Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916358775635968 |
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| author | Vogiatzi, Artemis A. |
| author_facet | Vogiatzi, Artemis A. |
| contents | We prove a sharp quartic curvature pinching for the mean curvature flow in $\mathbb{S}^{n+m}$, $m\ge2$, which generalises Pu's work on the convergence of submanifolds in $\mathbb{S}^{n+m}$ to a round point. Using a blow up argument, we prove a codimension and a cylindrical estimate, where in regions of high curvature, the submanifold becomes approximately codimension one, quantitatively, and is weakly convex and moves by translation or is a self shrinker. With a decay estimate, the rescaling converges smoothly to a totally geodesic limit in infinite time, without using Stampacchia iteration or integral analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08022 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere Vogiatzi, Artemis A. Differential Geometry Analysis of PDEs We prove a sharp quartic curvature pinching for the mean curvature flow in $\mathbb{S}^{n+m}$, $m\ge2$, which generalises Pu's work on the convergence of submanifolds in $\mathbb{S}^{n+m}$ to a round point. Using a blow up argument, we prove a codimension and a cylindrical estimate, where in regions of high curvature, the submanifold becomes approximately codimension one, quantitatively, and is weakly convex and moves by translation or is a self shrinker. With a decay estimate, the rescaling converges smoothly to a totally geodesic limit in infinite time, without using Stampacchia iteration or integral analysis. |
| title | Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2408.08022 |