Chern's Conjecture in the Dupin case
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917975786782720 |
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| author | Miyaoka, Reiko |
| author_facet | Miyaoka, Reiko |
| contents | Chern's conjecture states that a closed minimal hypersurface in the euclidean sphere is isoparametric if it has constant scalar curvature. When the number $g$ of distinct principal curvatures is greater than three, few satisfactory results have been known. We attack the conjecture in the Dupin hypersurface case. Our results are: A closed proper Dupin hypersurface with constant mean curvature is isoparametric (i) if $g=3$,
(ii) if $g=4$ and has constant scalar curvature, or (iii) if $g=4$ and has constant Lie curvature, and
(iv) if $g=6$ and has constant Lie curvatures. These cover all the non-trivial cases for a closed proper Dupin to be isoparametric since $g$ can take only values $1,2,3,4,6$. The originality of the proof is a use of topology and geometry,
which reduces assumptions needed in the algebraic argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02621 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chern's Conjecture in the Dupin case Miyaoka, Reiko Differential Geometry Chern's conjecture states that a closed minimal hypersurface in the euclidean sphere is isoparametric if it has constant scalar curvature. When the number $g$ of distinct principal curvatures is greater than three, few satisfactory results have been known. We attack the conjecture in the Dupin hypersurface case. Our results are: A closed proper Dupin hypersurface with constant mean curvature is isoparametric (i) if $g=3$, (ii) if $g=4$ and has constant scalar curvature, or (iii) if $g=4$ and has constant Lie curvature, and (iv) if $g=6$ and has constant Lie curvatures. These cover all the non-trivial cases for a closed proper Dupin to be isoparametric since $g$ can take only values $1,2,3,4,6$. The originality of the proof is a use of topology and geometry, which reduces assumptions needed in the algebraic argument. |
| title | Chern's Conjecture in the Dupin case |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2504.02621 |