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\begin{document}
\frontmatter
\title[Subcritical surgery and symplectic fillings]{Subcritical
contact surgeries and the~topology of symplectic fillings}
\author[\initial{P.} \lastname{Ghiggini}]{\firstname{Paolo} \lastname{Ghiggini}}
\address{
Laboratoire de Mathématiques Jean Leray \\
BP 92208,
2, Rue de la Houssinière,
F-44322 Nantes Cedex 03,
France}
\email{paolo.ghiggini@univ-nantes.fr}
\urladdr{http://www.math.sciences.univ-nantes.fr/~ghiggini/}
\author[\initial{K.} \lastname{Niederkr\"{u}ger}]{\firstname{Klaus} \lastname{Niederkr\"{u}ger}}
\address{
Alfréd Rényi Institute of Mathematics,
Hungarian Academy of Sciences,
POB 127,
H-1364 Budapest,
Hungary}
\address{
Institut de mathématiques de Toulouse,
Université Paul Sabatier -- Toulouse III\\
118 route de Narbonne,
F-31062 Toulouse Cedex 9,
France}
\email{niederkr@math.univ-toulouse.fr}
\urladdr{http://www.math.univ-toulouse.fr/~niederkr/}
\author[\initial{C.} \lastname{Wendl}]{\firstname{Chris} \lastname{Wendl}}
\address{
Department of Mathematics,
University College London \\
Gower Street,
London WC1E 6BT,
United Kingdom}
\email{c.wendl@ucl.ac.uk}
\urladdr{http://www.homepages.ucl.ac.uk/~ucahcwe/}
\thanks{The first and second authors were partially supported during this
project by ANR grant \textsl{ANR-10-JCJC 0102}.
The third author is partially supported by a Royal Society
\textsl{University Research Fellowship} and by EPSRC grant
\textsl{EP/K011588/1}.
Additionally, all three authors gratefully acknowledge the support of
the European Science Foundation's ``CAST'' research network.}
\begin{abstract}
By a result of Eliashberg, every symplectic filling of a three-dimensional contact connected sum is obtained by performing a boundary connected sum on another symplectic filling. We prove a partial generalization of this result for subcritical contact surgeries in higher dimensions: given any contact manifold that arises from another contact manifold by subcritical surgery, its belt sphere is zero in the oriented bordism group~$\Omega_*^{SO}(W)$ of any symplectically aspherical filling~$W$, and in dimension five, it will even be nullhomotopic. More generally, if the filling is not aspherical but is semipositive, then the belt sphere will be trivial in $H_*(W)$. Using the same methods, we show that the contact connected sum decomposition for tight contact structures in dimension three does not extend to higher dimensions: in particular, we exhibit connected sums of manifolds of dimension at least five with Stein fillable contact structures that do not arise as contact connected sums. The proofs are based on holomorphic disk-filling techniques, with families of Legendrian open books (so-called ``\textsf{Lob}{}s'') as boundary conditions.
\end{abstract}
\subjclass{57R17, 53D10, 32Q65, 57R65}
\keywords{Contact surgery, symplectic filling, holomorphic disks}
\alttitle{Chirurgies de contact sous-critiques et topologie des remplissages symplectiques}
\begin{altabstract}
Un résultat d'Eliashberg affirme que tout remplissage symplectique d'une somme connexe de contact en dimension $3$ est obtenu par somme connexe au bord d'un autre remplissage symplectique. Nous montrons une généralisation partielle de ce résultat pour les chirurgies de contact sous-critiques en dimension supérieure: étant donnée une variété de contact obtenue à partir d'une autre par chirurgie sous-critique, la cosphère de l'anse est nulle dans le groupe de bordisme orienté $\Omega_*^{SO}(W)$ de tout remplissage symplectiquement asphérique~$W$. En dimension~$5$, elle est même homotope à zéro. Plus généralement, si le remplissage n'est pas asphérique mais semi-positif, alors la cosphère de l'anse est triviale dans $H_*(W)$. Nous montrons aussi, en utilisant des méthodes similaires, que la décomposition en somme connexe de contact pour les structures de contact tendues en dimension $3$ ne s'étend pas en dimension supérieure. Nous exhibons en particulier des sommes connexes de variétés de dimension au moins $5$ qui portent une structure de contact Stein remplissable qui ne peut pas se mettre sous la forme d'une somme connexe de contact. Les démonstrations s'appuient sur les techniques de remplissage par disques holomorphes avec, pour conditions au bord, des familles de livres ouverts legendriens (que l'on abrège par \og \textsf{Lob}\fg).
\end{altabstract}
\altkeywords{Chirurgie de contact, remplissage symplectique, disques holomorphes}
\maketitle
\tableofcontents
\mainmatter
\section{Introduction}
\subsection{The main result and corollaries}
\label{sec:main}
The idea of constructing contact manifolds as boundaries of symplectic
$2n$-manifolds by attaching handles of index at most $n$ goes back to
Eliashberg \cite{Eliashberg_Stein} and Weinstein
\cite{WeinsteinHandlebodies}.
In this context, a special role is played by \emph{subcritical}
handles, \ie handles with index strictly less than~$n$.
One well-known result on this topic concerns subcritical Stein
fillings, which are known to be \emph{flexible} in the sense that
their symplectic geometry is determined by homotopy theory, see
\cite{CieliebakEliashberg}.
There are also known restrictions on the topological types of
subcritical fillings, \eg by results of M.-L.\,Yau
\cite{YauSubcritical} and Oancea-Viterbo
\cite[Prop.\,5.7]{OanceaViterbo}, the homology of a subcritical filling
with vanishing first Chern class is uniquely determined by its contact
boundary; this result can be viewed as a partial generalization of the
Eliashberg-Floer-McDuff theorem \cite{McDuff_contactType} classifying
symplectically aspherical fillings of standard contact spheres up to
diffeomorphism.
In dimension three, there is a much stronger result due to Eliashberg
\cite{Eliashberg_filling,CieliebakEliashberg}: in this case every
subcritical surgery is a connected sum, and the result states
that if $(M',\xi')$ is a closed contact $3$-manifold obtained from
another (possibly disconnected) contact manifold $(M,\xi)$ by a
connected sum, then every symplectic filling of $(M',\xi')$ is
obtained by attaching a Weinstein $1$-handle to a symplectic filling
of $(M,\xi)$.
This implies that symplectic fillings of subcritically fillable
contact $3$-manifolds are unique up to symplectic deformation equivalence
and blowup---in particular, their Stein fillings are unique up to
symplectomorphism.
The present paper was motivated by the goal of generalizing
Eliashberg's connected sum result to higher dimensions.
The natural question in this setting is the following:
\begin{question}\label{question:main}
Given a closed contact manifold $(M',\xi')$ that is obtained from
another contact manifold $(M,\xi)$ by subcritical contact surgery,
is every (symplectically aspherical) filling of $(M',\xi')$
obtained by attaching a subcritical Weinstein handle to a symplectic
filling of $(M,\xi)$?
\end{question}
Classifying fillings up to symplectomorphism as suggested in this
question would be far too ambitious in higher dimensions, \eg the
strongest result known so far, the Eliashberg-Floer-McDuff theorem, is
essentially a classification of fillings up to \emph{homotopy type}
(the $h$-cobordism theorem then improves it to a classification up to
diffeomorphism).
Our objective in this paper will therefore be to understand the main
\emph{homotopy-theoretic} obstruction to an affirmative answer to
Question~\ref{question:main}.
To state the main result, let us first recall some basic notions.
Given an oriented $(2n-1)$-dimensional manifold $M$, a (positive,
co-oriented) \defin{contact structure} on $M$ is a hyperplane
distribution of the form $\xi = \ker \alpha$, where the \defin{contact
form} $\alpha$ is a smooth $1$-form satisfying $\alpha \wedge
(d\alpha)^{n-1} > 0$, and the co-orientation of~$\xi$ is determined by
$\alpha > 0$.
In this paper, contact structures will \emph{always} be assumed to be
both positive (with respect to a given orientation on~$M$) and
co-oriented, and all contact forms will be assumed compatible with the
co-orientation.
A compact symplectic $2n$-manifold $(W,\omega)$ with oriented boundary
$M = \p W$ carrying a contact structure $\xi$ is called a
\defin{strong symplectic filling} of $(M,\xi)$ if $\xi$ admits a
contact form~$\lambda$ that extends to a primitive of $\omega$ on a
neighborhood of~$\p W$.
It is equivalent to say that the boundary is \defin{symplectically
convex}, as the vector field $\omega$-dual to $\lambda$ is then a
Liouville vector field pointing transversely outward at~$\p W$.
More generally, we say that $(W,\omega)$ is a \defin{weak symplectic
filling} of $(M,\xi)$ if $\xi$ is the bundle of complex tangencies
for some $\omega$-tame almost complex structure near $\p W$ that makes
the boundary pseudoconvex (see \cite{WeafFillabilityHigherDimension}).
Recall that if $(M,\xi)$ contains a $(k-1)$-dimensional isotropic
sphere $\attachingSphere{k-1}$ with trivial normal bundle, then one
can perform a \defin{contact surgery of index~$k$} on $(M,\xi)$ by
attaching to $(-\epsilon,0] \times M$ a handle of the form $\DD^k
\times \DD^{2n-k}$ along a neighborhood of~$\attachingSphere{k-1}$.
The new contact manifold $(M',\xi')$ then contains the
$(2n-k-1)$-dimensional coisotropic sphere $\beltSphere{2n-k-1} = \{0\}
\times \p \DD^{2n-k}$, which we call the \defin{belt sphere} of the
surgery.
This surgery operation was first introduced by Weinstein
\cite{WeinsteinHandlebodies}, and we will give a more precise
description of it in Section~\ref{sec: description handle and
surgery}.
A Weinstein handle yields a symplectic cobordism that can be attached
to any weak filling $(W,\omega)$ of $(M,\xi)$ for which $\omega$ is
exact along $\attachingSphere{k-1}$; the result is a weak filling of
$(M',\xi')$ in which the belt sphere is necessarily contractible
(Figure~\ref{fig:trivial}).
Our main result is the following.
\begin{theorem}\label{thm: main theorem}
Suppose $(M',\xi')$ is a closed contact manifold of dimension~\hbox{$2n-1\!\ge\! 3$} that has been obtained from a manifold $(M,\xi)$ by a contact
surgery of index \hbox{$k \!\le\! n-1$}, with belt sphere $\beltSphere{2n-k-1}
\subset M'$.
Assume $(W',\omega')$ is a weak symplectic filling of $(M',\xi')$.
\begin{enumeratea}
\item
If ($W',\omega')$ is semipositive, then the belt sphere
represents the trivial homology class in $H_{2n-k-1}(W'; \ZZ)$.
\item
If $(W',\omega')$ is symplectically aspherical, then the
belt sphere represents the trivial element in the oriented
bordism group $\Omega_{2n-k-1}^{SO}(W')$.
If additionally either (1)~$M'$ is $5$-dimensional, or (2)~$M'$ is
$7$-dimensional and $k=3$, then $\beltSphere{2n-k-1}$ is
contractible in $W'$, that is, it represents the trivial class in
$\pi_{2n-k-1}(W')$.
\end{enumeratea}
\end{theorem}
\begin{figure}[htbp]
\centering
\includegraphics[height=2.5cm,keepaspectratio]{handle_sketch.pdf}
\caption{\label{fig:trivial} The belt sphere bounds an embedded
disk inside the handle.}
\end{figure}
We now state two related results that follow via the same techniques.
Recall that in dimension three, convex surface theory gives rise to a
contact prime decomposition theorem, implying \eg that every tight
contact structure on a closed $3$-manifold of the form $M_0 \# M_1$
arises as a contact connected sum of tight contact structures on $M_0$
and $M_1$ (see \cite{ColinConnectedSum}, or \cite[\S 4.12]{GeigesBook}
for more details).
Some evidence against a higher-dimensional generalization of this
result appeared in the recent work of Bowden, Crowley and Stipsicz
\cite{BowdenCrowleyStipsicz2}, providing also a negative answer to a
topological version of Question~\ref{question:main}:
namely, there exist pairs of closed oriented manifolds $M_0, M_1$ such
that $M_0 \conSum M_1$ admits a Stein fillable contact structure but
$M_0$ and~$M_1$ do not.
This did not imply an actual answer to Question~\ref{question:main},
however, as it was unclear whether the contact structures on $M_0
\conSum M_1$ in the examples of \cite{BowdenCrowleyStipsicz2} could
actually be \emph{contact} connected sums, \ie whether they arise
from contact structures~$\xi_0$ on $M_0$ and $\xi_1$ on $M_1$ by
performing index~$1$ contact surgery.
The following result gives a negative answer to the latter question,
and shows that there is no hope of extending the contact prime
decomposition theorem to higher dimensions.
The theorem applies in particular whenever $M$ is an almost contact
$\SS^{n-1}$-bundle over~$\SS^n$ that is not a homotopy sphere, so for
instance $M$ could be $\SS^{n-1} \times \SS^n$ or---as in
\cite{BowdenCrowleyStipsicz2}---the unit cotangent bundle of a sphere.
\begin{theorem}\label{thm: almost contact connSum not genuine}
Suppose $M$ is a closed oriented manifold of dimension $2n-1 \ge 5$
that is not a homotopy sphere but admits an almost contact structure
$\Xi$ and a Morse function with unique local maxima and minima and
otherwise critical points of index $n-1$ and $n$ only.
