A Hartogs domain with a strictly plurisubharmonic exhaustion

Put $E:=e^{2\mathrm{\mid }z{\mathrm{\mid }}^2}$, so that $s=\mathrm{\mid }w{\mathrm{\mid }}^{2}E$ is $C^{\mathrm{\infty }}$ on ${\mathbb{C}}^2$; the Wirtinger operators of [F5] give ${\mathrm{\partial }}_{z}s=2\bar{z}s$, ${\mathrm{\partial }}_{\bar{z}}s=2zs$, ${\mathrm{\partial }}_{w}s=\bar{w}E$, ${\mathrm{\partial }}_{\bar{w}}s=wE$, and differentiating once more gives ${\mathrm{\partial }}^{2}s\mathrm{/}\mathrm{\partial }z\mathrm{\partial }\bar{z}=2s(1+2\mathrm{\mid }z{\mathrm{\mid }}^2)$, ${\mathrm{\partial }}^{2}s\mathrm{/}\mathrm{\partial }w\mathrm{\partial }\bar{w}=E$, ${\mathrm{\partial }}^{2}s\mathrm{/}\mathrm{\partial }z\mathrm{\partial }\bar{w}=2\bar{z}wE$, ${\mathrm{\partial }}^{2}s\mathrm{/}\mathrm{\partial }w\mathrm{\partial }\bar{z}=2z\bar{w}E$.

$\mathrm{\Omega }=\{s<1\}$ is open and nonempty ($s(0,0)=0<1$), and it is star-shaped about the origin: if $(z,w)\in \mathrm{\Omega }$ and $0<t\le 1$, then $\mathrm{\mid }w\mathrm{\mid }<e^{-\mathrm{\mid }z{\mathrm{\mid }}^2}$ gives $\mathrm{\mid }tw\mathrm{\mid }=t\mathrm{\mid }w\mathrm{\mid }<t{e}^{-\mathrm{\mid }z{\mathrm{\mid }}^2}\le e^{-t^2\mathrm{\mid }z{\mathrm{\mid }}^2}$, since $t\le 1$ and $t^2\mathrm{\mid }z{\mathrm{\mid }}^2\le \mathrm{\mid }z{\mathrm{\mid }}^2$, so $(tz,tw)\in \mathrm{\Omega }$; at $t=0$ the point is $(0,0)\in \mathrm{\Omega }$. Thus each radial segment $t\mapsto (tz,tw)$ lies in $\mathrm{\Omega }$ and joins $(z,w)$ to the origin, so $\mathrm{\Omega }$ is path-connected and, by [F6], connected. Hence $\mathrm{\Omega }$ is a domain.

The function $h(t):=(1-t{)}^{-1}$ is $C^{\mathrm{\infty }}$ and satisfies $h^{\mathrm{\prime }}(t)=(1-t{)}^{-2}>0$ and $h^{\mathrm{\prime }\mathrm{\prime }}(t)=2(1-t{)}^{-3}>0$ on $(-\mathrm{\infty },1)$, so $h$ is strictly increasing and convex there; since $s<1$ on $\mathrm{\Omega }$, the function $\psi =\mathrm{\mid }z{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2+h(s)$ is $C^{\mathrm{\infty }}$ and real-valued on $\mathrm{\Omega }$.

For a real-valued $C^2$ function $u$ and a $C^2$ function $\varphi$ of one real variable one has $\mathrm{\partial }\bar{\mathrm{\partial }}(\varphi \circ u)={\varphi }^{\mathrm{\prime }}(u)\mathrm{\partial }\bar{\mathrm{\partial }}u+{\varphi }^{\mathrm{\prime }\mathrm{\prime }}(u)\mathrm{\partial }u\wedge \bar{\mathrm{\partial }}u$, hence at every point ${\mathcal{L}}_{\varphi \circ u}(a;\xi )={\varphi }^{\mathrm{\prime }}(u(a)){\mathcal{L}}_u(a;\xi )+{\varphi }^{\mathrm{\prime }\mathrm{\prime }}(u(a))\mid {\sum }_j{\mathrm{\partial }}_{z_j}u(a){\xi }_j{\mid }^2$.

