Optimal quantitative stability of the Möbius group of the sphere in all dimensions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914639677227008 |
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| author | Guerra, André Lamy, Xavier Zemas, Konstantinos |
| author_facet | Guerra, André Lamy, Xavier Zemas, Konstantinos |
| contents | In any dimension $n\geq 3$, we prove an optimal stability estimate for the Möbius group among maps $u\colon \mathbb S^{n-1} \to \mathbb R^n$, of the form $\inf_{λ>0,ϕ\in \mathrm{Möb}(\mathbb S^{n-1})} \int_{\mathbb S^{n-1}}\left|\frac 1λ\nabla_{T} u -\nabla_{ T}ϕ\right|^{n-1} d\mathcal H^{n-1} \leq C_n \mathcal E_{n-1}(u).$ Here, $\mathcal E_{n-1}(u)$ is a conformally invariant deficit which measures simultaneously lack of conformality and the deviation of $u(\mathbb S^{n-1})$ from being a round sphere in an isoperimetric sense. This entails in particular the following qualitative statement: sequences with vanishing deficit, once appropriately normalized by the action of the Möbius group, are compact. Both the qualitative and the quantitative results are new for all dimensions $n\geq 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06593 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal quantitative stability of the Möbius group of the sphere in all dimensions Guerra, André Lamy, Xavier Zemas, Konstantinos Differential Geometry Analysis of PDEs In any dimension $n\geq 3$, we prove an optimal stability estimate for the Möbius group among maps $u\colon \mathbb S^{n-1} \to \mathbb R^n$, of the form $\inf_{λ>0,ϕ\in \mathrm{Möb}(\mathbb S^{n-1})} \int_{\mathbb S^{n-1}}\left|\frac 1λ\nabla_{T} u -\nabla_{ T}ϕ\right|^{n-1} d\mathcal H^{n-1} \leq C_n \mathcal E_{n-1}(u).$ Here, $\mathcal E_{n-1}(u)$ is a conformally invariant deficit which measures simultaneously lack of conformality and the deviation of $u(\mathbb S^{n-1})$ from being a round sphere in an isoperimetric sense. This entails in particular the following qualitative statement: sequences with vanishing deficit, once appropriately normalized by the action of the Möbius group, are compact. Both the qualitative and the quantitative results are new for all dimensions $n\geq 4$. |
| title | Optimal quantitative stability of the Möbius group of the sphere in all dimensions |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2401.06593 |