Uniqueness of Semigraphical Translators

Abstract

We prove a conjecture by Hoffman, White, and the first author regarding the uniqueness of pitchfork and helicoid translators of the mean curvature flow in $\mathbb{R}^3$. We employ an arc-counting argument motivated by Morse-Rad\'o theory for translators and a rotational maximum principle. Applications to the classification of semigraphical translators in $\mathbb{R}^3$ and their limits are discussed, strengthening compactness results of the first author with Hoffman-White and with Gama-Moller.