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The Chern–Ricci flow on complex surfaces

Part of: Differential operators in several variables Global differential geometry

Published online by Cambridge University Press:  02 December 2013

Valentino Tosatti  and
Ben Weinkove
Valentino Tosatti
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA email tosatti@math.northwestern.edu
Ben Weinkove
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA email weinkove@math.northwestern.edu
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Abstract

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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.

Keywords

Chern–Ricci flowcompact complex surfaceHermitian metric

MSC classification

Secondary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 53C55: Hermitian and Kählerian manifolds 32W20: Complex Monge-Ampère operators
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 12 , December 2013 , pp. 2101 - 2138
Copyright
© The Author(s) 2013 

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