Which shapes can appear in a Curve Shortening Flow Singularity?
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arXiv
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| Autori principali: | , , , , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913265983946752 |
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| author | Angenent, Sigurd Davis, Evan Patrick DeCleene, Ellie Ellingson, Paige Feng, Ziheng Gevorgyan, Edgar Lemmenes, Aris Moon, Alex Tommasi, Tyler Joseph Zhou, Yamin |
| author_facet | Angenent, Sigurd Davis, Evan Patrick DeCleene, Ellie Ellingson, Paige Feng, Ziheng Gevorgyan, Edgar Lemmenes, Aris Moon, Alex Tommasi, Tyler Joseph Zhou, Yamin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09876 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Which shapes can appear in a Curve Shortening Flow Singularity? Angenent, Sigurd Davis, Evan Patrick DeCleene, Ellie Ellingson, Paige Feng, Ziheng Gevorgyan, Edgar Lemmenes, Aris Moon, Alex Tommasi, Tyler Joseph Zhou, Yamin Differential Geometry Analysis of PDEs 53E10, 35B40, 35K55 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. |
| title | Which shapes can appear in a Curve Shortening Flow Singularity? |
| topic | Differential Geometry Analysis of PDEs 53E10, 35B40, 35K55 |
| url | https://arxiv.org/abs/2403.09876 |