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Ancient solutions of the mean curvature flow (1308.4095v1)
Published 19 Aug 2013 in math.DG
Abstract: 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 Sj x Rn-j and near the tips they have asymptotic translators modeled on Bowlj+1 x Rn-j-1. We also give a characterization of the round shrinking sphere among ancient alpha-Andrews flows. Our proofs are based on the recent estimates of Haslhofer-Kleiner.