The Affine Shape of a Figure-Eight under the Curve Shortening Flow
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866911956087078912 |
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| author | Coiculescu, Matei P. Schwartz, Richard Evan |
| author_facet | Coiculescu, Matei P. Schwartz, Richard Evan |
| contents | We consider the curve shortening flow applied to a class of figure-eight curves: those with dihedral symmetry, convex lobes, and a monotonicity assumption on the curvature. We prove that when (non-conformal) linear transformations are applied to the solution so as to keep the bounding box the unit square, the renormalized limit converges to a quadrilateral which we call a bowtie. Along the way we prove that suitably chosen arcs of our evolving curves, when suitably rescaled, converge to the Grim Reaper Soliton under the flow. Our Grim Reaper Theorem is an analogue of a theorem of S. Angenent, which is proven in the locally convex case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_09213 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Affine Shape of a Figure-Eight under the Curve Shortening Flow Coiculescu, Matei P. Schwartz, Richard Evan Analysis of PDEs Differential Geometry We consider the curve shortening flow applied to a class of figure-eight curves: those with dihedral symmetry, convex lobes, and a monotonicity assumption on the curvature. We prove that when (non-conformal) linear transformations are applied to the solution so as to keep the bounding box the unit square, the renormalized limit converges to a quadrilateral which we call a bowtie. Along the way we prove that suitably chosen arcs of our evolving curves, when suitably rescaled, converge to the Grim Reaper Soliton under the flow. Our Grim Reaper Theorem is an analogue of a theorem of S. Angenent, which is proven in the locally convex case. |
| title | The Affine Shape of a Figure-Eight under the Curve Shortening Flow |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2106.09213 |