Collapsing of the Chern-Ricci flow on elliptic surfaces

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.

46 pages, final version with minor changes to the presentation, to appear in Math. Ann

References in corpus (2)

• The Kähler-Ricci flow on surfaces of positive Kodaira dimension
• The Chern-Ricci flow on complex surfaces

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• Stability and regularity of solutions of the Monge-Ampère equation on Hermitian manifolds
• The Chern-Ricci flow on Oeljeklaus-Toma manifolds
• The Chern-Ricci flow and holomorphic bisectional curvature
• Regularizing properties of Complex Monge-Ampère flows II: Hermitian manifolds
• A parabolic Monge-Ampère type equation of Gauduchon metrics
• Hermitian curvature flow on complex locally homogeneous surfaces
• Leafwise flat forms on Inoue-Bombieri surfaces
• The continuity equation for Hermitian metrics: Calabi estimates, Chern scalar curvature and Oeljeklaus-Toma manifolds