Communications in Analysis and Geometry

Volume 24 (2016)

Number 3

Ancient solutions of the mean curvature flow

Pages: 593 – 604



Robert Haslhofer (Department of Mathematics, University of Toronto, Ontario, Canada)

Or Hershkovits (Courant Institute of Mathematical Sciences, New York University, New York, N.Y., U.S.A.)


In this short article, we prove the existence of ancient solutions of the mean curvature flow that for $t \to 0$ collapse to a round point, but for $t \to -\infty$ become more and more oval: near the center they have asymptotic shrinkers modeled on round cylinders $S^j \times \mathbb{R}^{n-j}$ and near the tips they have asymptotic translators modeled on $\mathrm{Bowl}^{j+1}\times \mathbb{R}^{n-j-1}$. We also obtain a characterization of the round shrinking sphere among ancient $\alpha$-Andrews flows, and logarithmic asymptotics.

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