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 \(\mathbb R^3\).