On the evolution of a Hermitian metric by its Chern-Ricci form

We consider the evolution of a Hermitian metric on a compact complex manifold by its Chern-Ricci form. This is an evolution equation first studied by M. Gill, and coincides with the Kähler-Ricci flow if the initial metric is Kähler. We find the maximal existence time for the flow in terms of the initial data. We investigate the behavior of the flow on complex surfaces when the initial metric is Gauduchon, on complex manifolds with negative first Chern class, and on some Hopf manifolds. Finally, we discuss a new estimate for the complex Monge-Ampère equation on Hermitian manifolds.