Uniqueness of Semigraphical Translators

TL;DR

This paper proves the uniqueness of certain mean curvature flow translators in three-dimensional space, using Morse-Radó theory and maximum principles, and discusses their classification and limits.

Contribution

It establishes the first proof of the uniqueness conjecture for pitchfork and helicoid translators, advancing the understanding of their classification.

Findings

Proved the uniqueness of pitchfork and helicoid translators.

Strengthened classification and compactness results for semigraphical translators.

Applied Morse-Radó theory and maximum principles in the proof.

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.