Canonical foliation of bubblesheets
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915029865988096 |
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| author | Lagacé, Jean Lynch, Stephen |
| author_facet | Lagacé, Jean Lynch, Stephen |
| contents | We introduce a new curvature condition for high-codimension submanifolds of a Riemannian ambient space, called quasi-parallel mean curvature (QPMC). The class of submanifolds with QPMC includes all CMC hypersurfaces and submanifolds with parallel mean curvature. We use our notion of QPMC to prove that certain kinds of high-curvature regions which appear in geometric flows, called bubblesheets, can be placed in a suitable normal form. This follows from a more general result asserting that the manifold $\mathbb{R}^k \times \mathbb{S}^{n-k}$, equipped with any metric which is sufficiently close to the standard one, admits a canonical foliation by embedded $(n-k)$-spheres with QPMC. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14340 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Canonical foliation of bubblesheets Lagacé, Jean Lynch, Stephen Differential Geometry Analysis of PDEs 53C21, 53C42, 53E10, 53E20 We introduce a new curvature condition for high-codimension submanifolds of a Riemannian ambient space, called quasi-parallel mean curvature (QPMC). The class of submanifolds with QPMC includes all CMC hypersurfaces and submanifolds with parallel mean curvature. We use our notion of QPMC to prove that certain kinds of high-curvature regions which appear in geometric flows, called bubblesheets, can be placed in a suitable normal form. This follows from a more general result asserting that the manifold $\mathbb{R}^k \times \mathbb{S}^{n-k}$, equipped with any metric which is sufficiently close to the standard one, admits a canonical foliation by embedded $(n-k)$-spheres with QPMC. |
| title | Canonical foliation of bubblesheets |
| topic | Differential Geometry Analysis of PDEs 53C21, 53C42, 53E10, 53E20 |
| url | https://arxiv.org/abs/2411.14340 |