Regularizing properties of Complex Monge-Amp\`ere flows

TL;DR

This paper investigates how complex Monge-Ampère flows on Kähler manifolds regularize initial data with zero Lelong number, proving immediate smoothness, uniqueness, and stability of solutions.

Contribution

It establishes the immediate regularization, uniqueness, and stability of solutions to complex Monge-Ampère flows with specific initial conditions on Kähler manifolds.

Findings

Solutions become immediately smooth

Uniqueness of solutions is proven

Stability of solutions is established

Abstract

We study the regularizing properties of complex Monge-Amp\`ere flows on a K\"ahler manifold $(X,\omega)$ when the initial data are $\omega$-psh functions with zero Lelong number at all points. We prove that the general Monge-Amp\`ere flow has a solution which is immediately smooth. We also prove the uniqueness and stability of solution.