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, analogous to some known results for the Kähler-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-Kähler surfaces. On Hopf surfaces and Inoue surfaces these solutions, appropriately normalized, collapse to a circle in the sense of Gromov-Hausdorff. For non-Kähler 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.
Original language | English (US) |
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Pages (from-to) | 2101-2138 |
Number of pages | 38 |
Journal | Compositio Mathematica |
Volume | 149 |
Issue number | 12 |
DOIs | |
State | Published - 2013 |
Keywords
- Chern-Ricci flow
- Hermitian metric
- compact complex surface
ASJC Scopus subject areas
- Algebra and Number Theory