On a modified parabolic complex Monge-Amp\`{e}re equation with applications

TL;DR

This paper investigates a modified parabolic complex Monge-Ampère equation on noncompact Kähler manifolds, establishing short-term existence, estimates, and long-term convergence of the Kähler-Ricci flow to a metric satisfying Ric = Ω.

Contribution

It extends the existence and convergence results of the Kähler-Ricci flow to noncompact manifolds under new assumptions, generalizing prior compact cases.

Findings

Proved short-time existence and basic estimates for the modified Monge-Ampère equation.

Established long-time existence and convergence of the Kähler-Ricci flow to a Ricci form equal to Ω.

Extended previous results to the setting of noncompact Kähler manifolds.

Abstract

We study a parabolic complex Monge-Amp\`{e}re type equation of the form \eqref{MA} on a complete noncompact \K manifold. We prove a short time existence result and obtain basic estimates. Applying these results, we prove that under certain assumptions on a given real and closed (1,1) form $\Omega$ and initial \K metric $g_0$ on $M$, the modified \KR flow $g'=-\Ric+\Omega$ has a long time smooth solution converging to a complete \K metric such that $\Ric=\Omega$, which extends the result in [1] to non-compact manifolds. We will also obtain a long time existence result for the \KR flow which generalizes a result [5].