text:
\begin{document}
\title[Levi flat hypersurfaces]{Non existence of Levi flat hypersurfaces
with positive normal bundle in compact K\"{a}hler manifolds of dimension $
\geqslant 3$}
\author{S\'{e}verine Biard}
\address{Institut de Math\'{e}matiques \\
UMR 7586 du CNRS, case 247 \\
Universit\'{e} Pierre et Marie-Curie \\
4 Place Jussieu \\
75252 Paris Cedex 05\\
France}
\email{biard@math.jussieu.fr}
\author{Andrei Iordan}
\address{Institut de Math\'{e}matiques \\
UMR 7586 du CNRS, case 247 \\
Universit\'{e} Pierre et Marie-Curie \\
4 Place Jussieu \\
75252 Paris Cedex 05\\
France}
\email{iordan@math.jussieu.fr}
\date{July, 30, 2014}
\subjclass{}
\keywords{Levi flat hypersurface, weighted $\overline{\partial }$-equation}

\begin{abstract}
Let $X$ be a compact K\"{a}hler manifold of dimension $\geqslant 3$ and $L$
a $C^{\infty }$ Levi flat hypersurface in $X$. Then the normal bundle to the
Levi foliation does not admit a Hermitian metric with positive curvature
along the leaves. This represents an answer to a conjecture of Marco
Brunella.
\end{abstract}

\maketitle

\section{\protect\bigskip Introduction}

A classical theorem of Poincar\'{e}-Bendixson \cite{Poincare1881}, \cite
{Poincare1882}, \cite{Bendixon1901} states that every leaf of a foliation of
the real projective plane accumulates on a compact leaf or on a singularity
of the foliation. As a holomorphic foliation $\mathcal{F}$ of codimension $1$
of $\mathbb{CP}_{n}$, $n\geqslant 2$, does not contain any compact leaf and
its singular set $Sing~\mathcal{F}$ is not empty, a major problem in
foliation theory is the following: can $\mathcal{F}$ contain a leaf $F$ such
that $\overline{F}\cap Sing~\mathcal{F}=\emptyset $? If this is the case,
then there exists a nonempty compact set $K$ called exceptional minimal,
invariant by $\mathcal{F}$ and minimal for the inclusion such that $K\cap
Sing~\mathcal{F}=\emptyset $. The problem of the existence of an exceptional
minimal in $\mathbb{CP}_{n}$, $n\geqslant 2$ is implicit in \cite{Camacho88}.

In \cite{Cerveau93} D. Cerveau proved a dichotomy under the hypothesis of
the existence of a holomorphic foliation $\mathcal{F}$ of codimension $1\ $
of $\mathbb{CP}_{n}$ which admits an exceptional minimal $\mathfrak{M}$: $
\mathfrak{M}$ is a real analytic Levi flat hypersurface in $\mathbb{CP}_{n}$
(i. e. $T\left( \mathfrak{M}\right) \cap JT\left( \mathfrak{M}\right) $ is
integrable, where $J$ is the complex structure of $\mathbb{CP}_{n}$), or
there exists $p\in \mathfrak{M}$ such that the leaf through $p$ has a
hyperbolic holonomy and the range of the holonomy morphism is a linearisable
abelian group. This gave rise to the conjecture of the non-existence of
smooth Levi flat hypersurface in $\mathbb{CP}_{n}$, $n\geqslant 2$.

The conjecture was proved for $n\geqslant 3$ by A.\ Lins Neto \cite{LinsNeto}
\ for real analytic Levi flat hypersurfaces and by Y.-T.\ Siu \cite{Siu00}
for $C^{12}$ smooth Levi flat hypersurfaces. The methods of proofs for the
real analytic case are very different from the smooth case.

A real hypersurface of class $C^{2}$ in a complex manifold is Levi flat if
its Levi form vanishes or equivalently, it admits a foliation by complex
hypersurfaces. We say that a (non-necessarly smooth) real hypersurface $L$
in a complex manifold $X$ is Levi flat if $X\backslash L$ have two
pseudoconvex connected components. An example of (non-smooth) Levi flat
hypersurface in $\mathbb{CP}_{2}$ is $L=\left\{ \left[ z_{0},z_{1},z_{2}
\right] :\ \left\vert z_{1}\right\vert =\left\vert z_{2}\right\vert \right\}
$, where $\left[ z_{0},z_{1},z_{2}\right] $ are homogeneous coordinates in $
\mathbb{CP}_{2}$ (see \cite{Henkin00}).

In \cite{Iordan08} Iordan and Matthey proved the non existence of Lipschitz
Levi flat hypersurfaces in $\mathbb{CP}_{n}$, $n\geqslant 3$, which are of
Sobolev class $W^{s}$, $s>9/2$. A principal element of the proof is that the
Fubini-Study metric induces a metric of positive curvature on any quotient
of the tangent space.

In \cite{Brunella08}, M. Brunella proved that the normal bundle to the Levi
foliation of a closed real analytic Levi flat hypersurface in a compact K
\"{a}hler manifold of dimension $n\geqslant 3$ does not admit any Hermitian
metric with leafwise positive curvature. The real analytic hypothesis may be
relaxed to the assumption of $C^{2,\alpha }$, $0<\alpha <1$, such that the
Levi foliation extends to a holomorphic foliation in a neighborhood of the
hypersurface.

The main step in his proof is to show that the existence of a Hermitian
metric with leafwise positive curvature on the normal bundle to the Levi
foliation of a compact Levi flat hypersurface $L$ in a Hermitian manifold $X$
, implies that $X\backslash L$ is strongly pseudoconvex, i.e. there exists
on $X\backslash L$ an exhaustion function which is strongly plurisubharmonic
outside a compact set. This was generalized in \cite{Brunella2010} for
invariant compact subsets of a holomorphic foliation of codimension one. Of
course, if $X$ is the complex projective space, then every proper
pseudoconvex domain in $X$ is Stein \cite{Takeushi67}.

Brunella stated also the following conjecture \cite{Brunella08}: Let $X$ be
a compact K\"{a}hler manifold of dimension $n\geqslant 3$ and $L$ a $
C^{\infty }$ compact Levi flat hypersurface in $X$. Then the normal bundle
to the Levi foliation does not admit any Hermitian metric with leafwise
positive curvature.

The assumption $n\geqslant 3$ is necessary in this conjecture (see Example
4.2 of \cite{Brunella08}).

In \cite{Brunella2011} Brunella and Perrone proved that every leaf of a
holomorphic foliation $\mathcal{F}$ of codimension one of a projective
manifold $X$ of dimension at least $3$ and such that $Pic\left( X\right) =
\mathbb{Z}$ accumulates on the singular set of the foliation. In this case
the normal bundle to the foliation is ample.

In \cite{Ohsawa2013}, T. Ohsawa considered a $C^{\infty }$ Levi flat compact
hypersurface $L$ in a K\"{a}hler manifold $X$ such that the normal bundle to
the Levi foliation admits a fiber metric whose curvature is semipositive of
rank$\geqslant k$ on the holomorphic tangent space to the leaves and proved
that $X\backslash L$ admits an exhaustion plurisubharmonic function of
logarithmic growth which is strictly $k$-convex. Then, if $\dim X\geqslant 3$
, he proved that there are no Levi flat real analytic hypersurfaces such
that the normal bundle to the Levi foliation admits a fiber metric whose
curvature is semipositive of rank$\geqslant 2$ on the holomorphic tangent
space to $L$.$\ $The same is also true if $L$ is of class $C^{\infty }$ and
the normal bundle to the Levi foliation extends to a holomorphic line bundle
\cite{Ohsawa2013}.

In this paper we solve the above mentioned conjecture of Brunella for
compact K\"{a}hler manifolds of dimension $\geqslant 3$. The principal
ingredient of the proof is a refinement of the proof of Brunella \cite
{Brunella08} of the strong pseudoconvexity of $X\backslash L$ : we show that
there exist a neighborhood $U$ of $L$ and a function $v$ on $U$ vanishing on
$L$, such that $-i\partial \overline{\partial }\ln v\geqslant c\omega $ on $
U\backslash L$, where $c>0$ and $\omega $ is the $\left( 1,1\right) $-form
associated to the K\"{a}hler metric. Then we use the $L^{2}$ estimates of H
\"{o}rmander for the $\overline{\partial }$-equation \cite{HO65} and methods
developped in \cite{Henkin00}.

