Classification of bubble-sheet ovals in $\mathbb{R}^{4}$
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866912351269158912 |
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| author | Choi, Beomjun Daskalopoulos, Panagiota Du, Wenkui Haslhofer, Robert Sesum, Natasa |
| author_facet | Choi, Beomjun Daskalopoulos, Panagiota Du, Wenkui Haslhofer, Robert Sesum, Natasa |
| contents | In this paper, we prove that any bubble-sheet oval for the mean curvature flow in $\mathbb{R}^4$, up to scaling and rigid motion, either is the $\textrm{O}(2)\times \textrm{O}(2)$-symmetric ancient oval constructed by Hershkovits and the fourth author, or belongs to the one-parameter family of $\mathbb{Z}_2^2\times \textrm{O}(2)$-symmetric ancient ovals constructed by the third and fourth author. In particular, this seems to be the first instance of a classification result for geometric flows that are neither cohomogeneity-one nor selfsimilar. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_04931 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Classification of bubble-sheet ovals in $\mathbb{R}^{4}$ Choi, Beomjun Daskalopoulos, Panagiota Du, Wenkui Haslhofer, Robert Sesum, Natasa Differential Geometry Analysis of PDEs 53E10 In this paper, we prove that any bubble-sheet oval for the mean curvature flow in $\mathbb{R}^4$, up to scaling and rigid motion, either is the $\textrm{O}(2)\times \textrm{O}(2)$-symmetric ancient oval constructed by Hershkovits and the fourth author, or belongs to the one-parameter family of $\mathbb{Z}_2^2\times \textrm{O}(2)$-symmetric ancient ovals constructed by the third and fourth author. In particular, this seems to be the first instance of a classification result for geometric flows that are neither cohomogeneity-one nor selfsimilar. |
| title | Classification of bubble-sheet ovals in $\mathbb{R}^{4}$ |
| topic | Differential Geometry Analysis of PDEs 53E10 |
| url | https://arxiv.org/abs/2209.04931 |