Theorem 18.47.31D58A228EF5A

Proof. For an arbitrary competitor $v$, lemma 18.47.2 gives, in terms of the affine coordinate $\xi $ of $\mathbb P^1$,

On the interior of $K$, the obstacle condition, which holds quasi-everywhere, becomes an inequality everywhere, by the submean inequality and the continuity of $w+\varphi _ X$. Evaluating at $x_1=x_2=s/2$, with $s=\log |\xi |^2$, gives

for every argument of $\xi $. At $s=0$ this bounds $\chi $ by zero on $|\xi |=1$, and the maximum principle for the subharmonic function $\chi $ bounds it by zero on the whole disk $|\xi |\leq 1$. At infinity use the local potential

The obstacle at $s=2$ bounds it by $-1$ on $|\eta |=\mathrm e^{-1}$, and the same maximum principle gives $\chi (\xi )\leq s-1$ for $s\geq 2$. Together with the obstacle for $0\leq s\leq 2$, this proves $\chi \leq g$, without any invariance assumption on the competitor.

Conversely, $g$ is convex with slopes in $[0,1]$, so $g(\log |\xi |^2)-\log (1+|\xi |^2)$ is an $\omega _{\rm FS}$-plurisubharmonic function on $\mathbb P^1$. By lemma 18.47.2, it defines the global function $P$. Directly,

Thus $P$ belongs to the competitor class of lemma 18.47.1 and satisfies the obstacle condition everywhere on $K$. Since it dominates every competitor, it is the envelope, including its values $-\infty $ at the poles. $\square$