Computationally-assisted proof of a novel $\mathsf{O}(3)\times \mathsf{O}(10)$-invariant Einstein metric on $S^{12}$
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929610473603072 |
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| author | Buttsworth, Timothy Hodgkinson, Liam |
| author_facet | Buttsworth, Timothy Hodgkinson, Liam |
| contents | We prove existence of a non-round Einstein metric $g$ on $S^{12}$ that is invariant under the usual cohomogeneity one action of $\mathsf{O}(3)\times\mathsf{O}(10)$ on $S^{12}\subset \mathbb{R}^{13}= \mathbb{R}^3\oplus \mathbb{R}^{10}$. The proof involves using several rigorous numerical analysis techniques to produce a Riemannian metric $\hat{g}$ which approximately satisfies the Einstein condition to known high precision, and then demonstrating that $\hat{g}$ can be perturbed into a true Einstein metric $g$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01184 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computationally-assisted proof of a novel $\mathsf{O}(3)\times \mathsf{O}(10)$-invariant Einstein metric on $S^{12}$ Buttsworth, Timothy Hodgkinson, Liam Differential Geometry 53C25, 65D15 We prove existence of a non-round Einstein metric $g$ on $S^{12}$ that is invariant under the usual cohomogeneity one action of $\mathsf{O}(3)\times\mathsf{O}(10)$ on $S^{12}\subset \mathbb{R}^{13}= \mathbb{R}^3\oplus \mathbb{R}^{10}$. The proof involves using several rigorous numerical analysis techniques to produce a Riemannian metric $\hat{g}$ which approximately satisfies the Einstein condition to known high precision, and then demonstrating that $\hat{g}$ can be perturbed into a true Einstein metric $g$. |
| title | Computationally-assisted proof of a novel $\mathsf{O}(3)\times \mathsf{O}(10)$-invariant Einstein metric on $S^{12}$ |
| topic | Differential Geometry 53C25, 65D15 |
| url | https://arxiv.org/abs/2412.01184 |