Fully Non-linear Elliptic Equations on Compact Manifolds with a Flat Hyperkähler Metric

Abstract

Mainly motivated by a conjecture of Alesker and Verbitsky, we study a class of fully non-linear elliptic equations on certain compact hyperhermitian manifolds. By adapting the approach of Székelyhidi (J Differ Geom 109(2):337–378, 2018) to the hypercomplex setting, we prove some a priori estimates for solutions to such equations under the assumption of existence of $${\mathcal {C}}$$ -subsolutions. In the estimate of the quaternionic Laplacian, we need to further assume the existence of a flat hyperkähler metric. As an application of our results we prove that the quaternionic analogue of the Hessian equation and Monge–Ampère equation for $$(n-1)$$ -plurisubharmonic functions can always be solved on compact flat hyperkähler manifolds.