Solvability of the quaternionic Monge-Ampere equation on compact manifolds with a flat hyperKaehler metric

TL;DR

This paper proves the solvability of a quaternionic Monge-Ampere equation on certain compact hyperKaehler manifolds with flat metrics, extending the quaternionic Calabi problem.

Contribution

It provides the first solution to the quaternionic Monge-Ampere equation under the assumption of a flat hyperKaehler metric on compact manifolds.

Findings

The quaternionic Monge-Ampere equation is solvable on flat hyperKaehler HKT manifolds.

The proof employs the continuity method and a priori estimates.

This advances the understanding of quaternionic geometric analysis.

Abstract

A quaternionic version of the Calabi problem was recently formulated by M. Verbitsky and the author. It conjectures a solvability of a quaternionic Monge-Ampere equation on a compact HKT manifold (HKT stays for HyperKaehler with Torsion). In this paper this problem is solved under an extra assumption that the manifold admits a flat hyperKaehler metric compactible with the underlying hypercomplex structure. The proof uses the continuity method and a priori estimates.