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arXiv · 2609.37606

Affine balancing of complex dilatations: affine invariants and optimal distortion of harmonic quasiconformal mappings

Zhi-Gang Wang·Deguang Zhong
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Abstract

For orientation-preserving planar harmonic mappings, postcomposition by real-affine mappings induces automorphisms on the space of complex dilatations. Motivated by this fact and the lack of natural affine-invariant geometric quantities in classical distortion theory, we introduce the affine circumradius and affine diameter associated with the image of the complex dilatation. The exponentiated affine circumradius gives a characterization of minimal quasiconformal distortion under affine normalization. Optimal affine balancing is unique up to similarity and yields canonical harmonic mappings with centrally symmetric dilatation. Using a three-point support principle and a sharp hyperbolic Jung theorem, we establish universal sharp two-sided bounds for these invariants and provide their pseudohyperbolic reformulations. For canonically balanced mappings, we prove sharp second-order estimates for pre-Schwarzian derivatives, showing that affine normalization cancels leading-order conformal discrepancies. We develop an affine-stable hierarchical classification for families of harmonic quasiconformal mappings, including normality and an invariant ordering. This framework unifies extremal distortion problems and connects classical analytic function theory with the distortion theory of harmonic quasiconformal mappings.

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Zhi-Gang Wang, Deguang Zhong. 2026-09-29. Affine balancing of complex dilatations: affine invariants and optimal distortion of harmonic quasiconformal mappings. https://arxiv.org/abs/2609.37606

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