Compactness of singular solutions to the sixth order GJMS equation
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866918219447533568 |
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| author | Andrade, João Henrique Ò, João Marcos do Ratzkin, Jesse Wei, Juncheng |
| author_facet | Andrade, João Henrique Ò, João Marcos do Ratzkin, Jesse Wei, Juncheng |
| contents | We study compactness properties of the set of conformally flat singular metrics with constant, positive sixth order Q-curvature on a finitely punctured sphere. Based on a recent classification of the local asymptotic behavior near isolated singularities, we introduce a notion of necksize for these metrics in our moduli space, which we use to characterize compactness. More precisely, we prove that if the punctures remain separated and the necksize at each puncture is bounded away from zero along a sequence of metrics, then a subsequence converges with respect to the Gromov--Hausdorff metric. Our proof relies on an upper bound estimate which is proved using moving planes and a blow-up argument. This is combined with a lower bound estimate which is a consequence of a removable singularity theorem. We also introduce a homological invariant which may be of independent interest for upcoming research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_05770 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Compactness of singular solutions to the sixth order GJMS equation Andrade, João Henrique Ò, João Marcos do Ratzkin, Jesse Wei, Juncheng Differential Geometry Analysis of PDEs We study compactness properties of the set of conformally flat singular metrics with constant, positive sixth order Q-curvature on a finitely punctured sphere. Based on a recent classification of the local asymptotic behavior near isolated singularities, we introduce a notion of necksize for these metrics in our moduli space, which we use to characterize compactness. More precisely, we prove that if the punctures remain separated and the necksize at each puncture is bounded away from zero along a sequence of metrics, then a subsequence converges with respect to the Gromov--Hausdorff metric. Our proof relies on an upper bound estimate which is proved using moving planes and a blow-up argument. This is combined with a lower bound estimate which is a consequence of a removable singularity theorem. We also introduce a homological invariant which may be of independent interest for upcoming research. |
| title | Compactness of singular solutions to the sixth order GJMS equation |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2302.05770 |