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.