Fully nonlinear elliptic equations on compact manifolds with a flat hyperK\"ahler metric

TL;DR

This paper extends the theory of fully nonlinear elliptic equations to compact hyperhermitian manifolds, establishing a priori estimates and solving quaternionic Hessian and Monge-Ampère equations on flat hyperkähler manifolds.

Contribution

It adapts Székelyhidi's approach to the hypercomplex setting and proves solvability of key quaternionic equations on flat hyperkähler manifolds.

Findings

Established a priori estimates for solutions under certain conditions.

Proved existence of solutions to quaternionic Hessian and Monge-Ampère equations.

Demonstrated solvability on compact flat hyperkähler manifolds.

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 Sz\'{e}kelyhidi 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 hyperk\"ahler metric. As an application of our results we prove that the quaternionic analogue of the Hessian equation and Monge-Amp\`ere equation for $(n-1)$-plurisubharmonic functions can always be solved on compact flat hyperk\"ahler manifolds.