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Definition 17.17.27D4FF1072CEF

Definition 17.17.2.7D4FF1072CEF An envelope test curve with initial potential $\varphi $ consists of a real number $c$ and potentials $\Gamma _\tau \in \mathrm{PSH}(X,\theta )$ for $\tau <c$ such that

  1. $\tau \mapsto \Gamma _\tau $ is pointwise concave and decreasing;

  2. $P_\theta [\Gamma _\tau ]=\Gamma _\tau $ for every $\tau <c$;

  3. $(\sup _{\tau <c}\Gamma _\tau )^*=\varphi $.

Its cutoff is $c$. We extend it by

\[ \Gamma _ c=\lim _{\tau \uparrow c}\Gamma _\tau , \qquad \Gamma _\tau \equiv -\infty \quad (\tau >c). \]
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