# Convergence of the weak K\"ahler-Ricci Flow on manifolds of general type

## TL;DR

This paper proves that the normalized K"ahler-Ricci flow on compact K"ahler manifolds with big canonical bundle exists for a long time, remains continuous on a large set, and converges to a unique singular K"ahler-Einstein metric, using a new viscosity theory for degenerate Monge-Ampère flows.

## Contribution

Develops a viscosity theory for degenerate complex Monge-Ampère flows in big classes, extending previous approaches, and applies it to analyze the convergence of the K"ahler-Ricci flow on manifolds of general type.

## Key findings

- Long-time existence of the flow in the viscosity sense
- Flow converges to a unique singular K"ahler-Einstein metric
- Continuity of the flow on a Zariski open set

## Abstract

We study the K\"ahler-Ricci flow on compact K\"ahler manifolds whose canonical bundle is big. We show that the normalized K\"ahler-Ricci flow has long time existence in the viscosity sense, is continuous in a Zariski open set, and converges to the unique singular K\"ahler-Einstein metric in the canonical class. The key ingredient is a viscosity theory for degenerate complex Monge-Amp\`ere flows in big classes that we develop, extending and refining the approach of Eyssidieux-Guedj-Zeriahi.