A parabolic approach to the Calabi-Yau problem in HKT geometry

TL;DR

This paper extends the parabolic Monge-Ampère equation to HKT geometry, proving short-time existence and long-time convergence under certain conditions, providing an alternative proof of a known theorem.

Contribution

It introduces a parabolic approach to the quaternionic Monge-Ampère equation in HKT geometry and establishes existence and convergence results.

Findings

Short-time existence of solutions in compact HKT manifolds.

Long-time convergence to solutions in locally flat hyperkähler manifolds.

Provides an alternative proof of Alesker's theorem.

Abstract

We consider the natural generalization of the parabolic Monge-Amp\`ere equation to HKT geometry. We prove that in the compact case the equation has always a short-time solution and when the hypercomplex manifold is locally flat and admits a hyperk\"ahler metric, then the equation has a long-time solution whose normalization converges to a solution of the quaternionic Monge-Amp\`ere equation introduced by Alesker and Verbitsky. The result gives an alternative proof of a theorem of Alesker.