Proof of the Toponogov Conjecture on Complete Surfaces
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2020
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| Acceso en línea: | |
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| _version_ | 1866929518658191360 |
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| author | Guilfoyle, Brendan Klingenberg, Wilhelm |
| author_facet | Guilfoyle, Brendan Klingenberg, Wilhelm |
| contents | We prove a conjecture of Toponogov on complete convex planes, namely that such planes must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value problem and an existence result for holomorphic discs with Lagrangian boundary conditions, both of which apply to a putative counterexample.
Corollaries of the main theorem include a Hawking-Penrose singularity-type theorem, as well as the proof of a conjecture of Milnor's from 1965 in the convex case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_12787 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Proof of the Toponogov Conjecture on Complete Surfaces Guilfoyle, Brendan Klingenberg, Wilhelm Differential Geometry 53A05 We prove a conjecture of Toponogov on complete convex planes, namely that such planes must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value problem and an existence result for holomorphic discs with Lagrangian boundary conditions, both of which apply to a putative counterexample. Corollaries of the main theorem include a Hawking-Penrose singularity-type theorem, as well as the proof of a conjecture of Milnor's from 1965 in the convex case. |
| title | Proof of the Toponogov Conjecture on Complete Surfaces |
| topic | Differential Geometry 53A05 |
| url | https://arxiv.org/abs/2002.12787 |