Weak Solutions to the complex Monge-Amp\`ere flows on compact K\"ahler manifolds : general measures on the right-hand side

TL;DR

This paper establishes the existence and uniqueness of bounded solutions to the complex Monge-Ampère flow on compact Kähler manifolds with general measure data, and demonstrates regularity properties of these solutions.

Contribution

It introduces new existence and comparison principles for solutions with measure data dominated by Monge-Ampère measures, extending previous results in complex geometry.

Findings

Existence of bounded solutions for the Monge-Ampère flow with general measure data.

Local Hölder continuity of the solution slices on the ample locus.

A comparison principle ensuring uniqueness of solutions.

Abstract

We show the existence of a bounded solution to the Cauchy problem for the complex Monge-Amp\`ere flow on a compact K\"ahler manifold, with the right-hand side of the form $dt \wedge d\mu$ where $d\mu$ is dominated by a Monge-Amp\`ere measure of a H\"older continuous quasi-plurisubharmonic function. We also prove that for a given semi-positive big from $\theta$, the $t$-slice of the solution is locally H\"older continuous on $\rm{Amp(\theta)}$ for all $t \in (0, T)$. Next, we prove a comparison principle when $d\mu$ is dominated by a Monge-Amp\`ere measure of a bounded quasi-plurisubharmonic function, which implies the uniqueness of the solution.