Bergman metric on a Stein manifold with nonpositive constant holomorphic sectional curvature
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912884816084992 |
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| author | Li, Xiaojun Huang and. Song-Ying |
| author_facet | Li, Xiaojun Huang and. Song-Ying |
| contents | We prove that the Bergman space of a Stein manifold separates points whenever its Bergman metric is well defined and has non-positive constant holomorphic sectional curvature. This, combined with earlier proved results, shows that a Stein manifold cannot admit a well-defined flat Bergman metric, and that it has a well-defined Bergman metric with negative constant holomorphic sectional curvature if and only if it is biholomorphic to the unit ball of the same dimension possibly with a pluripolar set removed. The proof is based on the Hormander L2-estimate for d-bar equations; and the curvature condition together with Calabi's rigidity and extension theorems is used to construct the required bounded strictly plurisubharmonic functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06813 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bergman metric on a Stein manifold with nonpositive constant holomorphic sectional curvature Li, Xiaojun Huang and. Song-Ying Complex Variables Primary: 32H15, Secondary: 32Q15, 53C55 We prove that the Bergman space of a Stein manifold separates points whenever its Bergman metric is well defined and has non-positive constant holomorphic sectional curvature. This, combined with earlier proved results, shows that a Stein manifold cannot admit a well-defined flat Bergman metric, and that it has a well-defined Bergman metric with negative constant holomorphic sectional curvature if and only if it is biholomorphic to the unit ball of the same dimension possibly with a pluripolar set removed. The proof is based on the Hormander L2-estimate for d-bar equations; and the curvature condition together with Calabi's rigidity and extension theorems is used to construct the required bounded strictly plurisubharmonic functions. |
| title | Bergman metric on a Stein manifold with nonpositive constant holomorphic sectional curvature |
| topic | Complex Variables Primary: 32H15, Secondary: 32Q15, 53C55 |
| url | https://arxiv.org/abs/2602.06813 |