The Monge–Ampère equation for $$(n-1)$$ -quaternionic PSH functions on a hyperKähler manifold

Abstract

We prove the existence of a unique smooth solution to the quaternionic Monge–Ampère equation for \((n-1)\)-quaternionic plurisubharmonic (psh) functions on a compact hyperKähler manifold and thus obtain solutions to the quaternionic form-type equation. We derive the \(C^0\) estimate by establishing a Cherrier-type inequality as in Tosatti and Weinkove (J Am Math Soc 30(2):311–346, 2017). By adopting the approach of Dinew and Sroka (Geom Funct Anal 33(4):875–911, 2023) to our context, we obtain the \(C^1\) and \(C^2\) estimates without assuming the flatness of underlying hyperKähler metric comparing to the previous result Gentili and Zhang (J Geom Anal 32:9, 2022).