The Chern-Ricci flow on complex surfaces

Valentino Tosatti, Ben Weinkove

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)2101-2138
Number of pages38
JournalCompositio Mathematica
Issue number12
StatePublished - 2013


  • Chern-Ricci flow
  • Hermitian metric
  • compact complex surface

ASJC Scopus subject areas

  • Algebra and Number Theory


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