Weak solutions to degenerate complex Monge-Amp\'ere flows I

TL;DR

This paper develops a viscosity theory for degenerate complex Monge-Ampère flows in complex domains, addressing the long-term behavior of the Kähler-Ricci flow on mildly singular varieties.

Contribution

It introduces a novel viscosity framework for weak solutions of degenerate parabolic complex Monge-Ampère equations in complex Euclidean domains.

Findings

Established a viscosity solution theory for degenerate complex Monge-Ampère flows.

Provided tools for analyzing the long-term behavior of Kähler-Ricci flows.

Laid groundwork for future studies on singular varieties.

Abstract

Studying the (long-term) behavior of the K\"ahler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Amp\'ere equations. The purpose of this article, the first of a series on this subject, is to develop a viscosity theory for degenerate complex Monge-Amp\`ere flows in domains of $\C^n$.