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\begin{document}
\title{Convexity in Contact Topology}
\author{Emmanuel Giroux \\ (Translated by Daniel Mathews)}%
\maketitle
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\tableofcontents
\section*{Translator's note}
Hopefully most of this is accurate. Thanks to Patrick Massot for several corrections and suggestions. All translation errors and inaccuracies, however, should be ascribed to me alone. All footnotes are mine. Figures and references can be found in the original.
\section*{Introduction}
This article tackles the study of convexity in contact geometry, as
it has been defined in [EG]: a structure, symplectic or contact, is
called convex if it is conformally invariant under the gradient flow
of a proper Morse function. For symplectic manifolds, this property
plays the same role as that of strict pseudo-convexity in complex
analytic manifolds. It can only, for example, be shown on open
manifolds having the homotopy type of polyhedra of half dimension
and, in [EG], Ya. Eliashberg and M. Gromov show how it tempers
the geometry and forbids certain exotic phenomena (see also [Gr] and
[El1]). In contact geometry, the situation presents itself
differently. First of all, the usual structures, on jet spaces of
order 1, spheres and manifolds of contact elements, are all convex
(see I.4.C). Then, the results obtained here show that in dimension
3, there exist numerous convex contact manifolds. In particular,
certain exotic structures discovered by T. Erlandsson and D.
Bennequin (see [Be]) are convex; in fact, we do not know any
examples of non-convex structures.
The approach adopted is the following: given a proper Morse function
$f$ on a 3-dimensional manifold $V$, one tries to construct on $V$ a
contact structure $\xi$ which is invariant under the flow of a
gradient $X$ of $f$. The study of contact fields (i.e. fields
preserving the contact structure) shows that, if this structure
$\xi$ exists, the surface $C$ of points of $V$ where $X$ is tangent
to $\xi$ must satisfy, vis-\`{a}-vis $f$, the following conditions
(Proposition I.4.5):
\begin{enumerate}
\item
$f|_C$ is a proper Morse function;
\item
the critical points of $f$ are all on $C$ and are exactly the
critical points of $f|_C$;
\item
$f$ and $f|_C$ have the same local extrema.
\end{enumerate}
A Morse function does not always admit surfaces satisfying these
properties (see IV.1.B). Nevertheless, one can modify it, adding
only several critical points of indices $1$ or $2$ in a position of
elimination, so that such a surface $C$ exists (Theorem IV.2.7).
Also, being given $C$ allows one effectively to construct the
desired contact structure $\xi$ (Theorem III.1.2). To obtain this,
one puts on each handle an induced structure via immersion in a
well-chosen model on $\R^3$. The difficulty is to adjust these
immersions in order to be able to glue the pieces: this problem is
localised along certain faces of the handles. Yet, in the
neighbourhood of a surface, a contact structure is entirely
described by the (singular) dimension-1 foliation that it traces on
the surface. Moreover, each surface considered here, corresponding
to a regular level set of $f$, is found, by construction, to be
transverse in $\R^3$ to a vector field that preserves the model
structure and holds the role of gradient of $f$. The crucial point
then is to understand how, when one moves the surface by an isotopy
keeping it transversal to this field, one modifies its foliation
(Proposition II.3.6). At this point, a convex contact structure
appears geometrically describable by a finite number of these
foliations, carried by the different regular level sets of the
function and determined only up to prior modifications.
Among these possible modifications of the foliation is the
elimination of pairs of singularities (Lemma II.3.3). One can thus
extend a result of Ya. Eliashberg that permits the elimination of
certain complex points on a surface contained in the pseudo-convex
boundary of a holomorphic domain (see [El1], Theorem 6.1 and [El2]).
For that, in place of the theory of holomorphic curves in
4-dimensional symplectic manifolds, we use the following remarkable
fact (Proposition II.2.6): in a contact 3-dimensional manifold, a
surface generically possesses a transverse contact vector field.
Thanks to this property of invariance, the problem of elimination
reduces to the symplectic geometry of surfaces.
The problems studied in this article have been expounded to me by
Yasha Eliashberg during some conversations which were marvellously
enriching for me; I thank him heartily. I equally thank Fran\c{c}ois
Laudenbach and Jean-Claude Sikorav for their numerous remarks and
pertinent suggestions regarding this text.
\section{The notion of convexity}
\subsection{Preliminary definitions}
\subsubsection{Symplectic and contact structures}
A \emph{symplectic structure on a vector space} $V$ of dimension
$2n$ is an exterior $2$-form $\omega$ whose exterior $n$'th power
is nonzero. The \emph{orthogonal complement} of a subspace $W$ of $V$
is the subspace $\{v \in V \; | \; \forall w \in W, \; \omega(v,w) =
0 \}$.
One says that $W$ is \emph{coisotropic} if it contains its
orthogonal complement. Note that, if $c$ is a nonzero real number,
$c\omega$ is still a symplectic form and the orthogonal complement
of $W$ is the same for $\omega$ and $c\omega$.
A \emph{symplectic structure on a vector bundle} of even rank is a
field of symplectic forms on its fibres.
A \emph{symplectic structure on a manifold $V$} of dimension $2n$ is
a closed differential 2-form $\omega$ which induces on each tangent
space a symplectic form.
A \emph{contact structure} on a manifold $V$ of dimension $2n+1$ is
a completely non-integrable hyperplane field $\xi$, that is,
defined locally by a 1-form $\alpha$ such that $\alpha \wedge
(d\alpha)^n$ is nowhere vanishing. In other words, $d\alpha|_\xi$ is at
every point a symplectic form. Multiplying $\alpha$ by a nowhere
vanishing function $f$ changes $d\alpha|_\xi$ to $f \cdot
d\alpha|_\xi$, so that $\xi$ is furnished with a \emph{conformal
symplectic structure}. We also remark that, if $n$ is even, $\xi$ is
naturally oriented while, if $n$ is odd, $V$ is naturally oriented.
In either case, any orientation transverse to $\xi$ (which exists if
and only if $\xi$ admits a global equation $\alpha = 0)$ orients at
the same time $\xi$ and $V$.
\subsubsection{Singular foliations of dimension $1$}
In this text, a \emph{singular foliation} (of dimension 1) on a
manifold $M$ of dimension $m$ is a foliation $\F$ defined by an
atlas $\{U_i, X_i\}$ where: $\{U_i\}$ is a covering of $M$, $X_i$ a
vector field on $U_i$ and, for every $(i,j)$, there exists a
nowhere vanishing function $f_{ij}$ on $U_i \cap U_j$ such that $X_i =
f_{ij} X_j$.
\begin{Remark} If each $U_i$ is furnished with a volume form
$\theta_i$, the data of $X_i$ is equivalent to being that of the
$(m-1)$-form $i(X_i) \theta_i$ (interior product of $\theta$ with
$X_i$).
\end{Remark}
We say that a vector field $X$ on $M$ \emph{directs} $\F$ if, for
all $i$, there exists a nowhere vanishing function $f_i$ on $U_i$ such
that $X = f_i X_i$; we say that $\F$ is \emph{orientable} if such a
field exists.
\subsubsection{Characteristic foliation of a hypersurface}
Let $S$ be a hypersurface in a contact manifold $(V, \xi)$ of
dimension $2n+1$. The trace on $\xi$ of the tangent bundle of $S$
determines a distribution (of non-constant rank) of coisotropic
subspaces in $\xi|_S$. The orthogonal distribution, via the
conformal symplectic structure on $\xi|_S$, is of rank 0 on the
singular locus $\Sigma$ where $\xi$ is tangent to $S$, and of rank
$1$ otherwise. It defines a singular foliation, in the sense of B,
that we call the characteristic foliation of $S$. Locally, if
$\theta$ is a volume form on $S$ and $\beta$ the 1-form induced by
an equation for $\xi$, the characteristic foliation is defined by
the vector field $X$ such that $i(X) \theta = \beta \wedge
(d\beta)^{n-1}$. One easily verifies that the characteristic foliation
of $S$ is orientable if and only if the normal bundle of $S$ is
isomorphic to the quotient bundle $(TV/\xi)|_S$.
\begin{Remark}
Outside the singular locus $\Sigma$, the characteristic
foliation $\F$ of $S$ has a transverse contact structure, $(\xi
\cap TS)/\F$, invariant under holonomy (see [McD]). On $\Sigma$,
$TS|_\Sigma = \xi|_\Sigma$ has a conformal symplectic structure,
invariant under local vector fields which direct $\F$ (see 2.C).
\end{Remark}
\subsection{Characteristic hypersurface of a contact vector field}
\subsubsection{Contact vector field}
Let $(V, \xi)$ be a contact manifold.
\begin{defn}
A \emph{contact vector field} on $(V, \xi)$ is a vector field
whose flow preserves $\xi$.
\end{defn}
It is well known (see [A]) that:
\begin{prop}
The contact vector fields on $(V, \xi)$ are in bijective
correspondence with the sections of the normal bundle to $\xi$,
$TV/\xi$. In other words, any section of this quotient lifts to
a unique contact vector field.
\end{prop}
\begin{cor}
Any contact vector field given locally can be extended globally.
\end{cor}
\noindent \emph{Remark.}
In the presence of an equation for $\xi$, i.e. of a trivialisation
of $TV/\xi$, a section of $TV/\xi$ is nothing other than a
function, called the Hamiltonian of the corresponding contact vector field.
\medskip
\subsubsection{Characteristic hypersurface}
Let $X$ be a contact vector field on $(V, \xi)$.
\begin{defn}
The \emph{characteristic hypersurface} of $X$ is the set $C =
C(X)$ of points where $X$ is tangent to $\xi$.
\end{defn}
On the space of vector fields (furnished with the $C^\infty$
topology), the property of having a reduction modulo $\xi$
transverse to the zero section of $TV/\xi$ is generic. In this case,
by abuse of language, we will say that the vector field is
\emph{generic}. Its characteristic hypersurface is then regular.
\begin{prop}
\label{2.5}
If $X$ is generic, $X$ is tangent to its characteristic
hypersurface $C$ and directs the characteristic foliation on it.
\end{prop}
\begin{Proof}
The flow of $X$ preserves $X$ and $\xi$, therefore $C$, so that
$X$ is tangent to $C$.
Let now $x$ be a point of $C$ and $\alpha$ a local equation for
$\xi$ near $x$. The hypersurface $C$ is defined locally by the
equation $i(X) \alpha = 0$ (regular since $X$ is generic). Also,
as $X$ is contact, the Lie derivative of $\alpha$ satisfies:
$L_X \alpha = g \alpha$ for some function $g$. For $v \in T_x C
\cap \xi_x$, we then have:
\begin{align*}
d\alpha(x) \cdot (X(x), v) &= (L_X \alpha) (x) \cdot v - (
di(X) \alpha)(x) \cdot v \\
&= (g\alpha)(x) \cdot v - (di(X)\alpha)(x) \cdot v = 0
\end{align*}
since the two terms are zero. Thus $X(x)$ is orthogonal to $T_x
C \cap \xi_x$.
Moreover, if $X(x) = 0$, we have:
\[
(L_X \alpha )(x) = (g\alpha)(x) = (di(X)\alpha)(x).
\]
Therefore $\xi$ is tangent to $C$ at $x$.
\end{Proof}
\emph{Remark.}
If $\xi$ is transversally orientable, there exist contact vector
fields $X$ with empty characteristic hypersurface; there are
vector fields transverse to $\xi$, i.e. Reeb fields associated
to various equations for $\xi$.
\begin{Example}
\label{1.2.6}
Any contact vector field $X$ which is nonsingular or has
nondegenerate singularities is generic.
\end{Example}
\begin{Proof}
Let $\alpha$ be a local equation for $\xi$; we want to show
that $d(i(X)\alpha)$ is nonzero at every point where
$i(X)\alpha$ is zero. As $X$ preserves $\xi$, $L_X \alpha = g
\alpha$ for some function $g$. Then $di(X)\alpha = g\alpha -
i(X) d\alpha$.
If $X$ is nonsingular at $x \in C$, $(i(X) d\alpha)(x)$ is not
proportional to $\alpha(x)$ since $d\alpha(x)$ is nondegenerate
on $\xi_x$. Thus, considering the expression for the Lie
derivative, $di(X)\alpha$ is nonzero at $x$.
Now, if $X$ has at $x$ a nondegenerate singularity, its
linearisation $A_x: T_x V \To T_x V$ is invertible. Then the
form $(di(X)\alpha)(x)$, which is equal to $\alpha(x) \circ
A_x$, is nonsingular.
\end{Proof}
\subsubsection{Singularities of contact vector fields}
\emph{Remarks.}
\begin{enumerate}
\item
The singularities of a contact vector field lie on its
characteristic hypersurface.
\item
The divergence of a vector field at a singular point does not
depend on the local volume form with which it is calculated: it
is the trace of the linearisation.
\end{enumerate}
\begin{prop}
\label{1.2.7}
Let $(V, \xi)$ be a contact manifold of dimension $2n+1$ and let
$X$ be a generic contact vector field. To each singularity $x$
of $X$ is associated a nonzero real number $c = c(x)$ (the
coefficient of contraction) having the following properties:
\begin{enumerate}
\item
$(n+1)c$ (resp. $nc$) is the divergence of $X$ at $x$
(resp. of $X|_C$ at $x$);
\item
for any local equation $\alpha$ of $\xi$, which induces
a form $\beta$ on $C$, we have:
\[
(L(X)\alpha)(x) = c \alpha(x) \quad \text{and} \quad
(L(X|_C) d\beta)(x) = c \; d\beta(x).
\]
\end{enumerate}
\end{prop}
\begin{Proof}
As $X$ is contact, $L(X)\alpha = g\alpha$ for some function $g$;
thus, if it exists, the desired coefficient is $c = g(x)$. But,
as $X$ is generic, $g(x)$ is nonzero. Then:
\[
L(X) d\alpha = dL(X) \alpha = dg \wedge \alpha + g \;
d\alpha.
