# Honda-Huang’s work on contact convexity revisited

Yakov Eliashberg  Department of Mathematics
Stanford University
Stanford, CA 94305 USA<EMAIL_ADDRESS>and  Dishant Pancholi  The
Institute of Mathematical Sciences
Taramani, Chennai 600 113, Tamil Nadu, India<EMAIL_ADDRESS>
###### Abstract.

Following the overall strategy of the paper [14] by Ko Honda and Yang Huang on
contact convexity in high dimensions, we present a simplified proof of their
main result.

###### 1991 Mathematics Subject Classification:

Primary: 57R17; Secondary: 37C10

YE was partially supported by NSF grant DMS-2104473 and a grant from the
Institute for Advanced Study

DP was partially supported by NSF grant DMS-1926686 during his stay at the
Institute for Advanced Study

##  1\. Introduction

A hypersurface in a contact manifold is said to be convex if it admits a
transverse contact vector field (see Section 3.3 below for precise
definitions). The central result of the article _“Convex hypersurfaces in
contact topology”_ by Ko Honda and Yang Huang is the following:

######  Theorem 1.1 (Ko Honda and Yang Huang, [14]).

Let $(M,\xi_{M})$ be a manifold with a co-orientable contact structure and
$\Sigma\subset M$ a co-oriented hypersurface. Then there exists a
$C^{0}$–small isotopy sending $\Sigma$ to a convex hypersurface
$\widetilde{\Sigma}$.

If $\dim M=2$ then Theorem 1.1 holds, according to a classical result of
Emmanuel Giroux, [9], in a stronger form, with a $C^{\infty}$-small isotopy
instead of a $C^{0}$-small isotopy. The purpose of this article is to provide
a more accessible proof of Theorem 1.1. While the proof follows the overall
strategy of [14] it is significantly different in its implementation. In
particular, we do not use any contact open book techniques. Besides Theorem
1.1 we do not discuss in this paper any other results formulated in [14].

Acknowledgements. The current paper was originated while the authors visited
the Institut for Advanced Study at Princeton, and completed when the first
author visited the Institut for Theoretical Study at the ETH, Zurich. The
authors thank both Institutes for their hospitality. The paper grew up from a
seminar at the IAS, where the second author discussed Honda-Huang’s work. The
results of this paper were presented by the first author at seminars at
Stanford, Neuchâtel, Brussels and the ITS ETH, Zurich. The authors are very
grateful to participants of all these seminars for questions and comments. Our
special thanks to Dietmar Salamon for asking many pointed questions and for
his proposed simplifications of several proofs presented in Section 3 of this
paper. In particular, we use his proof for Lemma 3.10. We thank François-Simon
Fauteux-Chapleau for clarifying some of the aspects of Honda-Huang’s paper, Ko
Honda for answering many questions concerning their proof, and Emmanuel Giroux
for his comments on the preliminary version of this paper. We are grateful to
N. Mishachev for making Fig. 5.

##  2\. Dynamics of vector fields

This section and Section 3 contains some background material which is mostly
well-known.

###  2.1. Lyapunov functions

An isolated zero $p$ of a vector field $X$ on an $m$-dimensional manifold
$\Sigma$ is called non-degenerate if $d_{p}X$ is non-degenerate, and it is
called an embryo or death-birth singularity if the corank of its linearization
$d_{p}X$ is equal to 1 and the quadratic differential
$d^{2}_{p}X:\mathrm{Ker\,}d_{p}X\to\mathrm{Coker\,}d_{p}X$, which is defined
up to scaling by a non-zero coefficient, does not vanish. We will call a non-
degenerate or death-birth zero hyperbolic if $d_{p}X$ has no pure imaginary
(non-zero) eigenvalues.

Let $X$ be a vector field on a compact manifold $\Sigma$. Let us endow
$\Sigma$ with a Riemannian metric. A function $f:\Sigma\to\mathbb{R}$ is
called Lyapunov for $X$ if $df(X)\geq C(||X||^{2}+||df||^{2})$ for a positive
constant $C$. Equivalently, one says that $X$ is a gradient like vector field
for $f$.

It is a standard fact that isolated hyperbolic zeroes, non-degenerate or
embryos, admit local Lyapunov function, e.g. see [1]. The stable manifold of a
non-degenerate zero is diffeomorphic to $\mathbb{R}^{k}$ for some
$k=0,\dots,m$ in the non-degenerate case, and to $\mathbb{R}^{k}_{+}$ in the
case of an embryo. The dimension $k$ of the stable manifold of a non-
degenerate hyperbolic zero $O$ is called its index and denoted by
$\mathrm{ind}(O)$. For an embryo the index is usually defined to be equal to
$k-\frac{1}{2}$.

Figure 1.

######  Lemma 2.1.

Let $X$ be a vector field with isolated hyperbolic zeroes which are non-
degenerate or of embryo type on a closed $m$-dimensional manifold $\Sigma$.
Then $X$ admits a Lyapunov function if and only if the following conditions
are satisfied:

  * (L1)

every trajectory of $X$ originates and terminates at a zero of $X$;

  * (L2)

there exists an ordering $O_{1},\dots,O_{N}$ of zeroes such that there are no
trajectories of $X$ which originate at $O_{i}$ and terminate in $O_{j}$ if
$i>j$.

###### Proof.

If $X$ admits a Lyapunov function then both conditions (L1) and (L2) are
straightforward. Suppose that these conditions are satisfied. We construct a
Lyapunov function $f:\Sigma\to\mathbb{R}$ by extending it inductively to
neighborhoods of stable manifolds of zeroes $O_{j}$.

We start with a local Lyapunov function near $O_{1}$ (which has to be of index
$0$) and set $f(O_{1})=1$. We assume that $\Sigma_{1}:=\\{f\leq\frac{3}{2}\\}$
is a small ball surrounding $O_{1}$, with boundary transverse to $X$.

Suppose that we already constructed $f$ on a domain $\Sigma_{k-1}:=\\{f\leq
k-\frac{1}{2}\\}$, $2\leq k\leq N$, such that zeroes $O_{1},\dots,O_{k-1}$ and
their stable manifolds are contained in $\mathrm{Int\,}\Sigma_{k-1}$ and
$f(O_{k-1})=k-1$. The stable manifold $P_{k}$ of $O_{k}$ transversely
intersects $\partial\Sigma_{k-1}$. Denote
$\widetilde{P}_{k}:=P_{k}\setminus\mathrm{Int\,}\Sigma_{k-1}$. Then
$\widetilde{P}_{k}$ is an embedded disk (or a half-disc, if $O_{k}$ is an
embryo) of dimension $\mathrm{ind}(O_{k})$ with boundary transverse to
$\partial\Sigma_{k-1}$. Extend $f$ to a neighborhood
$U_{k}\supset\Sigma_{k-1}\cup\widetilde{P}_{k}$ as a Lyapunov function for $X$
such that $f(O_{k})=k$ and the regular level set $\\{f=k+\frac{1}{2}\\}$ is
compact and is contained in $U_{k}$, see Figure 1 and [4] for details. Denote
$\Sigma_{k}=\\{f\leq k+\frac{1}{2}\\}$. For $k=N$ we have $\Sigma_{N}=\\{f\leq
N\\}=\Sigma$, and this completes the construction. ∎

Note that condition (L2) is guaranteed by the Morse-Smale property, i.e.
transversality of stable and unstable manifolds for any pair of zeros. While
the Morse-Smale property can be arranged by a $C^{\infty}$-small perturbation,
it is not clear to us whether this perturbation can be always done without
destroying property (L1).

Lemma 2.1 can be extended to 1-parametric families.

######  Lemma 2.2.

Any family of vector fields $X_{s}$, $s\in[0,1]$, which satisfy conditions
(L1) and (L2) admits a family of Lyapunov functions.

###### Proof.

The space of Lyapunov functions for a given vector field $X$ is contractible,
because a convex linear combination of two Lyapunov functions for $X$ is again
a Lyapunov function for $X$. Also note that a Lyapunov function $f_{s_{0}}$
for $X_{s_{0}}$ can always be included into a family $f_{s}$ of Lyapunov
functions for $X_{s}$ for $s$ close to $s_{0}$. Hence, the projection of the
space of pairs ((L1)+(L2) field, Lyapunov function) to the space of (L1)+(L2)
fields is a micro-fibration with a (non-empty!) contractible fiber, and hence,
it is a Serre fibration, see [11, 18]. ∎

Let us also formulate a version of Lemma 2.1 for a (trivial) cobordism. Let
$W$ be an $(m-1)$-dimensional manifold with boundary and
$\Sigma:=W\times[0,1]$. Denote by $y$ the coordinate which corresponds to the
second factor. Let $X$ be a vector field on $\Sigma$ which coincides with
$\frac{\partial}{\partial y}$ near $\partial\Sigma$.

######  Lemma 2.3.

Suppose that

  * (L$1^{\prime}$)

every trajectory of $X$ originates and terminates at a zero of $X$ or at a
point of $\partial\Sigma$;

  * (L2)

there exists an ordering $O_{1},\dots,O_{N}$ of zeroes such that there are no
trajectories of $X$ which originate at $O_{i}$ and terminate in $O_{j}$ if
$i>j$.

Then $X$ admits a Lyapunov function which is equal to $y$ near
$\partial\Sigma$.

###### Proof.

We construct $f$ by the process described in the proof of Lemma 2.1 with
$W\times 0$ and $W\times 1$ playing the role of the first and last zeroes,
$O_{0}$ and $O_{N+1}$. We then adjust $f$ near $\partial W\times[0,1]$, by
making it linear with respect to $y$ and then scaling it to make equal to $1$
on $W\times 1$. ∎

###  2.2. Blocking collections

The material of this section is fairly standard and its various versions
appear in many places (e.g. see [12, 19]). In particular, Lemma 2.5 is a
corollary of [19, Lemma 2].

A non-vanishing vector field $X$ in a neighborhood of a hypersurface $V$ in an
$m$-dimensional manifold $\Sigma$ is called in general position with respect
to $V$ if it has Thom-Boardman-Morin tangency singularities of type
$\Sigma^{1,\ldots,1}$, see [17, 3]. Let us fix a Riemannian metric on
$\Sigma$. Arguing by induction over strata of tangency singularity, it is
straightforward to prove the following statement (e.g. it is a corollary of
Morin’s normal forms [15] for $\Sigma^{1,\ldots,1}$-singularities).

######  Lemma 2.4.

Suppose that $X$ is in general position with respect to $V$. Then there exists
$\epsilon_{0}>0$ such that for any $\epsilon\in(0,\epsilon_{0})$ there exists
$\delta>0$ such that any connected trajectory arc of length $\epsilon$
contains a connected sub-arc of length $C(m)\epsilon$ which does not intersect
the $\delta$-neighborhood of $V$. Here $C(m)$ denotes a constant which depends
only on the dimension $m$

###### Proof.

We can assume that the vector field $X$ on a neighborhood of $V$ has a unit
length. We will be measuring below arcs $\gamma$ of $X$-trajectories by the
flow-parameter. This measurement, which we call length is equivalent to the
diameter of $\gamma$ for sufficiently short arcs.

For any point $p\in V$ and $\epsilon>0$ denote
$\gamma_{\epsilon}(p)=\bigcup\limits_{u\in[-\epsilon,\epsilon]}X^{u}(p)$.
There exists $\epsilon_{0}>0$ such that for each point $p\in V$ the arc
$\gamma_{3\epsilon_{0}}(p)$ intersects $V$ at no more than $m$ points, and
moreover $V\cap\gamma_{3\epsilon_{0}}(p)\subset\gamma_{\epsilon_{0}}(p).$
Given $\delta>0$ denote by $N_{\delta}(V)$ the $\delta$-tubular neighborhood
of $V$. If For any $\epsilon<\epsilon_{0}$ there exists $\delta>0$ such that
for every $p\in V$ the intersection $\gamma_{3\epsilon_{0}}(p)\cap
N_{\delta}(V)$ consists of no more than $m$ components of length
$<\frac{\epsilon}{2m+2}$. Any trajectory arc $\sigma$ of length $\epsilon$
which intersects $N_{\delta}(V)$ is contained in $\gamma_{3\epsilon_{0}}(p)$
for some $p\in V$. Hence $N_{\delta}(V)\cap\sigma$ consists of no more than
$m$ arcs of length $<\frac{\epsilon}{2m+2}$. Thus, the complement
$\sigma\setminus N_{\delta}(V)$ contains an arc of length
$>\frac{\epsilon}{m+1}-\frac{\epsilon}{2m+2}=\frac{\epsilon}{2m+2}.$ ∎

Let $X$ be a vector field on a compact $m$-dimensional manifold $\Sigma$,
possibly with boundary. Given $\epsilon>0$, a finite collection
$\\{D_{j}\\}_{1\leq j\leq K}$ of transverse to $X$ embedded into
$\mathrm{Int\,}\Sigma$ codimension one discs of diameter $<\epsilon$ is called
$\epsilon$-blocking  if any connected trajectory arc of diameter $>\epsilon$
intersects $\bigcup\limits_{1}^{K}\mathrm{Int\,}D_{j}$.

######  Lemma 2.5.

Let $X$ be a vector field on a compact $m$-dimensional manifold $\Sigma$,
possibly with boundary. Suppose that all zeros of $X$ are in
$\mathrm{Int\,}\Sigma$, isolated and hyperbolic (non-degenerate or embryos).
Suppose that $X$ is in general position with respect to $\partial\Sigma$. Then
for any $\epsilon>0$ the field $X$ admits an $\epsilon$-blocking collection.

###### Proof.

Part I. Suppose first that the vector field $X|_{\Sigma}$ admits a Lyapunov
function $f:\widetilde{\Sigma}\to\mathbb{R}$. without critical points. Suppose
that $\min f=0,\max f=1$ and choose $N$ large enough to guarantee that any
connected arc of an $X$-trajectory in $\\{\frac{j}{N}\leq
f\leq\frac{j+1}{N}\\}$ has its diameter $<\frac{\epsilon}{2}$,
$j=0,\dots,N-1$. Suppose that $\epsilon$ is chosen $<\epsilon_{0}$ from Lemma
2.4 and $\delta$ is chosen so small that any connected trajectory arc of
length $\epsilon$ contains a connected sub-arc of length
$\frac{\epsilon}{C(m)}$ which does not intersect the $\delta$-neighborhood of
$V$. Here $C(m)$ is the constant from Lemma 2.4. Choose an interior tubular
collar $\partial\Sigma\times[-1,0]\subset\Sigma$ such that
$\partial\Sigma=\partial\Sigma\times 0$ and $\partial\Sigma\times(-1)$ is at a
distance $\delta$ from $\partial\Sigma$. Denote
$\Sigma_{0}:=\Sigma\setminus\left(\partial\Sigma\times(-1,0]\right)$. For each
$j=1,\dots,N-1$ choose finitely many closed discs of radius $\epsilon$ in
$\mathrm{Int\,}\\{f=\frac{j}{N}\\}$ whose interiors cover
$\\{f=\frac{j}{N}\\}\cap\Sigma_{0}$. By shifting these discs to disjoint level
sets $\\{f=t_{j,k}\\},\;t_{j,k}\in(\frac{2j-1}{2N},\frac{2j+1}{2N}),$ we get
the required $\epsilon$-blocking collection.

Part II. In the general case let us choose any smooth function
$f:\Sigma\to\mathbb{R}$. Let us surround zeroes of $X$ by the union $B$ of
disjoint closed $\epsilon$-balls. We can assume that $X$ is in general
position with respect to $\partial B$. Denote
$\widetilde{\Sigma}:=\Sigma\setminus\mathrm{Int\,}B$.

Denote

$\widetilde{\Sigma}_{+}:=\\{df(X)\geq
0\\},\;\widetilde{\Sigma}_{-}:=\\{df(X)\leq
0\\},\;V:=\\{df(X)=0\\}=\widetilde{\Sigma}_{+}\cap\widetilde{\Sigma}_{-}.$

By $C^{\infty}$-perturbing $f$, if necessary, we can arrange that $V$ is a
codimension 1 submanifold, and $X$ is in general position with respect to $V$.
Let us assume that $\delta>0$ is chosen in such a way that any connected
trajectory arc of length $\epsilon$ contains a connected sub-arc of length
$\frac{\epsilon}{C(m)}$ which does not intersect the $\delta$-neighborhood of
$V$. Consider a $\delta$-tubular neighborhood $N\supset
V,N\subset\mathrm{Int\,}\Sigma$. Denote
$\widehat{\Sigma}_{\pm}:=\widetilde{\Sigma}_{\pm}\setminus\mathrm{Int\,}N$. We
can assume that $\partial\widehat{\Sigma}_{\pm}$ is in general position with
respect to $X$. By applying Part 1 we can construct
$\frac{\epsilon}{2}$-blocking collections for $\widehat{\Sigma}_{+}$ and
$\widehat{\Sigma}_{+}$. The union of these collections is the required
$\epsilon$-blocking collection for $X$ on $\Sigma$.

∎

Note that a compact arc $\gamma$ of a non-constant trajectory $X$ has a flow-
box neighborhood $U=D\times[0,c]$ such that $D\times 0$ is an embedded
transverse disc, and $x\times[0,c],x\in D$ are trajectories of $X$. Denote by
$\pi_{U}:U\to D$ the projection of the flow-box neighborhood to the first
factor. We will call an $\epsilon$-blocking collection $\\{D_{j}\\}$ generic,
if for any flow-box $U$ projections $\pi_{U}|_{\partial D_{j}\cap U}\to D$ are
transverse to each other. Any $\epsilon$-blocking collection can be made
generic by a $C^{\infty}$-perturbation.

###  2.3. Plugs

Given an $\epsilon$-blocking collection $\\{D_{j}\\}$, let us thicken discs
$D_{j}$ to disjoint flow-boxes $Q_{j}=D_{j}\times[0,a]$ such that intervals
$x\times[0,a]$, $x\in D_{j}$ are time $a$ trajectories of $X$ originated at
$x\in D_{j}=D_{j}\times 0$. We will assume that $a$ is chosen small enough to
guarantee that flow-boxes $Q_{j}$ have diameter $<2\epsilon$.

Let $D$ be an $(m-1)$-dimensional disc. A vector field $Y$ on $D\times[0,a]$
is called a $\sigma$-plug if the following conditions are satisfied:

  * P1.

$Y$ coincides with $\frac{\partial}{\partial y}$ on $\partial Q$, where $y$ is
the coordinate on $D\times[0,a]$ corresponding to the second factor;

  * P2.

