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

The results of the present work have been included in arXiv:2112.12934