text:
\section*{Introduction}
In \cite{biran3}, Biran proved a decomposition theorem for rational k\"ahler  manifolds which has proved useful in many situations
such as symplectic packings \cite{biran2,moi3}, Lagrangian embeddings \cite{bici1}
  \dots This paper tries to add  symplectic isotopies to this list of applications.
In polarized symplectic manifolds - triples $(M,\omega,\Sigma)$  where $\omega\in H^2(M,\mathbb{Z})$ and $\Sigma$ is a symplectic hypersurface
Poincaré-dual to a (necessarily positive) multiple $k\omega$ of the symplectic form -, this decomposition result may be expressed as :
\begin{thm} [Biran]\label{biran} In a polarized closed symplectic manifold $(M,\omega,\Sigma)$, there exists a zero-volume closed skeleton in $M$
whose complement is a standard symplectic disc bundle supported by $\Sigma$.
\end{thm}
 In fact, polarizations of  sufficiently high degree exist on all closed ({\it i.e.} compact, without boundary) rational symplectic manifolds \cite{donaldson}. These manifolds split into a {\it standard symplectic part} - an explicit disc bundle
 over $\Sigma$ - and a {\it negligible skeleton} (in the K\"ahler case it is even isotropic).
Although no kind of uniqueness can be expected (see \cite{biran3} for a discussion on the skeleton for instance), our
next theorem explains how to construct many such decompositions - {\it i.e.} with enough freedom - all of whose symplectic parts are
isotopic. In order to state it, let us mention that both $M\backslash \Sigma$ and the complement of the zero-section
$\mathcal{L}_0$ in the symplectic disc bundle  $\text{\small SDB}(\Sigma,k)$ are exact symplectic manifolds, so they admit Liouville
forms. In the statement below,  $\lambda_0$ is a distinguished such  form on $\text{\small SDB}(\Sigma,k)\backslash \mathcal{L}_0$ (see section \ref{sdbsec}).
Also, by a local embedding of $(X,M)$ into $(Y,N)$ - where $X\subset M$ and $Y\subset N$ - we always mean an embedding of a
neighbourhood  of $X$ in $M$ into $N$  which induces a diffeomorphism from $X$ to $Y$.
\begin{thm}\label{flex2}
Let $\phi$ be a local symplectic embedding of $\big(\text{\small SDB}\,(\Sigma,k),\mathcal{L}_0\big)$ into $(M,\Sigma)$. Any Liouville form $\beta$ on  $M\backslash \Sigma$  which  differs from $\phi_*\lambda_0$ by a smooth $1$-form on $M$ gives rise to a unique symplectic embedding $
\Phi:\text{\small SDB}(\Sigma,k)\hookrightarrow ~M$ such that $\Phi_*\lambda_0=\beta$. The embeddings obtained in this way form a contractible space.
\end{thm}
 The symplectic disc bundles naturally contain ellipsoids or ball packings \cite{moi3}. Theorem \ref{flex2} can be seen as a flexibility statement
 for embeddings of such objects, which are restricted to a jet condition on a hypersurface. In full generality the flexibility of this condition
 seems hardly tractable, but in dimension $4$ hypersurfaces are symplectic curves and pseudo-holomorphic techniques come to our
 rescue. We get for instance :
\begin{thm}\label{isoell}
The space of symplectic embeddings of (closed) balls of fixed radius or ellipsoids $\mathcal E(a_+,a_-)$ with $a_+<\pi$  in $\P^2$
is connected.
\end{thm}
It reproves in particular McDuff's connectedness results on  the space of symplectic  balls of $\P^2$ \cite{mcduff1}, and of
symplectic ellipsoids \cite{mcduff4} for not too long ellipsoids.  Her approach {\it via} blow-up and inflation is more efficient :
her results are valid in more general manifolds than only $\P^2$, for more than one ball alone \cite{mcduff2,mcduff3} and
for any ellipsoid. The present technique may simply be considered as another attempt to approach this still open question
of symplectic isotopy. In one respect at least it shows interest : compared to the inflation process, the isotopies we produce are more
explicit.  As such, we can  follow them enough  to achieve results for maximal objects. I proved for instance in \cite{moi3}
that all smooth maximal packings of $\P^2$ by two balls - {\it i.e.} smooth embeddings of two closed balls of maximal radii
($r_1^2+r_2^2=1$) whose images have disjoint interior - have basically the same intersections. In fact, these maximal packings
are unique up to symplectic isotopies.
\begin{thm}\label{isomax}
The space of smooth maximal symplectic packings of $\P^2$ by two balls of fixed radii is connected.
\end{thm}
We do not intend to get the most general results in this paper. For instance theorem \ref{isomax} can be
 easily adapted to five balls and theorem \ref{isoell} to other ellipsoids ({\it e.g.}  $(a_+,a_-)\approx (2\pi,\frac{\pi}{2})$), other spaces such as $S^2\times S^2$ with split symplectic forms (see \cite{moi6} theorem 10) or to results in the directions of Lalonde-Pinsonnault \cite{lapi,lapian}. Our aim is rather
to present a method, fundamentally based on the fact that since open domains can be made out of symplectic hypersurfaces, these two
classes of  objects should share some rigidity or flexibility properties. Each  piece of information on the flexibility of some class of
symplectic hypersurfaces is liable to have an interpretation in terms of packings.

The paper is organized as follows.  We first review Biran's description of the standard disc bundles. We then carry out an alternative proof of theorem \ref{biran} following the lines suggested  in \cite{biran3}. Although this proof gives less information on the skeleton, it is purely symplectic  and it clearly  shows how much flexibility sould be expected for Biran decompositions. It leads to a weak form
of theorem \ref{flex2},  enough for all subsequent applications.  We come to symplectic isotopies in section \ref{isosec},
where we let aside the most delicate point of the method - finding convenient symplectic curves. We deal with this last point in section
\ref{hardsec} with classical techniques of pseudo-holomorphic curves \cite{gromov,mcsa2}  as well as SFT-like techniques \cite{elgiho,behwz}  similar in spirit to those used by Hind-Kerman \cite{hike}. We finally prove theorem \ref{flex2} in appendix \ref{flex2sec}.

{\small \paragraph{Aknowledgements.} I wish to thank C. Viterbo, F. Bourgeois and A. Oancea  for explaining me how Symplectic Field Theory can be useful  in the kind of problems we are  led to here. This
paper owes a lot to D. McDuff who  pointed out (and sometimes even
fixed) inaccuracies in the first versions.}

\section{Symplectic disc bundles}\label{sdbsec}
\subsection{Description.}
Let $(\Sigma,\tau)$ be a closed  symplectic manifold with $\tau\in H^2(\Sigma,\mathbb{Z} )$. Let $\pi:\mathcal{L} \longrightarrow \Sigma$ be a line bundle on $\Sigma$ with first chern class $c_1(\mathcal{L})=[k\tau]$ endowed with a hermitian metric $g$ and a connection $\nabla$ with curvature
$R^\nabla=2i\pi k\tau$. Define the transgression $1$-form on the complement of the zero-section $\mathcal{L}_0$ by :
$$
\begin{array}{l}
\displaystyle \alpha(\frac{\partial}{\partial r})=0, \;\alpha(\frac{\partial}{\partial \theta})=\frac{1}{2\pi},\\
\displaystyle \alpha_{|H^\nabla}=0,
\end{array}
$$
where $r$ is the radial coordinate, $\partial /\partial \theta$ is the infinitesimal generator of the $S^1$-action $e^{i\theta}\cdot$ on $\mathcal{L}$ and $H^\nabla$ is the horizontal distribution of the connection $\nabla$. With this normalization, a simple computation shows that $d\alpha=-k\pi^*\tau$. Out of this $1$-form, one can make an obviously closed
$2$-form on $\mathcal{L}$ :
$$
\omega_0:=\pi^*\tau+d(r^2\alpha)=(1-kr^2)\pi^*\tau+dr^2\wedge \alpha.
$$
This $2$-form degenerates on the circle bundle $\{r^2=1/k\}$, and therefore restricts to a symplectic form on the $\pi/k$-disc
bundle on $\Sigma$ (that is the set $\{r^2<1/k\}$). Here, $\pi/k$  refers  to  area rather than radius. This disc bundle together with the form $\omega_0$ will be called the {\it standard symplectic disc bundle of degree $k$} and denoted by $\text{\small SDB}(\Sigma,k,\tau)$.
Of course, this construction relies on a choice of a hermitian metric on $L$, but it can be shown that any such choice
leads to the same symplectic manifold. This fact justifies the wording "standard" and the omission of the metric on the notation $\text{\small SDB}(\Sigma,k,\tau)$.
 Let us point out that these bundles appear naturally in symplectic geometry by Weinstein's theorem \cite{weinstein} :
\begin{thm*}[Weinstein]
Let $(M,\omega)$ be a closed rational symplectic manifold and $\Sigma$ a  symplectic hypersurface of $M$ which is Poincaré-dual to
an integer $k\omega$. Then a neighbourhood  of $\Sigma$ in $M$ is symplectomorphic to a neighbourhood  of $\mathcal{L}_0$ in $\text{\small SDB}(\Sigma,k,\omega_{|\Sigma})$.
\end{thm*}
When it is clear enough from the context that $\Sigma$ is a hypersurface of a symplectic manifold, we may write $\text{\small SDB}(\Sigma,k)$ in place
of $\text{\small SDB}(\Sigma,k,\omega_{|\Sigma}) $.

\subsection{Liouville forms.}  \label{liouvillesec}
The symplectic disc bundles above retract onto their zero-section, so the symplectic form $\omega_0$ is exact on the complement of $\mathcal{L}_0$. Our present aim is to describe some features of their primitives, called Liouville forms, and of their associated vector fields. Throughout this paper, a Liouville form for a symplectic structure $\omega$ is a $1$-form $\lambda$ such that $d\lambda=-\omega$. Recall that the Liouville vector field associated to  $\lambda$ is defined by $\omega(X_\lambda,\cdot)=\lambda$ and is a contracting vector field for $\omega$ : $L_{X_\lambda}\omega=-\omega$, so ${\Phi_{X_\lambda}^t}^*\omega=e^{-t}\omega$. Let us now fix a symplectic manifold $(\Sigma,\tau)$ and its symplectic disc bundle $\text{\small SDB}(\Sigma,k,\tau)$ with form $\omega_0=(1-kr^2)\pi^*\tau+dr^2\wedge\alpha$.

