Volume forms on balanced manifolds and the Calabi-Yau equation
Mathew George
TL;DR
This work extends the framework of volume-form geodesics from Kähler to balanced Hermitian geometry by introducing mixed-volume form spaces $\mathcal{V}_p$ on balanced manifolds and endowing them with an $L^2$ metric. It derives the geodesic equation in this setting, recasts it as a degenerate $2$-Hessian equation on a higher-dimensional domain, and establishes crucial $C^0$ and boundary estimates under a positivity-related assumption, paving the way for weak solutions. The second major contribution is a Calabi–Yau-type theorem for balanced metrics: under $\partial\bar{\partial}\alpha^{n-2}\le0$, one can prescribe a Chern–Ricci form $\Psi$ and find a balanced metric whose $(n-1)$st power matches the prescribed volume form, with $C^0$ control achieved via ABP-type methods or auxiliary Monge–Ampère techniques. Combined, these results advance non-Kähler canonical metric theory by linking geodesic geometry in form spaces with balanced CY-type equations and their solvability under structural sign conditions. The findings have implications for complex geometry and string-theoretic contexts where balanced metrics play a role beyond Kähler geometry.
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.