Simply connected translating solitons contained in slabs

TL;DR

This paper proves that certain 2D translating solitons in a slab with low entropy are convex and classifies them, highlighting the necessity of the entropy bound through new examples.

Contribution

It establishes convexity of low-entropy, simply connected translating solitons in slabs and classifies all such convex solutions, extending previous results.

Findings

Low-entropy solitons are mean convex and convex.

Complete classification of convex, simply connected solitons in slabs.

Existence of examples showing the entropy bound is sharp.

Abstract

In this work we show that $2$-dimensional, simply connected, translating solitons of the mean curvature flow embedded in a slab of $\mathbb{R}^3$ with entropy strictly less than $3$ must be mean convex and thus, thanks to a result by J. Spruck and L. Xiao, are convex. Recently, such $2$-dimensional convex translating solitons have been completely classified by Hoffman, Ilmanen, Mart\'in and White, up to an ambient isometry, as vertical plane, (tilted) grim reaper cylinders, $\Delta$-wings and bowl translater. These are all contained in a slab, except for the rotationally symmetric bowl translater. New examples show that the bound on the entropy is necessary.