The form $\lambda_0:=(1-kr^2)\alpha/k$  is a Liouville form, whose associated vector field is $X_0:=\displaystyle\frac{1-kr^2}{2kr}\frac{\partial}{\partial r}$. This vector field is radial, forward complete (its flow is defined for any positive time), but it explodes at $\mathcal{L}_0$, where it points outwards. An easy computation shows that if $p\in \text{\small SDB}(\Sigma,k,\tau)$ is at distance
$r(p)$ from $\mathcal{L}_0$, then $\Phi^t_{X_0}(p)$ goes to the boundary of the disc bundle when the time goes forward to $+\infty$
while it reaches $\mathcal{L}_0$ in finite negative time $\tau(p)=-\ln(1-kr(p)^2)$.
Notice now that if $\vartheta$ is any closed $1$-form on $\Sigma$, the form $\lambda_\vartheta:=\lambda_0+\pi^*\vartheta$ is also a Liouville
form. Since $\pi^*\vartheta$ vanishes on the fiber, the radial component of the Liouville vector field $X_\vartheta$ associated to
$\lambda_\vartheta$ is  $X_0$. Moreover, the invariance of $\pi^*\vartheta$ with respect to $r$ ensures that $X_0$ and $X_\vartheta-X_0$ commute. Finally,
since $\Sigma$ is compact,  $X_\vartheta-X_0$ is complete so the above completeness properties also hold for $X_\vartheta$. In fact - but we will not need this precise statement in the sequel -
$X_\vartheta$ differs from $X_0$ by a horizontal vector field which projects by $\pi$ to the symplectic vector field  $\tau$-dual to $\vartheta$.
Let us summarize this discussion.
\begin{prop}\label{lvf} Let $(\Sigma,\tau)$ be a closed symplectic manifold with $\tau\in H^2(\Sigma,\mathbb{Z})$. Any closed
$1$-form $\vartheta$ on $\Sigma$ gives rise to a Liouville form $\lambda_\vartheta:=\lambda_0+\pi^*\vartheta$ on $\text{\small SDB}(\Sigma,k)\backslash \mathcal{L}_0$ ($\lambda_0:=(1-kr^2)\alpha/k$) with corresponding vector field $X_\vartheta$ which is forward complete and whose flow satisfies the
following :
\begin{equation}\label{stop}
\Phi^t_{X_\vartheta}(p)  \longrightarrow
\left\{\begin{array}{l}
 \partial \text{\small SDB}(\Sigma,k) =\{r^2=1/k\}, \text{ when } t\longrightarrow +\infty\\
 \mathcal{L}_0 =\{r=0\}, \text{ when } t\longrightarrow\tau(p):=\ln(1-kr(p)^2).
\end{array} \right.
\end{equation}
\end{prop}
We finish this paragraph with the definition of a convex domain relative to a Liouville form.
\begin{definition}\label{convexity}
We say that a smoothly bounded compact subset $U\subset \text{\small SDB}(\Sigma,k)$ is $\lambda_\vartheta$-convex if
$X_\vartheta$ points outward along $\partial U$.
 \end{definition}
The relevance of this definition lies in that a convex domain $U$ can be recovered from $U\cap \mathcal{L}_0$ only, by "inflation" along
$X_\vartheta$. Moreover this inflation process only depends on $X_\vartheta$ inside $U$. For this reason, we call $\mathcal{L}_0$ a supporting hypersurface
of $U$.
\subsection{Symplectic disc bundles and ellipsoids.}\label{sdbell}
The role that symplectic disc bundles plays in the applications of this paper comes from their links with ellipsoids. This link was noticed in \cite{moi3}, but we recall it now for self-containedness. Call $\mathcal E(a_1,a_2,\dots,a_n)$
the closed ellipsoid defined in the following way :
$$
\mathcal E(a_1,\dots,a_n)=\left\{ (z_1,\dots,z_n)\in \mathbb{C}^n\;  |\; \pi\frac{|z_1|^2}{a_1}+\dots+\pi\frac{|z_n|^2}{a_n}\leq1 \right\}.
$$
With this definition, the $a_i$ are the areas of the "complex axes" of the ellipsoid, and in symplectic terms they are Eckeland-Hofer capacities of the ellipsoid. A symplectic embedding of an ellipsoid into a symplectic manifold will be simply  called a symplectic ellipsoid. Unless explicitly stated, the ellipsoids are {\it closed} in all the paper. The following proposition shows how they get into the picture (see \cite{moi3} for a proof):
\begin{prop}\label{ellinsdb}
The restriction of a symplectic disc bundle $\text{\small SDB}(\Sigma,k)$ to a symplectic ellipsoid $\mathcal E(a_1,\dots,a_n)$ of the base is
an open symplectic ellipsoid $\rond{\mathcal E}(a_1,\dots,a_n,1/k)$.
\end{prop}
In dimension $4$, that is when $\Sigma$ is of dimension $2$, an ellipsoid of the base is simply an embedded disc in $\Sigma$.
The proof of proposition \ref{ellinsdb} goes by finding an explicit mapping $\Psi$ between the disc bundle and an ellipsoid. It is easy to prove that smaller {\it straight ellipsoids} (or balls) are identified through $\Psi$ to $\lambda_0$-convex sets. By a   straight  ellipsoid of $\mathcal E(a_1,\dots,a_n)\subset \mathbb{C}^{n}$ we mean the  image of the unit
ball by a diagonal complex map $(z_1,\dots,z_n)\mapsto (\lambda_1z_1,\dots,\lambda_nz_n)$.
\begin{prop}\label{ballconv}
Let $\mathcal E$ be a symplectic ellipsoid in $\Sigma$ and $\Psi:\mathcal E'\tilde{\longrightarrow} \pi^{-1}(\mathcal E)$ be the natural parametrisation of  the restriction of $\text{\small SDB}(\Sigma,k,\tau)$ to $\mathcal E$ by an ellipsoid.
Then the image of any straight ellipsoid of $\mathcal E'\subset \mathbb{C}^n$ by $\Psi$ is $\lambda_0$-convex as well as $\lambda_\vartheta$-convex provided that the $1$-form $\vartheta$ vanishes  on $\mathcal E$ (this condition is not of cohomological nature).
\end{prop}

\section{Biran decomposition theorem.}\label{biransec}
\subsection{Proof of theorem \ref{biran}.}
 Let $(M,\omega)$ be a closed rational symplectic manifold with
a symplectic hypersurface $\Sigma$  which is Poincaré-dual to  $k\omega$. By Weinstein's neighbourhood  theorem, there is an embedding $\phi$ between open neighbourhoods  $\mathcal U,\mathcal V$ of $\mathcal{L}_0\subset \text{\small SDB}(\Sigma,k)$ and $\Sigma\subset M$.
Observe that being Poincaré-dual to $k\omega$ implies that $M\backslash \Sigma$ is
an exact symplectic manifold because $\omega$ vanishes on any $2$-cycle of $M\backslash \Sigma$. The following lemma ensures that one of the $\lambda_\vartheta$ defined in the previous section is compatible with a Liouville form on $M\backslash \Sigma$. Precisely,
\begin{lemma}\label{compatibility}
There exists a $1$-form $\vartheta$ on $\Sigma$ such that $\phi_*\lambda_\vartheta$ - defined on $\mathcal V\backslash \Sigma$ -
extends to a Liouville form  on $M\backslash \Sigma$.
\end{lemma}
\noindent Let us first explain why theorem \ref{biran} is a consequence of this lemma. First, restrict $\mathcal U$ to a
$\lambda_\vartheta$-convex
domain if necessary. Let $\beta$ be an extension of $\phi_*\lambda_\vartheta$
to a Liouville form of $M\backslash \Sigma$ and $X_\beta$ its dual vector field. Since it coincides with $\phi_*X_\vartheta$ on
$\mathcal V\backslash \Sigma$, $X_\beta$ points outward along $\partial\mathcal V$ so it is forward complete by compactness of $M$, exactly as $X_\vartheta$ is.  Define therefore the map \fonction{\Phi}{\text{\small SDB}(\Sigma,k)}{M}{x}{\left \{\begin{array}{ll} \phi(x)  & \text{if } x\in \mathcal U, \\ \Phi_{X_\beta}^\tau\circ \phi\circ \Phi_{X_\vartheta}^{-\tau}(x) & \text{if }\Phi_{X_\vartheta}^{-\tau}(x)\in \mathcal U.\end{array}\right.}
Since $\Phi$ transports $X_\vartheta$ to $X_\beta$, it commutes with the flows of these two vector fields inside $\mathcal U$, so there is in fact no need to specify which value of $\tau$ should be considered above : thanks to the convexity of $\mathcal U$, any choice gives the same point  for $\Phi(x)$. Notice
moreover that the procedure above actually defines $\Phi$ on $\text{\small SDB}(\Sigma,k)$ because of property (\ref{stop}) : the flow of
$X_\vartheta$ expands $\mathcal U$ and its image eventually covers the whole of $\text{\small SDB}(\Sigma,k)$. The map $\Phi$ is obviously a symplectic embedding.
Finally, notice that $M\backslash \textnormal{Im}\, \Phi$ is endowed with a  backward complete (volume-contracting)  Liouville vector field  so its volume must
vanish by compacity of $M$.\hfill $\square$ \vspace{0.1cm}\\
Let us finally turn back to the missing lemma.\\
\noindent {\it Proof of lemma \ref{compatibility} :} Pick first a Liouville form $\beta$ on $M\backslash \Sigma$.
If we can find a closed $1$-form $\vartheta$ on $\Sigma$
such that $\beta-\phi_*\lambda_\vartheta$ is exact on $\mathcal V\backslash \Sigma$, then the lemma follows. Indeed, $\phi_*\lambda_\vartheta$
then coincides with $\beta+dh$ where $h$ is a smooth function defined on $\mathcal V\backslash \Sigma$, and any extension of $h$
to $M\backslash \Sigma$ provides an extension of $\phi_*\lambda_\vartheta$ to a Liouville form on $M\backslash \Sigma$.

In order to construct the form $\vartheta$, consider a generating family $\{\gamma_0,(\gamma_i)\}$ of the $1$-dimensional homology of $\mathcal V\backslash \Sigma$ where $\gamma_0$ is "the small loop around $\Sigma$" parameterized by $\{\phi(e^{i\theta}\cdot p),\; \theta\in[0,2\pi]\}$ and $\gamma_i$  project by $\pi\circ \phi^{-1}$ to simple closed loops $\gamma_i'$
 which span $H_1(\Sigma)$.
Let us define $\vartheta$ by requiring that $ [\vartheta]\cdot \gamma_i'=\int_{\gamma_i}(\beta-\phi_*\lambda_0)$. Then $\beta-\phi_*\lambda_\vartheta$ vanishes on each class $[\gamma_i]\in H_1(\mathcal V\backslash \Sigma)$ because
$$
\begin{array}{ll}
\displaystyle \int_{\gamma_i}\beta-\phi_*\lambda_\vartheta & = \displaystyle \int_{\gamma_i}\beta - \int_{\gamma_i} \phi_*\lambda_\vartheta \\
 & = \displaystyle \int_{\gamma_i} \beta -\int_{\gamma_i}\phi_*(\lambda_0+\pi^*\vartheta) \\
 & = \displaystyle   \int_{\gamma_i}\big(\beta-\phi_*\lambda_0\big)- \int_{\gamma_i}\phi_*\pi^*\vartheta \\
 & =\displaystyle   \int_{\gamma_i}\big(\beta-\phi_*\lambda_0\big) - \int_{\gamma_i'}\vartheta  = 0 \text{ by construction.}
\end{array}
$$
Observe now that $\gamma_0$ is a torsion element in $H_1(\mathcal V\backslash \Sigma)$. Indeed, $[\gamma_0]$ represents the class of the fiber in the $S^1$-bundle $P:=\partial \mathcal V$ over $\Sigma$. Since the Euler class of this bundle is $k[\omega]$, the image of $\partial_*$ in the long exact sequence
$$
\dots\longrightarrow H_2(\Sigma) \overset{\partial_*}{\longrightarrow} H_1(S^1)\overset{i_*}{\longrightarrow} H_1(P)\longrightarrow \dots,
$$
  is $[k\omega]\big(H_2(\Sigma)\big)\neq \{0\}$. Thus $[\gamma_0]\in H_1(P)=H_1(\mathcal V\backslash \Sigma)$ has finite order as claimed and $\beta-\phi_*\lambda_\vartheta[\gamma_0]=0$. Having no period, this one form  is exact in $\mathcal V\backslash\Sigma$. \hfill $\square$
\subsection{Comments.}
The proof prompts two remarks. First, we were only able to produce a map which pulls back $\beta$ to some $\lambda_\vartheta$
but not necessarily to $\lambda_0$, which seems in contradiction with theorem \ref{flex2}. The reason for this restriction is
however purely technical and we get rid of it in the appendix.
The second remark is that a partial flexibility proceeds already from this  proof of theorem \ref{biran}.
\begin{cor}\label{corflex}
Let $(M,\omega,\Sigma)$ be a polarized symplectic manifold, $\phi$ a symplectic embedding of $\text{\small SDB}(\Sigma,k)$ into $M$ and
  $\vartheta$ a 1-form on $\Sigma$ such that $\phi_*\lambda_\vartheta$ extends to a Liouville form on $M$.
The set of symplectic embeddings of a $\lambda_\vartheta$-convex domain $U\subset \text{\small SDB}(\Sigma,k)$ into $M$ which coincide with $\phi$ near $\mathcal{L}_0$ is contractible.
\end{cor}
\noindent {\it Proof :} It is enough to prove the statement for those maps which coincide with $\phi$ on a fixed neighbourhood  of $\mathcal{L}_0$. Then we may as well assume that $U$ contains all $\mathcal{L}_0$, simply extending all these maps by $\phi$ on some neighbourhood  of $\mathcal{L}_0$.
Consider now an element $\psi$ of
$$
E:=\{\psi:U\longrightarrow M\, |\, \psi^*\omega=\omega_0, \psi=\phi \text{ on a neighbourhood  of } \mathcal{L}_0\}.
$$
Since $U$ is $\lambda_\vartheta$-convex, it retracts on $\mathcal{L}_0$ so the cohomological condition on $\vartheta$ implies that $\psi_*\lambda_\vartheta$
extends to a Liouville form $\beta$ on $M \backslash \Sigma$.  Arguing as in the proof of theorem \ref{biran}, we get a symplectic
embedding $\Psi_\beta:\text{\small SDB}(\Sigma,k)\hookrightarrow M$ with $\Psi_\beta^*\beta=\lambda_\vartheta$. Moreover, because of the $\lambda_\vartheta$-convexity of $U$, the restriction
of $\Psi_\beta$ to $U$ is $\psi$, whatever extension $\beta$ of $\psi_*\lambda_\vartheta$ we have chosen. This means that
we have a surjective, obviously continuous map from $E'$ onto $E$, where
$$
E':=\{\beta\, |\, d\beta=-\omega\, ,\, \beta\equiv \phi_*\lambda_\vartheta \text{ on a neighbourhood  of } \mathcal{L}_0\}.
$$
A right-inverse for this map is easily constructed by hand. Since its fibers together with $E'$ are contractible as affine spaces,
$E$ also is contractible. \hfill $\square$

