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17.17.1 Envelope test curves and rays22C33F2E9103

The members of a test curve become more singular as the parameter increases. The cutoff of the curve becomes the slope of the transformed ray, while its limit as the parameter tends to $-\infty $ gives the initial potential.

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

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

Each $\Gamma _\tau $ is an envelope fixed point and hence has supremum zero, so theorem 5.25.2, corollary 5.25.3 give $\Gamma _ c\in \mathrm{PSH}(X,\theta )$.

Definition 17.17.3.ECDB430D0021 An envelope ray with initial potential $\varphi $ and slope $c$ is a subgeodesic $u_ t$, $t>0$, such that

  1. $u_ t\to \varphi $ in $L^1(X)$ as $t\downarrow 0$, and, with $u_0=\varphi $, the restriction of the ray to every interval $(a,b)$ with $0\leq a<b<\infty $ is the Perron segment from $u_ a$ to $u_ b$;

  2. $P_\theta [u_ t]=\varphi $ for every $t>0$;

  3. $\sup _ Xu_ t=ct$ for every $t>0$.

For an envelope test curve, write $U_ t$ for its forward transform (5.41.6), the pointwise supremum of $\Gamma _\tau +t\tau $ over $\tau <c$. For a family $(u_ t)_{t>0}$, write $\widehat u_\tau $ for its inverse transform (5.41.7), the pointwise infimum of $u_ t-t\tau $ over $t>0$. The forward transform is a subgeodesic by theorem 5.41.9; it remains to show that its finite segments are Perron segments.

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