An all-dimensional Lewy theorem for pluriharmonic mappings via Milnor monodromy
Abstract
Hengartner asked whether a pluriharmonic mapping between equidimensional complex Euclidean domains is locally one-to-one precisely at the points where its real Jacobian is nonzero. Naser had earlier claimed the conclusion in complex dimension two, and a recent theorem of Kalaj supplies a complete proof in that dimension. We prove the result in every complex dimension. The main ingredient is a singularity-theoretic characterization which is of independent interest: if q:(Cn,0)→(C,0) is a nonconstant holomorphic germ, then dq(0)≠0 if and only if one (equivalently, every) phase hypersurface {Repart(e−iθq)=0} is a rational homology (2n−1)-manifold at the origin. The difficult direction combines local conical structure, Alexander duality, the Milnor fibration, and A'Campo's vanishing theorem for the Lefschetz number of local monodromy. The argument requires neither an isolated critical point nor a reducedness hypothesis. It follows that the real critical set of any pluriharmonic self-dimensional mapping agrees exactly with its local noninjectivity set. As a quantitative application, combining our theorem with Kalaj's boundary regularity theorem shows that, in every complex dimension, a quasiconformal pluriharmonic homeomorphism of the unit ball onto a bounded C1-Dini domain is bi-Lipschitz.
Related
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- Siu's analyticity theorem for positive pluriharmonic currents
- A p-adic monodromy theorem for curves
- A new degenerated second main theorem for meromorphic mappings with hypersurfaces
- A sharp stability inequality of Liouville's theorem for quasiregular mappings on bounded domains
BibTeX
@misc{zhong2026alldimensional,
title = {{An all-dimensional Lewy theorem for pluriharmonic mappings via Milnor monodromy}},
author = {Deguang Zhong and Zhi-Gang Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31212},
eprint = {2609.31212},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31212},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An all-dimensional Lewy theorem for pluriharmonic mappings via Milnor monodromy
AU - Deguang Zhong
AU - Zhi-Gang Wang
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31212
UR - https://arxiv.org/abs/2609.31212
AB - Hengartner asked whether a pluriharmonic mapping between equidimensional complex Euclidean domains is locally one-to-one precisely at the points where its real Jacobian is nonzero. Naser had earlier claimed the conclusion in complex dimension two, and a recent theorem of Kalaj supplies a complete proof in that dimension. We prove the result in every complex dimension. The main ingredient is a singularity-theoretic characterization which is of independent interest: if q:(C^n,0)→(C,0) is a nonconstant holomorphic germ, then dq(0)≠0 if and only if one (equivalently, every) phase hypersurface {Repart(e^−iθq)=0} is a rational homology (2n−1)-manifold at the origin. The difficult direction combines local conical structure, Alexander duality, the Milnor fibration, and A'Campo's vanishing theorem for the Lefschetz number of local monodromy. The argument requires neither an isolated critical point nor a reducedness hypothesis. It follows that the real critical set of any pluriharmonic self-dimensional mapping agrees exactly with its local noninjectivity set. As a quantitative application, combining our theorem with Kalaj's boundary regularity theorem shows that, in every complex dimension, a quasiconformal pluriharmonic homeomorphism of the unit ball onto a bounded C^1-Dini domain is bi-Lipschitz.
ER -