High Codimension Curve Shortening Flow with Free Boundary
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910031517057024 |
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| author | Nguyen, Huy The Vogiatzi, Artemis A. |
| author_facet | Nguyen, Huy The Vogiatzi, Artemis A. |
| contents | We study curve shortening flow in high codimension for arcs with free boundary meeting a fixed smooth barrier orthogonally. We prove dilation-invariant curvature and higher-derivative estimates up to the boundary using a Stahl-type localised maximum principle and an adapted cut-off. Using a reflected Gaussian entropy and blow-up analysis, Type I boundary singularities yield a shrinking semicircle model after reflection. Type II blow-ups give a Grim Reaper translator, which is ruled out under a free-boundary entropy bound $<2$. Hence in the low-entropy regime the flow either converges to the orthogonal chord or has only semicircle boundary singularities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20865 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | High Codimension Curve Shortening Flow with Free Boundary Nguyen, Huy The Vogiatzi, Artemis A. Differential Geometry Analysis of PDEs We study curve shortening flow in high codimension for arcs with free boundary meeting a fixed smooth barrier orthogonally. We prove dilation-invariant curvature and higher-derivative estimates up to the boundary using a Stahl-type localised maximum principle and an adapted cut-off. Using a reflected Gaussian entropy and blow-up analysis, Type I boundary singularities yield a shrinking semicircle model after reflection. Type II blow-ups give a Grim Reaper translator, which is ruled out under a free-boundary entropy bound $<2$. Hence in the low-entropy regime the flow either converges to the orthogonal chord or has only semicircle boundary singularities. |
| title | High Codimension Curve Shortening Flow with Free Boundary |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2602.20865 |