The Dirichlet problem for quaternionic Monge--Ampère equations on flat quaternionic quotients

Abstract

We prove smooth solvability of the Dirichlet problem for quaternionic Monge--Ampère equations on smooth relatively compact domains in ${\mathbb H}^m/Γ$, where $m\ge2$ and $Γ$ is a discrete additive subgroup: for a smooth positive density, the existence of a smooth strictly plurisubharmonic subsolution guarantees the existence and uniqueness of the smooth admissible solution. The technique of the proof is a simple adaptation from the Ph.D. Thesis of Jingyong Zhu [25] in 2017, supervised by Xinan Ma, which is a joint work of Ma and Zhu. The claim [22, Theorem A.1] by OpenAI on Sogge's local smoothing conjecture is a special case of our theorem.