We make a systematic study of (quasi-)plurisubharmonic envelopes on compact K\"ahler manifolds, as well as on domains of $\mathbb{C}^n$, by using and extending an approximation process due to Berman [Ber13]. We show that the quasi-psh envelope of a viscosity super-solution is a pluripotential super-solution of a given complex Monge-Amp\`ere equation. We use these ideas to solve complex Monge-Amp\`ere equations by taking lower envelopes of super-solutions.

中文翻译：

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多次谐波包络和超解

我们通过使用和扩展 Berman [Ber13] 的近似过程，对紧致 K\"ahler 流形以及 $\mathbb{C}^n$ 域上的（准）多次谐波包络进行了系统研究。我们证明了粘度超解的准 psh 包络是给定复杂 Monge-Amp\`ere 方程的多势超解。我们使用这些想法通过采用下包络来求解复杂的 Monge-Amp\`ere 方程的超级解决方案。