Uniqueness of the $[\varphi,\vec{e}_{3}]$-catenary cylinders by their asymptotic behaviour

TL;DR

This paper proves the uniqueness of certain $[,oldsymbol{e}_3]$-catenary cylinders based on their asymptotic behavior, extending known results for grim reaper translating solitons in mean curvature flow.

Contribution

It extends the uniqueness results for $[,oldsymbol{e}_3]$-catenary cylinders by applying the moving plane method and maximum principles, broadening the class of asymptotic behaviors considered.

Findings

Uniqueness of $[,oldsymbol{e}_3]$-catenary cylinders established.

Extended the class of asymptotic behaviors where uniqueness holds.

Applied Alexandrov's moving plane method and maximum principles effectively.

Abstract

We establish a uniqueness result for the $[\varphi,\vec{e}_{3}]$-catenary cylinders by their asymptotic behaviour. Well known examples of such cylinders are the grim reaper translating solitons for the mean curvature flow. For such solitons, F. Mart\'in, J. P\'erez-Garc\'ia, A. Savas-Halilaj and K. Smoczyk proved that, if $\Sigma$ is a properly embedded translating soliton with locally bounded genus, and $\mathcal{C}^{\infty}$-asymptotic to two vertical planes outside a cylinder, then $\Sigma$ must coincide with some grim reaper translating soliton. In this paper, applying the moving plane method of Alexandrov together with a strong maximum principle for elliptic operators, we increase the family of $[\varphi,\vec{e}_{3}]$-minimal graphs where these types of results hold under different assumption of asymptotic behaviour.