Compactness of Conformal Metrics with \(L^p\)-Bounded \(Q\)-Curvature on Closed Smooth Riemannian Manifolds
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908960196395008 |
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| author | Mcheik, Zeinab |
| author_facet | Mcheik, Zeinab |
| contents | Let \((M^n,g)\) be a smooth closed Riemannian manifold of dimension \(n \ge 5\) with positive Yamabe invariant and semi-positive \(Q\)-curvature. We establish a precompactness result in the \(C^α\)-Hölder topologie on the space of Riemannian metrics, for some \(α>0\), for the set of metrics \(\tilde{g}\) conformal to \(g\), with volume equal to that of the standard sphere \(\mathbb{S}^n\), whose \(Q\)-curvature is nonnegative and uniformly bounded in \(L^p(M,\tilde{g})\) for some \(p > \frac{n}{4}\), and whose first positive eigenvalue of the Laplace-Beltrami operator satisfies \( λ_1(M,\tilde{g}) \ge n + \frac{1}Λ \) for some positive constant \(Λ\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_11638 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Compactness of Conformal Metrics with \(L^p\)-Bounded \(Q\)-Curvature on Closed Smooth Riemannian Manifolds Mcheik, Zeinab Differential Geometry Let \((M^n,g)\) be a smooth closed Riemannian manifold of dimension \(n \ge 5\) with positive Yamabe invariant and semi-positive \(Q\)-curvature. We establish a precompactness result in the \(C^α\)-Hölder topologie on the space of Riemannian metrics, for some \(α>0\), for the set of metrics \(\tilde{g}\) conformal to \(g\), with volume equal to that of the standard sphere \(\mathbb{S}^n\), whose \(Q\)-curvature is nonnegative and uniformly bounded in \(L^p(M,\tilde{g})\) for some \(p > \frac{n}{4}\), and whose first positive eigenvalue of the Laplace-Beltrami operator satisfies \( λ_1(M,\tilde{g}) \ge n + \frac{1}Λ \) for some positive constant \(Λ\). |
| title | Compactness of Conformal Metrics with \(L^p\)-Bounded \(Q\)-Curvature on Closed Smooth Riemannian Manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2604.11638 |