Solvability of the quaternionic Monge–Ampère equation on compact manifolds with a flat hyperKähler metric

A quaternionic version of the Calabi problem was formulated by Alesker and Verbitsky (2010) [6]. It conjectures a solvability of a quaternionic Monge–Ampère equation on a compact HKT manifold (HKT stays for HyperKähler with Torsion). In this paper this problem is solved under the extra assumption that the manifold admits a flat hyperKähler metric compatible with the underlying hypercomplex structure. The proof uses the continuity method and a priori estimates.