\]
As $\beta(x) = 0$, we clearly have: $(L(X|_C)d\beta)(x) = c \;
d\beta(x)$. Now, to see that $c$ does not depend on the choice
of $\alpha$ it suffices to show (i). But as $\alpha \wedge
(d\alpha)^n$ and $(d\beta)^n$ are local volume forms on $V$ and
$C$ respectively, (i) follows from the expressions above by
derivation of a product. For example:
\begin{align*}
L(X)(\alpha \wedge (d\alpha)^n) &= (L(X)\alpha) \wedge
(d\alpha)^n + \alpha \wedge L(X)(d\alpha)^n \\
&= g\alpha \wedge (d\alpha)^n + n \alpha \wedge L(X) d\alpha
\wedge (d\alpha)^{n-1} = (n+1) g\alpha \wedge (d\alpha)^n.
\end{align*}
\end{Proof}
\begin{cor}
\label{1.2.8}
If $x$ is a singularity of a generic contact vector field, the
eigenvalue transverse to $C$ (the tangent space to $C$ is
stable) is equal to $c$.
\end{cor}
\begin{cor}
\label{1.2.9}
Suppose that $X$ is, for a certain metric, the gradient of a
function $f$ which has at $x$ a Morse critical point of index
$i$. If $c(x)$ is positive (respectively negative), then $i$ is
at most equal to $n$ (resp. at least equal to $n+1$).
\end{cor}
\begin{Proof}
Let $\alpha$ be an equation of $\xi$ near $x$ and $\beta$ the
form induced on $C$.
The form $d\beta(x)$ is a symplectic form on $T_x C$. If $c(x)$
is positive, the tangent space at $x$ to the stable manifold of
$X|_C$ has dimension $i$, since the transverse eigenvalue is
positive; moreover it is necessarily isotropic, that is,
contained in its symplectic orthogonal complement (see
remark 4.3). Then, $i$ is at most equal to $n$.
By similar reasoning on the unstable manifold, we see that,
if $c(x) < 0$, then $i \geq n+1$.
\end{Proof}
\subsection{Convex hypersurfaces}
\subsubsection{Definition, example}
\begin{defn}
\label{1.3.1}
We say that a hypersurface $S$ embedded
in a contact manifold $(V, \xi)$ is \emph{convex}
if there exists a contact vector field transverse to $S$.
\end{defn}
A convex hypersurface is therefore transversally orientable, that is,
its tubular neighbourhoods are diffeomorphic to $S \times
\R$. Also, any germ of a contact vector field along $S$, which is
transverse to $S$, extends to a global contact vector field. The
study of convex hypersurfaces is therefore closely linked to that of
contact structures on $S \times \R$ invariant under the vertical
vector field $\partial/\partial t$, where $t$ denotes the coordinate
on $\R$.
\begin{Example}
\label{1.3.2} (Contactization of an exact symplectic manifold). We
say that a symplectic manifold $(W, \omega)$ is \emph{exact} if
$\omega$ is the differential of a 1-form $\beta$ called the
Liouville form. By symplectic duality, it is equivalent to say that
there exists on $W$ a vector field $X$, called the Liouville vector
field, whose flow dilates $\omega$ exponentially: $L(X) \omega =
\omega$. If $(W, \omega = d\beta)$ is an exact symplectic manifold,
the form $\beta + dt$ defines on $W \times \R$ a vertically
invariant contact structure. Moreover, the Liouville field $X$,
$\omega$-dual to $\beta$, directs the characteristic foliation on
the hypersurfaces $W \times \{t\}$, $t \in \R$.
\end{Example}
\begin{Remark}
\label{1.3.3}
\begin{enumerate}
\item
The contact structure thus obtained depends not only on the
symplectic structure $\omega$ but also on the primitive $\beta$
chosen. We observe however that, if we change $\beta$ to $\beta +
dh$, where $h$ is a function on $W$, the diffeomorphism $\phi: W
\times \R \To W \times \R$, given by $\phi(x,t) = (x, t+h(x))$,
satisfies $\phi^*(\beta + dt) = (\beta + dh) + dt$. This
therefore establishes an isomorphism between the two contact
structures.
\item
If $H \subset W$ is a hypersurface transverse to $X$, the form
induced by $\beta$ on $H$ is contact. Indeed, $\beta \wedge
(d\beta)^{n-1} = (1/n) \; i(X) \omega^n$ induces on $H$ a volume
form.
\end{enumerate}
\end{Remark}
\subsubsection{Vertically invariant contact structures}
Let $S$ be a manifold of dimension $2n$. A contact structure $\xi$,
transversally orientable and vertically invariant, on the cylinder
$S \times \R$, can be defined by a global equation $\beta + u \; dt
= 0$, where $\beta$ and $u$ are respectively a 1-form and a function
on $S$ such that:
\begin{eqnarray}
\text{the form } \theta = (d\beta)^{n-1} \wedge (u \; d\beta + n
\beta \wedge du) \text{ does not vanish on $S$}.
\label{part1star}
\end{eqnarray}
In fact $\theta \wedge dt = (\beta + u \; dt ) \wedge (d(\beta + u
\; dt))^n.$
We observe that:
\begin{enumerate}
\item
The set $\Sigma$ where $u=0$ is the trace on $S \times \{0\}$ of
the characteristic hypersurface of the vector field
$\partial/\partial t$; it's a regular hypersurface on which
$\beta$ induces a contact form since, along $\Sigma$, \ref{part1star}
is written $(d\beta)^{n-1} \wedge \beta \wedge du \neq 0$.
\item
On the open set $\Omega = S \backslash \Sigma$, $\xi$ is still
defined by $\beta/u + dt = 0$ and we have: $\theta = u^{n+1}
(d(\beta/u))^n$. Thus $(\Omega \times \R, \xi)$ is the
contactization of the exact symplectic manifold $(\Omega,
d(\beta/u))$.
Let $Y$ be the vector field tangent to $S$ defined by:
\begin{equation}
\label{starstar}
\beta \wedge (d\beta)^{n-1} = i(Y) \theta.
\end{equation}
This vector field directs the characteristic foliation of $S
\times \{0\}$ (see 1.C) and satisfies the relations below.
\item
On $\Sigma$: $Y \cdot u = -1/n$. Indeed:
\[
\beta \wedge (d\beta)^{n-1} = -n \; i(Y) [du \wedge \beta
\wedge (d\beta)^{n-1}] = -n(Y \cdot u) \beta \wedge
(d\beta)^{n-1},
\]
since $i(Y)[\beta \wedge d\beta^{n-1}] = 0$.
\item
On $\Omega$, let $X$ be the Liouville vector field of $\beta/u$
defined by $\beta/u = i(X) d(\beta/u)$; we have:
\[
X = n u Y.
\]
Indeed:
\[
i(X) \theta = u^{n+1} i(X) \left( d \left( \frac{\beta}{u}
\right) \right)^n = nu^{n+1} \frac{\beta}{u} \wedge \left( d
\left( \frac{\beta}{u} \right) \right)^{n-1} = nu\beta \wedge
(d\beta)^{n-1}.
\]
\end{enumerate}
\begin{prop}
\label{1.3.4} Let $S$ be a closed manifold of dimension $2n$ and let
$\F$ be a singular foliation of dimension $1$ on $S$ (see 1.B). There
exists on $S \times \R$ a vertically invariant contact structure
which induces $\F$ as characteristic foliation on $S \times \{0\}$
if and only if there exists in $S$ a hypersurface $\Sigma$
transverse to $\F$ (in particular, avoiding the singularities of
$\F$) such that:
\begin{enumerate}
\item
the complement $S'$ of an open tubular neighbourhood of
$\Sigma$, whose fibres are in $\F$, is an exact
symplectic manifold whose Liouville field directs $\F$ and
exits transversally on the boundary;
\item
the involution of the double cover $\partial S' \To \Sigma$,
obtained by following the leaves of $\F$ across the tube,
preserves the contact structure induced on $\partial S'$ (see
remark 3.3b) but reverses its transverse orientation.
\end{enumerate}
\end{prop}
\begin{Proof}
We suppose first of all that there exists on $S \times \R$ a
vertically invariant contact structure $\xi$ which induces $\F$ as
characteristic foliation on $S \times \{0\}$. The intersection
$\Sigma$ of $S$ with the characteristic hypersurface of the vector
field $\partial/\partial t$ is a hypersurface of $S$ transverse to
$\F$ (see (1) and (3) above). On $\Omega = S \backslash \Sigma$, the
vertical vector field is transverse to $\xi$, so $\xi$ is transversally
orientable and defined by a unique equation $\beta + dt = 0$, where
$\beta$ is necessarily a Liouville form on $\Omega$. Using local
equations near $\Sigma$ and the relations (3) and (4) above, we see
that the Liouville field $X$ associated to $\beta$ exits along
$\partial S'$, if $S'$ is chosen as in the statement. Finally, the
contact structure $\xi'$ defined by $\beta$ on $\partial S'$ is the
trace on $\partial S'$ of the contact structure transverse to $\F$
and invariant under the holonomy of $\F$. It follows that the
involution of the double cover $\partial S' \To \Sigma$ preserves
$\xi'$; but, as $X$ changes the direction of crossing $\Sigma$,
the transverse orientation of $\xi'$ is reversed.
Conversely, we now suppose that conditions (i) and (ii) are
satisfied. We denote by $d\beta$ the exact symplectic structure on
$S'$ whose Liouville field $X$ directs $\F$ and exits along
$\partial S'$.
\begin{lem}
\label{1.3.5}
We can suppose that: (ii)' the involution of the
double cover $\partial S' \To \Sigma$ reverses the form induced
by $\beta$ on $\partial S'$.
\end{lem}
\begin{Proof}
Let $\bar{S}'$ be the manifold obtained as follows: we glue to
$S'$ the cylinder $\partial S' \times [0, \infty)$ along $\partial
S' =
\partial S' \times \{0\}$, connecting $X$ with the vector field
$\partial/\partial r$ where $r$ is the coordinate on $[0, \infty)$;
we still denote the extended vector field $X$. If $\eta$
denotes the 1-form induced by $\beta$ on $\partial S'$, we extend
$\beta$ to $\bar{S'}$ by setting $\beta = e^r \eta$ on $\partial S'
\times [0, \infty)$. Then $(\bar{S'}, d\beta)$ is an exact
symplectic manifold whose Liouville field is $X$.
Now let $\tau$ be the involution of the double cover $\partial S'
\To \Sigma$; by hypothesis, there exists a function, negative on
$\partial S'$, denoted $-e^h$, satisfying $\tau^* \eta = -e^h \eta$;
as $\tau^2$ is the identity, we have: $\tau^* h = -h$. Let $h_0$ be
a minimum of $h$ on $\partial S'$ and let
\[
S'_0 = S' \cap \{ (y,r) \in \partial S' \times [0, \infty) \; |
\; r \leq \frac{1}{2} [h(y) - h_0] \}.
\]
Then the form induced by $\beta$ on $\partial S'_0 \cong \partial
S'$ is: $\eta_0 = e^{(h-h_0)/2} \eta$; by the following:
\[
\tau^* \eta_0 = e^{\tau^*(h-h_0)/2} \tau^* \eta = -
e^{-(h+h_0)/2} e^h \eta = - \eta_0.
\]
Finally we have an isotopy which sends $S'_0$ to $S'$ and respects
the foliation by the orbits of $X$, which proves the lemma.
\end{Proof}
On $S' \times \R$, the equation $\beta + dt = 0$ defines a
vertically invariant contact structure $\xi$ which we seek to extend
over $\Sigma \times \R$. For this, we suppose first of all that
$\Sigma$ is transversally orientable, and we take a
split neighbourhood, $U \cong \Sigma \times (-1-\epsilon,
1+\epsilon)$, in which the leaves of $\F$ are the segments $\{pt\}
\times (-1-\epsilon, 1+\epsilon)$. We choose the parametrisation so
that:
\begin{itemize}
\item
$\Sigma \cap U = \Sigma \times \{0\}$ and $\partial S' \cap U =
\Sigma \times \{-1,1\}$;
\item
on $S' \cap U$, $X$ has the expression $-s(\partial/\partial
s)$, where $s$ is the coordinate on the interval $(-1-\epsilon,
1+\epsilon)$.
\end{itemize}
The relations $L(X) \beta = \beta$, $i(X) \beta = 0$ and
property (ii)' show that $\beta|_{S' \cap U} = (1/s)\gamma$, where
$\gamma$ is a contact form on $\Sigma \times \{1\}$. Then the form
$\gamma + s \; dt$ defines on $U$ a contact structure which
coincides with $\xi$ on $(U \cap S') \times \R$.
Finally, if $\Sigma$ is not transversally orientable, we pass to a
cover of $S$ in which it becomes so and we perform the preceding
construction in an equivariant manner.
\end{Proof}
\begin{Remark*}[F. Laudenbach]
If $n$ is even and if $S$ is orientable, the hypersurface $\Sigma$ separates. Indeed, $\xi$ is then orientable, thus transversely orientable, since $S \times \R$ is orientable. Then, the two sides of $\Sigma$ are given by the sign of $\partial/\partial t$ relative to this transverse orientation.
\end{Remark*}
\subsection{Convex contact structures}
\subsubsection{Pseudo-gradients of a Morse function}
\begin{defn}
\label{1.4.1} Let $f: M \To \R$ be a Morse function, that is, a
function all of whose critical points are nondegenerate. We say
that a vector field $X$ is a \emph{pseudo-gradient} of $f$ if there
exists on $M$ a Riemannian metric and a positive function $\rho$
such that, everywhere in $M$, we have $X \cdot f \geq \rho || df
||^2$. We then have a similar relation for any other Riemannian
metric. For example: the gradient of $f$ for a given metric verifies
this inequality.
