Fully non-linear elliptic equations on compact hyperk\"ahler manifolds

TL;DR

This paper establishes $C^{2,eta}$ a priori estimates for solutions to a broad class of fully non-linear elliptic equations, including quaternionic Monge-Ampère equations, on compact hyperk"ahler manifolds, advancing understanding of their regularity.

Contribution

It extends regularity results to a wide class of non-linear elliptic equations on hyperk"ahler manifolds, including quaternionic Monge-Ampère and Hessian equations.

Findings

Solutions satisfy $C^{2,eta}$ a priori estimates under certain conditions.

Includes equations like quaternionic Monge-Ampère and Hessian equations.

Provides a unified approach to regularity for these equations.

Abstract

We consider a general class of elliptic equations on hypercomplex manifolds which includes the quaternionic Monge-Amp\`ere equation, the quaternionic Hessian equation and the Monge-Amp\`ere equation for quaternionic $(n-1)$-plurisubharmonic functions. We prove that under suitable assumptions the solutions to these equations on hyperk\"ahler manifolds satisfy a $C^{2,\alpha}$ a priori estimate.