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# Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow

• 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.
Mathematics Subject Classification: Primary: 53C44, 35K59; Secondary: 35C20.

 Citation:

•  [1] S. B. Angenent and J. J. L. Velázquez, Degenerate neckpinches in mean curvature flow, J. Reine Angew. Math., 482 (1997), 15-66. [2] Sigurd Angenent, Cristina Caputo and Dan Knopf, Minimally invasive surgery for Ricci flow singularities, J. Reine Angew. Math., 672 (2012), 39–87(to appear). [3] Sigurd Angenent, Shrinking doughnuts, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), Progr. Nonlinear Differential Equations Appl., 7 Birkhäuser Boston, Boston, MA, (1992), 21-38. [4] Panagiota Daskalopoulos, Richard Hamilton and Natasa Sesum, Classification of compact ancient solutions to the curve shortening flow, J. Differential Geom., 84 (2010), 455-464. [5] M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.