\end{defn}
We recall that a singularity $x$ of a vector field $X$ is
\emph{hyperbolic} if the linearisation $A_x$ of $X$ at $x$ is
hyperbolic, i.e., has no eigenvalue with zero real part. In this
case, the theorem of the stable manifold asserts that the points
having $x$ for $\omega$-limit (resp. $\alpha$-limit) form an
immersed submanifold called the stable manifold (resp. unstable);
its tangent space at $x$ is the stable (resp. unstable) manifold of the
linearised field $A_x$. It is well known that:
\begin{prop}
\label{1.4.2} Let $f$ be a Morse function on a manifold $M$ of
dimension $m$ and let $X$ be a pseudo-gradient of $f$. Then:
\begin{enumerate}
\item
the singularities of $X$ are hyperbolic and are exactly the
critical points of $f$;
\item
at a critical point of $f$ of index $i$, the stable
(resp. unstable) manifold of $X$ has dimension $i$ (resp. $m-i$).
\end{enumerate}
\end{prop}
\begin{Remark}
\label{1.4.3} Let $A$ be a hyperbolic endomorphism of $\R^{2n}$ and
let $\omega$ be a linear symplectic form on $\R^{2n}$. If
$(e^{tA})^* \omega = e^{ct} \omega$ for $c$ a positive constant and
for all real $t$, then the stable manifold $W^s$ of the linearised
field $A$ is isotropic (i.e. contained in its symplectic orthogonal
complement). Indeed, for $v,w \in W^s$, $\omega(v,w) = e^{-ct} \;
\omega(e^{tA} v, e^{tA} w)$ tends towards $0$ when $t$ tends towards
$+\infty$, thus is zero. This allows us to extend corollary 2.9 to
the case where $X$ is a pseudo-gradient of a Morse function.
\end{Remark}
\subsubsection{Notion and condition of convexity for a contact
structure}
In [EG], Ya. Eliashberg and M. Gromov propose the following
definition.
\begin{defn}
\label{1.4.4} We say that a contact structure $\xi$ on a manifold
$V$ is \emph{convex} if there exists a proper Morse function $f: V
\To [0, \infty)$ having a complete pseudo-gradient which preserves
$\xi$.
\end{defn}
The regular levels of $f$ are then convex hypersurfaces. Moreover it
follows from 2.C and 4.A that:
\begin{prop}
\label{1.4.5} Let $(V, \xi)$ be a contact manifold and $f: V \To [0,
\infty)$ a proper Morse function. If $\xi$ is preserved by a
pseudo-gradient of $f$, the characteristic hypersurface $C$ of this
vector field satisfies the following:
\begin{enumerate}
\item
$f|_C$ is a proper Morse function;
\item
the critical points of $f$ are on $C$ and are exactly the
critical points of $f|_C$;
\item
a critical point of index $i$ for $f$ gives, for $f|_C$, a
critical point of index $i$ if $i \leq n$ and of index $i-1$ if
$i \geq n+1$.
\end{enumerate}
\end{prop}
In part III, we will show how to construct, conversely, convex
contact structures on a 3-dimensional manifold $V$, starting from a
Morse function and a surface in $V$ satisfying the above conditions.
\subsubsection{Examples of convex contact structures}
\begin{Example}
(Contactization of a Weinstein manifold)
\begin{defn}[Ya. Eliashberg and M. Gromov, EG ] We say that a
symplectic manifold $(W, \omega)$ is \emph{Weinstein} if there
exists a proper Morse function $f_0: W \To [0, \infty)$ having a
complete pseudo-gradient $X_0$ which dilates $\omega$ exponentially:
$L(X_0) \omega = \omega$. Such a symplectic manifold is therefore
exact since, as $\omega$ is closed, we have $\omega = d\beta$ where
$\beta = i(X_0) \omega$.
\end{defn}
Under these conditions, the contact structure $\xi$ defined on $W
\times \R$ by the equation $\beta + dt = 0$ is convex. Indeed,
the field $X = X_0 + t(\partial/\partial t)$ preserves $\xi$ since
$L(X)(\beta + dt) = \beta + dt$. Moreover, $X$ is a complete
pseudo-gradient for the proper Morse function $f: W \times \R \To
[0, \infty)$ given by $f(x,t) = f_0(x) + t^2$.
A typical example of a Weinstein manifold is the cotangent space (of
any manifold whatsoever) furnished with its canonical symplectic structure
$\omega$. In this case, we can choose $X_0$ so that $\beta = i(X_0)
\omega$ differs from the canonical Liouville form by the
differential of a function. The contactization of $\beta$ is then
isomorphic to the canonical contact structure on the space of 1-jets
of functions (see remark 3.3a (\ref{1.3.3}, (i))): this structure is
consequently convex.
\end{Example}
\begin{Example}
The contact structure given on $S^{2n+1}$ by the complex tangencies
of the unit sphere in $\C^{n+1}$ is convex. Indeed, if $z_j = x_j
+ i y_j$, $1 \leq j \leq n+1$, are the coordinates, this structure
has for example the form induced by $-\sum y_j \; dx_j$; we check
then that the contact vector field associated to the hamiltonian
$x_k$ is a pseudo-gradient of the function $y_k$. The characteristic
hypersurface of this vector field is the equatorial sphere with
equation $x_k = 0$.
\end{Example}
\begin{Example}(Canonical structure on the manifold of contact
elements.) Let $\pi: V \To M$ be the fibre bundle of contact
elements on a manifold $M$ of dimension $n+1$. Then the canonical
contact structure on $V$ (see [A]) is convex.
\emph{Argument.} Given a proper Morse function $f_0: M \To [0,
\infty)$, we choose a complete pseudo-gradient $X_0$ of $f_0$ having
the following property: at each critical point of $f_0$, the
eigenvalues of $X_0$ are real and distinct. Like all vector fields
on $M$, the vector field $X_0$ lifts naturally to a contact vector
field $X$ on $V$. It turns out that $X$ is a complete
pseudo-gradient for some proper Morse function $f: V \To [0,
\infty)$. We obtain $f$ by perturbing by perturbing the
function $f_0 \circ \pi$ above a neighbourhood of the critical
points of $f_0$ as follows: above such a point $x$, the vector field $X$ is
tangent to the fibre $F = \pi^{-1}(x)$ and is none other than the
vector field induced naturally by the linearisation of $X_0$ on the
projective cotangent space; as the eigenvalues of $X_0$ are real and
distinct, $X|_F$ is the gradient of a Morse function $g: F \To [0,
\infty)$ having exactly $n+1$ critical points with distinct indices;
it is this function $g$, properly weighted and extended, that we add
to $f_0 \circ \pi$.
\end{Example}
\begin{Remark} The characteristic hypersurface of the vector field
$X$ above is the conormal of the vector field $X_0$.
\end{Remark}
\section{On the characteristic foliation of surfaces in dimension 3}
\subsection{Properties of characteristic foliations}
We are interested here in singular foliations on a surface $S$ which
can be realised as characteristic foliations by embedding $S$ in a
3-dimensional contact manifold. In such a manifold, naturally
oriented, the normal bundle of $S$ is isomorphic to the bundle
$\wedge^2 TS$; this allows us to speak of germs of contact
structures along $S$ without specifying the ambient manifold.
\subsubsection{General form of characteristic foliations}
\begin{defn}
\label{2.1.1} We say that a singularity $x$ of a vector field $Y$ is
\emph{isochore}\footnote{Straight from the French!} if the
divergence of $Y$ at $x$ is zero. An isochore singularity of $Y$ is
then an isochore singularity of $f \cdot Y$ for any function $f$;
this notion is therefore well defined for singular foliations in the
sense of 1.1.B.
\end{defn}
\begin{prop}
\label{2.1.2} Let $\F$ be a singular foliation on a surface $S$. We
fix an orientation on the manifold $\wedge^2 TS$ and we are
interested only in germs of contact structures along $S$ which give
this orientation.
\begin{enumerate}
\item
$\F$ is the characteristic foliation induced on $S$ by a germ of
contact structures if and only if $\F$ is without isochore
singularities.
\item
If $S$ is closed, two germs of contact structures which induce
the same characteristic foliation $\F$ are isomorphic: they are
conjugate by a germ of a diffeomorphism which is isotopic to the
identity through diffeomorphisms preserving $\F$.
\end{enumerate}
\end{prop}
\begin{Proof}
\begin{enumerate}
\item
The absence of isochore singularities is necessary; indeed,
if $\alpha$ is a contact form which induces on $S$ a form
$\beta$ which is null at $x$, the form $d\beta(x)$, which is none other
than $d\alpha(x)|_{\text{Ker } \alpha(x)}$, is nondegenerate; in
other words, the vector field $Y$ given near $x$ by $\beta =
i(Y) d\beta$ has nonzero divergence at $x$.
The converse and (ii) rest on the following fact: let $S_0$ be
an orientable surface; a 1-form $\alpha = \beta_t + u_t \; dt$
on $S_0 \times \R$ is contact if and only if:
\begin{eqnarray}
\label{part2star}
u_t \; d\beta_t + \beta_t \wedge \left( du_t - \frac{\partial
\beta_t}{\partial t} \right) \quad \text{is nowhere zero.}
\end{eqnarray}
In particular, $\beta_0$ being given, the pairs $(u_0, (\partial
\beta_t/\partial t)|_{y=0})$ which satisfy this inequality for
$t=0$, with a fixed sign, form a convex set; yet these pairs are
those which determine a contact structure. We now suppose that
$S$ is orientable and we take on $S$ an area form $\omega$ such
that $\omega \wedge dt$ gives the chosen orientation on
$\wedge^2 TS \cong S \times \R$. We suppose additionally that
$\F$ is transversally orientable, that is, given by an
equation $\beta = 0$, where $\beta$ is a 1-form on $S$. We
denote by $u$ the function defined on $S$ by $d\beta = u
\omega$, and we take a 1-form $\gamma$ on $S$ such
that the 2-form $\beta \wedge \gamma$ is positive or zero
with respect to $\omega$, and strictly positive outside the singular
locus of $\beta$. We then set $\beta_t = \beta + t(du - \gamma$.
The condition \ref{part2star} shows immediately that the 1-form
$\beta_t + u \; dt$ defines a contact structure near $S \times
\{0\}$ in $S \times \R$; indeed:
\[
u \; d\beta + \beta \wedge \left( du - \frac{\partial
\beta_t}{\partial t}|_{t=0} \right) = u^2 \omega + \beta
\wedge \gamma.
\]
Yet, as $\F$ has no isochore singularities, $u$ is nonzero at
each point of the singular locus of $\beta$.
Finally, if $S$ is not orientable or if $\F$ is not transversally
orientable, we remedy this problem by passing to a cover
of order 2 or 4 on which we carry out the preceding construction in
an invariant manner.
\item
Passing if necessary to a double cover of $S$ on which $\F$
becomes transversally orientable, we reduce to the case
where the two germs of contact structures are transversally
orientable. They then admit equations $\alpha_0$ and $\alpha_1$
which induce on $S$ the same form. The formula \ref{part2star}
shows that the kernel $\xi_s$ of $\alpha_s = (1-s)\alpha_0 +
s \alpha_1$ is, near $S$, a contact structure for all $s \in
[0,1]$.
We now seek, by J. Moser's method, an isotopy $\varphi_s$, $s \in
[0,1]$, which takes $\xi_0$ to $\xi_s$, i.e. satisfies:
$\alpha_0 \wedge \varphi_s^* \alpha_s = 0$. This condition shows
that the path $s \mapsto \varphi_s^* \alpha_s$ remains on the ray
$\{r \alpha_0, r > 0 \}$ in the space of 1-forms; in other
words:
\[
\varphi_s^* \alpha_s \wedge \frac{\partial}{\partial
s}(\varphi_s^* \alpha_s) = 0 \quad \text{ for all $s$}.
\]
Denoting by $X_s$ the infinitesimal generator of $(\varphi_s)$, this
relation can be written:
\[
(L(X_s)\alpha_s)|_{\xi_s} = - \frac{\partial
\alpha_s}{\partial s}|_{\xi_s}.
\]
We take for $X_s$ the vector field satisfying at the same time
\[
i(X_s)\alpha_s = 0 \quad \text{and} \quad (i(X_s)
d\alpha_x)|_{\xi_S} = - \frac{\partial \alpha_s}{\partial
s}|_{\xi_s}.
\]
This system has a unique solution by definition of a contact
structure. Moreover, if $v$ is a vector of $\xi_x \cap TS =
\xi_0 \cap TS$, we have $(\partial \alpha_s / \partial s)(v) =
0$, then $d\alpha_s(X_s, v) = 0$. This shows that $X_s$ is
tangent to $\F$ along $S$. We finally use that $S$ is closed to
integrate $X$ to an isotopy.
\end{enumerate}
\end{Proof}
\subsubsection{Generic properties of characteristic foliations}
The space of singular foliations on a surface $S$ (in the sense of
I.1.B) has a natural topology as the quotient of the space of plane
fields by the null section in $\wedge^2 TS$. If now $S$ is embedded
in an oriented 3-dimensional manifold $V$, the function which to a
plane field on $V$ associates the induced foliation on $S$ is open.
As the set of contact structures forms an open set, its image is an
open set in the space of singular foliations of $S$. Also, contact
structures being locally stable by a theorem of J. Gray [G], we
see:
\begin{lem}
\label{2.1.3} Let $\P$ be a property of singular foliations which is
$C^\infty$-generic and let $S$ be a surface embedded in a contact
manifold $(V, \xi)$. We can move $S$ by a $C^\infty$-small isotopy
so that its characteristic foliation satisfies $\P$.
\end{lem}
\begin{Example}
Recall that a vector field on a closed surface is said to be Morse-Smale if it
satisfies the three following properties:
\begin{enumerate}
\item
the singularities and the periodic orbits of $X$ are hyperbolic;
\item
the $\alpha$-limit set (resp. $\omega$-limit) of every point is
a singularity or a limit cycle;
\item
there are no connections between saddles.
