Paper · Nov 2, 2026

Rigidity of shrinking Ricci solitons with low-dimensional focal sets

Matheus Andrade Ribeiro de Moura Horácio

math.DGarXiv:2611.02740
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Abstract

We prove that, in every dimension, the focal set of the potential function of a complete gradient shrinking Ricci soliton with constant scalar curvature is totally geodesic whenever it has dimension two or three, and that every complete five-dimensional gradient shrinking Ricci soliton with constant scalar curvature is rigid. The first result follows from a Bochner identity for the covariant derivative of the Ricci tensor at the focal set. For the second, when the scalar curvature is twice the soliton constant, we prove that total geodesy implies rigidity through an integral identity on the level hypersurfaces; the case of scalar curvature three times the soliton constant is treated by a separate argument. As further applications, we combine our methods with the rigidity theorem in the recent preprint of Li, Li, Ou, Qu, Wu and Zhou to obtain rigidity in arbitrary dimension when the scalar curvature is twice or three times the soliton constant, and under integral Weyl pinching for four-dimensional focal sets.

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Submitted
Nov 2, 2026
Subjects
math.DG

Comments: 28 pages

Ricci solitonsCurvature pinching4-manifoldsScalar curvatureStability & uniqueness

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  1. Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set

    Fengjiang Li, Wenqi Li, Jianyu Ou et al.

    Topics Scalar curvature, Stability & uniqueness, Ricci solitons · Shared terms “focal set”, “constant scalar”, “totally geodesic”

  2. Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature

    Matheus Horácio, Wenqi Li, Jianyu Ou et al.

    Topics Curvature pinching, Scalar curvature, Stability & uniqueness · Same author Matheus Andrade Ribeiro de Moura Horácio · Shared terms “constant scalar”, “Ricci tensor”

  3. Gradient Shrinking Ricci Solitons and Modified Sectional Curvature

    Xiaodong Cao, Ernani Ribeiro and Hosea Wondo

    Topics Curvature pinching, 4-manifolds, Scalar curvature · Shared terms “shrinking Ricci”, “gradient shrinking”, “dimensional gradient”

  4. On Ricci solitons whose level hypersurfaces have parallel second fundamental form

    Matheus Andrade Ribeiro de Moura Horácio

    Topics Scalar curvature, Ricci solitons · Same author Matheus Andrade Ribeiro de Moura Horácio · Shared terms “level hypersurfaces”, “shrinking Ricci”, “gradient shrinking”

  5. A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu

    Topics 4-manifolds, Scalar curvature, Stability & uniqueness · Shared terms “constant scalar”, “shrinking Ricci”, “Zhou”

  6. Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al.

    Topics Scalar curvature, Stability & uniqueness, Ricci solitons · Shared terms “five dimensional”, “constant scalar”, “shrinking Ricci”

  7. Rigidity of nonsteady gradient Kähler-Ricci solitons with constant scalar curvature

    Matheus Andrade Ribeiro de Moura Horácio

    Topics Scalar curvature, Stability & uniqueness, Ricci solitons · Same author Matheus Andrade Ribeiro de Moura Horácio · Shared terms “constant scalar”, “Ricci tensor”

  8. A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu

    Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “constant scalar”, “shrinking Ricci”, “Zhou”

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.

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) 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) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 6A.9
    Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives

    Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.

  2. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.

  3. 11A.10
    Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles

    Why high curvature in dimension 3 is almost positive.