The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampere flows

TL;DR

This paper develops a new pluripotential theory for degenerate parabolic complex Monge-Ampère equations in strongly pseudoconvex domains, establishing existence, uniqueness, and regularity of solutions modeled on the Kähler-Ricci flow.

Contribution

It introduces the first steps of a parabolic pluripotential theory in bounded strongly pseudoconvex domains, extending the understanding of complex Monge-Ampère flows with applications to algebraic geometry.

Findings

The envelope of pluripotential subsolutions is semi-concave in time.

The solution is continuous in space.

Uniqueness of the pluripotential solution is established.

Abstract

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{\`e}re equations, modelled on the K{\"a}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural assumptions on the Cauchy-Dirichlet boundary data, we show that the envelope of pluripotential subsolutions is semi-concave in time and continuous in space, and provides the unique pluripotential solution with such regularity.