$Y$ satisfies the Morse-Smale condition and admits a Morse Lyapunov function;

  * P3.

for any point $p\in D$ with $\mathrm{dist}(p,\partial D)>\sigma$ the
trajectory of $Y$ through $\ p\times 0$ converges to a critical point of $Y$;

  * P4.

given any point $p\in D$ the trajectory of $Y$ through $p\times 0$ either
converges to a critical point of $Y$, or exit $Q$ at a point $p^{\prime}\times
a$ where $\mathrm{dist}(p^{\prime},p)<\sigma$.

######  Lemma 2.6.

Let $\Sigma$ be a closed manifold of dimension $m$, and $X$ a vector field on
$\Sigma$ with non-degenerate hyperbolic zeroes. Let $\\{D_{j}\\}$ be a generic
$\epsilon$-blocking collection, and $\\{Q_{j}\\}$ a collection of their
disjoint flow-boxes of diameter $<2\epsilon$. Then there exists $\sigma>0$
such that by replacing for each $j$ the vector field $X|_{Q_{j}}$ by a
$\sigma$-plug $Y$ one gets a vector field $\widehat{X}$ which satisfies
condition (L1) and such that all its trajectories has diameter $<3\epsilon$.
Moreover, property (L1) survives a sufficiently small $C^{1}$-perturbation $X$
away from flow-boxes and neighborhoods of zeroes of $X$.

###### Proof.

By assumption any point of $p$ belongs to a connected arc $\gamma$ of an
$X$-trajectory of diameter $<\epsilon$, which has either both ends at
$\mathrm{Int\,}D_{i}$ and $\mathrm{Int\,}D_{j}$ for some $i,j$, or it limits
at one of the end to a zero and the other end is at $\mathrm{Int\,}D_{j}$, or
both ends it limits to zeroes. By compactness argument we can find smaller
closed discs $D_{j,-}\subset\mathrm{Int\,}D_{j}$ such that in the above
condition one can replace $\mathrm{Int\,}D_{j}$ by $D_{j,-}$.

The genericity property for the blocking collection $D_{j}$ implies that if an
arc $\gamma$ of an $X$-trajectory does not intersect
$\bigcup\limits_{1}^{K}\mathrm{Int\,}D_{j}$ then it cannot have more than
$m-1$ intersection points with $\partial D_{j}$. Hence, we can choose the
discs $D_{j,-}$ so close to $D_{j}$ that any arc $\gamma$ which does not
intersect $\bigcup\limits_{1}^{K}D_{i,-}$ intersects no more than $m-1$ annuli
$A_{j}:=D_{j}\setminus\mathrm{Int\,}D_{j,-}$. Choose
$\sigma<\frac{1}{m+1}\min\limits_{j}(\mathrm{dist}(\partial D_{j,-}.\partial
D_{j})$. We claim that the vector field $\widehat{X}$ obtained by replacing
$X|_{Q_{j}}$ by $\sigma$-plugs has the required properties. Indeed, consider
any trajectory $\widehat{\gamma}$ of $\widehat{X}$ which enters a flow-box
$Q_{i}$ through a point $p\in D_{i,-}=D_{i,-}\times 0$. Let $\gamma$ be an
$X$-trajectory through $p$ which intersect $D_{j,-}$ at a point $p^{\prime}$,
and does not contain any other points from $\bigcup\limits_{1}^{K}D_{k,-}$.
Then if $\widehat{\gamma}$ is not locked in $Q_{i}$, or any other of $<m$
plugs $Q_{k}$ for discs $D_{k}$ which intersect $\gamma$, then it enters
$Q_{j}$ through a point $p^{\prime\prime}$ with
$\mathrm{dist}(p^{\prime},p^{\prime\prime})<m\sigma$, and thus
$\mathrm{dist}(p^{\prime\prime},\partial D_{i})>\sigma$. But this means that
the trajectory $\widehat{\gamma}$ converges to a zero of $\widehat{X}$ in
$Q_{j}$. Analysis of trajectories with limiting at one of the ends to a zero
of $X$ is similar. Moreover, the trajectories of the vector field
$\widehat{X}$ have their diameter bounded by $3\epsilon$. It remains to
observe that the above analysis remains valid if $X$ is perturbed by a
sufficiently $C^{1}$-small homotopy outside flow-boxes.

∎

Figure 2.  Blocking discs with their flow-boxes.

##  3\. Contact convexity

###  3.1. Characteristic foliation

Let $M$ be a contact manifold of dimension $2n+1$ with a co-oriented contact
structure $\xi=\\{\alpha=0\\}$. The volume form
$\mu:=\alpha\wedge(d\alpha)^{n}$ defines an orientation of $M$, and
$(d\alpha)^{n}|_{\xi}$ defines an orientation of $\xi$. If $n=2k+1$ then the
former orientation, and if $n=2k$ than the latter orientation, depends only on
$\xi$.

Let $\Sigma\subset M$ be a co-oriented hypersurface. If $\nu$ is a vector
field defining its co-orientation then the orientation of $\Sigma$ is given by
the $2n$-form $\iota(\nu)\mu$. At any point $p\in\Sigma$ where
$\xi_{p}\pitchfork T_{p}\Sigma,$ there is defined a characteristic line
$\ell_{p}:=\mathrm{Ker\,}\\{d\alpha|_{\xi_{p}\cap
T_{p}\Sigma}\\}\subset\xi_{p}\cap T_{p}\Sigma$. Note that the co-orientation
of $\Sigma$ defines a co-orientation of $\xi_{p}\cap T_{p}\Sigma$ in
$\xi_{p}$. We orient $\ell_{p}$ by a vector $X_{p}\in\ell_{p}$ such that the
1-form $\iota(X_{p})d\alpha|_{\xi_{p}}$ defines that co-orientation.

The line field $\ell$, which is defined in the complement of the tangency
locus $T$ between $\xi$ and $\Sigma$, integrates to a singular foliation on
$\Sigma$ with singularities at the points of $T$. We will keep the notation
$\ell$ for this foliation, and write $\ell_{\Sigma}$, $\ell_{\xi}$ or
$\ell_{\xi,\Sigma}$ when it is important to stress the dependence of $\ell$ on
$\Sigma,\xi$, or both.

The singular locus $T$ splits as a union of disjoint closed subsets,
$T=T_{+}\cup T_{-}$, where $T_{+}$ (resp. $T_{-}$) consists of positive (resp.
negative) points, where the orientations of $\xi_{p}$ and $T_{p}(\Sigma)$
coincide (resp. opposite). On neighborhoods $U_{\pm}\supset T_{\pm}$,
$U_{\pm}\subset\Sigma$, the form $d\beta$, $\beta=\alpha|_{\Sigma}$, is
symplectic. We define a vector field $X$ on $\Sigma$ which directs $\ell$ as
equal to the Liouville field $d\beta$-dual to $\beta$ on $U_{+}$, and as a
vector field $d\beta$-dual to $-\beta$ on $U_{-}$, and extend it to the rest
of $\Sigma$ as any non-vanishing vector field. The following lemma is due to
E. Giroux in [9] and was pointed out to us by D. Salamon. It provides an
equivalent characterization of a vector field directing the characteristic
foliation. Choose a positive volume form $\rho$ on $\Sigma$ equal to
$(d\beta)^{n}$ on $U_{+}$ and to $-(d\beta)^{n}$ on $U_{-}$.

######  Lemma 3.1.

The vector field $X$ defined by the equation

(1) $\displaystyle\iota(X)\rho=n\beta\wedge(d\beta)^{n-1}.$

directs the characteristic foliation $\ell$.

###### Proof.

On $U_{\pm}$ equation (1) is equivalent to $\iota(X)d\beta=\pm\beta$, i.e. $X$
coincides with the Liouville field dual to $\beta$ on $U_{+}$ and to the
$d\beta$-dual to $-\beta$ vector field on $U_{-}$. Elsewhere, $X\neq 0$ and
$\iota(X)(\beta\wedge(d\beta)^{n-1})=0.$ If $n=1$ this implies that
$\beta(X)=0$ and hence, $X\in\ell$. For $n\geq 2$ we have

$\displaystyle
0=\iota(X)(\beta\wedge(d\beta)^{n-1})=\beta(X)(d\beta)^{n-1}+(n-1)\left(\iota(X)d\beta\right)\wedge\beta\wedge
d\beta^{n-2}.$

By restricting to $\mathrm{Ker\,}\beta$ we conclude that $\beta(X)=0$, and
hence, $\left(\iota(X)d\beta\right)\wedge\beta\wedge d\beta^{n-2}=0.$ But then
$\left(\iota(X)d\beta\right)|_{\mathrm{Ker\,}\beta}=0$. Indeed, the form
$d\beta$ descends as a symplectic form to the $(2n-2)$-dimensional quotient
space $Q_{p}:=(\xi_{p}\cap T_{p}\Sigma)/T\ell_{p}$ as a symplectic form.
Hence, the multiplication by $d\beta^{n-2}$ defines an isomorphism between
$1$\- and $(2n-3)$-forms on $Q_{p}$, and the claim follows.

∎

Let us recall that the contact structure on a neighborhood of a hypersurface
$\Sigma$ is determined by its restriction to the hypersurface.

######  Proposition 3.2 (A. Givental, [2]).

Let $\xi=\mathrm{Ker\,}\alpha,\xi^{\prime}=\mathrm{Ker\,}\alpha^{\prime}$ be
two contact structures defined on a neighborhood of $\Sigma=\Sigma\times
0\subset\Sigma\times\mathbb{R}$. Suppose that
$\alpha|_{\Sigma}=h\alpha^{\prime}|_{\Sigma^{\prime}}$ for a positive function
$h:\Sigma\to\mathbb{R}$. Then there exists a diffeomorphism
$g:{\mathcal{O}}{\it p}\,\Sigma\to{\mathcal{O}}{\it p}\,\Sigma$ which is fixed
on $\Sigma$ and such that $dg(\xi)=\xi^{\prime}$.

In fact, the statement formulated in [2] is slightly weaker. We thank D.
Salamon for providing the details of the proof of the above result.

All singularities of a vector field $X$ directing the characteristic foliation
$\ell$ can be made non-degenerate and hyperbolic by a $C^{\infty}$-small
perturbation of $\Sigma$, see e.g. [4]. For a generic 1-parametric family of
characteristic foliations, the directing vector field $X_{s}$ can also have
(hyperbolic) embryo singularities for isolated values of the parameter $s$.

We will need the following version of Lemma 2.5 for a vector field $X$
directing a characteristic foliation. Define the standard contact
$(2n-1)$-disc $(D=D^{2n-1},\xi_{\mathrm{st}})$ as contactomorphic to the
hemisphere $D=S^{2n-1}_{+}:=S^{2n-1}\cap\\{y_{n}\geq 0\\}$ endowed with the
contact structure
$\xi_{\mathrm{st}}:=\left\\{\sum\limits_{1}^{n}(x_{j}dy_{j}-y_{j}dx_{j})|_{S^{2n-1}}=0\right\\}$.

######  Lemma 3.3.

Let $X$ be a vector field directing a characteristic foliation on a closed
hypersurface $\Sigma$ in a contact manifold of dimension $2n+1$. Suppose that
all zeros of $X$ are isolated and hyperbolic (non-degenerate or embryos). Then
for any $\epsilon>0$ the field $X$ admits an $\epsilon$-blocking collection
$\\{D_{j}\\}$ which consists of standard contact $(2n-1)$-dimensional discs.

###### Proof.

We only need to ensure that discs forming the blocking collections can be
chosen contactomorphic to the standard contact disc. We recall that in the
proof of Lemma 2.5 discs $D_{j}$ arise as elements of a covering of a
transverse contact hypersurface. But the covering can always be chosen to be
formed by standard small Darboux balls. ∎

###  3.2. Lyapunov functions for characteristic foliations

For a vector field $X$ directing a characteristic foliation $\ell$ on a
hypersurface $\Sigma$ stable manifolds of positive zeroes are isotropic with
respect to $d\beta$, while unstable are coisotropic, see [4]. Near negative
zeroes the field $-X$ is Liouville, and thus stable manifolds of negative
zeroes are co-isotropic while unstable are isotropic, In particular, a local
Lyapunov function on ${\mathcal{O}}{\it p}\,(T_{+}\cup T_{-})$, have critical
points of index $\leq n$ at the positive points, and of index $\geq n$ in the
negative ones. It is important to note that critical points of index $n$ can
be either negative or positive. The stable manifold of a positive (resp.
negative) embryo is an isotropic (resp. co-isotropic) half-space.

We call a Lyapunov function $f:\Sigma\to\mathbb{R}$ for $X$ good if there
exists a regular value $c$ such that all positive zeroes of $X$ are in
$\\{f>c\\}$ and all negative ones are in $\\{f<c\\}$. Sometimes we will call
$f$ a Lyapunov function for $\ell$, rather than $X$.

Following Giroux, we call a trajectory $\gamma$ of $X$ a retrograde connection
if it originates at a negative point of $X$ and terminates at a positive one.
Lemma 2.1 implies

######  Corollary 3.4.

Suppose that a vector field $X$ satisfies conditions (L1) and (L2) from Lemma
2.1. Then it admits a good Lyapunov function if and only if it has no
retrograde connections. In particular, any $X$ which satisfies (L1) and the
Morse-Smale condition admits a good Lyapunov function.

###### Proof.

If there are no retrograde connections, then one can always order zeroes in
such a way that positive zeroes go first, and hence the construction in 2.1
yields a good Lyapunov function. The necessity of the absence of retrograde
connections for existence of good Lyapunov function is straightforward. ∎

Similarly, using Lemma 2.2 we get a parametric version of this statement.

######  Corollary 3.5.

Any family $X_{s}$, $s\in[0,1]$, which satisfy (L1) and (L2) and have no
retrograde connections admits a family of good Lyapunov functions.

###### Proof.

An additional observation which is needed for the proof, in addition to the
argument in 2.2, is that the space of good Lyapunov functions is contractible.
Indeed, if we normalize Lyapunov functions by the condition that the $0$ level
is separating positive and negative points, then their convex linear
combination is again a good Lyapunov function. ∎

###  3.3. Flavors of contact convexity

The notion of contact convexity was first defined in [7], and then explored by
Emmanuel Giroux, see [9], Ko Honda, see [13], and others.

######  Definition 3.6.

  1. (1)

A hypersurface $\Sigma\subset(M,\xi)$ is called convex if it admits a
transverse contact vector field $\Upsilon$.

  2. (2)

A hypersurface $\Sigma$ is called Weinstein convex if its characteristic
foliation $\ell$ admits a good Lyapunov function.

As we will see below in Lemma 3.10, Weinstein convexity is a stronger
condition which implies convexity.

E. Giroux proved in [9] that for $2$-dimensional surfaces contact convexity
can be achieved by a $C^{\infty}$-perturbation.

Using Corollary 3.4 we can equivalently characterize Weinstein convexity by
conditions (L1) and (L2) (or, equivalently, existence of any Lyapunov
function) for $X$ and absence of retrograde connections. As it was pointed out
above, condition (L2) is implied by the Morse-Smale condition, which is
generic for individual hypersurfaces.

Following Giroux, the set $S:=\\{x\in\Sigma;\Upsilon(x)\in\xi_{x}\\}$ is
called the dividing set of $\Sigma$.

######  Lemma 3.7 (E. Giroux, [9]).

Suppose $X$ is a contact vector field transverse to a hypersurface $\Sigma$
and $S$ the corresponding dividing set. Let $t$ be the flow coordinate such
that $\Sigma=\\{t=0\\}$ and $X=\frac{\partial}{\partial t}$. Then $\xi$ on
${\mathcal{O}}{\it p}\,\Sigma$ can be defined by a contact 1-form
$f(x)dt+\beta$, where $f:\Sigma\to\mathbb{R}$ is a function transversely
changing sign across $S$.

Note that the contact condition implies that $df\neq 0$ along $S$, and
$\alpha|_{S}$ is a contact form. In particular, the characteristic foliation
$\ell_{\Sigma}$ transverse to $S$.

Hence, we have the following:

######  Lemma 3.8 (E. Giroux, [9]).

Dividing set $S$ is a smooth submanifold, which is transverse to the
characteristic foliation, and independent of the choice of a contact vector
field transverse to $\Sigma$, up to an isotopy transverse to the
characteristic foliation.

Indeed, the space of contact vector fields transverse to $\Sigma$ is convex
subset of the vector space of all contact vector fields, and hence,
contractible.

The dividing hypersurface $S\subset\Sigma$ divides $\Sigma$ into
$\Sigma_{+}:=\\{f>0\\}$, $\Sigma_{-}:=\\{f<0\\}$. The form
$\alpha=(f(x)dt+\beta)|_{\Sigma\setminus S}$ can be divided by $f$,

$\frac{\alpha}{f}=dt+\frac{\beta}{f}.$

Denote $\lambda_{\pm}:=\frac{\beta}{f}|_{\Sigma_{\pm}}$. The contact condition
then is equivalent to $(d\lambda_{\pm})^{n}\neq 0$. In other words,
$\lambda_{\pm}$ are Liouville forms on $\Sigma_{\pm}$. Note that the
corresponding Liouville fields $Z_{\pm}$ directs the characteristic foliation
on $\Sigma$. Indeed,
$\lambda_{\pm}\wedge\iota(Z_{\pm})d\lambda_{\pm}=\lambda_{\pm}\wedge\lambda_{\pm}=0.$

######  Lemma 3.9.

Let $\Sigma\subset(M,\xi=\mathrm{Ker\,}\alpha)$ be a co-oriented hypersurface.
Denote $\beta:=\alpha|_{\Sigma}$. Then $\Sigma$ is convex if and only if there
exists a function $f:\Sigma\to\mathbb{R}$ such that the form $\beta+fdt$ is
contact on $\Sigma\times\mathbb{R}$.

###### Proof.

The necessity is a reformulation of Lemma 3.7. To see the sufficiency we
observe that $\Sigma=\Sigma\times
0\subset(\Sigma\times\mathbb{R},\mathrm{Ker\,}(\beta+fdt))$ is convex because
the field $\Upsilon:=\frac{\partial}{\partial t}$ is manifestly contact. On
the other hand, by Proposition 3.2 neighborhoods of $\Sigma$ in $(M,\xi)$ and
$\Sigma\times 0$ in $(\Sigma\times\mathbb{R},\mathrm{Ker\,}(\beta+fdt))$ are
contactomorphic. ∎

######  Lemma 3.10.

Any Weinstein convex hypersurface is convex.

