As on arXiv
Abstract
We compare the Chern holonomy of the first and second Chern-Ricci flows on compact complex manifolds. For the first flow, we construct an explicit immortal pluriclosed solution on a complex two-torus with full and restricted holonomy , except at one finite time when both groups are ; its smooth limit has holonomy . The solution is uniformly nondegenerate and has bounds of every fixed differential order for all time. In contrast, the second Chern-Ricci flow preserves full and restricted Chern holonomy under the natural unitary identification, including the initial time and nonclosed groups. The proof combines preservation of a connection reduction, positive-time analyticity, and a path-transport comparison. As an application, the optimal minimum mean curvature on a holomorphic bundle is unchanged when metrics are restricted to any realizable full and restricted holonomy class. An explicit projected-curvature source explains the failure of the corresponding Ricci-flow holonomy argument.
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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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The Chern-Ricci flow
Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “complex manifold”
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The behavior of the second Ricci flow on complex parallelizable manifolds
Topics Generalized & G₂ flows · Shared terms “Chern”, “compact complex”, “complex manifold”
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Singularity Types for Long-Time Chern-Ricci Flow
Topics Generalized & G₂ flows, Long-time existence & convergence, Singularities · Shared terms “Chern Ricci”
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Chern-Ricci flow on Kato surfaces
Topics Generalized & G₂ flows, Long-time existence & convergence · Shared terms “Chern Ricci”, “immortal”
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Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow
Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “complex manifold”, “positive time”
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Singularities of the Chern-Ricci flow
Topics Generalized & G₂ flows, Singularities · Shared terms “Chern Ricci”, “finite time”
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Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “pluriclosed solution”, “preserves”
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Pluripotential Chern-Ricci Flows
Topics Generalized & G₂ flows, Singularities · Shared terms “Chern Ricci”
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.