\section{Applications to isotopy problems in $\P^2$}\label{isosec}
The goal of this section is to exploit corollary \ref{corflex} in some isotopy problems in symplectic geometry. Let us first describe a recipe. Let $(M,\Sigma)$ be a polarized   symplectic manifold $M$ with an embedding of $\text{\small SDB}(\Sigma,k)$ into $M$ as constructed above.  Let $U$ be a $\lambda_0$-convex domain  of $\text{\small SDB}(\Sigma,k)$, $\phi:U\hookrightarrow M$
be the induced embedding and $\psi:U\hookrightarrow M$ be another symplectic embedding. In order to isotop $\psi$ to $\phi$, proceed as follows : \vspace{,3cm} \\
{\tiny $\bullet$ }\ Find a closed symplectic hypersurface $\Sigma'\subset M$ symplectomorphic to $\Sigma$,
 whose intersection with $\psi(U)$ coincides with $\psi(U\cap \mathcal{L}_0)$. We say that $\Sigma'$ is a supporting hypersurface of $\psi(U)$.
 \vspace{,1cm} \\
{\tiny $\bullet$ }\ Isotop $\Sigma'$ to $\phi(\mathcal{L}_0)$.\vspace{,1cm} \\
{\it Without further information on $\psi$, both previous steps need "hard" techniques, and for this reason the
present receipe only works in dimension $4$.}\vspace{,1cm} \\
{\tiny $\bullet$ }\ Isotop $\psi$ to $\phi$ on $\Sigma\cap U$, and then even on a neighbourhood  of $\Sigma\cap U$ in $U$. This means finding
local normal forms, which is often tractable in the symplectic world.\vspace{,1cm} \\
{\tiny $\bullet$ }\  Use finally corollary \ref{corflex} to isotop $\psi$  to $\phi$.\\

Roughly speaking,  the above method reduces the problems of isotopies of open domains from a question on symplectic foliations to a question on one single symplectic hypersurface. Indeed, when isotoping an open domain
it is {\it a priori} necessary to move full coordinate charts. For instance, Gromov's isotopies between the compactly supported symplectic transformations  of the bidisc  are done through deformations of the images of the grid $\{z_2=c\}$, $\{z_1=c\}$.
McDuff's proofs that the space of embeddings of one ball
in $\P^2$ is connected consists in straightening the foliation of $\psi(B^4)$ by Hopf discs (the images by $\psi$ of the intersections of $B^4$ with complex lines).
By contrast, theorem \ref{biran} and its corollary \ref{corflex} show that the data of a single "hard object" - {\it one} symplectic hypersurface - together with  "soft objects" - Liouville forms - provide symplectic coordinates which can already be deformed rather freely.

In this section, we first focus on the soft part : local isotopies.  We then reprove Gromov's result on the connectedness of the space of symplectic automorphisms of the ball which are the identity near the boundary. Although this result is well-known, we hope that its proof
will illustrate clearly the above receipe. We finally prove theorem \ref{isoell}  and \ref{isomax}, postponing  however the proof of the needed hard results  to the next section.
\subsection{Preliminaries : some local normal forms.}
\paragraph{Local ellipsoids.} The first proposition gives local isotopies between symplectic embeddings  of an ellipsoid $\mathcal E:=\mathcal E(1,a)$ which coincide on the Hopf disc $D:=\{z_2=0\}$. A local embedding of $(\mathcal E_a,D)$ is simply an embedding of a neighbourhood  of $D$ in $\mathcal E_a$.
\begin{prop}\label{groloc}
Let $\psi:(\mathcal E_a,D)\to \mathbb{C}^2$ be a local symplectic embedding with $\psi_{|D}=\text{Id}\,$. Then there is a symplectic isotopy $\Phi_t$
which is the identity on $D$ and which isotops $\psi$ to the identity (that is $\Phi_1\circ \psi=\text{Id}\,$). Moreover, the relative version holds : if $\psi_{|\partial \mathcal E_a}=\text{Id}\,$ then one can impose $\Phi_{t|\partial \mathcal E_a}\equiv \text{Id}\,$.
\end{prop}
\noindent {\it Proof :} Since $\psi_{|D}=\text{Id}\,$ the map $\psi$ writes
$$
\psi(z_1,z_2)=(z_1,A(z_1)z_2) + o(|z_2|),
$$
where  $A(z_1)$ is a symplectic linear map. Consider  a path of maps $A_t:D\to \sp(4)$ such that $A_0=A$ and $A_1=\text{Id}\,$. Impose also that  $A_{t|\partial D}\equiv \text{Id}\,$ when $\psi_{|\partial \mathcal E_a}\equiv \text{Id}\,$, which is possible because $\pi_2(\sp(4))=0$.
Then there is a path of diffeomorphisms $f_t:(\mathcal E_a,D)\to \mathbb{R}^4$ such that
\begin{enumerate}
\item $f_0=\phi$, $f_1=\text{Id}\,$,
\item $f_t(z_1,z_2)=(z_1,A_t(z_1)z_2) + o(|z_2|)$,
\item $f_{t|\partial \mathcal E_a}\equiv \text{Id}\,$ if $\psi_{|\partial \mathcal E_a}\equiv \text{Id}\,$.
\end{enumerate}
Since $f_{t*}$ is symplectic on $D$ (and $\partial \mathcal E_a$ in the relative situation), $f_{t*}\omega$ differs from $\omega$ by a small exact $2$-form near $D$, which can even be chosen to vanish on $D$ (and $\partial \mathcal E_a$). Then Moser's method gives maps $h_t$ which are the identity on $D$ (and $\partial \mathcal E_a$) such that $\phi_t:=h_t\circ f_t$ is a path of symplectic diffeomorphisms. Moreover,
$h_0=h_1=\text{Id}\,$ because $f_0$ and $f_1$ are symplectic.  \hfill $\square$
\paragraph{The kissing of two spheres along a characteristic.} The next proposition gives a local model, which can be reached
{\it via} isotopies, for a configuration of two spheres which intersect exactly along one characteristic. It will only be used in the proof of theorem \ref{isomax}. Before stating the result, let
us describe one such configuration. Consider the two surfaces $S_\pm:=\{|z_1|^2=1\pm |z_2|^2\}\subset \mathbb{C}^2$. Then $S_-$
is the $3$-sphere of radius $1$, $S_+\cap S_-=C:=\{(e^{i\theta},0)\}\subset \mathbb{C}^2$ and in fact $S_+$ is also a symplectic sphere locally near $C$.
Consider indeed the complex valued function defined in a neighbourhood  of the unit circle by $\rho(re^{i\theta}):=\sqrt{2-r^2}e^{-i\theta}$. It is an area-preserving involution that preserves the foliation by circles and exchanges the interior and exterior of the unit disc.  Then the
map  $(\rho(z_1),z_2)$ is a symplectomorphism
- only defined locally in a neighbourhood  of $C$ -
 which sends $S_-$ to $S_+$.  Let us denote by $\phi_1:=\text{Id}\,$ and $\phi_2:=(\rho(z_1),z_2)$ the local parametrization of $S_-$ and $S_+$ by $S^3\subset \mathbb{C}^2$.
 \begin{prop}\label{kiss}
Let $(\psi_1,\psi_2)$ be two local symplectic embeddings of $(S^3,C)$ into $(\mathbb{C}^2,C)$ with the properties that
$\psi_{1|C}=\text{Id}\,$, $\psi_{2|C}(e^{i\theta},0)=(e^{-i\theta},0)$ and $\psi_1(S^3)\cap\psi_2(S^3)=C$. Then there exist symplectic isotopies
$\Phi_1^t, \Phi_2^t$ with supports in a  neighbourhood  of $C$ such that $\Phi_i^1\circ \psi_i=\phi_i$ near $C$ and
$\Phi_1^t\circ \psi_1(S^3)\cap\Phi_2^t\circ \psi_2(S^3)=C$ for all time.
\end{prop}
\noindent{\it Proof :} Throughout this proof, we will denote by $B_i:=\psi_i(B\cap \mathcal U)$ and $\partial B_i:=\psi_i(S^3)$ where $\mathcal U$ is the neighbourhood  of $C$ on which the $\psi_i$ are defined.
Consider the orthogonal projections $(D_{\theta})_{\theta \in[0,2\pi[}$ of the discs $\{z_{1}=e^{i\theta},|z_2|<\varepsilon\}$ to the sphere $S^3$. It defines a local foliation $\mathcal D$ of $S^3$ around $C$ by symplectic discs, which
are tangent to the vertical lines $\{z_{1}=e^{i\theta}\}$. Notice that this foliation is invariant by the diagonal $S^1$-action
on $\mathbb{C}^2$, whose infinitesimal generator $X$ is tangent to the characteristic foliation of $S^3$. Denote also by $\mathcal D_i=(D_{\theta \,i})$,
$\mathcal D_i'=(D_{\theta\, i}')$, $X_i$ and $X_i'$ the images of  $\mathcal D$ and  $X$ by $\psi_{i}$ and $\phi_{i}$ respectively. \vspace{,1cm}

\noindent {\it Step 1 : Making all the foliations $\mathcal D_i,\mathcal D_i'$ tangent along $C$.} Since all the discs $D_{\theta,i}, \,D_{\theta,i}'$ are transverse to
$C$, there exists isotopies of foliations $(\mathcal D_i^s)_{s\in [0,1]}$ of $\partial B_i$ whose discs $(D_{\theta,i}^s)$ verify
$$
\left\{
\begin{array}{l}
D_{\theta,i}^0=D_{\theta,i}, \\
D_{\theta,i}(s)\pitchfork C=\{(e^{i\theta},0)\} \hspace{,5cm} \forall s\in[0,1],\\
\mathcal D_i^1 \text{ is tangent to } \mathcal D_i' \text{ along } C.
\end{array}
\right.
$$
Define now an area preserving map $\Phi_i^s:D_{i,0}\to D_{i,0}^s$ and extend it to a self-map of $\partial B_i$ by requiring that $\Phi_i^s$ sends $D_{\theta,i}$ to $D_{\theta,i}^s$ and  preserves the  characteristic foliation  of $\partial B_i$. It is then easy to see that
$\Phi_i^s$ extends to a symplectomorphism between neighbourhoods  of $\partial B_i$. This symplectic isotopy preserves $\partial B_i$
and precisely brings $\mathcal D_i$ to a foliation of $\partial B_i$ which  is tangent to $\mathcal D_i'$. Henceforth, we assume that the four foliations
$\mathcal D_1,\mathcal D_2,\mathcal D_1',\mathcal D_2'$ are tangent along $C$. \vspace{,1cm}

\noindent{\it  Step 2: Making the local {\bf  images} coincide.}
Thanks to the first step,  all the discs are now tangent so there exists a local diffeomorphism $f$ around $C$ which verifies the following~ :
\begin{enumerate}
\item $f$ is tangent to the identity on $C$ (that is $f_{*|C}=\text{Id}\,$),
\item $f:D_{0,i}\to D_{0,i}'$ is an area-preserving map,
\item  $f_{*}X_i=X_i'$ (then $f:D_{\theta \, i}\to D_{\theta \, i}'$ is also area-preserving),
\item $f_*$ is symplectic on $\partial B_i$. This is a requirement on the normal derivative of $f$ which is compatible with the condition ($1$),
\end{enumerate}
Now since $f$ is tangent to the identity on $C$, $f_*\omega$ differs from $\omega$ only by a small exact two form which vanishes
on $\partial B_i$, provided that we restrict it to a sufficiently small neighbourhood  of  $C$. Moser's path method therefore gives a "correcting map" $h$ which is the identity on $\partial B_1$, $\partial B_2$ and such
that $h_*f_{*}\omega=\omega$. The map $\Phi:=h\circ f$ is a symplectic map which transforms $\partial B_i$ to $S_\pm$, so $\Phi\circ \psi_i$ has the same image as $\phi_i$.

