A remark on the quaternionic Monge-Amp\`ere equation on foliated manifolds

TL;DR

This paper investigates the quaternionic Monge-Ampère equation on HKT manifolds with specific foliations, proving existence and uniqueness of solutions and extending the analysis to the SU(3) case.

Contribution

It establishes the existence and uniqueness of solutions to the quaternionic Monge-Ampère equation on certain foliated HKT manifolds, including the case of SU(3).

Findings

Unique solutions exist for every basic datum on the studied manifolds.

The approach applies to the case of SU(3).

Provides new insights into quaternionic Monge-Ampère equations on foliated manifolds.

Abstract

We study the quaternionic Monge-Amp\`ere equation on HKT manifolds admitting an HKT foliation having corank 4. We show that in this setting the quaternionic Monge-Amp\`ere equation has always a unique solution for every basic datum. This approach includes the study of the equation on SU(3).