A characterization of the grim reaper cylinder

TL;DR

This paper characterizes certain translating solitons in three-dimensional space, showing they are either flat or identical to the grim reaper cylinder under specific geometric conditions.

Contribution

It provides a classification theorem for properly embedded translating solitons with bounded genus and specific asymptotic behavior, identifying them as either flat or grim reaper cylinders.

Findings

Connected, properly embedded solitons with bounded genus are either flat or grim reaper cylinders.

Such solitons asymptotic to two planes outside a cylinder are classified.

The result advances understanding of translating solitons in geometric analysis.

Abstract

In this article we prove that a connected and properly embedded translating soliton in $\mathbb{R}^3$ with uniformly bounded genus on compact sets which is $C^1$-asymptotic to two planes outside a cylinder, either is flat or coincides with the grim reaper cylinder.