Corollary 11.19.36CD37D30AF59

Proof. Choose a smooth convex increasing function $\eta $ on $\mathbb R$ equal to $-\log (-t)$ for $t\leq -1$. On $E\setminus Z$ put

The added forms are nonnegative, so this is Kähler. Since $\Psi \geq 0$ is plurisubharmonic, $2\pi \mathrm{dd}^{\mathrm{c}}(\Psi ^2)\geq 2\mathrm i\partial \Psi \wedge \bar\partial \Psi $. Thus $\Psi $ has bounded gradient for $\widehat\Omega $, and a curve of finite length stays in a compact sublevel of $\Psi $ in $E$. The only possible escape from that compact set within $E\setminus Z$ is approach to $Z$. Where $\log q\leq -1$, set $\tau =\log (-\log q)$. Convexity and the chain rule give

Hence $\tau $ also has bounded gradient there, while $\tau \to +\infty $ on approach to $Z$. A finite-length curve cannot escape in this way either. The criterion that every escaping curve has infinite length proves completeness. $\square$