A Lewy theorem for harmonic quasiregular mappings in three-space

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 \[ f:Ω\subset \mathbb R^3\to \mathbb R^3 \] is nonconstant, sense-preserving, quasiregular, and harmonic componentwise, then \(J_f>0\) throughout \(Ω\). 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 \(\mathbb R^3\).

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“ontained in the manuscript. Conflicts of interest. The authors declare that they have no conflicts of interest regarding the publication of this paper. Declaration of generative AI use. During the preparation of this work, the authors used OpenAI’s ChatGPT to assist with language editing, exposition, and LaTeX formatting. After using this tool, the authors reviewed and edited the content as needed, verified the mathematical arguments and references, and take full responsibility for the conte”

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2026-08-24T00:34:05.733Z

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