Volume forms on balanced manifolds and the Calabi-Yau equation

We introduce the space of mixed-volume forms on a balanced manifold endowed with an L² metric. A geodesic equation can be derived in this space that has an interesting structure and extends the equation of Donaldson and Chen-He in the space of volume forms on a Riemannian manifold. This nonlinear PDE is studied in detail and the existence of weak solution is shown for the Dirichlet problem, under a positivity assumption. Later we study the Calabi-Yau equation for balanced metrics and introduce a geometric criteria for prescribing volume forms that is closely related to the positivity assumption above. By deriving C⁰ a priori estimates, we show that the existence of solutions can be established under this assumption