The structure of translating surfaces with finite total curvature

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.