\end{enumerate}
After a theorem of M. Peixoto, a vector field on a closed orientable
surface is $C^\infty$-generically Morse-Smale.
Let then $S$ be a closed orientable surface in a contact manifold
$(V, \xi)$. If $\xi$ is transversally orientable, the characteristic
foliation of $S$ is directed by a vector field that we can make
Morse-Smale by a $C^\infty$-small isotopy of $S$ in $V$.
\end{Example}
\subsection{Convex surfaces}
\subsubsection{Dividing set of a convex surface}
Recall that a surface $S$, embedded in a 3-dimensional contact
manifold $(V, \xi)$, is called convex if there exists a contact vector
field transverse to $S$. Such a surface is transversally orientable,
therefore orientable. It follows immediately from propositions I.3.4
and II.1.2(b) that:
\begin{prop}
\label{2.2.1} Let $(V, \xi)$ be a 3-dimensional contact manifold,
$S$ a closed orientable surface embedded in $V$ and $\F$ its
characteristic foliation. Then the surface $S$ is convex if and only
if there exists on $S$ a curve $\Gamma$ transverse to $\F$, in
general disconnected, which decomposes $S$ into subsurfaces where
$\F$ can be directed by a dilating vector field, for a certain area
form, and exiting through the boundary.
\end{prop}
\noindent \emph{Remark.}In particular, if $S$ is convex, all leaves
of $\F$ cut $\Gamma$ at most once.
\medskip
In the following, we will say that $\Gamma$ is the \emph{dividing
set} of $S$ (see remark 2.3 (\ref{2.2.3})). The data of a contact vector
field $X$ transverse to $S$ realises this dividing set as the
curve of points of $S$ where $X$ is tangent to $\xi$.
\begin{prop}
\label{2.2.2}
\begin{enumerate}
\item
Let $S$ be a closed surface. Two vertically invariant contact
structures on $S \times \R$ which define the same orientation
and induce the same characteristic foliation $\F$ on $S \times
\{0\}$ are isotopic: they are conjugate by a product
diffeomorphism $\varphi \times Id$, where $\varphi$ is isotopic
to the identity through diffeomorphisms which preserve $\F$.
Moreover, if the dividing set of $S$ associated to the vertical
vector field is the same for the two structures, it is preserved
all along the isotopy.
\item
For $i=0,1$ let $S_i$ be a convex surface in a contact manifold
$(V_i, \xi_i)$; let $\F_i$ be the characteristic foliation and
$X_i$ a contact vector field transverse to $S_i$ (the data of $X_i$
orients $S_i$). If $S_0$ and $S_1$ are closed (compact without
boundary) and there exists a diffeomorphism from $S_0$ to $S_1$
which respects orientations and sends $\F_0$ to $\F_1$, then
there exists a germ of a contact diffeomorphism, from $(V_0,
S_0)$ to $(V_1, S_1)$, which sends $X_0$ to $X_1$.
\end{enumerate}
\end{prop}
\begin{Proof}
(ii) follows immediately from (i) which is proved exactly like (ii)
in proposition 1.2. The isotopy consists of sliding along the leaves
of $\F$ to make one of the contact structures turn into the other.
This is possible in general only if $S$ is closed.
\end{Proof}
\begin{Remark}
\label{2.2.3} Proposition 2.2 shows that the characteristic
foliation $\F$ of a convex surface $S$ totally determines, up to
isotopy through curves transverse to $\F$, the dividing set
$\Gamma$, that is the trace on $S$ of the characteristic
surface of a transverse contact field. In paragraph 3, we will see
to what extent this curve reveals the geometry of the characteristic
foliation of $S$. But first we give geometric criteria for convexity and
non-convexity and we show in particular that an orientable surface
is generically convex. This genericity, exceptional, is related to
the fact that every open connected set in $\R$ (respectively, in
$\C$) is convex (respectively pseudo-convex): in dimension 3, the
minimal dimension of contact manifolds, convexity is a degenerate
property.
\end{Remark}
\subsubsection{Examples of non-convex surfaces}
A contact structure on $S \times \R$ invariant under
$\partial/\partial t$ is (locally) defined by equations of the type
$\beta + u \; dt = 0$ where $\beta$ and $u$ are respectively a
1-form and a function on (a neighbourhood of) $S$ such that:
\begin{eqnarray}
\label{part2starstar} u \; d\beta + \beta \wedge du \quad \text{ is
nowhere zero}.
\end{eqnarray}
The characteristic foliation $\F$ of $S$ is then defined by $\beta =
0$. If $\omega$ is an area form on $S$ and if $Y$ is the vector
field which directs $\F$ defined by $i(Y) \omega = \beta$, then
condition \ref{part2starstar} says
\begin{eqnarray}
\label{part2starstarstar}
u \text{div}_\omega(Y) - Y \cdot u \neq 0.
\end{eqnarray}
This shows immediatlely that the characteristic foliation of a
closed convex surface $S$ cannot be defined by a closed
(nonsingular) form. For example, the invariant tori of the Hopf
fibration in $S^3$ are not convex for the standard structure. We see
similarly that, if $S$ is convex, its characteristic foliation $\F$
possesses no closed leaf having a first return map tangent to the identity. Indeed, in a neighbourhood of one such leaf $F$, the foliation $\F$
admits an equation $\beta = 0$ where $d\beta|_F$ is identically
zero; it is then impossible to find a function $u$ such that $u \;
d\beta + \beta \wedge du$ is nonzero on $F$ since $u|_F$ necessarily
has critical points. Finally, convexity forbids certain connections
of saddles; to be precise, we say:
\begin{defn}
\label{2.2.4} Let $x$ be a non-isochore singularity of a singular
foliation $\F$. We say that we \emph{positively orient} $\F$ at $x$
when we choose, to direct $\F$ near $x$, a vector field for which
the divergence at $x$ is positive.
\end{defn}
If $S$ is convex, no leaf of its characteristic foliation joins two saddles while being a stable separatrix for both when they are positively oriented. This results for example from \ref{part2starstarstar}:
if, near such a leaf $F$, we orient the foliation by a vector field
$Y$ directed from the saddle $x_0$ towards the saddle $x_1$, we must
have $u(x_0)$ negative and $u(x_1)$ positive. Yet, by
\ref{part2starstarstar} $u$ can be zero only when decreasing in the
direction of $Y$.
\subsubsection{Examples of convex surfaces}
\begin{defn}
\label{2.2.5} We say that a singular foliation $\F$ on a closed
surface $S$ is \emph{Morse-Smale} if it satisfies the following
conditions:
\begin{enumerate}
\item
the singularities and the closed leaves of $\F$ are hyperbolic;
\item
the limit set of each half-leaf is a singularity or a closed
leaf;
\item
$\F$ has no connections between saddles.
\end{enumerate}
We say that $\F$ is \emph{almost Morse-Smale} if it satisfies (i),
(ii) and: \\
(iii)' when we orient $\F$ positively near saddles, the associated
stable manifolds do not intersect.
\end{defn}
\begin{prop}
\label{2.2.6} Let $S$ be an orientable closed surface embedded in a
contact manifold $(V, \xi)$. If the characteristic foliation $\F$ of
$S$ is almost Morse-Smale, then $S$ is convex.
\end{prop}
\begin{Proof}
By (b) of proposition 1.2 it suffices to construct on $S \times
\R$ a contact structure invariant under $\partial/\partial t$ which
makes $\F$ the characteristic foliation on $S \times \{0\}$. Around
each closed leaf (resp. each focus), we take an annulus (resp. a
disk) with boundary transverse to $\F$. Near saddles, we orient $\F$
positively. Using (ii) of definition 2.5, we place bands around
their stable manifolds so that the union of these annuli, discs and
bands forms a surface $S_0$ with boundary transverse to $\F$ (see
Figure 1). By construction, using (iii)' of definition 2.5, on a
neighbourhood $U$ of $S_0$, $\F$ is directed by a vector field $Y$ exiting
along $\partial S_0$ and for which singularities have positive
divergence. There then exists an area form $\omega$ on $S$ such that
$\text{div}_\omega (Y) > 0$ on $U$. We set $u=1$, then $u
\text{div}_\omega (Y) - Y \cdot u > 0$ on $U$.
On the surface with boundary $S' = \text{Cl}(S \backslash S_0)$,
$\F$ is a nonsingular foliation transverse to the boundary and
without closed leaves.\footnote{Cl here denotes closure.}
By (ii), as $S$ is orientable, $S'$ is a union of annuli foliated
by segments going from one boundary to another. We can then conclude
by using proposition 2.1 or the following elementary reasoning. We
choose on $S'$ a nonsingular vector field $Y'$ directing $\F$ and
coinciding with $\pm Y$ on a collar neighbourhood $U'$ of $\partial
S'$ in $U \cap S'$. We set $u' = \pm 1$ on $U'$ accordingly as $Y' =
\pm Y$; we then seek to extend to $S'$ the germ of $u'$ on the
boundary, so as to have: $u' \text{div}_\omega (Y') - Y' \cdot u' >
0$ on $S'$. This extension follows immediately from the following
remark.
\begin{Remark}
\label{2.2.7} Let $h: [0,1] \To \R$ be a function positive at $0$
and negative at $1$. There exists a function $v: [0,1] \To \R$ equal
to $1$ near $0$ and equal to $-1$ near $1$ such that $vh -
dv/d\theta$ is positive; we take $v(\theta) = w(\theta) \exp (
\int_0^\theta h(\sigma) d\sigma)$ where $w: [0,1] \To \R$ is a
suitable decreasing function.
\end{Remark}
\end{Proof}
\subsection{Deformations of characteristic foliations}
\subsubsection{A reduced form for characteristic foliations}
Let $S$ be a closed orientable surface embedded in a 3-dimensional
contact manifold $(V, \xi)$ with a Morse-Smale characteristic
foliation $\F$. By proposition 2.6, there exists a germ of a
contact vector field transverse to $S$. Given any neighbourhood $U$
of $S$, it's easy to extend this germ to a contact vector field for
which the flow defines an embedding $S \times \R \To V$ with image
$V_0 \subset U$. On $V_0 \cong S \times \R$, $\xi_0 = \xi|_{V_0}$ is
a contact structure invariant under $\partial/\partial t$ and the
characteristic surface of this contact vector field is a cylinder
$\Gamma \times \R$ where $\Gamma$ decomposes $S = S \times \{0\}$ as
indicated in 2.1. Then any function $h: S \To \R$ has for its graph
a convex surface $S_h$ contained in $V_0$ having the ``same dividing
set $\Gamma$" as $S$.
\begin{prop}
\label{2.3.1} There exists a function $h: S \To \R$ such that the
characteristic foliation $\F_h$ of $S_h$ is Morse-Smale and gives,
on each component $S'$ of the surface obtained by decomposing $S_h$
along $\Gamma$, the following:
\begin{enumerate}
\item
if $S'$ is a disk, $\F_{h|_{S'}}$ has a unique singularity which
is a focus and has no closed leaves: topologically it is a radial
foliation;
\item
if $S'$ is not a disk, $\F_{h|_{S'}}$ has exactly one closed
leaf and only has saddles for singularities.
\end{enumerate}
Moreover we can choose $h$ non-positive.
\end{prop}
We will show this proposition in C; it also follows from proposition 3.6.
\subsubsection{Elimination of singularities}
\begin{defn}
Given a singular foliation without isochore singularities on a
surface, we say that a focus $x_0$ and a saddle $x_1$ are in
\emph{simple elimination position} (resp. in \emph{cyclic
elimination position}) if when we positively orient the foliation
near $x_1$, one and only one stable separatrix comes from $x_0$
(resp. two stable separatrices come from $x_0$).
\end{defn}
\begin{lem}[Elimination lemma](see [El1] theorem 6.1 and
[El2]). \label{2.3.3} With the notation and the hypotheses of 3.A,
let $x_0$ and $x_1$ be a focus and a saddle of $\F$ in simple or
cyclic elimination position.
\begin{enumerate}
\item
There exists in $S$ an annulus $A$ disjoint from $\Gamma$ and
satisfying:
\begin{itemize}
\item
the only singularities of $\F$ on $A$ are $x_0$ and $x_1$;
\item
$\F|_A$ has no closed leaf;
\item
$\F$ is transverse to the boundary of $A$.
\end{itemize}
The two configurations are shown in figures 2 and 3.
\item
There exists a function $k: A \To (-\infty, 0]$ with support in
the interior of $A$ and such that the characteristic foliation
on the graph of $k$ has no singularity.
\end{enumerate}
\end{lem}
\begin{Proof}
\begin{enumerate}
\item
Let $S'$ be the connected component of $x_1$ in the surface
obtained by decomposing $S$ along $\Gamma$. There exists on $S'$
a vector field which directs $\F$, exiting along $\partial S'$
and dilating some area form on $S'$. In particular this vector
field positively orients $\F$ near $x_1$ and the stable manifold
$W^s(x_1)$ lies in $S'$.
If $x_0$ and $x_1$ are in cyclic elimination position, we choose
for $A$ an annular neighbourhood of the union $\{x_0\} \cup
W^s$.
If $x_0$ and $x_1$ are in simple elimination position, choose
one of the two: either the other branch of $W^s$ comes from a
focus $x_2$, or it comes from a closed leaf $F$ necessarily
disjoint from $\Gamma$. In the first case, we take for $A$ a
disk neighbourhood of the union $\{x_0, x_2\} \cup W^s$, minus a
disk around $x_2$. In the second case, we first take an annulus
$A'$ around $F$ with boundary transverse to $\F$; the branch of
$W^s$ which comes from $F$ then cuts $\partial A'$ in a point
$x$; we take for $A$ a neighbourhood of the union of the arc
which joins $x$ to $x_0$ in $W^s$ and of the component of $x$ in
$\partial A'$.