###### Proof.

According to Lemma 3.9 it is sufficient to find a function
$f:\Sigma\to\mathbb{R}$ such that the form $\widetilde{\alpha}:=\beta+fdt$ is
contact. We claim that in turn this condition is equivalent to the inequality

(2) $\displaystyle f(d\beta)^{n}+n\beta\wedge(d\beta)^{n-1}\wedge df>0.$

Indeed,

$\displaystyle\widetilde{\alpha}\wedge(d\widetilde{\alpha})^{n}=(\beta+fdt)\wedge(d\beta+df\wedge
dt)^{n}=fdt\wedge(d\beta)^{n}+n\beta\wedge(d\beta)^{n-1}\wedge df\wedge dt$
$\displaystyle=dt\wedge\left(f(d\beta)^{n}+n\beta\wedge(d\beta)^{n-1}\wedge
df\right),$

and hence, the inequality
$\widetilde{\alpha}\wedge(d\widetilde{\alpha})^{n}>0$ is equivalent to (2).

Suppose $\rho$ is a volume form on $\Sigma$ chosen as in Lemma 3.1, and the
vector field $X$ directing $\ell$ satisfies equation (1). Let
$h:\Sigma\to\mathbb{R}$ be a good Lyapunov function on $\Sigma$. We will
assume that $S:=\\{h=0\\}$ is a regular level set of $h$ separating values in
negative and positive zeroes of $X$. In particular,
$h|_{U_{+}}<0,h|_{U_{-}}>0$ for neighborhoods $U_{\pm}\supset T_{\pm}$ of the
singular point loci.

Define a function $g:\Sigma\to\mathbb{R}$ by the equation
$(d\beta)^{n}=g\rho$. We have $g|_{U_{+}}>0$ and $g|_{U_{-}}<0$. Hence, $gh<0$
on $U:=U_{+}\cup U_{-}$. We have $dh(X)>0$ on $\Sigma\setminus((T:=T_{+}\cup
T_{-})\cup S)$. Furthermore, $dh(X)-hg>0$ on a neighborhood of $S=\\{h=0\\}$,
and for a sufficiently large constant $C>0$ we have

$dh(X)-hg+Ch^{2}dh(X)>0$

everywhere on $\Sigma$. The function $f:\Sigma\to\mathbb{R}$, which satisfies
(2) can now be defined by the formula $f:=-e^{\frac{Ch^{2}}{2}}h.$ Indeed, we
have $df=-e^{\frac{Ch^{2}}{2}}(1+Ch^{2})dh$ and therefore,

$\displaystyle fd\beta^{n}+n\beta\wedge(d\beta)^{n-1}\wedge
df=fg\rho+n(\iota(X)\rho)\wedge df.$

But for any 1-form $\gamma$ we have
$\iota(X)\rho\wedge\gamma=-\iota(X)(\rho\wedge\gamma)-\gamma(X)\rho=\gamma(X)\rho,$
and hence

$\displaystyle fg\rho+n(\iota(X)\rho)\wedge df=(fg-
df(X))\rho=-e^{\frac{Ch^{2}}{2}}\left(hg-(1+Ch^{2})dh(X)\right)\rho$
$\displaystyle=e^{\frac{Ch^{2}}{2}}(dh(X)-hg+Ch^{2}dh(X))>0.$

∎

Corollary 3.4 and Lemma 3.10 reduce Theorem 1.1 (in a stronger form replacing
convexity by Weinstein convexity) to

######  Theorem 3.11 (Honda-Huang,[14]).

Let $\Sigma\subset(M,\xi)$ be a co-orientable hypersurface in a contact
manifold with a co-orientable contact structure. Then there exists a
$C^{0}$-small isotopy deforming $\Sigma$ into
$\widetilde{\Sigma}\subset(M,\xi)$ such that the characteristic foliation
induced on $\widetilde{\Sigma}$ satisfies condition (L1) and the Morse-Smale
property.

For the case $\dim M=2$ this result can be deduced from E. Giroux’s theorem
about $C^{\infty}$-genericity of contact convexity in $3$-dimensional contact
manifolds, [9].

##  4\. Construction of plugs

###  4.1. Main proposition

In view of Lemmas 2.6 and 3.3, the proof of Theorem 3.11 will be completed if
for any $\sigma>0$ one can create a $\sigma$-plug by a $C^{0}$-small isotopy
of the flow box of a standard, transverse to the flow contact disc. The next
proposition asserts that this is possible. As the statement will be proven by
induction, we need more properties of a $\sigma$-plug in order for the
induction to go through.

Let $(D^{2n-1},\alpha_{st})$ be the standard contact disc. Choose $c,b>0$ and
consider $U^{b}:=D\times T^{*}[0,b]$ endowed with a contact form
$\alpha_{\mathrm{st}}+xdy$. Denote

$Q^{b}:=\\{x=0\\}\subset U=D\times[0,b],\;U^{b}_{c}:=\\{|x|\leq c\\}\subset
U^{b}.$

We will omit the superscript $b$ when $b=1$.

######  Proposition 4.1.

For any positive $\epsilon$ and $\sigma\ll\epsilon$ there exists an isotopy
$h_{s}:Q\to U_{\epsilon}$, $s\in[0,1]$, which is fixed on ${\mathcal{O}}{\it
p}\,\partial Q$, begins with the inclusion $h_{0}:Q\hookrightarrow
U_{\epsilon}$ and has the following properties:

  * a)

$(Q,X_{1})$ is a $\sigma$-plug, where we denoted by $X_{s}$ the vector field
directing the characteristic foliation $\ell_{s}$ induced by
$h_{s}^{*}(\alpha_{\mathrm{st}}+xdy)$;

  * b)

for any $\sigma_{1}\ll\sigma$ there exists a family of compact manifolds with
boundary $C^{+}_{s}\subset\mathrm{Int\,}Q$ and
$C^{-}_{s}\subset\mathrm{Int\,}Q,$ and an extension of the isotopy $h_{s}$ to
a 2-parametric isotopy $h_{s,t},\;s,t\in[0,1]$, such that

    * (i)

$h_{s,0}=h_{s},\;h_{0,t}=h_{0}$ for all $s,t\in[0,1]$;

    * –

the foliation $\ell_{s,1}$ induced on $h_{s,1}(Q)$ has no singular points;

    * (ii)

for any fixed $s\in[0,1]$ the isotopy $h_{s,t}$, $t\in[0,1]$, is
$\sigma_{1}$-small in the $C^{0}$-sense and supported in a
$\sigma_{1}$-neighborhood of $C_{s}$;

    * (iii)

For each $s\in[0,1]$ the submanifold $C^{+}_{s}$ (resp. $C_{s}^{-}$) contains
all positive (resp. negative) singularities of $X_{s},$ $C_{s}^{+}\cap
C_{S}^{-}=\emptyset$ and $C^{+}_{s}$ (resp. $C^{-}_{s}$) is invariant with
respect to the backward (resp. forward) flow of $X_{s}$;

    * (iv)

there exists a family of generalized Morse Lyapunov functions
$\psi_{s,t}:Q\to\mathbb{R}$ for $X_{s,t}$ such that
$\psi_{s,t}|_{{\mathcal{O}}{\it p}\,\partial Q}=y$;

    * (v)

there exists a stratified $(n-1)$-dimensional subset $E\subset Q\cap\\{y=0\\}$
which contains all the intersection points of $Q\cap\\{y=0\\}$ with stable
manifolds of positive singular point of $X_{s,t}$ for all $s,t\in[0,1]$.

We will refer to the statement of Proposition 4.1 as an installation of a
$\sigma$-plug of height $\epsilon$ over $Q=D^{2n-1}\times[0,1]$, where
$(D^{2n-1},\alpha_{\mathrm{st}})$ is the standard contact disc. The same
statement with $Q,U_{c}$ are replaced by $Q^{b}$ and $U^{b}_{c}$ will be
referred as an installation over $Q^{b}$. Note that the contactomorphism
$U^{b}_{a}\to U^{1}_{ab}$ induced by the linear map
$(x,y)\mapsto(bx,\frac{y}{b})$ of the second factor always allows us to reduce
the installation to the case $b=1$.

If we further replace $(D,\alpha_{\mathrm{st}})$ in the statement of 4.1 by
any compact $(2n-1)$-dimensional manifold $V$ manifold with boundary (and
possibly with corners) and with a fixed contact form $\alpha$, we will say
that we are installing a $\sigma$-plug of height $\epsilon$ over
$V\times[0,b]$.

###  4.2. Plan of the proof of Proposition 4.1

We begin the proof in Section 5 by showing that Proposition 4.1 can be deduced
from a weaker Proposition 5.1, where the required isotopy $h_{s}$ is
constructed in $U_{K}$ for a large $K$ which may depend on $\sigma$, rather
than $U_{\epsilon}$ for an arbitrary small $\epsilon$. This is done by a
scaling argument. One of the subtleties here is that contact scalings are
better adjusted to Carnot-Caratheodory type metrics, rather than Riemannian
ones. Thus, we have to analyze separately an effect of the scaling on
measurements in directions tangent and transverse to contact planes.

The continuation of the proof is by induction on dimension $2n-1$. Lemma 6.4
in Section 6.2 serves as the base of the induction for $n=2$, as well as an
important ingredient in the proof of the induction step. The Giroux-Fuchs
creation-elimination construction, which we recall in Section 6.1, is an
essential ingredient to the proof of Lemma 6.4.

By taking a product of the two-dimensional plug constructed in Lemma 6.4 with
a $(2n-2)$-dimensional Weinstein domain $(W,\lambda)$ and appropriately
adjusting the product over ${\mathcal{O}}{\it p}\,\partial W$ we construct in
Section 7.1 a $2n$-dimensional preliminary plug over
$((W\times[0,1])\times[0,1]$. We call the contact domain
$(W\times[0,1],\lambda+dz)$ a Weinstein cylinder. Similar to a $\sigma$-plug,
a preliminary plug blocks all trajectories entering $W\times[0,1]\times 0$ at
a distance $\geq\sigma$ from the boundary of the Weinstein cylinder. However,
one has a much weaker control of the dynamics of the trajectories entering
near the boundary. Constructions of 2-dimensional and preliminary plugs are
variations of similar constructions in [14].

Next, we show in Section 7.4 that by a special arrangement of Weinstein
cylinders $(V_{1}=W_{1}\times[0,1],\dots,V_{k}=W_{k}\times[0,1])$, $k\geq 2$,
see the definition of a good position in Section 7.3, and by composing
preliminary plugs over $V_{j}\times[\frac{j-1}{k},\frac{j}{k}]$,
$j=1,\dots,k$, we create, see Lemma 7.8, a $\sigma$-quasi-plug over
$(\widehat{V}:=\bigcup\limits_{1}^{k}V_{j})\times[0,1]$ which blocks
trajectories entering at a distance $\geq\sigma$ from
$\partial\widehat{V}\times 0$, while a non-blocked trajectory which enters at
a point $(p_{0},0)\in\widehat{V}\times 0$ with
$\mathrm{dist}(p_{0},\partial\widehat{V})<\sigma$ exits at a point
$p_{1}\in\widehat{V}\times 1$ which satisfies the following condition: there
exist points $p_{0}^{\prime},p_{1}^{\prime}\in\partial\widehat{W}$ such that
$\mathrm{dist}(p_{0},p_{0}^{\prime}),\mathrm{dist}(p_{1},p_{1}^{\prime})<\sigma$
and $p_{1}^{\prime}$ belongs to the forward trajectory of $p_{0}^{\prime}$ for
a vector field directing the characteristic foliation
$\ell_{\partial\widehat{V}}$ on $\partial\widehat{V}$. We note that if the
characteristic foliation $\ell_{\partial\widehat{V}}$ is a $\sigma$-short,
then any $\sigma$-quasi-plug is automatically a $3\sigma$-plug.

Crucial Proposition 7.6 asserts that if a contact domain $V$ with a Weinstein
convex boundary $\partial V$ and a dividing set $S\subset\partial V$ can be
$C^{0}$-approximated by standard contact balls which coincide with $V$ in
${\mathcal{O}}{\it p}\,S$ then $V$ can be approximated by 3 Weinstein
cylinders in a good position.

In Section 8.2 we use the induction hypothesis to show that the standard
contact ball $D^{2n-1}$ can be be deformed by a $\sigma$-small in the
$C^{0}$-sense isotopy to a ball $\widetilde{D}$ with Weinstein convex boundary
and a dividing set $S\subset\partial\widetilde{D}$ such that the
characteristic foliations $\ell_{\partial\widetilde{D}}$ is $\sigma$-short and
$\widetilde{D}$ can be $C^{0}$-approximated by standard contact balls which
coincide with $\widetilde{D}$ on a neighborhood of $S$.

Together with Proposition 7.6 and Lemma 7.8 this leads in Section 8.3 to a
proof of Proposition 5.1, and with it, of all main results of the paper.

##  5\. Reducing the height of a $\sigma$-plug

The goal of this section is to reduce Proposition 4.1 to the following weaker
statement.

######  Proposition 5.1.

For any $\sigma>0$ there exists $K=K(\sigma)$ and an isotopy $h_{s}:Q\to
U_{K}$, $s\in[0,1]$, which satisfy properties a) and b) from Proposition 4.1.
In other words, one can install a $\sigma$-plug over $Q=D^{2n-1}\times[0,1]$
of height $K$ which may depend on $\sigma$.

###  5.1. Changing the base

Let $(V_{1},\mathrm{Ker\,}\alpha_{1}),(V_{2},\mathrm{Ker\,}\alpha_{2})$ be two
contact manifolds with boundary with corners endowed with contact forms and
Riemannian metrics. Any contactomorpohism $f:V_{1}\to V_{2}$ can be extended
to a contactomorphism

$F:(V_{1}\times T^{*}\mathbb{R},\mathrm{Ker\,}(\alpha_{1}+xdy))\to(V_{2}\times
T^{*}\mathbb{R},\mathrm{Ker\,}(\alpha_{2}+xdy)$

by the formula $F(v,x,y)=(f(v),g(v)x,y),\;v\in V_{1},\;x,y\in\mathbb{R},$
where the function $g$ is defined by the equation
$f^{*}\alpha_{2}=g\alpha_{1}$.

######  Lemma 5.2.

Let $h_{s,t}:V_{1}\times[0,1]\to V_{1}\times T^{*}[0,1]$ be an isotopy
installing a $\sigma$-plug of height $\epsilon$ over $V_{1}\times[0,1]$.
Denote $C_{1}:=\mathop{\max}\limits_{v\in V_{1}}||d_{v}f||$,
$C_{2}:=\max\limits_{V_{1}}g$. Then the isotopy $\widehat{h}_{s}:=F\circ
h_{s}\circ F^{-1}:V_{2}\times[0,1]\to V_{2}\times T^{*}[0,1]$ is installing a
$C_{1}\sigma$-plug of height $C_{2}\epsilon$ over $V_{2}\times[0,1]$.

###### Proof.

First, we note that $\widehat{h}_{s}(V_{2}\times[0,1])\subset\\{|x|\leq
C_{2}\epsilon\\}$ because
$h_{s}(V_{1}\times[0,1])\subset\\{|x|\leq\epsilon\\}$ by assumption. If
$X_{1}^{1}$ is the vector field on $V_{1}\times[0,1]$ directing the
characteristic foliation of the form $h_{s}^{*}(\alpha_{1}+xdy)$ then the
vector field $X_{1}^{2}:=df(X_{1}^{1})$ on $V_{2}\times[0,1]$ directs the
characteristic foliation defined by the form $h_{s}^{*}(\alpha_{2}+xdy)$. Let
$N^{1}_{a}$ and $N^{2}_{a}$ denote metric $a$-neighborhoods of $\partial
V_{1}$ in $V_{1}$ and $\partial V_{2}$ in $V_{2}$, respectively. Then
$f(N^{1}_{\sigma})\subset N^{2}_{C_{1}\sigma}$. Hence, all trajectories of
$X_{1}^{2}$ originated in $V_{2}\setminus N^{2}_{C_{1}\sigma}$ are blocked. On
the other hand, the non-blocked trajectories originated in
$N^{2}_{C_{1}\sigma}$ exit with a distortion for no more than $C_{1}\sigma$.
All other properties of a $C_{1}\sigma$-plug installation isotopy listed in
Proposition 4.1 are straightforward. ∎

Consider a class $\mathcal{D}$ of $(2n-1)$-dimensional compact manifolds with
boundary (and possibly with corners) which are contactomorphic to a domain in
the standard contact $(\mathbb{R}^{2n-1},dz+\lambda_{\mathrm{st}})$ with
boundary transverse to the contact vector field
$\Upsilon=2\frac{\partial}{\partial
z}+\sum\limits_{1}^{n-1}x_{j}\frac{\partial}{\partial
x_{j}}+y_{j}\frac{\partial}{\partial y_{j}}.$ For instance, the standard
$(2n-1)$-dimensional contact ball belongs to $\mathcal{D}$.

######  Lemma 5.3.

If there exists a domain $V\in\mathcal{D}$ such that for any $\sigma>0$ one
can install a plug of height $K=K(\sigma,V)$ over $V\times[0,1]$, then for any
domain $V^{\prime}\in\mathcal{D}$ and any $\sigma>0$ one can install a plug of
height $K^{\prime}:=K(\sigma,V^{\prime}))$ over $V^{\prime}\times[0,1]$. If
there exists a domain $V\in\mathcal{D}$ such that for any $\sigma>0$ and any
$\epsilon>0$ one can install a plug of height $\epsilon$ over $V\times[0,1]$,
then the same is true for any domain $V^{\prime}\in\mathcal{D}$.

###### Proof.

For any domain $(V,\mathrm{Ker\,}\alpha)\in\mathcal{D}$ its interior
$\mathrm{Int\,}V$ is contactomorphic to the standard contact
$\mathbb{R}^{2n-1}$, see [6]. Besides, the boundary $\partial V$ is convex,
and hence $\mathrm{Int\,}V=\bigcup V_{j}$, where $V_{j}$ is contactomorphic to
$V$, Hence, for any $V,V^{\prime}\in\mathcal{D}$
$\mathrm{Int\,}V^{\prime}=\bigcup V_{j}$, where $V_{j}$ is contactomorphic to
$V$. Hence, the statement follows from Lemma 5.2. ∎

###  5.2. Scaling

For $a,b>0$ denote $R_{a,b}:=\\{|x_{j}|,|y_{j}|\leq a,|z|\leq
b,\;j=1,\dots,n-1\\}\subset(\mathbb{R}^{2n-1},dz+\lambda_{\mathrm{st}})$. Note
that $R_{a,b}\in\mathcal{D}$.

