§ 5.33 Ancient ovals with reduced symmetry37474A54C83A

For $n\ge 3$ and $2\le k\le n-1$, Du–Haslhofer construct $(k-1)$-parameter families of compact ancient noncollapsed mean-convex hypersurfaces in $\mathbb R^{n+1}$ with $\mathbb Z_2^ k\times O(n+1-k)$ symmetry. The parameters encode independent elongations in the $k$ axial directions. The fully symmetric limiting choices overlap the previously recorded ancient ovals; the reduced-symmetry members are the additional cases here. The slices are smooth embedded convex hypersurfaces with topology $S^ n$, complete and proper. After extinction normalization their time interval is $(-\infty ,0)$. The backward parabolic model is $\mathbb R^ k\times S^{n-k}$; this does not determine every finite-time slice or establish uniqueness from this limit alone. The smallest case is a three-dimensional hypersurface in $\mathbb R^4$, not a surface in $\mathbb R^3$. The drawing is an axial section with two distinct elongations and a labeled spherical-fiber radius. Its ellipsoidal outline is schematic and does not satisfy the ancient-flow PDE. Entropy is finite for each compact slice; no numerical value is assigned.

Sources: [DuHaslhofer2021].