Ancient mean curvature flows out of polytopes (Q2105819)

Ancient solutions for the mean curvature flow are those solutions which have existed for an infinite amount of time in the past. In the present paper, the authors construct a new large class of solutions emerging from circumscribed polytopes at time \(-\infty\). They confirm a well-known conjecture attributed to R. S. Hamilton (see [\textit{G. Huisken} and \textit{C. Sinestrari}, J. Differ. Geom. 101, No. 2, 267--287 (2015; Zbl 1332.53085)]). A conjecture of \textit{B. White} [J. Am. Math. Soc. 16, No. 1, 123--138 (2003; Zbl 1027.53078)] is resolved in the negative. A partial classification of convex ancient solutions in slab regions is obtained. Furthermore, a new rigidity result for translators is presented.