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1 results

  • The Chern–Ricci flow on complex surfaces

  • Part of
  • Valentino Tosatti, Ben Weinkove
  • Journal:
    Compositio Mathematica / Volume 149 / Issue 12 / December 2013
    Published online by Cambridge University Press:
    02 December 2013, pp. 2101-2138
    Print publication:
    December 2013
    • Article
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  • 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.