# Geometric pluripotential theory on K\"ahler manifolds

## TL;DR

This survey explores the development of finite energy pluripotential theory on Kähler manifolds, highlighting its metric geometry and variational approach to solving complex Monge-Ampère equations.

## Contribution

It provides a comprehensive overview of recent advances in the metric and variational aspects of pluripotential theory on Kähler manifolds.

## Key findings

- Potential spaces have rich metric structures.
- Variational methods lead to existence results for Monge-Ampère equations.
- Infinite dimensional convex optimization frameworks are effective.

## Abstract

Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Amp\`ere type arising in K\"ahler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of infinite dimensional convex optimization, yielding existence results for solutions of the underlying complex Monge-Amp\`ere equations. The purpose of this survey is to describe these developments from basic principles.