# Plurisubharmonic envelopes and supersolutions

## TL;DR

This paper systematically studies plurisubharmonic envelopes on compact Kähler manifolds and domains in complex space, extending approximation methods to connect viscosity super-solutions with pluripotential solutions of complex Monge-Ampère equations.

## Contribution

It introduces a method to relate viscosity super-solutions to pluripotential super-solutions, enabling new approaches to solving complex Monge-Ampère equations.

## Key findings

- Quasi-psh envelope of a viscosity super-solution is a pluripotential super-solution.
- Envelopes can be used to solve complex Monge-Ampère equations.
- Extension of Berman's approximation process to broader contexts.

## Abstract

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.