In order to see that $\Phi$ can be obtained by a symplectic isotopy, we use Moser's method once again. Consider a path $(f_t)_{t\in[0,1]}$
of (germs of) diffeomorphisms  near $C$ between $\text{Id}\,$ to $\Phi$ such that all $f_t$ are tangent to the identity on $C$.
Again, $f_{t*}\omega$ is close to $\omega$ near $C$ thanks to condition iii), so we can apply Moser's method and get a path of correction
maps $h_t$ such that $h_t\circ f_t$ is a symplectic isotopy between $\Phi$ and the identity  - recall that $\text{Id}\,$ and $\Phi$ are symplectic
so $h_0=h_1=\text{Id}\,$.  Notice also that since $h_t$ is a global isotopy, it obviously preserves the feature
$\partial B_1\cap \partial B_2=C$.  \vspace{.1cm} \\
{\it Step 3 : Reparametrization.} We are now in the situation where $\psi_i(S^3)=\phi_i(S^3)$, $\psi_i(D_\theta)=\phi_i(D_\theta)$
 and $\psi_{i*}X=\phi_{i*}X$. Observe that the
condition $\psi_i(S^3)=\phi_i(S^3)$ together with the fact that these two maps coincide on $C$ impose that
 their derivatives  only differ by rotations along $C$ :  $\psi_{*i}(p)=\phi_{*i}(p)\circ A^i_p$, where
$A^i_p(z_1,z_2)=(z_1,e^{i\theta^i_p}z_2)$. Hence rotating the balls allows that $\psi_{*i}(1,0)=\phi_{*i}(1,0)$. Either being more cautious in the previous step, or applying it again at this point of the proof, we can even achieve that $\psi_{i|D_0}=\phi_{i|D_0}$. Since $\Phi^t_X(D_0)=D_t$ and $\psi_{i*}X=\phi_{i*}X$  we thus get $\psi_{i|D_t}=\phi_{i|D_t}$, so $\psi_{i|S^3}=\phi_{i|S^3}$.
Arguing exactly as for the proof of proposition \ref{groloc}, we get symplectic isotopies from $\psi_i$ to $\phi_i$ which are the
identity on $\partial B_i$, so they preserve the intersection property $\partial B_1\cap \partial B_2=C$. \hfill $\square$
\subsection{Gromov's theorem}
We now prove Gromov's theorem cited above.  Figure \ref{figgro} illustrates the proof.
\begin{thm*}[Gromov] The space of compactly supported symplectic automorphisms of $B^4(1)$ is connected.
\end{thm*}
\noindent {\it Proof :} Let $\psi:B^4(1)\,\text{\raisebox{,3cm}{\rotatebox{-90}{$\curvearrowleft$}}}$ be a symplectic automorphism which is the identity near $\partial B^4(1)$. Gromov's proof shows
that the disc $\psi(D):=\psi(\{z_2=0\})$ is symplectic isotopic to $D$ {\it via} a symplectic isotopy with compact support, so we can
assume that $\psi(D)=D$. Moreover, a Hamiltonian isotopy with support in $D$ transforms $\psi_{|D}$ to the identity (this is an easy $2$-dimensional problem). This isotopy obviously extends to a compactly supported symplectic isotopy, achieving
$\psi_{|D}=\text{Id}\,$. By proposition \ref{groloc}, we can even assume that $\psi\equiv \text{Id}\,$ on a neighbourhood  of $D$. Since the restriction
of $\text{\small SDB}(S^2,1)$ to $S^2\backslash\{N\}$ ($N$ is the north pole) is symplectomorphic to $B^4(1)$, we can see $\psi$ as a transformation
of $\text{\small SDB}(S^2,1)$ with compact support which is the identity near $\mathcal{L}_0$ and the fiber over $N$. As in corollary \ref{corflex},
moving $\psi_*\lambda_0$ to $\lambda_0$ {\it via} a path of Liouville forms which fits with $\lambda_0$ wherever $\psi_*\lambda_0=\lambda_0$ gives a path of symplectic automorphisms of $B^4(1)$ which are the identity on $\partial B^4(1)$ and connect
$\psi$ to the identity.~\hfill $\square$ \vspace{0.1cm}\\

\begin{figure}[h!]
\begin{center}
\input gromov2.pstex_t
\caption{Proof of Gromov's theorem.}
\label{figgro}
\end{center}\vspace{-.7cm}
\end{figure}

\subsection{Isotopies of ellipsoids : proof of theorem \ref{isoell}.}\label{isoellsec}
We isotop a symplectic embedding $\psi:\mathcal E(a_+,a_-)\hookrightarrow \P^2$ with $a_-\leq a_+<\pi$ to a model embedding
which we describe now. Given a projective line $L$
in $\P^2$, there is an explicit embedding $\Phi$ of $\text{\small SDB}(S^2,1)$  into $\P^2$, whose complement is
one point. The restriction of the symplectic disc bundle to the  disc $S^2\backslash\{N\}$  is
symplectomorphic to $B^4(1)$. Our reference embeddings $\phi$ is simply the restriction
of $\Phi\circ s$  to the standard inclusion $\mathcal E(a_+,a_-)\subset B^4(1)$, where $s$ is the coordinate exchange
$s(z_1,z_2)=(z_2,z_1)$ in $B^4(1)$. Notice that $L$ intersects $\textnormal{Im}\, \phi$ through the small axis $\phi(\{z_1=0\})$
of the ellipsoid. Our first task  is to find a
supporting line for $\psi$ also, that is  a symplectic line of $\P^2$
which intersects $\textnormal{Im}\, \psi$ exactly through $\psi(\{z_1=0\})$.
\begin{lemma} \label{hard}
Every symplectic embedding $\psi:\mathcal E(a_+,a_-)\hookrightarrow\P^2$ with $a_-\leq a_+<\pi$
has a supporting surface of symplectic area $\pi$.
\end{lemma}
This lemma is the hardest point of the proof, and it is postponed to the next section. Now since
 any two symplectic lines  are isotopic,  we can assume that the supporting surface is $L$. Moreover, any two discs of the same area in
 $L$ are isotopic by an ambient Hamiltonian, so we can also assume that $\psi_{|\{z_1=0\}}=\phi_{|\{z_1=0\}}$. Proposition \ref{groloc} actually shows that these embeddings can even fit as germs around $\{z_1=0\}$. By corollary
 \ref{corflex}, we thus get that $\psi$ is  isotopic to $\phi$.  \hfill $\square$

\subsection{Isotopies of maximal symplectic packings.}
We now explain theorem \ref{isomax}. Let us first recall the setting. A symplectic packing of $\P^2$ by $k$ balls of
radii $r_1,\dots,r_k$ is a symplectic embedding $\phi:\rond{B}^4(r_1)\sqcup \dots \sqcup \rond{B}^4(r_k)\hookrightarrow \P^2$. Is is called smooth if
each $\phi_i:=\phi_{|\rond{B}(r_i)}$ extends to a smooth embedding of the closed ball. It is called maximal if none of the $\phi_i$
can be extended to a larger ball while still defining a packing (that is $\textnormal{Im}\, \phi_i\cap \textnormal{Im}\, \phi_j=\emptyset$).
Finally, we say that two packings $\phi,\, \psi$ are isotopic if there exist paths of symplectomorphisms $(\Phi_i^t)$
such that
\begin{itemize}
\item[{\tiny $\bullet$ }] $\Phi_0^i=\text{Id}\,$, $\Phi_i^1\circ \phi_i=\psi_i$
 \item[{\tiny $\bullet$ }] the embeddings $\phi_i^t:=\Phi^t_i\circ \phi_i$ define a packing for each $t\in[0,1]$.
\end{itemize}
When the packing is not maximal and the closure of the balls are disjoint, it is easy to see that this notion of isotopy coincides with
the notion of ambiant isotopy. However, when the closed balls are not disjoint, this notion allows for more freedom :  breaking intersections or reparametrizing only one of the balls for instance.
Symplectic packings were first introduced by Gromov, who showed that a symplectic packing of $\P^2$ by two balls
is subject to a symplectic obstruction : $r_1^2+r_2^2\leq 1$. In \cite{mcpo}, it was proved that this is the only obstruction
and Karshon constructed an example of smooth maximal $2$-packings. Theorem \ref{isomax} explains that all smooth maximal symplectic packing
 are isotopic to this particular one. We will only address below the problem for two balls of the same capacity $\pi/2$, but the
 generalization is completely straightforward.

 Since Karshon's example is the packing to which all the others will be isotoped, we describe it briefly now (the description is not Karshon's one). We already pointed out that the restriction of the bundle $\text{\small SDB}(S^2,1)$ to a disc of area $\pi/2$ is an ellipse $\mathcal E(\pi/2, \pi)$. Covering $S^2$  with two such discs, we get a packing
 of $\text{\small SDB}(S^2,1)$ by two ellipsoids $\mathcal E(\pi/2,\pi)$, which contain balls $\mathbb{\mathcal B}_1,\mathbb{\mathcal B}_2$ of capacity $\pi/ 2$. The embedding $\Phi$ of $\text{\small SDB}(S^2,1)$ around the line  $L$ already used in the previous section therefore gives a smooth maximal symplectic packing  when restricted to $\mathbb{\mathcal B}_1,\mathbb{\mathcal B}_2$.  Note that the line $L$ is a supporting line of the packing ({\it i.e.} for both balls) and that the two balls intersect along a characteristic circle $C$ of their boundaries (we also write a {\it Hopf circle} in the following).  We denote by $\phi=\{\phi_1,\phi_2\}:=\{\Phi_{|\mathbb{\mathcal B}_1},\Phi_{|\mathbb{\mathcal B}_2}\}$ this reference maximal packing and by $C_i\subset \partial \mathbb{\mathcal B}_i$ the circle $\phi_i^{-1}(C)$.
\begin{figure}[h!]
\begin{center}
\input karshon.pstex_t
\caption{Karshon's maximal packing.}
\label{figkar}
\end{center}\vspace{-.7cm}
\end{figure}

\noindent {\it Proof of theorem \ref{isomax} :} Let $\Psi=\{\psi_1,\psi_2\}$ be a smooth maximal symplectic packing  of $\P^2$ by two balls of
capacity $\pi/ 2$. By \cite{moi3}, we know that $\Psi$ is isotopic to a packing whose balls intersect exactly along one Hopf circle
of their boundaries. Taking this circle to $C\subset L$ and performing yet another packing isotopy, we can assume that
$\textnormal{Im}\, \psi_1\cap \textnormal{Im}\, \psi_2=C$, $\psi_i(C_i)=C$ and even that $\psi_{i|C_i}=\phi_{i|C_i}$.  By proposition \ref{kiss}, we can
even assume that $\psi_i=\phi_i$ near $C$. The intersections $L\cap \psi_i(\mathbb{\mathcal B}_i)$ are therefore symplectic discs which coincide
 with the images $\psi_i(\mathcal{L}_0)$ of the zero section $\mathcal{L}_0\subset \text{\small SDB}(S^2,1)$ on their boundary. By Gromov's result,
$\psi_i^{-1}(L\cap \psi_i(\mathbb{\mathcal B}_i))$ can be isotoped to $\mathcal{L}_0$ {\it via} a symplectic isotopy which is the identity on the
boundary of the ball. So we can safely assume that $\psi_i$ and $\phi_i$ coincide on $\mathcal{L}_0$. By the relative
version of proposition \ref{groloc},  we can achieve that $\psi_i=\phi_i$
on a neighbourhood  of $\mathcal{L}_0$. Then $\{\psi_1,\psi_2\}$ can be seen as an embedding $\Psi:\mathbb{\mathcal B}_1\cup\mathbb{\mathcal B}_2\subset \text{\small SDB}(S^2,1)\hookrightarrow \P^2$ which coincides with $\Phi=\{\phi_1,\phi_2\}$ on a neighbourhood  of $\mathcal{L}_0$. By corollary \ref{corflex}, $\Psi$ and $\Phi$ are
symplectic isotopic.
\hfill $\square$

\section{Supporting surfaces of embedded ellipsoids.}\label{hardsec}
We now turn back to the proof of  lemma \ref{hard}.

