On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions

Zhengnan Chen

Paper · Sep 16, 2025

On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions

Zhengnan Chen

math.DGarXiv:2509.13183

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Abstract

For all dimensions $n \geq 5$ , let $(M, g, f)$ be a $n -$ dimensional shrinking gradient Ricci soliton with strictly positive isotropic curvature (PIC). Suppose furthermore that $\nabla^{2} f$ is $2 -$ nonnegative and the curvature tensor is WPIC1 at some point $x ˉ \in M$ . Then $(M, g)$ must be a quotient of either $S^{n}$ or $S^{n - 1} \times R$ . Our result partially extends the classification result for 4-dimensional PIC shrinking Ricci solitons established in [LNW16] to highter dimensions. Combining the pinching estimates deduced in [Chen24] we also extend the result in [CL23] to dimensions $n \geq 9$ . Namely that a complete ancient solution to the Ricci flow of dimension $n \geq 9$ with uniformly PIC must be weakly PIC2.

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Submitted: Sep 16, 2025
Subjects: math.DG

Ricci flow Ricci solitons Ancient solutions Curvature pinching 4-manifolds Sphere theorems

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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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

11A.7
Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles
Scalar and tensor maximum principles, and preserved curvature conditions.
11A.11
Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles
Positive Ricci curvature in dimension 3 flows to a round metric.
12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

[1] 11A.7
Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles
Scalar and tensor maximum principles, and preserved curvature conditions.

[2] 11A.11
Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles
Positive Ricci curvature in dimension 3 flows to a round metric.

[3] 12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions

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