Translators asymptotic to cylinders

TL;DR

This paper proves the uniqueness of the Bowl soliton as a translating solution to mean curvature flow with cylindrical asymptotics and shows that generically, all non-static translating limits are the Bowl soliton, without assuming convexity.

Contribution

It establishes the uniqueness of the Bowl soliton under weak asymptotic conditions and without convexity assumptions, extending understanding of translating solutions in mean curvature flow.

Findings

Bowl soliton is the unique translating solution with cylindrical asymptotics.

Generic mean curvature flows have the Bowl soliton as their translating limit.

No global convexity assumption is needed for the main results.

Abstract

We show that the Bowl soliton in $\mathbb{R}^3$ is the unique translating solutions of the mean curvature flow which has the family of shrinking cylinders as an asymptotic shrinker at $-\infty$. As an application, we show that for a generic mean curvature flow, all (non-static) translating limit flows are the bowl soliton. The crucial point is that we do not make any global convexity assumption, while as the same time, the asymptotic requirement is very weak.