Figure 3. The arrangement of blocks $Q^{i,0}$, $i=0,\pm 1,\pm 2$, and
$Q^{i,1}$, $i=-2,\dots,1,$ for the case $N=2$.

######  Lemma 5.4.

Choose $\sigma>0$. Suppose that one can install a $\sigma$-plug of height $K$
over $R_{1,1}\times[0,1]$. Then for any integer $N\geq 0$ one can install a
$\sigma$-plug of height $K$ over $R_{1,2N+1}\times[0,2]$.

###### Proof.

Denote $\widetilde{Q}:=R_{1,2N+1}\times[0,2]$, $\widehat{U}:=R_{1,2N+1}\times
T^{*}[0,2]$. We assume $\widehat{U}$ is endowed with the contact form
$dz+\lambda_{\mathrm{st}}+xdy$. Furthermore, for $i=0,\pm 1,\dots,\pm N$
denote $Q^{i,0}:=\\{z\in[2i-1,2i+1],y\in[0,1]\\}\subset\widehat{Q}$,
$U^{i,0}:=\\{z\in[2i-1,2i+1],y\in[0,1]\\}\subset\widehat{U}$, and for
$i=-N,\dots,N-1$ denote
$Q^{i,1}:=\\{z\in[2i,2i+2],y\in[1,2]\\}\subset\widehat{Q}$,
$U^{i,0}:=\\{z\in[2i,2i+2],y\in[1,2]\\}\subset\widehat{U}$, see Fig. 3. Note
that $Q^{0,0}=R_{1,1}\times[0,1]$ and $U^{0,0}=R_{1,1}\times T^{*}[0,1]$. The
diffeomorphisms $(z,y)\mathop{\mapsto}\limits^{\Pi^{i,0}}(z+2i,y)$ and
$(z,y)\mathop{\mapsto}\limits^{\Pi^{i,1}}(z+1+2i,y+1)$ preserves the contact
form $dz+\lambda_{\mathrm{st}}+xdy$ and identify $(U:=R_{1,1}\times
T^{*}[0,1],Q:=R_{1,1}\times[0,1])$ with $(U^{i,0},Q^{i,0})$ and
$(U^{i,1},Q^{i,1})$, respectively. Let $h_{s}:Q\to U$ be an isotopy installing
a plug of height $K$ over $Q=R_{1,1}\times[0,1]$. Then the isotopy
$g_{s}:\widehat{Q}\to\widehat{U}$ which is equal to $\Pi^{i,0}\circ
g_{s}\circ(\Pi^{i,0})^{-1}$ on $Q^{i,0}$, $i=0,\dots,\pm N$, and to
$\Pi^{i,1}\circ g_{s}\circ(\Pi^{i,1})^{-1}$ on $Q^{i,1}$, $i=-N,\dots,N-1$ is
installing the required $\sigma$-plug of height $K$ over
$R_{1,2N+1}\times[0,2]$. ∎

######  Lemma 5.5.

If one can install a $\sigma$-plug of height $K$ over $R_{1,1}\times[0,1]$
then for any integer $N\geq 0$ one can install a $\frac{\sigma}{2N+1}$-plug of
height $\frac{2K}{(2N+1)^{2}}$ over $R_{\frac{1}{2N+1},1}\times[0,1]$.

###### Proof.

By applying Lemma 5.4 we install a $\sigma$-plug of height $K$ over
$R_{1,(2N+1)^{2}}\times[0,2]$. This is equivalent to a $\sigma$-plug of height
$2K$ over $R_{1,(2N+1)^{2}}\times[0,1]$ . Let $h_{s}$ be an isotopy which
installs this plug. Consider a contactomorphism

$\displaystyle(x_{1},y_{1},\dots,x_{n-1},y_{n-1},x,y,z)$
$\displaystyle\mathop{\mapsto}\limits^{f}\left(\frac{x_{1}}{2N+1},\frac{y_{1}}{2N+1},\dots,\frac{x_{n-1}}{2N+1},\frac{y_{n-1}}{2N+1},\frac{x}{(2N+1)^{2}},y,\frac{z}{(2N+1)^{2}}\right).$

Then the isotopy $f\circ h_{s}\circ(f)^{-1}$ is installing the required
$\frac{\sigma}{2N+1}$-plug of height $\frac{2K}{(2N+1)^{2}}$ over
$R_{\frac{1}{2n+1},1}\times[0,1]$. ∎

Lemma 5.5 and Proposition 5.1 imply

######  Corollary 5.6.

For any $\sigma,\epsilon>0$ and $p\in D^{2n-2}$ there exists $N_{0}$ such that
for any $N\geq N_{0}$ one can install a $\frac{\delta}{N}$-plug of height
$\epsilon$ over $R_{\frac{1}{N},1}\times[0,1]$.

For a point $p=(a_{1},b_{1},\dots,a_{n-1},b_{n-1})\subset\mathbb{R}^{2n-2}$
consider a map

$\tau_{p}:\mathbb{R}^{2n-1}=\mathbb{R}^{2n-2}\times\mathbb{R}\to\mathbb{R}^{2n-2}\times\mathbb{R}$

given by the formula

$\displaystyle\tau_{p}(x_{1},y_{1},\dots,x_{n-1},y_{n-1},z)$
$\displaystyle=\left(x_{1}+a_{1},y_{1}+b_{1},\dots,x_{n-1}+a_{n-1},y_{n-1}+b_{n-1},z-\sum\limits_{1}^{n-1}a_{j}y_{j}-b_{j}x_{j}\right).$

Note the $\tau_{p}$ preserves the contact form $\lambda_{\mathrm{st}}+dz$:

$\tau_{p}^{*}(\lambda_{\mathrm{st}}+dz)=\lambda_{\mathrm{st}}+dz.$

Fix an integer $N>1$. Given an integer vector

$I:=(i_{1},j_{1},\dots,i_{n-1},j_{n-1})\in[1-N,N-1]^{2n-2}$

denote $p_{I}=\frac{I}{N},\;\delta_{N}:=\frac{2n-2}{N}$ and

$\displaystyle
P_{I}:=\tau_{p_{I}}\left(R_{\frac{1}{N},1}\right),\;P_{I,-}:=\tau_{p_{I}}\left(R_{\frac{2}{3N},1-\delta_{N}}\right),\;\;\hbox{see
Fig. \ref{fig:scaling2}.}$

Figure 4. Domain $R_{\frac{1}{N},1}$ and its image $P_{1,1}$ under the shear
$\tau_{p_{11}}$.

Note that we have

######  Lemma 5.7.

$\mathrm{Int\,}\left(R_{1-\frac{1}{3N},1-\frac{2n-2}{N}}\right)\subset\bigcup\limits_{I\in[1-N,N-1]^{2n-2}}\mathrm{Int\,}\left(P_{I,-}\right))\subset\mathrm{Int\,}\left(R_{1,1}\right)$

and the multiplicity of the covering
$\bigcup\limits_{I\in[1-N,N-1]^{2n-2}}\mathrm{Int\,}\left(P_{I,-}\right))\supset\mathrm{Int\,}\left(R_{1-\frac{1}{3N},1-\frac{2n-2}{N}}\right)$
is equal to $2^{2n-2}$.

###### Reduction of Proposition 4.1 to Proposition 5.1.

Suppose that for any $\sigma>0$ one can install a $\sigma$-plug of some height
$K=K(\sigma)$ over $D^{n-1}\times[0,1]$. As the domain $R_{1,1}$ belongs to
the class $\mathcal{D}$ it follows that for any $\sigma>0$ one can install a
$\sigma$-plug of some height $K^{\prime}=K^{\prime}(\sigma)$ over
$R_{1,1}\times[0,1]$.

We can assume $\sigma<\frac{1}{4}.$ Set
$\widehat{\sigma}:=\frac{\sigma}{2^{2n-2}\sqrt{2n}},\widehat{\epsilon}:=\frac{\epsilon}{2^{2n-2}}$.
Let $N$ be the integer provided by Corollary 5.6 for the pair
$(\widehat{\sigma},\widehat{\epsilon})$. In other words, one can install a
$\frac{\widehat{\sigma}}{N}$-plug of height $\widehat{\epsilon}$ over
$R_{\frac{1}{N},1}\times[0,1]$. Choosing $N$ large enough we will ensure that
$\frac{2n}{N}<\sigma$ which implies that $\mathrm{dist}(\partial
R_{1-\frac{1}{3N},1-\frac{2n-2}{N}},\partial R_{1,1})<\sigma$. Note that
$||d\tau_{p}||\leq\sqrt{2n}$ for any
$p=(a_{1},b_{1},\dots,a_{n-1},b_{n-1})\in[-1,1]^{2n-2}$. Hence, we can apply
Lemma 5.2 to install a $\frac{\sqrt{2n}\widehat{\sigma}}{N}$-plug of height
$\widehat{\epsilon}$ over $P_{I}\times[0,1]$ for any $I$.

Let us partition the set $\mathcal{I}=[1-N,N-1]^{2n-2}$ of all indices into
$2^{2n-2}$ subsets $\mathcal{I}_{A}$ indexed by subsets
$A\subset\\{1,\dots,2n-2\\}$: the subset $\mathcal{I}_{A}$ consists of
$I=(i_{1},j_{1},\dots,i_{n-1},j_{n-1})$ which have odd entries at positions of
the subset $A$, and even at other places. For instance, for $A=\varnothing$
the set $\mathcal{I}_{\varnothing}$ consists of
$I=(i_{1},j_{1},\dots,i_{n-1},j_{n-1})$, where all $i_{k},j_{k}$ are even.
Note that for $I,I^{\prime}\in\mathcal{I}_{A}$, $I\neq I^{\prime}$ we have
$\mathrm{Int\,}P_{I}\cap\mathrm{Int\,}P_{I^{\prime}}=\varnothing$. We
enumerate all subsets $A\subset\\{1,\dots,2n-2\\}$ as
$A_{1},\dots,A_{2^{2n-2}}$, and write $\mathcal{I}_{j}$ instead of
$\mathcal{I}_{A_{j}}$.

We claim that by installing for each $I\in\mathcal{I}_{j}$,
$j=1,\dots,2^{2n-2}$ a $\frac{\sqrt{2n}\widehat{\sigma}}{N}$-plug of height
$2^{2n-2}\widehat{\epsilon}=\epsilon$ over
$P_{I}\times[\frac{j-1}{2^{2n-2}},\frac{j}{2^{2n-2}}]$ we construct the
required $\sigma$-plug of height $\epsilon$ over $R_{1,1}\times[0,1]$. Indeed,
let $h_{s}:R_{1,1}\times[0,1]\to R_{1,1}\times T^{*}[0,1]$ be the resulting
isotopy. First, note that $h_{s}(R_{1,1}\times[0,1])\subset\\{|x|\leq
2^{2n-2}\widehat{\epsilon}=\epsilon$. Let us verify that the vector field
$X_{1}$ directing the characteristic foliation $\ell_{1}$ induced by
$h_{1}^{*}(\alpha_{\mathrm{st}}+xdy)$ is a $\sigma$-plug. As there are
$2^{2n-2}$ layers of plugs, each trajectory $\gamma$ of $X_{1}$ beginning at
$(p,0)\in R_{1,1}\times 0$ intersects no more than $2^{2n-2}$ plugs
$P_{I}\times[\frac{j-1}{2^{2n-2}},\frac{j}{2^{2n-2}}]$. Each of these plugs
either blocks $\gamma$, or displaces it for no more than
$\frac{\sqrt{2n}\widehat{\sigma}}{N}$. Hence if $\gamma$ exits through a point
$(p^{\prime},1)\in R_{1,1}\times 1$ then
$\mathrm{dist}(p,p^{\prime})<2^{2n-2}\sqrt{2n}\frac{\widehat{\sigma}}{N}=\frac{\sigma}{N}$.
On the other hand, if $\mathrm{dist}(p,\partial
R_{1,1})\geq\sigma>\frac{2n}{N}$ then $p\in P_{I,-}$ for a multi-index
$I\in\mathcal{I}_{j}$ for some $j=1,\dots,2^{2n-2}$. If the trajectory
$\gamma$ originates at $(p,0)$ and it is not blocked by any of the plugs on
the layers $[\frac{i-1}{2^{2n-2}},\frac{i}{2^{2n-2}}]$ for $i<j$ then by the
above argument it enters the plug
$P_{I,-}\times[\frac{j-1}{2^{2n-2}},\frac{j-1}{2^{2n-2}}]$ through a point
$(p^{\prime},\frac{j-1}{2^{2n-2}})$ with
$\mathrm{dist}(p,p^{\prime})<\frac{\sigma}{N}$. Hence,
$\mathrm{dist}(p^{\prime},\partial P_{I})>\mathrm{dist}(\partial
P_{I},\partial
P_{I,-})-\frac{\sigma}{N}>\frac{1}{3N}-\frac{1}{4N}>\frac{\sigma}{3N}>\frac{\sqrt{2n}\widehat{\sigma}}{N}$.
But $(P_{I,-}\times[\frac{j-1}{2^{2n-2}},\frac{j-1}{2^{2n-2}}],X_{1})$ is a
$\frac{\sqrt{2n}\widehat{\sigma}}{N}$-plug, and therefore, the trajectory
$\gamma$ is blocked inside
$P_{I,-}\times[\frac{j-1}{2^{2n-2}},\frac{j-1}{2^{2n-2}}]$. This verifies the
property a) of Proposition 4.1. Property b) follows from the fact that it
holds for plugs
$(P_{I,-}\times[\frac{j-1}{2^{2n-2}},\frac{j-1}{2^{2n-2}},X_{1})]$ for each
$I\in\mathcal{I}$ and transversality arguments.

Finally, we again apply Lemma 5.3 to conclude that the installation for any
$\sigma,\epsilon$ of a $\sigma$-plug of height $\epsilon$ over $R_{1,1}$ is
equivalent to the installation for any $\sigma,\epsilon$ of a $\sigma$-plug of
height $\epsilon$ over $D^{2n-1}\times[0,1]$, because both domains $D^{2n-1}$
and $R_{1,1}$ belong to the class $\mathcal{D}$. ∎

######  Remark 5.8.

Note that the above proof of the height-reduction for $\sigma$-plugs
significantly simplifies for $n=1$, i.e. when the plug is 2-dimensional.
Indeed, in this case $D^{2n-1}=R_{1,1}=R_{\frac{1}{N},1}=[-1,1]$, and the
claim follows directly from Corollary 5.6.

##  6\. The 2-dimensional case

We will prove in this section Proposition 5.1 (and hence, Proposition 4.1) in
the 2-dimensional case. In fact, we will establish a stronger statement, Lemma
6.4, which will enable us to continue the construction by induction on
dimension of the plug.

###  6.1. Creation and elimination of singularities of a 2-dimensional
characteristic foliation

The following statement is a slight modification of the Giroux-Fuchs
elimination lemma, see [9].

######  Lemma 6.1.

Let $\Sigma$ be a 2-dimensional surface in a contact $3$-manifold
$(M,\xi=\mathrm{Ker\,}\alpha)$. Let $p\in\Sigma$ be a non-singular point of
the characteristic foliation $\ell=\ell_{\Sigma,\xi}$. Let $\gamma\ni p$ be an
arc of the leaf of $\ell$ through $p$. Suppose that $(d\alpha)_{p}>0$ Then for
any positive $\epsilon$ and $\sigma\ll\epsilon$ there exists an
$\epsilon$-small 2-parametric isotopy $\phi_{t,s}:\Sigma\to(M,\xi)$ supported
in an $\epsilon$-neighborhood of $p\in\Sigma$ with the following properties.
Denote $\beta_{s,t}:=\phi_{s,t}^{*}\alpha$ and let $\ell_{s,t}$ be the
characteristic foliation defined by $\beta_{s,t}$, and $X_{s,t}$ the vector
field directing $\ell_{s,t}$.

  * •

$\phi_{0,t}=\phi_{0,0}$ is the inclusion $\Sigma\hookrightarrow M$;

  * -

the $1$-form $\beta_{s,1}$ has no zeros for all $s\in[0,1]$;

  * -

the $1$-form $\beta_{1,0}$ has exactly two zeros, one positive elliptic and
one hyperbolic on the arc $\gamma$;

  * -

the arc $\gamma$ is tangent to $X_{s,t}$ for all $s,t\in[0,1]$.

  * -

$d\beta_{s,t}=d\beta_{0,0}$ for all $s,t\in[0,1]$;

  * -

If $\ell$ admits a Lyapunov function $f:\Sigma\to\mathbb{R}$ then for all
$s,t\in[0,1]$ the characteristic foliation $\ell_{s,t}$ admits a Lyapunov
function $f_{s,t}$ which coincides with $f$ outside an $\epsilon$-neighborhood
of $p$.

We split the proof into two parts.

######  Lemma 6.2.

Consider the form $\delta=dz+dx+xdy$ in $\mathbb{R}^{3}$. Under assumptions of
Lemma 6.1 there is a neighborhood $U\ni p$ in $M$, a neighborhood $U^{\prime}$
of $0$ in $\mathbb{R}^{3}$ and a contactomorphism
$h:(U^{\prime},\delta)\to(U,\alpha)$ such that $h^{-1}(\Sigma\cap
U)=\mathbb{R}^{2}\cap U^{\prime}$.

###### Proof.

We first find a diffeomorphism $h_{1}:{\mathcal{O}}{\it
p}\,_{\mathbb{R}^{2}}(0)\to{\mathcal{O}}{\it p}\,_{\Sigma}(p)$ such that
$h_{1}^{*}d\alpha=dx\wedge dy$, and then compose it with a symplecomorphism to
equate the pull-back of $\alpha$ with $dx+xdy$. Finally, we evoke Proposition
3.2 to conclude that two contact structures with the same restriction to a
surface are contactomorphic via a contactomorphism fixed on the surface. ∎

######  Lemma 6.3.

For any $\epsilon\gg\sigma>0$ there exists a 2-parametric family of
$C^{\infty}$-functions $G_{s,t}:\mathbb{R}^{2}\to\mathbb{R}$, $s,t\in[0,1]$
which are supported in $\\{|x|,|y|<\epsilon\\}$ which have the following
properties. Denote $\alpha_{s,t}:=xdy+dx+dG_{s,t}.$

  * -

$G_{0,0}=0$;

  * -

$G_{s,t}-G_{s,0}$ is supported in $\\{x<\sigma,y<\epsilon\\}$.

