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

TL;DR

This paper investigates a class of fully non-linear elliptic equations on compact hyperhermitian manifolds, establishing a priori estimates and solving the quaternionic Hessian equation under flat hyperk"ahler conditions.

Contribution

It adapts Székelyhidi's approach to hypercomplex manifolds and proves solvability of the quaternionic Hessian equation on compact flat hyperk"ahler manifolds.

Findings

Established a priori estimates for solutions to elliptic equations on hyperhermitian manifolds.

Proved the solvability of the quaternionic Hessian equation on compact flat hyperk"ahler manifolds.

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.