A Finiteness Theorem Via the Mean Curvature Flow with Surgery

In this article, we use the recently developed mean curvature flow with surgery for 2-convex hypersurfaces to prove certain isotopy existence and finally extrinsic finiteness results (in the spirit of Cheeger’s finiteness theorem) for the space of 2-convex closed embedded hypersurfaces in \({\mathbb {R}}^{n+1}\).