The parabolic quaternionic Calabi-Yau equation on hyperkähler manifolds

Abstract

We show that the parabolic quaternionic Monge-Ampère equation on a compact hyperkähler manifold has always a long-time solution which once normalized converges smoothly to a solution of the quaternionic Monge-Ampère equation. This is the same setting in which Dinew and Sroka prove the conjecture of Alesker and Verbitsky. We also introduce an analogue of the Chern-Ricci flow in hyperhermitian manifolds.

This is a new version of the paper "The Calabi-Yau theorem on Hypercomplex manifolds", which was withdrawn due to a crucial mistake in a key Lemma. In this new version we fix the mistake by introducing the extra assumption of the existence of a background hyperkähler metric