The Structure of Translating Surfaces with Finite Total Curvature

TL;DR

This paper characterizes translating surfaces in three-dimensional space with finite total curvature, showing they are planes or asymptotic to planes, and provides detailed asymptotic behavior of their ends.

Contribution

It proves that finite total curvature translators with one end are planes, and describes the asymptotic structure of multiple-ended translators.

Findings

Translators with one end are planes.

Multiple-ended translators are asymptotic to a plane.

Quantitative estimates for asymptotic behavior of ends.

Abstract

In this paper, we prove that any mean curvature flow translator $\Sigma^2 \subset \mathbb{R}^3$ with finite total curvature and one end must be a plane. We also prove that if the translator $\Sigma$ has multiple ends, they are asymptotic to a plane $\Pi$ containing the direction of translation and can be written as graphs over $\Pi$. Finally, we determine that the ends of $\Sigma$ are strongly asymptotic to $\Pi$ and obtain quantitative estimates for their asymptotic behavior.