On the Bergman metric of Cartan-Hartogs domains
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915538405425152 |
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| author | Loi, Andrea Mossa, Roberto ZUddas, Fabio |
| author_facet | Loi, Andrea Mossa, Roberto ZUddas, Fabio |
| contents | We study the Bergman metric and introduce the Bergman dual on Cartan-Hartogs (CH) domains. For a bounded domain D in C^n with Bergman kernel K_D, we define the Bergman dual of (D, g_D) as (D*, g_D*), where D* is the maximal domain on which the modified kernel K_D*(z, zbar) = K_D(z, -zbar) is positive, and g_D* is the Kahler metric obtained from K_D*. For a Cartan-Hartogs domain M_{Omega, mu} we prove the equivalence of: (i) M_{Omega, mu} is biholomorphic to the unit ball; (ii) its Bergman metric is a Kahler-Ricci soliton; (iii) after rescaling by a constant factor, the Bergman dual is finitely projectively induced. Conditions (i) and (ii) are Bergman-metric analogues of classical rigidity for Kahler-Einstein metrics (related to Yau's problem and Cheng's conjecture) and to recent rigidity for Kahler-Ricci solitons. Condition (iii) emphasizes the duality viewpoint, inspired by bounded symmetric domains and their compact duals. We also compare our results with other canonical metrics on CH domains, namely g_{Omega, mu} and hat g_{Omega, mu}, and discuss open problems about the maximal domain on which the Bergman dual is defined. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_06405 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Bergman metric of Cartan-Hartogs domains Loi, Andrea Mossa, Roberto ZUddas, Fabio Complex Variables Differential Geometry We study the Bergman metric and introduce the Bergman dual on Cartan-Hartogs (CH) domains. For a bounded domain D in C^n with Bergman kernel K_D, we define the Bergman dual of (D, g_D) as (D*, g_D*), where D* is the maximal domain on which the modified kernel K_D*(z, zbar) = K_D(z, -zbar) is positive, and g_D* is the Kahler metric obtained from K_D*. For a Cartan-Hartogs domain M_{Omega, mu} we prove the equivalence of: (i) M_{Omega, mu} is biholomorphic to the unit ball; (ii) its Bergman metric is a Kahler-Ricci soliton; (iii) after rescaling by a constant factor, the Bergman dual is finitely projectively induced. Conditions (i) and (ii) are Bergman-metric analogues of classical rigidity for Kahler-Einstein metrics (related to Yau's problem and Cheng's conjecture) and to recent rigidity for Kahler-Ricci solitons. Condition (iii) emphasizes the duality viewpoint, inspired by bounded symmetric domains and their compact duals. We also compare our results with other canonical metrics on CH domains, namely g_{Omega, mu} and hat g_{Omega, mu}, and discuss open problems about the maximal domain on which the Bergman dual is defined. |
| title | On the Bergman metric of Cartan-Hartogs domains |
| topic | Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2510.06405 |