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 $q\colon(\C^n,0)\to(\C,0)$ is a nonconstant holomorphic germ, then $dq(0)\ne0$ if and only if one (equivalently, every) phase hypersurface \( \{\Repart(e^{-i\theta}q)=0\} \) 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.