Infinitely many non-collapsed steady Ricci solitons on complex line bundles

Hanci Chi

Paper · Dec 22, 2024

Infinitely many non-collapsed steady Ricci solitons on complex line bundles

Hanci Chi

math.DGarXiv:2412.16907v3

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Abstract

We construct a continuous 3-parameter family of non-shrinking Ricci solitons complex line bundles $O (k)$ over $CP^{2 m + 1}$ , where the base space is not necessarily Kähler–Einstein. Each $O (k)$ with $k \in [3, 2 m + 1]$ admits at least one asymptotically conical (AC) Ricci-flat metric in this family. For each $O (k)$ with $k \geq 3$ , the family includes infinitely many asymptotically paraboloidal (AP) steady Ricci soliton.

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Submitted: Dec 22, 2024
Last revised: Jul 27, 2026 (v3)
Subjects: math.DG

Comments: Typos fixed. Section 5 has been restructured for better exposition

Ricci solitons Einstein metrics Ricci-flat & Calabi–Yau (Non)collapsing Kähler geometry Boundaries & non-compact Kähler–Einstein & K-stability

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

9B.3
Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.
12A.4
κ-NoncollapsingBook 12A · Entropy and Noncollapsing
The theorem that excludes the cigar.
12A.8
Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing
Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.

[1] 9B.3
Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

[2] 12A.4
κ-NoncollapsingBook 12A · Entropy and Noncollapsing
The theorem that excludes the cigar.

[3] 12A.8
Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing
Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.

Infinitely many non-collapsed steady Ricci solitons on complex line bundles

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