\cap \{ \mu = \widehat{ c}\}$.

mu-M-X-sing.pstex\_t

§4. Holomorphic extension to $D_{\rm reg}$ {#holomorphic-extension-to-d_rm-reg .unnumbered}
==========================================

For $c\in {\mathbb{R}}$, we introduce $$X_{\mu>c}:= 
\{z\in X:\,
\mu(z)
>
c\}.$$ This open set is contained in $X_{\rm reg}$, since $X_{\rm sing} = \{
\mu = - \infty\}$. For every connected component $M_{\mu > c}'$ of $$M_{\mu>c}
:=
M\cap
X_{\mu>c}= 
M\cap
\{\mu>c\},$$ we want to fill in (by means of a finite number of families of analytic discs) a certain domain $Q_{ \mu > c}'$ which is enclosed by $M_{ \mu > c}'$ inside $\{ \mu > c\}$. Similarly as in Proposition 5.3 of [@mp2006b], we must consider [*all*]{} the connected components $M_{\mu > c}'$ and analyze the combinatorics of how they merge or disappear.

Let $\mathcal{ V} ( M)$ be a thin tubular neighborhood of $M$, whose thinness shrinks to zero while approaching $X_{\rm sing}$. For every connected component $M_{ \mu > c}'$ of $M_{ \mu > c}$, we denote by $\mathcal{ V} \big( M_{\mu > c}' \big)_{ \mu > c}$ the part of $\mathcal{ V} ( M)$ around $M_{ \mu > c}'$ again intersected with $\{
\mu > c\}$. It is a connected tubular neighborhood of $M_{ \mu > c}'$ inside $\{ 