Collapsing of the Chern-Ricci flow on elliptic surfaces

TL;DR

This paper studies the behavior of the Chern-Ricci flow on elliptic surfaces, showing it collapses fibers and converges to a Kähler-Einstein metric, with implications for non-Kähler surfaces.

Contribution

It demonstrates the collapsing behavior of the Chern-Ricci flow on elliptic surfaces and establishes convergence to a Kähler-Einstein metric, extending understanding beyond Kähler cases.

Findings

Flow collapses elliptic fibers

Metrics converge to pullback of Kähler-Einstein metric

Results apply to non-Kähler surfaces of Kodaira dimension one

Abstract

We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics generalizing the Kahler-Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a Kahler-Einstein metric from the base. Some of our estimates are new even for the Kahler-Ricci flow. A consequence of our result is that, on every minimal non-Kahler surface of Kodaira dimension one, the Chern-Ricci flow converges in the sense of Gromov-Hausdorff to an orbifold Kahler-Einstein metric on a Riemann surface.