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).

References

It is promising to have a positive answer to the following conjecture: A mean curvature flow with generic initial data in $R{n+1}$ (or more generally, a general complete manifold with bounded geometry), only develops nondegenerate cylindrical singularities or spherical singularities in finite time.

— Passing through nondegenerate singularities in mean curvature flows  (2501.16678 - Sun et al., 28 Jan 2025) in Introduction, Why nondegenerate cylindrical singularities? (Conjecture \ref{conj:genericMCF})

The type-I assumption is a technical requirement in our proof. We expect that it is not necessary, but we do not have a proof here.

— Mean curvature flows with cylindrical singularities II: stability and genericity  (2609.19390 - Sun et al., 16 Sep 2026) in Section 1, immediately following Theorem 1.4 (Denseness of nondegenerate singularities)

White Section 5 conjectured that mean curvature flows in $R3$ can only have the singular set consisting of isolated singularities and curves.

— Mean curvature flows with cylindrical singularities II: stability and genericity  (2609.19390 - Sun et al., 16 Sep 2026) in Section 1, paragraph following the thin-tube discussion

The firecracker singular set exists in mean curvature flow, but does not exist generically.

— Mean curvature flows with cylindrical singularities II: stability and genericity  (2609.19390 - Sun et al., 16 Sep 2026) in Section 1, subsection beginning “Finally, we discuss a possibly existing conjectural example called firecracker”