Plurisubharmonic envelopes and supersolutions (Q2331238)

The authors present a systematic study of (quasi-)plurisubharmonic envelopes on compact Kähler manifolds. Let \((X,\omega)\) be an \(n\)-dimensional compact Kähler manifold and let \(\theta\) be a closed smooth \((1,1)\)-form on \(X\). The main results are the following three theorems. \begin{itemize} \item Let \(v\) be a viscosity supersolution to a complex Monge-Ampère equation \((\theta+dd^cu)^n=e^ufdV\). Then the (quasi-)plurisubharmonic envelope \(P(v):=(\sup\{u\in\mathcal{PSH}(X, \theta): u\leq h\})^\ast\) is a pluripotential supersolution to this equation. \item Let \(\mu\) be a non-pluripolar positive measure and let \(v:X\longrightarrow\mathbb{R}\) be a bounded Borel measurable function. Let \(\varphi_j\in\mathcal E^1(X,\omega)\) be the unique solution to the complex Monge-Ampère equation \((\omega+dd^c\varphi_j)^n = e^{j(\varphi_j-v)}\mu\). Then \((\varphi_j)_{j=1}^\infty\) converges in capacity to the \((\omega,\mu)\)-envelope \(P_{\omega,\mu}(v):=(\sup\{\varphi\in\mathcal{PSH}(X, \omega) : \varphi\leq v\; \mu\text{-a.e. on }X\})^\ast\). \item Let \(\mu\) be a non-pluripolar Radon measure in some open set \(\Omega\subset X\). Assume there exists a finite energy subsolution \(u_0\in\mathcal E(X,\omega)\), \((\omega+dd^cu_0)^n\geq e^{u_0}\mu\) in \(\Omega\). Then the envelope \(\varphi:=P\big(\inf\{\psi\in\mathcal E(X, \omega): (\omega+dd^c\psi)^n\leq\boldsymbol{1}_\Omega e^\psi\mu\}\big)\) is the unique pluripotential solution of \((\omega + dd^c\varphi)^n=\boldsymbol{1}_\Omega e^{\varphi}\mu\). \end{itemize}