Critical behavior of collapsing surfaces

TL;DR

This paper investigates the critical phenomena in the mean curvature flow of rotationally symmetric surfaces, revealing a threshold for singularity formation akin to gravitational collapse, and models the limiting flow with a translating soliton.

Contribution

It identifies a critical initial surface leading to a degenerate neckpinch and models the singularity's limiting flow using a rotationally symmetric translating soliton.

Findings

Detection of a critical initial surface causing a degenerate neckpinch

Numerical evidence of Type II singularity formation

Modeling of the singularity's limiting flow with a translating soliton

Abstract

We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the existence of a critical initial surface that develops a degenerate neckpinch. The limiting flow of the Type II singularity is accurately modeled by the rotationally symmetric translating soliton.