\item
As $A$ is disjoint from $\Gamma$, the contact structure on $A
\times \R$ is the contactisation of a Liouville form $\beta$ on
$A$; in other words, it has an equation of the form $\beta + dt
= 0$. Thus, for any function $k: A \To \R$ the characteristic
foliation on the graph $A_k$ of $k$ is defined by $\beta + dk =
0$.
Let then $\omega$ be any area form on $A$ and $Y$ the vector
field given by $i(Y) \omega = \beta$. We seek to add to $Y$ the
$\omega$-Hamiltonian $Y_k$ of a function $k$ with support in the
interior of $A$ such that $Y+Y_k$ is nonsingular (here $Y_k$ is
defined by $i(Y_k)\omega = dk$. For this we take, on $A$, a
foliation by circles parallel to the boundary which we denote
$\mathcal{G}$. On a neighbourhood $B$ of $\partial A$ in $A$,
$\F$ and $\mathcal{G}$ are transverse. On $A \backslash B$, the
vector field $Y$ is bounded. We
therefore choose a function $k: A \To (-\infty, 0]$, with
support in the interior of $A$, constant on the leaves of
$\mathcal{G}$, and for which the $\omega$-Hamiltonian $Y_k$ is
very large on $A \backslash B$. Then $Y + Y_k$ is nonzero on $A
\backslash B$. On $B$, $Y$ is nonsingular and is transverse to
$Y_k$ there or $Y_k$ is nonzero. Then $Y + Y_k$ is everywhere
nonzero.
\end{enumerate}
\end{Proof}
\begin{Remark}
In the case where $x_0$ and $x_1$ are in cyclic elimination
position, we thus create a closed leaf.
In the case where $x_0$ and $x_1$ are in simple elimination
position all leaves go from one boundary to the other of the
annulus.
We can easily check that this construction preserves the
Morse-Smale character of the foliation.
\end{Remark}
\subsubsection{End of the proof of proposition 3.1}
Let $S'$ be a component of the surface obtained by decomposing $S$
along $\Gamma$. On $S'$, we choose a vector field $Y$ which directs
$\F$ and which dilates a given area form; the foci and the closed
orbits of $Y$ are then repulsive.
\begin{enumerate}
\item
We suppose that $S'$ is a disk. Then $S'$ does not contain any
closed orbits since $Y$ is dilating. If $x_1 \in S'$ is a saddle
of $Y$, its stable manifold lies in $S'$, therefore $x_1$ is in
elimination position with a focus. When we have eliminated all
the saddles, there remains only one focus.
\item
We now suppose that $S'$ is not a disk. The orbits of $Y$ which
leave from a focus $x_0 \in S'$ cannot go across a closed orbit
$F \subset S'$. Nor can they all exit since $S'$ is not a
disk. It follows that at least one goes
towards a saddle $x_1 \in S'$ such that we can eliminate all the
foci of $S'$. Now, as the $\alpha$-limit set of every point
in $S'$ is in $S'$, $S'$ contains at least one closed orbit. If
it contains only one, we stop. If it contains two, $F$ and $F'$,
then $S'$ is not an annulus and there exists at least one saddle
$x$ in $S'$ for which one and only one separatrix comes from
$F'$. Indeed, if not, let $y_1, \ldots, y_p$ be the saddles for which
one separatrix (and in fact the whole stable manifold) comes
from $F'$; the set of points of $S'$ which have for
$\alpha$-limit one of the $y_i$, or $F'$, is a connected
component of $S'$, but $S'$ is connected. By the inverse
procedure to cyclic elimination, we replace $F'$ by a focus
$x_0$ and a saddle $x_1$ in cyclic elimination position. The
separatrix of $x$ which came from $F'$ now comes from $x_0$ so
that $x_0$ and $x$ are in simple elimination position.
\end{enumerate}
\qed
\subsubsection{Foliations adapted to a given dividing set}
Let $S$ be a convex closed surface in a 3-dimensional contact
manifold $(V, \xi)$, and let $X$ be a contact vector field
transverse to $S$ whose flow defines an embedding $S \times
\R \To V$. We denote by $\Gamma$ the dividing set of $S$ associated
to $X$, the curve consisting of points of $S$ where $X$ is tangent
to $\xi$, and we denote by $S_\Gamma$ the compact surface with
boundary obtained by decomposing $S$ along $\Gamma$.
\begin{defn}
\label{2.3.5}
\begin{enumerate}
\item
An \emph{admissible isotopy} of $S$ in $V$ is an isotopy of $S$
through surfaces transverse to $X$, which in particular avoid
singularities of $X$.
\item
A singular foliation on $S$ is \emph{adapted} to $\Gamma$ if the
foliation induced on $S_\Gamma$ is directed by a vector field
which dilates an area form and which exits transversely through
the boundary $\partial S_\Gamma$.
\end{enumerate}
\end{defn}
\begin{prop}
\label{2.3.6} Let $\F$ be a foliation on $S$ adapted to $\Gamma$.
Then there exists an admissible isotopy $\delta_s: S \To V$, $s \in
[0,1]$, such that the characteristic foliation on $\delta_1 S$ is
$\delta_1 \F$. Moreover, for all $s \in [0,1]$, the dividing set of
$\delta_s S$ associated to $X$ is $\delta_s \Gamma$.
\end{prop}
\begin{Proof}
We denote by $\F_0$ the characteristic foliation of $S$ and by
$\xi_0$ the vertically invariant contact structure induced on $S
\times \R$ by the flow of $X$, $\psi: S \times \R \To V$. We take an
area form $\omega$ on $S$ such that $\omega \wedge dt$ orients $S
\times \R$ like $\xi_0$; finally we take a closed tubular
neighbourhood $A$ of $\Gamma$, small enough so that $\F$ and $\F_0$
foliate it by segments from one boundary to the other.
On $(S \backslash \text{int} A) \times \R$, $\xi_0$ admits a unique
equation of the type $i(Y_0) \omega + dt = 0$, where $Y_0$ is a
vector field on $S \backslash \text{int} A$ which directs $\F_0$ and
which dilates $\omega$. Also, as $\F$ is adapted to $\Gamma$ there
exists on $S \backslash \text{int} A$ a vector field $Y$ which
directs $\F$ and which dilates a certain area form; observing that
$\text{div}_{\pm e^g \omega} (Y) = e^{-g} \text{div}_\omega (e^g
Y)$, we replace $Y$ with a vector field $Y_1$ which dilates $\omega$.
For $s \in [0,1]$, we set $Y_s = (1-s) Y_0 + s Y_1$. Then, the
equation $i(Y_s) \omega + dt = 0$ defines, for each $s$ in $[0,1]$,
a vertically invariant contact structure $\xi_s$ on $(S \backslash
\text{int} A ) \times \R$. Now, on a small neighbourhood $U$ of $A$
in $S$, we take vector fields $Y'_0$ and $Y'_1$ which respectively
direct $\F_0$ and $\F$ and which coincide with $\pm Y_0$ and $\pm
Y_1$ on $U \cap (S \backslash \text{int} A)$; for $s \in [0,1]$, we
again set $Y'_s = (1-s) Y'_0 + s Y'_1$. On $U \times \R$, the
contact structure $\xi_0$ is defined by a unique equation of the
type $i(Y'_0) \omega + u_0 dt$; the function $u_0$ is zero on
$\Gamma$, is equal to $\pm 1$ wherever $Y'_0 = \pm Y_0$ and
satisfies on $U$: $u_0 \text{div}_\omega (Y'_0) - Y'_0 \cdot u_0 >
0$.
Then, using remark 2.7, we form a family $u_s$ of functions on $U$
such that, for all $s \in [0,1]$, we have $u_s \text{div}_\omega
(Y'_s) - (Y'_s \cdot u_s) > 0$, with $u_s = \pm 1$ wherever $Y'_s =
\pm Y_s$. We thus obtain on $S \times \R$ a family still denoted
$\xi_s$, $s \in [0,1]$, of vertically invariant contact structures;
by construction, the characteristic surface of the vertical vector
field is $\Gamma \times \R$ for each structure $\xi_s$, and the
characteristic foliation induced by $\xi_1$ on $S \times \{0\}$ is
none other than $\F$.
J. Moser's method (see the proof of proposition 1.2) then
provides a family of vertically invariant vector fields on $S \times
\R$ which, since $S$ is closed, integrates to an isotopy
$\varphi_s$ satisfying $\varphi_s^* \xi_s = \xi_0$; moreover the
diffeomorphisms $\varphi_s: S \times \R \To S \times \R$ preserve
$\partial/\partial t$ therefore $\Gamma \times \R$; it follows that
$\varphi_s^{-1} (S \times \{0\})$ is always transverse to
$\partial/\partial t$ and is decomposed by its intersection with
$\Gamma \times \R$. Composing with a vertical translation, we can
arrange that $\varphi_s^{-1}(X \times \{0\})$ extends to $S \times
(-\infty, 0]$. We then set $\delta_s = \psi \circ \varphi_s^{-1}|_{S
\times \{0\}}$.
\end{Proof}
\emph{Remark.} The previous proposition allows us, with lemma 3.3,
to eliminate singularities and prove proposition 3.1. Equally it
gives other reduced forms for the characteristic foliation of convex
surfaces; for example:
\begin{Example}
(Foliation associated to a handle decomposition). Let $(S, X,
\Gamma, S_\Gamma)$ be as above. By \emph{handle decomposition} of
$S_\Gamma$, we mean a finite collection of arcs $\gamma_1, \ldots,
\gamma_r$ , disjoint in $S_\Gamma$, going from boundary to boundary,
and such that the complement in $S_\Gamma$ of a regular
neighbourhood $\Omega$ of $\partial S_\Gamma \cup \gamma_1 \cup
\cdots \cup \gamma_r$ is a disjoint union of disks $\Delta_1,
\ldots, \Delta_q$.
To any handle decomposition of $S_\Gamma$, we associate a singular
foliation of $S_\Gamma$, unique up to homeomorphism, in the
following manner: on each disk $\Delta_i$, we put a radial foliation
and, on $\Omega$, we take the foliation described in Figure 4; this
foliation is directed by a vector field exiting through $\partial
S_\Gamma$, entering through $\partial \Omega \backslash \partial
S_\Gamma$, which has no closed orbits and which has for
singularities precisely $r$ saddles with positive divergence for
which the unstable manifolds are the $\gamma_j$; note that the
stable manifolds of these saddles come from the centres of the disks
$\Delta_i$. By gluing, we construct on $S$ foliations adapted to
$\Gamma$, without closed leaves.
\end{Example}
\section{Construction of convex contact structures in dimension 3}
\subsection{Convex contact structures and essential surfaces}
\subsubsection{Existence results}
\begin{defn}
\label{3.1.1}
\begin{enumerate}
\item
Let $V$ be a 3-dimensional manifold and $f: V \To [0, \infty)$ a
proper Morse function. We say that a surface $C$
embedded in $V$, not necessarily
connected, is \emph{$f$-essential} if it satisfies the
following three properties:
\begin{enumerate}
\item
$f|_C$ is a proper Morse function;
\item
all critical points of $f$ are on $C$ and are exactly the
critical points of $f|_C$;
\item
a critical point of index 1 or 2 for $f$ is of index 1 for
$f|_C$; equivalently $f$ and $f|_C$ have the same local
extrema.
\end{enumerate}
\item
We say that a contact structure on an oriented 3-dimensional
manifold is positive if it induces the given orientation.
\end{enumerate}
\end{defn}
\begin{thm}[Existence theorem]
\label{3.1.2} Let $V$ be a 3-dimensional oriented manifold and $f: V
\To [0, \infty)$ a proper Morse function. There exists on $V$ a
contact structure which is preserved under a complete
pseudo-gradient of $f$ if and only if there exists in $V$ an
$f$-essential surface $C$.
\end{thm}
\emph{Remark.} In I.4, we saw that the existence of an $f$-essential
surface is necessary; we will prove that this is also sufficient.
The problem of existence of essential surfaces for a given function
will be discussed in part IV; from this discussion will follow a
version of the theorem of R. Lutz and J. Martinet (see [Ma]) for
convex contact structures, namely:
\begin{thm}
\label{3.1.3} Any oriented 3-dimensional manifold carries a positive
convex contact structure.
\end{thm}
\begin{defn}
\label{3.1.4} (Ya. Eliashberg [El3]). We say that a contact
structure on a 3-dimensional manifold $V$ is \emph{overtwisted} if
there exists a 2-dimensional disk embedded in $V$, for which the
characteristic foliation has a limit cycle (with exactly one
singularity in the interior according to [El3], but the arguments of
II.3 show that this condition adds nothing).
\end{defn}
R. Lutz has described a procedure for constructing on any
3-dimensional manifold an overtwisted contact structure [Lu]; we
will give a ``convex" version showing that:
\begin{cor}
\label{3.1.5} Any oriented 3-dimensional manifold carries a positive
convex overtwisted contact structure.
\end{cor}
By a theorem of M. Gromov and Ya. Eliashberg (see [Gr] and
[El1]), overtwisted contact structures are not symplectically
fillable (see [El1] and [EG] for the definition). It follows that there exist
contact structures which are convex but not symplectically fillable,
answering a question of [EG].
\subsubsection{Scheme of the proof of theorem 1.2}
Let $a_0 < a_1 < \cdots$ be the critical values of $f$, which we
suppose are distinct (only to simplify the exposition), and let $b_0
< b_1 < \cdots$ be intermediate regular values, i.e. so that $a_0 <
b_0 < a_1 < b_1 < \cdots$. We set $V_k = \{x \in V \; | \; f(x) \leq
b_k \}$ and $C_k = C \cap V_k$.
