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

We show existence of unique smooth solutions to the Monge-Ampère 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-Kähler manifolds: Those satisfying the Jost-Yau condition known as Astheno-Kähler. Gauduchon conjectured in 1984 that a Calabi-Yau theorem for Gauduchon metrics holds on all compact complex manifolds. We discuss another Monge-Ampère 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.