Corollary 15.17.464893D9DABFE

Proof. For $g\in G$, pullback by $g$ preserves $\theta $-plurisubharmonicity, and $V_\theta \circ g=V_\theta $ by the definition of the envelope. Pullback also preserves the positive-exponent equation, so its uniqueness gives $u_\lambda \circ g=u_\lambda $, and the normalization preserves this equality. Pullback by a fixed smooth biholomorphism is continuous on $L^1(X)$, since its Jacobian relative to the fixed smooth volume is bounded. Hence every cluster limit is invariant almost everywhere, and therefore everywhere for the canonical representatives. This argument does not require $g$ to preserve the barrier $\rho $ or the set $\Omega $.

If the invariant solution is unique, every cluster limit is the same potential. Relative compactness then implies convergence of the whole family for uniform convergence of differences on $\Omega $: a sequence staying at a fixed positive distance from that potential would have a convergent subsequence, a contradiction. The interpolation argument above upgrades this to the asserted local $C^\beta $ convergence. $\square$