I will present new results on the classification problem for self-translating hypersurfaces for the mean curvature flow. Such surfaces show up as Type II singularity models in the flow, and have been studied since the first examples were found by Mullins in 1956.

Examples from gluing constructions show that one cannot easily classify such solitons, nor can one classify their projections to one dimension lower, nor their convex hulls. But if one does both of these “forgetful” operations, the list becomes short, coinciding with the one given by Hoffman-Meeks in 1990 for minimal submanifolds. This also implies some known obstructions for existence, e.g. for convex self-translating solitons. Our proofs are based on an Omori-Yau maximum principle at infinity.