The Chern–Ricci flow on complex surfaces
By Valentino Tosatti, B. Weinkove. Compositio Mathematica, 2012
74 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/8dfe28108d03c0953d39996812bf1e9b46c14963
How researchers see this paper
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In Applied Mathematics
- Xiaokui Yang
- Valentino Tosatti
- Dat T. To
- Kefeng Liu
- Duong H. Phong
- B. Weinkove
- G'abor Sz'ekelyhidi
- Fang Zheng
In Geometry and Topology
- Xiaokui Yang
- Valentino Tosatti
- Dat T. To
- Kefeng Liu
- Duong H. Phong
- B. Weinkove
- G'abor Sz'ekelyhidi
- Fang Zheng
Citations in the corpus
- Non-Kähler Calabi-Yau manifolds (2014, methodology): “Furthermore, as shown by Gill [34], on such manifolds the Chern-Ricci flow [73, 74, 77] deforms any given Hermitian metric to a Chern-Ricci flat one.”
- Non-K\"ahler Calabi-Yau manifolds (2014, methodology): “Furthermore, as shown by Gill [34], on such manifolds the Chern-Ricci flow [73, 74, 77] deforms any given Hermitian metric to a Chern-Ricci flat one.”
- Ricci Curvatures on Hermitian manifolds (2014, methodology): “[45, 46, 47], [39], [24, 51, 52, 55, 25, 26]) by using the well-studied Hamilton’s Ricci flow.”
- Scalar curvature on compact complex manifolds (2017, background): “This example is studied with details in [15, 16, 21, 22, 23, 26, 27].”
- Fully non-linear parabolic equations on compact Hermitian manifolds (2017, methodology): “It is known as the Chern-Ricci flow, with the Chern-Ricci tensor Ric(ω) = −i∂∂̄ log ω playing the role of the Ricci tensor in the Kähler-Ricci flow (we refer to [54, 55, 56, 59] and references therein).”