Positive smooth collars for strictly plurisubharmonic negative sets

Put $F=\bar{D}$. Enumerate all rational centers $q_j\in {\mathbb{Q}}^{2n}$ and positive rational radii $r_j$ for which $\bar{B}(q_j,2r_j)\cap F=\mathrm{\varnothing }$. Their inner balls cover ${\mathbb{C}}^n\setminus F$: openness of the complement supplies a sufficiently small ball, then a rational center and radius. Set $b_j(x)=b((x-q_j)\mathrm{/}{r}_j)$, $M_j={\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le j}{\sup }_{{\mathbb{R}}^{2n}}\mathrm{\mid }{D}^{\alpha }{b}_j\mathrm{\mid },a_j=\frac{2^{-j}}{1+M_j},\beta ={\sum }_{j\ge 1}{a}_j{b}_j\mathrm{.}$ Each $M_j$ is finite since the derivatives have compact support. For every fixed multi-index $\alpha$, the tail with $j\ge \mathrm{\mid }\alpha \mathrm{\mid }$ is bounded termwise by $2^{-j}$, so the series of derivatives converges uniformly. Apply [F2] on coordinate segments in closed boxes, successively to every derivative: the limit is $C^{\mathrm{\infty }}$ with $D^{\alpha }\beta ={\sum }_j{a}_j{D}^{\alpha }{b}_j$. Every derivative vanishes on $F$, while at any point outside $F$ some $b_j$ is $1$; hence $\beta \ge 0$ and ${\beta }^{-1}(0)=F$. This argument proves smoothness across $F$, without assuming local finiteness of the bumps there.

Choose a compact neighborhood of $\mathrm{\partial }D$ contained in the strict Levi collar of $\rho$ and in $U$. Compactness of this neighborhood times the unit sphere gives a uniform positive lower Levi bound for $\rho$ and a finite upper absolute Levi bound for $\beta$. Thus for some $t>0$, $\psi :=\rho +t\beta$ is strictly plurisubharmonic on an open neighborhood $V$ of $\mathrm{\partial }D$. On $F$ it agrees with $\rho$, so it is negative on $D$ and zero on $\mathrm{\partial }D$; on $U\setminus F$ both $\rho \ge 0$ and $\beta >0$, so $\psi >0$. In particular $\{\psi <0\}=D$ and the zero set of $\psi$ in $U$ is exactly $\mathrm{\partial }D$.

Choose a bounded open $W$ with $F\subset W$ and $\bar{W}\subset U$. Then $\mathrm{\partial }W$ is compact and disjoint from $F$, so ${\min }_{\mathrm{\partial }W}\psi >0$. On the compact set $\bar{W}\setminus O$ the function $\psi$ is positive whenever that set is nonempty. On $\bar{W}\setminus V$ the compact subset where $\psi \ge 0$ also has a positive minimum if nonempty, since its zero set would lie in $\mathrm{\partial }D\subset V$. Choose a positive regular value $\epsilon$ smaller than all these positive minima, using [F3]. Then $S=\{x\in W:\psi (x)<\epsilon \}$ contains $F$, has $\bar{S}\subset O\cap W$, and its boundary lies in $V\cap \{\psi =\epsilon \}$. Regular-level charts show that $\mathrm{\partial }S$ is smooth and that the inside half of each such chart is connected. Consequently every connected component of $S$ has smooth boundary locally defined by $\psi -\epsilon$, with strictly positive tangential Levi form by step 2.1.

Components of the open set $S$ are open and cover the compact set $F$, so finitely many distinct components $G_i$ cover $F$. Their union $G$ satisfies the required compact containment. In each $G_i$ choose an open neighborhood $N_i$ of $\mathrm{\partial }{G}_i$ with $\overline{N_i}\subset V$. Put $f=-\log (\epsilon -\psi )$ on $G_i$. It is smooth and plurisubharmonic near $\mathrm{\partial }{G}_i$ by [F5], and tends to $+\mathrm{\infty }$ there. The set $\overline{G_i}\setminus N_i$ is a compact subset of $G_i$, so choose $A_i$ greater than its maximum of $f$. The function $E_i=\max \{f,A_i\}+\mathrm{\mid }z{\mathrm{\mid }}^2$ is continuous and plurisubharmonic: near the boundary both terms in the maximum are psh; near every point outside $N_i$ the maximum is the constant $A_i$; these descriptions agree on their overlap. Its sublevels are closed in $G_i$ and stay away from the boundary, hence are compact in the bounded $G_i$. Thus it is an exhaustion.

The domains $G_i$ constructed in steps 3.1–4.1 have all the properties in the Statement, including when the original boundary has critical points or the original $\rho$ has zeros outside $\bar{D}$.