Rotational symmetry of solutions of mean curvature flow coming out of a double cone II

TL;DR

This paper proves that rotational symmetry is preserved in certain mean curvature flows originating from symmetric double cones with low entropy, and constructs examples of non-self-similar flows with finite-time singularities.

Contribution

It establishes symmetry preservation for mean curvature flows from symmetric cones and provides explicit examples of non-self-similar flows with singularities.

Findings

Rotational symmetry is maintained in flows with entropy ≤ 2 from symmetric cones.

Existence of non-self-similar flows with finite-time singularities.

Construction of explicit examples demonstrating flow behaviors.

Abstract

We show that any integral Brakke flow coming out of a rotationally symmetric double cone with entropy at most two must stay rotationally symmetric for all time, provided the flow is smooth for a short time. We also show the existence of a non-self-similar flow coming out of a double cone with entropy at most two, and give an example of such a flow with a finite time singularity.