Hermitian metrics, (n-1, n-1) forms and Monge-Ampère equations

Abstract

We show existence of unique smooth solutions to the Monge-Ampere equation for (n-1)-plurisubharmonic functions on Hermitian manifolds, generalizing previous work of the authors. As a consequence we obtain Calabi-Yau theorems for Gauduchon and strongly Gauduchon metrics on a class of non-Kahler manifolds: those satisfying the Jost-Yau condition known as Astheno-Kahler. Gauduchon conjectured in 1984 that a Calabi-Yau theorem for Gauduchon metrics holds on all compact complex manifolds. We discuss another Monge-Ampere equation, recently introduced by Popovici, and show that the full Gauduchon conjecture can be reduced to a second order estimate of Hou-Ma-Wu type.

37 pages, v2 some corrections to computations in Section 3

References in corpus (1)

• The Monge-Ampère equation for (n-1)-plurisubharmonic functions on a compact Kähler manifold

Cited by in corpus (5)

• On a class of Hessian type equations on Riemannian manifolds
• On the Alesker-Verbitsky conjecture on hyperKähler manifolds
• Transverse Fully Nonlinear Equations on Sasakian Manifolds and Applications
• A parabolic Monge-Ampère type equation of Gauduchon metrics
• On canonical metrics of complex surfaces with split tangent and related geometric PDEs