UC San Diego

We prove C ∞ convergence for suitably normalized solutions of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds. This provides a parabolic proof of a recent result of Tosatti-Weinkove. Additionally, let X = M x E where M is an m-dimensional Kähler manifold with negative first Chern class and E is an n-dimensional complex torus. We obtain C ∞ convergence of the normalized Kähler-Ricci flow on X to a Kähler-Einstein metric on M. This strengthens a convergenc°e result of Song-Weinkove and confirms their conjecture