Paper · Oct 21, 2026

Fixed geodesics of involutions on -spheres

Bin Shen, Hui Zhang

math.DGarXiv:2610.29796
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Abstract

Let an isometric involution act on a smooth Riemannian -sphere with sectional curvature at least one. We prove that every one-dimensional connected component of its fixed-point set has length at most , and that the bound is sharp. The proof realizes the fixed circle as the link of a complete totally geodesic surface in a positively curved expanding Ricci soliton and applies Bishop–Gromov comparison to its area growth. The analytic input is an equivariant version of Deruelle's smoothing of positively curved cones. We use Gianniotis–Schulze's equivariant local deformation and explain a marked compactness argument for arbitrary finite isotropy. In particular, the end chart retains the specified group action, rather than only an unspecified conjugate.

Ricci solitonsLimit spaces & RCD

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    Bin Shen PDF

    Topics Limit spaces & RCD, Ricci solitons · Same author Bin Shen · Shared terms “Gromov”

  2. Ricci flow from spaces with edge type conical singularities

    Lucas Lavoyer PDF

    Topics Limit spaces & RCD, Ricci solitons · Shared terms “positively curved”, “curved cone”, “curved expanding”

  3. Li-Yau Estimates for a Nonlinear Parabolic Equation on Finsler Manifolds

    Bin Shen and Yuhan Zhu PDF

    Topics Limit spaces & RCD, Ricci solitons · Same author Bin Shen · Shared terms “comparison”

  4. Rigidity of the round cylinders in Ricci shrinkers

    Yu Li and Bing Wang PDF

    Topics Limit spaces & RCD, Ricci solitons · Shared terms “Gromov”

  5. Rigidity of complex projective spaces in Ricci shrinkers

    Yu Li and Wenjia Zhang PDF

    Topics Limit spaces & RCD, Ricci solitons · Shared terms “Gromov”

  6. Long-Time Behavior of the -Ricci Flow on the 2-Torus

    Yifan Guo and Xiaokang Wang PDF

    Topics Limit spaces & RCD, Ricci solitons · Shared terms “Gromov”

  7. Notes on scalar curvature lower bounds of steady gradient Ricci solitons

    Shota Hamanaka PDF

    Topics Limit spaces & RCD, Ricci solitons · Shared terms “Gromov”

  8. Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition

    Ling Wu PDF

    Topics Limit spaces & RCD, Ricci solitons

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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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.