Ancient and Eternal Solutions to Mean Curvature Flow from Minimal Surfaces

Abstract

We construct embedded ancient solutions to mean curvature flow related to certain classes of unstable minimal hypersurfaces in $\mathbb{R}^{n+1}$ for $n \geq 2$. These provide examples of mean convex yet nonconvex ancient solutions that are not solitons, meaning that they do not evolve by rigid motions or homotheties. Moreover, we construct embedded eternal solutions to mean curvature flow in $\mathbb{R}^{n+1}$ for $n \geq 2$. These eternal solutions are not solitons, are $O(n)\times O(1)$-invariant, and are mean convex yet nonconvex. They flow out of the catenoid and are the rotation of a profile curve which becomes infinitely far from the axis of rotation. As $t \to \infty$, the profile curves converge to a grim reaper for $n \geq 3$ and become flat for $n=2$. Concerning these eternal solutions, we also show they are asymptotically unique up to scale among the embedded $O(n)\times O(1)$-invariant, eternal solutions with uniformly bounded curvature and a sign on mean curvature.