\subsection{The case of a ball.}
The assertion of lemma \ref{hard} when $a_-=a_+=a$ is weaker than what is established in \cite{mcduff1}, namely that {\it all} Hopf discs of
a ball close to supporting lines. We explain however the proof briefly for the convenience of the reader. It
 relies on the blow-up procedure introduced by McDuff (see \cite{mcsa}), which we recall now.
Let $\hat\mathbb{C}^2_1:=\{(p,d)\in \mathbb{C}^2\times \P^1,\; p\in d\}$ together with its projections $\pi_1$ on $\mathbb{C}^2$ and $\pi_2$
on $\P^1$ (so that $\pi_1:\hat\mathbb{C}^2_1\to \mathbb{C}^2$ is the classical blow-up of $\mathbb{C}^2$ at the origin). Put $E:=\pi_1^{-1}(\{0\})$,
 $L_\delta:=\pi_1^{-1}\big(B^4(\delta)\big)$ and define the symplectic form $\omega_\lambda:=\pi_1^*\omega_\text{st}+\frac{\lambda}{\pi}\pi_2^*\omega_\text{FS}$ on $\hat\mathbb{C}^2_1$. Denote finally by $g:\mathbb{C}^2\backslash B^4(\lambda)\to \mathbb{C}^2\backslash \{0\}$ the map defined by
$g(z):=\sqrt{|z|^2-\frac{\lambda}{\pi}}\frac{z}{|z|}$.    Then,
\begin{prop} \label{blowup}
The map $\Phi:=\pi_1^{-1}\circ g$ is a symplectomorphism between $\big(B^4(\lambda+\delta)\backslash B^4(\lambda),\omega_\text{st}\big)$ and $(L(\delta)\backslash L_0,\omega_\lambda)$. Moreover,
\begin{enumerate}
\item[i)] $\Phi^*i$ is $\omega_\text{st}$-compatible,
\item[ii)] A disc $D$ intersects $E$ transversally at exactly one point $\xi$
if and only if $\Phi^{-1}(D)$ is an annulus one of whose boundary coincides with the circle
$\Phi^{-1}(\xi)\in \partial B^4(\lambda)$.
\end{enumerate}
\end{prop}
The blow-up procedure can now be summed-up as follows. Given a symplectic embedding $\psi$ of a ball of capacity (slightly larger than) $\lambda$ in $\P^2$, one can use the map $\Phi$ to replace the image of $\psi$ by a neighbourhood  of $L_0$ in $\hat\mathbb{C}^2_1$.
The resulting symplectic manifold is diffeomorphic to the standard complex blow-up of $\P^2$, denoted by $\hat \P^2_1$. Thus the
embedding $\psi$ defines a symplectic form $\omega_\psi$ on $\hat \P^2_1$. It is also possible to extend the standard K\"ahler
structure $i$ defined in a neighbourhood  of $L_0$ to a $\omega_\psi$-compatible almost complex structure $J_\psi$ on $\hat \P^2_1$. Notice that the exceptional divisor $E$ is automatically a $J_\psi$-curve. \vspace{,2cm}

\noindent{\it Proof of lemma \ref{hard} ($a_-=a_+=a$):} First blow up  the ball $\psi(B^4(a))$ and consider the induced stuctures
$\omega_\psi,J_\psi$ on $\hat \P^2_1$. In view of the positivity of
intersection between holomorphic curves and the point ii) above, it is  enough to establish the existence of
a $J_\psi$-holomorphic curve of $\hat P^2_1$ in the class $L-E$ (where $L$ stands for the projective line and $E$ for the
exceptional divisor). Indeed, such a curve has area $\pi-a$ and intersects $E$ precisely at one point $\xi$, so its lift by $\Phi$
is a symplectic disc in $\P^2$ with boundary $\partial D=\Phi^{-1}(\xi)$. Gluing the Hopf disc attached to $\Phi^{-1}(\xi)$, we get a topological supporting surface - it may not be smooth along $\Phi^{-1}(\xi)$ - of area $\pi$. Such a topological
symplectic curve can easily be perturbed to a smooth supporting line for $\psi(B^4(a))$ (\cite{moi3}, lemma 5.2).

 Let $(\omega_t,J_t)_{t\in]0,1]}$ be the symplectic and almost-complex structures on $\hat\P^2_1$ given by the blown-up procedure when carried on for $\psi\big(B^4(ta)\big)$. Denote by $\mathcal M_t$ the moduli space of interest for us :
$$
\mathcal M_t:=\{u:\P^1\longrightarrow\hat P^2_1\; |\; du\circ i=J_t\circ du , \; [u]=L-E,\; p\in \textnormal{Im}\, u\},
$$
where $p$ is a fixed point of $\hat \P^2_1$ not on the exceptional divisor, and where the moduli-space has to be
understood modulo reparametrization. Let also be $\mathcal M_t^\varepsilon$ be the same moduli space but for a generic $\varepsilon$-perturbation $J_t^\varepsilon$ of the path $J_t$.
Our task is to understand that $\mathcal M_1$ is not empty. By Alexander's trick, we know that if $\delta$ is small enough, $\psi(B(\delta a))$ is symplectically isotopic to a standard ball in $\P^2$. We can thus assume that $\psi$ and $\phi$ (the standard embedding described in paragraph \ref{isoellsec}) coincide on $B(\delta a)$, which implies that $J_\delta$ is the standard complex structure on $\hat P^2_1$.
Thus $J_\delta$ is generic for this problem and it is easy to see
that $\mathcal M_\delta$ consists of exactly one element (the strict transform of a line passing through $p$ and the center of the ball
in $\P^2$).  If $\mathcal M_1$ is empty, then the space $\{\mathcal M_t^\varepsilon,\; t\in[0,1]\}$ cannot be compact,  so that a bubbling appears by Gromov's  compactness theorem \cite{gromov}. Taking the perturbation $\varepsilon$ small enough, we even see that the bubbling has to appear at some unperturbed almost-complex structure $J_{t_0}$, where $t_0\in ]0,1]$. It means that there is a non trivial decomposition
$$
L-E=\sum_i A_i\, ,
$$
where $A_i=k_iL+l_i E$ are homology classes of $\hat\P^2_1$ which are represented by  $J_{t_0}$-holomorphic spheres.
Since holomorphic curves intersect transversally, have positive symplectic area, and noting that $E$ is always $J_t$-holomorphic, we easily see that the decomposition must be of the form
$$
\begin{array}{ll}
a_0=L-kE& \text{ where } k\geq 2,\\
a_i=k_i E, & \text{ where } k_i\geq 0 \text { and }\sum_{i\geq 1}k_i = k-1.
\end{array}
$$
But we claim that the class $A_0$ is not represented by a $J_{t_0}$-holomorphic curve. Indeed, such a curve would lift to
a symplectic curve of area $\pi$ in $\P^2$ which coincides with $k\geq 2$ Hopf discs inside the ball $\psi\big(B(t_0 a)\big)$. This symplectic line would therefore have self-intersections which is prohibited by the adjunction inequality. This argument shows that  $\mathcal M_1$ consists of exactly one point. \hfill $\square$

\subsection{Symplectic Field Theory and supporting surfaces.}
This paragraph is aimed at setting the framework for completing the proof of lemma \ref{hard} {\it via} SFT-type arguments. Let
us first explain the broad idea. Lemma \ref{hard} asserts that one can find a supporting surface of our ellipsoids in the class of a
symplectic line (of area $\pi$). Following Gromov, we look for this curve among the pseudo-holomorphic spheres for tame almost complex structures. In fact, if we consider a sequence $J_\tau$ of almost complex structures which "stretches the neck" around the boundary of our ellipsoid, the $J_\tau$-holomorphic curves passing through the ellipsoid tend to intersect its boundary along
closed characteristics when $\tau$  goes to $+\infty$. These curves are therefore good candidates for providing the supporting curves we are looking for. We now review the construction of Symplectic Field Theory  relevant to us and the results we need to prove lemma \ref{hard}.

Let $\psi:\mathcal E(a_+,a_-)\hookrightarrow \P^2$  be a symplectic embedding of an irrational ellipsoid (that is $a_+,a_-$ are independent
over $\mathbb{Q}$) and  $\mathcal E$ its image. Then $V:=\partial \mathcal E$ is a hypersurface of contact type in $\P^2$, which has a standard neighbourhood  of the form $(V\times ]-\delta,\delta[,d(e^t\lambda))$ where $\lambda$ is a contact form on $V$ (with associated plane field $\xi$). The Reeb vector field defined on $V$ by $d\lambda(R,\cdot)=0$, $\lambda(R)=1$ and extended by $t$-invariance to $\partial \mathcal E\times ]-\delta,\delta[$ is easily seen to be Hamiltonian. One defines an $\omega$-compatible
almost complex structure on $V\times ]-\delta,\delta[$ by
\begin{equation}\tag{$*$}
\left\{
\begin{array}{l}
J:\xi\longrightarrow \xi \text{ is $\omega$-compatible}, \\
\displaystyle J\frac{\partial}{\partial t}=R,
\end{array}
\right.
\end{equation}
and it can be extended to an $\omega$-compatible almost complex structure $J_0$ on $\P^2$. Note that ($*$) can be used to define
an almost-complex structure $\wdt J$ on $V\times \mathbb{R}$. Pulling back $\wdt J$ to $V\times]-\delta,\delta[$ by a map of the form \fonction{\Phi}{V\times]-\delta,\delta[}{V\times]-\tau,\tau[}{(x,t)}{(x,f(t))} where $f'(t)=1$ near $-\delta$ and $\delta$, we get an almost complex structure $J_\tau$ which extends $J_0$ and remains $\omega$-compatible. The process of stretching the neck consists in letting $\tau$ go to $\infty$ in this construction. From the complex structure point of view, $\P^2$ then breaks in (at least, see proposition \ref{compactsft}) two pieces bounded by $V$ (the ellipsoid and its complement in our context), both equipped with the almost complex structure $J_\infty$ only defined on the {\it interior} of each piece. What makes this process interesting is a compactness result for sequences of $J_\tau$-holomorphic
curves due to Bourgeois-Eliashberg-Hofer-Wizocki-Zehnder \cite{behwz}. We sum it up briefly (and not in a strictly rigorous way)
in our context : $V\subset \P^2$ is the boundary of an irrational ellipsoid and the curves are $J_\tau$-holomorphic lines (that is homological to a projective line). Recall that since bubbling cannot appear for this special class of curves, there exists exactly one $J_\tau$-holomorphic line passing through two fixed points for every $\tau<+\infty$. These curves are moreover smoothly embedded spheres.
\begin{prop}\label{compactsft}
Let $\psi:\mathcal E(a_+,a_-)\hookrightarrow \P^2$ a symplectic embedding of an irrational ellipsoid, $\mathcal E:=\textnormal{Im}\, \psi$, $V:=\partial \mathcal E$,
$\gamma_{\pm}:=\psi(\{z_{2/1}=0\})\cap \partial \mathcal E$ and $J_\tau$ the almost-complex structures defined above. Fix two points in $\P^2$ (at least one lying inside $\mathcal E$), and denote by $S_\tau$ the $J_\tau$-holomorphic line passing through these two points. Then :
\begin{enumerate}
\item[i)] $S_\tau\cap \mathcal E$ converges on any compact set to a collection of $J_\infty$-holomorphic punctured spheres, each of whose ends is asymptotic to some positive multiple of $\gamma_+$ and $\gamma_-$.
\item[ii)] The same happens for $S_\tau\cap {}^c \mathcal E$, except that the ends are asymptotic to negative multiples of $\gamma_+$ and $\gamma_-$.
\item[iii)] As a whole, in a sense which we do not make precise here (see \cite{behwz}), $S_\tau$ converges to a so-called
holomorphic building  : the main layers are the ones described in i) and ii) above. The intermediate layers are $\wdt J$-holomorphic punctured spheres on $V\times \mathbb{R}$ asymptotic to positive multiples of $\gamma_1$ and $\gamma_2$ near
$V\times \{+\infty\}$ and negative multiples near $V\times \{-\infty\}$. All the layers glue together to form a topological sphere.
\end{enumerate}
\end{prop}

\begin{figure}[h!]
\begin{center}
\input sft.pstex_t
\caption{A holomorphic building.}
\label{holobuilding}
\end{center}\vspace{-.7cm}
\end{figure}

