Volume forms on balanced manifolds and the Calabi-Yau equation

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.