| Résume | We will discuss rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we will show that any free boundary minimal hypersurface of Morse index $I$ admits an $I$-parameter family of ancient solutions that emanate from it. Moreover, among ancient solutions that backward converge exponentially fast to the minimal hypersurface, these exhaust all possibilities. Additionally, we will show how to construct a smooth free boundary mean convex foliation around an unstable free boundary minimal hypersurface. As an application, we use it to provide a more detailed geometric description of mean-convex ancient solutions that backward converge to that minimal surface. This is joint work with Giada Franz. |
