Grauert’s theory shows how to jump across them with $\overline{ \partial}$ techniques, and we summarize how we can proceed here[^9], using only analytic discs in Levi-Hartogs figures.

Consider a point $\widehat{ p} \in X_{\rm reg}$ which is critical: $d\mu ( \widehat{ p}) = 0$, and set $\widehat{c} := \mu ( \widehat{
p})$. The Morse lemma provides local real coordinates centered at $\widehat{ p}$ in which $\mu = x_1^2 + \cdots + x_k^2 - y_1^2 - \cdots
- y_{ 2n - k}^2$, for some $k$. Since $i\, \partial \overline{
\partial}\, \mu$ has at least $2$ positive eigenvalues everywhere, $k$ is $\geqslant 2$. This is a crucial fact, because this implies that super-level sets $\{ \mu > \widehat{ c} + \delta \}$ are all connected[^10] in a neighborhood of $\widehat{ p}$, for every $\delta \in {\mathbb{R}}$ close to $0$, and moreover, that these domains grow regularly and continuously as $\delta$ decreases from positive values to negative values.

double-excised.pstex\_t

Next, we fix a ball $\widehat{ B}$ centered at $\widehat{ p}$ and we cut out a small neighborhood $\widehat{ U}
\subset \widehat{ B}$ of $\widehat{ p}$. If $\widehat{ V} \subset \widehat{ U}$ is a small neighborhood, we consider the $\mathcal{ C}^\infty$ hypersurface: $$\big\{
\mu 