Then $M\conSum (-M)$ admits a Stein fillable contact structure that
is homotopic to the almost contact structure $\Xi \conSum
\overline{\Xi}$ but is not isotopic to $\xi_1 \conSum \xi_2$ for any
contact structures $\xi_1$ and $\xi_2$ on $M$ and $-M$ respectively.
\footnote{Given an oriented manifold~$M$ with almost contact
structure~$\Xi$, we denote by $-M$ the same manifold with reversed
orientation, and let $\overline{\Xi}$ denote the almost contact
structure on $-M$ obtained by inverting the co-orientation
of~$\Xi$.}
\end{theorem}
Note that the contact structures in the above statement are
necessarily tight in the sense of Borman-Eliashberg-Murphy
\cite{BormanEliashbergMurphy_wow}; this follows from Stein
fillability, using \cite{NiederkrugerPlastikstufe} and the observation
in \cite{BormanEliashbergMurphy_wow} that any overtwisted contact
structure is also PS-overtwisted.
The holomorphic disk techniques developed in this article can also be
used as in the work of Hofer \cite{HoferWeinstein} to prove the
Weinstein conjecture for a wide class of contact manifolds obtained by
subcritical surgery.
The following theorem, proved in Section~\ref{sec:Weinstein}, is
related to the well-known result that every subcritically Stein
fillable contact form admits a contractible Reeb orbit.
(A similar result specifically for index~$1$ surgeries appeared
recently in \cite{GeigesZehmisch}.)
\begin{theorem}\label{thm:Weinstein}
Assume $(M',\xi')$ is the result of performing a contact surgery of
index $k \le n-1$ on a closed contact manifold $(M,\xi)$ of
dimension $2n-1 \ge 3$, with belt sphere $\beltSphere{2n-k-1}
\subset M'$, and suppose at least one of the following conditions
holds:
\begin{enumerate}
\item $[\beltSphere{2n-k-1}] \ne 0$ in $\Omega_{2n-k-1}^{SO}(M')$;
\item $[\beltSphere{2n-k-1}] \ne 0$ in $\pi_{2n-k-1}(M')$ and
either $\dim M' = 5$ or $\dim M' = 7$ with $k=3$;
\item $\dim M' = 5$ and $(M',\xi')$ is a contact connected sum
$(M_0,\xi_0) \conSum (M_1,\xi_1)$ with the following two
properties:
\begin{enumerate}
\item Neither $M_0$ nor $M_1$ is homeomorphic to a sphere;
\item If $M_0$ and $M_1$ are both rational homology spheres, then
either both are not simply connected or at least one of them has
infinite fundamental group.
\end{enumerate}
\end{enumerate}
Then every contact form for $\xi$ admits a contractible Reeb orbit.
\end{theorem}
Before discussing the proofs, some further remarks about the main
theorem are in order.
\begin{remark}
We do not know whether the dimensional restriction for the
contractibility result in part~(b) of Theorem~\ref{thm: main
theorem} is essential, but given the wide range of known contact
geometric phenomena that can happen \emph{only} in sufficiently high
dimensions, we consider it plausible that the contractibility
statement could be false without some restriction of this type (thus
implying a definitively negative answer to
Question~\ref{question:main} in general).
It is apparent in any case that our method will not work in all
dimensions, as the improvement from ``null-bordant'' to
``nullhomotopic'' involves subtle topological difficulties that
increase with the dimension; see the beginning of Section~\ref{sec:
surgery on the moduli space} for more discussion of this.
\end{remark}
\begin{remark}
It is clear that nothing like Theorem~\ref{thm: main theorem} can be
true for \emph{critical surgeries} in general, \ie the case $k=n$.
There are obvious counterexamples already in dimension three, as any
Legendrian knot $L \subset (M',\xi')$ can be viewed as the belt
sphere arising from a critical contact surgery---take $(M,\xi)$ in
this case to be the result of a Legendrian $(+1)$-surgery along~$L$.
It is certainly not true in general that arbitrary Legendrian knots
are nullhomologous in every filling of $(M,\xi)$!
\end{remark}
\begin{remark}\label{remark:aspherical}
The semipositivity assumption in part~(a) of Theorem~\ref{thm: main
theorem} is there for technical reasons and could presumably be
lifted using more advanced technology (\eg polyfolds, see
\cite{HoferWZ_GW}).
In contrast, symplectic asphericity in part~(b) is a geometrically
meaningful condition that, while not needed for Eliashberg's
three-dimensional version of this result, cannot generally be
removed in higher dimensions; see Example~\ref{ex:aspherical} below.
The answer to Question~\ref{question:main} thus becomes negative
without this assumption.
\end{remark}
\begin{example}\label{ex:aspherical}
The blowup of the total space of the rank~$2$ holomorphic vector
bundle ${\mathcal O}(-2) \oplus {\mathcal O}$ over $\CC P^1$ at the
zero section can be viewed as a (not symplectically aspherical) weak
filling of a subcritically Stein fillable contact manifold
$(M',\xi')$ containing a belt sphere that is homotopically
nontrivial in the filling.
This is a special case of the following construction, which gives
examples with subcritical handles of any even index $k=2m \ge 2$ in
any dimension $2n \ge 2k+2 \ge 6$.
Choose integers $m, \ell \ge 1$ and set $n = 2m + \ell$, and
suppose $(\Sigma,\sigma)$ is a $2m$-dimensional closed symplectic
manifold.
Then consider the $2n$-dimensional Weinstein manifold $T^*\Sigma
\times \CC^\ell$, \ie the $\ell$-fold stabilization of $T^*\Sigma$
with its standard Weinstein structure, and denote its ideal contact
boundary by $(M',\xi')$.
Any Morse function on~$\Sigma$ gives rise to a Weinstein handle
decomposition of $T^*\Sigma \times \CC^\ell$, such that the
function's maximum $q \in \Sigma$ corresponds to an
$(n-\ell)$-handle whose belt sphere $\beltSphere{n+\ell-1}$ is
isotopic to the unit sphere in $T_q^*\Sigma \times \CC^\ell$.
Let $\Sigma \subset T^*\Sigma$ denote the zero section, so $\Sigma
\times \{0\}$ is an isotropic submanifold in $T^*\Sigma \times
\CC^\ell$, and denote by $\pi \colon T^*\Sigma \times \CC^\ell \to
\Sigma \times \{0\}$ the obvious projection.
Then for any $\epsilon > 0$ sufficiently small, adding $\epsilon
\pi^*\sigma$ to the natural exact symplectic form on $T^*\Sigma
\times \CC^\ell$ gives a weak filling of $(M',\xi')$ with $\Sigma
\times \{0\}$ as a symplectic submanifold.
We can then blow up along this submanifold, as explained in
\cite[\S7.1]{McDuffSalamonIntro}.
This produces a new weak filling $(W',\omega')$ of $(M',\xi')$, in
which the belt sphere $\beltSphere{n+\ell-1} \subset M'$ is
nullhomologous but homotopically nontrivial:
indeed, every fiber $T_q^*\Sigma \times \CC^\ell$ has now been
replaced by its blowup at the point $(0,0)$, which can be viewed as
the tautological line bundle over $\CC P^{m+\ell-1}$, so the bundle
projection sends $\beltSphere{n+\ell-1}$ to a generator of
$\pi_{2(m+\ell)-1}(\CC P^{m+\ell-1}) \cong \ZZ$.
The special case with $\Sigma = \SS^2$ and $\ell=1$ gives the
construction described at the beginning of this example, because the
total space of ${\mathcal O}(-2)$ is a deformation of $T^*\SS^2$.
\end{example}
The following represents another easy application of Theorem~\ref{thm:
main theorem}.
\begin{example}
Suppose $(M_1,\xi_1)$ is a contact $5$-manifold obtained by a
subcritical surgery of index~$2$ on a sphere $(\SS^5,\xi)$, where
$\xi$ is any contact structure.
Then $M_1$ is diffeomorphic to either $\SS^2 \times \SS^3$ or
$\SS^2\mathbin{\tilde\times} \SS^3$, \ie the trivial or nontrivial
$3$-sphere bundle over the $2$-sphere.
Indeed, closed loops in $\SS^5$ are automatically unknotted, and the
possible framings of the surgery are classified by the elements of
$\pi_1\bigl(\SO(3)\bigr) \cong \ZZ_2$.
If $(W,\omega)$ is any symplectically aspherical weak filling of
$(M_1,\xi_1)$, then by Theorem~\ref{thm: main theorem}, the fiber
$\{p\}\times \SS^3$ is a contractible $3$-sphere in $W$.
Now take $(M_2,\xi_2)$ to be the unit cotangent bundle of the
$3$-sphere or, more generally, any contact manifold supported by a
contact open book with page $T^*\SS^2$ and monodromy isotopic to a
$2k$-fold product of Dehn twists for some integer $k \ge 1$.
Then $M_2$ will be diffeomorphic to $\SS^2\times \SS^3$, but
$(M_2,\xi_2)$ admits a Stein filling that contracts to a bouquet of
$2k-1$ three-dimensional spheres (see
\eg \cite{vanKoertNiederkruger2005}).
We conclude that whenever $(M_1,\xi_1)$ admits a symplectically
aspherical weak filling, it is not contactomorphic to $(M_2,\xi_2)$.
This implies for instance that the contact structures induced on the
ideal contact boundaries of $T^*\SS^3$ and $T^*\SS^2 \times \CC$
(\cf Remark~\ref{remark:aspherical}) are not isomorphic.
There are presumably other ways to distinguish $\xi_1$ and $\xi_2$
in many cases, \eg using Symplectic Homology, but the technique
described above is much more topological.
\end{example}
\subsection{Idea of the proof}
\label{sec:proof}
Our proof of Theorem~\ref{thm: main theorem} is based on a
higher-dimensional analogue of the disk-filling methods underlying
Eliashberg's result in dimension three \cite{Eliashberg_filling}.
Such methods work whenever one can find a suitable submanifold to
serve as a boundary condition for holomorphic disks, and the natural
object to consider in this case is known as a \emph{Legendrian open
book} or ``\LOB{}''.
Let us recall the definition, which is due originally to the second
author and Rechtman \cite{NiederkrugerRechtman}.
\begin{definitionNumbered}\label{defn:Lob}
A \LOB{} in a $(2n-1)$-dimensional contact manifold $(M,\xi)$ is a
closed $n$-dimensional submanifold $L \subset M$ equipped with an
open book decomposition $\pi \colon L \setminus B \to \SS^1$ whose
binding $B \subset L$ is an $(n-2)$-dimensional isotropic
submanifold of $(M,\xi)$, and whose pages $\pi^{-1}(*)$ are each
Legendrian submanifolds of $(M,\xi)$.
\end{definitionNumbered}
The simplest interesting example of a \LOB{} occurs at the center of
the ``neck'' in any $3$-dimensional contact connected sum:
here we find a $2$-sphere $S \subset M$ on which the characteristic
foliation $\xi \cap TS$ traces an $\SS^1$-family of longitudes
connecting the north and south poles, so we can regard the longitudes
as pages of an open book with the poles as binding.
Such spheres were used as totally real boundary conditions for
holomorphic disks in \cite{BedfordGaveau, Gromov_HolCurves,
Eliashberg_filling}, and similarly in Hofer's proof of the Weinstein
conjecture \cite{HoferWeinstein} for contact $3$-manifolds $(M,\xi)$
with $\pi_2(M) \ne 0$.
In higher dimensions, a \LOB{} $L \subset (M,\xi)$ with binding $B
\subset L$ similarly defines a totally real submanifold $\{0\} \times
(L \setminus B)$ in the symplectization $\RR \times M$ of $(M,\xi)$,
and thus serves as a natural boundary condition for pseudoholomorphic
disks.
\begin{figure}[htbp]
\centering
\includegraphics[height=3cm,keepaspectratio]{sketch_Bishop.pdf}
\caption{A schematic picture of the Bishop family around the binding
of a \LOB~$L$.}\label{fig: schematic Bishop family}
\end{figure}
Moreover, for a suitably ``standard'' choice of almost complex
structure near the binding, a \LOB{} always gives rise to a canonical
family of holomorphic disks near $\{0\} \times B$ whose boundaries
foliate a neighborhood of $B$ in $L \setminus B$ (see Figure~\ref{fig:
schematic Bishop family}).
This is the so-called \defin{Bishop family} of holomorphic disks, and
it has the useful property that no other holomorphic curve can enter
the region occupied by the Bishop disks from outside.
For a unified treatment of the essential analysis for Bishop disks
with boundary on a \LOB{}, see \cite{NiederkrugerHabilitation}.
As in the $3$-dimensional case, we will see that the belt sphere of a
surgery of index $n-1$ on a contact $(2n-1)$-manifold is also
naturally a \LOB{}, so there is again a natural moduli space of
holomorphic disks that fill the belt sphere, implying that it is
nullhomologous.
For surgeries of index $k < n-1$, the belt sphere has dimension
$2n-k-1 > n$, and thus cannot be a \LOB{}, but we will show that after
a suitable deformation, the belt sphere can be viewed as a
parametrized \emph{family} of \LOB{}s, giving rise to a well-behaved
moduli space of disks with moving boundary condition.
It should now be clear why this method cannot work for critical
surgeries: the belt sphere in this case has dimension $n-1$, so it is
too small to define a totally real boundary condition.
The construction of the family of \LOB{}s foliating a general
subcritical belt sphere is somewhat less than straightforward: as we
will see in Section~\ref{sec: description handle and surgery}, the
standard model for a contact form after surgery does not lend itself
well to this construction, but a natural family of \LOB{}s can be
found after deforming to a different model of the belt sphere as
piecewise smooth boundary of a poly-disk.
