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

2021/12/23 by Gentili, Giovanni
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.12691

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.

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• Fully non-linear elliptic equations on compact hyperkähler manifolds
• Fully non-linear parabolic equations on compact manifolds with a flat hyperkähler metric
• Fully non-linear elliptic equations on compact Hermitian manifolds
• Fully non-linear elliptic equations on noncompact complex manifolds
• Fully non-linear elliptic equations on compact almost Hermitian manifolds