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arXiv · 2406.00995

Volume forms on balanced manifolds and the Calabi-Yau equation

Mathew George
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Abstract

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.

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Mathew George. 2024-06-03. Volume forms on balanced manifolds and the Calabi-Yau equation. https://arxiv.org/abs/2406.00995

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