Let us now discuss the topological reasons why the family of \LOB{}s
foliating $\beltSphere{2n-k-1} \subset (M',\xi')$ in Theorem~\ref{thm:
main theorem} places constraints on the filling $(W',\omega')$.
We focus for now on the case where $(W',\omega')$ is symplectically
aspherical, which rules out bubbling of holomorphic spheres.
For a suitable choice of tame almost complex structure~$J$
on~$(W',\omega')$, the Bishop families associated to
$\beltSphere{2n-k-1}$ generate a compactified moduli space~$\overline
\mM$ of $J$-holomorphic disks in $W'$ with one marked point, whose
boundaries are mapped to $\beltSphere{2n-k-1}$.
In light of the marked point, this moduli space is necessarily
diffeomorphic to a manifold with boundary and corners of the form
\begin{equation*}
\overline \Sigma \times \DD^2,
\end{equation*}
where $\overline \Sigma$ is a smooth, compact, connected and oriented
manifold with boundary and corners, whose boundary is a sphere.
Furthermore, the natural evaluation map
\begin{equation*}
\ev \colon \bigl(\overline\mM, \p\overline \mM\bigr)
\to \bigl(W',\beltSphere{2n-k-1}\bigr)
\end{equation*}
is smooth, and its restriction
\begin{equation*}
\restricted{\ev}{\p \overline \mM} \colon
\p\overline \mM \to \beltSphere{2n-k-1}
\end{equation*}
is a continuous map of degree~$\pm 1$.
The latter follows readily from the properties of the Bishop family,
which provide a nonempty open subset $U\subset W'$ that intersects
$\beltSphere{2n-k-1}$ and is the diffeomorphic image of $\ev^{-1}(U)
\subset \overline\mM$.
This description of the evaluation map $\ev \colon \overline{\mM} \to
W'$ already implies the homological part of Theorem~\ref{thm: main
theorem}, \ie that $[\beltSphere{2n-k-1}] = 0 \in H_{2n-k-1}(W')$.
To deduce stronger constraints, we will apply two further techniques.
The first consists in performing surgery on the moduli space
$\overline{\mM}$ to simplify its topology and suitably extending the
evaluation map in order to prove $[\beltSphere{2n-k-1}] = 0 \in
\pi_{2n-k-1}(W')$.
This method works when the dimension of $\overline{\mM}$ is not too
large.
The second method is relevant in particular to the case $k=0$ of the
main result, as well as to Theorem~\ref{thm: almost contact connSum
not genuine}, and is based on the following topological lemma.
\begin{lemma}\label{lemma: map of product manifold induces
contractibility}
Let $X,Y$ be smooth orientable compact manifolds with boundary and
corners such that $\p Y$ is homeomorphic to a sphere and $\dim X + 2
= \dim Y \ge 3$.
Write $X' = X\times \DD^2$, and assume that
\begin{equation*}
f\colon (X',\p X')\to (Y,\p Y)
\end{equation*}
is a continuous map that is smooth on the interior of $X'$, and for
which we find an open subset~$U\subset \mathring Y$ such that
$\restricted{f}{f^{-1}(U)} \colon f^{-1}(U) \to U$ is a
diffeomorphism.
Then $Y$ is contractible.
\end{lemma}
While it will not be essential to most of our arguments, note that the
$h$-cobordism theorem implies:
\begin{corollary}\label{coro: h-cobordism}
If $\dim Y \ge 5$ in the lemma, then $Y$ is diffeomorphic to a ball.
\end{corollary}
The $k=0$ case of Theorem~\ref{thm: main theorem} is the case where
$(M',\xi')$ is the standard contact sphere and the belt sphere is the
entirety of~$M'$.
In this setting, applying the above lemma to the evaluation map $\ev
\colon (\overline{\mM} = \overline{\Sigma} \times
\DD^2,\p\overline{\mM}) \to (W',\SS^{2n-1})$ implies that~$W'$ must be
diffeomorphic to a ball, hence this reproves the
Eliashberg-Floer-McDuff theorem.
We will explain this argument in more detail in Section~\ref{sec:EFM},
including the proof of the lemma (see
Lemma~\ref{lemma:KlausTopology}).
In another context, we will also apply the lemma in Section~\ref{sec: almost
contact surgery not genuine surgery} to demonstrate that the contact
structures arising on the boundaries of certain Stein domains which
look topologically like connected sums cannot arise from index~$1$
contact surgery, thus proving Theorem~\ref{thm: almost contact connSum
not genuine}.
Here is a brief outline of the paper.
In Section~\ref{sec:EFM}, we provide a foretaste of the methods in the
rest of the paper by using them to give an alternative proof of the
Eliashberg-Floer-McDuff theorem.
Section~\ref{sec: description handle and surgery} then explains the
general case of the family of \LOB{}s associated to a subcritical belt
sphere.
In Section~\ref{sec:moduliSpace}, we define the relevant moduli space
of holomorphic disks and establish its basic properties, leading to
the proof of the homological part of Theorem~\ref{thm: main theorem}.
Section~\ref{sec: surgery on the moduli space} then improves this to a
homotopical statement in cases where the moduli space has sufficiently
low dimension.
Finally, in Section~\ref{sec: almost contact surgery not genuine surgery}
and Section~\ref{sec:Weinstein} respectively we prove Theorems~\ref{thm:
almost contact connSum not genuine} and~\ref{thm:Weinstein} on
contact connected sums and contractible Reeb orbits.
The paper concludes with a brief appendix addressing the technical
question of orientability for our moduli space of holomorphic disks.
\subsubsection*{Acknowledgments}
The authors would like to thank Yasha Eliashberg, François Laudenbach,
Patrick Massot, Emmy Murphy, Otto van Koert and András Stipsicz for
useful conversations.
They are also very grateful to the referees for careful reading and
suggesting many improvements.
\section{The Eliashberg-Floer-McDuff theorem revisited}
\label{sec:EFM}
In this section we modify slightly the proof of the
Eliashberg-Floer-McDuff theorem \cite[Th.\,1.5]{McDuff_contactType}
in order to illustrate the methods that will be applied in the rest of
the article.
The original argument worked by capping off the symplectic filling and
then sweeping through it with a moduli space of holomorphic spheres.
Our version will be the same in many respects, but has more in common
with the $3$-dimensional argument of Eliashberg in
\cite{Eliashberg_filling}: instead of spheres, we use holomorphic
disks attached to a family of \LOB{}s.
\begin{theorem}[Eliashberg-Floer-McDuff]\label{theorem: ElFlMD
revisited}
Let $\SS^{2n-1} \subset \CC^n$ be the unit sphere with its standard
contact structure~$\xi_0$ given by the complex tangencies to the
sphere, that~is,
\begin{equation*}
\xi_0 = T\SS^{2n-1} \cap \bigl(i\cdot T\SS^{2n-1}\bigr).
\end{equation*}
Every symplectically aspherical filling of $\bigl(\SS^{2n-1},
\xi_0\bigr)$ is diffeomorphic to the $(2n)$-ball.
\end{theorem}
Let $\z = \x + i\y = \bigl(x_1+iy_1, \dotsc, x_n+iy_n\bigr)$ be the
coordinates of $\CC^n$.
The function $f\colon \CC^n \to [0,\infty)$ given by
\begin{equation*}
f(\z) = \sum_{j=1}^n (x_j^2 + y_j^2)
\end{equation*}
is plurisubharmonic, and the unit sphere is the boundary of the ball
\begin{equation*}
\DD^{2n} = \bigl\{ \z \in \CC^n\bigm|\, f(\z) \le 1 \bigr\}.
\end{equation*}
The main geometric ingredient needed for our proof is a foliation of
$\SS^{2n-1}$ (minus some singular subset) by a family of \LOB{}s, but
this idea does not seem to work when applied directly to the unit
sphere $f^{-1}(1)$.
Instead, we will deform the sphere to a different shape, which does
contain a suitable family of \LOB{}s that will suffice for our
purposes.
Define two functions $g_A, g_B\colon \CC^n \to [0,\infty)$ by
\begin{align*}
g_A(\z) &= y_1^2 + \dotsm + y_{n-1}^2 \\
g_B(\z) &= x_1^2 + \dotsm + x_{n-1}^2 + x_n^2 + y_n^2.
\end{align*}
Note that $g_B$ is strictly plurisubharmonic and $g_A$ is weakly
plurisubharmonic as
\begin{align*}
-dd^c g_A &= 2\, dx_1 \wedge dy_1 + \dotsm + 2\,dx_{n-1} \wedge dy_{n-1} \\
-dd^c g_B &= 2\, dx_1 \wedge dy_1 + \dotsm + 2\,dx_{n-1} \wedge
dy_{n-1} + 4\, dx_n \wedge dy_n.
\end{align*}
We will now consider the subset (see Fig.\,\ref{fig: convex deformation
sphere})
\begin{equation*}
\widehat \DD^{2n} = \bigl\{ \z \in \CC^n\bigm|\, g_B(\z) \le 1 \bigr\}
\cap \bigl\{ \z \in \CC^n\bigm|\, g_A(\z) \le 1 \bigr\}.
\end{equation*}
Up to reordering the coordinates, $\widehat \DD^{2n}$ is a
bi-disk~$\DD^{n+1}\times \DD^{n-1} \subset \RR^{2n}$, which clearly
contains the unit ball.
Its boundary is not a smooth manifold, but is nonetheless homeomorphic
to the unit sphere.
It decomposes as
\begin{equation*}
\p \widehat \DD^{2n} \cong
\DD^{n+1} \times \bigl(\p \DD^{n-1}\bigr) \,\cup \,
\bigl(\p \DD^{n+1}\bigr) \times \DD^{n-1}
= S_A \cup S_B,
\end{equation*}
where we have used the notation
\begin{align*}
S_A &:= \bigl\{g_A = 1\bigr\} \cap \p \widehat \DD^{2n} \cong
\DD^{n+1} \times \SS^{n-2}
\intertext{and}
S_B &:= \bigl\{g_B = 1\bigr\} \cap \p \widehat \DD^{2n} \cong \SS^n
\times \DD^{n-1}.
\end{align*}
\begin{figure}[htbp]
\centering
\includegraphics[height=5cm,keepaspectratio]{family_LOBs2.pdf}
\caption{We find a family of \LOB{}s by deforming the sphere to the
boundary of a bi-disk. One of the two parts of the boundary,
which we denote by $S_B$, will then be foliated by
\LOB{}s.}\label{fig: convex deformation sphere}
\end{figure}
Let now $(W,\omega)$ be a symplectically aspherical filling of
$\bigl(\SS^{2n-1}, \xi_0\bigr)$.
If it is only a weak filling, we can extend it by attaching a
symplectic collar to obtain a strong symplectic filling of the sphere
\cite[Rem.\,2.11]{WeafFillabilityHigherDimension} because
$\restricted{\omega}{T \SS^{2n-1}}$ is exact.
This filling is diffeomorphic to the initial one, and it is also still
symplectically aspherical, because any $2$-sphere can just be pushed
by a homotopy entirely into the old symplectic filling.
After rescaling the symplectic form, the extended symplectic manifold
will be a strong symplectic filling of the unit sphere.
Remove now the interior~$\mathring \DD^{2n}$ of the unit ball from
$\widehat \DD^{2n}$, and glue $\widehat \DD^{2n} \setminus \mathring
\DD^{2n}$ symplectically onto the filling~$W$.
Denote this new symplectic manifold by $(\widehat W, \widehat
\omega)$.
Clearly $\widehat W$ is homeomorphic to $W$.
Using holomorphic disks, we will show as in the original paper by
McDuff that $\widehat W$ is contractible, so that the $h$-cobordism
theorem \cite{SmaleHCobordism, MilnorHCobordism} implies that $W$ must
be diffeomorphic to $\DD^{2n}$ whenever $2n-1 \ge 5$.
To study $\widehat W$ using holomorphic curves, choose first an almost
complex structure~$J$ on $\widehat W$ that is tamed by $\widehat
\omega$ and that agrees on a small neighborhood of $\p \widehat
\DD^{2n}$ in $\widehat W$ with the standard complex structure~$i$ on
$\CC^n$.
The holomorphic curves we are interested in are attached to a family
of \LOB{}s, which we will introduce now.
Let $\Psi\colon \SS^n \times \DD^{n-1} \to S_B\subset \p\widehat W$ be
the embedding into the boundary of $\widehat W$ given by
\begin{equation*}
\bigl((a_1,a_2,\dotsc,a_{n+1});(b_1,\dotsc,b_{n-1})\bigr) \mto
\bigl(a_1+ i b_1,\dotsc,a_{n-1} + i b_{n-1},a_n + i a_{n+1}\bigr).
\end{equation*}
The image of $\Psi$ lies in $S_B \subset \p \widehat W$, and the
$J$-complex tangencies on the corresponding part of $\p \widehat W$
are the kernel of the $1$-form
\begin{equation*}
- d^c g_B = 2x_1\, dy_1 + \dotsm + 2x_{n-1}\, dy_{n-1} +
2\,\bigl(x_n\, dy_n - y_n\,dx_n\bigr).
\end{equation*}
We obtain for the pull-back
\begin{equation*}
\Psi^*\bigl(- d^c g_B\bigr) =
2a_1\, db_1 + \dotsm + 2a_{n-1}\, db_{n-1} +
2\,\bigl(a_n\, da_{n+1} - a_{n+1}\,da_n\bigr),
\end{equation*}
so that the restriction of $\Psi^*\bigl(- d^c g_B\bigr)$ to each of
the spheres
\[
\SS^n\times\bigl\{(b_1,\dotsc, b_{n-1}) = \mathrm{const}\bigr\}
\]
gives
\begin{equation*}
2\,\bigl(a_n\, da_{n+1} - a_{n+1}\,da_n\bigr).
