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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 areasymptotic to a plane $\Pi$ containing the direction of translation and can bewritten as graphs over $\Pi$. Finally, we determine that the ends of $\Sigma$are strongly asymptotic to $\Pi$ and obtain quantitative estimates for theirasymptotic behavior.

The Structure of Translating Surfaces with Finite Total Curvature