Degenerate complex Monge-Amp\`ere flows on strictly pseudoconvex domains

TL;DR

This paper investigates a complex Monge-Ampère flow equation on strictly pseudoconvex domains, exploring its connection to the Kähler-Ricci flow with smooth boundary conditions and bounded initial data.

Contribution

It provides new insights into the behavior of degenerate complex Monge-Ampère flows in pseudoconvex domains, extending understanding of their boundary and initial conditions.

Findings

Established existence and regularity results for the flow

Connected the flow to Kähler-Ricci flow dynamics

Analyzed boundary behavior under smooth conditions

Abstract

We study the equation $\dot{u}=\log\det (u_{\alpha\bar{\beta}})+f(t,z,u)$ in domains of $\mathbb C^n$. This equation has a close connection with the K\"ahler-Ricci flow. In this paper, we consider the case of the boundary conditions are smooth and the initial conditions are bounded.