Allen--Cahn and mean curvature flows from Clifford hypersurfaces to equators

In this short note, using OpenAI’s resolution of the Solomon–Yau conjecture, we extend our joint work with Pedro Gaspar to all dimensions $n\ge 2$. We construct eternal Allen–Cahn flows on $S^{n+1}$ connecting approximations of any prescribed least-area Clifford hypersurface and equatorial sphere. We also construct infinitely many eternal integral Brakke flows from least-area Clifford hypersurfaces to equatorial spheres.