  * -

the $1$-form $\alpha_{s,1}$ has no zeros for all $s\in[0,1]$;

  * -

the $1$-form $\alpha_{1,0}$ has exactly two positive zeroes, one elliptic and
one hyperbolic, on the line $\\{x=0\\}$.

  * -

for all $s,t\in[0,1]$ the vector field $Y_{s,t}$ directing the characteristic
foliation on $\\{x=0\\}$ generated by $\alpha_{s,t}$ admits a Lyapunov
function which is equal to $y$ outside of a compact set.

Figure 5. Creating and eliminating zeroes.

###### Proof.

Consider an even function $\theta:\mathbb{R}\to\mathbb{R}_{+}$,
$\theta(u)=\theta(-u)$, and for any $c>0$ consider an odd functions
$\eta_{c}:\mathbb{R}\to\mathbb{R},\;\eta_{c}(u)=-\eta_{c}(u)$ such that the
following properties are satisfied:

  * -

$\theta(u)=u^{2}-1-\frac{\epsilon^{2}}{9}$ on $u\in[-\epsilon/2,\epsilon/2]$,
$\theta(u)=0$ for $|u|\geq\epsilon$ and $0\leq\theta^{\prime}(u)$ for $u\geq
0$;

  * -

$\eta_{c}(u)=u$ for $u\in[-c\sigma,c\sigma]$, $\eta_{c}(u)=0$ for $|u|\geq
c\epsilon$, $|\eta_{c}(u)|\leq|u|$ and
$-\frac{2\sigma}{\epsilon}\leq\eta^{\prime}(u)\leq 1$ for $u\geq 0$.

Consider a family of functions $G_{s},\;s\in[0,1]$ by the formula

$\displaystyle G_{s}(x,y)=s\theta(y)\eta_{1}(x)$

and a family of 1-forms

$\displaystyle\alpha_{s}$
$\displaystyle=dx+xdy+dG_{s}=(1+s\theta(y)\eta_{1}^{\prime}(x))dx+(x+s\theta^{\prime}(y)\eta_{1}(x))dy=f_{s}(x,y)dx+g_{s}(x,y)dy.$

Let us check that the form $\alpha_{1}$ have a hyperbolic $0$ at the points
$(x=0,y=-\frac{\epsilon}{3})$, an elliptic zero at
$(x=0,y=\frac{\epsilon}{3})$ and no other zeroes. We have

(3)
$\begin{split}&|g_{1}(x,y)|=|x+2y\eta_{1}(x)|\geq|x|(1-2|y|)\geq(1-\epsilon)|x|\;\;\hbox{for}\;\;|y|\leq\frac{\epsilon}{2};\\\
&f_{1}(x,y)=1+\left(y^{2}-1-\frac{\epsilon^{2}}{9}\right)\eta_{1}^{\prime}(x)\geq\frac{5\epsilon^{2}}{36}>0,\;\;|y|\geq\frac{\epsilon}{2}.\end{split}$

Hence $\alpha_{1}$ has only zeros along the interval
$\\{x=0,-\frac{\epsilon}{2}<y<\frac{\epsilon}{2}\\}$. In the neighborhood of
this interval we have $\alpha_{1}=(y^{2}-\frac{\epsilon^{2}}{9})dx+(1+2y)xdy$,
which has 2 zeroes, elliptic and hyperbolic, respectively at the points
$(x=0,y=\frac{\epsilon}{3})$ and $(x=0,y=-\frac{\epsilon}{3})$.

Let us now extend the family $G_{s}$ to the 2-parametric family of functions
$G_{s,t}:\mathbb{R}^{2}\to\mathbb{R}$ by setting

$G_{s,t}=H_{s}(x,y)-st\theta(y)\eta_{\sigma}(x)=f_{s,t}(x,y)dx+f_{s,t}(x,y)dy.$

Let us verify that the form $\alpha_{s,1}=dx+xdy+dG_{s,1}$ has no zeros.
First, note that for $|x|\leq\frac{\sigma^{2}}{\epsilon},$ we have $f_{s,1}=1$
and $g_{s,1}(x,y)=x$.

Similarly to the above estimates (3) for $f_{s}$ and $g_{s}$ we conclude that
$f_{s,1}\neq 0$ for $|y|\geq\frac{\epsilon}{2}$ and $x\neq 0$ and $g_{s,1}\neq
0$ $|y|\leq\frac{\epsilon}{2}$.

It remains to show existence of a family of Lyapunov functions for the family
of vector fields $Y_{s,t}.$ According to Corollary 3.4 it is sufficient to
verify for $Y_{s,t}$ the property (L1) and the Morse-Smale condition. Because
it is the 2-dimensional case then by Poincaré-Bendixson’s theorem it is
sufficient to show that there are no periodic orbits. But any periodic orbit
in $R$ bounds a disc and the sum of indices of singular points in this disc
should be equal to $1$. On the other hand, the only 2 singular points of
$Y_{s,t}$ are connected by a separatrix trajectory, and hence the disc bounded
by a periodic orbit must enclose both singular points, whose sum of indices is
equal to $0$. ∎

###  6.2. Special 2-dimensional plug

Figure 6. Graphs of functions $\Theta$, $\Psi_{1}$, and $\Psi_{2}$.

We construct in this section a special 2-dimensional plug. In the contact
space $(\mathbb{R}^{3},\mathrm{Ker\,}\\{dz+xdy\\})$ consider

$O:=\\{0\leq y,z\leq 1,-4\leq x\leq 0\\},\;\;R:=O\cap\\{x=0\\}.$

For a sufficiently small $\epsilon$ let us choose non-decreasing
$C^{\infty}$-functions $\psi,\theta:[0,1]\to\mathbb{R}$ such that

  * •

$\theta(y)=\epsilon$ for $y\in[0,\frac{1}{3}]$; $\theta(y)=1-\epsilon$ for
$y\in[\frac{2}{3},1]$ and $0<\theta^{\prime}(y)<4$ for
$y\in(\frac{1}{3},\frac{2}{3})$.

  * •

$\psi(0)=0$, $\psi_{1}(1)<\frac{\epsilon}{2}$;

  * •

$\psi$ has vanishing derivatives of all orders at the points $0$ and $1$, and
$0<\psi^{\prime}(y)<\epsilon$ for $y\in(0,1)$.

Denote

$\psi_{1}:=\psi+\frac{\epsilon}{2},\;\psi_{2}:=-\psi+(1-\psi(1)-\frac{\epsilon}{2}).$

and consider the graphs $\Theta,\Psi_{1},\Psi_{2}$ of the functions
$\theta,\psi_{1}$ and $\psi_{2}$:

$\displaystyle\Theta:=\\{(y,\theta(y));\;y\in[0,1]\\},\;\Psi_{1}:=\\{(y,\psi_{1}(y));\;y\in[0,1]\\},$
$\displaystyle\Psi_{2}:=\\{(y,\psi_{2}(y));\;y\in[0,1]\\}\subset R.$

Denote

$R_{+}:=\\{(y,z)\in R,z\leq\theta(y)\\},\;R_{-}:=\\{(y,z)\in
R,z\geq\theta(y)\\}.$

######  Lemma 6.4.

For any $\epsilon>0$ there exists an isotopy $h_{s}:R\to O$, $s\in[0,1]$,
which is fixed together with its $\infty$-jet along $\partial R$, constant for
$s\in[0,\frac{1}{8}]\cup[\frac{7}{8},1]$, and such that the following
properties a)–i) are satisfied. Denote $\beta_{s}:=h_{s}^{*}(dz+xdy)$. Let
$Y_{s}$ be the vector field directing the characteristic foliation $\ell_{s}$
of $\beta_{s}$.

  * a)

$d\beta$ restricted to the interior $\mathrm{Int\,}R_{+}$ of $R_{+}$ is
positive, and $d\beta$ restricted to the interior $\mathrm{Int\,}R_{-}$ of
$R_{-}$ is negative for all $s\in(\frac{1}{8},1]$;

  * b)

$\beta_{s}$ has

    * –

no zeros for $s<\frac{3}{4}$,

    * –

a positive and negative embryos
$o_{+}:=(\frac{1}{6},\psi_{1}(\frac{1}{6})),o_{-}=(\frac{5}{6},\psi_{2}(\frac{5}{6}))$
for $s=\frac{3}{4}$,

    * –

a pair

$\displaystyle
e_{+}(s)=(e_{+}^{1}(s),\psi_{1}(e_{+}^{1}(s))),\hbar_{+}(s)=(\hbar_{+}^{1}(s),\psi_{1}(\hbar_{+}^{1}(s))),\;0<e_{+}^{1}(s)<\hbar_{+}^{1}(s)<\frac{1}{3}$

of positive elliptic and hyperbolic points, and a pair

$\displaystyle
e_{-}(s)=(e_{-}^{1}(s),\psi_{1}(e_{-}^{1}(s))),\hbar_{-}(s)=(\hbar_{-}^{1}(s),\psi_{1}(\hbar_{-}^{1}(s))),\;1>e_{+}^{1}(s)>\hbar_{+}^{1}(s)>\frac{2}{3}$

of negative elliptic and hyperbolic points for $s>\frac{3}{4}$.

  * c)

the incoming separatrices of $\hbar_{+}(s)$ and $o_{+}$ for $Y_{s}$, are
contained in $\Psi_{1}$, and outgoing separatrices of $\hbar_{-}(s)$ and
$o_{-}$ for $Y_{s}$ are contained in $\Psi_{2}$, $s\geq\frac{3}{4}$;

  * d)

there exists $\epsilon_{1}\in(0,\epsilon)$ such that the outgoing separatrices
of $\hbar_{+}(s)$ for $Y_{1}$ terminate at $e_{-}$ and $(1,\epsilon_{1})$, and
the incoming separatrices of $\hbar_{-}(s)$ for $Y_{1}$ originate at $e_{+}$
and $(0,1-\epsilon_{1})$;

  * f)

$Y_{s}$ for $s\in[\frac{1}{2},1]$ is outward transverse to the graph $\Theta$,
viewed as a part of the boundary of the domain $R_{+}$;

  * g)

$Y_{s}$ admits a family of good Lyapunov function $\psi_{s}:R\to\mathbb{R}$
such that $\psi_{s}|_{{\mathcal{O}}{\it p}\,\partial R}=y$;

  * h)

$\beta_{s}(\frac{\partial}{\partial z})>0$ everywhere in $R$ for
$s\in[\frac{1}{2},1]$;

  * i)

for any $\sigma>0$ the isotopy $h_{s}$, $s\in[0,1]$, can be extended to a
2-parametric isotopy $h_{s,t},0\leq s,t\leq 1$, such that

    * –

$h_{s,0}=h_{s}$, $h_{0,t}=h_{0}$ for all $s,t\in[0,1]$;

    * –

$h_{s,t}=h_{s}$ for $s\leq\frac{3}{4}-\sigma,t\in[0,1]$;

    * –

for each $s\in(\frac{3}{4},1]$ the isotopy $h_{s,t},t\in[0,1]$, is supported
in a $\sigma$-neighborhood of the separatrices connecting $\hbar_{\pm}(s)$
with $e_{\pm}(s)$; for each $s\in[\frac{3}{4}-\sigma,1]$ $h_{s,t}$ is
supported in a $\sigma$-neighborhood of $o_{\pm}$;

    * –

$h_{s,t}$ is $\sigma$-close in the $C^{0}$-sense to $h_{s,0}$ for all
$s,t\in[0,1]$;

    * –

the family of vector fields $Y_{s,t}$ directing the characteristic foliations
$\ell_{s,t}$ of $\beta_{s,t}:=h_{s,t}^{*}\alpha$ admits a family good Lyapunov
functions $\psi_{s,t}:R\to\mathbb{R}$ such that $h_{s,t}|_{{\mathcal{O}}{\it
p}\,\partial R}=y$;

    * –

$Y_{s,t}$ has a pair of positive elliptic and hyperbolic zeroes at
$e_{\pm}(s(1-2t)+\frac{3t}{2})$ and $\hbar_{\pm}(s(1-2t)+\frac{3t}{2})$ for
$s>\frac{3}{4},t<\frac{1}{2}$, pairs of embryos at $o_{\pm}$ for
$s=\frac{3}{4},t=\frac{1}{2}$ and no zeroes otherwise;

    * –

$h_{s,t}=h_{0}$ for $s\in[0,\frac{1}{8}],t\in[0,1],$ and $h_{s,t}=h_{1,t}$ for
$s\in[\frac{7}{8},1]$ $t\in[0,1]$;

    * –

$X_{s,t}$ is outwardly transverse to $\Theta$ for all
$s\in[\frac{1}{2},1],\;t\in[0,1]$.

Figure 7. The characteristic foliation on $h_{\frac{4}{5}}(R).$ The red curve
is the dividing set $\Gamma=\\{d\mu=0\\}$ while blue curves depict
separatrices connecting $e^{\pm}$ to $h^{\mp}$.

###### Proof.

Choose a function $H:R\to\mathbb{R}$ such that

  * (C1)

$H$ vanishes on $\partial R$ together with all its derivatives;

  * (C2)

$0\geq H(y,z)\geq-4$, $y,z\in[0,1]$;

  * (C3)

$H(y,\psi_{1}(y))=H(y,\psi_{2}(y))=-\psi^{\prime}(y).$

  * (C4)

$\frac{\partial H}{\partial z}(y,z)=\begin{cases}<0,&(y,z)\in R_{+}\\\
>0,&(y,z)\in R_{-};\end{cases}$

  * (C5)

$H(y,\theta(y))\leq-\theta^{\prime}(y)$.

An additional property (C6) will be imposed later.

Define an isotopy $h_{s}:R\to\mathbb{R}$, $s\in[0,1]$, as follows. For
$s\in[0,1/2]$ we define

$h_{s}(y,z):=(y,z,2sH(y,z)),\;\;(y,z)\in R.$

Let $\ell_{s}$ be the characteristic foliation defined by
$h_{s}^{*}\widehat{\mu}$ on $R$. Leaves of the characteristic foliation on
$\ell_{s}$ are graphs of solutions of the equation

(4) $\frac{dz}{dy}=-2sH(y,z).$

For $\zeta\in[0,1]$ we denote by $\ell_{\zeta}$ and $\ell^{\zeta}$ the
solutions of (4) with the initial data $\ell_{\zeta}(0)=\zeta$ and
$\ell^{\zeta}(3)=\zeta$, respectively. Condition (C3) ensures that
$\ell_{\frac{\epsilon}{2}}=\psi_{1}$, $\ell^{\frac{\epsilon}{2}}=\psi_{2}$.

For $s\in[1/2,1]$ we use Lemma 6.1 to create for $s>\frac{3}{4}$ pairs of
elliptic-hyperbolic positive and negative points at
$e_{\pm}(s),\hbar_{\pm}(s)$ through embryos at $o_{\pm}$ for $s=\frac{3}{4}$.
The isotopy can be constructed arbitrary $C^{0}$-small and supported in a
neighborhood of separatrices connecting $e_{\pm}$ and $\hbar_{\pm}$. It can
also be arranged that the isotopy also fixes the leaves $\Psi_{1}$ and
$\Psi_{2}$ of the foliation $\ell_{\frac{1}{2}}$, so that these leaves become
broken leaves of the characteristic foliation $\ell_{s}$ for
$s\in[\frac{3}{4},1]$. In particular, these curves contain, respectively, the
incoming separatrix of $\hbar_{+}(s)$ and outgoing separatrix of
$\hbar_{-}(s)$. It then follows that one of the outgoing separatrices of
$\hbar_{+}(s)$ terminates at $(1,\epsilon_{1})$ for $\epsilon_{1}<\epsilon$,
and it could be arranged that one of the incoming separatrices of $\hbar_{-}$
originates at $(1,1-\epsilon_{1})$.

Suppose that the second outgoing separatrix of $\hbar_{+}(1)$ intersects the
line $y=\frac{1}{3}$ at a point $(\frac{1}{3},a)$,
$a\in(\psi_{1}(\frac{1}{3}),\theta(\frac{1}{3}))$, while the second incoming
separatrix of $\hbar_{-}(1)$ intersects the line $y=\frac{2}{3}$ at a point
$(\frac{2}{3},b)$, $b\in(\theta(\frac{2}{3}),\psi_{2}(\frac{2}{3}))$. We now
impose the remaining condition on the function $H$:

  * (C6)

$H(y,z)<-3$ for $y\in[1,2],z\in[a,b]$;

This guarantees that one of the outgoing separatrices of $\hbar_{+}(1)$
terminates at $e_{-}(1)$, and one of the incoming separatrices of
$\hbar_{-}(1)$ originates at $e_{+}(1)$.

Using the extension to the 2-parametric isotopy in Lemma 6.1 we extend the
isotopy $h_{s}$ to a 2-parametric isotopy $h_{s,t}$ for $s,t\in[0,1]$ with the
required properties. ∎

Let us denote by $\Gamma_{\pm}(s)$, $s\in[\frac{3}{4},1]$ the (closure of the)
trajectory of $Y_{s}$ connecting $e_{\pm}(s)$ and $\hbar_{\pm}(s)$ and by
$\Delta_{+}(s)$ (resp. $\Delta_{-}(s))$ the (closure of the) incoming (resp.
outgoing) separatrix of $\hbar_{+}(s)$ (resp. $\hbar_{-}(s)$). For
$s=\frac{3}{4}$ we assume that
$e_{\pm}(\frac{3}{4})=\hbar_{\pm}(\frac{3}{4})=o_{\pm}$. We extend the
definition of $\Gamma_{\pm}(s)$ and $\Delta_{\pm}(s)$ to all $s\in[0,1]$ by
setting $\Gamma_{\pm}(s)=\Delta_{\pm}(s)=\varnothing$ for $s<\frac{3}{4}$.

##  7\. Preliminary and quasi-plugs

###  7.1. Preliminary plug

Let $(W,\lambda)$ be a Weinstein domain. We denote by $Z$ the Liouville field
dual to $\lambda$. Consider an interior boundary collar $C:=\partial
W\times[1-\epsilon,1]$ such that $\partial W\times 1=\partial W$ and
$\lambda|_{C}=\tau\gamma$, $\tau\in[1-\epsilon,1],$ for a contact form
$\gamma=\lambda|_{\partial W}$. Denote $W_{0}:=W\setminus C$. Furthermore,
denote by $\mathrm{Skel}(W,Z)$ the skeleton of $W$, i.e. the union of stable
manifolds of zeroes of $Z$. Alternatively,
$\mathrm{Skel}(W,Z)=\bigcap\limits_{s\in[0,\infty)}Z^{-s}(W)$. Here we denote
by $Z^{-s}$ the flow of $-Z$, which is defined for all $s\geq 0$.

