Singular Rotational Self-Similar Tori for Odd ${\sigma }_k$-Curvature Flows

For every pair of integers $3\le k<n$ with $k$ odd, we construct a compact embedded rotational torus in ${\mathbb{R}}^{n+1}$ whose homothetic dilations satisfy the unnormalised ${\sigma }_k$-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has Hölder regularity $C^{1,1/k}$ and Sobolev regularity $W^{2,p}$ for every $1\le p<k/(k-1)$. Away from two singular latitudes the torus is smooth; globally, the flow equation is interpreted using the weak shape operator of the associated Lipschitz boundary. Under rotational symmetry, the self-similar equation $⟨X,\nu ⟩=-{\sigma }_k$, where $X$ is the position vector and $\nu$ is the unit normal, reduces to a degenerate profile system. We solve this system by combining an odd-power desingularisation, a shooting argument, uniform radial and axial bounds, and a strict gap between the shooting parameters and the cylindrical radius. No classical $C^2$ rotational torus can satisfy the soliton equation, so the loss of regularity is unavoidable within the rotational toroidal class.