Fully non-linear elliptic equations on compact hyperhermitian manifolds with a flat hyperkähler metric

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 Székelyhidi 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 hyperkähler metric. As an application of our results we prove that the quaternionic analogue of the Hessian equation can always be solved on compact flat hyperkähler manifolds.

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