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Ancient collapsed solutions of mean curvature flow

Theodora Bourni, University of Tennessee
Tuesday, May 14, 2019 - 1:30pm
PDL C-401

We will focus on ancient compact convex solutions of mean curvature flow that lie in slab regions. In particular we will show the existence and uniqueness of an ancient pancake, that is, a compact, convex ancient solution of mean curvature flow in \$\mathbb{R}^{n+1}\$ with \$O(1)\times O(n)\$ symmetry that lies in a slab of width \$\pi\$. Moreover, we will discuss the role of translating solutions in constructing ancient solutions and show, in all dimensions \$n\geq 2\$, the existence of convex translators lying in slabs of width \$\pi\sec\theta\$ in \$\mathbb{R}^{n+1}\$ (and in no smaller slab) for \$\theta\in[0,\frac{\pi}{2}]\$. Finally, in the case of curve shortening flow, we will show that our techniques can be used to classify all convex ancient solutions. In the compact case, this yields a different proof of the classification theorem of Hamilton, Daskalopoulos and Sesum that states that the only compact convex ancient solutions are the shrinking circle and the Angenent oval. This work is joint with Mat Langford and Giuseppe Tinaglia.

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