Lemma 17.15.1477520F48ECEB

Proof. To simplify the notation, we write

for all $\tau \in \mathbb {R}$. Thanks to proposition 21.11.36, for any fixed $x\in X$, we have

for all $t>0$ as long as $\Gamma _{\tau }(x)\neq -\infty $ and $\Gamma '_{\tau }(x)\neq -\infty $ for some $\tau \in \mathbb {R}$. Otherwise, assume for instance that $\Gamma _{\tau }(x)=-\infty $ for all $\tau \in \mathbb {R}$; then by definition, $\psi _{\tau }(x)=\Gamma '_{\tau }(x)$ for all $\tau \in \mathbb {R}$. Therefore, $\Gamma ^*_ t(x)=-\infty $ for all $t>0$ and (17.15.15) continues to hold. The case where $\Gamma '_{\tau }(x)=-\infty $ for all $\tau $ is symmetric. Therefore, we have shown that

It follows from proposition 17.10.13 that $(\psi ^*_ t)_{t\in [a,b]}$ is a subgeodesic for any $0< a<b$.

Next we observe that $\psi _{\bullet }$ is closed by definition. So it follows from proposition 21.11.36 and proposition 5.14.15 that

Due to proposition 17.12.16 and proposition 21.11.12, there is a pluripolar set $Z\subseteq X$ such that for $x\in X\setminus Z$, we have

for all $\tau <\Gamma _{\max }\lor \Gamma '_{\max }$. It follows from proposition 5.14.6 that

for all $\tau <\Gamma _{\max }\lor \Gamma '_{\max }$.

We now check that $\Gamma \lor \Gamma '$ is a test curve. The function $\rho \mapsto \Gamma _\rho \lor \Gamma '_\rho $ is decreasing, hence its upper concave envelope $\psi _\bullet $ is decreasing. The monotonicity of the $P$-envelope proposition 16.4.2 3 gives the decreasingness of $\Gamma \lor \Gamma '$. Since $\psi _\bullet $ is concave by construction, proposition 16.4.2 gives the concavity of $\Gamma \lor \Gamma '$.

Fix $\tau <\Gamma _{\max }\lor \Gamma '_{\max }$. Then either $\tau <\Gamma _{\max }$ or $\tau <\Gamma '_{\max }$. Since $\psi _\bullet $ is a concave majorant of $\rho \mapsto \Gamma _\rho \lor \Gamma '_\rho $, we have

Together with the fact that $\psi _\tau \in \mathrm{PSH}(X,\theta )\cup \{ -\infty \} $, these inequalities show that $\psi _\tau \in \mathrm{PSH}(X,\theta )$. It follows from 17.12.6 and the monotonicity theorem theorem 13.13.11 that $\psi _\tau \in \mathrm{PSH}(X,\theta )_{>0}$. By corollary 16.3.20,

is a model potential. Finally, $(\Gamma \lor \Gamma ')_{-\infty }\in \mathrm{PSH}(X,\theta )$ by proposition 5.14.2; since it dominates $(\Gamma \lor \Gamma ')_\tau $ for any $\tau <\Gamma _{\max }\lor \Gamma '_{\max }$, it has positive mass by another application of the monotonicity theorem theorem 13.13.11. Thus $\Gamma \lor \Gamma '\in \mathrm{TC}(X,\theta )_{>0}$.

It follows from (17.15.16) that $\psi _{\tau }$ is $\mathcal{I}$-good if $\Gamma ,\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$, thanks to proposition 18.1.17 and proposition 18.1.19; the family in the upper envelope is bounded above by $0$. Since $(\Gamma \lor \Gamma ')_\tau =P_\theta [\psi _\tau ]$, we conclude that $\Gamma \lor \Gamma '$ is $\mathcal{I}$-model in this case. $\square$