On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914913221345280 |
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| author | Chang, Shu-Cheng Li, Fengjiang Lin, Chien Wu, Chin-Tung |
| author_facet | Chang, Shu-Cheng Li, Fengjiang Lin, Chien Wu, Chin-Tung |
| contents | In this paper, we derive the uniform L^{4}-bound of the transverse conic Ricci curvature along the conic Sasaki-Ricci flow on a compact transverse log Fano Sasakian manifold M of dimension five and the space of leaves of the characteristic foliation is not well-formed. Then we first show that any solution of the conic Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular orbifold conic Sasaki-Ricci soliton on M_{infinite} which is a S^{1}-orbibundle over the unique singular conic Keahler-Ricci soliton on a log del Pezzo orbifold surface. As a consequence, there exists a Keahler-Ricci soliton orbifold metric on its leave space which is a log del Pezzo orbifold surface. Second, we show that the conic Sasaki-Ricci soliton is the conic Sasaki-Einstein if M is transverse log K-polystable. In summary, we have the existence theorems of orbifold Sasaki-Ricci solitons and Sasaki-Einstein metrics on a compact quasi-regular Sasakian manifold of dimension five. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16430 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five Chang, Shu-Cheng Li, Fengjiang Lin, Chien Wu, Chin-Tung Differential Geometry In this paper, we derive the uniform L^{4}-bound of the transverse conic Ricci curvature along the conic Sasaki-Ricci flow on a compact transverse log Fano Sasakian manifold M of dimension five and the space of leaves of the characteristic foliation is not well-formed. Then we first show that any solution of the conic Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular orbifold conic Sasaki-Ricci soliton on M_{infinite} which is a S^{1}-orbibundle over the unique singular conic Keahler-Ricci soliton on a log del Pezzo orbifold surface. As a consequence, there exists a Keahler-Ricci soliton orbifold metric on its leave space which is a log del Pezzo orbifold surface. Second, we show that the conic Sasaki-Ricci soliton is the conic Sasaki-Einstein if M is transverse log K-polystable. In summary, we have the existence theorems of orbifold Sasaki-Ricci solitons and Sasaki-Einstein metrics on a compact quasi-regular Sasakian manifold of dimension five. |
| title | On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2406.16430 |