The Monge-Amp\`{e}re equation for $(n-1)$-quaternionic PSH functions on a hyperK\"{a}hler manifold

TL;DR

This paper proves the existence and uniqueness of smooth solutions to the quaternionic Monge-Ampère equation on hyperKähler manifolds, extending previous results by establishing new estimates without flatness assumptions.

Contribution

It introduces a novel approach to obtain $C^1$ and $C^2$ estimates for quaternionic Monge-Ampère equations on hyperKähler manifolds, removing the flatness requirement.

Findings

Existence and uniqueness of smooth solutions established.

New $C^0$, $C^1$, and $C^2$ estimates derived.

Solutions obtained for quaternionic form type equations.

Abstract

We prove the existence of unique smooth solutions to the quaternionic Monge-Amp\`{e}re equation for $(n-1)$-quaternionic plurisubharmonic functions on a hyperK\"{a}hler manifold and thus obtain solutions for the quaternionic form type equation. We derive $C^0$ estimate by establishing a Cherrier-type inequality as in Tosatti and Weinkove [22]. By adopting the approach of Dinew and Sroka [9] to our context, we obtain $C^1$ and $C^2$ estimates without assuming the flatness of underlying hyperK\"{a}hler metric comparing to previous results [14].