\section{Preliminaries}

Let $X$ be a $n$-dimensional K\"{a}hler manifold , $\Omega $ a domain in $X$
and $\sigma $ a positive function on $\Omega $. For $\alpha \in \mathbb{R}$
denote
\begin{equation*}
L_{(p,q)}^{2}(\Omega ;\sigma ^{\alpha })=\left\{ f\in L_{\left( p,q\right)
loc}^{2}\left( \Omega \right) :\int_{\Omega }\left\vert f\right\vert
^{2}\sigma ^{2\alpha }dV<\infty \right\}
\end{equation*}
endowed with the norm
\begin{equation*}
N_{\alpha }(f)=\left( \int_{\Omega }\left\vert f\right\vert ^{2}\sigma
^{2\alpha }dV\right) ^{1/2}.
\end{equation*}
We consider the unbounded operator
\begin{equation*}
\overline{\partial }=\overline{\partial }_{\alpha }:L_{(p,q)}^{2}(\Omega
;\sigma ^{\alpha })\rightarrow L_{(p,q+1)}^{2}(\Omega ;\sigma ^{\alpha }).
\end{equation*}

\begin{definition}
Let $\alpha \geqslant 0$ and $f\in L_{(p,q)}^{2}(\Omega ;\sigma ^{-\alpha })$
. We say that $f$ verifies the moment condition of order $\alpha $ if $
\int_{\Omega }f\wedge h=0$ for every $\overline{\partial }$-closed form $
h\in L_{(n-p,n-q)}^{2}(\Omega ;\sigma ^{\alpha })$.
\end{definition}

For a subset $A$ of $\partial \Omega $, we will denote by $\delta _{A}$ the
geodesic distance to $A$.

Let $\Omega $ be a pseudoconvex domain in $\mathbb{CP}_{n}$ and $\delta
_{\partial \Omega }$ the geodesic distance to the boundary for the
Fubini-Study metric. By using the $L^{2}$ estimates for the $\overline{
\partial }$-operator of H\"{o}rmander with the weight $e^{-\varphi }$, $
\varphi =-\alpha \log \delta _{\partial \Omega }$ which is strongly
plurisubharmonic by a theorem of Takeuchi \cite{Takeushi67}, Henkin and
Iordan proved in \cite{Henkin00} the existence and regularity of the $
\overline{\partial }$ equation for $\overline{\partial }$-closed forms in $
L_{(p,q)}^{2}(\Omega ;\delta _{\partial \Omega }^{-\alpha })$ verifying the
moment condition. This gives the regularity of the $\overline{\partial }$
-operator in pseudoconcave domains with Lipschitz boundary \cite{Henkin00}
and, by using a method of Siu \cite{Siu00}, \cite{Siu02}, the non existence
of smooth Levi flat hypersurfaces in $\mathbb{CP}_{n}$, $n\geqslant 3$
follows (see \cite{Iordan08}). These techniques will be used in the 4th
paragraph.

We will use also the following theorem of regularity of $\overline{\partial }
$ equation of Brinkschulte \cite{Brinkschulte2004}:

\begin{theorem}
\cite{Brinkschulte2004}\label{Brinkschulte} Let $\Omega $ be a relatively
compact domain with Lipschitz boundary in a K\"{a}hler manifold and set $
\delta _{\partial \Omega }$ the geodesic distance to the boundary of $\Omega
$. Let $f\in L_{(p,q)}^{2}(\Omega ;\delta _{\partial \Omega }^{-k})\cap
C_{(p,q)}^{k}\left( \overline{\Omega }\right) \cap C_{(p,q)}^{\infty }\left(
\Omega \right) $, $q\geqslant 1$, $k\in \mathbb{N}$ and $u\in
L_{(p,q-1)}^{2}(\Omega ;\delta _{\partial \Omega }^{-k})$ such that $
\overline{\partial }u=f$ and $\overline{\partial }_{-k}^{\ast }u=0$, where $
\overline{\partial }_{-k}^{\ast }$ is the Hilbert space adjoint of the
unbounded operator $\overline{\partial }_{-k}:L_{(p,q-1)}^{2}(\Omega ;\delta
_{\partial \Omega }^{-k})\rightarrow L_{(p,q)}^{2}(\Omega ;\delta _{\partial
\Omega }^{-k})$. Then for $k$ big enough $u\in C_{(p,q-1)}^{s\left( k\right)
}\left( \overline{\Omega }\right) $ where $s\left( k\right) \underset{
k\rightarrow \infty }{\thicksim }\sqrt{k}$.
\end{theorem}

\section{Strong pseudoconvexity of the complement of a Levi flat hypersurface
}

As was mentioned in \cite{Brunella08} and in \cite{Ohsawa2013} by taking a
double covering, we can assume that $L$ is orientable and the complement of
a Levi flat hypersurface $L$ has two connected components. This will be
always supposed in the sequel and for an open neighborhood $U$ of $L$ we
will denote by $U^{+}$ and $U^{-}$ the two connected components of $
U\backslash L$.

In \cite{Brunella08} Brunella proved that the complement of a closed Levi
flat hypersurface in a compact Hermitian manifold of class $C^{2,\alpha }$, $
0<\alpha <1$, having the property that the Levi foliation extends to a
holomorphic foliation in a neighborhood of $L$ and the normal bundle to the
Levi foliation admits a $C^{2}$ Hermitian metric with leafwise positive
curvature is strongly pseudoconvex. The following proposition strenghtens
this result:

\begin{proposition}
\label{Strongly pseudoconvex}Let $L$ be a compact $C^{3}$ Levi flat
hypersurface in a Hermitian manifold $X$ of dimension $n\geqslant 2$, such
that the normal bundle $\mathcal{N}_{L}^{1,0}$ to the Levi foliation admits
a $C^{2}$ Hermitian metric with leafwise positive curvature. Then there
exist a neighborhood $U$ of $L$, $c>0$ and a positive function $v\in
C^{2}\left( U\right) $, vanishing on $L$ such that $-i\partial \overline{
\partial }\ln v\geqslant c\omega $ on $U\backslash L$, where $\omega $ is
the $\left( 1,1\right) $-form associated to the metric. Moreover, there
exists a nonvanishing continuous function $g$ in a neighborhood of $L$ such
that $v=g\delta ^{2}$, where $\delta $ is the geodesic distance to $L$.
\end{proposition}

\begin{proof}
Let $z_{0}\in L$. There exist holomorphic coordinates $z=\left( z_{1},\cdot
\cdot \cdot ,z_{n-1},z_{n}\right) =\left( z^{\prime },z_{n}\right) $ in a
neighborhood of $z_{0}$ such that the local parametric equations for $L$ are
of the form
\begin{equation*}
z_{j}=w_{j},\ j=1,...,n-1,\ z_{n}=\varphi \left( w^{\prime },t\right)
\end{equation*}
where $\varphi $ is of class $C^{3}$ on a neighborhood of the origin in $
\mathbb{C}^{n-1}\times \mathbb{R}$, holomorphic in $w^{\prime }$ and $\frac{
\partial \varphi }{\partial t}\left( z_{0}\right) \in \mathbb{R}^{\ast }$.
We consider a $C^{3}$ extension $\psi =\left( \psi _{1},...,\psi _{n}\right)
$ of $\varphi $ on a neighborhood of the origin in $\mathbb{C}^{n-1}\times
\mathbb{C}$, $\psi \left( w^{\prime },t+is\right) =\left( w^{\prime
},\varphi \left( w^{\prime },t\right) +is\right) $. Then $\psi $ is a $C^{3}$
diffeomorphism in a neighborhood of $z_{0}$ and holomorphic in $w^{\prime }$
. It follows that
\begin{equation*}
L=\left\{ \left( z^{\prime },z_{n}\right) :\rho \left( z^{\prime
},z_{n}\right) =0\right\} .
\end{equation*}
where $ \rho =\func{Im}\left( \psi ^{-1}\right) _{n}$. We denote $f=\left(
\psi ^{-1}\right) _{n}\left( z^{\prime },z_{n}\right) $. Since $\overline{
\partial }_{z^{\prime }}f=0$ on $L$, there exists an extension $\widetilde{f}
$ of class $C^{3}$ in a neighborhood of $z_{0}$ such that $\overline{
\partial }\widetilde{f}$ vanishes to order greater than $2$ on $L$, i.e. $
D^{l}\overline{\partial }\widetilde{f}=0$ for $\left\vert l\right\vert
\leqslant 2$ on $L$.