Then $V_{k+1}$ is obtained from $V_k$ by attaching a single handle of index
equal to the index of $f$ at the critical point $x_{k+1}$ of value
$a_{k+1}$. As $C$ is $f$-essential, $C_{k+1}$ is obtained
simultaneously from $C_k$ by attaching a handle of index equal to
the index of $f|_C$ at $x_{k+1}$. Precisely, let $H_i = D^i \times
D^{3-i}$ be a handle of index $i=0,1,2,3$; the attachment of $H_i$
to $V_k$ is given by an embedding $\varphi: \partial D^i \times
D^{3-i} \To \partial V_k$; the pair $(V_{k+1}, V_k)$ only depends on
the isotopy class of $\varphi$. For $j \leq i$, let $D^j$ be the
sub-dis $D^j \times \{0\}$ contained in $D^i$; then for an
appropriate choice of $\varphi$, the handle that we attach to $C_k$
is $D^j \times D^{2-j}$ with $j=0,1,1,2$ when $i=0,1,2,3$; we glue it
along the restriction of $\varphi$ to $\partial D^j \times D^{2-j}
\subset \partial D^i \times D^{3-i}$.
By induction on $k$, we will construct on $V_k$ a positive contact
structure $\xi_k$, as well as a pseudo-gradient $X_k$ of $f_k =
f|_{V_k}$, which preserves $\xi_k$ and for which the characteristic
surface is $C_k$. For this we distinguish four cases corresponding
to the different possible indices. It is not necessary to worry
about the problem of completeness since we can always handle it afterwards; indeed:
\begin{Remark}
\label{3.1.6} Let $c$ be a given positive number, $S$ a closed
surface and $\xi$ a vertically invariant contact structure on $S
\times [0,1]$. Then there exists on $S \times [0,1]$ a contact
structure $\xi'$ having the following properties:
\begin{enumerate}
\item
$\xi'$ coincides with $\xi$ near the boundary;
\item
$\xi'$ is preserved by a vector field $X'$ which is equal to
$\partial/\partial t$ near the boundary, and whose
orbits are the segments $\{ \cdot \} \times [0,1]$ covered in time
$c$.
\end{enumerate}
\end{Remark}
\begin{Proof}
We extend $\xi$ to a vertically invariant contact structure on $S
\times \R$ and we choose a diffeomorphism $\rho: [0,c] \To [0,1]$
which coincides with the identity near $0$ and with a translation
near $c$; we then take for $\xi'$ and $X'$ the images under $Id
\times \rho$ of $\xi$ and $\partial/\partial t$.
\end{Proof}
\subsection{Attachment of handles of index 0 and 3}
\subsubsection{The model}
On $\R^3$ oriented by $dx \wedge dy \wedge dz$, the plane field with
equation $dz + uy \; dx + vx \; dy = 0$, $u,v \in \R$, is a positive
contact structure if and only if $v-u > 0$. This plane field is
preserved by all vector fields of the form
\[
ax \frac{\partial}{\partial x} + by \frac{\partial}{\partial y}
+ cz \frac{\partial}{\partial z}, \quad a,b,c \in \R, \quad
\text{with } c = a+b;
\]
indeed, their flow at time $t$ is given by $(x,y,z) \mapsto
(e^{at} x, e^{bt} y, e^{ct} z)$. Finally, for $v-u>0$ and $c=a+b$,
the characteristic surface of the contact vector field so defined
has equation: $cz + (au+bv)xy = 0$.
Let $\zeta_0$ be the contact structure with equation $dz - y \; dx +
x \; dy = 0$. The contact vector fields
\[
Z_0 = x \frac{\partial}{\partial x} + y \frac{\partial}{\partial
y} + 2z \frac{\partial}{\partial z} \quad \text{ and } Z_3 = -Z_0
\]
both have for characteristic surface the plane $\{z=0\}$ are
pseudo-gradients respectively of $g_0 = x^2 + y^2 + z^2$ and $g_3 =
-g_0$.
We denote by $H_3$ the handle of index $3$: $\{(x,y,z) \in \R^3 \; |
\; x^2 + y^2 + z^2 \leq 1\}$ which we orient by $\zeta_0$; we denote
by $F_3$ the boundary of $H_3$ furnished with the
orientation induced by the entering vector field $Z_3$: this orientation
is opposite to the usual orientation of the unit sphere in $\R^3$ as
the boundary of the ball.
\subsubsection{Handles of index 0}
As $a_0$ is the minimum of $f$, there exists a diffeomorphism of
$V_0$ onto the closed ball $B^3 = \{(x,y,z) \in \R^3 \; | \; x^2 + y^2
+ z^2 \leq 1 \}$ which respects orientations, which sends $C_0$ to
$B^3 \cap \{z=0\}$ and which, up to an affine transformation of
$\R$, conjugates $f_0 = f|_{V_0}$ to $x^2 + y^2 + z^2$. Then the
inverse of this diffeomorphism transforms $\zeta_0$ into a contact
structure $\xi_0$ on $V_0$ and sends the vector field $Z_0$ to a
pseudo-gradient $X_0$ of $f_0$; by construction, this
pseudo-gradient preserves $\xi_0$ and has $C_0$ as its
characteristic surface.
Any ``attachment" of a handle of index 0 can be dealt with in the
same manner.
\subsubsection{Handles of index 3}
On $V_k$ we have, by the inductive hypothesis, a contact structure
$\xi_k$, as well as a pseudo-gradient $X_k$ of $f_k = f|_{V_k}$ which
preserves $\xi_k$ and has characteristic surface $C_k$.
\begin{defn}
Let $S \subset \partial V_k$ be a surface. We will say that an
isotopy $\delta_s$ of embeddings of $S$ in $V_k$ is \emph{admisible}
if, for all $s$, $\delta_s S$ is transverse to $X_k$ in $V_k$ and
cuts $C_k$ along $\delta_s (S \cap C_k)$.
\end{defn}
It is clear that such an isotopy extends to an isotopy of embeddings
$\bar{\delta}_s: V_k \To V_k$ which are \emph{admissible} in the
following sense:
\begin{itemize}
\item
for all $s$, $\bar{\delta}_s$ sends $C_k$ to $C_k$;
\item
for all $s$, $\bar{\delta}^*_s$ is still a pseudo-gradient of
$f_k$ and evidently preserves the positive contact structure
$\bar{\delta}^*_s \xi_k$.
\end{itemize}
We now suppose that $V_{k+1}$ is obtained from $V_k$ by attaching a
handle of index $3$, that is, by gluing a ball onto a spherical
component $S$ of $\partial V_k$. Simultaneously $C_{k+1}$ is
obtained by attaching to $C_k$ a disk along $S \cap \partial C_k$;
this intersection is tthus a connected curve $\Gamma$. We denote by
$\phi: F_3\To S$ a gluing diffeomorphism which respects
orientations and sends $F_3 \cap \{z=0\}$ onto $\Gamma$.
\begin{lem}
We can find an admissible isotopy $\delta_s: S \To V_k$, $s \in
[0,1]$, such that there exists a germ of a diffeomorphism $\psi:
(H_3, F_3) \To (V_k, \delta_1 S)$ having the following properties:
\begin{enumerate}
\item
$\psi|_{F_3}$ is isotopic to $\delta_1 \phi$ through
diffeomorphisms of $F_3$ in $\delta_1 S$ which send $F_3 \cap
\{z = 0\}$ to $\delta_1 \Gamma$;
\item
$\psi$ takes $\zeta_0$ to $\xi_k$ and $Z_3$ to $X_k$.
\end{enumerate}
\end{lem}
\begin{Proof}
By proposition II.2.2, it suffices to find an admissible isotopy
$\delta_s$ for which the diffeomorphism $\delta_1 \phi: F_3 \To
\delta_1 S$ respects orientations and sends the characteristic
foliation induced by $\zeta_0$ to that induced by $\xi_k$. This
isotopy is immediately given by proposition II.3.6 since the
foliation obtained on $S$ by transporting via $\phi$ the
characteristic foliation on $F_3$ is adapted to $\Gamma$.
\end{Proof}
Now let $\bar{\delta}_s$ be an admissible isotopy of embeddings $V_k
\To V_k$ which extends the isotopy $\delta_s$ of the above lemma (see
2.1). We can attach $H_3$ to $V_k$ by gluing on the one hand
$\bar{\delta}^*_1 (\xi_k)$ to $\zeta_0$, and on the other
$\bar{\delta}_1^* (X_k)$ to $Z_3$. We then extend $f_k$ to this
manifold via a function on $H_3$ which admits $Z_3$ as a
pseudo-gradient and equals $(a_{k+1} - x^2 - y^2 - z^2)$ near the
origin. On the other components of $\partial V_k$, we add an
exterior collar up to the level $b_{k+1}$; there, we extend $X_k$
trivially then $\xi_k$ in an invariant manner.
\subsection{Attachment of handles of index 1 and 2}
\subsubsection{The model}
On $\R^3$ oriented by $dx \wedge dy \wedge dz$, let $\zeta_1$ be the
positive contact structure with equation $dz + y \; dx + 2x \; dy =
0$. The contact vector fields
\[
Z_1 = 2x \frac{\partial}{\partial x} - y
\frac{\partial}{\partial y} + z \frac{\partial}{\partial z}
\quad \text{and} \quad Z_2 = - Z_1
\]
have as characteristic surface the plane $\{z=0\}$ and are
pseudo-gradients respectively of $g_1 = x^2 - y^2 + z^2$ and $g_2 =
-g_1$.
Given $\epsilon>0$, we denote by $H_1 = H_1(\epsilon)$ the handle of
index 1 $\{(x,y,z) \in \R^3 \; | \; x^2 + z^2 \leq \epsilon^2, \;
y^2 \leq 1 \}$ and we denote by $F_1 = F_1(\epsilon)$ the surface $H_1
\cap \{y = \pm 1\}$. The data of $\zeta_1$ and $Z_1$ orient $H_1$
and $F_1$. Similarly, we denote by $H_2$ the handle of index 2
$\{(x,y,z) \in \R^3 \; | \; y^2 \leq \epsilon^2, \; x^2 + z^2 \leq 1
\}$ and we denote by $F_2$ the surface $H_2 \cap \{x^2 + z^2 = 1\}$;
$H_2$ and $F_2$ are oriented by the data of $\zeta_1$ and $Z_2$.
If we parametrise $F_2$ by $(\theta,y) \mapsto (x = \sin \theta, \;
y, \; z = \cos \theta)$, $\theta \in [0, 2\pi]$, the orientation
described previously is given by $d\theta \wedge dy$. In addition the
characteristic foliation induced by $\zeta_1$ has equation: $(y \cos
\theta - \sin \theta) \; d\theta + 2 \; \sin \theta \; dy = 0$; this
appears therefore as in figure 5.
We can easily show that:
\begin{lem}
For $i=1,2$ and $\epsilon>0$ given, let $h_i$ be a non-singular
germ of a function along $F_i$, equal to a negative constant on
$F_i$. Then $h_i$ extends to a function on $H_i$ which coincides
with $g_i$ near the origin, for which $Z_i$ is a
pseudo-gradient.
\end{lem}
\subsubsection{Handles of index 2}
We suppose that $V_{k+1}$ (resp. $C_{k+1}$) is obained by attaching
to $V_k$ (resp. to $C_k$) a handle of index 2 (resp. of index 1).
This attachment is given by an embedding $\phi: F_2 \To S = \partial
V_k$ which respects orientations and which meets $\Gamma = \partial
C_k$ exactly along $F_2 \cap \{z=0\}$. The attaching curve
$\Theta$, the image under $\phi$ of $F_2 \cap \{y=0\}$, therefore cuts
$\Gamma$ in two points and is thus divided into two arcs
denoted $\Theta_+$ and $\Theta_-$. Finally, we denote by $S_\Gamma$
the surface obtained by decomposing $S$ along $\Gamma$. To construct
the contact structure $\xi_{k+1}$ and the vector field $X_{k+1}$ on
$V_{k+1}$ it suffices, by lemma 3.1, to show that:
\begin{lem}
We can find an admissible isotopy $\delta_s: S \To V_k$, $s \in
[0,1]$, such that there exists an annulus $A$ around $\Theta$, and a
germ of a diffeomorphism $\psi: (H_2, F_2) \To (V_k, \delta_1 A)$
having the following properties:
\begin{enumerate}
\item
$\psi|_{F_2}$ is isotopic to $\delta_1 \phi$ through embeddings of
$F_2$ in $\delta_1 A$ which meet $\delta_1 \Gamma$ exactly along
$F_2 \cap \{z = 0\}$;
\item
$\psi$ takes $\zeta_1$ to $\xi_k$ and $Z_2$ to $X_k$.
\end{enumerate}
\end{lem}
\begin{Proof}
We begin by building an admissible isotopy $\delta'_s: S \To
V_k$, $s \in [0,1]$, such that there exists an annulus $A$ around
$\Theta$, and a diffeomorphism $\psi': F_2 \To \delta'_1 A$ which
respects orientations, which meets $\delta'_1 \Gamma$ exactly along
$F_2 \cap \{z=0\}$ and which conjugates the characteristic
foliations induced respectively by $\zeta_1$ and $\xi_k$. For this, we take
two arcs $\gamma_+$ and $\gamma_-$ having their endpoints on
$\Gamma$ and satisfying the following conditions (see figure 6):
\begin{itemize}
\item
$\gamma_+$ and $\gamma_-$ are contained in a tubular
neighbourhood $\Omega$ of $\Theta$ in $S$ and are respectively
isotopic to $\Theta_+$ and $\Theta_-$ in $\Omega$; moreover they
do not cut $\Gamma$ in their interiors;
\item
$\gamma_\pm$ crosses $\Theta_\pm$ at a single point $m_\pm$;
\item
in $\Omega$, $\Theta$ cuts $\Gamma$ between $\gamma_+$ and
$\gamma_-$.
