§ 5.26 Non-self-similar flows emerging from conesEA5C07CB7D0F

Let $\Sigma $ be a smooth asymptotically conical self-expander with positive finite Morse index for the expander functional, under the weighted decay hypotheses in Chen's Theorem 1.1. The source constructs an index-parameter family of solutions ancient in expander-rescaled time $s$, converging to $\Sigma $ as $s\to -\infty $. Their small graph functions lie in the source's weighted Hölder and spectral spaces; a generic chosen perturbation of an expander is not asserted to be ancient. With ordinary time $t=e^ s$, the associated MCF is $M_ t=\sqrt t\, \widetilde M_{\log t}$ on $0<t<T$ for a suitable $T>0$. It emerges from the same asymptotic cone at $t=0$ and need not be self-similar. Ordinary negative time is not part of this assertion, nor is global existence for all positive time. In the smooth graphical interval the slices retain the topology and embeddedness of the underlying embedded expander. The picture separates the limiting cone, an expander profile, and a small unstable-mode displacement. The displacement is a schematic mode, not a computed nonlinear flow or a universal growth rate.

Sources: [Chen2026].