So there exists an open finite covering $\left( \widetilde{U_{j}}\right)
_{j\in J}$ by holomorphic charts of $L$ such that $\widetilde{U_{j}}
\backslash L=\widetilde{U_{j}^{+}}\cup \widetilde{U_{j}^{-}}$ such that $
U_{j}=L\cap \widetilde{U_{j}}=\left\{ z\in \widetilde{U_{j}}:\func{Im}
\widetilde{f_{j}}=0\right\} $, where $\overline{\partial }\widetilde{f_{j}}$
vanishes to order greater than $2$ on $L$ and the Levi foliation is given on
$U_{j}$ by $\left\{ z\in U_{j}:\ \widetilde{f_{j}}\left( z\right)
=c_{j}\right\} $, $c_{j}\in \mathbb{R}$ . Thus $d\widetilde{f_{j}}=\partial
\widetilde{f_{j}}$ is a non vanishing section of $\mathcal{N}_{L}^{1,0}$ on $
U_{j}$ and by shrinking $\widetilde{U_{j}}$, we may consider that $d
\widetilde{f_{j}}\neq 0$ on $\widetilde{U_{j}}$.

We may suppose that $\mathcal{N}_{L}^{1,0}$ is represented by a cocycle $
\left\{ g_{jk}\right\} $ of class $C^{2}$ subordinated to the covering $
\left( U_{j}\right) _{j\in J}$ and there exist closed $\left( 1,0\right) $
-forms $\alpha _{j}$ of class $C^{2}$ on $U_{j}$ holomorphic along the
leaves such that $T^{1,0}\left( U_{j}\right) =\ker \alpha _{j}$ for every $
j\in J$ and $\alpha _{j}=g_{jk}\alpha _{k}$ on $U_{j}\cap U_{k}$. So $\left(
\alpha _{j}\right) _{j\in J}$ defines a global form $\alpha $ on $L$ with
values in $\mathcal{N}_{L}^{1,0}$ such that locally on $U_{j}$ we have $
\alpha \left( z\right) =\alpha _{j}\left( z\right) \otimes \alpha _{j}^{\ast
}\left( z\right) $ where $\alpha _{j}^{\ast }$ is the dual frame of $\alpha
_{j}$. In particular we have $\alpha _{k}^{\ast }=g_{jk}\alpha _{j}^{\ast }$.

Let $h$ be a $C^{2}$ Hermitian metric with positive leafwise curvature $
\Theta _{h}\left( \mathcal{N}_{L}^{1,0}\right) $ on $\mathcal{N}_{L}^{1,0}$.
$h$ is defined on each $U_{j}$ by a $C^{2}$ function $h_{j}=$ $\left\vert
\alpha _{j}^{\ast }\right\vert ^{2}$ such that $h_{k}=\left\vert
g_{jk}\right\vert ^{2}h_{j}$ on $U_{j}\cap U_{k}$.

Since $\alpha _{j}=\eta _{j}d\widetilde{f_{j}}$ on $U_{j}$ for every $j$,
where $\eta _{j}$ are nowhere vanishing functions of class $C^{2}$ on $U_{j}$
holomorphic along the leaves and
\begin{equation*}
\frac{1}{\eta _{k}}\left( d\widetilde{f_{k}}\right) ^{\ast }=\frac{1}{\eta
_{j}}g_{jk}\left( d\widetilde{f_{j}}\right) ^{\ast }
\end{equation*}
on $U_{j}\cap U_{k}$, it follows that
\begin{equation*}
\left\vert g_{jk}\left( z\right) \right\vert ^{2}=\left\vert \frac{\eta
_{j}\left( z\right) }{\eta _{k}\left( z\right) }\right\vert ^{2}=\frac{
h_{k}\left( z\right) }{h_{j}\left( z\right) },\ z\in U_{j}\cap U_{k}.
\end{equation*}

So
\begin{equation*}
h_{j}\left\vert \eta _{j}\right\vert ^{2}\left( \func{Im}\widetilde{f_{j}}
\right) ^{2}-h_{k}\left\vert \eta _{k}\right\vert ^{2}\left( \func{Im}
\widetilde{f_{k}}\right) ^{2}
\end{equation*}
vanishes to order greater than $2$ on $U_{j}\cap U_{k}$ and $\left(
h_{j}\left\vert \eta _{j}\right\vert ^{2}\left( \func{Im}\widetilde{f_{j}}
\right) ^{2}\right) _{j\in J}$ defines a jet of order $2$ on $L$. By Whitney
extension theorem there exists a $C^{2}$ function $v$ on $X$ such that $
v-h_{j}\left\vert \eta _{j}\right\vert ^{2}\left( \func{Im}\widetilde{f_{j}}
\right) ^{2}$ vanishes to order $2$ on $U_{j}$ for every $j\in J$. Let $
\widetilde{\eta _{j}},\widetilde{h_{j}}$ be $C^{2}$ extensions of $\eta
_{j},h_{j}$ on $\widetilde{U_{j}}$ and set $\widetilde{\alpha _{j}}=
\widetilde{\eta _{j}}d\widetilde{f_{j}}$, $\widetilde{v}=\widetilde{h_{j}}
\left\vert \widetilde{\eta _{j}}\right\vert ^{2}\left( \func{Im}\widetilde{
f_{j}}\right) ^{2}$.

For $z\in \widetilde{U_{j}}$ denote $E_{z}^{\prime }=\left\{ V^{\prime }\in
T_{z}^{1,0}\left( X\right) :\ \left\langle \partial \func{Im}\widetilde{f_{j}
},V^{\prime }\right\rangle =0\right\} $ and $E_{z}^{^{\prime \prime }}$ the
orthogonal of $E_{z}^{\prime }$ in $T_{z}^{1,0}\left( X\right) $. Then for
every $V\in T_{z}^{1,0}\left( X\right) $ there exists $V^{\prime }\in
E_{z}^{\prime },V^{\prime \prime }\in E_{z}^{\prime \prime }$ such that $
V=V^{\prime }+V^{\prime \prime }$. The curvature form $\Theta \left(
\mathcal{N}_{L}^{1,0}\right) $ is represented by $-i\partial \overline{
\partial }\ln \left( h_{j}\left\vert \alpha _{j}\right\vert ^{2}\right) $ on
$U_{j}$, so by shrinking $\widetilde{U_{j}}$ we may suppose that there
exists $\beta >0$ such that $\left( -i\partial \overline{\partial }\ln
\left( \widetilde{h_{j}}\left\vert \widetilde{\alpha _{j}}\right\vert
^{2}\right) \right) \left( V^{\prime },\overline{V^{\prime }}\right)
\geqslant \beta \omega \left( V^{\prime },\overline{V^{\prime }}\right) $
for every $z\in \widetilde{U_{j}}$ and $V\in T_{z}^{1,0}\left( X\right) $.

