Mean curvature flow self-translating solitons are minimal hypersurfaces for a certain incomplete conformal background metric, and are among the possible singularity models for the flow. In the collapsed case, they are confined to slabs in space. The simplest non-trivial such example, the grim reaper curve $\Gamma$ in $\mathbb{R}^2$, has been known since 1956, as an explicit ODE-solution, which also easily gave its uniqueness. We consider here the case of surfaces, where the rigidity result for $\Gamma\times\mathbb{R}$ that we'll show this: The grim reaper cylinder is the unique (up to rigid motions) finite entropy unit speed self-translating surface which has width equal to $\pi$ and is bounded from below. (Joint w/ Impera \& Rimoldi.) Time permitting, we'll also discuss recent uniqueness results in the collapsed simply-connected low entropy case (joint w/ Gama \& Martín), using Morse theory and nodal set techniques, which extend Chini's classification.

We show that a wedge theorem (also called a bi-halfspace theorem) holds for properly immersed ancient solutions to the mean curvature flow in n-dimensional Euclidean space. This adds to a long story, as it generalizes our own wedge theorem for self-translators from 2018, which implies the minimal surface case by Hoffman-Meeks (1990) that in turn contains the classical cone theorem by Omori (1967). Another application of the wedge theorem is to classify the convex hulls of the sets of reach of all proper ancient flows, hence posing restrictions on the possible singularities that can occur in mean curvature flow. The proof uses a parabolic Omori-Yau maximum principle for proper ancient flows. This is joint work with Francesco Chini (Univ. Copenhagen).