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 ≥ 1, q ≥ 2 we find a formal ancient solution which is a small perturbation of an ellipsoid. For t→ −∞ the solution becomes increasingly astigmatic: q of its major axes have length ≈ √ 2(q − 1)(−t), while the other p axes have length ≈ √ −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. In honor of Hiroshi Matano’s sixtieth birthday and a belated thank you for his absolutely wonderful lectures given in Amsterdam, 1983.