As on arXiv
Abstract
We study the nature of finite-time singularities for the Chern-Ricci flow, partially answering a question of Tosatti-Weinkove. We show that a solution of degenerate parabolic complex Monge-Ampère equations starting from arbitrarily positive (1,1)-currents are smooth outside some analytic subset, generalizing works by Di Nezza-Lu. We extend Guedj-Lu's recent approach to establish uniform a priori estimates for degenerate complex Monge-Ampère equations on compact Hermitian manifolds. We apply it to studying the Chern-Ricci flows on complex log terminal varieties starting from an arbitrary current.
Comments: final version, to appear in Analysis & PDE
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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Pluripotential Chern-Ricci Flows
Topics Analytic minimal model program, Generalized & G₂ flows, Kähler geometry · Same author Quang-Tuan Dang · Shared terms “Chern Ricci”, “complex Monge”, “degenerate parabolic”
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Regularity of solutions to Monge–Ampère equations on compact Hermitian manifolds
Topics Generalized & G₂ flows, Kähler geometry · Same author Quang-Tuan Dang · Shared terms “Ampère equation”, “compact Hermitian”, “Hermitian manifolds”
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Hermitian null loci
Topics Analytic minimal model program, Generalized & G₂ flows, Singularities · Same author Quang-Tuan Dang · Shared terms “partially answering”, “Tosatti”, “Chern”
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The Chern-Ricci flow
Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “Hermitian”
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Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces
Topics Analytic minimal model program, Kähler geometry, Singularities · Shared terms “complex Monge”, “Ampère equation”, “Monge Ampère”
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Convergence of the Chern-Ricci flow on complex minimal surfaces of general type
Topics Analytic minimal model program, Generalized & G₂ flows, Kähler geometry · Shared terms “Chern Ricci”, “Tosatti Weinkove”, “Hermitian”
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Singularity Types for Long-Time Chern-Ricci Flow
Topics Generalized & G₂ flows, Kähler geometry, Singularities · Shared terms “Chern Ricci”
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The estimate for parabolic complex Monge-Ampère equations
Topics Kähler geometry · Shared terms “complex Monge”, “parabolic complex”, “Ampère equation”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.4Blow-Ups, Blow-Downs and Splitting at Infinity
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.
- 11B.6Rotationally Symmetric Flows and the Standard Solution
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.