A gluing construction of singular solutions for a fully non-linear equation in conformal geometry
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912949278343168 |
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| author | Espinal, María Fernanda González, María del Mar |
| author_facet | Espinal, María Fernanda González, María del Mar |
| contents | In this paper we study the $σ_2$--Yamabe equation, $n>4$, for solutions with a prescribed singular set $Λ$ given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than $(n-\sqrt{n}-2)/2$. The $σ_2$--curvature in conformal geometry is defined as the second elementary symmetric polynomial of the eigenvalues of the Schouten tensor, which yields a fully non-linear PDE for the conformal factor. We show that the classical gluing method, used by Mazzeo-Pacard (JDG 1996) for the scalar curvature problem, can be used in the fully non-linear setting. This is a consequence of the conformal properties of the $σ_2$ equation, which imply that the linearized operator has good mapping properties in weighted spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05965 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A gluing construction of singular solutions for a fully non-linear equation in conformal geometry Espinal, María Fernanda González, María del Mar Differential Geometry In this paper we study the $σ_2$--Yamabe equation, $n>4$, for solutions with a prescribed singular set $Λ$ given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than $(n-\sqrt{n}-2)/2$. The $σ_2$--curvature in conformal geometry is defined as the second elementary symmetric polynomial of the eigenvalues of the Schouten tensor, which yields a fully non-linear PDE for the conformal factor. We show that the classical gluing method, used by Mazzeo-Pacard (JDG 1996) for the scalar curvature problem, can be used in the fully non-linear setting. This is a consequence of the conformal properties of the $σ_2$ equation, which imply that the linearized operator has good mapping properties in weighted spaces. |
| title | A gluing construction of singular solutions for a fully non-linear equation in conformal geometry |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2404.05965 |