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Octonionic and Monogenic Completions of Harmonic Functions
Abstract
Let Ω⊂ℝn be a domain, with n≥2. We characterize the real-valued harmonic functions that can occur as prescribed real components of monogenic functions. For any fixed nonzero real Clifford module M and any fixed unit vector s∈ M, a smooth real-valued function h can be written as h=⟨ F,s⟩ with F monogenic if and only if the (n−1)-form *dh is exact. Consequently, every real-valued harmonic function on Ω admits such a representation if and only if HdRn−1(Ω)=0. In dimension eight, the same exactness criterion characterizes the real parts of octonionic regular functions and yields the corresponding equivalence with HdR7(Ω)=0. The proofs use global solvability of the Poisson equation and elementary identities for differential forms.
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BibTeX
@misc{li2026octonionic,
title = {{Octonionic and Monogenic Completions of Harmonic Functions}},
author = {Yong Li and Yuchen Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30648},
eprint = {2610.30648},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30648},
}
RIS
TY - UNPB
TI - Octonionic and Monogenic Completions of Harmonic Functions
AU - Yong Li
AU - Yuchen Zhang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30648
UR - https://arxiv.org/abs/2610.30648
AB - Let Ω⊂ℝ^n be a domain, with n≥2. We characterize the real-valued harmonic functions that can occur as prescribed real components of monogenic functions. For any fixed nonzero real Clifford module M and any fixed unit vector s∈ M, a smooth real-valued function h can be written as h=⟨ F,s⟩ with F monogenic if and only if the (n−1)-form *dh is exact. Consequently, every real-valued harmonic function on Ω admits such a representation if and only if H_dR^n−1(Ω)=0. In dimension eight, the same exactness criterion characterizes the real parts of octonionic regular functions and yields the corresponding equivalence with H_dR^7(Ω)=0. The proofs use global solvability of the Poisson equation and elementary identities for differential forms.
ER -