Corollary 13.28.23017BC9CAC896

Proof. The classifications of ordinary and ideal models in lemma 13.28.1, lemma 20.3.20 show that $P_\theta [u]=P_\theta [u]_{\mathcal I}$ if and only if $\mu _ u=D_ u$. Step 1 of the proof of proposition 13.28.22 shows that this is equivalent to equality of the envelopes at every smooth obstacle. Membership of $u$ in $\mathcal E_{\mathrm{env}}(Y,\theta ,q)$ implies the latter by proposition 18.12.2.

Conversely, assume that $\mu _ u=D_ u$. By lemma 20.3.20, $\mu _ q=D_ u$. Ordinary equivalence preserves polar curvature by corollary 5.27.15, so every potential equivalent to $q$ has non-pluripolar mass $A-D_ u(Y)$, by Stokes' theorem. Hence

If $D_ u(Y)<A$, then $P_\theta [u]=P_\theta [u]_{\mathcal I}$, and proposition 13.24.2 puts $u$ in $\mathcal E_{\mathrm{env}}(Y,\theta ,q)$.

If $D_ u(Y)=A$, the positive curvature measures $T_ u$ and $T_ q$ both dominate $D_ u$ and both have total mass $A$. Hence $T_ u=T_ q=D_ u$. The distribution $u-q$ is harmonic, and therefore constant on the compact connected curve. Thus $u=q+C$ for a constant $C$, and proposition 16.28.62 gives $u\in \mathcal E_{\mathrm{env}}(Y,\theta ,q)$ also in this case. $\square$