\end{equation*}
This means that the projection
\begin{equation*}
\bigl((a_1,a_2,\dotsc,a_{n+1});(b_1,\dotsc,b_{n-1})\bigr) \mto
\arg (a_n + i a_{n+1}) \in \SS^1
\end{equation*}
defines for each $(b_1,\dotsc,b_{n-1})$ a \LOB with the $(n-1)$-ball
as pages and trivial monodromy.
From now on we denote the points in $\DD^{n-1}$ by $\bbf = (b_1,
\dotsc, b_{n-1})$, and write for the \LOB
\begin{equation*}
L_\bbf = \Psi\bigl(\SS^n \times \{\bbf\} \bigr),
\end{equation*}
and $B_\bbf$ for its binding.
For the technical details of the following part, we refer to
Section~\ref{sec: topology moduli space}.
We will study the space
\begin{equation*}
\widetilde{\mM}_\star = \Bigl\{(\bbf, u, z_0)\Bigm|\, \bbf \in \DD^{n-1},\,
u\colon (\DD^2, \p \DD^2) \ra \bigl(\widehat W, L_\bbf\bigr),\,
z_0 \in \DD^2 \Bigr\}
\end{equation*}
of nonconstant holomorphic maps from a disk, equipped with one marked
point $z_0$, and with boundary sent into one of the \LOB{}s~$L_\bbf$.
Additionally, we require that $u$ is homotopic to a Bishop disk as an
element in $\pi_2\bigl(\widehat W, L_\bbf \setminus B_\bbf\bigr)$, and
we denote the corresponding subset by $\widetilde \mM$.
Next we divide $\widetilde \mM$ by the action of the group
$\Aut(\DD^2)$ of biholomorphic transformations on $\DD^2$, where
$\varphi \in \Aut(\DD^2)$ acts on $\widetilde \mM$ via
\begin{equation*}
\varphi \cdot (\bbf,u,z_0) =
\bigl(\bbf, u\circ \varphi^{-1}, \varphi(z_0)\bigr).
\end{equation*}
We denote the moduli space $\widetilde \mM / \Aut(\DD^2)$ by $\mM$.
Note that for every class~$[\bbf, u, z]$ in~$\mM$, we can fix a unique
representative $(\bbf, u_0, z_0)$ by choosing a parametrization of $u$
such that
\begin{equation}
u(z) \in
\begin{cases}
\text{the $0$ degree page of the \LOB, } & \text{if $z = 1$,} \\
\text{the $\pi/2$ degree page of the \LOB, } & \text{if $z = i$,} \\
\text{the $\pi$ degree page of the \LOB, } & \text{if $z = -1$}. \\
\end{cases}\label{eq: standard parametrization of hol disks}
\end{equation}
A corollary of this is that the moduli space~$\mM$ (before the
compactification, see below) is a \emph{trivial} disk bundle over the
space of unmarked disks.
This is the key fact that will allow us to ``push'' the topology of
$W$ into its boundary (which is the geometric analogue of the
algebraic argument given in \cite{McDuff_contactType} and
\cite{OanceaViterbo}).
Next, we need to understand the compactification of $\mM$.
Note first that typical holomorphic disks are surrounded by a
neighborhood of other typical holomorphic disks, that is, they
represent smooth points of the interior of the moduli space~$\mM$.
With ``typical'', we mean smooth holomorphic disks whose interior
points are mapped to the interior of $\widehat W$, and whose boundary
sits on a \LOB~$L_\bbf$ that is not a boundary \LOB, \ie for which
$\norm{\bbf} < 1$, and such that the disk does not touch the
binding~$B_\bbf$ of the \LOB.
Let us now consider the remaining cases. The boundary of $\widehat W$
consists of $S_A \subset\bigl\{g_A = 1\bigr\}$ and $S_B
\subset\bigl\{g_B = 1\bigr\}$, which are weakly and strongly
plurisubharmonic hypersurfaces respectively.
A disk touching $S_B$ with one of its interior points will
automatically be constant.
If the disk touches $S_A$ instead, then it needs to be entirely
contained in this hypersurface, and in particular its boundary will
lie on a \LOB with $\norm{\bbf} = 1$; below we will explain how to
understand the disks in this second case explicitly.
For every \LOB~$L_\bbf$, there is a certain neighborhood of its
binding~$B_\bbf$ that is only intersected by Bishop disks.
Since there is exactly one disk meeting every point of this
neighborhood, that is, the \defin{evaluation map}
\begin{equation*}
\ev\colon \mM \to \widehat W, \quad [\bbf, u,z_0] \mto u(z_0)
\end{equation*}
restricts close to $B_\bbf$ to a diffeomorphism, it follows that the
compactification $\overline{\mM}$ contains disks that collapse to a
point in the binding.
In \cite{NiederkrugerRechtman} it was shown that adding these constant
disks to $\mM$, corresponds to adding points which lie on the smooth
boundary of the compactification~$\overline{\mM}$.
Before understanding the bubbling, we will discuss disks whose
boundary lies in a \LOB~$L_\bbf \subset S_A$.
\begin{lemma}\label{lemma: disks with boundary in S_A}
Suppose $u \in \overline{\mM}$ maps $\p \DD^2$ to a \LOB{}~$L_\bbf$
such that $\norm{\bbf}=1$.
Then the image of $u$ is completely contained in $S_A$, and
moreover, it is obtained by the intersection of a complex line
parallel to the $z_n$-plane with $S_A$.
\end{lemma}
\begin{proof}
Parametrize the disk by polar coordinates~$r e^{i\phi}$.
By acting on the coordinates $z_1, \dotsc, z_{n-1}$ with a matrix in
$\SO(n-1)$ (regarded as an element of $\SU(n-1)$ with real entries),
we can assume without loss of generality that the \LOB~$L_\bbf$
corresponds to the parameter $\bbf = (1,0,\dotsc,0)$, as the
functions~$g_A$ and $g_B$ are invariant under such an action.
In particular it follows that the $y_1$-coordinate of $u$ has its
maximum on the boundary of $u$.
The $x_1$-coordinate of $\restricted{u}{\p\DD^2}$ is bounded, and
hence there is an angle~$e^{i\phi_0}$ at which the derivative
\begin{equation*}
\restricted{\frac{d}{d\phi}}{\phi = \phi_0}
x_1\bigl(u(e^{i\phi})\bigr) = 0
\end{equation*}
is zero.
Complex multiplication gives $i\cdot\partial_r = \partial_\phi$,
hence
\begin{equation*}
dy_1\bigl(Du\cdot\partial_r\bigr)
= dy_1\bigl(Du\cdot(-i\cdot\partial_\phi)\bigr)
= - dy_1\bigl(i\cdot Du\cdot\partial_\phi\bigr)
= - dx_1 \bigl(Du\cdot\partial_\phi\bigr) = 0.
\end{equation*}
It follows that the outward derivative of the $y_1$-coordinate
vanishes at the point $e^{i\phi_0} \in\nobreak \DD^2$, so that according to
the boundary point lemma, $y_1$ must equal the constant~$1$ on the
whole disk, and as a consequence $u$ lies entirely in $S_A$.
The $y_2$- to $y_{n-1}$-coordinates are all $0$ on the boundary of
the disk, and hence by the maximum principle, they need to be both
maximal and minimal on all of the disk.
With the Cauchy-Riemann equation we obtain that the $x_1$- up to
$x_{n-1}$-coordinates of $u$ need all to be constant on $u$ (for
more details read Section~\ref{sec: bishop disks at the boundary}).
\end{proof}
As explained in Section~\ref{sec: topology moduli space}, no bubbling
can occur under our assumptions, and hence $\overline{\mM}$ will be a
compact manifold with boundary and corners (the boundary is smooth
everywhere with the exception of the disks corresponding to the edges
of $\widehat W$).
Moreover, the moduli space is orientable (see Appendix~\ref{sec:
orientability of the moduli space}) and the evaluation map
\begin{equation*}
\ev\colon \bigl(\overline{\mM}, \p \overline{\mM} \bigr) \to
\bigl(\widehat W, \p \widehat W\bigr), \quad
[\bbf, u,z_0] \mto u(z_0)
\end{equation*}
is a degree~$1$ map, that is, it maps the fundamental class
$[\overline{\mM}] \in H_{2n}\bigl(\overline{\mM}, \p \overline{\mM};
\ZZ\bigr)$ onto the fundamental class $[\widehat W] \in
H_{2n}\bigl(\widehat W, \p \widehat W; \ZZ\bigr)$.
We are therefore in a position to apply the following topological
result, which was stated as Lemma~\ref{lemma: map of product manifold
induces contractibility} in the introduction.
\begin{lemma}\label{lemma:KlausTopology}
Let $X,Y$ be smooth orientable compact manifolds with boundary and
corners such that $\p Y$ is homeomorphic to a sphere and $\dim X + 2
= \dim Y \ge 3$.
Write $X' = X\times \DD^2$, and assume that
\begin{equation*}
f\colon (X',\p X')\to (Y,\p Y)
\end{equation*}
is a continuous map that is smooth on the interior of $X'$, and for
which we find an open subset~$U\subset \mathring Y$ such that
$\restricted{f}{f^{-1}(U)} \colon f^{-1}(U) \to U$ is a
diffeomorphism.
Then $Y$ is contractible.
\end{lemma}
\begin{proof}
Note that by Whitehead's theorem, it suffices to show that $Y$ is
weakly contractible, that is, $\pi_j(Y) = 0$ for all $j > 0$.
Using Hurewicz's theorem we will show instead that $Y$ is simply
connected and satisfies $H_j(Y ; \ZZ) = 0$ for all $j > 0$.
\begin{enumeratei}
\item
We will first consider the fundamental group of $Y$.
Choose the base point \hbox{$p_0 \in U\subset \mathring Y$}.
Let $\gamma$ be a smooth, embedded loop representing a class in
$\pi_1\bigl(Y, p_0\bigr)$ that lies in the interior of $Y$.
After a perturbation, we can assume that $\gamma$ is transverse to
the map~$f$, so $f^{-1}\bigl(\gamma\bigr)$ will be a finite
collection of loops~$\Gamma_0,\dotsc,\Gamma_N$ in $X'$.
There is one loop, say $\Gamma_0$, that is mapped to $\gamma$ with
degree one.
The reason for this is that $\gamma$ runs through $U$, where $f$ is
a diffeomorphism.
Then the loop $f\circ \Gamma_0$ is homotopic to $\gamma$, and thus
represents the same class in $\pi_1\bigl(Y, p_0\bigr)$.
Using the fact that $X'$ is diffeomorphic to a trivial disk bundle,
we may shift $\Gamma_0$ into the boundary of $X'$ (just by moving it
inside the $\DD^2$-factor).
In particular, this shows that $[\gamma] = [f\circ \Gamma_0]$ can be
represented by a loop that lives in the boundary of~$Y$, and is thus
contractible.
\item
Next we need to compute the homology of $Y$.
It is easy to see that the image~of
\begin{equation*}
f_*\colon H_*\bigl(X'; \ZZ\bigr) \to H_*\bigl(Y; \ZZ\bigr)
\end{equation*}
is trivial.
Indeed, all homology groups $H_k\bigl(X'; \ZZ\bigr)$ with $k \ge
\dim X$ are trivial, so that we only need to study $k < \dim X <
\dim \p Y$.
Let $A\in H_k(Y; \ZZ)$ be a homology class that lies in the image of
$f_*$ so that there is a $B \in H_k(X'; \ZZ)$ with $A = f_*B$.
Since $X' = X \times \DD^2$ where $\DD^2$ is contractible, $B$ can
be represented by a cycle in $X \times \{p\}$ for any point $p \in
\DD^2$; in particular we are free to choose $p \in \p\DD^2$, hence
$B$ is represented by a cycle in~$\p X'$.
This implies that the class~$A$ is homologous to a cycle in the
sphere~$\p Y$, which shows that $A$ must be trivial.
We will now show that $f_*\colon H_k\bigl(X'; \ZZ\bigr) \to
H_k\bigl(Y; \ZZ\bigr)$ is surjective for every $k \le\nobreak
\frac{1}{2}\,\dim Y$.
Assume for now that $k< \frac{1}{2}\,\dim Y$ and that we already
have shown for every $\ell 0$ is a
constant, \ie $H_r$ is the intersection of the two subsets
\begin{equation*}
\bigl\{\norm{\y^-}^2 \le 1\bigr\} \,\cap \,
\bigl\{\norm{\x^-}^2 + \norm{\z^+}^2 + \abs{z^\circ}^2 \le r^2\bigr\}.
\end{equation*}
We denote
\begin{equation*}
\p_- H_r = \p \DD^k \times \DD^{2n-k}_r, \quad
\p_+ H_r = \DD^k \times \p \DD^{2n-k}_r.
\end{equation*}
The core of $\p_-H_r$ is the $(k-1)$-sphere
\begin{align*}
S_- &:= \bigl\{\norm{\y^-}^2 = 1,\, \x^- = \0, \, \z^+ = \0,\, z^\circ =
0\bigr\},
\intertext{which will be identified with the attachment sphere in a
contact manifold; the core of $\p_+H_r$ is the $(n+m)$-sphere
(note that $n+m = 2n -k -1$)}
S_+ &:= \bigl\{\y^- = \0,\, \norm{\x^-}^2 + \norm{\z^+}^2 +
\abs{z^\circ}^2 = r^2\bigr\}.
