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arXiv · 2112.12691

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

Giovanni Gentili
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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.

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Giovanni Gentili. 2021-12-23. Fully non-linear elliptic equations on compact hyperhermitian manifolds with a flat hyperk\"ahler metric. https://arxiv.org/abs/2112.12691

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