Fully non-linear elliptic equations on compact hyperhermitian manifolds with a flat hyperk\"ahler metric

Abstract

Mainly motivated by a conjecture of Alesker and Verbitsky, we study a class of elliptic equations on compact hyperhermitian manifolds. By adapting the approach of Sz\'ekelyhidi to the hypercomplex setting, we prove some a priori estimates for solutions to such equations. In the estimate of the Laplacian we 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 can always be solved on compact flat hyperk\"ahler manifolds.