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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2402.17060 |
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| _version_ | 1866910807630020608 |
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| author | Guckenheimaer, John |
| author_facet | Guckenheimaer, John |
| contents | Umbilics are points of a surface embedded in three space where normal curvatures are independent of direction. The (in)famous Carathéodory Conjecture states that a compact simply connected embedded surface has at least two umbilic points. A counterexample to this conjecture would be a surface whose principal foliation has index two at a single umbilic. All (purported) proofs of the Carathéodory Conjecture are based on analyses of the index of an umbilic, concluding that it is at most one. This investigation gives a much simpler geometric argument that the index of an umbilic on an analytic surface cannot be an integer larger than one, providing new insight into the Carathéodory Conjecture. The results also establish lower bounds for the index of an umbilic based on its degeneracy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17060 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Degenerate Umbilic Points of Analytic Surfaces Guckenheimaer, John Differential Geometry 53A05 Umbilics are points of a surface embedded in three space where normal curvatures are independent of direction. The (in)famous Carathéodory Conjecture states that a compact simply connected embedded surface has at least two umbilic points. A counterexample to this conjecture would be a surface whose principal foliation has index two at a single umbilic. All (purported) proofs of the Carathéodory Conjecture are based on analyses of the index of an umbilic, concluding that it is at most one. This investigation gives a much simpler geometric argument that the index of an umbilic on an analytic surface cannot be an integer larger than one, providing new insight into the Carathéodory Conjecture. The results also establish lower bounds for the index of an umbilic based on its degeneracy. |
| title | Degenerate Umbilic Points of Analytic Surfaces |
| topic | Differential Geometry 53A05 |
| url | https://arxiv.org/abs/2402.17060 |