On the evolution of a Hermitian metric by its Chern-Ricci form

Abstract

We consider the evolution of a Hermitian metric on a compact complex manifold by its Chern-Ricci form. This is an evolution equation first studied by M. Gill, and coincides with the Kahler-Ricci flow if the initial metric is Kahler. We find the maximal existence time for the flow in terms of the initial data. We investigate the behavior of the flow on complex surfaces when the initial metric is Gauduchon, on complex manifolds with negative first Chern class, and on some Hopf manifolds. Finally, we discuss a new estimate for the complex Monge-Ampere equation on Hermitian manifolds.

37 pages, v3 corrected typos, final version to appear in J. Differential Geom

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Cited by in corpus (58)

• The Monge-Ampère equation for (n-1)-plurisubharmonic functions on a compact Kähler manifold
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• Non-Kähler Calabi-Yau manifolds
• The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits
• KAWA lecture notes on the Kähler-Ricci flow
• Hermitian metrics, (n-1, n-1) forms and Monge-Ampère equations
• Hessian equations on closed Hermitian manifolds
• Inoue surfaces and the Chern-Ricci flow
• Local Calabi and curvature estimates for the Chern-Ricci flow
• Ricci flow and birational surgery
• Stability and regularity of solutions of the Monge-Ampère equation on Hermitian manifolds
• A parabolic flow of balanced metrics
• Geometric flows and their solitons on homogeneous spaces
• The Chern-Ricci flow on Oeljeklaus-Toma manifolds
• The Chern-Ricci flow on smooth minimal models of general type
• Regularizing properties of Complex Monge-Ampère flows II: Hermitian manifolds
• The Chern-Ricci flow and holomorphic bisectional curvature
• The Anomaly flow and the Fu-Yau equation
• Kähler manifolds with almost non-negative curvature
• Scalar curvature on compact complex manifolds
• Chern-Ricci flows on noncompact complex manifolds
• A parabolic Monge-Ampère type equation of Gauduchon metrics
• New curvature flows in complex geometry
• The Chern-Ricci flow
• Instantaneously complete Chern-Ricci flow and Kähler-Einstein metrics
• Griffiths positivity for Bismut curvature and its behaviour along Hermitian Curvature Flows
• Generalized complex Monge-Ampère type equations on closed Hermitian manifolds
• On a class of fully nonlinear elliptic equations on closed Hermitian manifolds
• Hermitian curvature flow on complex homogeneous manifolds
• Hermitian curvature flow on complex locally homogeneous surfaces
• Ricci Curvatures on Hermitian manifolds
• Long time existence of the (n-1)-plurisubharmonic flow
• A flow of conformally balanced metrics with Kähler fixed points
• Deformed Hermitian-Yang-Mills Equation on Compact Hermitian Manifolds
• Hermitian manifolds with semi-positive holomorphic sectional curvature
• The parabolic flows for complex quotient equations
• Levi-Civita Ricci-flat metrics on compact complex manifolds
• Minimal complex surfaces with Levi-Civita Ricci-flat metrics
• Regularity of A Complex Monge-Ampère Equation on Hermitian Manifolds
• The Dirichlet Problem for a Complex Monge-Ampere Type Equation on Hermitian Manifolds
• Geometric formalities along the Chern-Ricci flow
• Parabolic Flow for Generalized complex Monge-Ampère type equations
• Weak Solutions of the Chern-Ricci flow on compact complex surfaces
• Leafwise flat forms on Inoue-Bombieri surfaces
• Lie-Algebraic Curvature Conditions preserved by the Hermitian Curvature Flow
• The transverse Chern-Ricci flow
• Chern-Ricci curvatures, holomorphic sectional curvature and Hermitian metrics
• Parabolic complex Monge-Ampère equations on compact Hermitian manifolds
• Kahler-Ricci flow on blowups along submanifolds
• The continuity equation, Hermitian metrics and elliptic bundles
• Hermitian manifolds with quasi-negative curvature
• Integral inequalities for holomorphic maps and applications
• The continuity equation for Hermitian metrics: Calabi estimates, Chern scalar curvature and Oeljeklaus-Toma manifolds
• Lower order tensors in non-Kähler geometry and non-Kähler geometric flow
• The Kähler-Ricci flow on compact Kähler manifolds
• Deformation of Hermitian metrics
• Hermitian Curvature Flow on compact homogeneous spaces
• The complex Monge-Ampère equation on some compact Hermitian manifolds