Abstract
By the Poisson integral formula, each self-homeomorphism of the unit circle extends to a complex-valued harmonic function that maps the unit disc onto itself. The Radó–Kneser–Choquet theorem guarantees the injectivity of this harmonic extension. We show that this result fails in Rnn≥3, by constructing a self-homeomorphism of the sphere whose harmonic extension fails to be injective. Even worse, this homeomorphism can be chosen arbitrarily close to the identity in the uniform norm. The homeomorphism provides also an example of a non-injective conformally natural extension. The question of injectivity remains open for pluriharmonic extensions in Cn.
Published Version
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