Translating Annuli for Mean Curvature Flow

TL;DR

This paper constructs a new family of complete, embedded annular translators for mean curvature flow, characterized by specific asymptotic behaviors and widths, expanding understanding of translating solutions in geometric flows.

Contribution

It introduces a novel family of annular translators with prescribed widths and asymptotics, and establishes existence results for widths greater than or equal to π/2.

Findings

Existence of translators with inner width ≥ π/2 and specified neck size.

No such translators exist with inner width less than π/2.

Constructed translators are asymptotic to four vertical planes at minus infinity.

Abstract

We construct a family of complete, properly embedded, annular translators $M$ such that $M$ lies in a slab and is invariant under reflections in the vertical coordinate planes. Each translator in the family is asymptotic as $z\to -\infty$ to four vertical planes $\{y= \pm b\}$ and $\{y= \pm B\}$, where $0<b \le B < \infty$. We call $b$ and $B$ the inner width and the (outer) width of the translator. We show that for each $b \ge \pi/2$ and each $s>0$, there is a translator in the family with inner width $b$ and with necksize $s$. (We also show that there are no translators with inner width $<\pi/2$ having the properties of the examples we construct.)