\end{align*}
Choose on $\CC^n$ the symplectic form
\begin{equation*}
\omega = 2\, \sum_{s=1}^k dx^-_s \wedge dy^-_s
+ 4\, \sum_{t=1}^m dx^+_t \wedge dy^+_t
+ 4\, dx^\circ \wedge dy^\circ.
\end{equation*}
It admits the Liouville form
\begin{equation*}
\lambda = 2\, \sum_{s=1}^k \bigl(2x^-_s\, dy^-_s + y^-_s \, dx^-_s\bigr)
+ 2\, \sum_{t=1}^m \bigl(x^+_t \, dy^+_t - y^+_t \, dx^+_t\bigr)
+ 2\, (x^\circ \, dy^\circ - y^\circ \, dx^\circ)
\end{equation*}
that is associated to the Liouville vector field
\begin{multline*}
X_L = \sum_{s=1}^k \Bigl(2x_s^-\, \frac{\partial}{\partial
x_s^-} - y_s^-\, \frac{\partial}{\partial y_s^-} \Bigr) \\[-5pt]
+ \frac{1}{2}\, \sum_{t=1}^m \Bigl(x_t^+\,
\frac{\partial}{\partial x_t^+} + y_t^+\, \frac{\partial}{\partial
y_t^+}\Bigr) +
\frac{1}{2}\,\Bigl(x^\circ\,\frac{\partial}{\partial x^\circ} +
y^\circ\, \frac{\partial}{\partial y^\circ}\Bigr).
\end{multline*}
The field~$X_L$ points outward through $\p_+H_r$ and inward at
$\p_-H_r$, so that both $\p_+H_r$ and $\p_-H_r$ are contact type
hypersurfaces with the corresponding coorientations.
The core~$S_- \subset \p_-H_r$ is an isotropic sphere with trivial
conformal symplectic normal bundle.
\begin{figure}[htbp]
\centering
\includegraphics[height=5cm,keepaspectratio]{unmodified_handle2.pdf}
\caption{The handle can be glued onto a contact
manifold.}\label{fig: unmodified handle}
\end{figure}
Let now $(M,\xi)$ be a given contact manifold and let
$\attachingSphere{k-1} \subset M$ be a $(k-\nobreak1)$\nobreakdash-dimen\-sional isotropic
sphere with trivial conformal symplectic normal bundle that will serve
as the \defin{attaching sphere} of the $k$-handle~$H_r$.
Fixing $r > 0$ small enough, $\p_- H_r$, endowed with the contact
structure induced by $\lambda$, is contactomorphic to a
neighborhood~$\nN(\attachingSphere{k-1}) \subset (M,\xi)$ of
$\attachingSphere{k-1}$.
We choose a contact form $\alpha$ for $\xi$ on $M$ such that
$\restricted{\alpha}{\nN(\attachingSphere{k-1})}$ can be glued to
$\restricted{\lambda}{T(\p_-H_r)}$ and define the Liouville manifold
\begin{equation*}
(W_0,\lambda_0) := \bigl((-\epsilon,0]\times M, e^t\alpha\bigr)
\cup_{\p_-H_r} (H_r,\lambda).
\end{equation*}
The positive boundary of $W_0$ (denoted $\p_+ W_0$) has two smooth
faces $M \setminus \nN(\attachingSphere{k-1})$ and $\p_+H_r$, meeting
along a corner which is the image of the corner $\p \DD^k \times \p
\DD^{2n-k}_r$ in $\p H_r$, see Fig.\,\ref{fig: unmodified handle}.
Fix a small neighborhood~$\uU$ of the corner, and choose a smooth
hypersurface~$M'$ that matches $\p_+ W_0$ outside of $\uU$, and is
transverse to $X_L$ in $\uU$.
Denote the induced contact structure on $M'$ by $\xi' = TM' \cap \ker
\lambda_0$.
Note that the constant~$r > 0$ can be made arbitrarily small, without
changing the isotopy class of the contact structure on $M'$; we can
shrink the size of the handle continuously (including the smoothing)
which allows us to apply Gray stability to obtain an isotopy with
support in the model neighborhood.
The belt sphere~$\beltSphere{2n-k-1} = \beltSphere{n + m}$ of the
$k$-handle is the core~$S_+$ of $\p_+ H_r$.
\subsection{Families of $\mathsf{Lob}$s on a deformed subcritical handle}
\label{sec: deformed subcritical handles}
To find the desired family of \LOB{}s, we will now modify the contact
structure in a neighborhood of the belt sphere in two steps.
The first deformation is borrowed from the recent article
\cite{GeigesZehmisch}; it replaces a technically more complicated
method that was used in an earlier version of this paper.
Consider again a ``thin'' handle $H_r = \DD^k \times \DD^{2n-k}_r
\subset \CC^n$ with $r \ll 1$ as used above.
Suppose that the rounding of the corners has been performed for values
of $\norm{\y^-}$ in the interval~$[1 - \epsilon, 1]$.
The part of $\p_+ H_r$ outside the smoothing region lies in the level
set~$\{f = r^2\}$ of the function
\begin{equation*}
f(\z^-,\z^+, z^\circ) = \norm{\x^-}^2 + \norm{\z^+}^2 + \abs{z^\circ}^2.
\end{equation*}
We would like to modify the Liouville field on a neighborhood of the
belt sphere so that the induced contact structure coincides with the
field of complex hyperplanes on the boundary.
For this, add the Hamiltonian vector field~$X_H$ of a function
$H\colon \CC^n \to \RR$ to $X_L$, since then $\hat X_L := X_L + X_H$
will still be a Liouville field.
Let $\rho\colon [0,\infty) \to [-1,0]$ be a smooth function that is
equal to $-1$ on the interval $[0, \sqrt{1-2\epsilon}]$, equal to $0$
on the interval $[\sqrt{1-\epsilon}, 1]$ and that increases
monotonically in between.
Define the Hamiltonian function
\begin{equation*}
H(\z^-, \z^+, z^\circ) := 2\, \langle\x^-, \y^-\rangle\,
\rho\bigl(\norm{\y^-}^2\bigr).
\end{equation*}
The Hamiltonian vector field corresponding to $H$ is
\begin{equation*}
X_H = - \sum_{s=1}^k y^-_s\, \rho\bigl(\norm{\y^-}^2\bigr)\,
\frac{\partial}{\partial y^-_s} +
\sum_{s=1}^k \left(2\,\langle\x^-, \y^-\rangle\, y^-_s\,
\rho'\bigl(\norm{\y^-}^2\bigr)
+ x^-_s\, \rho\bigl(\norm{\y^-}^2\bigr) \right)\,
\frac{\partial}{\partial x^-_s}.
\end{equation*}
The vector field $\hat X_L$ agrees outside the support of $\rho$ with
$X_L$, and it is everywhere transverse to $\p_+ H_r$ as can be seen
from
\begin{equation*}
\begin{split}
\lie{\hat X_L} f &= \lie{X_L} f + \lie{X_H} f \\
& = 4\, \norm{\x^-}^2 + \norm{\z^+}^2 + \abs{z^\circ}^2 +
\lie{X_H} \norm{\x^-}^2 \\
& = (4 + 2\, \rho)\, \norm{\x^-}^2 + \norm{\z^+}^2 +
\abs{z^\circ}^2 + 4\,\langle\x^-, \y^-\rangle^2\, \rho' > 0,
\end{split}
\end{equation*}
because $\rho \ge -1$, and $\rho' \ge 0$.
It follows that $\lambda$ and $\hat \lambda := \iota_{\hat X_L}
\omega$ induce isotopic contact structures on $M'$.
The contact structure on the domain $\p_+ H_r \cap \{\norm{\y^-}^2 \le
1 - 2\epsilon\}$ is the kernel of the Liouville form
\begin{equation*}
\hat \lambda = \lambda + dH = 2\, \sum_{s=1}^k x^-_s\, dy^-_s
+ 2\, \sum_{t=1}^m \bigl(x^+_t \, dy^+_t - y^+_t \, dx^+_t\bigr)
+ 2\, (x^\circ \, dy^\circ - y^\circ \, dx^\circ).
\end{equation*}
\begin{remark}\label{rmk: always long cylinder}
This first deformation shows that the surgered manifold contains a
neighborhood of the belt sphere that is contactomorphic to a
cylinder
\[
\bigl\{f = r^2\} \cap \bigl\{\norm{\y^-}^2 \le 1/2\bigr\}
\subset \CC^n,
\]
with $r$ arbitrarily small and a contact structure
given as kernel of $\hat \lambda$.
Note that $\hat \lambda$ on the domain under consideration is equal
to the differential~$- d^c f$, \ie the contact structure on our
domain coincides with the complex tangencies.
This is the key fact that we will exploit in the second deformation
below.
\end{remark}
To continue, we consider the setting of Theorem~\ref{thm: main
theorem}, in which $(M',\xi')$ was a fillable contact manifold
obtained by subcritical surgery.
Since this will be the main object of study from now on, it will be
convenient to simplify the notation, hence we assume (unlike in the
statement of Theorem~\ref{thm: main theorem}) that $(M,\xi)$ is a
closed contact manifold of dimension $2n-1$ that has been obtained by
a surgery of index $k \in \{1,\dotsc,n-1\}$ from another contact
manifold, and let $(W, \omega)$ be a weak symplectic filling of $(M,
\xi)$.
Since the theorem in dimension three already follows from the much
stronger result of Eliashberg \cite{Eliashberg_filling}, we are free
to assume $n \ge 3$. The belt sphere then has dimension $2n-k-1 \ge n
\ge 3$, hence the restriction of $\omega$ to $\beltSphere{2n-k-1}$ is
automatically exact.
It follows (using \cite[Rem.\,2.11]{WeafFillabilityHigherDimension})
that $\omega$ can be deformed in a collar neighborhood of $\p W$ so
that an outwardly transverse Liouville vector field exists near
$\beltSphere{2n-k-1}$, and we are therefore free to pretend in the
following discussion that $(W,\omega)$ is a \emph{strong} filling of
$(M,\xi)$.
In particular, we may assume that the symplectic structure on a collar
neighborhood close to the belt sphere looks like the symplectic
structure on the boundary of the model of the handle, and we may
identify both.
Let $f\colon \CC^n \to [0,\infty)$ be again the plurisubharmonic
function
\begin{equation*}
f(\z^-,\z^+, z^\circ) = \norm{\x^-}^2 + \norm{\z^+}^2 + \abs{z^\circ}^2.
\end{equation*}
By the explanations above (see Remark~\ref{rmk: always long
cylinder}), the belt sphere~$\beltSphere{n + m}$ has a
neighborhood~$\uU_M \subset M$ that is contactomorphic to the cylinder
\begin{equation*}
C_r := \bigl\{(\z^-, \z^+, z^\circ) \in \CC^n\bigm|\,
f(\z^-,\z^+, z^\circ) = r^2, \, \norm{\y^-}^2 < 1/2 \bigr\}
\end{equation*}
for arbitrarily small $r \ll 1$ with contact structure~$\hat{\xi}$
given as the kernel of the Liouville form~$\hat{\lambda} =
-\restricted{d^cf}{TC_r}$, and since the cylinder is a level set of
$f$, this also means that~$\hat{\xi}$ are the complex tangencies of
$C_r$.
We denote by $\uU_W$ a small neighborhood of $\uU_M$ in~$W$ that is
symplectomorphic to the subset
\begin{equation*}
\bigl\{(\z^-, \z^+, z^\circ) \in \CC^n\bigm|\,
f(\z^-,\z^+, z^\circ) \in (r^2-\delta,r^2], \, \norm{\y^-}^2 < 1/2 \bigr\}
\end{equation*}
with symplectic form $\omega = -dd^c f$.
Using the embedding of $\uU_W$ into our model, we can extend the
symplectic filling $(W, \omega)$ by attaching the following compact
symplectic subdomain of $\CC^n$:
replace $f$ by $G := \max\{g_A, g_B\}$ which is obtained as the
maximum of the two functions:
\begin{align*}
g_A(\z^-,\z^+, z^\circ) &:= \norm{\y^+}^2 \\
\tag*{and}
g_B(\z^-,\z^+, z^\circ) &:= \norm{\x^-}^2 + \norm{\x^+}^2 +
\psi\bigl(\norm{\y^-}\bigr) \cdot \norm{\y^+}^2 + \abs{z^\circ}^2
,
\end{align*}
where $\psi$ is a cut-off function that vanishes close to $0$, and
increases until it reaches $1$ close to $\norm{\y^-} = 1/\sqrt{2}$.
Clearly $g_A$ is a weakly plurisubharmonic function.
The function~$g_B$ is strictly plurisubharmonic on a neighborhood of
$\{\y^+ = \0\}$ because the last term of
\begin{equation*}
-dd^cg_B = 2\, \sum_{s=1}^k dx^-_s \wedge dy^-_s
+ 2\, \sum_{t=1}^m dx^+_t \wedge dy^+_t + 4\, dx^\circ \wedge dy^\circ
- dd^c \bigl(\psi(\norm{\y^-}) \cdot \norm{\y^+}^2 \bigr)
\end{equation*}
simplifies along this subset to
\begin{equation*}
\begin{split}
-dd^c \bigl(\psi(\norm{\y^-}) \cdot \norm{\y^+}^2 \bigr)
&= - \psi(\norm{\y^-}) \, dd^c\norm{\y^+}^2
- d\norm{\y^+}^2\wedge d^c\psi(\norm{\y^-}) \\
&\qquad\qquad - d\psi(\norm{\y^-}) \wedge
d^c\norm{\y^+}^2 -\norm{\y^+}^2\, dd^c\psi(\norm{\y^-}) \\
&= 2 \psi(\norm{\y^-}) \, \sum_{t=1}^m dx^+_t \wedge
dy^+_t,
\end{split}
\end{equation*}
which is weakly plurisubharmonic.
