PreprintarXiv licencearXiv
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 ℍm/Γ, where m≥2 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.
Related
- Continuous solutions to the quaternionic Monge-Ampère equation on locally flat HKT manifolds
- Uniqueness results for the exterior Dirichlet and Neumann problem in two dimensions for the Euler-Lagrange equations of variational integrals with linear growth
- The exterior Dirichlet problem for special Lagrangian equations
- Existence of Admissible Subsolutions to the Dirichlet Problem for Symmetric Augmented k-Hessian Type Equations in Bounded Domains
- Regularity of Free Boundary for Monge-Ampère Obstacle Problem up to the Boundary
BibTeX
@misc{ma2026dirichlet,
title = {{The Dirichlet problem for quaternionic Monge--Ampère equations on flat quaternionic quotients}},
author = {Xi-Nan Ma and Jingyong Zhu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03214},
eprint = {2611.03214},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03214},
}
RIS
TY - UNPB
TI - The Dirichlet problem for quaternionic Monge--Ampère equations on flat quaternionic quotients
AU - Xi-Nan Ma
AU - Jingyong Zhu
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03214
UR - https://arxiv.org/abs/2611.03214
AB - We prove smooth solvability of the Dirichlet problem for quaternionic Monge--Ampère equations on smooth relatively compact domains in ℍ^m/Γ, where m≥2 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.
ER -