Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds

TL;DR

This paper introduces a quaternionic Monge-Ampere equation on hypercomplex manifolds with HKT-metrics, establishing uniqueness and a priori estimates, and reformulating it as a complex Hessian equation, advancing the quaternionic Calabi problem.

Contribution

It formulates a quaternionic Monge-Ampere equation on hypercomplex manifolds, proves uniqueness and estimates, and connects it to complex Hessian equations, proposing a new direction in quaternionic geometry.

Findings

Proved uniqueness of solutions up to a constant.

Established zero order a priori estimates.

Reformulated the quaternionic equation as a complex Hessian equation.

Abstract

A quaternionic version of the Calabi problem on Monge-Ampere equation is introduced. It is a quaternionic Monge-Ampere equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For a hypercomplex manifold with holonomy in SL(n;H), uniqueness (up to a constant) of a solution is proven, as well as the zero order a priori estimate. The existence of solution is conjectured, similar to Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of a complex Hessian equation, making sense on any complex manifold.