This implies that if the chosen handle~$C_r$ is thin enough, that is
if $r > 0$ has been chosen sufficiently small, then $g_B$ will be
strictly plurisubharmonic on its neighborhood.
For large values of $\norm{\y^-}^2$, the cut-off function is equal to
$1$ and $g_B$ agrees with $f$.
Since it also dominates $g_A$, the level set~$\{G = r^2\}$ glues
smoothly to the given contact manifold, and it bounds a symplectic
manifold
\begin{equation*}
\widehat W = W \cup \bigl\{ (\z^-,\z^+, z^\circ) \in \CC^n\bigm|\,
f(\z^-,\z^+, z^\circ) \ge r^2 \text{ and }
G (\z^-,\z^+, z^\circ) \le r^2\bigr\}
\end{equation*}
obtained from the given symplectic filling~$W$ by attaching to it the
symplectic domain lying in our model between the level sets~$\{f =
r^2\}$ and $\{G = r^2\}$, see Figures~\ref{fig: deformed handle} and~\ref{fig: deformed handle cut open}.
Note that the boundaries of $W$ and $\widehat W$ are continuously
isotopic.
We also write $\Wmodel$ for the subdomain $\uU_W \cup \bigl(\widehat W
\setminus W\bigr)$ that lies entirely in $\CC^n$.
We decompose the boundary of $\widehat W$ into three domains, which we
denote by $\Mreg$,~$M_A$ and $M_B$.
Here $M_A$ and $M_B$ are the parts of $\p \widehat W$ that lie in the
level set $\{g_A = r^2\}$ or $\{g_B = r^2\}$ respectively, and satisfy
$\norm{\y^-}^2 < 1/2$; $\Mreg$ is the remaining part of the boundary
of $\widehat W$, \ie the part that is disjoint from the boundary of
the deformed handle.
The boundary of the handle contains a deformation of the belt sphere,
which we will still write as $\beltSphere{n + m} = \{\y^- = \0\}$ even
though it has edges.
The cut-off function~$\psi$ vanishes on a neighborhood of
$\beltSphere{n+m}$, so that $G$ simplifies to
\begin{equation*}
G (\z^-,\z^+, z^\circ) := \max\Bigl\{\norm{\y^+}^2,\,
\norm{\x^-}^2 + \norm{\x^+}^2 + \abs{z^\circ}^2 \Bigr\}.
\end{equation*}
It follows that $\beltSphere{n+m}$ is the boundary of the poly-disk
\begin{equation*}
\Bigl\{\norm{\y^+}^2 \le r^2, \, \y^- = \0\Bigr\} \cap
\Bigl\{ \norm{\x^-}^2 + \norm{\x^+}^2 + \abs{z^\circ}^2 \le r^2,
\y^- = \0\Bigr\} \cong \DD^m \times \DD^{n+1}.
\end{equation*}
\begin{figure}[htbp]
\centering
\includegraphics[height=5cm,keepaspectratio]{deformed_handle2.pdf}
\caption{We need to deform the handle to find a suitable family
of \LOB{}s in the belt sphere.
The new handle will have edges.
The green area in the picture represents the boundary~$M_A$, the
yellow one is the boundary~$M_B$, and the grey part is
$\Mreg$.
The belt sphere~$S_A\cup S_B$ corresponding to the deformed
handle also has edges.
The part~$S_B$ is foliated by \LOB{}s.}\label{fig: deformed handle}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[height=5cm,keepaspectratio]{deformed_handle_cut2.pdf}
\caption{The deformed handle differs from the original one by
the attachment of a symplectic cobordism.
This cobordism can also be added to any other symplectic filling
of the surgered contact manifold.
The figure shows a cut through this cobordism. }\label{fig:
deformed handle cut open}
\end{figure}
We can decompose the boundary of the poly-disk $\DD^m \times
\DD^{n+1}$ as a union of two smooth parts
\begin{equation*}
\p \bigl(\DD^m \times \DD^{n+1}\bigr) =
\SS^{m-1} \times \DD^{n+1} \cup \DD^m \times \SS^n.
\end{equation*}
We will denote the first part of the belt sphere by
\begin{align*}
S_A &:= \Bigl\{\y^- = \0,\, \norm{\y^+}^2 = r^2,\, \norm{\x^-}^2 +
\norm{\x^+}^2 + \abs{z^\circ}^2 \le r^2 \Bigr\} \cong \SS^{m-1} \times
\DD^{n+1},
\intertext{but for now, we will be mostly interested in the second
part}
S_B &:= \Bigl\{\y^- = \0,\, \norm{\y^+}^2 \le r^2,\, \norm{\x^-}^2 +
\norm{\x^+}^2 + \abs{z^\circ}^2 = r^2 \Bigr\} \cong \DD^m \times
\SS^n.
\end{align*}
It lies in the $i$-convex hypersurface $M_B := \bigl\{g_B = r^2 \bigr\}$,
whose complex tangencies are the kernel of the $1$-form $-
\restricted{\bigl(d^c g_B\bigr)}{TM_B}$.
Close to $\{\y^- = \0\}$, we compute
\begin{align*}
-d^c g_B &= 2\, \sum_{s=1}^k x_s^-\, dy_s^- + 2\, \sum_{t=1}^m
x_t^+\, dy_t^+ + 2\,\bigl(x^\circ\,dy^\circ - y^\circ \, d
x^\circ\bigr),
\intertext{which simplifies on $S_B \subset \{\y^- = \0\}$ further
to}
-d^c g_B &= 2\, \sum_{t=1}^m x_t^+\, dy_t^+ +
2\,\bigl(x^\circ\,dy^\circ - y^\circ \, d x^\circ\bigr).
\end{align*}
The submanifold $S_B \cong \DD^m \times \SS^n$ can be foliated by the
$n$-spheres with constant $\y^+$-value.
For every fixed value of $\y^+ = \bbf^+ \in \DD^m_r$ we write the
corresponding leaf as
\begin{equation*}
L_{\bbf^+}
= \Bigl\{(\y^-,\y^+) = (\0, \bbf^+),\, \norm{\x^-}^2
+ \norm{\x^+}^2 + \abs{z^\circ}^2 = r^2 \Bigr\}.
\end{equation*}
The restriction of $-d^c g_B$ to each of these spheres is
\begin{equation*}
2\,\bigl(x^\circ\,dy^\circ - y^\circ \, d x^\circ\bigr),
\end{equation*}
so that $L_{\bbf^+}$ is actually a spherical \LOB in the (strictly)
$i$-convex level set~$M_B$.
The binding~$B_{\bbf^+}$ of the \LOB{}~$L_{\bbf^+}$ is given by the set of
points where $z^\circ$ vanishes, \ie $B_{\bbf^+} \cong \SS^{n-2}$; the
pages of the open book are the fibers of the map
\begin{equation*}
\vartheta\colon L_{\bbf^+} \setminus B_{\bbf^+} \to \SS^1,\,
(\z^-, \z^+, z^\circ) \mto \frac{z^\circ}{\abs{z^\circ}}.
\end{equation*}
In the following sections we will study holomorphic disks that each
have boundary on one of the \LOB{}s~$L_{\bbf^+}$.
\section{The space of holomorphic disks attached to the
belt sphere}
\label{sec:moduliSpace}
We will now construct the moduli space of pseudoholomorphic disks
needed for the proof of Theorem~\ref{thm: main theorem} and show that
it is a smooth manifold with boundary.
We continue with the setup and notation used in Section~\ref{sec: deformed
subcritical handles} above.
\begin{assumptions}\label{assumptions: choice of J}
Choose an almost complex structure~$J$ on $\widehat W$ with the
following properties:
\begin{itemize}
\item $J$ is tamed by $\omega$;
\item $J$ agrees on $\Wmodel$ with the standard complex
structure~$i$;
\item the unmodified domain~$\Mreg$ in $\p W$ is $J$-convex, and its
$J$-complex tangencies agree with $\xi$.
\end{itemize}
\end{assumptions}
\subsection{The top stratum of the moduli space}
We define $\widetilde\mM\bigl(\widehat W, S_B; J\bigr)$ as the moduli
space of ``parametrized'' curves $\bigl(\bbf^+, u, z_0\bigr)$, where:
\begin{enumeratei}
\item
$\bbf^+ \in \DD^m_r$ is a point in the $m$-disk
parametrizing the \LOB{}s~$L_{\bbf^+} \subset S_B$ described in the
previous section;
\item
$u\colon \bigl(\DD^2, \p \DD^2\bigr) \to \bigl(\widehat W,
L_{\bbf^+} \setminus B_{\bbf^+})$ is a $J$-holomorphic map which is
trivial in $\pi_2(\widehat W, L_{\bbf^+})$; and
\item
$z_0$ is a marked point in the closed unit disk~$\DD^2$.
\end{enumeratei}
\noindent
Additionally we require that
\begin{equation*}
\restricted{\bigl(\vartheta\circ u\bigr)}{\p \DD^2}\colon
\SS^1 \to \SS^1
\end{equation*}
is a degree~$1$ map, that is, the boundary of each disk makes one turn
around the binding of the open book.
Since $L_{\bbf^+}$ lies in the strictly convex hypersurface~$M_B$, the
map $\restricted{\bigl(\vartheta\circ u\bigr)}{\p \DD^2}$ is a
diffeomorphism, \ie the disk intersects every page of the
\LOB~$L_{\bbf^+}$ precisely once;
see~\cite[Cor.\,II.1.11]{NiederkrugerHabilitation}.
It is very easy to deduce from this that the disks in
$\widetilde\mM\bigl(\widehat W, S_B; J\bigr)$ are somewhere injective
near the boundary.
However, since we are not really free to choose the almost complex
structure near the boundary, we need more to achieve transversality.
We say that a $J$-holomorphic disk $u$ is \emph{simple} if it is
somewhere injective in an open dense set of $\DD^2$.
For closed holomorphic curves, simple means not multiply covered, but
for disks the situation is more complicated; see
\cite{LazzariniSimpleDisks}.
However our situation is not the most general one, and it is possible
to adapt the arguments of \cite[Prop.\,2.5.1]{McDuffSalamonJHolo}
to prove that all disks in $\widetilde\mM\bigl(\widehat W, S_B;
J\bigr)$ are simple.
\begin{lemma}\label{lemma: our disks are simple}
Every disk $u \in \widetilde\mM\bigl(\widehat W, S_B; J\bigr)$ is
simple.
\end{lemma}
\begin{proof}
We know $u \colon \DD^2 \to W$ is embedded near the boundary.
Let $X$ denote the set of points $z \in \DD^2$ such that either of
the following is true:
\begin{enumeratei}
\item
$D_zu = 0$, or
\item
There exists a different point $z' \in \DD^2$ such that
$u$ restricted to disjoint neighborhoods of $z$ and $z'$ has an
isolated intersection $u(z) = u(z')$.
\end{enumeratei}
\noindent
(Recall that either $u(z) = u(z')$ is an isolated intersection, or
there exists neighbourhoods of $z$ and $z'$ with the same image.)
Standard local results plus the fact that $u$ is embedded at $\p
\DD^2$ tell us that $X$ is a finite set of interior points.
The image $u(\DD^2 \setminus X)$ is then a smoothly embedded
$J$-holomorphic submanifold of $W$; in particular, it is a Riemann
surface $\dot \Sigma$ with connected boundary and finitely many
punctures.
The inclusion of $\dot \Sigma$ into $W$ is then a $J$-holomorphic
embedding, and it extends over the punctures to a $J$-holomorphic
map $v \colon \Sigma \to W$, which is not necessarily an embedding
but has only finitely many critical points and self-intersections.
At this point we don't know the topology of $\Sigma$, except that it
has connected boundary.
But the original map $u$, restricted to $\DD^2 \setminus X$, defines
a holomorphic map to $\dot \Sigma$, which then extends by removal of
singularities to a holomorphic map $\varphi \colon \DD^2 \to \Sigma$
such that $u = v \circ \varphi$.
Given the properties of $u$ at the boundary, $\varphi$ must restrict
to a diffeomorphism $\p \DD^2 \to \p \Sigma$, and it maps interior
to interior.
So it has degree one, and is therefore biholomorphic.
\end{proof}
Lemma~\ref{lemma: our disks are simple} will allow us to use the
following transversality result.
\begin{proposition}\label{prop: smoothness of moduli space}
Let $(W,J)$ be an almost complex manifold, and let $\DD^m_\epsilon
\times L \subset W$ be a submanifold for which every slice $L_\x :=
\{\x\} \times L$ is a totally real submanifold.
For generic choices of $J$ satisfying Assumptions~\ref{assumptions:
choice of J}, the following holds.
Suppose $u_0\colon \bigl(\DD^2, \p \DD^2\bigr) \to \bigl(W,
L_\0\bigr)$ is any $J$-holomorphic map such that
\begin{itemize}
\item the interior points of $u_0$ do not touch the boundary of $W$;
\item the boundary of $u_0$ lies in the interior of $L_\0$;
\item the disk~$u_0$ is \emph{simple}.
\end{itemize}
Let $\widetilde\mM$ be the space of all $J$-holomorphic maps
\begin{equation*}
u\colon \bigl(\DD^2, \p \DD^2\bigr) \to \bigl(W, L_\x\bigr)
\end{equation*}
for all $\x \in \DD^m_\epsilon$.
