# Analysis of Vel$\acute{a}$zquez's solution to the mean curvature flow with a type $\mathrm{II}$ singularity

## TL;DR

This paper proves that the rescaled mean curvature flow near a type II singularity converges smoothly to a minimal hypersurface, providing detailed blow-up rates and confirming the singularity model proposed by Velázquez.

## Contribution

It establishes local smooth convergence of the rescaled flow to the minimal hypersurface and refines the understanding of the singularity's blow-up behavior.

## Key findings

- Rescaled flow converges smoothly to the minimal hypersurface.
- Mean curvature blows up at a rate slower than the second fundamental form.
- Confirms the singularity model of Velázquez with detailed convergence analysis.

## Abstract

J.J.L. Vel$\acute{a}$zquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type $\mathrm{II}$ singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled flow converges in the $C^{0}$ sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this paper, we prove that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type $\mathrm{II}$ singularity. In addition, we show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form.