§ 5.45 Ancient flows from unstable free-boundary minimal hypersurfaces1897F09E9330

Let $M^{m+1}$ be a compact Riemannian manifold with strictly convex boundary, and let $\Sigma ^ m\subset M$ be a smooth properly embedded two-sided free-boundary minimal hypersurface of Morse index $I>0$. Bourni–Franz's 2026 preprint constructs an $I$-parameter family of ancient free-boundary mean curvature flows backward converging to $\Sigma $. The construction uses adapted normal graphs that preserve the nonlinear orthogonality condition; merely adding an eigenfunction to the position vector is not an exact free-boundary deformation. The uniqueness theorem additionally imposes the specified small graph and decay conditions; its generality must not be inferred solely from the index count. A mean-convex ancient branch is obtained from the first eigenfunction, with further uniqueness under nondegeneracy. Smooth graphical slices preserve the topology and embeddedness of $\Sigma $ and meet the fixed ambient boundary orthogonally. The figure shows a critical-catenoid-type free-boundary background and a schematic unstable displacement, not a computed ancient flow. The preprint's existence and classification are attributed results, with no formal proof verification recorded.

Sources: [BourniFranz2026].