Then the space of solutions in $\widetilde\mM$ close to $u_0$ forms
a smooth ball that has $u_0$ as its center and whose dimension is
\begin{equation*}
\dim \widetilde\mM = \frac{1}{2} \dim W + \mu\bigl(u_0^*T W, u_0^*T L_\0\bigr) + m,
\end{equation*}
where $\mu\bigl(u_0^*T W, u_0^*T L_\0\bigr)$ denotes the Maslov
index of the disk~$u_0$.
\end{proposition}
\begin{proof}
The result is standard if $m=0$, in which case $\frac{1}{2} \dim W +
\mu(u_0^*TW,u_0^*TL_0)$ is the Fredholm index of the linearized
Cauchy-Riemann operator on a suitable Banach space of sections of
$u_0^*TW$ with totally real boundary condition; see
\cite[\S3.2]{McDuffSalamonJHolo}.
For $m > 0$, the linearized problem is the same as that of the $m=0$
case, but with an extra $m$-dimensional space of smooth sections
added to the domain in order to allow for the moving boundary
condition, \cf \cite[\S 4.5]{Wendl_thesis}.
Thus the Fredholm index becomes larger by~$m$.
Given the corresponding enlargement of the nonlinear configuration
space, the proof of transversality for generic $J$ works as in the
standard case by defining a suitable universal moduli space and
applying the Sard-Smale theorem, see
\eg \cite[Chap.\,3]{McDuffSalamonJHolo}.
\end{proof}
Recall from Section~\ref{sec: deformed subcritical handles} that the
family of \LOB{}s is parametrized by a disk $\DD^{m}_r$ of some fixed
radius $r \ll 1$.
We define
\begin{equation*}
\wtmMint\bigl(\widehat W, S_B; J\bigr) =
\bigl\{ \bigl(\bbf^+, u, z_0 \bigr) \in
\widetilde\mM\bigl(\widehat W, S_B; J\bigr) \bigm|\, \norm{\bbf^+}
< r \bigr\}.
\end{equation*}
Since $g_B$ is plurisubharmonic and $g_A$ is weakly plurisubharmonic,
this subspace consists of $J$-holomorphic disks that map the interior
of $\DD^2$ to the interior of $\widehat W$; see also
Proposition~\ref{prop: only Bishop disks touch boundary at interior
points}.
\begin{corollary}\label{cor: boh}
The subspace $\wtmMint\bigl(\widehat W, S_B; J\bigr)$ of
the parametrized moduli space is a smooth manifold with boundary,
and its dimension is $2n-k+3$, where $k$ is the index of the
surgery.
Its boundary consists of triples $\bigl(\bbf^+, u, z_0 \bigr)$ with
$z_0 \in \p \DD^2$.
\end{corollary}
Note that triples $\bigl(\bbf^+, u, z_0 \bigr)$ with $\norm{\bbf^+} =
r$ do not belong to $\wtmMint\bigl(\widehat W, S_B; J\bigr)$,
and therefore are not points of its boundary.
\begin{proof}
By definition the elements of $\wtmMint\bigl(\widehat W,
S_B; J\bigr)$ satisfy the hypotheses of Proposition~\ref{prop:
smoothness of moduli space}, and therefore every $J$-holomorphic
disk $u \in \wtmMint\bigl(\widehat W, S_B; J\bigr)$ has an
open neighbourhood which is diffeomorphic to a ball of dimension
$\frac 12 \dim \widehat{W} + \mu \bigl(u^*T\widehat W, u^*
TL_{\bbf^+}\bigr) +m +2$ --- the presence of the marked point adds
$2$ to the index.
A simple computation shows that $\mu \bigl(u^*T\widehat W, u^*
TL_{\bbf^+}\bigr) =2$ for all $(\bbf^+, u, z_0)$, so that
$\wtmMint\bigl(\widehat W, S_B; J\bigr)$ is a smooth
manifold with boundary of dimension
\begin{equation*}
\dim \wtmMint\bigl(\widehat W, S_B; J\bigr) = n+m+4.
\end{equation*}
(The boundary points are those with the mark point in $\partial
\DD^2$.)
Since $m=n-k-1$, we obtain the desired formula for the dimension.
\end{proof}
In the next subsection we will analyze what happens when
$\norm{\bbf^+} =r$, and we will also show that
$\widetilde\mM\bigl(\widehat W, S_B; J\bigr)$ is non-empty.
To consider geometric disks instead of parametrized ones, we divide
$\widetilde \mM\bigl(\widehat W, S_B; J\bigr)$ by the group of
biholomorphic reparametrizations of $\DD^2\subset \CC$.
We define the moduli space of ``unparametrized'' curves:
\begin{equation*}
\mM \bigl(\widehat W, S_B; J\bigr) =
\widetilde \mM\bigl(\widehat W, S_B; J\bigr) / \sim\,,
\end{equation*}
where $\bigl(\bbf^+, u, z_0\bigr) \sim \bigl(\widetilde \bbf^+,
\widetilde u, \widetilde z_0\bigr)$ if and only if $\bbf^+ =
\widetilde \bbf^+$ and there exists a transformation
$\varphi \in \Aut(\DD^2)$ with $u = \widetilde u\circ
\varphi^{-1}$ and $z_0 = \varphi(\widetilde z_0)$.
The action of the reparametrization group $\Aut(\DD^2)$ preserves
$\wtmMint\bigl(\widehat W, S_B; J\bigr)$; we denote its
quotient by $\mMint\bigl(\widehat W, S_B; J\bigr)$.
\begin{proposition}\label{prop: topology of interior moduli space}
The subspace $\mMint\bigl(\widehat W, S_B; J\bigr)$ of the moduli
space is a smooth manifold with boundary of dimension $2n-k$.
\end{proposition}
\begin{proof}
The map $\vartheta \colon S_B \setminus \{ z^\circ =0 \} \to
\SS^1$ is globally defined for all \LOB{}s in the family.
Therefore we can define the subset
\begin{equation*}
\widetilde \mM_0\bigl(\widehat W, S_B; J\bigr) \subset \wtmMint
\bigl(\widehat W, S_B; J\bigr)
\end{equation*}
consisting of triples $\bigl(\bbf^+, u, z_0 \bigr)$ such that
\begin{equation}\label{eq: slice}
\vartheta\bigl(u(z)\bigr) =
\begin{cases}
1 & \text{if $z = 1$,} \\
i & \text{if $z = i$,} \\
-1 & \text{if $z = -1$}. \\
\end{cases}
\end{equation}
We know that $\widetilde \mM_0\bigl(\widehat W, S_B; J\bigr)$ is a
submanifold of $\wtmMint\bigl(\widehat W, S_B; J\bigr)$
because $\vartheta \circ \restricted{u}{\p\DD^2}$ is a
diffeomorphism and the biholomorphism group of the disk is triply
transitive on $\p\DD^2$.
Then the subset $\widetilde \mM_0\bigl(\widehat W, S_B; J\bigr)$
provides a global slice for the action of $\Aut(\DD^2)$ on
$\wtmMint\bigl(\widehat W, S_B; J\bigr)$.
\end{proof}
\subsection{The Bishop disks}
\label{sec: bishop disks at the boundary}
In this section, we want to study a certain class of disks in $\mM
\bigl(\widehat W, S_B; J\bigr)$ that lie entirely in the model
neighborhood~$\Wmodel$ and that can be described explicitly.
A \defin{Bishop disk} is a disk that we obtain by intersecting a
$z^\circ$-plane in $\CC^n$ with constant $(\z^-, \z^+)$-coordinates
with the model neighborhood~$\Wmodel$.
A possible way to parametrize it is as a map
\begin{equation*}
u\colon \bigl(\DD^2, \p\DD^2\bigr) \to \bigl(W, L_{\bbf^+}\bigr),
\end{equation*}
with constant coordinates $(\y^-,\y^+) = (\0, \bbf^+)$, constant
$\x^-$ and $\x^+$-coordinates, so we write
\begin{equation*}
u(z) = \bigl(\x^-; \x^+ + i\,\bbf^+; Cz\bigr),
\end{equation*}
where $C = \sqrt{r^2 - \norm{\x^-}^2 - \norm{\x^+}^2}$.
The Bishop disks are the buds from which the moduli space will grow,
and it is therefore important to establish that they are Fredholm
regular, meaning that their linearized Cauchy-Riemann operators are
surjective.
This is ensured by the following ``automatic'' transversality lemma
(see~\cite[\S III.1.3]{NiederkrugerHabilitation}).
\begin{lemma}\label{lemma: Bishop disks are regular}
Let $u \colon \DD^2 \to \widehat W$ be a Bishop disk with image in
$\Wmodel$ and boundary mapped to the \LOB~$L_{\bbf^+}$.
Then its linearized Cauchy-Riemann operator~$D_{(\bbf^+, u)}$,
defined on suitable Banach space completions with totally real
boundary condition determined by the \LOB~$L_{\bbf^+}$, is
surjective.
\end{lemma}
\begin{corollary}\label{cor: Bishop disks are regular}
The triples $(\bbf^+, u, z_0)$ where $u$ is a Bishop disk with image
in $\Wmodel$ are regular points of the moduli space $\widetilde \mM
\bigl(\widehat W, S_B; J\bigr)$.
\end{corollary}
\begin{proof}
The relevant linearized operator is the same as $D_{(\bbf^+, u)}$ in
Lemma~\ref{lemma: Bishop disks are regular}, except that the moving
boundary condition satisfied by $J$-holomorphic maps in $\widetilde
\mM\bigl(\widehat W, S_B; J\bigr)$ means that this domain must be
enlarged by some finite-dimensional space of smooth sections,
allowing the boundary to move to different \LOB{}s in the family
(see the appendix for more details).
The target of the operator remains the same, so surjectivity of
$D_{(\bbf^+, u)}$ in Lemma~\ref{lemma: Bishop disks are regular}
immediately implies surjectivity on the enlarged domain.
\end{proof}
The rest of this subsection will be concerned with the proof that the
Bishop disks are the only holomorphic disks in $\Wmodel$.
\begin{proposition}\label{prop: only Bishop disks touch boundary at
interior points}
If a holomorphic disk
\begin{equation*}
u\colon \bigl(\DD^2, \p\DD^2\bigr) \to \bigl(\widehat W, L_{\bbf^+}\bigr)
\end{equation*}
touches the boundary of $\widehat W$ at an interior point of
$\DD^2$, then either it is constant or it is a multiple cover of a
Bishop disk that is completely contained in $S_A \subset M_A \cap
\beltSphere{n+m}$.
\end{proposition}
\begin{proof}
Let $z_0 \in \mathring \DD^2$ be a point in the interior of the disk
at which $u$ touches $M_A$, $M_B$, or $\Mreg$.
We will obtain the desired statement by using the maximum principle;
we only need to be a bit more careful compared with the standard
situation, because the boundary of $\widehat W$ is defined piecewise
as a union of level sets of different plurisubharmonic functions.
Assume first that $u(z_0)$ touches $M_B$.
The function~$g_B$ is not defined on the whole symplectic filling,
but we may nonetheless assume that $g_B$ exists on a small
neighborhood of $u(z_0)$, hence we find an open subset $U \subset
\mathring \DD^2$ containing $z_0$ such that
\begin{equation*}
\restricted{\bigl(g_B\circ u\bigr)}{U}\colon U \to \RR
\end{equation*}
is a plurisubharmonic function having a maximum at $z_0$.
It follows from the maximum principle that $\restricted{g_B\circ
u}{U}$ is constant, and due to strong convexity it even follows
that the holomorphic map $\restricted{u}{U}$ itself must be
constant.
This implies that the open set~$U$ chosen above can in fact be
extended to the whole disk, and $u$ will be a constant disk.
Note that this argument also remains valid if $u(z_0)$ lies in the
edge where $M_A$ and~$M_B$ meet.
The disk lies in the model locally in the domain with $g_B \le r^2$,
and thus $\restricted{g_B\circ u}{U}$ still has a local maximum at
$z_0 \in \mathring\DD^2$, as used previously.
Similarly, the argument can be used verbatim for disks that touch
$\Mreg$, and this implies in fact that there are no disks at all
touching $\Mreg$ at interior points, because a constant disk must
lie in $L_{\bbf^+} \subset \beltSphere{n+m}$, which is disjoint from
$\Mreg$.
Let us now assume that the disk~$u$ touches the hypersurface~$M_A$
at $z_0$.
Again, we find an open subset~$U \subset \mathring \DD^2$ containing
$z_0$ for which\enlargethispage{.5\baselineskip}%
\begin{equation*}
\restricted{\bigl(g_A\circ u\bigr)}{U}\colon U \to \RR
\end{equation*}
is defined and has a maximum at $z_0$.
By weak plurisubharmonicity, this function must be constant.
Now it is easy to see that we can choose $U$ to be the whole
disk~$\DD^2$, because by continuity, the image of every point~$z \in
\overline{U}$ lies in $\p\widehat W$.
If $z$ is an interior point of the disk, and if $u(z)$ is an
interior point of $M_A$, \ie it does not lie in $M_A\cap M_B$, then
we can extend $U$ to a larger open domain that contains $z$ in its
interior.
If $z$ is an interior point but $u(z)$ \emph{does} lie in $M_A\cap
M_B$, then we know by the first part of the proof that~$u$ must be a
constant map.
In both cases the whole disk lies in $M_A$.
It remains to see that a nonconstant holomorphic disk lying in
$M_A$ must be a Bishop disk (or a multiple cover).