\end{itemize}
We then extend the data of $\gamma_+$ and $\gamma_-$ to a handle
decomposition of $S_\Gamma$ (see example II.3.7). The associated
foliation induces on $S$ a foliation $\F$ adapted to $\Gamma$
(Definition II.3.5) which, on an annulus $A$ around $\Theta$, is
conjugate to the germ of the characteristic foliation of $F_2$ along
the circle $\{y=0, x^2 + z^2 = 1\}$. Proposition II.3.6 provides an
admissible isotopy $\delta'_s: S \To V_k$ such that $\delta'_1 A$
has characteristic foliation $\delta'_1(\F)$. We thus obtain the
desired diffeomorphism $\psi': F_2 \To \delta'_1 A$.
Now, we extend $\psi'$ to a germ of a diffeomorphism, still denoted
$\psi'$, $(H_2, F_2) \To (V_k, \delta'_1 A)$, which sends $Z_2$ to
$X_k$. Thus, $\xi_k$ (resp. $\psi'_* \zeta_1$) induces on $S
\times \R$, via $\delta'_1$ and the flow of $X_k$, a vertically
invariant contact structure $\eta_0$ (resp. $\eta$). It then
suffices to establish the following fact:
\begin{lem}[Sub-lemma]
We can extend $\eta$ to $S \times \R$ as a vertically invariant
contact structure $\eta$ giving on $S \times \{0\}$ the same
characteristic foliation and the same dividing set as $\eta_0$.
\end{lem}
\emph{Proof that 3.3 implies 3.2.} As $S$ is closed, we can now
argue from the uniqueness of vertically invariant contact structure
which induce a given characteristic foliation on $S \times \{0\}$
(proposition II.2.2): there exists an isotopy $\varphi_s: S \times
\R \To S \times \R$, which preserves at once $\partial/\partial S$
and the levels $S \times \{t\}$, such that $\varphi_1$ straightens
$\eta_1$ to $\eta_0$. We then obtain an admissible isotopy
$\delta_s$ and the desired diffeomorphism $\psi$ by correcting by
$\varphi_1$ the isotopy $\delta'_s$ and the diffeomorphism $\psi'$.
\end{Proof}
\begin{Proof}[of 3.3]
As $\psi'$ meets $\delta'_1 \Gamma$ exactly along $F_2 \cap
\{z=0\}$, there exists a function $h: A \To \R$ such that, if
$\eta_0$ is defined near a point of $A \times \R$ by an equation
$\beta + u \; dt = 0$, then $\eta_1$ is defined near this point by
$\beta + e^h u \; dt = 0$. We extend $h$ arbitrarily in a
neighbourhood of $\Gamma$.
On $S_\Gamma \times \R$, $\eta_0$ induces a vertically invariant
contact structure, globally defined by an equation of the form $i(Y)
\omega + u \; dt = 0$ where:
\begin{itemize}
\item
$\omega$ is an area form on $S_\Gamma$;
\item
$Y$ is a vector field which exits along $\partial S_\Gamma$ and
which directs the foliation on $S_\Gamma$ induced by $\F$, that
is, the foliation associated to the handle decomposition
chosen on $S_\Gamma$;
\item
$u$ is a positive function on the interior of $S_\Gamma$, zero
at the boundary and satisfying $u \; \Div_\omega (Y) - Y \cdot u >
0$.
\end{itemize}
Let $A_\Gamma$ be the part of $S_\Gamma$ corresponding to $A$; on
$A_\Gamma \times \R$, $\eta$ induces a contact structure with
equation $i(Y) \omega + e^h u \; dt = 0$. From which: $u (\Div_\omega (Y)
- Y \cdot h) - Y \cdot u > 0$, in other words:
\begin{eqnarray}
\label{part4star}
u \left( Y \cdot h \right) < u \; \Div_\omega (Y) - Y \cdot u.
\end{eqnarray}
We must therefore extend $h$ to $S_\Gamma$ preserving this
inequality. We observe that, on a sufficiently small neighbourhood
of $\partial S_\Gamma$, the function $h$ given arbitrarily satisfies
\ref{part4star} since $u$ is zero on $\partial S_\Gamma$. The fact
the we can then extend $h$ results from the two following remarks:
\begin{itemize}
\item
On the interior of $S_\Gamma$, where $u>0$ \ref{part4star} says
$Y \cdot h < \Div_\omega (Y) - Y \cdot \log u$. Yet each orbit
of $Y$ which exits $A_\Gamma$ goes in finite time to $\partial
S_\Gamma$ without cutting $A_\Gamma$ again. On such a segment
of the orbit, $h$ is given near its endpoints, but the variation
of $-\log u$ is infinite and $\Div_\omega Y$ is bounded; we can
therefore extend $h$ over the segment.
\item
An orbit of $Y$ which enters into $A_\Gamma$ comes to a focus
without cutting $A_\Gamma$ first. On a time interval
of the type $(-\infty, \tau_0]$, $h$ is only given
near $\tau_0$. The condition \ref{part4star}, which bounds
its derivative from above by a quantity which is strictly positive and bounded from below, does not prevent us from extending $h$ to a function with
compact support.
\end{itemize}
\end{Proof}
\subsubsection{Handles of index 1}
We suppose that $V_{k+1}$ (resp. $C_{k+1}$) is obtained from $V_k$
(resp. $C_k$) by attaching a handle of index 1 to two points $p$ and
$q$ of $\Gamma = \partial C_k \subset S = \partial V_k$. We denote by
$p_0$ and $q_0$ the points with coordinates $(0,1,0)$ and $(0,-1,0)$
in $\R^3$. By lemma 3.1, it suffices to establish the following
fact:
\begin{lem}
There exists a germ of a diffeomorphism $(V_k,p,q) \To (H_1, p_0,
q_0)$ which sends $\xi_k$ to $\zeta_1$ and $X_k$ to $Z_1$.
\end{lem}
It is in this lemma, whose proof is easy, that the
orientability of $V$ is needed.
\section{Construction of essential surfaces}
In this part, we give methods for constructing, on 3-dimensional
manifolds, Morse funcitons having essential surfaces (see definition
III.1.1). I have had the pleasure of discussing this question with
several people, in particular Slava Kharlamov, Fran\c{c}ois
Laudenbach, Christine Lescop and Alexis Marin; I thank them for
their suggestions and remarks.
\subsection{Some examples}
\subsubsection{Examples of essential surfaces}
\begin{Example}
\label{4.1.1} (F. Laudenbach). Let $V_0$ be a compact 3-dimensional
manifold with connected boundary $C = \partial V_0$, and let $f_0:
V_0 \To \R$ be a function having the following properties:
\begin{enumerate}
\item
$f_0$ is nonsingular and its restriction to $C$ is a Morse
function;
\item
any local minimum (resp. local maximum) of $f_0|_C$ is a local
minimum (resp. local maximum) of $f_0$ on $V_0$.
\end{enumerate}
Then there exists on the double $V = V_0 \cup_C V_0$ of $V_0$ a Morse
function $f$ for which $C$ is an essential surface.
\end{Example}
\begin{Remark}
We will see later (lemma 2.2) that, if $V_0$ possesses a function $f_0$
satisfying (i) and (ii) then $V_0$ is a handlebody.
\end{Remark}
\begin{Proof}
A simple way to construct the double $V$ of $V_0$ is the following:
we take a Morse function $g_0: (V_0, C) \To ([0,1], 1)$ without
singularities near the boundary. We take on $V_0 \times [-1,1]$ the
function $g(x,s) = g_0(x) + s^2$ and we set $V = \{g=1\} \subset V_0
\times [-1,1]$. We have a smooth manifold which is identified
with the double of $V_0$ via the two functions $V_0 \To V$, $x
\mapsto (x, \pm (1 - g_0(x))^{1/2})$, which send $C$ to $C \times
\{0\} \subset V$.
Now let $\pi$ be the projection $V_0 \times [-1,1] \To V_0$ and let
$f$ be the restriction to $V$ of $f_0 \circ \pi$. As the kernel of
$d(f_0 \circ \pi)$ contains at each point $\partial/\partial s$ at
as the tangent space to $V$ is defined by $d(g_0 \circ \pi) + 2s \;
ds = 0$, we see that the critical points of $f$ all lie on
$C \times \{0\} = V \cap (V_0 \times \{0\})$ and correspond exactly
to the critical points of $f_0|_C$. Moreover condition (ii)
follows since each local minimum (resp. maximum) of $f|_C$ is a minimum
(resp. maximum) of $f$.
\end{Proof}
\begin{Example}
\label{4.1.2} (V.M. Kharlamov). Let $\Gamma$ be a link in $S^3$ and
$\pi: V \To S^3$ a branched double cover over $\Gamma$. We suppose
there exists a Seifert surface $C_0$, bounded by $\Gamma$, and a
Morse function $f_0$ on $S^3$ satisfying the following conditions:
\begin{enumerate}
\item
the critical points of $f_0$ lie on $C_0 \backslash \Gamma$ and
are exactly the critical points of $f_0|_{C_0}$;
\item
$f_0|_{C_0}$ has no local minimum nor local maximum on $\Gamma$.
\end{enumerate}
Then $C = \pi^{-1}(C_0)$ is an essential surface for $f = f_0 \circ
\pi$.
\end{Example}
\begin{Remark}
For several links, we can find a Seifert surface satisfying (i) and
(ii) with $f_0$ the standard height function on $S^3$.
\end{Remark}
\begin{Proof}
The critical points of $f$ (resp. of $f|_C$) are of two types:
\begin{itemize}
\item
the preimages under $\pi$ of critical points of $f_0$ (resp. of
$f_0|_{C_0})$;
\item
the preimages under $\pi$ of critical points of $f_0|_\Gamma$.
For $f$, such a point $x \in V$ is of index 1 or 2 accordingly
as $f_0|_\Gamma$ has at $\pi(x)$ a minimum or maximum; for
$f|_C$, such a point is always of index 1, by (ii).
\end{itemize}
\end{Proof}
\subsubsection{An example of a function having no essential surface
(constructed with C. Lescop)}
\begin{Example}
\label{4.1.3} Let $p,q$ be relatively prime integers with $0 \leq q
\leq p-1$. The oriented lens space $L(p,q)$ possesses a ``canonical"
Morse function $f$ which is ordered and has exactly one Morse critical point of
each index 0,1,2,3. If this function has an essential surface $C$
then either $q=1$, or $q=p-1$, or $q$ is odd and $p=2(q \pm 1)$.
\end{Example}
\begin{Proof}
Let $b$ be a regular value of $f$ between critical values of index 1
or 2. We set $C_0 = C \cap \{f \leq b\}$, $\Gamma = \partial C_0
\subset \{f=b\}$ and we denote by $\Theta$ the attaching curve of
the handle of index 2 on the surface $\{f=b\}$, which is an oriented
torus. Finally, we denote by $\mu$ an oriented meridian of this
torus ($\mu$ bounds a disk in $\{f \leq b\}$) and by $\lambda$ the
oriented curve determined by the 2 following conditions:
\begin{itemize}
\item
the intersection number with $\mu$ is $+1$: $[\lambda] \cdot
[\mu] = +1$;
\item
for a good orientation of $\Theta$, $[\Theta] = q[\mu] + p
[\lambda]$.
\end{itemize}
We distinguish two cases accordingly as $C_0$ is orientable or not.
\begin{enumerate}
\item
If $C_0$ is orientable, it's an annulus and the curve $\Gamma$
has two isotopic components, $\Gamma_0$ and $\Gamma_1$, which
cut $\mu$ once each. Orienting them appropriately, we have,
for $i=0,1$:
\[
[\Gamma_i] = m[\mu] + [\lambda],
\]
therefore
\[
[\Theta] \cdot [\Gamma_i] = pm - q, \quad \text{where }m \in
\Z.
\]
Thus, $\Gamma$ cuts $\Theta$ in at least $2|pm-q|$ points; yet,
since $C$ exists, $\Theta$ cuts $\Gamma$ in exactly two points,
so $pm-q=0,1$ or $-1$. It follows that either $m=0$ and $q=1$ (unless
$q=0$ and $p=1$) or $m=1$ and $q=p-1$.
\item
If $C_0$ is not orientable, it's a Mobius strip and $\Gamma$ is
connected. With the appropriate orientation, we have $[\Gamma] =
m[\mu] + 2 [\lambda]$, where $m$ is an odd integer.
The same argument as before shows that we must have $mp-2q = 0,
2$ or $-2$; then, $m=1$ and $p=2(q \pm 1)$, with $q$ odd since
$p$ and $q$ are relatively prime.
\end{enumerate}
\end{Proof}
\begin{Remark}
\label{4.1.4} Keeping in mind proposition I.4.5, this example shows
that there exist vector fields which, for global reasons, do not
preserve any contact structure.
\end{Remark}
\subsection{A general method for constructing essential surfaces}
\subsubsection{Splitting along an essential surface}
\begin{defn}
\label{4.2.1} Let $S$ be a surface and $\Gamma$ a closed curve on
$S$, in general disconnected. We will say that $\Gamma$
\emph{divides $S$ ``equitably''} if we can recover $S$ from two
subsurfaces, in general non connected, which are both bounded by
$\Gamma$ and have the same Euler--Poincar\'{e} characteristic.
\end{defn}
\begin{lem}
\label{4.2.2} Let $V$ be a 3-dimensional manifold, $f: V \To [0,
\infty)$ a proper Morse function (with distinct critical values)
and $C$ an $f$-essential surface transversely orientable in
$V$. Then:
\begin{enumerate}
\item
$C$ separates $V$ into handlebodies;
\item
$C$ cuts each regular level set of $f$ along a curve which
divides the level set equitably.
\end{enumerate}
\end{lem}
We recall that a compact handlebody is a compact 3-dimensional
manifold with boundary obtained by attaching to a ball handles of
index 1; in the non-compact case, a handlebody is a direct
limit of compact handlebodies.
\begin{Proof}
We choose a transvese orientation on $C$ and we take two regular
values of $f$, $b_0 < b_1$, between which $f$ takes exactly one
critical value. For $i = 0,1$, we set $V_i = \{f \leq b_i \}$, $C_i
= C \cap V_i$, $S_i = \{f = b_i\}$ and $\Gamma_i = C \cap S_i$.