On $\widetilde{U_{j}}\backslash L$ we have
\begin{eqnarray}
-i\partial \overline{\partial }\ln \widetilde{v} &=&-i\partial \overline{
\partial }\ln \left( \widetilde{h_{j}}\left\vert \frac{\widetilde{\alpha _{j}
}}{d\widetilde{f_{j}}}\right\vert ^{2}\left( \func{Im}\widetilde{f_{j}}
\right) ^{2}\right)  \label{A} \\
&=&-i\partial \overline{\partial }\ln \widetilde{h_{j}}\left\vert \widetilde{
\alpha _{j}}\right\vert ^{2}+i\partial \overline{\partial }\ln \left\vert d
\widetilde{f_{j}}\right\vert ^{2}-i\partial \overline{\partial }\ln \left(
\func{Im}\widetilde{f_{j}}\right) ^{2}.  \notag
\end{eqnarray}
Let $z\in \widetilde{U_{j}}$ and $V\in T_{z}^{1,0}\left( X\right) $. Then $
V=V^{\prime }+V^{\prime \prime }$, $V^{\prime }\in E_{z}^{\prime }$ and $
V"\in E_{z}^{\prime \prime }$ and
\begin{eqnarray*}
-i\partial \overline{\partial }\ln \widetilde{h_{j}}\left\vert \widetilde{
\alpha _{j}}\right\vert ^{2}\left( V,\overline{V}\right) &=&\left(
-i\partial \overline{\partial }\ln \left( \widetilde{h_{j}}\left\vert
\widetilde{\alpha _{j}}\right\vert ^{2}\right) \right) \left( V^{\prime },
\overline{V^{\prime }}\right) \\
&&+2\func{Re}\left( -i\partial \overline{\partial }\ln \left( \widetilde{
h_{j}}\left\vert \widetilde{\alpha _{j}}\right\vert ^{2}\right) \left(
V^{\prime },\overline{V^{\prime \prime }}\right) \right) \\
&&+\left( -i\partial \overline{\partial }\ln \left( \widetilde{h_{j}}
\left\vert \widetilde{\alpha _{j}}\right\vert ^{2}\right) \right) \left(
V^{\prime \prime },\overline{V^{\prime \prime }}\right)
\end{eqnarray*}
For every $\varepsilon >0$
\begin{equation*}
2\left\vert \func{Re}\left( -i\partial \overline{\partial }\ln \left(
\widetilde{h_{j}}\left\vert \widetilde{\alpha _{j}}\right\vert ^{2}\right)
\left( V^{\prime },\overline{V^{\prime \prime }}\right) \right) \right\vert
\leqslant \varepsilon \omega \left( V^{\prime },\overline{V^{\prime }}
\right) +\frac{1}{\varepsilon }\omega \left( V^{\prime \prime },\overline{
V^{\prime \prime }}\right) ,
\end{equation*}
so
\begin{eqnarray}
-i\partial \overline{\partial }\ln \widetilde{h_{j}}\left\vert \widetilde{
\alpha _{j}}\right\vert ^{2}\left( V,\overline{V}\right) &\geqslant &\beta
\omega \left( V^{\prime },\overline{V^{\prime }}\right) -\varepsilon \omega
\left( V^{\prime },\overline{V^{\prime }}\right) -\frac{1}{\varepsilon }
\omega \left( V^{\prime \prime },\overline{V^{\prime \prime }}\right)
\label{B} \\
&&-\left\Vert -i\partial \overline{\partial }\ln \widetilde{h_{j}}\left\vert
\widetilde{\alpha _{j}}\right\vert ^{2}\right\Vert _{\omega }\omega \left(
V^{\prime \prime },\overline{V^{\prime \prime }}\right)  \notag
\end{eqnarray}

Since $\overline{\partial }\widetilde{f_{j}}$ vanishes to order greater than
$2$ on $L$, for every $\gamma >0$ there exists a neighborhood of $L$ such
that
\begin{equation}
\left\vert i\partial \overline{\partial }\ln \left\vert d\widetilde{f_{j}}
\right\vert ^{2}\left( V,\overline{V}\right) \right\vert \leqslant \gamma
\omega \left( V,\overline{V}\right)  \label{C}
\end{equation}
and
\begin{equation}
\left\vert i\partial \overline{\partial }\func{Im}\widetilde{f_{j}}\left( V,
\overline{V}\right) \right\vert \leqslant \gamma \left( \func{Im}\widetilde{
f_{j}}\right) \omega \left( V,\overline{V}\right) .  \label{D}
\end{equation}
Let $z\in \widetilde{U_{j}}\backslash L$. By (\ref{D}) it follows that
\begin{eqnarray}
-i\partial \overline{\partial }\ln \left( \func{Im}\widetilde{f_{j}}\right)
^{2}\left( V,\overline{V}\right) &=&\left( -2\frac{i\partial \overline{
\partial }\func{Im}\widetilde{f_{j}}}{\func{Im}\widetilde{f_{j}}}+2i\frac{
\partial \func{Im}\widetilde{f_{j}}\wedge \overline{\partial }\func{Im}
\widetilde{f_{j}}}{\left( \func{Im}\widetilde{f_{j}}\right) ^{2}}\right)
\left( V,\overline{V}\right)  \notag \\
&\geqslant &-2\gamma \omega \left( V,\overline{V}\right) +2i\frac{\partial
\func{Im}\widetilde{f_{j}}\wedge \overline{\partial }\func{Im}\widetilde{
f_{j}}}{\left( \func{Im}\widetilde{f_{j}}\right) ^{2}}\left( V^{\prime
\prime },\overline{V^{\prime \prime }}\right)  \label{E} \\
&\geqslant &-2\gamma \omega \left( V,\overline{V}\right) +\frac{2\underset{U}
{\inf }\left\Vert \partial \func{Im}\widetilde{f_{j}}\right\Vert _{\omega
}^{2}}{\left( \func{Im}\widetilde{f_{j}}\right) ^{2}}\omega \left( V^{\prime
\prime },\overline{V^{\prime \prime }}\right) .  \notag
\end{eqnarray}
By using (\ref{B}),\ (\ref{C}) and (\ref{E}), from (\ref{A}) we obtain
\begin{eqnarray*}
\left( -i\partial \overline{\partial }\ln \widetilde{v}\right) \left( V,
\overline{V}\right) &\geqslant &\left( \beta -\varepsilon \right) \omega
\left( V^{\prime },\overline{V^{\prime }}\right) \\
&&+\left( \frac{2}{\left( \func{Im}f_{j}\right) ^{2}}\underset{U}{\inf }
\left\Vert \partial \func{Im}\widetilde{f_{j}}\right\Vert _{\omega }^{2}-
\frac{1}{\varepsilon }-\left\Vert -i\partial \overline{\partial }\ln
\widetilde{h_{j}}\left\vert \widetilde{\alpha _{j}}\right\vert
^{2}\right\Vert _{\omega }\right) \omega \left( V^{\prime \prime },\overline{
V^{\prime \prime }}\right) \\
&&-2\gamma \omega \left( V,\overline{V}\right) .
\end{eqnarray*}

By choosing $0<\varepsilon <\beta $ and by shrinking $\widetilde{U_{j}}$
such that $\frac{2}{\left( \func{Im}f_{j}\right) ^{2}}$ is big enough and $
\gamma $ small enough, we obtain that there exists $c>0$ such that $
-i\partial \overline{\partial }\ln \widetilde{v}\geqslant c\omega $ on $
\widetilde{U_{j}}$. Finally, since $v-\widetilde{v}$ vanishes to order
greater than $2$ on $L$, it follows that there exists a neighborhood $
U^{\prime }$ of $L$ such that $-\ln v$ is strongly plurisubharmonic on $
U^{\prime }\backslash L$. We can now take $U=\left\{ z\in U^{\prime }:\
v\left( z\right) <\mu \right\} $ for $\mu >0$ small enough.

$L$ is a $C^{3}$ manifold, so the signed distance function $\delta $ is a
defining function of class $C^{3}$ for $L$. Since $v$ is of class $C^{2}$ on
$U$ and vanishes to order greater than $2$ on $L$, we have $v=g\delta ^{2}$
with $g$ continuous in a neighborhood of $L$.

Suppose that there exists $x\in L$ such that $g\left( x\right) =0$. Then $
v=o\left( \delta ^{2}\right) $ in a neighborhood of $x$. But there exists $j$
such that $x\in U_{j}$ and $v=h_{j}\left\vert \eta _{j}\right\vert
^{2}\left( \func{Im}\widetilde{f_{j}}\right) ^{2}+o\left( \delta ^{2}\right)
$. Since $\func{Im}\widetilde{f_{j}}=0$ and $d\func{Im}\widetilde{f_{j}}\neq
0$ on $L$ it follows that $\left\vert \nabla ^{2}v\right\vert \left(
x\right) \neq 0$. This contradiction shows that $g\left( x\right) \neq 0$ on
$L$.
\end{proof}

\section{Weighted estimates for the $\overline{\partial }$-equation}

\begin{proposition}
\label{Solution a support compact}Let $L$ be a compact $C^{\infty }$ Levi
flat hypersurface in a compact K\"{a}hler manifold $\left( X,\omega \right) $
of dimension $n\geqslant 3$.We suppose that there exists a neighborhood $U$
of $L$ in $X$ and a positive $C^{2}$ function $v$ on $\overline{U}$ such
that $-i\partial \overline{\partial }\ln v\geqslant c\omega $ on $U$, $c>0$,
and $v=g\delta _{L}^{2}$ on $U$, where $g$ is continuous on $\overline{U}$
and $g\left( z\right) \neq 0$ for every $z\in \overline{U}$. Consider $a>0$
small enough such that $W=\left\{ z\in U:\ v<a\right\} \subset U$. Let $
\sigma $ be a $C^{2}$ function on $W$ such that $\sigma =v$ in a
neighborhood of $\left\{ z\in U:\ v=a\right\} $ and $\sigma =v^{-1}$ on a
neighborhood of $\left\{ z\in U:\ v=0\right\} \backslash L$. Then:

$1)\ $Let $f\in L_{\left( p,q\right) }^{2}\left( W,\sigma ^{k}\right) $, $
q\geqslant 1$, a $\overline{\partial }$-closed form. Then for $k$ big
enough, there exists $u\in L_{\left( p,q-1\right) }^{2}\left( W,\sigma
^{k}\right) $ such that $\overline{\partial }u=f$ and $N_{k}\left( u\right)
\preccurlyeq N_{k}\left( f\right) $, where $N_{k}\left( u\right)
\preccurlyeq N_{k}\left( f\right) $ means $N_{k}\left( u\right) \leqslant
C_{k}N_{k}\left( f\right) $, with $C_{k}>0$ independent of $f$.

