Non-quadratic solutions to the Monge-Ampère equation
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913806922285056 |
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| author | Pan, Yifei Zhang, Yuan |
| author_facet | Pan, Yifei Zhang, Yuan |
| contents | We construct ample smooth strictly plurisubharmonic non-quadratic solutions to the Monge-Ampère equation on either cylindrical type domains or the whole complex Euclidean space $\mathbb C^2$. Among these, the entire solutions defined on $\mathbb C^2$ induce flat Kahler metrics, as expected by a question of Calabi. In contrast, those on cylindrical domains produce a family of nowhere flat Kahler metrics. Beyond these smooth solutions, we also classify solutions that are radially symmetric in one variable, which exhibit various types of singularities. Finally, we explore analogous solutions to Donaldson's equation motivated by a result of He. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_17625 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-quadratic solutions to the Monge-Ampère equation Pan, Yifei Zhang, Yuan Complex Variables Primary 32W20, Secondary 32Q15 We construct ample smooth strictly plurisubharmonic non-quadratic solutions to the Monge-Ampère equation on either cylindrical type domains or the whole complex Euclidean space $\mathbb C^2$. Among these, the entire solutions defined on $\mathbb C^2$ induce flat Kahler metrics, as expected by a question of Calabi. In contrast, those on cylindrical domains produce a family of nowhere flat Kahler metrics. Beyond these smooth solutions, we also classify solutions that are radially symmetric in one variable, which exhibit various types of singularities. Finally, we explore analogous solutions to Donaldson's equation motivated by a result of He. |
| title | Non-quadratic solutions to the Monge-Ampère equation |
| topic | Complex Variables Primary 32W20, Secondary 32Q15 |
| url | https://arxiv.org/abs/2504.17625 |