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arXiv · 2611.03214

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

Xi-Nan Ma·Jingyong Zhu
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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.

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Xi-Nan Ma, Jingyong Zhu. 2026-11-02. The Dirichlet problem for quaternionic Monge--Ampère equations on flat quaternionic quotients. https://arxiv.org/abs/2611.03214

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