Nguyen's Tridents and the Classification of Semigraphical Translators for Mean Curvature Flow

TL;DR

This paper constructs a new family of periodic solutions to mean curvature flow that resemble grim reaper surfaces and provides a near-complete classification of semigraphical translators, advancing understanding of their structure.

Contribution

It introduces a one-parameter family of semigraphical translating solutions and offers a comprehensive classification of such translators.

Findings

Constructed a family of periodic translating solutions converging to a grim reaper and a plane.

Provided a near-complete classification of semigraphical translators.

Demonstrated properties of these solutions, including proper embedding and graph-like structure after removing vertical lines.

Abstract

We construct a one-parameter family of singly periodic translating solutions to mean curvature flow that converge as the period tends to $0$ to the union of a grim reaper surface and a plane that bisects it lengthwise. The surfaces are semigraphical: they are properly embedded, and, after removing a discrete collection of vertical lines, they are graphs. We also provide a nearly complete classification of semigraphical translators.