The CR Yamabe invariant and constant scalar curvature Sasaki metrics
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912562653691904 |
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| author | Lahdili, Abdellah Legendre, Eveline Scarpa, Carlo |
| author_facet | Lahdili, Abdellah Legendre, Eveline Scarpa, Carlo |
| contents | We propose a new approach to the existence of constant transversal scalar curvature Sasaki structures drawing on ideas and tools from the CR Yamabe problem, establishing a link between the CR Yamabe invariant, the existence of Sasaki structures of constant transversal scalar curvature, and the K-stability of Sasaki manifolds. Assuming that the Sasaki-Reeb cone contains a regular vector field, we show that if the CR Yamabe invariant of a compact Sasaki manifold attains a specific value determined by the geometry of the Reeb cone, then the Sasaki manifold is K-semistable. Under the additional assumption of non-positive average scalar curvature, the CR Yamabe invariant attains this topological value if the manifold admits approximately constant scalar curvature Sasaki structures, and we also show a partial converse. As an application, we provide a new numerical criterion for the K-semistability of polarised compact complex manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00743 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The CR Yamabe invariant and constant scalar curvature Sasaki metrics Lahdili, Abdellah Legendre, Eveline Scarpa, Carlo Differential Geometry Algebraic Geometry 53C25 (primary), 58E11, 53C18, 32Q15, 53D10 (secondary) We propose a new approach to the existence of constant transversal scalar curvature Sasaki structures drawing on ideas and tools from the CR Yamabe problem, establishing a link between the CR Yamabe invariant, the existence of Sasaki structures of constant transversal scalar curvature, and the K-stability of Sasaki manifolds. Assuming that the Sasaki-Reeb cone contains a regular vector field, we show that if the CR Yamabe invariant of a compact Sasaki manifold attains a specific value determined by the geometry of the Reeb cone, then the Sasaki manifold is K-semistable. Under the additional assumption of non-positive average scalar curvature, the CR Yamabe invariant attains this topological value if the manifold admits approximately constant scalar curvature Sasaki structures, and we also show a partial converse. As an application, we provide a new numerical criterion for the K-semistability of polarised compact complex manifolds. |
| title | The CR Yamabe invariant and constant scalar curvature Sasaki metrics |
| topic | Differential Geometry Algebraic Geometry 53C25 (primary), 58E11, 53C18, 32Q15, 53D10 (secondary) |
| url | https://arxiv.org/abs/2509.00743 |