Volume forms on balanced manifolds and the Calabi-Yau equation

TL;DR

This paper develops a geometric framework for balanced manifolds, introduces a geodesic equation extending previous volume form equations, and proves existence results for the Calabi-Yau equation under certain positivity conditions.

Contribution

It introduces a new space of mixed-volume forms with an $L^2$ metric, derives a geodesic equation extending Donaldson and Chen-He, and proves existence of solutions to the Calabi-Yau equation on balanced manifolds.

Findings

Derived a geodesic equation extending previous volume form equations.

Established $C^0$ a priori estimates for solutions.

Proved existence of solutions to the Calabi-Yau equation under positivity assumptions.

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.