The quaternionic Calabi conjecture on abelian hypercomplex nilmanifolds viewed as tori fibrations

TL;DR

This paper proves that the quaternionic Monge-Ampère equation can always be solved on certain 8-dimensional nilmanifolds with abelian hypercomplex structures, extending the quaternionic Calabi-Yau problem in HKT geometry.

Contribution

It demonstrates the solvability of the quaternionic Monge-Ampère equation on 8-dimensional abelian hypercomplex nilmanifolds with torus-invariant data.

Findings

Solution exists for all torus-invariant data

Extends quaternionic Calabi-Yau problem to specific nilmanifolds

Provides new insights into HKT geometry

Abstract

We study the quaternionic Calabi-Yau problem in HKT geometry introduced by Alesker and Verbitsky on 8-dimensional 2-step nilmanifolds with an abelian hypercomplex structure. We show that the quaternionic Monge-Amp\`ere equation on these manifolds can always be solved for every data which is invariant by the action of a 3-dimensional torus.