PomeCorollary 15.36.11.E75339160354 Every separated threshold subsolution $r=r(y)$ with $r\in \mathrm{PSH}(Y,\kappa )$ and $\mathrm{MA}^{\mathrm{ext}}_ H(r)\geq bq_\alpha \mu $ equals $u_\alpha +d$ for some constant $d$. In particular $u_\alpha $ is its unique supremum-zero member. If $\alpha <k/(k+1)$, then $u_\alpha -M_\alpha $ is the greatest such subsolution whose $q_\alpha $-mean on $Y$ is nonpositive.
15.36.10 Order in the separated classA874D9C61D74
We use here the closed-class uniqueness theorem: on a compact connected Kähler manifold, two potentials whose non-pluripolar measures have full mass and are equal differ by a constant. This is an external input; its source is identified in 25.13.
Proof. The product formula reduces the inequality to $\mathrm{MA}^{\mathrm{ext}}_\kappa (r)\geq q_\alpha \kappa ^ k$. Closed Stokes gives each bounded truncation mass one, so the extended measure of $r$ has mass at most one. The inequality therefore gives equality and full mass. Closed-class uniqueness yields $r=u_\alpha +d$. In the finite-moment range its mean is $M_\alpha +d$, which is nonpositive exactly when $d\leq -M_\alpha $. $\square$
At and beyond the moment boundary every constant translate has mean $-\infty $, so that mean condition no longer chooses a greatest translate. The order assertion concerns separated potentials. Selection for arbitrary integrable densities and uniqueness of arbitrary critical potentials on the nonclosed product remain the questions of 24.2.2 and section 15.34.
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