Multiple Blow-Up Phenomena for $Q$-Curvature in High Dimensions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912767195217920 |
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| author | Caju, Rayssa Santos, Almir Silva |
| author_facet | Caju, Rayssa Santos, Almir Silva |
| contents | Let $(M,g_0)$ be a closed Riemannian manifold of dimension $n \geq 25$ with positive Yamabe invariant $Y(M,g_0)>0$ and positive fourth-order invariant $Y_4(M,g_0)>0$. We show that, arbitrarily $C^1$-close to $g_0$, there exists a Riemannian metric such that, within its conformal class, one can find infinitely many smooth metrics with the same constant $Q$-curvature and arbitrarily large energy. Moreover, within this conformal class, there exists a sequence of smooth metrics with constant $Q$-curvature equal to $n(n^2-4)/8$ and unbounded volume. This extends to the $Q$-curvature setting the result previously obtained for the scalar curvature in Marques (2015) (see also Gond and Li (2025)). The proof is based on constructing small perturbations of multiple standard bubbles that are glued together. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13811 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiple Blow-Up Phenomena for $Q$-Curvature in High Dimensions Caju, Rayssa Santos, Almir Silva Differential Geometry 35B09, 53C21, 35J30, 35J60 Let $(M,g_0)$ be a closed Riemannian manifold of dimension $n \geq 25$ with positive Yamabe invariant $Y(M,g_0)>0$ and positive fourth-order invariant $Y_4(M,g_0)>0$. We show that, arbitrarily $C^1$-close to $g_0$, there exists a Riemannian metric such that, within its conformal class, one can find infinitely many smooth metrics with the same constant $Q$-curvature and arbitrarily large energy. Moreover, within this conformal class, there exists a sequence of smooth metrics with constant $Q$-curvature equal to $n(n^2-4)/8$ and unbounded volume. This extends to the $Q$-curvature setting the result previously obtained for the scalar curvature in Marques (2015) (see also Gond and Li (2025)). The proof is based on constructing small perturbations of multiple standard bubbles that are glued together. |
| title | Multiple Blow-Up Phenomena for $Q$-Curvature in High Dimensions |
| topic | Differential Geometry 35B09, 53C21, 35J30, 35J60 |
| url | https://arxiv.org/abs/2512.13811 |