Consider a contact manifold $(V:=W\times[0,1],\mathrm{Ker\,}(\lambda+dz))$,
and in $(V\times T^{*}[0,1],\mathrm{Ker\,}(\lambda+dz+xdy))$ take the domain
$U=\\{-4\leq x\leq 0\\}$ and a hypersurface $Q=\\{x=0\\}$. Note that
$Q=V\times[0,1]$, and we can naturally identify $V\times T^{*}[0,1]$ and $Q$
with $W\times O$ and $W\times R$ respectively, where we use the notation $O,R$
introduced above in Section 6.2.

Let $h_{s,t}:R\to O$ be the isotopy constructed in Lemma 6.4. Define an
isotopy

$g_{s,t}:Q=W\times R\to U=W\times O$

by the formula

$\displaystyle g_{s,t}(w,q)=\begin{cases}(w,h_{s,t}(q)),&w\in W_{0},q\in R\\\
(w,h_{s\overline{\tau},t}(q)),&w=(v,\tau)\in C=\partial
W\times[1-\epsilon.\epsilon],\;\overline{\tau}=\frac{1-\tau}{\epsilon}\end{cases}$

Denote $\widehat{\beta}_{s,t}:=g_{s,t}^{*}(\lambda+dz+xdy)$ and let $X_{s,t}$
be the vector field directing the characteristic foliation defined by the form
$\widehat{\beta}_{s,t}$. Set $\widehat{\beta}_{s}:=\widehat{\beta}_{s,0}$ and
$X_{s}:=X_{s,0}$. It follows from the corresponding properties of the isotopy
$h_{s,t}$ that for each fixed $s\in[0,1]$ the isotopy $g_{s,t}$, $t\in[0,1]$,
is $\sigma$-close to $g_{s}$ in the $C^{0}$-sense.

Denote

$\displaystyle{}^{in}V:=V\times 0\subset Q,\;\;{}^{out}V:=V\times 1\subset Q,$
$\displaystyle{}^{in}P:=W_{0}\times[\epsilon,1-\frac{\epsilon}{2}]\times
0\subset{}^{in}V,\;\;{}^{out}P:=W_{0}\times[\epsilon_{1},1-\frac{\epsilon}{2}]\times
1\subset{}^{out}V,$
$\displaystyle{}^{in}T:=W_{0}\times(1-\frac{\epsilon}{2},1]\times
0\subset{}^{in}V,\;\;{}^{out}T:=W_{0}\times[0,\epsilon_{1})\times
1\subset{}^{out}V.$

Furthermore, denote

$\widehat{\Gamma}_{\pm}:=W_{0}\times\Gamma_{\pm}(1)\cup\bigcup\limits_{u\in\partial
W,\;1-\epsilon\leq\tau\leq
1}(u,\tau)\times\Gamma_{\pm}({\overline{\tau}})\subset
Q,\;\overline{\tau}=\frac{1-\tau}{\epsilon}.$

Figure 8. Regions in the preliminary block with controlled dynamics.

The following proposition lists the necessary for our application properties
of the family of vector field $X_{s,t}$.

######  Proposition 7.1.

The vector field $X_{s,t}$ on $Q$, $s,t\in[0,1]$, has the following
properties.

  1. (1)

every trajectory of $X_{1}$

    1. (a)

which starts at ${}^{in}P$ converges to a zero of $X_{1}$;

    2. (b)

which starts at a point $(u,z,0)\in{}^{in}T$ exits at a point
$(u^{\prime},z^{\prime},1)\in V$, then $u^{\prime}=Z^{-a}(u),a>0$ and
$z^{\prime}\in(1-\frac{\epsilon}{2},1]$;

    3. (c)

which ends ${}^{out}P$ originates at a zero of $X_{1}$;

    4. (d)

which terminates at a point $(u,z,1)\in{}^{out}T$ begins at a point
$(u^{\prime},z^{\prime},0)$ such that $u^{\prime}=Z^{-a}(u),a>0$, and
$z^{\prime}\in[0,\frac{\epsilon}{2})$;

  2. (2)

vector fields $X_{s,1}$, $s\in[0,1]$, have no zeroes;

  3. (3)

$\widehat{\Gamma}_{+}$ is invariant with respect to the negative flow of
$X_{1}$, and $\widehat{\Gamma}_{-}$ is invariant with respect to the positive
one;

  4. (4)

All trajectories of $X_{s}$ which converge to positive singularities either do
not intersect $\partial Q$ and are contained in $\widehat{\Gamma}_{+}$, or
intersect it at points of
$\mathrm{Skel}(W)\times\frac{\epsilon}{2}\subset{}^{in}V$; all trajectories of
$X_{s}$ which originate at negative singularities either do not intersect
$\partial Q$ and are contained in $\widehat{\Gamma}_{-}$, or intersect it at
points of $\mathrm{Skel}(W)\times(1-\frac{\epsilon}{2})\subset{}^{out}V;$

  5. (5)

the family of vector fields $X_{s,t}$ admits a family of good Lyapunov
functions $\psi_{s,t}$, equal to $y$ on $\partial Q$;

We begin the proof with an explicit computation of the vector field $X_{s,t}$.
Let $Z$ denote the Liouville field on $W$ corresponding to the Liouville form
$\lambda$. Recall that $\lambda|_{C}=\tau\gamma$, $\tau\in[1-\epsilon,1]$,
where $\gamma$ is a contact form on $\partial W$. We have
$Z|_{C}=\tau\frac{\partial}{\partial\tau}.$ Let $R$ be the Reeb vector field
on $\partial W$ lifted to $C=\partial W\times[1-\epsilon,\epsilon]$ via the
projection to the first factor.

Let us view $Q=W\times R$ as the fiber bundle over $W$. The form
$\widehat{\beta}_{s,t}$ restricts to the fiber $w\times Q,\;w\in W$ as
$\beta_{s,t}$ if $w\in W_{0}$ and for $w=(u,\tau)\in C=\partial
W\times[1-\epsilon,1]$ as $\beta_{s\overline{\tau},t},$ where
$\overline{\tau}=\frac{1-\tau}{\epsilon}$. To simplify the notation we will
write $\widehat{\mu}$ instead of $\widehat{\beta}_{s,t}$, $X$ instead of
$X_{s,t}$, $\mu$ instead of $\beta_{s,t}$ and $\mu_{\tau}$ instead of
$\beta_{s\overline{\tau},t}$. Denote $\dot{\mu}:=\frac{d\mu_{\tau}}{d\tau}$.

For each $\tau\in[1-\epsilon,1]$ consider a form
$\kappa_{\tau}:=\mu_{\tau}-\tau\dot{\mu}_{\tau}$ on $Q$ and choose a vector
field $Y_{\tau}$ directing $\mathrm{Ker\,}\,\kappa_{\tau}$.

Consider a tangent to fibers $w\times Q,w\in W$, vector field $Y$ which is
equal to $X_{s,t}$ on $w\times Q$ for $w\in W_{0}$ and to $Y_{\tau}$ on
$w\times Q$ for $w=(u,\tau)\in C=\partial W\times[1-\epsilon,1]$.

######  Lemma 7.2.

We have

  * 1.

$X=Y+aZ$ over $W_{0}\times Q$, where the function $a:Q\to\mathbb{R}$ is
determined by the equation

(5) $\displaystyle a\mu=\iota(Y)d\mu.$

  * 2.

$X=Y+aZ+bR$ over $C\times Q$, where the functions $a,b:Q\to\mathbb{R}$ are
determined by the equations

(6) $\begin{split}&a\kappa_{\tau}=\iota(Y_{\tau})d\mu_{\tau},\\\
&b\tau+\mu_{\tau}(Y_{\tau})=0.\end{split}$

######  Remark 7.3.

Note that for any vector $v$ and a symplectic form $\omega$ we have
$\iota(v)\omega(v)=0$. Hence, the equation (5) and the first equation (6) are
always solvable for some function $a:Q\to\mathbb{R}$.

###### Proof.

1\. According to Remark 7.3 we can solve equation (5) with respect to a
function $a$. We have

$\iota(aZ+Y)d(\mu+\lambda)=\iota(Y)d\mu+a\iota(Z)d\lambda=a(\mu+\lambda).$

But this means that $aZ+Y$ is tangent to the characteristic foliation defined
by the form $\mu+\lambda=\widehat{\mu}$.

2\. Note that
$d\widehat{\mu}=d\mu_{\tau}+d(\tau\gamma)=d\tau\wedge\dot{\mu_{\tau}}+d\mu_{\tau}+d\tau\wedge\gamma+\tau
d\gamma.$ Hence, we get that

$\displaystyle\iota(Y+aZ+bR)d\widehat{\mu}=-(\dot{\mu}(Y_{\tau})+b)d\tau+a\mu+a\tau\gamma=a(\mu+\tau\gamma),$

where we used the second equation (6) to conclude that
$\dot{\mu}(Y_{\tau})+b=0$. But this implies that

$\widehat{\mu}\wedge\left(\iota(Y+aZ+bR)d\widehat{\mu}\right)=0,$

which means that $Y+aZ+bR$ generates the characteristic foliation on $C\times
Q$ defined by the form $\widehat{\mu}$, as required.

∎

###### Proof of Proposition 7.1.

(1a) When a trajectory of $X_{1}$ which enters at a point of ${}^{in}P$ it is
in the region where $X_{1}=Y_{1}+aZ$ with $a<0$. Since $a$ is negative the
trajectory continues to remain in the region where the plug is given by
$W_{0}\times R,$ and hence projects onto trajectories of $-Z$ and $Y$, when
projected to the corresponding factors. Moreover, the projection to $R$
remains in $R_{-}$, according to Property f) of Lemma 6.4, and therefore
remains in the region where the coefficient $a$ is negative. But in $R_{-}$
any trajectory of $Y_{1}$ entering at a point in
$[\epsilon,1-\epsilon_{1})\times 0$ converges to a negative zero of $Y_{1}$,
while every trajectory of $-Z$ converges to a zero of $Z$. Hence, any
trajectory of $Y_{1}$ entering through ${}^{in}P$ converges to a $0$ of the
vector field $X_{1}$.

(1b) If a trajectory enters at a $(u,z)\in{}^{in}T$ then similarly to 1a) we
have $X_{1}=Y_{1}+aZ$ for a negative coefficient $a$, and therefore, it
projects onto trajectories of $-Z$ and $Y_{1}$ in the factors $W$ and $R$. But
every trajectory of $Y_{1}$ which enters through $(1-\epsilon_{1},1]\times 0$
exits $R$ at a point of $[1-\frac{\epsilon}{2},1]$, and hence the
corresponding trajectory of $X_{1}$ exits ${}^{out}V$ at a point
$(u^{\prime}=Z^{-c}(u),z^{\prime})$ for some positive $c>0$ and
$z^{\prime}\in(-\frac{\epsilon}{2}.1]$.

(1c) and (1d) follows from the same arguments as, respectively, (1a) and (1b)
applied to the vector field $-X_{1}$.

Properties (2)-(4) are straightforward from the corresponding properties in
Lemma 6.4 and Lemma 7.2.

(5) According to Corollary 3.4 it is sufficient to verify that

  * •

each trajectory of $X_{s,t}$ either originates at $V\times 0$, or at a
critical point of $X_{s,t}$;

  * •

each trajectories of $X_{s,t}$ either terminates at $V\times 1$, or at a
critical point of $X_{s,t}$;

  * •

$X_{s,t}$ has no retrograde connections.

In addition to the Weinstein subdomain $W_{0}=W\setminus\partial
W\times(1-\epsilon,1]$ consider also a larger subdomain
$W_{1}=W\setminus\partial W\times(1-\frac{\epsilon}{2},1]$. Denote
$Q_{\pm}:=W_{1}\times R_{\pm},Q_{b}:=(W\setminus W_{1})\times R\subset Q.$ We
have $Q=Q_{+}\cup Q_{-}\cup Q_{b}$. Let us first analyze the forward
trajectory $X_{s,t}^{u}(p)=(w(u),r(u))$, $u\in\mathbb{R}_{+}$ of a point
$p=(w,r)\in Q$.

If $p\in Q_{-}$ and $r\in W_{0}$ then $w(u)$ belongs to the negative Liouville
trajectory $\bigcup\limits_{\tau\geq 0}Z^{-\tau}(w)$ as long as $r(u)\in
R_{-}$. Similarly, if $w\in W_{1}\setminus W_{0}=\partial
W\times[1-\epsilon,1-\frac{\epsilon}{2}]$ the second coordinate of $w$
decreases as long as $w(u)\in R_{-}$. But the $Q$-component $Y_{s,t}$ of
$X_{s,t}$ is by construction inwardly transverse to the boundary of $R_{-}$,
and hence, remains in $R_{-}$ for all $u\geq 0$. Therefore, the trajectory
either converges to a singular point of $X_{s,t}$, or exits through $V\times
1$.

Suppose $p\in Q_{b}$. Recall that the $Q$-component $Y_{s,t}$ of $X_{s,t}$ has
in $Q_{b}$ a positive projection to the $y$-direction. Hence, $X_{s,t}^{u}(p)$
either exits through $V\times 1$, or enters $Q_{+}\cup Q_{-}$. But in that
case it only can enter $Q_{-}$, and therefore, the analysis of the previous
case does apply.

Suppose now that $p\in Q_{+}$. If $w\in W_{0}$ then $w(u)$ moves along a
positive Liouville trajectory of $w$, and if $w\in\partial
W\times[1-\epsilon,1-\frac{\epsilon}{2}]$, then the second coordinate of
$w(u)$ increases as long as $r(u)\in R_{+}$. Hence, the trajectory either
exits through $V\times 1$, or enters either $Q_{b}$ or $Q_{-}$, and therefore,
the previous analysis applies.

The incoming trajectories could be analyzed similarly, with exchanging $Q_{+}$
and $Q_{-}$ cases. The absence of retrograde connections follow from (4).

∎

###  7.2. Approximating balls by Weinstein cylinders

A hypersurface $\Sigma\subset(M,\xi)$ is said to have an admissible corner
along a smooth hypersurface $S\subset\Sigma$ if

  * •

$S$ is a codimension $2$ contact submanifold of $(M,\xi)$;

  * •

$\Sigma\cap{\mathcal{O}}{\it p}\,S=\Sigma_{1}\cup\Sigma_{2}$, where
$\Sigma_{1}$ and $\Sigma_{2}$ are two manifolds with common boundary $S$ which
transversely intersect along $S$.

We will call the hypersurface $S\subset\Sigma$ the corner locus of $\Sigma$
and denoted by $\mathrm{Corner}(\Sigma)$.

Suppose $\widetilde{\Sigma}$ is a smooth hypersurface, let us choose its
tubular $\epsilon$-neighborhood $N$ and denote by $\widehat{\tau}$ the
hyperplane field on $N$ orthogonal to the fibers of the projection
$N\to\widetilde{\Sigma}.$ We say that a hypersurface with admissible corners
$\Sigma\subset N$ is $C^{1}$ $\epsilon$-close to $\widetilde{\Sigma}$ if all
its tangent planes do not deviate for more than $\epsilon$ from
$\widehat{\tau}$.

Given a Weinstein domain $(W,\lambda)$, a domain $U$ in a contact manifold
$(M,\xi)$ is called a Weinstein cylinder if there is given a contactomorphism
$\phi:(W\times[0,a],\mathrm{Ker\,}(\lambda+dz))\to(U,\xi|_{U})$. We already
encountered Weinstein cylinders in Proposition 7.1.

Note that the boundary of a Weinstein cylinder $\phi(W\times[0,a])$ is a
hypersurface with admissible corners along $\mathrm{Corner}(U)=\partial
W\times 0\cup\partial W\times a$. We denote $\partial_{-}U:=\phi(W\times
0),\;\partial_{+}U:=\phi(W\times a)$.

For a general $W$ the contact topology of the Weinstein cylinder
$W\times[0,a]$ is very sensitive to the value of the parameter $a$. However,
there is one exception (see e.g. [8]):

######  Lemma 7.4.

Let $D=D^{2n-2}$ be the unit ball on $\mathbb{R}^{2n-2}$ endowed with the
Liouville form
$\lambda_{\mathrm{st}}:=\sum\limits_{1}^{n-1}(x_{j}dy_{j}-y_{j}dx_{j})$. Then
for any $a>0$ there is a contactomorphism

$\Delta_{a}:=(D\times[0,a],\mathrm{Ker\,}(\lambda+dz))\to\Delta_{1}:=(D\times[0,1],\mathrm{Ker\,}(\lambda+dz)).$

Hence, we will use the notation $\Delta$ for any Weinstein cylinder of the
type $D\times[0,a].$

######  Lemma 7.5.

Let $D$ be the standard contact ball and $p_{\pm}\in\partial D$ its poles.
Then for any $\epsilon>0$ there exists a contact embedding $h:\Delta\to D$
such that $h(\partial_{+}\Delta)\subset\partial D$, $\partial D\setminus
h(\partial_{+}\Delta)$ is contained in an $\epsilon$-neighborhood of the pole
$p_{-}$ and $D\setminus\Delta$ is contained in the $\epsilon$-neighborhood of
$\partial D$.

