  The Chern-Ricci flow is an evolution equation of Hermitian metrics by their
Chern-Ricci form, first introduced by Gill. Building on our previous work, we
investigate this flow on complex surfaces. We establish new estimates in the
case of finite time non-collapsing, anologous to some known results for the
Kahler-Ricci flow. This provides evidence that the Chern-Ricci flow carries out
blow-downs of exceptional curves on non-minimal surfaces. We also describe
explicit solutions to the Chern-Ricci flow for various non-Kahler surfaces. On
Hopf surfaces and Inoue surfaces these solutions, appropriately normalized,
collapse to a circle in the sense of Gromov-Hausdorff. For non-Kahler properly
elliptic surfaces, our explicit solutions collapse to a Riemann surface.
Finally, we define a Mabuchi energy functional for complex surfaces with
vanishing first Bott-Chern class and show that it decreases along the
Chern-Ricci flow.
