We restrict the envelope test curves of definition 17.17.2 by requiring that each slice be fixed by the multiplier-ideal envelope and that the curve be constant for $\tau \leq a$. Constancy identifies the initial potential explicitly as $\Gamma _ a$, and it confines the associated ray between the translates of this potential by $at$ and by $ct$.
Definition 16.29.2.F0A985779829 Let $\mathscr T_{a,c}(\theta )$ be the set of families $\Gamma =(\Gamma _\tau )_{\tau <c}$ such that
$\Gamma _\tau \in \mathrm{PSH}(X,\theta )$ and $P_\theta [\Gamma _\tau ]_{\mathcal I}=\Gamma _\tau $ for every $\tau <c$;
the family is pointwise concave and decreasing in $\tau $;
$\Gamma _\tau =\Gamma _ a$ for $\tau \leq a$.
The cutoff of $\Gamma $ is $c$, and we extend $\Gamma $ by $\Gamma _ c=\lim _{\tau \uparrow c}\Gamma _\tau $ and $\Gamma _\tau =-\infty $ for $\tau >c$. Such curves are ordered by the pointwise order of their slices $\Gamma _\tau $, $\tau <c$.
Every slice has supremum zero by theorem 18.4.2. Hence, by theorem 5.25.2, corollary 5.25.3, the decreasing limit $\Gamma _ c$ belongs to $\mathrm{PSH}(X,\theta )$; in particular, it is not identically $-\infty $. It is also $P_\theta [\cdot ]_{\mathcal I}$-fixed. Indeed, $\Gamma _ c\leq 0$ is a competitor for its own envelope, and $\Gamma _ c\leq \Gamma _\tau $, so the order preservation of proposition 18.4.6 gives
\[ \Gamma _ c\leq P_\theta [\Gamma _ c]_{\mathcal I}\leq \Gamma _\tau \quad (\tau <c). \]
Taking the infimum over $\tau $ proves equality.
Proposition 16.29.3.160B15BF08FE A curve $\Gamma \in \mathscr T_{a,c}(\theta )$ is an envelope test curve with initial potential $\varphi =\Gamma _ a$. Its Legendre transform
\[ U_ t=\Bigl (\sup _{\tau <c}(\Gamma _\tau +t\tau )\Bigr )^*\qquad (t>0) \]
is an envelope ray with initial potential $\varphi $ and slope $c$, in the sense of definition 17.17.3, and satisfies
\[ \varphi +at\leq U_ t\leq \varphi +ct\qquad (t>0). \]
In particular $U_ t\sim \varphi $ for every $t>0$.
Proof.
By corollary 18.4.4, a $P_\theta [\cdot ]_{\mathcal I}$-fixed potential is also fixed by the ordinary envelope $P_\theta $. The regularized supremum of the curve is $\Gamma _ a$, since the curve is decreasing and constant for $\tau \leq a$. Hence all the conditions of definition 17.17.2 hold, and theorem 17.17.5 applies. The term $\tau =a$ in the transform gives the lower bound, and the inequalities $\Gamma _\tau \leq \varphi $ for $\tau <c$ give the upper bound.
$\square$
Pome
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