The previous proposition describes how look like the spheres $S_\tau$ when $\tau$ is large enough. We need to reduce further the possible shapes of these curves. The relevant ingredient is an index formula giving the virtual dimensions
of the moduli spaces of the involved curves. More specifically, denote by $\mathcal M_\mathcal E(\gamma_1,\dots,\gamma_n,k)$ the space of
$J_\infty$-spheres with n-punctures in $\mathcal E$, asymptotic to $\gamma_i=l_i \gamma_\pm$ ($l_i>0$) at these punctures, which pass through $k$ fixed points in $\mathcal E$ ($k=1$ or $2$ in our proof), and which are somewhere injective. Similarly, $\mathcal M_{\P^2\backslash \mathcal E}((\gamma_i)_{i\leq n},k)$
is the space of somewhere injective $J_\infty$-holomorphic spheres with $n$-punctures in $\P^2\backslash \mathcal E$ asymptotic to $\gamma_i=l_i\gamma_\pm$
($k_i<0$) in the homology class of a projective line (relative to $(\gamma_i)_{i\geq n}$) :
$$
\begin{array}{c}
\mathcal M_\mathcal E\big((\gamma_i),k\big):=\left\{
u:S^2\backslash\{(x_i)\}\to \mathcal E
 \left|\begin{array}{l}
 $u$ \text{ is somewhere injective} \\
  du\circ i=J_\infty\circ du \\
 u(\mathcal V(x_i)\backslash \{x_i\})\approx \gamma_i, \\   y_1,\dots, y_k\in \textnormal{Im}\, u
 \end{array}\right.\right\}\\
\mathcal M_{\P^2\backslash \mathcal E}\big((\gamma_i),k,mL\big):=\left\{u:S^2\backslash\{(x_i)\}\to \P^2\backslash\mathcal E
\left|\begin{array}{l}   $u$ \text{ is somewhere injective} \\ du\circ i=J_\infty\circ du\\ u(\mathcal V(x_i)\backslash \{x_i\})\approx \gamma_i, \\ y_1,\dots, y_k\in \textnormal{Im}\, u\\ u\in m[L] \text{ rel. } (\gamma_i)_i\} \end{array}\right.\right\}.
\end{array}
$$
We then have (see \cite{elgiho}, proposition 1.7.1 for the dimension of the moduli spaces and \cite{dragnev} for the regularity assertions) :
\begin{prop}\label{dim}
Let $\mu_\text{CZ}(\gamma_i)$ be the Conley-Zehnder index of $\gamma_i\subset \partial \mathcal E\subset \mathbb{C}^2$. Then :
\begin{enumerate}
\item[i)] $\text{VirtDim} \,\mathcal M_\mathcal E\big((\gamma_i),k\big)=-2+\sum (\mu_\text{CZ}(\gamma_i)+1)-2k$,
\item[ii)] $\text{VirtDim}\, \mathcal M_{\P^2\backslash \mathcal E}\big((\gamma_i),k,mL\big)=2C_1(mL)-2+\sum (-\mu_\text{CZ}(\gamma_i)+1)-2k=\sum (-\mu_\text{CZ}(\gamma_i)+1)+6m-2-2k$,
\item[iii)] The moduli space $\mathcal M_\mathcal E\big((\gamma_i),k\big)$ or $\mathcal M_{\P^2\backslash \mathcal E}\big((\gamma_i),k,mL\big)$ of negative virtual dimension are empty  for a generic choice of $J_0$.
\end{enumerate}
\end{prop}
\subsection{Proof of lemma \ref{hard}.}
Let $\psi:\mathcal E(a_+,a_-)\hookrightarrow \P^2$ be a symplectic embedding. Denote by $\mathcal E$ its image and $\gamma_+$, $\gamma_-$
 the closed characteristics of $\partial \mathcal E$ of $\omega$-area $a_+,a_-$ in $\mathcal E$. Let us fix two points $p_1\in\mathcal E$,
 $p_2\in{}^c\mathcal E$, consider a stretching of the neck $(J_\tau)$  defined as  above, and let $(S_\tau)$
be the path of $J_\tau$-holomorphic lines in $\P^2$ (of symplectic area $\pi$) passing through $(p_1,p_2)$.

When $\tau$ tends to $+\infty$, the spheres $S_\tau$ converge to a $J_\infty$-holomorphic building $S_\infty$ (in the sense of proposition \ref{compactsft}), whose main layers will be denoted in the sequel $S_\infty\cap \mathcal E$ and $S_\infty\cap {}^c\mathcal E$.
We know that $S_\tau\cap {}^c\mathcal E$ converges to $S_\infty\cap {}^c\mathcal E$ on all compact sets of $\P^2\backslash \mathcal E$.

\begin{lemma} \label{areares}
The surface $S_\infty\cap{}^c\mathcal E$ is made of exactly one connected component.
\end{lemma}
\noindent {\it Proof :} The numbers $n$ of these components is obvisouly  not zero since no compact surface of  $\mathcal E$ is symplectic. Moreover, each connected component of $S_\infty\cap{}^c\mathcal E$
contributes at least  by $\pi$ to the total area of $S_\tau$ for large enough $\tau$ ({\it i.e.} $\mathcal A_\omega(S_\tau)\geq n\pi$).
Indeed since $S_\tau$ is close to $S_\infty$ in ${}^c\mathcal E$, it has $n$ connected components, each symplectic. These components  thus have positive symplectic areas on one hand, and are  asymptotic to negative multiples
$-k_i\gamma_i$ of $\gamma_+,\gamma_-$ on the other hand, so they have area at least $\pi-k_ia_i$. Since the
complement of these components in $S_\tau$ belongs to a contractible set, the total symplectic area of $S_\tau$ verifies :
$$
\hspace{+2,5cm}\mathcal A_\omega(S_\tau) \geq\sum_{i=1}^n (\pi-k_ia_i)+ \sum_{i=1}^n k_ia_i=n\pi. \hspace{3cm} \square
$$
Call $\gamma_i$ the asymptotes of $S_\infty\cap{}^c\mathcal E$. The curve $S_\infty\cap{}^c\mathcal E$ belongs to the moduli space $\mathcal M_{\P^2\backslash\mathcal E}\big((\gamma_i),1\big)$ which must have non-negative virtual dimension by proposition \ref{dim} iv).
Thus,
$$
\sum (-\mu_\text{CZ}(\gamma_i)+1)+2\geq 0.
$$
Since the Conley-Zehnder indices of the $\gamma_i$ are all no less than $3$ (and $\gamma_-$ is the only orbit with index $3$), we conclude that $S_\infty\cap{}^c\mathcal E$ is a disk asymptotic to $\gamma_-$. This implies that for $\tau$ large enough,
a small Hamiltonian perturbation of $S_\tau\cap{}^c\mathcal E$ makes it a symplectic disk of $\P^2\backslash \mathcal E$
asymptotic to $\gamma_-$. Closing it by the Hopf disc $\psi(\{z_1=0\})$, we thus get a piecewise smooth symplectic
curve whose intersection with $\mathcal E$ is exactly $\psi(\{z_1=0\})$. The smoothening of this kind of singular symplectic surfaces is done in \cite{moi3}, lemma 5.2, and it produces a smooth supporting symplectic line for $\mathcal E$.  \hfill $\square$

\subsection{Further supporting polarizations.}
The proof of theorem \ref{isoell} went by finding a supporting line of an ellipsoid passing through his small axis (lemma \ref{hard}). It is however so specific to the situation $a_+<\pi$ that it would be almost dishonest to claim it is really relevant.
Commonly, the polarizations have high degree, so the ellipsoids which live in the associated disc bundles are very thin and excentric. The needed supporting surface must therefore intersect the ellipsoid we wish to isotop along its big axis instead of
the  small one. The aim of this paragraph is to show that although it is much harder this way round, finding these
supporting curves is not always hopeless. We focus on the following :
\begin{thm}\label{isoell2}
The space of symplectic embeddings of ellipsoids $\mathcal E(a_+,a_-)$ in $\P^2$ with $a_+\in]\pi,2\pi[$ and $4a_-<a_+<2\pi<5a_-$
is connected.
\end{thm}
Since all symplectic conics are isotopic, the same reasoning as in paragraph  \ref{isoellsec} shows that we only need to find a supporting conic for every embedded ellipsoid $\mathcal E:=\psi\big(\mathcal E(a_+,a_-)\big)$ which intersects $\mathcal E$ along its big axis.
\begin{lemma}\label{hard2} Every symplectic embedding $\psi:\mathcal E(a_+,a_-)\hookrightarrow \P^2$  with $a_+\in]\pi,2\pi[$ and
$\frac{2\pi}{5}<a_-<\frac{a_+}{4}$ admits a supporting conic, that is a symplectic curve of area $2\pi$ whose intersection
with $\textnormal{Im}\,\psi$ is exactly $\psi(\{z_2=0\})$.
\end{lemma}
\noindent {\it Proof :} Under the hypothesis of lemma \ref{hard2}, the Conley-Zehnder indices are~:
$$
\begin{array}{c}
\mu_\text{CZ}(\gamma_-)=3,\;\mu_\text{CZ}(2\gamma_-)=5,\;\mu_\text{CZ}(3\gamma_-)=7,\;\mu_\text{CZ}(4\gamma_-)=9,\\\mu_\text{CZ}(\gamma_+)=11,\;\mu_\text{CZ}(5\gamma_-)=13.
\end{array}
$$
Consider a stretching of the neck $(J_\tau)$ around $\partial \mathcal E$. Fix five generic points
$(p_1,\dots,p_5)$ inside $\mathcal E$. Since bubbling of spheres appear in codimension $2$, a choice of a generic almost-complex structure $J_0$ around $p_1$ insures that for each $\tau$ there is a   $J_\tau$-holomorphic sphere $S_\tau$ in the class of a conic passing through these five points. This sphere is even unique by positivity of intersection. When $\tau$ tends to infinity, $S_\tau$ converges in the sense of \cite{behwz} to a holomorphic building $S_\infty$. Call again $S_\infty\cap\mathcal E$ and $S_\infty\cap{}^c\mathcal E$ the main layers of the building
respectively inside and outside $\mathcal E$. Reasoning as in lemma \ref{areares} we see that $S_\infty\cap{}^c\mathcal E$ has one
or two connected components which are punctured spheres. Since $S_\infty$ is a topological sphere, we conclude that
\begin{itemize}
\item[{\tiny $\bullet$ }] either $S_\infty\cap {}^c\mathcal E$ has two components and the components of $S_\infty\cap \mathcal E$ are disks and
at most one annulus (the annulus required to connect the two outside connected components may lie in an intermediate
layer),
\item[{\tiny $\bullet$ }] or $S_\infty\cap{}^c\mathcal E$ has one component and all the components of $S_\infty\cap\mathcal E$ are disks.
\end{itemize}
Using a trick due to Hind-Kerman \cite{hike}, we can get a sharp energy-related restriction on $S_\infty\cap \mathcal E$ :
\begin{lemma}\label{gamma+} The curve $S_\infty\cap \mathcal E$ is a disk asymptotic to $\gamma_+$.
\end{lemma}
\noindent{\it Proof :} Since $a_-<a_+/5$, the open ellispoid $\rond{\mathcal E}(a_+,a_-)$ contains
five disjoint closed symplectic balls $\phi_i(B^4(a_+/5-\varepsilon))$, $i=1,\dots,5$.   Starting with an almost complex structure $J_0$
which coincides with ${\psi\circ\phi_i}_*i$ on the images of the $\phi_i$, and $p_1,\dots,p_5$ at the centers of
these balls, the area of any $J_\infty$-curve passing through $p_1,\dots,p_5$ is no less than $a_+-5\varepsilon$ which is  greater
than $4a_-$ provided that $\varepsilon$ is small enough. The asymptotes of $S_\infty\cap \mathcal E$ must therefore contain at least
"one $\gamma_+$" or "five $\gamma_-$". Since $\gamma_++\gamma_->5\gamma_->2\pi$, the only possibility is that
$S_\infty\cap \mathcal E$ has exactly one puncture, asymptotic to $\gamma_+$. \hfill $\square$ \vspace{0.1cm}\\

