§ 5.40 Eternal tori in perturbations of the round three-sphereEF033313C0BB

Magaño–Smith prove that there is a generic subset $\mathcal U\subset C^\infty (S^3)$ such that for each $u\in \mathcal U$ there is $\epsilon >0$ with the following property: for every $0<|\delta |<\epsilon $, the fixed metric $g_\delta =e^{2\delta u}g_{\mathrm{round}}$ admits a nontrivial eternal mean curvature flow of embedded tori. The flow parameter ranges over all $t\in \mathbb R$ and is distinct from the small metric parameter $\delta $. The construction perturbs trajectories of the area functional on the space of Clifford tori, using a Morse–Smale argument and a parabolic correction. It is not an explicit radius formula, and the theorem does not assert such a flow in the unperturbed round metric $\delta =0$. Each regular slice is a compact, complete, proper embedded surface of genus one. The figure depicts changing normal graphs over a Clifford-torus background in a labeled projection, with the ambient metric held fixed. The plotted deformations are schematic; they do not compute the generic conformal factor or solve the correction equation. No Euclidean Gaussian area or entropy is assigned.

Sources: [MaganoSmith2020].