$2)\ $Let $f\in L_{\left( p,q\right) }^{2}\left( W,\sigma ^{-k}\right) $, $
q\geqslant 1$, a $\overline{\partial }$-closed form verifying the moment
condition of order $k$. Then for $k$ big enough there exists $u\in L_{\left(
p,q-1\right) }^{2}\left( W,\sigma ^{-k}\right) $ such that $\overline{
\partial }u=f$ and $N_{-k}\left( u\right) \preccurlyeq N_{-k}\left( f\right)
$.
\end{proposition}

\begin{proof}
$1)$ Let $0<\varepsilon <a<1$ such that $W_{\varepsilon }=\left\{ z\in X:\
\varepsilon <v<a\right\} \subset W$. Then $\partial W_{\varepsilon
}=\partial W_{\varepsilon }^{1}\cup \partial W_{\varepsilon }^{2}$ where $
\partial W_{\varepsilon }^{1}=\left\{ z\in U:\ v\left( z\right) =a\right\}
=\left\{ z\in U:\ -\ln \sigma \left( z\right) =-\ln a\right\} $ is strongly
pseudoconcave and $\partial W_{\varepsilon }^{2}=\left\{ z\in U:\ v\left(
z\right) =\varepsilon \right\} =\left\{ z\in U:\ -\ln \sigma \left( z\right)
=-\ln \varepsilon \right\} $ is strongly pseudoconvex.

By the classical results of H\"{o}rmander \cite{HO65} and Kohn \cite{KO63},
\cite{KO73} and detailed in \cite{Folland72} the basic estimate holds for $
\left( p,q\right) $-forms if and only if the Levi form has at least $n-q$
positive eigenvalues or at least $q+1$ negative eigenvalues. This implies
that the $\overline{\partial }$-Neumann problem on $W_{\varepsilon }$
satisfies $\frac{1}{2}$-subelliptic estimates and that the harmonic space $
\mathcal{H}_{\left( p,q\right) }\left( W_{\varepsilon }\right) $ is finite
dimensional.

More generally, in \cite{Shaw85}, Shaw proved that the harmonic space $
\mathcal{H}_{\left( p,q\right) }\left( \Omega \right) $ where $\Omega
=\Omega _{1}\backslash \overline{\Omega _{2}}$, where $\Omega _{1},\Omega
_{2}$ are pseudoconvex domains with smooth boundary in $\mathbb{C}^{n}$, $
0\leqslant p\leqslant n$, $1\leqslant q<n-1$ is finite dimensional and in
\cite{Shaw2010} she proved that in fact $\mathcal{H}_{\left( p,q\right)
}\left( \Omega \right) =\left\{ 0\right\} $. The same is true in complex
manifolds under the hypothesis that there exists strongly plurisubharmonic
functions in a neighborhood of the boundaries.

As in the proof of Proposition 3.1 of \cite{Shaw85}, we consider a weight $
\sigma $ which is equal to $\ln v$ in a neighborhood of $\partial
W_{\varepsilon }^{1}$ and $-\ln v$ in a neighborhood of $\partial
W_{\varepsilon }^{2}$ and we obtain that for every $f\in L_{\left(
p,q\right) }^{2}\left( W_{\varepsilon },\sigma ^{k}\right) $, $q\geqslant 1$
and $k$ big enough
\begin{equation*}
N_{k}\left( f\right) ^{2}\leqslant C_{k}\left( N_{k}\left( \overline{
\partial }f\right) ^{2}+N_{k}\left( \overline{\partial }_{k}^{\ast }f\right)
^{2}\right) ,
\end{equation*}

where $C_{k}>0$ depends only on the smallest positive eigenvalue of $
-i\partial \overline{\partial }\ln \sigma $ which is $\geqslant c$ and
independent of $\varepsilon $. By taking $\varepsilon \rightarrow 0$ it
follows that for $k$ big enough, there exists $u\in L_{\left( p,q-1\right)
}^{2}\left( W,\sigma ^{k}\right) $ such that $\overline{\partial }u=f$ and $
N_{k}\left( u\right) \preccurlyeq N_{k}\left( f\right) $.

$2)$ The proof of this point is analoguous of those of \cite{Henkin00},
Proposition 5.3 and we give it for the convenience of the reader.By the
first point it follows that for $k$ big enough the range $\mathcal{R}
_{\left( p,q\right) }^{k}\left( W\right) $ of the operator $\overline{
\partial }_{k}:L_{\left( p,q-1\right) }^{2}\left( W,\sigma ^{k}\right)
\rightarrow L_{\left( p,q\right) }^{2}\left( W,\sigma ^{k}\right) $ is
closed and we can find a bounded operator $T_{\left( p,q\right) }^{k}:
\mathcal{R}_{\left( p,q\right) }^{k}\left( W\right) \rightarrow L_{\left(
p,q-1\right) }^{2}\left( W,\sigma ^{k}\right) $, such that $\overline{
\partial }T_{\left( p,q\right) }^{k}f=f$ for every $f\in \mathcal{R}_{\left(
p,q\right) }^{k}\left( W\right) $.

For every $\varphi \in \mathcal{R}_{\left( n-p,n-q+1\right) }^{k}\left(
W\right) $, define $\Phi _{f}\left( \varphi \right) =\int_{W}f\wedge
T_{\left( n-p,n-q+1\right) }^{k}\varphi $. $\Phi _{f}$ is a continuous
linear form on $\mathcal{R}_{\left( n-p,n-q+1\right) }^{k}\left( W\right) $.
By the Hahn-Banach theorem, we extend $\Phi _{f}$ as a linear form $
\widetilde{\Phi _{f}}$ on $L_{\left( n-p,n-q+1\right) }^{2}\left( W,\sigma
^{k}\right) $ such that $\left\Vert \widetilde{\Phi _{f}}\right\Vert
=\left\Vert \Phi _{f}\right\Vert $. Since $\left( L_{\left( n-p,n-q+1\right)
}^{2}\left( W,\sigma ^{k}\right) \right) ^{\prime }=L_{\left( p,q-1\right)
}^{2}\left( W,\sigma ^{-k}\right) $ by the pairing
\begin{equation*}
\left( \alpha ,\beta \right) =\int_{W}\alpha \wedge \beta ,\ \alpha \in
L_{\left( p,q-1\right) }^{2}\left( W,\sigma ^{-k}\right) ,\ \beta \in
L_{\left( n-p,n-q+1\right) }^{2}\left( W,\sigma ^{k}\right) ,
\end{equation*}
there exists $u\in L_{\left( p,q-1\right) }^{2}\left( W,\sigma ^{-k}\right) $
such that
\begin{equation*}
\widetilde{\Phi _{f}}\left( \varphi \right) =\int_{W}u\wedge \varphi
\end{equation*}
for every $\varphi \in L_{\left( n-p,n-q+1\right) }^{2}\left( W,\sigma
^{k}\right) $.

Since $f$ verifies the moment condition of order $k$, $\Phi _{f}\left(
\varphi \right) =\int_{W}f\wedge \psi $ for every $\psi \in L_{\left(
n-p,n-q\right) }^{2}\left( W,\sigma ^{k}\right) $ such that $\overline{
\partial }\psi =\varphi $. In particular
\begin{equation*}
\Phi _{f}\left( \varphi \right) =\int_{W}f\wedge \psi =\int_{W}u\wedge
\varphi =\int_{W}u\wedge \overline{\partial }\psi
\end{equation*}
for every $\psi \in L_{\left( n-p,n-q\right) }^{2}\left( W,\sigma
^{k}\right) $ such that $\overline{\partial }\psi =\varphi $. So $\left(
-1\right) ^{p+q+1}\overline{\partial }u=f$ and the proof is completed.
\end{proof}

\begin{lemma}
\label{Cond moment}Under the hypothesis of Proposition \ref{Solution a
support compact}, let $f\in C_{(p,q)}^{\infty }(W\backslash L)\cap L_{\left(
p,q\right) }^{2}\left( W,\delta _{\partial W}^{-k-\frac{1}{2}}\right) $ be a
$\overline{\partial }$-closed form, $1\leqslant q\leqslant n-1$. Then, for $
k $ big enough, $f$ verifies the moment condition of order $k$.
\end{lemma}

\begin{proof}
Let $h\in L_{(n-p,n-q)}^{2}(W;\sigma ^{k})$ be a $\overline{\partial }$
-closed form. Since $n-q\geqslant 1$, by Proposition \ref{Solution a support
compact}, for $k$ big enough there exists $g\in L_{(n-p,n-q-1)}^{2}(W;\sigma
^{k})$ such that $\overline{\partial }g=h$ and $N_{k}\left( g\right)
\preccurlyeq N_{k}\left( h\right) $.

