We show that mean curvature flow translators may exhibit non-removable singularities at infinity, due to jump discontinuities in their asymptotic profiles, and that oscillation can persist so as to yield a continuum of subsequential limit tangent planes. Nonetheless, we prove that as time $t\to\pm \infty$, any finite entropy, finite genus, embedded, collapsed translating soliton in $\mathbb{R}^3$ converges to a uniquely determined collection of planes. This requires global analysis of quasilinear soliton equations with non-perturbative drifts, which we analyze via sharp non-standard elliptic decay estimates for the drift Laplacian, implying improvements on the Evans-Spruck and Ecker-Huisken estimates in the soliton setting, and exploiting a link from potential theory of the Yukawa equation to heat flows with $L^\infty$-data on non-compact slice curves of these solitons. The structure theorem follows: such solitons decompose at infinity into standard regions asymptotic to planes or grim reaper cylinders. As one application, we classify collapsed translators of entropy two with empty limits as $t\to +\infty$. This is a joint work with E. S. Gama and F. Martín.

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).