2013, 8(1): 1-8. doi: 10.3934/nhm.2013.8.1

Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow

1. 

Department of Mathematics, UW Madison, Madison, WI53705, United States

Received  March 2012 Revised  October 2012 Published  April 2013

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.
Citation: Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 1-8. doi: 10.3934/nhm.2013.8.1
References:
[1]

S. B. Angenent and J. J. L. Velázquez, Degenerate neckpinches in mean curvature flow,, J. Reine Angew. Math., 482 (1997), 15.

[2]

Sigurd Angenent, Cristina Caputo and Dan Knopf, Minimally invasive surgery for Ricci flow singularities,, J. Reine Angew. Math., 672 (2012).

[3]

Sigurd Angenent, Shrinking doughnuts,, Nonlinear diffusion equations and their equilibrium states, 7 (1992), 21.

[4]

Panagiota Daskalopoulos, Richard Hamilton and Natasa Sesum, Classification of compact ancient solutions to the curve shortening flow,, J. Differential Geom., 84 (2010), 455.

[5]

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves,, J. Differential Geom., 23 (1986), 69.

show all references

References:
[1]

S. B. Angenent and J. J. L. Velázquez, Degenerate neckpinches in mean curvature flow,, J. Reine Angew. Math., 482 (1997), 15.

[2]

Sigurd Angenent, Cristina Caputo and Dan Knopf, Minimally invasive surgery for Ricci flow singularities,, J. Reine Angew. Math., 672 (2012).

[3]

Sigurd Angenent, Shrinking doughnuts,, Nonlinear diffusion equations and their equilibrium states, 7 (1992), 21.

[4]

Panagiota Daskalopoulos, Richard Hamilton and Natasa Sesum, Classification of compact ancient solutions to the curve shortening flow,, J. Differential Geom., 84 (2010), 455.

[5]

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves,, J. Differential Geom., 23 (1986), 69.

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