Proposition 11.45.3.DA11AEC1C7C7 The function $\phi _{a,d}$ is an ordinary model reference for $\theta $, with
\[ P_\theta [\psi ]=P_\theta [\phi _{a,d}]=\phi _{a,d},\qquad \psi \leq \phi _{a,d}\leq \psi +C_*,\qquad \sup _ X\phi _{a,d}=0. \]
For $d/(a+d)<\delta <1$, the envelope $P_{\delta \theta }(\phi _{a,d})$ is not identically $-\infty $. The model has finite positive Monge–Ampère measure
\[ \mathrm{MA}_\theta (\phi _{a,d}) =N b^ kK\mathbf1_{\{ |z|^2<R\} }\, \mathrm{d}W. \]
The difference $\phi _{a,d}-\psi $ extends continuously across $D$. If $(a_ j,d_ j)\to (a,d)$ in $(0,+\infty )^2$, these residual extensions converge uniformly.
Proof.
First take the ordinary envelope on the curve with background $K\omega _1$ and obstacle $\min (\psi +C,0)$. Rotation of $z$ preserves both the obstacle and the admissible family, so the envelope is rotation invariant. With $s=\log |z|^2$ and $F(s)=K\log (1+\mathrm{e}^ s)$, its profile plus $F$ is convex and lies below $F$. The comparison with $\psi +C$ bounds its final slope by $a$. At $s_*=\log R$, one has $F'(s_*)=a$. Above $s_*$, convexity and the final slope bound give
\[ u(s)+F(s)\leq F(s_*)+a(s-s_*), \]
and below $s_*$ the obstacle gives $u\leq 0$. The function in (11.45.2) attains both bounds. Its convex profile, including its logarithmic value at infinity, gives a curve potential, and direct calculation gives $\psi \leq \phi _{a,d}\leq \psi +C_*$. It is therefore a competitor for every $C\geq C_*$, so the curve envelope is exactly $\phi _{a,d}$.
A product competitor restricts to each curve $\mathbb P^1\times \{ y\} $ as either the constant $-\infty $ or a curve competitor. It is thus bounded above by $\phi _{a,d}$. Conversely, the curve model is a product competitor because the vertical form $b\kappa $ is positive. Hence the product envelope is the same function. Ordinary-type invariance of the singularity envelope gives its model property. Moreover,
\[ \delta \theta +\mathrm{dd}^{\mathrm{c}}\psi =(\delta K-d)\omega _1+\delta b\kappa +d[D]\geq 0, \]
which proves the smaller-background assertion.
Off $D$, the horizontal curvature of $\phi _{a,d}$ is $K\mathbf1_{\{ |z|^2<R\} }\omega _1$. The first derivatives of its convex profile agree at $R$, so there is no boundary-circle atom. Expanding the product off $D$ gives the displayed measure, and the non-pluripolar product assigns $D$ zero mass. Finally, in terms of $\ell $ the residual extension is
\[ \phi _{a,d}-\psi =\begin{cases} -d\log \ell ,& \ell \geq d/K,\\ a\log (1-\ell )+C_*,& \ell <d/K. \end{cases} \]
The formulas agree at the moving interface and extend to both endpoints. On compact positive parameter ranges each logarithm is used away from its singular endpoint. Joint continuity on the resulting compact set proves uniform variation.
$\square$
Pome
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