Corollary 18.20.31B6E50A3B70B

Proof. By the lemma and the normalization of the coordinates, equality of the envelopes below zero is equivalent to $q_ i=r_ i$ for every $i$. By corollary 13.29.2, this is equivalent to equality of the polar and atomic measures, and in that case $u_ i\in \mathcal E_{\mathrm{env}}(C_ i,\sigma _ i,r_ i)$. Choose stretching competitors in these curve classes. The sum of their pullbacks plus $\Psi $ has positive current, since the contributions of the curves, the form $\varpi ^*\alpha $ and the current of the divisor are all positive. Adding the convex stretching inequalities, which is possible also at poles, gives a stretching competitor relative to $R$ for every finite stretch, and adding the ordinary bounds gives $U\preceq R$. Thus $U\in \mathcal E_{\mathrm{env}}(M,\Theta _\alpha ,R)$, and proposition 18.12.2 gives equality below every smooth obstacle. Taking the zero obstacle recovers the first condition, which proves the equivalence. $\square$