High Codimension Mean Curvature Flow with Surgery
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866912756487159808 |
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| author | Lynch, Stephen Nguyen, Huy The |
| author_facet | Lynch, Stephen Nguyen, Huy The |
| contents | We construct a mean curvature flow with surgery for submanifolds of arbitrary codimension. The theory applies to closed submanifolds satisfying a natural quadratic pinching condition, which serves as the high-codimension analogue of 2-convexity and is preserved under the flow in dimensions $n \geq 8$. Our results therefore are in line with the current state-of-the-art in codimension one (where at present 2-convexity is required for surgery). Central to our analysis is a collection of new a priori estimates for the second fundamental form, uniform across surgeries, which yield a precise description of high-curvature regions and permit controlled surgeries. This provides the first notion of mean curvature flow through singularities with topological control in higher codimensions. As a consequence we obtain a sharp classification: Every closed quadratically 2-convexity submanifold is diffeomorphic either to $\mathbb{S}^n$ or to a finite connected sum of $\mathbb{S}^{n-1}$-bundles over $\mathbb{S}^1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_07163 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | High Codimension Mean Curvature Flow with Surgery Lynch, Stephen Nguyen, Huy The Differential Geometry 53C44 We construct a mean curvature flow with surgery for submanifolds of arbitrary codimension. The theory applies to closed submanifolds satisfying a natural quadratic pinching condition, which serves as the high-codimension analogue of 2-convexity and is preserved under the flow in dimensions $n \geq 8$. Our results therefore are in line with the current state-of-the-art in codimension one (where at present 2-convexity is required for surgery). Central to our analysis is a collection of new a priori estimates for the second fundamental form, uniform across surgeries, which yield a precise description of high-curvature regions and permit controlled surgeries. This provides the first notion of mean curvature flow through singularities with topological control in higher codimensions. As a consequence we obtain a sharp classification: Every closed quadratically 2-convexity submanifold is diffeomorphic either to $\mathbb{S}^n$ or to a finite connected sum of $\mathbb{S}^{n-1}$-bundles over $\mathbb{S}^1$. |
| title | High Codimension Mean Curvature Flow with Surgery |
| topic | Differential Geometry 53C44 |
| url | https://arxiv.org/abs/2004.07163 |