The sharp $σ_2$-curvature inequality on the sphere in quantitative form
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912159829590016 |
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| author | Frank, Rupert L. Peteranderl, Jonas W. |
| author_facet | Frank, Rupert L. Peteranderl, Jonas W. |
| contents | Among all metrics on $\mathbb S^d$ with $d>4$ that are conformal to the standard metric and have positive scalar curvature, the total $σ_2$-curvature, normalized by the volume, is uniquely (up to Möbius transformations) minimized by the standard metric. We show that if a metric almost minimizes, then it is almost the standard metric (up to Möbius transformations). This closeness is measured in terms of Sobolev norms of the conformal factor, and we obtain the optimal stability exponents for two different notions of closeness. This is a stability result for an optimization problem whose Euler-Lagrange equation is fully nonlinear. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12819 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The sharp $σ_2$-curvature inequality on the sphere in quantitative form Frank, Rupert L. Peteranderl, Jonas W. Analysis of PDEs Differential Geometry Functional Analysis Among all metrics on $\mathbb S^d$ with $d>4$ that are conformal to the standard metric and have positive scalar curvature, the total $σ_2$-curvature, normalized by the volume, is uniquely (up to Möbius transformations) minimized by the standard metric. We show that if a metric almost minimizes, then it is almost the standard metric (up to Möbius transformations). This closeness is measured in terms of Sobolev norms of the conformal factor, and we obtain the optimal stability exponents for two different notions of closeness. This is a stability result for an optimization problem whose Euler-Lagrange equation is fully nonlinear. |
| title | The sharp $σ_2$-curvature inequality on the sphere in quantitative form |
| topic | Analysis of PDEs Differential Geometry Functional Analysis |
| url | https://arxiv.org/abs/2412.12819 |