Rigidity and non-existence results for collapsed translators

TL;DR

This paper establishes a uniqueness result for certain mean curvature self-translating solitons, specifically identifying the grim reaper cylinder as the only finite entropy example under specific conditions.

Contribution

It provides a rigidity theorem characterizing the grim reaper cylinder among self-translating solitons with finite entropy in a bounded width slab.

Findings

The grim reaper cylinder is uniquely characterized as the only finite entropy self-translating 2-surface in the specified setting.

The proof utilizes parabolicity in a weighted context and a universally L-superharmonic function.

The result rules out the existence of other such solitons under the given conditions.

Abstract

We prove a rigidity result for mean curvature self-translating solitons, characterizing the grim reaper cylinder as the only finite entropy self-translating 2-surface in $\mathbb{R}^3$ of width $\pi$ and bounded from below. The proof makes use of parabolicity in a weighted setting applied to a suitable universally $L$-superharmonic function defined on translaters in such slabs.