A parabolic approach to the Calabi-Yau problem in HKT geometry

We consider the natural generalization of the parabolic Monge-Amp`ere equation to HKT geometry. We prove that in the compact case the equation has always a short-time solution and when the hypercomplex manifold is locally flat and admits a hyperkahler metric, then the equation has a long-time solution whose normalization converges to a solution of the quaternionic Monge-Amp`ere equation introduced by Alesker and Verbitsky. The result gives an alternative proof of a theorem of Alesker.