Entropy and codimension bounds for generic singularities

TL;DR

This paper establishes universal bounds on entropy and the dimension of the ambient space for generic singularities in higher codimension mean curvature flow, providing foundational limits in the study of such singularities.

Contribution

It introduces the first general bounds on entropy and subspace dimension for generic singularities in arbitrary codimension mean curvature flow.

Findings

All non-perturbable 2D singularities have bounded entropy.

Such singularities lie in low-dimensional linear subspaces.

Bounds depend linearly on the genus of the singularity.

Abstract

We show that all closed $2$-dimensional singularities for higher codimension mean curvature flow that cannot be perturbed away have uniform entropy bounds and lie in a linear subspace of small dimension. The entropy and dimension of the subspace are both $\leq C\,(1+\gamma)$ for some universal constant $C$ and genus $\gamma$. These are the first general bounds on generic singularities in arbitrary codimension.