A Lewy-type theorem for pluriharmonic mappings in $\C^2$ and bi-Lipschitz quasiconformal harmonic maps

Abstract

Lewy's theorem says that a one-to-one harmonic mapping between plane domains has nonvanishing Jacobian. In dimensions at least three this statement is false for general harmonic homeomorphisms. We prove a four-dimensional Lewy theorem under the additional complex-analytic assumption of pluriharmonicity. More precisely, if \[ F:\Omega\subset \mathbb C^2\to \mathbb C^2 \] is a \(C^2\) pluriharmonic mapping which is one-to-one in a neighborhood of a point \(p\), then the real Jacobian of \(F\) is nonzero at \(p\). The proof is local. If the Jacobian vanished, a nontrivial real linear projection of \(F\) would be the real part of a holomorphic function \(f\) with \(df(p)=0\). The level hypersurface of this projection would therefore be a real analytic germ of the form \[ \{\operatorname{Re} f=0\}. \] We prove that such a germ cannot be locally flat at a critical point of \(f\). The obstruction is topological: local flatness forces the local homology of the hypersurface germ to agree with that of a real hyperplane, and hence forces every sufficiently small admissible link to have the integral homology, in particular the Euler characteristic, of \(S^2\). In the reduced case the Milnor open book of the plane curve singularity \(f^{-1}(0)\) gives a contradictory Euler-characteristic formula, while in the nonreduced case the local normal form produces more than two local complementary components. We then prove a separate bi-Lipschitz criterion for harmonic quasiconformal mappings: a harmonic quasiconformal homeomorphism from the unit ball onto a bounded \(C^1\)-Dini domain is bi-Lipschitz, provided it is already a local \(C^1\)-diffeomorphism.