The Pluripotential Cauchy-Dirichlet problem for the Complex Monge-Amp\`ere flow with a general measure on the right-hand side

TL;DR

This paper proves the existence of solutions to the complex Monge-Ampère flow with a general measure on the right-hand side, extending previous results by removing positivity constraints and applying to parabolic pluripotential theory.

Contribution

It establishes solvability of the pluripotential Cauchy-Dirichlet problem for the Monge-Ampère flow with a broad class of measures, including non-strictly positive ones, and extends Kolodziej's subsolution theorem to the parabolic setting.

Findings

Solvability of the Monge-Ampère flow with general measures.

Removal of strict positivity assumption on the measure.

Extension of subsolution theorem to parabolic pluripotential theory.

Abstract

We show that the pluripotential Cauchy-Dirichlet problem for the complex Monge-Amp\`ere flow is solvable for the right-hand side of the form $dt \wedge d\mu$ where $d\mu$ is dominated by a Monge-Amp\`ere measure of a bounded plurisubharmonic function. In particular, we remove the strict positivity assumption on $d\mu$. We use this result to prove the parabolic version of the bounded subsolution theorem due to Kolodziej in pluripotential theory.