represented by an ambient function $\widetilde{ \rho}_{ j (
p)} : \widetilde{ \sf B}_{ j (p)} \to {\mathbb{R}}$ whose Levi-form has at least $N_{ j ( p)} - n + 2$ eigenvalues $>0$. By diagonalizing the Levi matrix $i\, \partial \overline{ \partial} \widetilde{ \rho}_{ j(p)}$ at the central point of $\widetilde{ \sf B}_{ j (p)}$, we may easily define, in some small open sub-ball $\widetilde{ \sf C}_{ j (p)}
\subset \widetilde{ \sf B}_{ j (p)}$ having the same center, a $\mathcal{ C}^\infty$ family $\widetilde{ q} \mapsto \widetilde{ F}_{
\widetilde{ q}}$ of complex $(N_{j(p)}-n+2)$-dimensional affine subspaces such that the Levi-form of $\widetilde{ \rho}_{ j(p)}$ is positive definite on every $\widetilde{ F}_{ \widetilde{ q}}$, for every $\widetilde{ q} \in
\widetilde{ \sf C}_{ j (p)}$.

Next, if we set $V_{ j(p)} := \varphi_{ j(p)}^{ - 1} \big(
\widetilde{\sf C}_{ j(p)} \big)$, which is an open subset of $U_{
j(p)}$, we can cover the compact set $\{ \rho = C\}$ by finitely many $V_{ j(p)}$, hence there is a finite number of points $p_a$, $a = 1,
\dots, A$, such that $$\{\rho=C\}
\subset
V_{j(p_1)}\cup\cdots\cup
V_{j(p_A)}.$$ According to , on each $\widetilde{ \sf C}_{ j( p_a)}$, $a = 1, \dots A$, we have: $$i\,\partial\overline{\partial}\,
\widetilde{\mu}_{j(p_a)}