We know that all coordinate functions are harmonic, and hence each
of them must attain both its maximum and its minimum at a point on
the boundary of the disk.
The boundary of $u$ lies in $L_{\bbf^+} \subset \bigl\{\y^- =
\0\bigr\}$, and hence it follows that all of the $\y^-$-coordinates
vanish on the disk.
From the Cauchy-Riemann equation, we then see that the
$\x^-$-coordinates of the disk will be constant.
Similarly, the $\y^+$-coordinates of the disk must all be equal to
$\bbf^+$, because $L_{\bbf^+} \subset \bigl\{\y^+ = \bbf^+\bigr\}$,
and again by the Cauchy-Riemann equation also the $\x^+$-coordinates
will be constant.
The only nonconstant coordinate functions of the disk are the
$z^\circ$-coordinate, and they span a round disk.
\end{proof}
Recall that
\begin{equation*}
B_{\bbf^+} = L_{\bbf^+} \cap \{z^\circ = 0\}
\end{equation*}
is the binding of the \LOB~$L_{\bbf^+}$.
\begin{proposition}\label{prop: bishop disks close to binding}
There exists an open subset $V \subset \Wmodel$, containing
$B_{\bbf^+}$ for every $\bbf^+ \in \DD^m_r$, such that every
holomorphic disk
\begin{equation*}
u\colon \bigl(\DD^2, \p\DD^2\bigr) \to \bigl(\widehat W,
L_{\bbf^+}\bigr)
\end{equation*}
in $\widetilde \mM\bigl(\widehat W, S_B; J\bigr)$ intersecting $V$
must be a Bishop disk up to reparametrization.
\end{proposition}
\begin{proof}
Note that
\begin{equation*}
h(\z^-,\z^+, z^\circ) = \norm{\x^-}^2 - \frac{1}{2} \, \norm{\y^-}^2
+ \norm{\x^+}^2
\end{equation*}
is a weakly plurisubharmonic function on $\Wmodel$.
Its value on the binding~$B_{\bbf^+}$ is $r^2$, and it decreases
along the \LOB.
If we choose a sufficiently small $\epsilon>0$, we can make sure
that $V := h^{-1}\bigl((r^2-\epsilon,+ \infty)\bigr) \cap \Wmodel$
is an open neighborhood of $B_{\bbf^+}$ with $\overline{V} \subset
\Wmodel$.
It follows in fact from $g_B \le r^2$ and $h > r^2 - \epsilon$ that
\begin{equation*}
g_B(\z^-,\z^+, z^\circ)-h(\z^-,\z^+, z^\circ)= \frac{1}{2}\, \norm{\y^-}^2
+ \psi\bigl(\norm{\y^-}\bigr)\cdot \norm{\y^+}^2
+ \abs{z^\circ}^2 < \epsilon,
\end{equation*}
so that both the $\y^-$ and the $z^\circ$-coordinates are small in
$V$, and in particular we can assume that $\psi=0$ on $V$.
On the other hand,
\begin{equation*}
\norm{\x^-}^2 + \norm{\x^+}^2 > r^2 - \epsilon
+ \frac{1}{2} \, \norm{\y^-}^2 \ge r^2 - \epsilon
\end{equation*}
implies that every point in $V$ lies in an arbitrarily small
neighborhood of $S_B$.
Let now $u\colon \bigl(\DD^2, \p\DD^2\bigr) \to \bigl(\widehat W,
L_{\bbf^+}\bigr)$ be a holomorphic disk whose image intersects~$V$.
Assume that $h\circ u$ is not constant: then we can choose by Sard's
theorem a slightly smaller number $\epsilon' < \epsilon$ for which
$r^2 - \epsilon'$ will be a regular value of $h\circ u$, so that the
subdomain
\begin{equation*}
G := \bigl\{z \in \DD^2\bigm|\, (h\circ u)(z) \ge r^2-\epsilon' \}
\end{equation*}
is compact and has piecewise smooth boundary, which we denote by
\begin{equation*}
\p G = \p_+G \cup \p_-G,
\end{equation*}
where $\p_+ G = G \cap \p\DD^2$ lies in the boundary of the unit
disk, and $\p_- G$ lies in the interior of the unit disk.
Denote the restriction
\begin{equation*}
\restricted{u}{G}\colon G \to \Wmodel
\end{equation*}
by $u_G$.
By the maximum principle, it follows that the maximum of $h\circ
u_G$ on each component of $G$ must lie on the boundary of that
component.
Clearly then the boundary of every component of $G$ must intersect
$\p_+G$, because otherwise $h\circ u_G$ would have an interior
maximum on that component, so it would be equal to $r^2-\epsilon'$,
but this contradicts the assumption that $r^2-\epsilon'$ is a
regular value.
It follows then that every component of $G$ must intersect
$\p\DD^2$, and since $h\circ u_G$ is minimal along $\p_- G$, the
maximum of $h\circ u_G$ must lie at a point $z_0 \in \p_+G \subset
\p\DD^2$.
By the boundary point lemma, a version of the maximum principle at
the boundary (see for example
\cite[Th.\,II.1.3]{NiederkrugerHabilitation}), the derivative of
$h\circ u_G$ at $z_0$ in the outward radial direction must be
strictly positive.
We choose polar coordinates $(r, \varphi)$ on $\DD^2$. Using the
fact that $u$ is $J$-holomorphic, we can write
\begin{equation*}
\begin{split}
\partial_r\bigl(h\circ u\bigr) &= dh\bigl(Du
\cdot \partial_r\bigr)
= dh \bigl(Du \cdot (-i\cdot\partial_\varphi) \bigr) \\
&= - dh\bigl(i \cdot Du \cdot \partial_\varphi\bigr) = - d^ch
\bigl(Du \cdot \partial_\varphi\bigr),
\end{split}
\end{equation*}
but note that
\begin{equation*}
- d^c h = \sum_{s=1}^k \Bigl(2 x_s^-\, dy_s^-
+ \ y_r^- \, dx_r^-\Bigr) + 2\,\sum_{t=1}^m x_t^+\, dy_t^+.
\end{equation*}
We obtain $- d^ch \cdot Du \cdot \partial_\varphi = 0$ along the
whole boundary of the disk, because the boundary of $u$ lies in the
\LOB~$L_{\bbf^+}$, which is a subset of $\{\y^- = \0, \y^+ =
\bbf^+\}$.
It follows that $\partial_r\bigl(h\circ u\bigr) = - d^ch \bigl(Du
\cdot \partial_\varphi\bigr)$ vanishes at $z_0$, and by the boundary
point lemma, the disk must be contained in one of the level sets of
$h$, so in particular it lies in $V \subset \Wmodel$.
The rest of the statement follows from standard arguments.
All of the coordinate functions on $\Wmodel$ are harmonic, hence
they must attain their maxima and minima on the boundary of the
disk.
Since $\y^- = \0$ along $\p\DD^2$, the $\y^-$-coordinates of $u$ are
zero on the whole disk, and using the Cauchy-Riemann equation, we
see that the $\x^-$\nobreakdash-coordinates must be constant on the disk.
Similar arguments work for $\y^+$ and~$\x^+$, and we finally
conclude that $u$ must be a Bishop disk.
\end{proof}
\begin{proposition}\label{prop: bishop disks on Lobs in MA}
Let $L_{\bbf^+}$ be a \LOB that lies in the hypersurface~$M_A$,
\ie $\bbf^+ \in \DD^m_r$ has been chosen such that $\norm{\bbf^+} =
r$.
Then up to parametrization, every holomorphic disk~$u$ in
$\widetilde \mM\bigl(\widehat W, S_B; J\bigr)$ whose boundary lies
in $ L_{\bbf^+}$ is a Bishop disk.
\end{proposition}
\begin{proof}
Note that $L_{\bbf^+}$ lies in the level set of the weakly
plurisubharmonic function $g_A\colon \widehat W \to \RR$.
It suffices to prove that the image of $u$ has to lie entirely in
$M_A \subset \Wmodel$, as this already implies the desired statement
by Proposition~\ref{prop: only Bishop disks touch boundary at
interior points}.
Since the whole boundary $u\bigl(\p\DD^2\bigr)$ lies in $M_A$, we
can find a closed annulus $G \subset \DD^2$ having $\p\DD^2$ as one
of its boundary components such that
\begin{equation*}
\restricted{\bigl(g_A\circ u\bigr)}{G}\colon G \to \RR
\end{equation*}
is defined and everywhere weakly plurisubharmonic, and it takes its
maximum along $\p\DD^2 \subset G$.
Assume first that the disk~$u$ is tangent to $M_A$ at one of its
boundary points.
We can apply the boundary point lemma around this point (see again
\cite[Th.\,II.1.3]{NiederkrugerHabilitation}) to deduce that
$\restricted{\bigl(g_A\circ u\bigr)}{G}$ has to be constant on all
of $G$.
In particular this implies that $u(G)$ lies in $M_A$, and $u$
touches $M_A$ also with one of its interior points.
Proposition~\ref{prop: only Bishop disks touch boundary at interior
points} then implies that $u$ is either constant or one of the
Bishop disks.
Conversely suppose that $u$ is everywhere transverse to $M_A$,
meaning that the function \hbox{$\partial_r \bigl(g_A\circ u\bigr)(z)$} is
\emph{strictly} positive for every $z \in \p\DD^2$.
The restriction $\restricted{u}{G}$ is a $J$-holomorphic map whose
image lies in $\Wmodel$; moreover, $g_A\circ u \equiv r^2$ on
$\p\DD^2$ and $g_A\circ u < r^2$ on the inner boundary of $G$.
Introduce on $G$ the polar coordinates $z= \rho e^{i\varphi}$.
Note that along $\p\DD^2$, all of the $\y^+$-coordinates are
constant in the $\varphi$-direction, because the boundary of the
disk lies in the \LOB~$L_{\bbf^+}$.
Multiplying the complex coordinates~$\z^+$ by a suitable
$\SO(m)$-matrix (the standard complex structure~$i$ and the
functions $g_A$, $g_B$ are invariant under such a multiplication),
we may assume that $\bbf^+ = (r,0,\dotsc,0)$.
It follows that the $y_1^+$-coordinate of $\restricted{u}{G}$ has
its maximum on $\p\DD^2$.
Note now that the $x_1^+$-coordinate of $\restricted{u}{\SS^1}$ is
bounded, and hence it necessarily must take a maximum at some point
$e^{i\varphi_0} \in \SS^1 = \p\DD^2$, so that
\begin{equation*}
\restricted{\frac{d}{d\varphi}}{\varphi = \varphi_0}
x_1^+ \bigl(u(e^{i\varphi}) \bigr) = 0.
\end{equation*}
Again, we can use complex multiplication to see
$i\cdot \partial_\rho = \partial_\varphi$, hence
\begin{equation*}
dy_1^+\bigl(Du \cdot \partial_\rho\bigr) =
dy_1^+\bigl(Du \cdot (-i\cdot\partial_\varphi)\bigr)
= - dy_1^+\bigl(i\cdot Du\cdot\partial_\varphi\bigr)
= - dx_1^+ \bigl(Du\cdot\partial_\varphi\bigr),
\end{equation*}
and in particular the radial derivative of $y_1^+$ vanishes at
$e^{i\varphi_0}$, so that by the boundary point lemma, $y_1^+$ must
be constant on all of $G$.
Using the fact that $r^2 = \abs{y_1^+}^2 \le g_A(\z^-,\z^+, z^\circ)
\le r^2$ everywhere on $G$, we deduce that all of $u(G)$ lies in
$M_A$.
In particular, $u$ touches $M_A$ at an interior point, which allows
us to conclude the proof by applying Proposition~\ref{prop: only
Bishop disks touch boundary at interior points}.
\end{proof}
We end this subsection with a description of the global topology of
the moduli spaces $\widetilde \mM\bigl(\widehat W, S_B; J\bigr)$ and
$\mM\bigl(\widehat W, S_B; J\bigr)$.
\begin{proposition}\label{prop: topology of the uncompactified moduli
space}
The parametrized moduli space $\widetilde \mM\bigl(\widehat W, S_B;
J\bigr)$ is a smooth \hbox{$(2n-k+3)$}-dimensional manifold with boundary
and corners.
Its boundary has two smooth strata, one corresponding to elements
$(\bbf^+, u, z_0)$ with $\norm{\bbf^+} =r$, and the other
corresponding to elements $(\bbf^+, u, z_0)$ with $\abs{z_0}=1$.
The moduli space $ \mM\bigl(\widehat W, S_B; J\bigr)$ is a smooth
$(2n-k)$-dimensional manifold with boundary and corners, which
decomposes as a product
\begin{equation*}
\mM\bigl(\widehat W, S_B; J\bigr) = \Sigma \times \DD^2,
\end{equation*}
where $\Sigma$ is a (non-compact) manifold with boundary.
\end{proposition}
\begin{proof}
Let $(\bbf^+,u,z_0)$ be an element of $\widetilde
\mMint\bigl(\widehat W, S_B; J\bigr)$.
Since $\wtmMint\bigl(\widehat W, S_B; J\bigr)$ is open in
$\widetilde \mM\bigl(\widehat W, S_B; J\bigr)$, it follows from
Proposition~\ref{prop: only Bishop disks touch boundary at interior
points} that the image of $u$ does not touch $\p \widehat W$ with
any interior point if $\norm{\bbf^+} 2$.
Let $\bB$ denote the space of pairs $(\bbf^+, u)$ where:
\begin{itemize}
\item $\bbf^+ \in \DD^m_r$ with $\norm{\bbf^+}