For $\varepsilon >0$ let $W_{\varepsilon }=\{z\in W|\delta _{L}(z)\geq
\varepsilon \}$ and $f_{\varepsilon }=\chi _{\varepsilon }f$, where $\chi
_{\varepsilon }$ is a $\mathcal{C}^{\infty }$ function on $W$ such that $
\chi _{\varepsilon }\equiv 1$ on $W_{2\varepsilon }$, $\chi _{\varepsilon
}\equiv 0$ on $W\backslash W_{\varepsilon }$, $0\leq \chi _{\varepsilon
}\leq 1$, $\left\vert D\chi _{\varepsilon }\right\vert \leq \frac{C}{
\varepsilon }$, $C>0$.

Since there exists $a,b>0$ such that $a\delta _{L}^{2}\leqslant \sigma
\leqslant b\delta _{L}^{2}$ on $W$,
\begin{eqnarray*}
N_{-k}(\overline{\partial }f_{\varepsilon }) &=&N_{-k}(\overline{\partial }
\chi _{\varepsilon }\wedge f)=\left( \int_{W_{\varepsilon }\backslash
W_{2\varepsilon }}\left\vert \overline{\partial }\chi _{\varepsilon }\wedge
f\right\vert ^{2}\sigma ^{-2k}dV\right) ^{\frac{1}{2}} \\
&\leqslant &\frac{C}{\varepsilon }\left( \int_{W_{\varepsilon }\backslash
W_{2\varepsilon }}\left\vert f\right\vert ^{2}\sigma ^{-2k}dV\right) ^{\frac{
1}{2}}\leqslant 2C\left( \int_{W_{\varepsilon }\backslash W_{2\varepsilon
}}\left\vert f\right\vert ^{2}\sigma ^{-2k}\delta _{L}^{-2}dV\right) ^{\frac{
1}{2}} \\
&\leqslant &2\sqrt{b}C\left( \int_{W_{\varepsilon }\backslash
W_{2\varepsilon }}\left\vert f\right\vert ^{2}\sigma ^{-2k-1}dV\right) ^{
\frac{1}{2}}\underset{\varepsilon \rightarrow 0}{\rightarrow }0.
\end{eqnarray*}

We consider an exhaustion $\left( W_{i}\right) _{i\in \mathbb{N}}$ of $W$
such that $g\in L_{(n-p,n-q-1)}^{2}(\partial W_{i};\sigma ^{k})$ and we have
\begin{equation*}
\int_{W}f_{\varepsilon }\wedge h=\underset{i\rightarrow \infty }{\lim }
\int_{W_{i}}f_{\varepsilon }\wedge \overline{\partial }g=\left( -1\right)
^{p+q+1}\underset{i\rightarrow \infty }{\lim }\left( \int_{W_{i}}\overline{
\partial }f_{\varepsilon }\wedge g+\int_{\partial W_{i}}f_{\varepsilon
}\wedge g\right)
\end{equation*}
Since $f=o\left( \delta _{\partial W}^{k}\right) $ on $\partial W$, it
follows that
\begin{equation*}
\int_{W}f_{\varepsilon }\wedge h=\int_{W}\overline{\partial }f_{\varepsilon
}\wedge g.
\end{equation*}
So
\begin{eqnarray*}
\left\vert \int_{W}f\wedge h\right\vert &=&\underset{\varepsilon \rightarrow
0}{\lim }\left\vert \int_{W}f_{\varepsilon }\wedge \overline{\partial }
g\right\vert =\underset{\varepsilon \rightarrow 0}{\lim }\left\vert \int_{W}
\overline{\partial }f_{\varepsilon }\wedge g\right\vert \\
&=&\underset{\varepsilon \rightarrow 0}{\lim }\left\vert \int_{W}\sigma ^{-k}
\overline{\partial }f_{\varepsilon }\wedge \sigma ^{k}g\right\vert \leqslant
\underset{\varepsilon \rightarrow 0}{\lim }N_{-k}(\overline{\partial }
f_{\varepsilon })N_{k}\left( g\right) =0.
\end{eqnarray*}
\end{proof}

\begin{corollary}
\label{Solution fi -k Levi plat}Let $L$ be a compact $C^{\infty }$ Levi flat
hypersurface in a K\"{a}hler manifold $X$ of dimension $n\geqslant 3$ such
that the normal bundle $\mathcal{N}_{L}^{1,0}$ to the Levi foliation admits
a $C^{2}$ Hermitian with leafwise positive curvature. Then for $k$ big
enough there exists a neighborhood $W$ of $L\ $such that for every $
\overline{\partial }$-closed$\ f\in C_{\left( p,q\right) }^{\infty }\left(
W\backslash L\right) \cap L_{\left( p,q\right) }^{2}\left( W\backslash
L,\delta _{\partial W}^{-k}\right) 1\leqslant q\leqslant n-1$, there exists $
u\in L_{\left( p,q-1\right) }^{2}\left( W\backslash L,\delta _{\partial
W}^{-k}\right) $ such that $\overline{\partial }u=f$ and $N_{-k}\left(
u\right) \preccurlyeq N_{-k}\left( f\right) $.
\end{corollary}

\begin{proof}
By Proposition \ref{Strongly pseudoconvex} there exists a neighborhood $U$
of $L$, $c>0$ and a positive function $v\in C^{2}\left( \overline{U}\right) $
, $v\thicksim \delta _{L}^{2}$, such that $-i\partial \overline{\partial }
\ln v\geqslant c\omega $ on $U\backslash L$. So for every $\alpha \in
\mathbb{R},$ $L_{\left( p,q\right) }^{2}\left( U\backslash L,\delta
_{L}^{\alpha }\right) =L_{\left( p,q\right) }^{2}\left( U\backslash
L,v^{\alpha /2}\right) $. By Lemma \ref{Cond moment}, $f$ verifies the
moment condition of order $k$ for each $k$ big enough and the Corollary
follows from Proposition \ref{Solution a support compact}.
\end{proof}

From Theorem \ref{Brinkschulte} and Corollary \ref{Solution fi -k Levi plat}
we obtain:

\begin{corollary}
\label{Solution d bar delta -alfa lisse}Under the hypothesis of Corollary
\ref{Solution fi -k Levi plat}, for $k$ big enough, there exist $u_{k}\in
C_{\left( p,q-1\right) }^{s\left( k\right) }\left( \overline{W}\right) \cap
L_{\left( p,q-1\right) }^{2}\left( W,\delta _{\partial W}^{-k}\right) $ such
that $\overline{\partial }u_{k}=f$ where $s\left( k\right) \underset{
k\rightarrow \infty }{\thicksim }\sqrt{k}$.
\end{corollary}

