>
\widehat{c}
+ 
{\textstyle 
\frac{\eta}{2}}\}\,
\big\backslash\,
\widehat{V}.$$ Placing finitely many $(n-1)$-concave Levi-Hartogs figures at points of this hypersurface, we get holomorphic or meromorphic extension to $\big\{ \mu > \widehat{ c} - \frac{ \eta}{ 2} \big\}\, \big
\backslash\, \widehat{ V}_1$, where $\widehat{ V}_1 \subset \widehat{
V}$ is slightly bigger than $\widehat{ V}$. Repeating the filling process finitely many times until $\big\{ \mu = \widehat{ c} - \frac{
k \eta}{ 2} \big\}$ does not intersect $\widehat{ B}$, where $k$ is an odd integer, we fill in $\widehat{ B} \big \backslash \widehat{ U}$. At each step, monodromy of the extension is assured thanks to connectedness of $\big\{ \mu > \widehat{ c} + \delta \big\}\, \big
\backslash\, \widehat{U}$, for every small $\delta \in {\mathbb{R}}$. However, we cannot fill in $\widehat{ U}$ directly this way.

The trick is to shift $\widehat{ p}$. One introduces a $\mathcal{
C}^\infty$ perturbation $\mu'$ of $\mu$ localized near $\widehat{ p}$ (namely $\mu ' = \mu$ elsewhere) such that $\mu'$ has another critical point $\widehat{ p}'$ (having the same Morse index of course), with corresponding neighborhoods disjoint: $\widehat{ U} \cap \widehat{ U}'
= \emptyset$ and both contained in $\widehat{ B} \cap \widehat{ B}'$. We repeat the Levi-Hartogs filling with $\mu'$, getting 