On an Iwaniec--Kovalev--Onninen Conjecture for harmonic quasiconformal annulus mappings
Abstract
In 2012, Iwaniec, Kovalev and Onninen proposed an upper Nitsche--Grötzsch type estimate for harmonic K-quasiconformal homeomorphisms between circular annuli: in normalized form, every such map h:A(1,s)→ A(1,S) should satisfy S ≤ (K+1)/2s−(K−1)/2s. The radial and spiral-radial one-mode models explain why this estimate is natural. We show that this one-mode evidence does not extend to the unrestricted non-radial class. For every 1<s<S, we construct a non-radial harmonic orientation-preserving diffeomorphism h:A(1,s)→ A(1,S) with normωh∞<(S−s)/(S−s−1), strictly below the conjectural threshold. Thus, the dilatation lower bound predicted by the conjecture fails, even for smooth harmonic diffeomorphisms with prescribed circular boundary components. The construction uses a small high-frequency reparametrization of the outer boundary: it lowers the first Fourier-mode dilatation by order t2, while the compensating high modes are exponentially damped at the inner boundary. We also record structured regimes in which the one-mode estimate survives, including an inner-boundary anti-conformal energy condition, a Fourier leakage criterion, and a low-frequency spectral stability result. In the minimal-surface interpretation, the upper radial model is helicoidal and vertical-periodic rather than single-valued catenoidal; accordingly, the counterexamples give vertical-periodic minimal annuli with slope below the expected helicoidal threshold, and lead to a non-radial extremal problem for harmonic annulus diffeomorphisms.
Related
- Harmonic Quasiconformal Mappings in the BMO-Teichmüller space
- Coefficients and Integral Mean Estimates for K-Quasiconformal Harmonic Mappings
- Affine balancing of complex dilatations: affine invariants and optimal distortion of harmonic quasiconformal mappings
- Holomorphic motions, Assouad dimension and quasiconformal mappings
- Sharp constant problems for harmonic mappings
BibTeX
@misc{kalaj2026iwanieckovalevonninen,
title = {{On an Iwaniec--Kovalev--Onninen Conjecture for harmonic quasiconformal annulus mappings}},
author = {David Kalaj and Jinsong Liu and Jian-feng Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29795},
eprint = {2609.29795},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29795},
}
RIS
TY - UNPB
TI - On an Iwaniec--Kovalev--Onninen Conjecture for harmonic quasiconformal annulus mappings
AU - David Kalaj
AU - Jinsong Liu
AU - Jian-feng Zhu
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29795
UR - https://arxiv.org/abs/2609.29795
AB - In 2012, Iwaniec, Kovalev and Onninen proposed an upper Nitsche--Grötzsch type estimate for harmonic K-quasiconformal homeomorphisms between circular annuli: in normalized form, every such map h:A(1,s)→ A(1,S) should satisfy S ≤ (K+1)/2s−(K−1)/2s. The radial and spiral-radial one-mode models explain why this estimate is natural. We show that this one-mode evidence does not extend to the unrestricted non-radial class. For every 1<s<S, we construct a non-radial harmonic orientation-preserving diffeomorphism h:A(1,s)→ A(1,S) with normω_h_∞<(S−s)/(S−s^−1), strictly below the conjectural threshold. Thus, the dilatation lower bound predicted by the conjecture fails, even for smooth harmonic diffeomorphisms with prescribed circular boundary components. The construction uses a small high-frequency reparametrization of the outer boundary: it lowers the first Fourier-mode dilatation by order t^2, while the compensating high modes are exponentially damped at the inner boundary. We also record structured regimes in which the one-mode estimate survives, including an inner-boundary anti-conformal energy condition, a Fourier leakage criterion, and a low-frequency spectral stability result. In the minimal-surface interpretation, the upper radial model is helicoidal and vertical-periodic rather than single-valued catenoidal; accordingly, the counterexamples give vertical-periodic minimal annuli with slope below the expected helicoidal threshold, and lead to a non-radial extremal problem for harmonic annulus diffeomorphisms.
ER -