Let us finally conclude the proof of lemma \ref{hard2}. Lemma \ref{gamma+} together with the fact
that  $a_++a_->2\pi$ shows that there can be no intermediate layer. Thus $S_\infty\cap{}^c\mathcal E$ is one disk
asymptotic to $-\gamma_+$, of area $2\pi-a_+$. Gluing the Hopf disc $\psi(\{z_2=0\})$ to $S_\infty\cap{}^c\mathcal E$, we get
again a singular symplectic curve, the kind of singularities of which we know how to deal with by \cite{moi3}. We can
smoothen this surface to a symplectic conic coinciding with $\psi(\{z_2=0\})$ inside $\mathcal E$. \hfill $\square$

redpajama_set_name: RedPajamaArXiv


text:
Q: How to prevent a process from spawning more childs I am working on an online judge for conducting ACM-ICPC like competitions on my college LAN.
For that I require that the judge may be safe enough to prevent malicious programs from executing themselves on my server.
(An example of such a program would be )
int main(){
    while(1) fork();
}

Lets call the executable of this program testcode.
This program would cause my server running the judge to freeze.
Obviously i dont want that to happen.So to prevent that i tried using ptrace.I came up with the following code:(Lets call the executable of this code monitor)
#include <sys/ptrace.h>
#include <sys/types.h>
#include <sys/wait.h>
#include <unistd.h>
#include <string.h>
#include <errno.h>
#include <sys/user.h>
#include <sys/syscall.h>
#include <sys/reg.h>
#include<stdio.h>
#include <sys/ptrace.h>
#include <sys/types.h>
#include <sys/wait.h>
#include <unistd.h>
#include <string.h>
#include <errno.h>
#include <sys/user.h>
#include <sys/syscall.h>
#include <sys/reg.h>
#include<stdio.h>
#include<signal.h>
#include<sys/prctl.h>
#include<stdlib.h>
#define NOBANNEDSYS 40
int bannedsys[50]={2,14,12,15,26,37,38,39,39,40,41,42,46,47,48,49,50,60,61,63,72,83,88,120,102,182,183,190};

int main(int argc,char **argv) {
                int insyscall=0;
                if(argc!=2) {
                    fprintf(stderr,"Usage: %s <prog name> ",argv[0]);
                    exit(-1);
                    }

  int status = 0;
  int syscall_n = 0;
  int entering = 1;
  int amp;
  struct user_regs_struct regs;
  int pid = fork();

  if ( !pid ) {
  prctl(PR_SET_PDEATHSIG, SIGKILL);
    ptrace( PTRACE_TRACEME, 0, 0, 0 );
    execlp( argv[1],argv[1], 0 );
    }

  else {

    //ptrace( PTRACE_SINGLESTEP ,pid, 0, 0 );
    // ptrace( PTRACE_SYSCALL, pid, 0, 0 );
    while (1) {

        wait( &amp);
            if ( WIFEXITED( amp ) ) break;

                //ptrace( PTRACE_SINGLESTEP ,pid, 0, 0 );

    if(insyscall==0){
        ptrace( PTRACE_GETREGS, pid, 0,&regs );
      int i=0;
      for(i=0;i<NOBANNEDSYS;i++) if(regs.orig_eax==bannedsys[i]) {  kill(pid,SIGKILL);
                                printf("%d killed due to illegal system call\n",pid);

                                    abort();
                                    }

        insyscall=1;
               }
       else insyscall=0;
      // ptrace(PTRACE_CONT,pid,0,0);
     // wait(&amp);

    ptrace( PTRACE_SYSCALL, pid, 0, 0 );
     // puts("Here");
//ptrace(PTRACE_CONT, pid, 0, 0);

    }

  }

  return 0;
}

This code works quite well in blocking system calls that may cause problem.But when the code to be monitored contains  fork calls in a loop like the testcode the machine freezes because of thrashing.The reason i could make out is while the original process is killed by the monitor code its child survives and continues to carry the fork bomb.
How to fix the monitor code so that it can be deployed successfully?
PS:Portability is not a concern.I am looking for a linux specific answer.
EDIT:
I used setrlimit for setting the maximum number of child processes to 0 before calling exec.This till now appears to be a a good solution.Though it would be nice to hear from the community if there are still loopholes in the monitor code.

A: You can stop the child and trace new processes:

PTRACE_O_TRACEFORK (since Linux 2.5.46) Stop the child at the next
  fork(2) call with (SIGTRAP | PTRACE_EVENT_FORK << 8) and automatically
  start tracing the newly forked process, which will start with a
  SIGSTOP. The PID for the new process can be retrieved with
  PTRACE_GETEVENTMSG.

You can have a working example with these changes:
/* ... */
  ptrace(PTRACE_SETOPTIONS,pid,NULL, PTRACE_SYSCALL | PTRACE_O_TRACEFORK) ;
  while (1) {
    printf("Waiting\n");
    pid = wait(&amp);
    printf("Waited %d\n", amp);
    if (WIFEXITED(amp)) {
      break;
    }
    if (WSTOPSIG(amp) == SIGTRAP)
    {
      int event = (amp >> 16) & 0xffff;
      if (event ==  PTRACE_EVENT_FORK) {
        printf("fork caught\n");
        pid_t newpid;
        ptrace(PTRACE_GETEVENTMSG, pid, NULL, (long) &newpid);
        kill(newpid, SIGKILL);
        kill(pid, SIGKILL);
        break;
      }
    }
    if (insyscall == 0) {
      ptrace(PTRACE_GETREGS, pid, 0, &regs);
      int i = 0;
      for (i = 0; i < NOBANNEDSYS; i++) if (regs.orig_eax == bannedsys[i]) {
        kill(pid, SIGKILL);
        printf("%d killed due to illegal system call\n", pid);

        abort();
      }

      insyscall = 1;
    } else {
      insyscall = 0;
    }
    ptrace(PTRACE_CONT, pid, NULL, 0);
  }

Reference

A: For what you're doing, I would strongly recommend investigating seccomp, which will allow you to disallow a process from using any system calls at all other than exit, read and write (only to file descriptors which are already open), and sigreturn.

redpajama_set_name: RedPajamaStackExchange


Endless summer. To many, it is as close to paradise on earth as they can get. The Exceptional Ms. E, for example, feels most at home during the dog days. Credit her upbringing in Venezuela and South American blood. In fact, the warmer it gets, the more she claims to be in the Christmas spirit.
As much as I enjoy summer's vacations, cookouts and cracking thunderstorms, the summer months hold less enjoyment than they did when I was in school. Primarily because I don't like to sweat unless I'm exercising.
That having been said, there are a few things that I love about the season.
I was once taught the wonderful phrase "summer rules" while at an engagement party on the Main Line. The happy couple were of an older generation, both widowed, and had decided to tie the knot. The party was wonderful, so different from what you see today. When I walked into the foyer, I was warmly greeted, led through the library (Wow!) and into the gardens. My host asked what I was drinking and handed over a cool G&T letting me know that "Summer rules apply…. I'll mix you the first. After that, you're free to help yourself now that you know where the bar is…." I'm not sure that I have ever been made to feel more comfortable, more quickly, at a party. I hope that they are still enjoying their slow boat to China.
But how does one stay cool at these affairs? Particularly when the typical male attire of t-shirt, cargo shorts and sneakers/flip-flops is even more out-of-place than usual?
Today we are under no such expectations and even fewer obligations. Most often to our collective embarrassment.
Still, there is something to be said for not surrendering one's elegance when the "thermometer goes way up." Consider two fabrics: seersucker and linen.
Seersucker is our interpretation of the Indian word "shir shakkar" meaning "milk and honey." The fabric is woven in such a way that crinkled stripes alternate with smooth stripes. Yes it looks a bit wrinkle-y. But those wrinkles help the suit to stand away from your skin allowing for more air-circulation. More air-circulation, more comfort (the same principle that works in your hi-tech sweat-wicking work-out clothing.) Wear your seersucker between Easter and Labor Day. Whether or not you accessorize it with white bucks, a button-down and a bowtie or brown loafers, and a black knit tie (or even a black polo shirt with no tie for very casual affairs) is up to you.
More about being wrinkled. For real casual elegance, you can't beat white linen trousers in my book. They (along with white tropical wool trousers and white jeans) are the staple of my summer wardrobe. Buy them in the heaviest weight that you can — they will tailor and drape better. I wear them as often as possible. Freshly pressed for a dinner party, wrinkled the next day. And even the next day, when I will roll them up a bit more and wear them with flip-flops or espadrilles. During the day pair them with a polo shirt or a butcher-stripe, open-collar dress shirts and Tod's suede driving shoes. For the evening, black Belgian loafers, a striped t-shirt, open collar blue button down and a blue blazer looks sharp. (Don't forget your white linen pocket square.) Yes you wrinkle, but so what? It's summer. And if you doubt that tailored clothing can look good under some pretty rough travel, check out the boys in the photo above (from a Daily Mail article about Darfur.) I wish I had better examples of casually elegant summer wear, but for now, it's up to you to lead the charge.
This entry was posted in Elegantology and tagged Elegantology, films, Hollywood. Bookmark the permalink.


This entry was posted on Friday, April 15th, 2016 at 6:28 am and is filed under Commercial Property tax.
All property owners dream of lower property taxes. But how many property owners will actually do something to make that dream a reality? Not nearly enough. There are two reasons why property owners complain about high property taxes but don't do anything about it. First, there are those property owners who simply don't realize that they have the legal right to challenge their property tax assessment. Second, even though most property owners actually do know that there is a process by which they can legally challenge their property tax assessment, they won't initiate the process because they think it's a losing fight.
To some extent, the latter group of property owners is correct. Anyone who's been through the property tax appeals process knows how exhausting it can be. After all, you're approaching your county government, who is likely hard up for cash, and telling them that you think you ought to pay them less money than what they requested. Do you think they're going to make this pleasant for you? Of course not.
Even if you survive the property tax appeals process, unless you're some sort of property tax expert, there is little chance that you'll save big on your property tax bill. You'll either save nothing at all or save so little that it was never worth the amount of time and frustration that you spent on the appeals process. So much for lower property taxes, right?
Fortunately, there is a solution. You really can transform your dream of lower property taxes into reality, if you get the right kind of help. The right kind of help can be found at Commercial Property Tax Professionals (CPTP), an Atlanta-based tax consulting service that has been helping property owners throughout the United States minimize their property tax burdens.
At CPTP, experienced property tax consultants will take on the appeals process for you. They will save you hours of time and frustration. Best of all, they will save you money on your property tax bill! They can get these results because they have the right expertise and experience.
Do you want to learn more about how Commercial Property Tax Professionals can help you minimize your property tax burden? Contact their office at (678) 466-6815 or email them at info@cptaxpro.com.


text:
Q: Error using Mockito and Specs2 I have the following spec:
import org.specs2.mock.Mockito
import org.specs2.mutable.Specification

class LinkUserServiceSpec extends Specification with Mockito {

  val linkUserService = mock[LinkUserService]

  "The 'LinkUserService' isUserLinked method" should {
      "return false when a previously unlinked userId is passed in for a given service" in {
          linkUserService.isUserLinked("nobody", "YT") returns false

          linkUserService.isUserLinked("nobody", "YT") must beFalse
      }
  }
}

And the following dependency in my build.sbt:
"org.specs2"              %%  "specs2"        % "2.2" % "test"
However I get this error when I type test into the sbt console:
[error] bad symbolic reference. A signature in MocksCreation.class refers to type MockSettings
[error] in package org.mockito which is not available.
[error] It may be completely missing from the current classpath, or the version on
[error] the classpath might be incompatible with the version used when compiling MocksCreation.class.
[error] bad symbolic reference. A signature in MockitoStubs.class refers to term stubbing
[error] in package org.mockito which is not available.
[error] It may be completely missing from the current classpath, or the version on
[error] the classpath might be incompatible with the version used when compiling MockitoStubs.class.
[error] two errors found
[error] (test:compile) Compilation failed
[error] Total time: 3 s, completed Sep 12, 2013 3:23:41 PM

Anyone know what this could be?

