Weak Solutions to the complex Monge-Ampère flows on compact Kähler manifolds : general measures on the right-hand side

Abstract

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

34 pages