The Chern-Ricci flow on complex surfaces

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.

45 pages

References in corpus (1)

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

Cited by in corpus (18)

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