Mean curvature flow with surgeries of two-convex hypersurfaces
Inspired by Hamilton's Ricci flow with surgery, the authors define a way to extend the mean curvature flow beyond singularities. The idea is to stop the flow close to the singularity time and replace the regions with high curvature by more regular ones, dropping in fact the maximum curvature by a fixed factor. Simultaneously, one also removes the connected components of the hypersurface which are diffeomorphic to \({\mathbb{S}}^n\) or \({\mathbb{S}}^{n-1} \times {\mathbb{S}}^1\). The main theorem shows that for an immersed smooth \(2\)-convex hypersurface of dimension three or greater, the mean curvature flow with surgery exists and terminates in a finite number of steps when the hypersurface is diffeomorphic to either \({\mathbb{S}}^n\) or to a finite connected sum of \({\mathbb{S}}^{n-1} \times {\mathbb{S}}^1\). Very rigourously, the paper treats convexity estimates under surgery, the replacement of necks, geometric cylinders, by two spherical caps and a needed gradient estimate. The latter differs from previous estimates through the fact that the constants do not depend on the maximum of the curvature on the hypersurface, despite its pointwise character. The existence of an algorithm describing the mean curvature with surgery, as well as the finite number of steps of the algorithm concluded by the main result, combine very technical parts.
scientific article; zbMATH DE number 5530518