Uniqueness and short time regularity of the weak K\"ahler-Ricci flow

TL;DR

This paper proves that the weak K"ahler-Ricci flow on compact K"ahler manifolds is smooth outside an analytic subset and establishes conditions for uniqueness and regularity based on initial current properties.

Contribution

It demonstrates the optimal regularity of the flow and establishes uniqueness when starting from currents with zero Lelong numbers.

Findings

Flow is smooth outside an analytic subset for arbitrary initial currents.

Flow has positive Lelong numbers for short time if initial currents do.

Uniqueness holds when initial currents have zero Lelong numbers.

Abstract

Let $X$ be a compact K\"ahler manifold. We prove that the K\"ahler-Ricci flow starting from arbitrary closed positive $(1,1)$-currents is smooth outside some analytic subset. This regularity result is optimal meaning that the flow has positive Lelong numbers for short time if the initial current does. We also prove that the flow is unique when starting from currents with zero Lelong numbers.