A Lewy theorem for harmonic quasiregular mappings in three-space

[2607.04720] A Lewy theorem for harmonic quasiregular mappings in three-space Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Complex Variables arXiv:2607.04720 (math) [Submitted on 6 Jul 2026 ( v1 ), last revised 15 Aug 2026 (this version, v3)] Title: A Lewy theorem for harmonic quasiregular mappings in three-space Authors: David Kalaj , Jian-Feng Zhu View a PDF of the paper titled A Lewy theorem for harmonic quasiregular mappings in three-space, by David Kalaj and 1 other authors View PDF HTML (experimental) Abstract: Lewy's classical theorem asserts that a one-to-one planar harmonic mapping has nonvanishing Jacobian. We prove a three-dimensional bounded-distortion analogue: if \[ f:\Omega\subset \mathbb R^3\to \mathbb R^3 \] is nonconstant, sense-preserving, quasiregular, and harmonic componentwise, then \(J_f>0\) throughout \(\Omega\). Thus harmonic quasiconformal mappings between domains in three-space are local harmonic diffeomorphisms. The new point is the Lewy-type differential conclusion \(J_f\neq0\), not merely topological local invertibility, which is already known for sufficiently smooth quasiregular mappings. The proof is by blow-up. A hypothetical zero of \(J_f\) produces a nonconstant homogeneous harmonic polynomial quasiregular mapping \(P:\mathbb R^3\to\mathbb R^3\) of degree \(m>1\). We exclude such homogeneous blow-ups by a second-order trace identity for \(J_P|_{S^2}\): after normalizing the first jet at a positive minimum, the identity gives a negative spherical trace, contradicting the maximum principle. We also derive an affine Liouville theorem for entire harmonic quasiregular mappings in \(\mathbb R^3\). Comments: 23 pages Subjects: Complex Variables (math.CV) Cite as: arXiv:2607.04720 [math.CV] (or arXiv:2607.04720v3 [math.CV] for this version) https://doi.org/10.48550/arXiv.2607.04720 Focus to learn more arXiv-issued DOI via DataCite Submission history From: David Kalaj [ view email ] [v1] Mon, 6 Jul 2026 06:46:46 UTC (1,015 KB) [v2] Tue, 7 Jul 2026 11:26:54 UTC (939 KB) [v3] Sat, 15 Aug 2026 07:49:27 UTC (1,018 KB) Full-text links: Access Paper: View a PDF of the paper titled A Lewy theorem for harmonic quasiregular mappings in three-space, by David Kalaj and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.CV < prev | next > new | recent | 2026-07 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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