On mean curvature flow translators with prescribed ends

Given a smooth closed embedded self-shrinker $S$ with index $I$ in ${\mathbb{R}}^n$, we construct an $I$-dimensional family of complete translators polynomially asymptotic to $S\times \mathbb{R}$ at infinity, which answers a long-standing question by Ilmanen. We further prove that ${\mathbb{R}}^{n+1}$ can be decomposed in many ways into a one-parameter family of closed sets ${\coprod }_{a\in \mathbb{R}}{T}_a$, and each closed set $T_a$ contains a complete translator asymptotic to $S\times \mathbb{R}$ at infinity. If the closed set $T_a$ fattens, namely it has nonempty interior, then there are at least two translators asymptotic to each other at an exponential rate, which can be viewed as a kind of nonuniqueness. We show that this fattening phenomenon is non-generic but indeed happens.