Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow
Sigurd Angenent
Networks & Heterogeneous Media 2013, 8(1): 1-8 doi: 10.3934/nhm.2013.8.1
We provide formal matched asymptotic expansions for ancient convex solutions to MCF. The formal analysis leading to the solutions is analogous to that for the generic MCF neck pinch in [1].
    For any $p, q$ with $p+q=n$, $p\geq1$, $q\geq2$ we find a formal ancient solution which is a small perturbation of an ellipsoid. For $t\to-\infty$ the solution becomes increasingly astigmatic: $q$ of its major axes have length $\approx\sqrt{2(q-1)(-t)}$, while the other $p$ axes have length $\approx \sqrt{-2t\log(-t)}$.
    We conjecture that an analysis similar to that in [2] will lead to a rigorous construction of ancient solutions to MCF with the asymptotics described in this paper.
keywords: Mean curvature asymptotic expansions. ancient solutions

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