Volume forms on balanced manifolds and the Calabi-Yau equation
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914167557980160 |
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| author | George, Mathew |
| author_facet | George, Mathew |
| contents | We introduce the space of mixed-volume forms endowed with a $L^2$ metric on a balanced manifold. A geodesic equation can be derived in this space that has an interesting structure and extends the equation of Donaldson \cite{Donaldson10} and Chen-He \cite{CH11} in the space of volume forms on a Riemannian manifold. This nonlinear PDE is studied in detail and we prove several estimates, under a positivity assumption. Later we study the Calabi-Yau equation for balanced metrics and introduce a geometric criterion for prescribing volume forms, that is closely related to the positivity assumption above. By deriving $C^0$ a priori estimates, we prove the existence of solutions on all such manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00995 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Volume forms on balanced manifolds and the Calabi-Yau equation George, Mathew Differential Geometry Analysis of PDEs Complex Variables 53C55 (Primary) 35R01 (Secondary) We introduce the space of mixed-volume forms endowed with a $L^2$ metric on a balanced manifold. A geodesic equation can be derived in this space that has an interesting structure and extends the equation of Donaldson \cite{Donaldson10} and Chen-He \cite{CH11} in the space of volume forms on a Riemannian manifold. This nonlinear PDE is studied in detail and we prove several estimates, under a positivity assumption. Later we study the Calabi-Yau equation for balanced metrics and introduce a geometric criterion for prescribing volume forms, that is closely related to the positivity assumption above. By deriving $C^0$ a priori estimates, we prove the existence of solutions on all such manifolds. |
| title | Volume forms on balanced manifolds and the Calabi-Yau equation |
| topic | Differential Geometry Analysis of PDEs Complex Variables 53C55 (Primary) 35R01 (Secondary) |
| url | https://arxiv.org/abs/2406.00995 |