Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.29795

On an Iwaniec--Kovalev--Onninen Conjecture for harmonic quasiconformal annulus mappings

David Kalaj·Jinsong Liu·Jian-feng Zhu
View original source ↗Read source document ↗Export metadata ↓

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 $h:A(1,s)\to A(1,S)$ should satisfy \[ S \leq \frac{K+1}{2}s-\frac{K-1}{2s}. \] 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 $h:A(1,s)\to A(1,S)$ with \[ \norm{ω_h}_{\infty}<\frac{S-s}{S-s^{-1}}, \] 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.

Explore related subjects

math.CVmath.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

David Kalaj, Jinsong Liu, Jian-feng Zhu. 2026-09-24. On an Iwaniec--Kovalev--Onninen Conjecture for harmonic quasiconformal annulus mappings. https://arxiv.org/abs/2609.29795

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Reciprocal Problem on Weighted Bergman Spaces

We study the reciprocal problem for weighted Bergman spaces: if $f\in A_α^p(\Bn)$ and $\inf_{\Bn}|f|>0$, does it follow that $1/f\in A_α^p(\Bn)$? We determine the range of parameters for which the answer is affirmative. In particular, we prove that functions in $A_α^p(\D)$ have the reciprocal property for all $α\in \R$ and $p\geq 1,$ where $\D$ denotes the unit disk in $\C.$ Moreover, functions in $A_α^p(\Bn)$ have the reciprocal property for all $α\in \R$ when $n=2$ and $p=2.$ In addition, we resolve the reciprocal problem in the three-dimensional Drury--Arveson space $H_3^2$ and obtain an equivalent condition in the four-dimensional space $H_4^2$. We also settle the reciprocal problem for general $A_α^p(\Bn)$ under some additional conditions.

math.CV↗
arXiv2026

Wiman-Valiron inequalities in the unit disk outside sets of finite logarithmic measure

We give affirmative answers to both parts of Question 2.6 posed by Grosse-Erdmann (2025) concerning Wiman--Valiron inequalities in the unit disk. For every unbounded analytic function in the disk, we establish the proposed iterated-logarithm inequalities outside exceptional sets of finite logarithmic measure. The corresponding power estimate is a corollary. Both conclusions follow from a variance bound for Khinchin families and a classical estimate for their largest atom. The multiplicative constants in the main inequalities can be chosen absolute. We also obtain a disk analogue of Rosenbloom's composition estimate, with an explicit boundary prefactor. The key step combines a boundary change of variable with a monotone auxiliary function whose derivative is exactly the variance of a rescaled member of the Khinchin family. A classical example shows that the leading logarithmic exponent $1/2$ cannot be decreased.

math.CV↗
arXiv2026

Minimal m-subharmonic functions with nonmaximal Hessian measures

For every $2\le m\le n$, we construct Hölder continuous functions on the closed unit ball that are minimal in the Cegrell class $\F_m$, although their Hessian measures are not maximal in the $m$-subharmonic ordering. This answers a question posed by the second author. We also obtain a minimality criterion based on partial pluricomplex energy and an explicit family of Monge--Ampère examples on a product domain. For $m=1$, minimality and maximality are equivalent on a ball.

math.CV↗