Ancient solutions of the mean curvature flow

In this short article, we prove the existence of ancient solutions of the mean curvature flow that for t -> 0 collapse to a round point, but for t -> -infinity become more and more oval: near the center they have asymptotic shrinkers modeled on round cylinders S-j x Rn-j and near the tips they have asymptotic translators modeled on Bowl(j+1) x Rn-j-1. We also obtain a characterization of the round shrinking sphere among ancient alpha-Andrews flows, and logarithmic asymptotics.