\section{Non existence of Levi flat hypersurfaces}

\begin{proposition}
\label{Extension}Let $L$ be a compact $C^{\infty }$ Levi flat hypersurface
in a K\"{a}hler manifold $X$ of dimension $n\geqslant 3$ such that the
normal bundle $\mathcal{N}_{L}^{1,0}$ to the Levi foliation admits a $C^{2}$
Hermitian with leafwise positive curvature. Let $u\in C_{\left( p,q\right)
}^{\infty }\left( L\right) $, $1\leqslant q\leqslant n-2$, such that $
\overline{\partial }_{b}u=0$. Then for every $k$ big enough there exists a
neighborhood $W$ of $L$ and a $\overline{\partial }$-closed extension $
U_{k}\in C_{\left( p,q\right) }^{k}\left( W\right) $ of $u$.
\end{proposition}

\begin{proof}
By Proposition \ref{Strongly pseudoconvex} there exist a neighborhood $U$ of
$L$, $c>0$ and positive function $v\in C^{2}\left( \overline{U}\right) $
vanishing on $L$ such that $v=g\delta _{L}^{2}$ and $-i\partial \overline{
\partial }\ln v\geqslant c\omega $ on $U\backslash L$. Let $\widetilde{u}\in
C_{\left( p,q\right) }^{\infty }\left( X\right) $ an extension of $u$ such
that $\overline{\partial }\widetilde{u}$ vanishes to infinite order on $L$
and $supp~\widetilde{u}$ is relatively compact in $U$. Then $\overline{
\partial }\widetilde{u}\in L_{\left( p,q+1\right) }^{2}\left( U;\delta
_{\partial U}^{-k}\right) $ for every $k\in \mathbb{N}$. Since $q+1\leqslant
n-1$, by Proposition \ref{Solution a support compact}, for $k$ big enough,
there exist a neighborhood $W$ and $h^{\pm }\in L_{\left( p,q\right)
}^{2}\left( W^{\pm },\delta _{\partial W}^{-k}\right) $, where $W^{\pm
}=\left\{ z\in U:\ v\gtrless 0\right\} $, such that $\overline{\partial }
h^{\pm }=\overline{\partial }\widetilde{u}$ on $W^{\pm }$. By Corollary \ref
{Solution d bar delta -alfa lisse} we can choose $h^{\pm }\in C_{\left(
p,q\right) }^{s\left( k\right) }\left( \overline{W^{\pm }}\right) $, $
s\left( k\right) \underset{k\rightarrow \infty }{\thicksim }\sqrt{k}$. This
means that for $k$ big enough, the form $h$ defined as $h^{\pm }$ on $
\overline{W^{\pm }}$ is of class $C^{k}$ on $\overline{W}$ and vanishes on $
L $. So $\widetilde{u}-h^{\pm }$ is a smooth $\overline{\partial }$-closed
form on $W$ which is an extension of $u$.
\end{proof}

\begin{theorem}
Let $X$ be a compact K\"{a}hler manifold of dimension $n\geq 3$ and $L$ a $
C^{\infty }$ compact Levi flat hypersurface. Then the normal bundle to the
Levi foliation does not admit any Hermitian metric of class $C^{2}$ with
leafwise positive curvature.
\end{theorem}

\begin{proof}
Suppose that the normal bundle $\mathcal{N}$ to the Levi foliation admits a
Hermitian metric of class $C^{2}$ with leafwise positive curvature. Since $
\mathcal{N}$ is topologically trivial, its curvature form $\Theta ^{\mathcal{
N}}$ for the K\"{a}hler metric of $X$ is $d$-exact. So there exists a $1$
-form $u$ of class $C^{\infty }$ on $L$ such that $du=\Theta ^{\mathcal{N}}$
; we may suppose that $u$ is real and $u=u^{0,1}+\overline{u^{0,1}}$, where $
u^{0,1}$ is the $\left( 0,1\right) $ component of $u$. Since $\Theta ^{
\mathcal{N}}$ is a $\left( 1,1\right) $-form, it follows that $\overline{
\partial }_{b}u^{0,1}=0$, where $\overline{\partial }_{b}$ is the tangential
Cauchy-Riemann operator. By Proposition \ref{Extension} there exists a $
C^{k} $-extension $U^{0,1}$ of $u^{0,1}$ to a neighborhood of $L$, $
k\geqslant 2$, such that $\overline{\partial }U^{0,1}=0$.

From now on, the proof follows \cite{Brunella08}.

Since $\dim X\geqslant 3$, $U^{0,1}$ admits a $\overline{\partial }$-closed
extension $\widetilde{U^{0,1}}$ to the full $X$. Indeed, consider a $C^{k}$
extension $V^{0,1}$ of $U^{0,1}$ to $X$. Since $X\backslash L$ is strongly
pseudoconvex by a theorem of Grauert and Riemenschneider \cite{Grauert70}, $
H^{n-2}\left( X\backslash L,K_{X}\right) =0$. By Serre duality it follows
the group of cohomology with compact support $H_{c}^{2}\left( X\backslash L,
\mathcal{O}\right) =0$. So there exists $G^{0,1}$ with compact support in $
X\backslash L$ such that $\overline{\partial }V^{0,1}=\overline{\partial }
G^{0,1}.$ It follows that $\widetilde{U^{0,1}}=V^{0,1}-G^{0,1}$ is a $
\overline{\partial }$-closed $C^{k}$ extension $U^{0,1}$ to $X$.

By Hodge symmetry and Dolbeault isomorphism $H^{0,1}\left( X,\mathbb{C}
\right) \thickapprox \overline{H^{1,0}\left( X,\mathbb{C}\right) }
\thickapprox \overline{H^{0}\left( X,\Omega _{X}^{1}\right) }$, where $
\Omega _{X}^{1}$ is the sheaf of holomorphic $1$-forms on $X$. So there
exists $\eta \in H^{0}\left( X,\Omega _{X}^{1}\right) $ and $\Phi \in
C^{k}\left( X\right) $ such that $\widetilde{U^{0,1}}=\overline{\eta }+
\overline{\partial }\Phi $. It follows that $\Theta ^{\mathcal{N}}=i\partial
_{b}\overline{\partial _{b}}\func{Im}\Phi $ on $L$ and this gives a
contradiction at the point of $L$ where $\func{Im}\Phi $ reaches its maximum.
\end{proof}

\begin{acknowledgement}
We would like to thank M. Adachi and the anonymous referee for very useful
remarks.
\end{acknowledgement}

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config: arxiv


TITLE: Two homotopy equivalent closed $n$-manifolds are homeomorphic
QUESTION [0 upvotes]: I heard somewhere that there is a wrong conjecture that states:

Two homotopy equivalent closed $n$-manifolds ($n\geq 3$) are homeomorphic.

and the lens spaces are the first counterexample to this conjecture. I want to know the author of this wrong conjecture, and a reference for it. I am also curious about other counterexamples? (Here is a related question)

REPLY [2 votes]: The right conjecture (due to Armand Borel, not Horowitz) is that any two homotopy-equivalent closed aspherical  manifolds are  homeomorphic, see here. (A manifold is aspherical if it is connected and has contractible universal covering space.) This conjecture is wide-open (in dimensions $\ge 4$) but is expected to be false. The conjecture is known to be true for Riemannian manifolds of negative curvature (very hard work of Farrel and Jones). It is also known to be true for 3-dimensional (aspherical) manifolds, which is primarily due to Perelman's solution of the Geometrization Conjecture.


TITLE: Continuous image of a cantor set and other space filling curves
QUESTION [1 upvotes]: I am studying the theory of space filling curves, more specifically looking at the continuous image of a Cantor set.
I have been given this definition to characterise the continuous image of a Cantor set.
"Every compact set is the continuous image of the Cantor set".
Does this refer to the compact set in the topological sense or the metric space sense?
In general when we are referring to the continuous image of a space filling curve, are we talking about them in the topological space or the metric space?
So would the following theorem (below) be valid in the theory of space filling curves?
Let $f$ be a function $f:X \to Y$ where $X$ and $Y$ are topological
 spaces. The function $f$ is continuous on X if and only if the
 pre-image of $f(V)$, where $V$ is a neighbourhood,  is open in $X$
 whenever $V$ is open in $Y$
I have learnt from the comments below that there is no difference between compact sets in the topological sense and compact sets in metric sense.
What about other topological concepts like connectedness etc...?

REPLY [2 votes]: The correct statement is that every nonempty compact metrizable space is a continuous image of the ternary Cantor set (Alexandroff-Hausdorff theorem).


TITLE: Choosing a Subset of Points that Best Fit a Polygonal Curve
QUESTION [1 upvotes]: Suppose you have a sequence of finite sets of 2D points, $\mathbf{S} = S_1, S_2, ..., S_n$, and you have a sequence of points which define a polygonal curve, $C = p_1, p_2, ..., p_n$ . The problem I am trying to solve is to match each $p_i$ with a $p'_i \in S_i$, so that you have a new curve, $C' = p'_1, p'_2, ..., p'_n$, that optimally approximates $C$ under a particular distance measure $d$.
I know that this depends strongly on $d$, but I am hoping to find out first if this is a known/formalized problem in any related context and if so what the problem is referred to as.

