The Chern-Ricci flow on complex surfaces

TL;DR

This paper studies the behavior of the Chern-Ricci flow on complex surfaces, establishing new estimates, explicit solutions, and a decreasing energy functional, revealing how the flow collapses certain surfaces to lower-dimensional spaces.

Contribution

It provides new estimates for the flow, explicit solutions for non-Kahler surfaces, and introduces a Mabuchi energy functional decreasing along the flow.

Findings

Flow collapses Hopf and Inoue surfaces to a circle.

Flow collapses non-Kahler properly elliptic surfaces to a Riemann surface.

Mabuchi energy decreases along the flow.

Abstract

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.