Classification of Semigraphical Translators

TL;DR

This paper completes the classification of semigraphical translators for mean curvature flow in three-dimensional space, establishing non-existence, uniqueness, and properties of specific solutions using advanced mathematical techniques.

Contribution

It proves the non-existence of certain boundary value solutions and establishes the uniqueness of pitchfork and helicoid translators in the classification.

Findings

No solutions with alternating infinite boundary values on the upper half-plane.

Uniqueness of pitchfork and helicoid translators.

Application of Morse-Radó theory and angular maximum principle.

Abstract

We complete the classification of semigraphical translators for mean curvature flow in $\mathbb{R}^3$ that was initiated by Hoffman-Mart\'in-White. Specifically, we show that there is no solution to the translator equation on the upper half-plane with alternating positive and negative infinite boundary values, and we prove the uniqueness of pitchfork and helicoid translators. The proofs use Morse-Rad\'o theory for translators and an angular maximum principle.