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 $t\to +\infty$.