On a modified parabolic complex Monge–Ampère equation with applications

Abstract

We study a modified parabolic complex Monge–Ampere type equation on a complete non-compact Kahler 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 Ω and initial Kahler metric g0 on M, the modified Kahler–Ricci flow g′ = −Ric + Ω has a long time solution converging to a complete Kahler metric such that Ric = Ω, which extends the result in Cao (Invent Math 81:359–372, 1985) to non-compact manifolds. We will also obtain a long time existence result for the Kahler–Ricci flow which generalizes a result (Chau et al. in Can J Math, to appear).