The Hermitian matrix of the coefficients of step 1.1 is $M=\left(\begin{array}{cc}2s(1+2\mathrm{\mid }z{\mathrm{\mid }}^2)& 2\bar{z}wE\\ 2z\bar{w}E& E\end{array}\right)$: its diagonal entries are nonnegative, its determinant is $2s(1+2\mathrm{\mid }z{\mathrm{\mid }}^2)E-4\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{\mid }w{\mathrm{\mid }}^2{E}^2=2\mathrm{\mid }w{\mathrm{\mid }}^2{E}^2\ge 0$, and for $w=0$ it is $\mathrm{diag}(0,E)$ with $E>0$; hence ${\mathcal{L}}_s(a;\xi )={\sum }_{j,k}{\xi }_j{M}_{jk}{\bar{\xi }}_k\ge 0$ for all $a,\xi$, so [F3] makes $s$ plurisubharmonic on ${\mathbb{C}}^2$, and at every point with $w\mathrm{\ne }0$ the matrix $M$ is even positive definite (trace at least $E>0$, determinant $2\mathrm{\mid }w{\mathrm{\mid }}^2{E}^2>0$).

The boundary of $\mathrm{\Omega }$ is $\{s=1\}$: a point with $s(p)<1$ lies in the open set $\mathrm{\Omega }$ and a point with $s(p)>1$ has a neighbourhood disjoint from $\mathrm{\Omega }$, so neither is a boundary point; and if $s(p)=1$ then $w\mathrm{\ne }0$, so ${\mathrm{\partial }}_{\bar{w}}s(p)=wE\mathrm{\ne }0$ and $ds(p)\mathrm{\ne }0$, and the points $p-t\mathrm{\nabla }s(p)$ for small $t>0$ satisfy $s(p-t\mathrm{\nabla }s(p))=1-t\mathrm{\mid }\mathrm{\nabla }s(p){\mathrm{\mid }}^2+O(t^2)<1$, hence lie in $\mathrm{\Omega }$ and converge to $p$, while $p\mathrm{\notin }\mathrm{\Omega }$; therefore $p\in \mathrm{\partial }\mathrm{\Omega }$.

Applying step 1.4 to $\varphi =h$ and $u=s$, and adding the strictly plurisubharmonic term $\mathrm{\mid }z{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2$ with ${\mathcal{L}}_{\mathrm{\mid }z{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2}(a;\xi )=\mathrm{\mid }{\xi }_1{\mathrm{\mid }}^2+\mathrm{\mid }{\xi }_2{\mathrm{\mid }}^2$, gives at every $a\in \mathrm{\Omega }$ and every $\xi \mathrm{\ne }0$ the bound ${\mathcal{L}}_{\psi }(a;\xi )=\mathrm{\mid }{\xi }_1{\mathrm{\mid }}^2+\mathrm{\mid }{\xi }_2{\mathrm{\mid }}^2+h^{\mathrm{\prime }}(s(a)){\mathcal{L}}_s(a;\xi )+h^{\mathrm{\prime }\mathrm{\prime }}(s(a))\mathrm{\mid }\mathrm{\partial }s(a;\xi ){\mathrm{\mid }}^2\ge \mathrm{\mid }\xi {\mathrm{\mid }}^2>0$, because $h^{\mathrm{\prime }}>0$, $h^{\mathrm{\prime }\mathrm{\prime }}>0$ and ${\mathcal{L}}_s\ge 0$ by step 2.1; hence $\psi$ is strictly plurisubharmonic on $\mathrm{\Omega }$ by [F1].

On $\{s=1\}$ the matrix $M$ of step 2.1 is positive definite, because $w\mathrm{\ne }0$ there; hence with the global defining function $\rho :=s-1$, the neighbourhood $U={\mathbb{C}}^2$ and ${\mathcal{L}}_{\rho }={\mathcal{L}}_s$ one has $\mathrm{\Omega }=\{\rho <0\}$, $d\rho (p)\mathrm{\ne }0$ and ${\mathcal{L}}_{\rho }(p;\xi )>0$ for every nonzero complex tangent vector $\xi$ at every boundary point $p$; in particular $\mathrm{\Omega }$ is Levi pseudoconvex in the sense of [F2].

For $c\le 0$ the sublevel set $\{\psi \le c\}$ is empty; for $c>0$ it is contained in $\{\mathrm{\mid }z{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2\le c\}\cap \{s\le 1-c^{-1}\}$, on which $\psi$ is continuous, so $\{\psi \le c\}$ is a closed subset of ${\mathbb{C}}^2$; it is bounded, and it lies in $\mathrm{\Omega }$ because $s\le 1-c^{-1}<1$; by [F8] it is a closed and bounded subset of ${\mathbb{R}}^4$, hence compact by [F7], and a compact subset of ${\mathbb{C}}^2$ contained in $\mathrm{\Omega }$ is compact in $\mathrm{\Omega }$. Thus every sublevel set of $\psi$ is compact, and with steps 1.2, 1.3 and 3.1 the function $\psi$ is a continuous strictly plurisubharmonic exhaustion of $\mathrm{\Omega }$ in the sense of [F4].