تقرير
Which shapes can appear in a Curve Shortening Flow Singularity?
| العنوان: | Which shapes can appear in a Curve Shortening Flow Singularity? |
|---|---|
| المؤلفون: | Angenent, Sigurd, Davis, Evan Patrick, DeCleene, Ellie, Ellingson, Paige, Feng, Ziheng, Gevorgyan, Edgar, Lemmenes, Aris, Moon, Alex, Tommasi, Tyler Joseph, Zhou, Yamin |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Differential Geometry, Mathematics - Analysis of PDEs, 53E10, 35B40, 35K55 |
| الوصف: | 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 $\mathbb R^3$. 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 $\{(x, y) : |x|\leq 1, y=\pm x^{n-1}\}\cup\{(\pm 1, y) : |y|\leq 1\}$. 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. Comment: 21 pages, 14 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2403.09876 |
| رقم الأكسشن: | edsarx.2403.09876 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |