On the stability of the Yamabe invariant of $S^3$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917580433784832 |
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| author | Mazurowski, Liam Yao, Xuan |
| author_facet | Mazurowski, Liam Yao, Xuan |
| contents | Let $g$ be a complete, asymptotically flat metric on $\mathbb{R}^3$ with vanishing scalar curvature. Moreover, assume that $(\mathbb{R}^3,g)$ supports a nearly Euclidean $L^2$ Sobolev inequality. We prove that $(\mathbb{R}^3,g)$ must be close to Euclidean space with respect to the $d_p$-distance defined by Lee-Naber-Neumayer. We then discuss some consequences for the stability of the Yamabe invariant of $S^3$. More precisely, we show that if such a manifold $(\mathbb{R}^3,g)$ carries a suitably normalized, positive solution to $Δ_g w + λw^5 = 0$ then $w$ must be close, in a certain sense, to a conformal factor that transforms Euclidean space into a round sphere. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00815 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the stability of the Yamabe invariant of $S^3$ Mazurowski, Liam Yao, Xuan Differential Geometry 53C21 Let $g$ be a complete, asymptotically flat metric on $\mathbb{R}^3$ with vanishing scalar curvature. Moreover, assume that $(\mathbb{R}^3,g)$ supports a nearly Euclidean $L^2$ Sobolev inequality. We prove that $(\mathbb{R}^3,g)$ must be close to Euclidean space with respect to the $d_p$-distance defined by Lee-Naber-Neumayer. We then discuss some consequences for the stability of the Yamabe invariant of $S^3$. More precisely, we show that if such a manifold $(\mathbb{R}^3,g)$ carries a suitably normalized, positive solution to $Δ_g w + λw^5 = 0$ then $w$ must be close, in a certain sense, to a conformal factor that transforms Euclidean space into a round sphere. |
| title | On the stability of the Yamabe invariant of $S^3$ |
| topic | Differential Geometry 53C21 |
| url | https://arxiv.org/abs/2402.00815 |