A free boundary Monge-Ampère equation and applications to complete Calabi-Yau metrics
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916126740447232 |
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| author | Collins, Tristan C. Tong, Freid Yau, Shing-Tung |
| author_facet | Collins, Tristan C. Tong, Freid Yau, Shing-Tung |
| contents | Let $P$ be a convex body containing the origin in its interior. We study a real Monge-Ampère equation with singularities along $\del P$ which is Legendre dual to a certain free boundary Monge-Ampère equation. This is motivated by the existence problem for complete Calabi-Yau metrics on log Calabi-Yau pairs $(X, D)$ with $D$ an ample, simple normal crossings divisor. We prove the existence of solutions in $C^{\infty}(P)\cap C^{1,α}(\overline{P})$, and establish the strict convexity of the free boundary. When $P$ is a polytope, we obtain an asymptotic expansion for the solution near the interior of the codimension $1$ faces of $\del P$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10111 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A free boundary Monge-Ampère equation and applications to complete Calabi-Yau metrics Collins, Tristan C. Tong, Freid Yau, Shing-Tung Differential Geometry Let $P$ be a convex body containing the origin in its interior. We study a real Monge-Ampère equation with singularities along $\del P$ which is Legendre dual to a certain free boundary Monge-Ampère equation. This is motivated by the existence problem for complete Calabi-Yau metrics on log Calabi-Yau pairs $(X, D)$ with $D$ an ample, simple normal crossings divisor. We prove the existence of solutions in $C^{\infty}(P)\cap C^{1,α}(\overline{P})$, and establish the strict convexity of the free boundary. When $P$ is a polytope, we obtain an asymptotic expansion for the solution near the interior of the codimension $1$ faces of $\del P$. |
| title | A free boundary Monge-Ampère equation and applications to complete Calabi-Yau metrics |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2402.10111 |