  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 hyperk\"ahler 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.

Differential Geometry