REPLY [0 votes]: After more digging, I have found what I was looking for. A similar problem, is to match a curve with a subset of points in $S$. This problem and several variants have been called the set-chain matching problems [1], or in other places Curve/Point-Set Matching problems (CPSM) [2]. Or in another case, matching a curve with edges in a Graph, has been termed the discrete map matching problem [3].
[1] Wylie, T., & Zhu, B. (2014). Following a curve with the discrete Fréchet distance. Theoretical Computer Science, 556, 34-44.
[2] Accisano, P., & Üngör, A. (2012). Hardness Results on Curve/Point Set Matching with Fr\'echet Distance. arXiv preprint arXiv:1211.2030.
[3]Chen, D., Driemel, A., Guibas, L. J., Nguyen, A., & Wenk, C. (2011, January). Approximate map matching with respect to the Fréchet distance. In 2011 Proceedings of the Thirteenth Workshop on Algorithm Engineering and Experiments (ALENEX) (pp. 75-83). Society for Industrial and Applied Mathematics.


TITLE: Determine if the sets of vectors are a basis for the indicated space
QUESTION [0 upvotes]: I am confused on how to determine if the sets of vectors are a basis of the space:
$a) \{(5,2,-1), (1,0,1), (3,-1,0)\}$ in $\mathbb R^3$ , and
$b) \{1,0,,-2,5), (4,4,-3,2), (0,1,0,-3), (1,3,3,-10)\}$ in $\mathbb R^4$.
Attempt:
$a)$ I put each of the vectors into an equation as follows:
$$5a+b+3c=0
\\
2a-c=0
\\
-a+b=0$$
and found out that the equations above will only be true if $a=b=c=0,$ and therefore they are linearly independent.  I also found out that the matrix is invertible through calculating the determinant (which equals $12$), and therefore these sets of vectors are a basis of $\mathbb R^3$.
$b)$ I put each of the vectors into a matrix, and after RREF I found out that there are solutions to each of the vectors, but do not know how to carry on.
Am I doing something wrong in my work or am I giving a false conclusion?

REPLY [0 votes]: For the first set of vectors the determinant is 6 (not 0) which indicates that the matrix is inversible, thus the vectors are linearly independent, and these 3 vectors FORM a base of $\mathbb R^3$.
For the set b, you can simply put the vectors in a 4x4 matrix and calculate the determinant, you will get 0 indicating that the vectors aren’t linearly independent and thus they can’t form a base of $\mathbb R^4$.
You can also proceed in the same manner of a, by setting the vectors in a system of 4 unknowns $(a,b,c,d)$ and solve it for $(0,0,0,0)$, you will definitely get an answer other than vector 0 indicating the dependence between these vectors.
Hope I helped.


TITLE: $E=mc^2$ - why can we say that mass and energy are equivalent?
QUESTION [0 upvotes]: Why is it that we can say mass and energy are "equivalent" if energy is equal to mass times the speed of light squared?

REPLY [4 votes]: Because speed of light (in vacuum) is a fundamental constant. You can think of $c^2$ as a constant of proportionality between mass and energy. In fact, in theoretical physics we often set $c=1$ and $\hbar=1$, which is called natural units. Then $E=m$.


TITLE: Arbitrary element in a tensor product is a finite sum.
QUESTION [2 upvotes]: Many books on algebra say that any arbitrary element in a tensor product is a finite sum of pure tensors, however, I cannot clearly see why this is true. Actually, if both of the spaces $U$ and $V$ are finidimensional, then $U \otimes V$ is finidimensional as well. Then it is really true: any element $x \in U \otimes V$ can be expressed as a finite linear combination of basis elements, and so it can be expressed as a finite sum of pure tensors:
$$
x = \sum_{i=1}^{n} \sum_{j=1}^{m} c_{ij} (u_i \otimes v_j) = \sum_{i=1}^{n} \sum_{j=1}^{m} ((c_{ij} u_i) \otimes v_j).
$$
But why is it true when $U$ and $V$ are not finidimensional? I clearly do not understand this, please give me a hint.

REPLY [2 votes]: I think your doubt can be expressed just the same as "Why in an infinite dimensional vector space any element in it is a finite linear combination of a (finite, of course) subset of a basis of the space?" .
This is so just definition based on the axioms of vector space (or module, or algebra or whatever): we know how to do finite sums, we do not how to do infinite sums unless some further structure kicks in, say a normed space.


TITLE: Rearrangement inequality in cyclic but non-symmetric inequality
QUESTION [1 upvotes]: Applying the rearrangement inequality usually involves introducing ordering in the variables of the inequality. This must be done without loss of generality. Now, I understand that if the inequality is symmetric ie unchanged by swapping any variables, then introducing any ordering does not lead to a loss of generality.
However, I have also seen the rearrangement inequality applied to inequalities that are cyclic but non-symmetric, for instance
$$\sum_{cyc} \frac{1}{b(a+b)} \geq \sum_{cyc}\frac{1}{b(a+c)}$$
Now, this is almost trivial if we are allowed to introduce ordering in $a, b, c$ that is assume $a\geq b\geq c$. But how is this not loss of generality?

REPLY [2 votes]: Let $x \le y \le z$ be an increasing permutation of the (positive) numbers $a, b, c$ and $s := a+b+c = x+y+z$.
 Then
$$
\frac 1z \le \frac 1y \le \frac 1x \text{ is an increasing permutation of} \frac 1b ,  \frac 1c ,  \frac 1a
$$
and
$$
 \frac 1{s-x} \ge \frac 1{s-y} \ge \frac 1{s-x}
\text{ is a decreasing permutation of}
\frac 1{s-c} ,  \frac 1{s-a} ,  \frac 1{s-b} \, .
$$
It follows from the rearrangement inequality that
$$
\sum_{cyc}\frac{1}{b(a+c)} = \sum_{cyc}\frac{1}{b(s-b)} =
 \frac{1}{x(s-x)} +  \frac{1}{y(s-y)} +  \frac{1}{z(s-z)} \\
 \le \frac{1}{b(s-c)} +  \frac{1}{c(s-a)} +  \frac{1}{a(s-b)} \\
 = \sum_{cyc} \frac{1}{b(s-c)} = \sum_{cyc} \frac{1}{b(a+b)}
$$


TITLE: Construct a field of 27 elements and find the structure of its additive group.
QUESTION [1 upvotes]: My attempt:
To construct a field of 27 elements. We need a 3 degree irreducible polynomial over $\mathbb F_{3}$. We know that such a polynomial $x^{3}+2x^{2}+1$ is irreducible over $\mathbb F_{3}$. Then we can construct a field
$\mathbb F_{27}  $is isomorphic to $\frac{. \mathbb F_{3}}{x^{3}+2x^{2}+1}$.
Is there is any way to construct irreducible polynomial?

REPLY [0 votes]: In a fixed algebraic closure, $\mathbf F_3$ admits a unique extension of degree $3$, which we can denote $\mathbf F_{27}$. Let us determine it using Artin-Schreier theory, which provides a canonical irreducible polynomial, see e.g. math.stackexchange.com/a/3462533/300700. Over a field $k$ of characterisic $p\neq 0$, the Artin-Schreier operator $P$ is defined by $P(x)=x^p-x$. Here the image $P(\mathbf F_3)$ consists only of $0$ and $P(-1)=-1$. Then $\mathbf F_{27}$ is the splitting field over $\mathbf F_3$ of the A.-S. polynomial $P(X)=X^3-X+1=X^3+2X+1$. Note that this is not the polynomial that you gave. As for the structure of the additive group, $\mathbf F_{p^n}$ is obviously a vector space of dimension $n$ over $\mathbf F_p$.


TITLE: Navier-Stokes Equation Derivation in "A Mathematical Introduction to Fluid Mechanics" (Stress Tensor)
QUESTION [1 upvotes]: My question relates to the attached pictures, which contain text from the book noted in the question title. In particular, my question is related to the first assumption made in the text (See bottom of attached picture). For context, these are assumptions made about the stress tensor describing a fluid as the author builds up to deriving the Navier-Stokes Equations. The assumption is that the stress tensor depends linearly on the velocity gradients.
Question - Can someone express mathematically what is meant by the first assumption? I understand physically what this assumption means but I think my understanding would be richer if I could see this assumption stated precisely. Does this just mean that the stress tensor is the Jacobian of u, the fluid velocity field, multiplied by some matrix A?

REPLY [1 votes]: Assumption 1) in the book means that each component $\sigma_{ij}$ is a linear and homogeneous function of each $\frac{\partial u_k}{\partial x_l}$.
To express that mathematically you need $3^4$ numbers say $A_{ijkl}$. You do not think of tensors if you don't want to
$$\sigma_{ij}=\sum_{kl} A_{ijkl}\frac{\partial u_k}{\partial x_l}$$