Smooth global defining functions for strongly pseudoconvex boundaries

Compactness of $\mathrm{\partial }D$ supplies finitely many local defining charts and smaller relatively compact neighborhoods covering $\mathrm{\partial }D$. Shrink them so each local differential stays nonzero on the boundary in its chart and each tangential Levi form stays positive there. By [F2] take nonnegative smooth bumps $b_i$ supported in the charts and equal to $1$ on the smaller neighborhoods. On a neighborhood $T$ of $\mathrm{\partial }D$ where $B={\sum }_i{b}_i>0$, put ${\lambda }_i=b_i\mathrm{/}B$ and $r_0={\sum }_i{\lambda }_i{r}_i$, extending each supported product by zero outside its chart. This is smooth and has the same negative, zero and positive sides as the local defining functions. At a boundary point, all active differentials are positive multiples of one outward conormal: they annihilate the common real tangent hyperplane and evaluate positively on an outward vector. Thus $d{r}_0={\sum }_i{\lambda }_{i}d{r}_i\mathrm{\ne }0$.

If $v$ is complex tangent at a boundary point, then $\mathrm{\partial }{r}_i(v)=0$ for all active charts and $r_i=0$. The product rule therefore gives ${\mathcal{L}}_{r_0}(v)={\sum }_i{\lambda }_i{\mathcal{L}}_{r_i}(v)>0(v\mathrm{\ne }0)\mathrm{.}$ Terms involving derivatives of the weights vanish because they contain either $r_i$ or a tangential first derivative of $r_i$. By [F2] choose $\eta \in C_c^{\mathrm{\infty }}(T,[0,1])$ equal to $1$ near $\mathrm{\partial }D$. Define $s=-1$ on $D$ and $s=1$ on ${\mathbb{C}}^n\setminus \bar{D}$, and define $r=\eta r_0+(1-\eta )s$ off $\mathrm{\partial }D$, with $r=r_0=0$ on the boundary. Here $\eta r_0$ is extended by zero off $T$, and $(1-\eta )s$ is zero near the boundary. Hence $r$ is globally smooth, negative exactly on $D$, positive outside $\bar{D}$, and agrees with $r_0$ near the boundary.

On the compact boundary put $a=\mathrm{\partial }r$. Its norm has a positive lower bound $m$, and the operator norm of the Levi matrix of $r$ has a finite bound $M$. For $n>1$ the tangential Levi form has a uniform positive bound $\lambda$. Write any $v=t+w$ with $t\in \ker a$ and $w\perp \ker a$. Then $\mathrm{\mid }a(v)\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }\ge m\mathrm{\mid }w\mathrm{\mid }$ and ${\mathcal{L}}_r(v)\ge \lambda \mathrm{\mid }t{\mathrm{\mid }}^2-2M\mathrm{\mid }t\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }-M\mathrm{\mid }w{\mathrm{\mid }}^2\ge \frac{\lambda }{2}\mathrm{\mid }t{\mathrm{\mid }}^2-(M+\frac{2M^2}{\lambda })\mathrm{\mid }w{\mathrm{\mid }}^2\mathrm{.}$ Choose $C$ with $C{m}^2>M+2M^2\mathrm{/}\lambda$. For $n=1$ the tangent space is zero and one instead chooses $C{m}^2>M$. In either case ${\mathcal{L}}_r(v)+C\mathrm{\mid }a(v){\mathrm{\mid }}^2>0$ for every nonzero $v$ on the boundary.

Set $\rho =e^{Cr}-1$. The chain rule gives ${\mathcal{L}}_{\rho }(v)=C{e}^{Cr}({\mathcal{L}}_r(v)+C\mathrm{\mid }\mathrm{\partial }r(v){\mathrm{\mid }}^2)\mathrm{.}$ Step 3.1 and compactness give strict positivity on a neighborhood of $\mathrm{\partial }D$. The function is $C^{\mathrm{\infty }}$ because the constructed $r$ is $C^{\mathrm{\infty }}$; its negative set is exactly $D$ and $d\rho =Cdr\mathrm{\ne }0$ on the boundary. Restricting to any neighborhood $U$ of $\bar{D}$ gives the Statement.