Thus, $V_1$ (resp. $C_1$) is obtained from $V_0$ (resp. from $C_0$)
by attaching a handle $H$ (resp. $K \subset H$: see the discussion
of II.1.B). We observe that $K$ separates $H$ into two
components; we denote them $H^-$ and $H^+$, $K$ being transversally
oriented from $H^-$ towards $H^+$.
If the critical value of $f$ between $b_0$ and $b_1$ is the absolute
minimum of $f$, $C_1$ is a disk which separates the ball $V_1$ into
two balls (with corners). Moreover, $\Gamma_1$ is circle and
therefore divides the sphere $S_1$ equitably.
We now suppose that $V_0$ is the union of two handlebodies,
possibly disconnected and with corners, which intersect each other exactly along
$C_0$. We denote them by $V_0^-$ and $V_0^+$, $C_0$ being
transversally oriented from $V_0^-$ towards $V_0^+$. We suppose
additionally that $\Gamma_0$ divides $S_0$ equitably.
As the attachment of $K$ to $C_0$ must respect the transverse
orientation, $C_1$ separates $V_1$ into $V_1^- = V_0^- \cup H^-$ and
$V_1^+ = V_0^+ \cup H^+$. Thus $C$ separates $V$ into two
submanifolds $V^-$ and $V^+$.
To show (i) and (ii), we observe that the boundary of $V_i^\pm$, for
$i=0,1$, decomposes into two parts: $C_i$ and $S_i^\pm = V_i^\pm
\cap S_i$. By hypothesis, $S_0^-$ and $S_0^+$ have the same Euler
characteristic. Yet:
\begin{itemize}
\item
If $H$ has index $j=0,1$, $V_i^\pm$ is obtained from $V_0^\pm$
by attaching a handle of index $j$. Similarly, $S_i^\pm$ is
obtained from $S_0^\pm$ by attaching a handle of index $j$,
hence:
\[
\chi(S_i^\pm) = (-1)^j + \chi(S_0^\pm),
\]
therefore
\[
\chi(S_1^+) = \chi(S_1^-).
\]
\item
If $H$ has index $j=2,3$, $V_i^\pm$ is homeomorphic to
$V_0^\pm$: we simply glue a ball along a disk contained in the
boundary. However, $S_i^\pm$ is obtained from $S_0^\pm$ by a
``half-surgery" of index $j$ (a surgery along an arc or
a disc lying on the boundary of $S_0^\pm$). We then have:
\[
\chi(S_i^\pm) = (-1)^j + \chi(S_0^\pm),
\]
therefore, as previously,
\[
\chi(S_1^+) = \chi(S_1^-).
\]
\end{itemize}
\end{Proof}
\subsubsection{The principal construction}
\begin{lem}
\label{4.2.3} Let $S$ be a closed surface, $\Gamma_0$ a closed curve
in $S$, not necessarily connected, and $\alpha$ a simple arc joining
in $S$ two points of $\Gamma_0$ without other intersection. Then
there exists a Morse function $f: S \times [0,1] \To [0,1]$
satisfying the following conditions:
\begin{enumerate}
\item
$f$ has exactly two ordered critical points with respective indices 1
and 2; moreover, for $t$ near $0$ or $1$, $f|_{(S \times t)} =
t$;
\item
$f$ has an essential surface which cuts $S \times \{0\}$ along
$\Gamma_0$ and $S \times \{1\}$ along the curve $\Gamma_1$,
drawn in figure 7 and obtained as follows: we add a small closed
component $\Gamma'$, on one side or the other of $\alpha$ and we
perform surgery on $\Gamma_0$ along $\alpha$ in a neighbourhood of
$\alpha$ avoiding $\Gamma'$.
\item
If $\Gamma_0$ divides $S$ equitably, $C$ is transversely
orientable, and $\Gamma_1$ divides $S$ equitably also.
\end{enumerate}
\end{lem}
\begin{Proof}
The method is the following: we realise $S \times [0,1]$ by
attaching successively a handle of index 1 to $S \times [0,
\epsilon]$, then a handle of index 2 in elimination position;
simultaneously, we attach two handles of index 1 to $\Gamma_0 \times
[0, \epsilon]$ so as to obtain the desired essential surface.
Let $\alpha_0, \alpha_1$ be elements of $\Gamma_0$ at the
endponints of $\alpha$. For $i=0,1$ we choose on $\alpha_i$ a basis
$(v_i, w_i)$ for the tangent space to $S$ having the following
properties:
\begin{enumerate}
\item
$v_0$ and $v_1$ are tangent to $\Gamma_0$ and on the same side
of $\alpha$;
\item
$w_0$ and $w_1$ are tangent to $\alpha$ and go inside
$\alpha$.
\end{enumerate}
We attach then a handle of index 1, $H_1 = \{(x,y,z) \in \R^3 \; |
\; 0 \leq x \leq 1, y^2 + z^2 \leq 1\}$, to $S \times \{\epsilon\}$,
as follows: we send $(i,0,0)$ to $\alpha_i$, $(\partial/\partial
y)(i,0,0)$ to $v_i$ and $(\partial/\partial z)(i,0,0)$ to $w_i$.
More precisely, the points $(i,y,0)$ with $-1 \leq y \leq 1$ go to
$\Gamma_0$ and the points $(i,0,z)$ with $0 \leq z \leq 1$ go to
$\alpha$. We thus attach $K_1 = H_1 \cap \{z = 0\}$ to $\Gamma_0
\times \{\epsilon\}$. We denote by $C_1$ the surface obtained and by
$\Gamma$ its upper boundary: $\Gamma =
\partial C_1 \backslash \Gamma_0$.
For $i=0,1$ we now denote by $\alpha'_i$ the $\alpha$ image of
$(i,0,1)$ and $\alpha'$ the sub-arc of $\alpha$ joining $\alpha'_0$
and $\alpha'_1$. In the lateral boundary of $H$, $H \cap \{y^2 + z^2
= 1\}$, we choose an arc $\alpha''$ transverse to the circles $\{x =
\text{const.}\}$, isotopic to the fixed endpoints of the segment
$\{(x,0,1) \; | \; 0 \leq x \leq 1 \}$ and which cuts
the set $\{(x,\pm 1, 0) \; | \; 0 \leq x \leq 1 \}$ in two points (see Figure 8).
We then attach a handle of index 2 along $\Theta = \alpha' \cup
\alpha''$. As $\alpha''$ is transverse to the circles $\{x =
\text{const.}\}$, the manifold thus obtained is diffeomorphic to $S
\times [0,1]$ by the elimination lemma of S. Smale [Mi].
Moreover, by construction, $\Theta$ cuts $\Gamma$ in two points in
such a way that we can attach (in a unique way) a handle of index 1
to $C_1$. We then see painlessly that the surface $C$ obtained
satisfies the stated conditions.
\end{Proof}
\begin{Example}
\label{4.2.4} If $S$ is the sphere $S^2$ and if $\Gamma_0$ is a
circle, the curve $\Gamma_1$ given by lemma 2.2 is composed of
three nested circles (i.e. for which the
complement is a disjoint union of two disks and two annuli).
\end{Example}
\begin{cor}
\label{4.2.5} There exists a Morse function $g: S^2 \times [0,2] \To
[0,2]$ satisfying the following conditions:
\begin{enumerate}
\item
for $t$ near $0$ or $2$, $g|_{S^2 \times \{t\}} = t$;
\item
$g$ possesses an essential surface $C$ which cuts $S^2 \times
\{0\}$ and $S^2 \times \{2\}$ along a circle, and which meets
$S^2 \times \{1\}$ along three nested circles.
\end{enumerate}
\end{cor}
\begin{Proof}
Let $f: S^2 \times [0,1] \To [0,1]$ be ``the" function given by
lemma 2.3 taking for $\Gamma_0$ a circle. We obtain $g$ by gluing
$f$ with the function: $S^2 \times [1,2] \To [1,2]$, $(x,y) \mapsto
(2 - f(x,2-t))$.
\end{Proof}
\begin{cor}
\label{4.2.6} (Convex version of the Lutz modifictaion [Lu]). Any
3-dimensional manifold which carries a convex contact structure
carries a convex overtwisted contact structure.
\end{cor}
\begin{Remark}
This corollary shows how to deduce corollary III.1.5 from theorem
III.1.3.
\end{Remark}
\begin{Proof}
Let $V$ be the manifold. If there exists on $V$ a convex contact
structure, then there exists, by proposition I.4.5, a proper Morse
function $f: V \To [0, \infty)$ possessing an essential surface $C$.
For a regular value $b$ of $f$, slightly larger than the absolute
minimum and for $\epsilon$ sufficiently small, the set $\{b-\epsilon
\leq f \leq b + \epsilon\}$ is a product cobordism $W \cong S^2
\times [0,1]$ which $C$ cuts along a cylinder with circular base
$\Gamma \times [0,1]$. Corollary 2.5 allows us to replace $f$ by a
proper Morse function $f': V \To [0, \infty)$ with an essential
surface $C'$ which cuts $S = \{f' = b\} \cong S^2$ along three
nested circles. Theorem III.1.2 gives a
positive contact structure $\xi'$ on $V$ which is invariant under a
pseudo-gradient $X'$ of $f'$ admitting $C'$ as characteristic
surface. Proposition II.3.1 shows that then, up to admissible
isotopy, the characteristic foliation of $S$ has two limit cycles,
each bounding by a disk with exactly one singularity in its
interior.
\end{Proof}
\subsubsection{An existence theorem}
\begin{thm}
\label{4.2.7} On any 3-dimensional manifold, there exists a positive
proper Morse function which admits a transversally orientable
essential surface.
\end{thm}
\begin{Remark}
Theorem 2.7, with theorem III.1.2, immediately implies theorem
III.1.3.
\end{Remark}
\begin{Proof}
Let $V$ be the manifold, and $f: V \To [0, \infty)$ a proper Morse
function, having only one maximum if $V$ is closed and no maximum if
$V$ is open. Let $b_0$ and $b_1$ be two regular values of $f$
between which $f$ takes only one critical value $a$. We set $V_i =
\{f \leq b_i \}$ for $i=0,1$ and $S = \{f = b_0\}$.
If $a$ is the absolute minimum of $f$, $f|_{V_1}$ possesses a
transversally orientable essential surface. We therefore now suppose
that $f|_{V_0}$ has a transversally orientable essential surface
$C_0$, with boundary $\Gamma_0$, and we distinguish three cases,
according to the index of the critical value $a$.
\emph{Index 1.} $V_1$ is obtained from $V_0$ by attaching a handle
$H$ of index 1. Changing the attachment of $H$ by isotopy, we can
simultaneously attach along $C_0$ a handle of index 1 so as to have,
for $f|_{V_1}$, a transversally orientable essential surface.
\emph{Index 2.} Byy lemma 2.2, $\Gamma_0$ divides $S$
equitably. It follows that the attaching curve $\Theta$ and the handle $H$ of
index $2$ cuts $\Gamma_0$ in an even number $2r$ of points. If
$r=1$, we can attach to $C_0$ a handle of index 1, $K \subset H$,
which gives for $f|_{V_1}$ a transversally orientable essential
surface. If $r=0$, we move $\Theta$ by an isotopy to create two
intersection points. Now if $r>1$, we apply lemma 2.3 to a sub-arc
$\alpha$ of $\Theta$ which joins two consecutive intersection points
with $\Gamma_0$. We thus eliminate these two points replacing
$f|_{V_0}$ with a function $f'$ which has two more critical points with
respective indices 1 and 2; we then have a new transversally
orientable essential surface $C'_0$ whose boundary
$\Gamma'_0$ still divides the surface $\{f' = b_0\} = \{f = b_0\}$
equitably. Repeating this operation several times, we reduce to the
case where $r=1$.
\begin{Remark}
For compact manifolds with boundary, the proof is finished; for open
and non-compact manifolds, we finish with a classic direct
limit argument.
\end{Remark}
\emph{Index 3.} As $f$ has a single maximum, the surface $S = \{f =
b_0\}$ is a sphere. If $\Gamma_0 \subset S$ is a circle, we can,
attaching a handle of index 3, reglue a disk to $C_0$, which gives
the sought transversally orientable essential surface. Now, if
$\Gamma_0$ is not connected, we proceed as follows.
By lemma 2.2, $\Gamma_0$ divides $S$ equitably. Then, there exists a
component $\Gamma$ of $\Gamma_0$ which satisfies the following
properties:
\begin{enumerate}
\item
$\Gamma$ does not bound a disk of $S \backslash \Gamma_0$;
\item
in one of the hemispheres of $S$ bounded by $\Gamma$, each
component of $\Gamma_0$ bounds a disk of $S \backslash
\Gamma_0$; we denote by $S'$ this hemisphere and $S''$ the
other.
\end{enumerate}
(To see that $\Gamma$ exists, we observe that, if no component of
$\Gamma$ satisfies (i), $S \backslash \Gamma_0$ is composed on the one
hand of a disjoint union of disks, and on the other of a disk
with holes. Consequently, $\Gamma_0$
does not divide $S$ equitably. To obtain (ii), we choose a component
$\Gamma$ satisfying (i) ``minimally".)
Now, we take a component $\Gamma'$ of $\Gamma_0$ in $S'$ and, in
$S''$, we choose a component $\Gamma''$ which we can connect to
$\Gamma$ by an arc $\alpha^*$ without recutting $\Gamma_0$. The
inverse construction to that of lemma 2.3 allows us to eliminate $\Gamma'$ by doing connected sum of $\Gamma$ and $\Gamma''$ along
$\alpha^*$. The curve thus obtained again divides $S$ equitably and has two fewer components. Repeating this operation, we render $\Gamma_0$ connected which ends
the proof.
\end{Proof}
\end{document}