###### Proof.

Consider a $(2n-2)$-dimensional open disc $B_{-}\subset\partial D$ of radius
$\epsilon$ centered at $p_{-}$ with boundary $\partial B_{-}$ transverse to
the characteristic foliation $\ell_{\partial D}$. Denote $D_{+}:=\partial
D\setminus B_{-}$ By scaling the contact form along $\partial D$ we can
arrange that $D_{+}$ with the resulted form is the standard Liouville ball,
and flowing for some time $\delta$ with the corresponding Reeb field
$\Upsilon$, inwardly transverse to $D_{+}$, we construct a Weinstein cylinder
$\Delta=\bigcup\limits_{t\in[0,\delta]}\Upsilon^{t}(D_{+})\subset D$. Let
$D^{\prime},D^{\prime\prime}\subset D$ be smaller standard contact balls such
that $D^{\prime}\subset\mathrm{Int\,}\Delta$,
$D^{\prime}\subset\mathrm{Int\,}D^{\prime\prime}$ and $D\setminus
D^{\prime\prime}$ is in an $\epsilon$-neighborhood of $\partial D$. Note that
the space of contact embeddings of a standard contact ball into any connected
contact manifold is connected. Hence, there exists a contact diffeotopy
$h_{t}:D\to D$, $t\in[0,1]$, which is fixed on ${\mathcal{O}}{\it p}\,\partial
D$, and such that $h_{1}(D^{\prime})=D^{\prime\prime}$. Then the Weinstein
cylinder $h_{1}(\Delta)$ has the required properties. ∎

###  7.3. Weinstein cylinders in a good position

We say that Weinstein cylinders $V_{1},\dots,V_{k}$ are in a good position,
see Fig. 9, if

  * •

$V_{j}\cap V_{j+2}=\varnothing$ for all $j=1,\dots,k-2$;

  * •

$\partial V_{1}\setminus V_{2}\subset\mathrm{Int\,}\,\partial_{-}V_{1}$;

  * •

for each $j=2,\dots,k$ we have $\partial V_{j}\setminus
V_{j-1}\subset\mathrm{Int\,}\,\partial_{+}U_{j}$;

  * •

for each $j=2,\dots,k-q$ we have $\partial V_{j}\setminus
V_{j+1}\subset\mathrm{Int\,}\,\partial_{+}V_{j}$;

  * •

$\partial V_{j}$ and $\partial V_{j+1}$, $j=1,\dots,k-1$, intersect
transversely along a codimension 2 contact submanifold $S_{j}$, and the
orientations induced on $S_{j}$ from $\partial
V_{j+1}\setminus\mathrm{Int\,}V_{j}$ and from $\partial
V_{j}\setminus\mathrm{Int\,}V_{j+1}$ are opposite.

Figure 9. Three Weinstein cylinders $V_{1}=W_{1}\times I,$ $V_{2}=W_{2}\times
I,$ and $V_{3}=W_{3}\times I$ in a good position. The blue region is
${}^{out}T_{1}$, while the green ones are ${}^{out}T_{2}$ and ${}^{out}T_{3}$.

Note that if $V_{1},\dots,V_{k}$ are in a good position then
$\partial\left(\bigcup\limits_{1}^{k}V_{i}\right)$ is a piecewise smooth
hypersurface with admissible corners, and it can be made Weinstein convex by a
$C^{\infty}$-small perturbation.

Let $V\subset(M,\xi)$ be a domain diffeomorphic to a closed ball with a
Weinstein convex boundary $\partial V$. Let $S\subset\partial V$ be a dividing
set. We say that $(V,S)$ can be approximated by standard contact balls if
there exists a neighborhoood $N$ of $S$ in $M$ such that for every $\sigma>0$
there is a (iso-)contact embedding $g:D\to(M,\xi)$ of the standard contact
ball such that

  * -

$g(\partial D)$ is contained in a $\sigma$-neighborhood of $\partial V$;

  * -

$g(\partial D)\cap N=\partial V\cap N$.

######  Proposition 7.6.

Let $V\subset(M,\xi)$ be a domain with a Weinstein convex boundary $\partial
V$ and $S\subset\partial V$ be a dividing set. Suppose that $(V,S)$ can be
approximated by standard contact balls. Then for any $\epsilon>0$ there exist
three Weinstein cylinders $V_{1},V_{2},V_{3}\subset V$ in a good position such
that a piecewise smooth hypersurface $\partial(V_{1}\cup V_{2}\cup V_{3})$ is
$C^{1}$ $\epsilon$-close to $\partial V$, see Fig. 10.

Figure 10. Three Weinstein cylinders in good position approximating a domain
which admits an approximation by standard balls.

###### Proof.

Choose an inwardly pointing transverse to $\Sigma:=\partial V$ contact vector
field and consider an interior collar $C:=\partial\Sigma\times[0,1]\subset V$
such that $\Sigma\times 0=\Sigma$, $x\times[0,1],x\in\Sigma$ are trajectories
of $\Upsilon$ and $\Upsilon^{t}(\Sigma)=\Sigma\times t$. By scaling $X$ we can
assume that $C$ is contained in an $\epsilon$-small neighborhood of $\Sigma$.
Choose an $\frac{\epsilon}{4}$-approximation of
$(X^{\frac{1}{2}}(V),X{\frac{1}{2}}(S))$ by a standard ball $h(D)$. By
assumption the standard sphere $\widetilde{\Sigma}:=h(\partial D)$ coincides
with $\Sigma_{\frac{1}{2}}:=\Sigma\times\frac{1}{2}$ along a tubular
neighborhood $N=S\times[-1,1]\subset\Sigma_{\frac{1}{2}}$. We can assume that
$S\times t\subset N$ is transverse to the characteristic foliation on
$\Sigma_{\frac{1}{2}}$ for all $t\in[-1,1]$. Take a $C^{\infty}$-function
$\theta:[0,1]\to\mathbb{R}$ which is equal to $1-u$ for
$u\in[\frac{1}{4},\frac{3}{4}]$, equal to $0$ near $1$ and has non-positive
derivative everywhere. Let $\Theta:\Sigma\to\mathbb{R}$ be a function
supported in $N$ and defined on $N$ by the formula
$\Theta(x,u)=\theta(|u|),\;(x,u)\in N=S\times[-1,1].$ For
$\sigma\in(0,\frac{1}{2})$ define an isotopy $g_{s}:\Sigma\to V$:

$g_{s}(w)=\Upsilon^{-s\Theta(u)}(u),\;u\in\Sigma,s\in[0,\sigma].$

If $\sigma$ is chosen sufficiently small then the spheres
$g_{s}(\widetilde{\Sigma})$, $s\in[0,\sigma]$, are almost standard, and hence,
by Lemma 8.2 a $C^{0}$-small adjustment near one of the poles makes them
standard. Assuming this is done we can extend the isotopy to a global
compactly supported contact diffeotopy $G_{s}:V\to V,s\in[0,\sigma].$

The dividing set $S\subset\Sigma_{\frac{1}{2}}$ divides $\Sigma$ into domains
$\Sigma_{\frac{1}{2},\pm}$ with the common boundary $S$. Denote

$\displaystyle\widehat{\Sigma}_{+}:=\Sigma_{\frac{1}{2},+}\setminus\left(S\times\big{(}-\frac{1}{2},0\big{]}\right);$
$\displaystyle\widehat{\Sigma}_{-}:=\Sigma_{\frac{1}{2},-}\setminus\left(S\times\big{[}0,\frac{1}{2}\big{)}\right).$

Consider two Weinstein cylinders:
$\widehat{W}_{\pm}=\widehat{\Sigma}_{\pm}\times[-\frac{\sigma}{2},\frac{\sigma}{2}]=\bigcup\limits_{|s|\leq\frac{\sigma}{2}}\Upsilon^{s}(\widehat{\Sigma}_{\pm}),$
see Fig. 10. We have
$\partial_{\pm}\widehat{W}_{+}=\Upsilon^{\mp\frac{\sigma}{2}}(\widehat{\Sigma}_{+})$
and
$\partial_{\pm}\widehat{W}_{-}=\Upsilon^{\pm\frac{\sigma}{2}}(\widehat{\Sigma}_{-})$.
Consider the standard ball $\widehat{D}:=G_{\sigma}(h(D))\subset V$. The
standard sphere $\partial\widehat{D}$ transversely intersects
$\partial\widehat{W}_{+}$ along
$\Upsilon^{\frac{\sigma}{2}}(S\times(-\frac{1}{2}))\in\partial_{+}\widehat{W}_{+}$,
and transversely intersects $\partial\widehat{W}_{-}$ along
$\Upsilon^{-\frac{\sigma}{2}}(S\times\frac{1}{2})\in\partial_{-}\widehat{W}_{-}$.
Note that the South pole of the sphere $\partial\widehat{D}$ is contained in
$\mathrm{Int\,}\widehat{W}_{-}$. Hence, using Lemma 7.5 we can
$C^{1}$-approximate $\widehat{D}$ by a contact embedding $f:\Delta=D\times
I\to\widehat{D}$ such that $f(\mathrm{Corner}(\partial\Delta))$ is contained
in $\mathrm{Int\,}\widehat{W}_{-}$. It then follows that the Weinstein
cylinders $\widehat{W}_{+},f(\Delta)$ and $\widehat{W}_{-}$ are in good
position, while the boundary $\widehat{W}_{+}\cup
f(\Delta)\cup\widehat{W}_{-}$ $C^{1}$-approximates $\Sigma$. ∎

###  7.4. Quasi-plugs

Let $V\subset(M,\xi)$ be a domain whose boundary $\partial U$ is a
hypersurface with admissible corners. Denote $Q:=V\times[0,a]$ and let $y$ be
the coordinate corresponding to the second factor. Given sufficiently small
$\sigma>0$ and a vector field $Y$ on $Q$, we call $(Q,Y)$ a $\sigma$-quasi-
plug if the following conditions are satisfied:

  * QP1.

$Y$ coincides with $\frac{\partial}{\partial y}$ on ${\mathcal{O}}{\it
p}\,\partial Q$;

  * QP2.

$Y$ admits a Morse Lyapunov function which is equal to $y$ on
${\mathcal{O}}{\it p}\,\partial Q$;

  * QP3.

for any point $p\in V$ with $\mathrm{dist}(p,\partial V)>\sigma$ the
trajectory of $Y$ through $p\times 0$ converges to a critical point of $Y$;

  * QP4.

given any point $p\in V$ with $\mathrm{dist}(p,\partial V)\leq\sigma$, there
exists a point $p^{\prime}\in\partial V$ and a positive $u(p)$ such that the
trajectory of $Y$ through a point $p\times 0$ either converges to a critical
point of $Y$, or exit $Q$ at a point $p^{\prime\prime}\times a$ with
$\mathrm{dist}(p^{\prime\prime},Y^{u(p)}p^{\prime})<\sigma.$

We will use quasi-plugs in combination with the following simple observation.

######  Lemma 7.7.

Suppose that the characteristic foliation on $\partial V$ is $\sigma$-short.
Then any $\sigma$-quasi-plug is a $3\sigma$-plug.

######  Lemma 7.8.

Let $V_{1},\dots,V_{k}$, $k\geq 2$ be $k$ Weinstein cylinders in a good
position. Denote $\widehat{V}=\bigcup\limits_{1}^{k}V_{j}$,
$\widehat{Q}:=\widehat{V}\times[0,k]$,
$Q_{j}:=V_{j}\times[j-1,j],\;j=1,\dots,k$. Denote by $Y_{j}$ the vector field
$X_{1}$ constructed in Proposition 7.1 and implanted to $Q_{j}$,
$j=1,\dots,k$. Let $Y$ be the resulted field on $\widehat{Q}$. Then
$(\widehat{Q},Y)$ is a $C\sigma$-quasi-plug for some constant $C$.

###### Proof.

For each $j=1,\dots,k$ we denote ${}^{in}P^{j},{}^{out}P^{j},{}^{in}U^{j}$,
${}^{out}U^{j}$, ${}^{in}T^{j}$ and ${}^{out}T^{j}$ the corresponding domains
defined in Proposition 7.1 for $Q_{j}$. Denote $\widehat{V}_{\leq
i}=\bigcup\limits_{1}^{i}V_{j}$, $\widehat{Q}_{\leq
i}:=\widehat{V}_{<k}\times[0,i]$. We will prove the following more precise
statement by induction in $i$:

$(\widehat{Q}_{\leq i},Y|_{\widehat{Q}_{\leq i}})$ is a $\sigma$-quasi-plug
for any $i\geq 2$. Moreover, any non-blocked trajectory entering
$\widehat{V}_{<k}\times 0$ exit through
${}^{out}U_{1}\cup\bigcup\limits_{2}^{i}T_{j}$.

Suppose $i=2$. For a sufficiently small $\sigma$ we have
${}^{out}V_{1}\setminus{}^{out}R_{1}\subset{}^{in}P_{2}$ and
${}^{in}V_{2}\setminus{}^{in}R_{2}\subset{}^{out}P_{1}$. This implies that not
blocked trajectories which enter ${}^{in}V^{1}\times 0$ either exit through
${}^{out}U^{1}\times 2$, or through ${}^{out}T^{2}\times 2$. In the former
case the exit point moves, possibly with a $\sigma$-error, in the positive
direction of the Liouville flow of $W_{1}$, and in the latter one in the
negative direction of the Liouville flow of $W_{2}$. which means that in both
cases they are moved forward along the characteristic foliation on
$\partial\widehat{V}$. Similarly, not blocked trajectories entering
$V^{2}\setminus V^{1}$ must enter through ${}^{in}U^{2}\times 0$ and exit
through ${}^{out}T_{2}\times 2$, and hence, also are moved forward by the
characteristic flow on $\partial\widehat{V}$.

Suppose now that the statement holds for $<i$ blocks. By the induction
assumption all the non-blocked trajectories which enter through $V_{\leq
i-1}\times 0$ exit through
$({}^{out}U_{1}\cup\bigcup\limits_{2}^{i-1}T_{j})\times(i-1)$. On the other
hand,
$V_{i}\cap(\bigcup\limits_{2}^{i-1}{}^{out}T_{j})\subset{}^{out}T_{i-1}\cap({}^{in}U_{i}\cup{}^{in}P_{i})$.
But all trajectories entering $Q_{i}$ through ${}^{in}P_{i}\times(i-1)$ are
blocked and those entering through ${}^{in}U_{i}\times(i-1)$ exit through
${}^{out}T_{i}$ and are pulled in the negative direction by the flow of
$Y_{i}$. Finally, trajectory entering $(V_{i}\setminus V_{\leq i-1})\times 0$
which are not blocked similarly exit through ${}^{out}T_{i}$ and are pulled in
the negative direction by the flow of $Y_{i}$.

∎

##  8\. From a quasi-plug to a $\sigma$-plug

As we already mentioned above, we prove Proposition 4.1 by induction on
dimension. Lemma 6.4 (together with the height reduction argument from Section
5, see also Remark 5.8) serves as the base of the induction for $2n=2$.
Suppose that Proposition 4.1 is already proven in dimension $<2n$.

###  8.1. Standard and almost standard spheres in a contact manifold

Recall that we defined the standard contact $(2n+1)$-ball as the upper
hemisphere $D:=S^{2n+1}_{+}=\\{y_{n+1}\geq 0\\}$ in the unit sphere
$S^{2n+1}=\left\\{\;\sum\limits_{1}^{n+1}x_{j}^{2}+y_{j}^{2}=1\right\\}\subset\mathbb{R}^{2n+2}$
endowed with the standard contact structure
$\xi_{\mathrm{st}}=\\{\sum\limits_{1}^{n+1}x_{j}dy_{j}-y_{j}dx_{j}=0\\}.$

We call a germ of a contact structure on $S^{2n}=\partial D$ standard if it
contactomorphic to the germ of of $\xi_{\mathrm{st}}$ along $\partial D$ and
coincides with $\xi_{\mathrm{st}}$ on a neighborhood of poles

$p_{\pm}:=\\{x_{n+1}=\pm
1,\;x_{j}=0\;\hbox{for}\;j=1,\dots,n,\;y_{j}=0\;\hbox{for}\;j=1,\dots,n+1\\}.$

A germ $\xi$ along a sphere is called almost standard if it coincides with
$\xi_{\mathrm{st}}$ on a neighborhood of poles $p_{\pm}$ and its is
characteristic foliation on $S^{2n}$ admits a Lyapunov Morse function with
exactly 2 critical points. Recall (see Proposition 3.2) that a germ of a
contact structure along a hypersurface $\Sigma$ is determined by its
restriction to $\Sigma$ up to a diffeomorphism fixed on $\Sigma$. Hence, we
will not distinguish below between the germs and their restrictions.

Recall that any linear (conformal) symplectic structure $\omega$ on a vector
space $E$ defines a canonical contact structure $\zeta_{\omega}$ on its sphere
at infinity $S(E)$, i.e. the space of oriented lines through the origin. The
group $Sp(E,\omega)$ of linear symplectic transformations acts by linear
projective contactomorphisms on $(S(E),\zeta_{\omega})$ and this
representation is faithful. Hence, we can view $Sp(E,\omega)$ as a subgroup of
the group of contactomorphisms of $(S(E),\zeta_{\omega})$. In particular,
given a hypersurface $\Sigma$ in a contact manifold $(M,\xi)$ the contact
structure $\xi$ defines a conformal symplectic structure on $T_{p}\Sigma$ for
each singularity $p$ of the characteristic foliation $\ell_{\Sigma,\xi}$. We
denote the corresponding contact structure on the sphere
$S_{p}=S(T_{p}\Sigma)$ by $\zeta_{p,\xi}$.

For the contact structure $\xi_{\mathrm{st}}$ along $S^{2n}=\partial D$ the
holonomy along the leaves of the characteristic foliation
$\ell_{\xi_{\mathrm{st}}}$ allows us to identify the contact spheres at
infinity
$(S_{\pm},\zeta_{\pm}):=\left(S\left(T_{p_{\pm}}(S^{2n})\right),\zeta_{p_{\pm}}\right).$
In turn, the contact sphere $(S_{+},\zeta_{+})$ can be canonically identified
with the standard contact sphere $(S^{2n-1},\xi_{\mathrm{st}})$ Hence, for any
almost standard germ $\xi$ along $S^{2n}$ the holonomy along the leaves of
$\ell_{\xi}$ can be viewed as a contactomorphism
$h_{\xi}:(S^{2n-1},\xi_{\mathrm{st}})\to(S^{2n-1},\xi_{\mathrm{st}})$. We will
call $h_{\xi}$ the clutching contactomorphism of an almost standard germ
$\xi$. Let $AlSt$ be the space of almost standard contact germs on $S^{2n}$
and $\mathcal{D}$ the group of contactomorphisms of the standard contact
sphere $(S^{2n-1},\xi_{\mathrm{st}})$. The image of the map
$\pi:AlSt\to\mathcal{D}$ is the subgroup $\mathcal{D}_{0}\subset\mathcal{D}$
which consists of contactomorphisms which are pseudo-isotopic to the identity.
Denote by $St$ the subspace of $AlSt$ which consists of standard contact
germs. For $\xi\in St$ we have
$h_{\xi}\in\mathcal{D}_{1}:=Sp(\mathbb{R}^{2n},\omega_{\mathrm{st}})\subset\mathcal{D}_{0}$.
The following lemma is straightforward.

######  Lemma 8.1.

The projection $\pi:AlSt\to\mathcal{D}_{0}$ is a Serre fibration. If
$h_{\xi}\in Sp(\mathbb{R}^{2n},\omega_{\mathrm{st}})$ then $\xi$ is standard,
i.e. $St=\pi^{-1}(\mathcal{D}_{1})$.

