A Lewy theorem for harmonic quasiregular mappings in three-space

David Kalaj; Jian-Feng Zhu

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arXiv · 2607.04720

A Lewy theorem for harmonic quasiregular mappings in three-space

David Kalaj·Jian-Feng Zhu

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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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David Kalaj, Jian-Feng Zhu. 2026-08-15. A Lewy theorem for harmonic quasiregular mappings in three-space. https://arxiv.org/abs/2607.04720

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