Fixed geodesics of involutions on $3$-spheres

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 $2\pi$, 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.