property, it is enough to check it for small elements of the
reduction. We can therefore assume that both $\Sigma$ and $\Sigma^{\prime}$
are $D^{2}(1)\times\mathbb{R}^{n-1}\subset\mathbb{C}^{n}$. We now argue by
contradiction. Assume that $\hat{h}$ is not area preserving, and considering
maybe $h^{-1}$ instead of $h$, we can assume that it expands the area of some
elements of the reduction. Then there exists a subdisc $D(R_{0})\Subset D(1)$
whose area is increased by $\hat{h}$. Moreover, there must exist a point $z\in
D(R_{0})$ and a decreasing sequence $\varepsilon_{n}>0$ that tends to $0$ such
that the area of $\hat{h}(D(z,\varepsilon_{n}))$ exceeds $\varepsilon_{n}$ for
all $n$. But this is precisely ruled out by theorem 2. $\square$

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#### Emmanuel Opshtein,

Institut de Recherche Mathématique Avancée
UMR7501, Université de Strasbourg et CNRS
7 rue René Descartes
67000 Strasbourg, FRANCE.
E-mail<EMAIL_ADDRESS>