Figure 11. Adjusting a characteristic foliation near a pole.

######  Lemma 8.2.

Let $\xi_{t}$ be a family of contact structures on ${\mathcal{O}}{\it
p}\,\partial D\subset(S^{2n+1},\xi_{\mathrm{st}})$ such that their germs along
$\partial D$ are almost standard. Suppose that for $t\in[0,\frac{1}{8}]$ the
germ $\xi_{t}$ is standard. Then there exists a diffeotopy
$g_{t}:{\mathcal{O}}{\it p}\,\partial D\to{\mathcal{O}}{\it p}\,\partial D$
supported in an arbitrarily small neighborhood of $p_{+}$ such that the germs
$g_{t}^{*}\xi_{t}$ along $\partial D$ are standard and $g_{0}$ is the
identity. See Fig. 11.

###### Proof.

Recall that by assumption $\xi_{t}=\xi_{0}=\xi_{\mathrm{st}}$ in a
neighborhood $U\supset\\{p_{+}\\},U\in{\mathcal{O}}{\it p}\,\partial D$. There
exists a smaller neighborhood $U^{\prime}\subset U$ such that the pair
$(U^{\prime},\partial D\cap U^{\prime};\xi_{\mathrm{st}})$ is contactomorphic
to $(D^{2n}\times(-\epsilon,\epsilon),D^{2n}\times
0;\mathrm{Ker\,}(\gamma:=\sum\limits_{1}^{n}x_{i}dy_{i}-y_{i}dx_{i}+dz)).$
Denote $u:=\sum\limits_{1}^{n}(x_{j}^{2}+y_{j}^{2})$, and consider the
splitting $D^{2n}\setminus 0=S^{2n-1}\times(0,1]$, given by the radial
projection to the unit sphere and the $u$-coordinate. With respect to this
splitting the form $\gamma$ can be written as $dz+u\alpha_{\mathrm{st}}$,
where $\alpha_{\mathrm{st}}$ is the standard contact form of the standard
contact $(2n-1)$-sphere.

Figure 12. The function $\theta(u)$.

Choose a positive $\sigma\ll\epsilon,$ and consider a $C^{\infty}$-function
$\theta:[0,\infty)\to[-3\sigma,0]$, see 12 which is supported in
$[0,6\sigma]$, and such that $\theta(u)=0$ for $u<\sigma$,
$\theta(u)=-3\sigma$ for $u\in[4\sigma,5\sigma],$ $\theta(u)=\sigma-u$ for
$u\in[2\sigma,3\sigma]$ and $\theta^{\prime}(u)\leq 0$ for $u\in[0,5\sigma]$.
Let us view $u\in[0,1],z\in(-\epsilon,\epsilon)$ and $w\in S^{2n-1}$ as
coordinates in the neighborhood $U^{\prime}$, so that the equation $z=0$
defines $\partial D\cap U^{\prime}$. Consider the family of hypersurfaces
$\Theta_{s}\subset{\mathcal{O}}{\it p}\,\partial D$ which coincides with
graphs $\\{z=s\theta(u)\\}$ in $U^{\prime}$ and equal to $\partial D$
elsewhere. Take a neighborhood
$U^{\prime\prime}:=\\{u<7\sigma,|z|<\sigma\\}\Subset U^{\prime}$. There exists
a supported in $U^{\prime\prime}$ diffeotopy $\phi_{s}:{\mathcal{O}}{\it
p}\,\partial D\to{\mathcal{O}}{\it p}\,\partial D$ such that
$\phi_{0}=\mathrm{Id}$, $\phi_{s}(\partial D)=\Theta_{s}$. Define a diffeotopy
$\psi_{t}:{\mathcal{O}}{\it p}\,\partial D\to{\mathcal{O}}{\it p}\,\partial D$
as $\psi_{t}:=\phi_{8t}$ for $t\in[0,\frac{1}{8}]$, $\psi_{t}=\phi_{1}$ for
$t\in[\frac{1}{8},\frac{7}{8}]$ and $\psi_{t}=\phi_{8-8t}$ for
$t\in[\frac{7}{8},1]$. Note that the germs of contact structures
$\widetilde{\xi}_{t}:=\psi_{t}^{*}\xi_{t}$ along $\partial D$ are almost
standard, and moreover, for each $t\in[0,1]$ the clutching diffeomorphism
$\widetilde{h}_{t}:=h_{\widetilde{\xi}_{t}}$ differs from $h_{\xi_{t}}$ by a
unitary rotation $w\mapsto e^{C(t)i\phi}w$ of the sphere
$S^{2n-1}\subset\mathbb{C}^{n}$. Consider the domain $\widehat{U}:=\\{u\leq
3\sigma,\;-2\sigma\leq z\leq-\sigma\\}\subset U^{\prime\prime}$. Note that for
$t\in[\frac{1}{8},\frac{7}{8}]$ we have $\psi_{t}(\partial
D)\cap\widehat{U}=\\{u=\sigma-z,\;-2\sigma\leq z\leq-\sigma\\}$. Consider the
space $\mathcal{K}$ of functions $K:[-2\sigma,-\sigma]\times
S^{2n-1}\to(0,3\sigma]$ which are equal to $\sigma-z$ near the boundary. Given
$K\in\mathcal{K}$ consider its graph
$\Gamma_{K}:=\\{u=K(z,w);\;(z,w)\in[-2\sigma,-\sigma]\times
S^{2n-1}\\}\subset\widehat{U}$. The contact structures $\xi_{t}$ in
$\widehat{U}$ is given by the form $\frac{1}{u}dz+\alpha_{\mathrm{st}}$ and
hence, the holonomy along the leaves of the characteristic foliation
$\ell_{\Gamma_{K}}$ is equal to the time $\sigma$ map of the contact flow of
the contact Hamiltonian $\frac{1}{K(z,w)}$. We view here
$z\in[-2\sigma,-\sigma]$ as the time parameter.

Define a contact isotopy
$g_{t}:=\widetilde{h}_{t}^{-1}=(h_{\widetilde{\xi}_{t}})^{-1}:S^{2n-1}\to
S^{2n-1}$. While its contact Hamiltonian $G_{t}:S^{2n-1}\to S^{2n-1}$ is not
necessarily positive, it can be made positive and even arbitrarily large by
composing $g_{t}$ with appropriate unitary rotations $w\mapsto
e^{\widetilde{C}(t)i\phi}$ of the sphere $S^{2n-1}\subset\mathbb{C}^{n}$. We
will keep the notation $g_{t}$ for the modified isotopy. Hence, there exists a
family of functions $K_{t}\in\mathcal{K}$, $t\in[\frac{1}{8},\frac{7}{8}]$
such that $K_{t}=\sigma-z$ for $t=\frac{1}{8},\frac{7}{8},$ and such that the
holonomy along the leaves of the characteristic foliation
$\ell_{\Gamma_{K_{t}}}$ coincide with $g_{t}$ up to a unitary rotation of the
sphere $S^{2n-1}$. Let us modify the diffeotopy $\psi_{t}$ for
$t\in[\frac{1}{8},\frac{7}{8}]$, keeping it supported in $\widehat{U}$, so
that $\psi_{t}(\partial D\cap\widehat{U})=\Gamma_{K_{t}}$. Denote
$\widehat{\xi}_{t}:=\psi_{t}^{*}\xi_{t}$. By construction the clutching
diffeomorphisms $h_{\widehat{\xi}_{t}}:S^{2n-1}\to S^{2n-1}$ are unitary
rotations, and hence, the germs of contact structures $\widehat{\xi}_{t}$
along $\partial D$ are standard.

∎

###  8.2. Making the characteristic foliation short

######  Proposition 8.3.

Suppose that Proposition 4.1 holds for plug installation over
$D^{2n-3}\times[0,1]$. Let $\Sigma$ be the standard $(2n-2)$-dimensional
sphere in $(M^{2n-1},\xi=\mathrm{Ker\,}\alpha)$. Then for any $\sigma>0$ there
exists an $\sigma$-small in the $C^{0}$-sense isotopy $f_{s}:\Sigma\to M$
starting with the inclusion $f_{0}:\Sigma\hookrightarrow M$ such that

  * a)

$f_{s}$ is fixed on a neighborhood of poles of the characteristic foliation
$\ell$ of $\Sigma$;

  * b)

the family of characteristic foliations $\ell_{s}$, $s\in[0,1]$, induced on
$\Sigma$ by $f_{s}^{*}\alpha$; admits a family of good Lyapunov functions
$F_{s}:\Sigma\to\mathbb{R}$;

  * c)

the characteristic foliation $\ell_{1}$ is $\sigma$-short;

  * d)

for any $\sigma>0$ the isotopy $f_{s}$ can be included into a 2-parametric
isotopy $f_{s,t},\;s,t\in[0,1]$, such that

    * (i)

$f_{s,0}=f_{s}$, $f_{0,t}=f_{0}$ for all $s,t\in[0,1]$;

    * (ii)

the spheres $f_{s,1}(\Sigma)$ are almost standard for all $s\in[0,1]$;

    * (iii)

$f_{s,t}$ is $\sigma$-close to $f_{s,0}$ for all $(s,t)\in[0,1]$;

    * (iv)

the isotopy $f_{1,t},t\in[0,1]$, is fixed on a neighborhood of a dividing set
$S_{1}$ of $\Sigma_{1}=f_{1}(\Sigma)$.

###### Proof.

Using Corollary 3.4 we can find an $\sigma$-blocking system $\\{D_{j}\\}$,
$j=1,\dots,N$, of transverse standard contact discs. We can assume that the
Lyapunov function $F$ for $X$ on $\Sigma$ is constant on each $D_{j}$. Denote
$c_{j}=F|_{D_{j}}$ and assume that $c_{1}<c_{2}<\dots<c_{N}$. Let
$Q_{j}\subset\Sigma$ be disjoint flow-boxes of $D_{j}$ of diameter $<2\sigma$.
Let $h_{s,t}:Q:=D\times[0,1]\to D\times T^{*}[0,1]$ be a $C^{0}$ small isotopy
constructed in Proposition 4.1. We define the required isotopy $f_{s,t}$ by
successively deforming flow-boxes $Q_{j}$. Denote
$\Delta_{j}:=[\frac{j-1}{N},\frac{j}{N}]\subset[0,1]$, $1\leq j\leq N$. For
$s\in\Delta_{j},t\in[0,1]$ define $f_{s,t}:=\Phi_{j}\circ
h_{Ns-j+1,t}\circ\Phi_{j}^{-1}$ on $Q_{j}$, $f_{s,t}=\Phi_{i}\circ
h_{1,t}\circ\Phi_{i}^{-1}$ on $\ Q_{i}$ for $i<j$, and fixed elsewhere on
$\Sigma$. Then Proposition 4.1 guarantees all the required properties of the
isotopy $f_{s,t}$ except b), d)(ii) and d)(iv). More precisely, for b) we
automatically get a family of Lyapunov functions for $X_{s}$, but not
necessarily good ones. Moreover, we get a family of Lyapunov functions for the
whole 2-parametric family $X_{s,t}$. In particular, Lyapunov functions for
$X_{s,1}$ have no critical points except the maximum and the minimum of the
original Lyapunov function $F$. This implies that the sphere $f_{s,1}(\Sigma)$
are almost standard. Recall that according to Corollary 3.5 it is sufficient
to ensure properties (L1) and (L2) and absence of retrograde connections.
According to property c)(v) of Proposition 4.1 trajectories of $X_{s,t}$
converging to positive zeroes enter each plug through the same
$(n-2)$-dimensional stratified subset $E_{j}\subset
D_{j}\subset\partial\widehat{Q}_{j}$. According to our staged construction of
the isotopy, the isotopy $f_{s,0}$ for $s\in[\frac{j-1}{N},\frac{j}{N}]$,
$j=1,\dots,N$, which creates a plug in $\widehat{Q}_{j}$ does not change
trajectories of $X_{s}=X_{s,0}$ in $F\leq c_{j}$. Hence, by a
$C^{\infty}$-small adjustment of embeddings $\phi_{j}$ before each step of the
isotopy we can arrange that the closure $\overline{G_{j}}$ of the negative
tail $\bigcup\limits_{u\geq 0}X_{\frac{j-1}{N}}(E_{j})$ of the set $E_{j}$
does not contain any negative zeroes of $X_{\frac{j-1}{N}}$, and hence, the
same property holds for all $s\geq\frac{j-1}{N}$. The deformation $f_{s,t}$
for a fixed $s\in\Delta_{j}$ changes the field $X_{s,t}$ only in an
arbitrarily small neighborhood of $\overline{G}_{j}$, and hence thanks to
compactness of the set of zeroes, one can arrange that $X_{s,t}$ have no
retrograde connections. Corollary 3.4 then guaranteed that $f_{s,t}(\Sigma)$
are Weinstein convex. It remains to satisfy property d)(iv). Consider the
dividing set $S$ for $X_{1}$. Using Property c)(iii) of Proposition 4.1 we can
find an isotopy of $\Sigma$ preserving leaves of $\ell_{1}$ with disjoins $S$
with compact subsets $C_{1}^{\pm}$, and hence with their neighborhoods
$U^{\pm}\supset C_{1}^{\pm}$. According to c)(ii) we can arrange the isotopy
$f_{1,t}$ to be supported in $U^{\pm}$, which implies property d)(iv).

∎

######  Proposition 8.4.

Suppose that Proposition 4.1 holds for plug installation over
$D^{2n-3}\times[0,1]$. Let $D=(D^{2n-1},\mathrm{Ker\,}\alpha_{\mathrm{st}})$
be the standard contact disc. Then for any $\sigma>0$ there exists a
$\sigma$-small in the $C^{0}$-sense isotopy $h_{s}:D\to D$ such that
$h_{s}(D)\subset\mathrm{Int\,}D$ for all $s>0$ and

  * (i)

the ball $\widetilde{D}:=h_{1}(D)$ has a Weinstein convex boundary
$\partial\widetilde{D}$ with the dividing set $S\subset\partial\widetilde{D}$;

  * (ii)

the characteristic foliation $\ell_{\partial\widetilde{D}}$ is $\sigma$-short;

  * (iii)

$(\widetilde{D},S)$ can be approximated by standard contact balls.

######  Remark 8.5.

In the case $n=2$ property (iii) is automatic from the classification of tight
contact structures on the $3$-ball, see [5].

###### Proof.

Let us first shrink $D\to\mathrm{Int\,}D$ by a $C^{\infty}$-small contracting
contact isotopy and then apply to the image $D^{\prime}$ Proposition 8.3. Let
us extend the constructed there isotopy $f_{s,t}:\partial D^{\prime}\to D$,
$s,t\in[0,1]$, to an isotopy $D^{\prime}\to D$. We will continue using the
notation $f_{s,t}$ for the extension. We claim that that the isotopy obtained
by concatenating the shrinking isotopy with the isotopy $f_{s,0}$ has the
required properties. Indeed, the balls $D^{\prime}_{s}:=f_{s,0}(D^{\prime})$
have Weinstein convex boundaries with dividing sets $S_{s}\subset\partial
D^{\prime}_{s}$ and the characteristic foliation on $\partial D^{\prime}_{1}$
is $\sigma$-short and the family of spheres $\Sigma_{s}:=f_{s,1}(\partial
D^{\prime})$, $s\in[0,1]$ are almost standard and hence, can be made standard
by an arbitrarily $C^{0}$-small adjustment away from the poles and dividing
sets by applying Lemma 8.2. Th sphere $\Sigma_{0}=\partial D^{\prime}$ bounds
the standard ball, and hence, the same holds, by continuation of the contact
isotopy argument, for $\Sigma_{1}$. But by construction $\Sigma_{1}$ coincides
with $\partial D^{\prime}_{1}$ on a neighborhood of the dividing set
$S_{1}\subset\partial D^{\prime}_{1}$ and can be made arbitrarily
$C^{0}$-close to $\partial D^{\prime}_{1}$, i.e. $(D^{\prime}_{1},S_{1})$ can
be approximated by standard contact balls. ∎

###  8.3. Proofs on main results

###### Proof of Proposition 5.1.

Applying the induction hypothesis and Proposition 8.4 we find a disc
$\widetilde{D}\subset D$ such that

  * -

$\widetilde{D}$ has a Weinstein convex boundary $\partial\widehat{D}$ with the
dividing set $S\subset\partial\widehat{D}$;

  * -

the characteristic foliation $\ell_{\partial\widetilde{D}}$ is $\sigma$-short;

  * -

$(\widetilde{D},S)$ can be approximated by standard contact balls;

  * -

$D\setminus f(\partial D)$ is contained in $\sigma$-small neighborhood of
$\partial D$,

Next, we apply Proposition 7.6 and find $3$ Weinstein cylinders
$V_{1},V_{2},V_{3}\subset f(D)$ such that a piecewise smooth hypersurface
$\partial(V_{1}\cup V_{2}\cup V_{3})$ is $C^{1}$ $\sigma$-close to $\partial
f(D)$. Applying now Lemma 7.8 we install into $D\times[0,1]$ a $\sigma$-quasi-
plug in $(V_{1}\cup V_{2}\cup V_{3})\times[0,1]\subset f(D)\times[0,1]$. But
the characteristic foliation on $f(\partial D)$ is $\sigma$-short, and hence
the constructed plug is a genuine $C\sigma$-plug for $D\times[0,1]$ for some
universal constant $C>0$.

This concludes the proof of Proposition 5.1, and hence, of Proposition 4.1. ∎

###### Proof of Theorem 3.11.

First, we adjust $\Sigma$ by a $C^{\infty}$-isotopy to make all singularities
of its characteristic foliation $\ell_{\Sigma}$ non-degenerate and hyperbolic.
Next we apply Lemma 3.3 to find a blocking collection of transverse standard
contact discs $D_{j}\subset\Sigma$. According to Lemma 2.6 there exists
$\sigma>0$ such that by installing $\sigma$-plugs instead of flow boxes one
can arrange the resulting flow to satisfy condition (L1). Proposition 4.1
asserts that such plugs can be installed by deforming flow-boxes via an
arbitrarily small in the $C^{0}$-sense isotopy. By an additional
$C^{\infty}$-perturbation of the hypersurface outside plugs we can satisfy the
Morse-Smale property, while still preserving condition (L1), see Lemma 2.6,
and hence, by Corollary 3.4 the resulting $\Sigma$ is Weinstein convex. This
concludes the proof of Theorem 3.11, and in combination with Lemma 3.10 of
Theorem 1.1. ∎

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