Which shapes can appear in a Curve Shortening Flow Singularity?
arxiv(2024)
摘要
We study possible tangles that can occur in singularities of solutions to
plane Curve Shortening Flow. We exhibit solutions in which more complicated
tangles with more than one self-intersection disappear into a singular point.
It seems that there are many examples of this kind and that a complete
classification presents a problem similar to the problem of classifying all
knots in ℝ^3. As a particular example, we introduce the so-called
n-loop curves, which generalize Matt Grayson's Figure-Eight curve, and we
conjecture a generalization of the Coiculescu-Schwarz asymptotic bow-tie
result, namely, a vanishing n-loop, when rescaled anisotropically to fit a
square bounding box, converges to a "squeezed bow-tie," i.e. the curve {(x,
y) : |x|≤ 1, y=± x^n-1}∪{(± 1, y) : |y|≤ 1}. As evidence in
support of the conjecture, we provide a formal asymptotic analysis on one hand,
and a numerical simulation for the cases n=3 and n=4 on the other.
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