Formal asymptotic expansions for symmetric ancient ovalsin mean curvature flow

Sigurd Angenent

doi:10.3934/nhm.2013.8.1

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2013, Volume 8, Issue 1: 1-8. Doi: 10.3934/nhm.2013.8.1

This issue Previous Article Preface Next Article A nonlinear partial differential equation for the volume preserving mean curvature flow

Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow

Sigurd Angenent^1,

1.
Department of Mathematics, UW Madison, Madison, WI53705

Received: February 29, 2012

Revised: September 30, 2012

Published: February 2013

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Abstract

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.

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

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References

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