Generic singularities in mean curvature flow

Establish whether, for mean curvature flow of hypersurfaces in Euclidean space R^{n+1} (or, more generally, on complete manifolds with bounded geometry), generic initial data produce only nondegenerate cylindrical singularities or spherical singularities in finite time.

Background

The paper proves detailed geometric and topological behavior for flows passing through nondegenerate cylindrical singularities and argues that such singularities are locally generic. Building on prior genericity results in low dimensions and under low-entropy assumptions, the authors propose extending this paradigm to fully generic initial data in all dimensions.

A positive resolution would considerably broaden the applicability of their main theorem, enabling topological conclusions without additional assumptions and aligning mean curvature flow singularity models with the most stable shrinkers (spheres and generalized cylinders).