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# Communications in Analysis and Geometry

## Volume 24 (2016)

### Number 3

### Ancient solutions of the mean curvature flow

Pages: 593 – 604

DOI: http://dx.doi.org/10.4310/CAG.2016.v24.n3.a6

#### Authors

#### Abstract

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.