An index formula for hemispheres of a $C^2$-regular convex closed surface in Euclidean $3$-space
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908746340368384 |
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| author | Ando, Naoya Umehara, Masaaki |
| author_facet | Ando, Naoya Umehara, Masaaki |
| contents | Carathéodory's conjecture has long been regarded as one of the central problems in the classical theory of convex surfaces. In this paper, we establish an index formula for hemispheres of convex closed surfaces under $C^2$-regularity. The proof is based on studying a vertical section of the null hypersurfaces in Lorentz--Minkowski $4$-space associated with the originally given convex surface. As a consequence, the conjecture is affirmatively solved in the $C^2$-case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23181 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An index formula for hemispheres of a $C^2$-regular convex closed surface in Euclidean $3$-space Ando, Naoya Umehara, Masaaki Differential Geometry Carathéodory's conjecture has long been regarded as one of the central problems in the classical theory of convex surfaces. In this paper, we establish an index formula for hemispheres of convex closed surfaces under $C^2$-regularity. The proof is based on studying a vertical section of the null hypersurfaces in Lorentz--Minkowski $4$-space associated with the originally given convex surface. As a consequence, the conjecture is affirmatively solved in the $C^2$-case. |
| title | An index formula for hemispheres of a $C^2$-regular convex closed surface in Euclidean $3$-space |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2512.23181 |