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arXiv · 2610.29796

Fixed geodesics of involutions on $3$-spheres

Bin Shen·Hui Zhang
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Abstract

Let an isometric involution act on a smooth Riemannian $3$-sphere with sectional curvature at least one. We prove that every one-dimensional connected component of its fixed-point set has length at most $2π$, 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.

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Bin Shen, Hui Zhang. 2026-10-21. Fixed geodesics of involutions on $3$-spheres. https://arxiv.org/abs/2610.29796

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