A: Apparently if you want to use Mockito with Specs2 you have to provide the dependency yourself, I added the following to my build.sbt and things started working:
"org.mockito"          %   "mockito-all"   % "1.9.5"


text:
Columbia 6/29/2020 8:27:44 PM
News / Fashion
Lil Wayne Planning A New Collab With Mannie Fresh
For anyone who even has passing interest in Lil Wayne, Young Money Radio is a must listen to. Widely regarded as one of the very best, fans are constantly yearning for new music to drop from the legend. It's no secret that his music is fire, but have you seen the new custom pendants HipHopBling.co's been putting out? Well worth your time, especially if you want to support George Floyd with an "I Can't Breathe" pendant.
On one of his most recent radio shows, it sounds as though he may be doing just that- on a call with Mannie Fresh Wayne revealed that everyone's always asking when they will collaborate on an album. Shortly after, Mannie speaks up saying that he already has "more than a handful of tracks put together," for their collab.
As for the release date of such an album? We'll probably need to wait up to 2021. "Before it's over with, 2021, after this whole coronavirus bullshit, we will drop the album and it's just gonna be solely me and him," said Fresh on Wayne. When you're looking to shine like Lil Wayne though, peep HipHopBling.com's new jewelry section for your next iced out piece.
HipHopBling.com is a jewelry wholesaler and retailer with tremendous experience and knowledge of the hip hop jewelry segment. Our company has been around for nearly 20 years offering the finest products to consumers and hip hop related retailers across the globe. We have grown from a small room during the 90s to a full blown 25000sqft warehouse in Columbia MD shipping hundreds of orders each day. Our mission is simple and we stand by it each and every day.
Mission: Find the best jewelry in the world and sell it at the lowest prices possible without compromising quality or service.
Ad@hiphopbling.com
More Stories www.findit.com/hiphopbling

redpajama_set_name: RedPajamaCommonCrawl


The Portland food industry is a titan; the foodies here run among the elite gourmand. The Portland food industry is also metamorphic; it is at once a science, a moral attitude, a political persuasion, and an art. The cult of gastronomic purists in Portland has taken the art of culinary masterpiece to the extreme (Dr. Sushi and BBQ?). For all of its myriad forms, the kind of eating we do in Portland is almost a singularly bourgeois past-time and yet Oregon also shamefully has the 2nd highest rate of hunger. In a town with an army of epicurists it's criminal to have so many who suffer from food insecurity. Many agencies have stepped up to diminish the yawning divide between the hungry and the well-fed.
One such agency, Urban Gleaners, has taken dumpster diving and wielded it as a sharp and effective tool for reducing waste and hunger. For many, when we think of dumpster diving there is a quick shiver of distaste, like you just swallowed a bad piece of sushi. But for many, dumpster diving, also known as gleaning, is a badge of honor and a way of life.
Urban Gleaners is trying to capitalize on the asymmetry between food production and consumption. They are a non-profit agency that creates relationships with local restaurants and farmers markets to gather up the excess food not used or sold. The gleaners then store their bounty in refrigerators that are strategically placed around Portland and eventually handout the food at certain dispensary sights around the city, usually schools.
Urban Gleaners was started by Tracy Oseran in 2005. Oseran was listening to an NPR broadcast of Here On Earth. The show was highlighting a nonprofit in Cambridge Massachusetts called Food For Free that provided low-income families with "recycled" food. Inspired, Oseran took her two teenage kids and canvassed the local Portland restaurant scene. Oseran was surprised to find that no parallel to Food For Free yet existed in Portland, the progressive pioneer. On her first whirlwind tour of Portland, Oseran visited Blue Hour, one of Portland's high end restaurants, and received their first food contract. Soon her efforts started to snowball. The early momentum has sustained the organization until today. Urban Gleaners, like Food For Free in Massachusetts, and City Harvest in New York, receives some outside funding (Amex contributes to Portland's effort), but relies heavily on the capable and free labor of its volunteers and donators.
After 6 years of operation, Urban Gleaners feeds 900 families and delivers food to 5 schools in low-income neighborhoods. In weight, that translates to 45,000 pounds of recycled food a month diverted from the dump. Their effort seems negligible compared to the 200,000 tons of food waste that is destined for landfill in Portland every year. But the human incentive is great and keeps Urban Gleaners rallying against the mountain of waste.
Despite some early hang-ups, friends and allies in the food industry are supportive. My contact at Urban Gleaners, Emily Kanter, believes that last year Trader Joe's donated almost $700,000 worth of food. Because of growing momentum the stigma around "recycled" food is dissolving. Good food is good food and Kanter often finds herself drooling over the delicacies that pass through the hands of her agency. In addition to Trader Joe's, Urban Gleaners receives food from Blue Hour, Simpatica, New Seasons, local farms and farmers markets. Eventually the gleaners hope to source directly from the farms.
Emily recognizes that the very service they provide supports and perpetuates, even validates, a system that produces too much waste – a system that isn't held accountable for its surplus and glut. But Urban Gleaners provides an interim service, much like the economic bailout, it's meant to float the food industry until it's independently viable. Emily dreams of a food system in which her organization is obsolete, but admits that a small amount of gleaning will probably always be necessary.
If you'd like to learn more, I'd first recommend watching the movie that sparked my interest, Gleaners and I. It does a wonderful job of articulating the anguish and the criminality of a system that perverts the true meaning of value. For those in Portland, check out the Portland Fruit Tree Project, an organization that gathers volunteers to harvest produce from people's yards. Individuals contact The Fruit Tree Project when they have too much produce (lemons on their lemon tree, raspberries, spinach in the garden) to pick themselves. Half of the loot goes to people who pick it and half goes back to the agency for donation.
This entry was posted in Food Warrior Interns and tagged Ava Mikolavich, blue hour, food warriors, gleaners and i, gleaning, portland, Real Time Farms, simpatica, trader joes, urban gleaners. Bookmark the permalink.


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Top Moments From Past MLB All-Star Games
By David Dwork July 14, 2015 at 4:11 pm
Filed Under:2015 MLB All-Star Game, Babe Ruth, Cal Ripken Jr., Comiskey Park, Dee Gordon, Fenway park, Giancarlo Stanton, John Kruk, Larry Walker, Major League Baseball, Miami Marlins, MLB, MLB All-Century Team, MLB All-Star Game, Pete Rose, Randy Johnson, Ted Williams
Cal Ripken Jr., of the Baltimore Orioles, watches his home run clear the fence in the third inning of the baseball All-Star game 10 July, 2001, at Safeco Field in Seattle, Washington. (Source: DAN LEVINE/AFP/Getty Images)
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The 2015 MLB All-Star game is finally upon us. While the first half of the season has not been too kind to our hometown Miami Marlins, the annual display of superstars is (almost) always fun to watch.
Fans in South Florida won't get to see the two Marlins All-Stars that were voted in as starters, as slugger Giancarlo Stanton and speedster Dee Gordon both sustained injuries that will keep them from playing in the game.
Now that the midsummer classic has some meaning to it, with the winning side getting home field advantage in the World Series, the game has the potential to be a bit more exciting and have some depth to it.
We can only hope that this year's edition provides some exciting and memorable moments, but for now let's take a look back at some of the biggest and most influential moments in All-Star Game history.
1933- The Babe Goes Deep
In the first All-Star Game ever played, the game held at Comiskey Park in Chicago was attached to the 1933 World's Fair and was intended to be a one-time affair.
In the third inning of a 1-0 game, Babe Ruth stepped up to the place to face St. Louis' Bill Hallahan. With a runner on first, Ruth crushed a ball into the right field stands giving the American League a lead they would not relinquish.
Ruth's home run was the first ever in an All-Star game and the event was such a great success that the decision came down to hold the event every year.
1999- Ted Williams Returns To Fenway
The 70th MLB All-Star Game would be the last one played in the 20th century and was fittingly played at the oldest ballpark in the league, Fenway Park.
Prior to the game there was an on-field ceremony to recognize the MLB All-Century team. When the time came to introduce Ted Williams, the atmosphere inside Fenway became electric. Williams was brought out on a golf cart and as he was driven towards home plate, the standing ovation from the players and fans grew louder and louder.
Many of the players had tears in their eyes, as did so many of the faithful Red Sox fans in attendance. When Williams arrived at the pitcher's mound, everyone in uniform, every player and coach, surrounded the legend and took their turns wishing him well and thanking him for all he had done.
The shots of Williams fighting back tears, somewhat unsuccessfully, is an image that will never be forgotten and will long be remembered as one of the most touching moments in baseball history.
1970- Exhibition or Not, Pete Rose Always Gives His All
In the 41 All-Star Games played before 1970 only five had gone into extra innings. This would be the sixth to go beyond nine innings and the fans inside Cincinnati's Riverfront Stadium did not leave disappointed.
In the 12th inning of a 4-4 game, Cincinnati Reds All-Star Pete Rose came to the plate with none on and two outs. Rose singled to extend the inning and then advanced to second when the Dodgers' Billy Grabarkewitz followed up Rose's base hit with one of his own.
The next batter was Chicago's Jim Hickman and he smacked a single to center field. Rose turned on the jets in an attempt to score the winning run from second base but was beaten to the plate by a great throw from Royals outfielder Amos Otis.
Showing no hesitation, Rose lowered his shoulder and prepared for impact as he crashed into Indians catcher Ray Fosse. The collision jarred the ball loose and gave the National League a 5-4 win in front of Rose's home fans who erupted in joy over their star player giving his all in a game that ultimately meant nothing.
The violent play at the plate dislocated Fosse's shoulder but Rose showed no remorse after the game. "I just want to get to that plate as quickly as I can," explained Rose. "Besides, nobody told me they changed it to girls' softball between third and home."
2001- Ripken Goes Deep In Final All-Star Game
Similar to what baseball fans saw during the 2014 All-Star Game for Derek Jeter, when Cal Ripken Jr. played in his final All-Star Game in 2001 he was also given a respectful tribute from his peers.
When the American League team took the field in the top of the first inning, starting shortstop Alex Rodriguez walked over to Ripken at third base and insisted that they swap positions. Ripken had spent almost all of his career as a shortstop and Rodriguez wanted to give him the opportunity to play his natural position in an All-Star Game one final time.
As much as the fans enjoyed the grand gesture from Rodriguez and his American League teammates, Ripken still had a little magic left to share with everyone.
In the third inning Ripken came to the place with the game still scoreless. He took the first pitch he saw and sent it into the outfield bleachers to give the AL a 1-0 lead and the fans went crazy. Ripken would be named the All-Star MVP for the second time in his career as the American League went on to win the game 4-1.
1993 & 1997- Randy Johnson Announces His Presence With Authority
Throughout his 22-year career Randy Johnson kept batters on their toes with great command and a fierce fastball. That was especially true for left-handed hitters and one in particular got quite the scare during the 1993 Midsummer Classic.
Philadelphia's John Kruk came to plate to face Johnson and received the scare of his life when the Big Unit sent a fastball screaming over Kruk's head. That was all that Phillies All-Star needed to see as for the remainder of the at-bat Kruk stood on the back end of the batter's box with no intention of digging in against Johnson. A few embarrassing swings later Kruk gladly accepted the strikeout and got a friendly wink from Johnson on his way back to the dugout.
A similar situation came about during the 1997 All-Star game when Johnson was once again on the mound facing a great left-handed National League hitter. This time it was Colorado's Larry Walker at the plate but the at-bat began the same way that Kruk's did four years prior with Johnson rocketing a fastball over Walker's head.
Walker didn't shy away like Kruk did though. Instead, Walker took off his batting helmet and turned it around to wear backwards before stepping into the right side batter's box. The fans in attendance at Jacobs Field in Cleveland gave Walker a boisterous cheer while the players and coaches on both sides had a good laugh.
Walker switched back to the left side of the plate one pitch later and ended up drawing a walk from Johnson, a far better result then what Kruk experienced.
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